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CORNELL
UNIVERSITY
LIBRARY
DEPARTMENT OF COMMERCE AND LABOR
COAST AND GEODETIC SURVEY
oO. H. TITTIMANN
SUPERINTENDENT
GEODESY
THE EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION
UPON THE INTENSITY OF GRAVITY
BY
JOHN F. HAYFORD
Formerly Inspector of Geodetic Work and Chief of the Computing Division
AND
WILLIAM BOWIE
Inspector of Geodetic Work and Chief of the Computing Division
Assistant, Coast and Geodetic Survey
SPECIAL PUBLICATION No. 10
WASHINGTON: a3
GOVERNMENT PRINTING OFFICE wey
1912
gi
fea,
CONTENTS.
Page
Generalistatementsai cosa cel. agen ease ee Gane at oc uated Cum east Mi re aah Ltd 5
TeOStASY: CONN ed shea sve Pe cee ese ue alihus Stein eda apatiaieaiany Lo dee eenl a eRe 2 ened ies aida oo 6
Assumptions as to isostasy..... 2.22.00 002 eee ee ee cee eee cece eee eceeeeeeeeeseeesreeeess 10
Formulas: Sinneaciodec ocasatnacives-smaaumec ke oa esac netnel sah dade semstred videlsaatekteoscudate: Guioe side 12
Division of the surface of the earth into zones and compartments..............2.0-0222.0ec eee e cee eeeeeeeeees 17
Computation of reduction tables for near zones............... 20000202 eee eee eee cee eee cee eee e eee cesses 19
Computation of reduction tables for distant zones...........2. 0.000200 e cece eee cence ee eee cece ee ences 23
Explanation of reduction tables.c;éscocqosesecteci ccmeseca ds tac deeremewe vy sve exec eieaiea' ales Sabdetouin daw eile 28
Reduction tables for lettered zones............. 00020202 c eee ene eee eee nee eee e eee e een eee 30
Reduction tables for numbered zones......... 22.200 eee eee eee een cece teen eee eeeees 44,
Special reduction tables for sea stations..............222.0 0.0220 o eee ee eee ee ee eee reese 46
FU seson templates 5.222 cic sictecdsaiics de rane 2 at LAG Oe ie Bat emt ced banat Whe sanmund Sako L ane Oe aes AT
Examples of computations of corrections. ..............0.. 20202020 eee ee eee eee eee eee nee eens 48
Corrections for topography and isostatic compensation, separate ZoneS...--.-......022-2 22 eee eee eee 53
Initérpolation:for Outer: 2ON eS! 23 456.02 vevesensnen od sx Sede e 22+ 5 exept eete ds Se oy eee de yer el Seek ee vosies 58
Method of interpolation for outer zones.......-.----.-.-22-2-020 02222 eee eee Beige s tee Sages 60
Criteria of accepted interpolations for outer zones........-..-.. 2.222202 2 eee eee ence eee eee 63
Saving by interpolation for outer zones.........-..2.22- 2222022 e eee eee eee eee eee eect teense 64
Change of sign due to distance 22.0. ee cvs cds hens wanes poss Smueieiimeidie See ed onaionoele See ba dhciamelee ds 26 65
Distant topography necessarily considered.............2.2.2.2. 222020222 cece cece cece eee ete e nee neee 71
Curvatite:must be considered 22. essa. sceccccedes Peseeecuica y Soe SA eeee eae See see eeenioes Sak ysl megie es eis s 71
Principal facts for 89 stations in the United States.................. atta Ma tates alas leita Aah SrA i eaetun a has 72
Correction to Helmert’s formula of 1901..............0..20 2220202 e eee eee eee eee eee eeee 75
Comparison of apparent anomalies by new and old methods...........-... 22. 2.222222 202002 c ee eee eee eee eee 75
Possible relations of anomalies to topography.........-...2.-.22-- 22-2222 eee eee eee eee shee 34 Eaaeeaedunne tes 1T
Comparison of Bouguer anomalies with new-method anomalies................-.2..2-2-2020 2022s cece ee eee ee 79°
Comparison of free-air anomalies with new-method anomalies................2-2-.-0. 202.22 e cece eee eee eee 80
Test by stations not in the United States...........-.22. 222222222 e eee 81
VISCO UBSLOLN OL: CEN OMS 25555 ve 2 Son avy veuZue ced: S55 ocd aeuesevesonteas Se seve ep eeeaag eset e oe e Gein cacnanns ize oa aS Beans hehe as eds Stubbs ay SR SAA 86.
HrrorsOHOMSePVatODN ic iste ec ee c wee tae oA es 2s HE Oten Reds Zr esete oedaneeeeee see 87
Errors of computations... 220 sean ocivinnewudese Seles egioetees bt Pe eee eee teas Ee RU RIES Oe ctemictigeeey ee 88
Nature of apparentianomallesncccciscsaeeeeeccene yo. sy acter des oc se eden dey esas eyeweGee ese eeeeerRe Less 94
The method not subject to hidden errors........... 2.2.2.0 202222 e eee eee eee ee 95°
Effects of topography and compensation—why combined............. 2.2.2.2. 2 020 eee eee ee eee eee 97
Regional versus local distribution of compensation.....-..--.....---++-.222 02022022 e eee ee eee eee eee 98
Test of depth of compensation ..o0)c2.sless' ot tgeeieiew es oes oeeaes eyes agen see neem ede te das aende eee ees 103
Graphical comparison of three kinds of anomalies.........--..----- 22-222 0eeeee eee eee eee ec ee eee eee eee 106
Interpretation of anomalies in terms of masses.........--.----- +02 +2202 eee eee eee eee eee eee eee ee 108
Possible relation of new-method anomalies to other things..-........-.-.-..--.-2----2--2- 2222 e eee eee seegieiae's 112
Relation between new-method anomalies and geologic formations...........-...-.-2-2-2-2-2020 eee eee eee eee 113
Discussion of other regional peculiarities. ............---- 20-20-22 eee eee eee ee 117
Hypothesis of horizontal displacement of compensation............---- +. 22-00-2052 cee ee eee e eee e ee eee 121
Comment on Bouguer and free-air anomalies...........-.---- 2-2-2202 c ee ee eee eee eee eee eee eee 122
Comment on Faye method of reduction.............----. 2-2-2222 eee ere eee 125,
Summary...-.-----2--- 2-22 ee ee eee ee nent ete tee eee eee eee eee ee 126
BEE eee eB ee ee
CONBNMARWNESOS
{LLUSTRATIONS.
Page
. Three unit columns showing ideal depth of isostatic compensation...........2-2.22 0-0-0022 e cece reece eee 7
. Three unit columns showing approximate depth of isostatic compensation as used in computations......... 10
. Showing station and elementary mass at same elevation...........---.-------2200e eee eee eee eee eee eee 15
Showing elementary mass at greater elevation than station..........-.2----- 0-202 eee eee eee eee eee eters 16
. Showing elementary mass at less elevation than station.............-----2---2--e ee eee cece ee eee cere ee eees 17
. Showing topography in land zones—three cases.............0-2- 220-22 e eee eee eee eee eee eee ee eeee 20
. Graphical computation of reduction table for Zone E.................-0-- 20222 eee eee eee eee eee 22
. Graphical representation of values of E for various depths.......-...---.--.----+ 2-20-02 e eee eee ee eee eee 23
. Graphical representation of values of Eg..............2 2-0-2222 eee cee eee eee eee 24
. (a) Template for maps of scale 1/10000 (reduced)...........-.---2- 22-2222 eee e ce ee ee eee ee eee eee eens 48
. (b) Template for maps of scale 1/6018500 (reduced)......-.1.--. 222-022 eee eee eee eee eee eee eee eee eee 48
. Overlapping of corresponding zones for two stations.........-...-.- 222-222 c eee eee eee eee eee tect eeee 58
. Graphical illustration of interpolation...............0-....-. 2202-2 e eee ee ee ee eee eee eeee 60
. Map showing location of gravity stations used in the investigation .......-..--.---.---2---eee eee e eee In pocket
. Showing topography and compensation near station...........------- 20022 - eee eee eee eee eee ees 66
. Showing distant topography and compensation...........-...----.---+--+--- waUspe ee 6 Ree Sttaere 2 Srepcreeteds ee 67
. Lines of equal anomaly for new method of reduction ............-...-2-2 22-2202 ee teen e eee eee ee In pocket
. Lines of equal anomaly for Bouguer method of reduction ...........-.----.-2-22--2-2 2220-222 e eee ee In pocket
. Lines of equal anomaly for free-air method of reduction .......-.....-- 2222-2220 ee eee eee eee eee eee In pocket
. Illustration from Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, showing resid-
uals of Solution H, all stations, with areas of excessive and defective density, and showing also all gravity
stations with new-method anomalies -.._......... 2.22.22 2 222s eee eee cece eee eee eens In pocket
4
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION UPON
THE INTENSITY OF GRAVITY.
BY
Joun F. Hayrrorp,* formerly Inspector of Geodetic Work and Chief of the Computing Division,
AND
Witu1am Bowis, Inspector of Geodetic Work and Chief of the Computing Division, Assistant, Coast and Geodetic Survey
GENERAL STATEMENT.
In the United States the assumption of isostasy in a definite and reasonable form has been
introduced into the computations of the figure and size of the earth from the observed deflections
of the vertical. These computations have shown that the assumption of isostasy is substan-
tially correct. They have shown that a close approach to perfect isostatic compensation exists
under the United States and adjacent areas. This is important to geology and geophysics.
They have also shown that the proper recognition of isostasy in making computations of the
figure and size of the earth from observed deflections of the vertical has about doubled the
accuracy of such computations by reducing errors of both the accidental and the systematic
classes in such work. “This increase in accuracy is important to geodesy. These computations
and the investigations of which they form a part have been published in full.t
As soon as it was evident that the proper recognition of isostasy in connection with com-
putations of the figure and size of the earth from observed deflections of the vertical would
produce a great increase in accuracy, it appeared to be very probable that a similar recognition
of isostasy in connection with computations of the shape of the earth from observations of the
intensity of gravity would produce a similar increase of accuracy. Logically the next step to
be taken was therefore to introduce such a definite recognition of isostasy into gravity compu-
tations. Moreover, it appeared that if this step were taken it would furnish a proof of the
existence of isostasy independent of the proof furnished by observed deflections of the vertical,
and would therefore be of great value in supplementing the deflection investigations and in
testing the conclusions drawn from them. In other words, the effects of isostasy upon the
direction of' gravity at various stations on the earth’s surface having been studied, it then
appeared to be almost equally important to investigate the effects of isostasy upon the intensity
of gravity.
It was evident from the beginning that to properly take into account the possible existence
of isostasy in connection with computations of the intensity of gravity a rather extensive revi-
sion of formule and methods of computation would be necessary, and that the computations
must be thorough and must involve a considerable number of gravity stations if the results were
to be convincing. Thus it was realized that the problem was both a large and a difficult one.
Partly for this reason, Mr. Hayford, as inspector of geodetic work, recommended frequently from
1900 to 1908 that the Coast and Geodetic Survey confine its energy in geodetic observations and
* Now Director, College of Engineering, Northwestern University, Evanston, Tl.
+ The Figure of the Earth and Isostasy from Measurements in the United States, by John F. Hayford, published in 1909 by the Coast and
Geodetic Survey, and Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, by John F. Hayford, published in 1910 by the
Coast and Geodetic Survey. [ach of these is a separate publication not included in the annual reports of the survey. They may be obtained by
interested parties on application to the Superintendent of the Coast and Geodetic Survey, Washington. D. C.
5
6 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
investigations mainly to deflections of the vertical until that part of the field of investigation
had been well covered and reasonably safe conclusions reached, and that then, and not till then,
should much energy be expended in gravity observations and the corresponding investigations.
This policy was adopted and adhered to.
In the summer of 1908 Mr. Hayford began an extensive study of the theoretical side of the
investigation, the revision of formule and of methods of computation.
Early in 1909 a long, continuous series of gravity observations with the half-second pen-
dulum apparatus at various stations in the United States was commenced. This series is still
in progress. In this publication there are used 89 stations, including those of this series which
are available at this time. :
In January, 1909, Mr. Bowie became closely associated with Mr. Hayford at the Coast and
Geodetic Survey office and was brought into close touch with the investigation set forth in this
publication. In October, 1909, he assumed his present position, and has since that time been
in charge of the gravity observations and computations of gravity made in the Coast and Geo-
detic Survey, of which many are utilized in this publication. In certain lines he has extended
the investigation beyond its former limits. In the preparation of this publication the two
authors have cooperated. They are jointly responsible for the opinions expressed and the
statement of conclusions reached.
Miss Sarah Beall, computer, efficiently supervised much of the computing in connection
with this investigation, and especially the computation of the reduction tables, the most diffi-
cult part of the work. To her and to the various members of the computing division who
assisted, the credit is largely due for the unusual rapidity and success of the computations.
In September, 1909, Mr. Hayford presented to the International Geodetic Association at
London a paper bearing the same title as the present publication. It has been printed as
pages 365-389 of Volume I of the Report of the Sixteenth General Conference of the Inter-
national Geodetic Association, held at London and Cambridge in September, 1909.
The present investigation is in many respects a counterpart of the previous investigations
based on deflections of the vertical, to which reference has already been made. It supplements
those investigations, and therefore the three should be studied together to obtain their full force.
The computations of the present investigation have been based upon certain assumptions
as to the existence of the condition called isostasy which are substantially identical with the
assumptions in the previous investigations involving deflections of the vertical. It is important
to the reader to understand clearly the meaning of the word isostasy and of certain related
phrases, as otherwise he may fail to understand, or may misunderstand, many statements in
this publication. These definitions are given below in substantially the same words as were
used in connection with the previous investigations.
ISOSTASY DEFINED.
If the earth were composed of homogeneous material, its figure of equilibrium, under the
influence of gravity and its own rotation, would be an ellipsoid of revolution.
The earth is composed of heterogeneous material which varies considerably in density.
If this heterogeneous material were so arranged that its density at any point depended simply
upon the depth of that point below the surface, or, more accurately, if all the material lying
at each equipotential surface (rotation considered) was of one density, a state of equilibrium
would exist, and there would be no tendency toward a rearrangement of masses. The figure of
the earth in this case would be a very close approximation to an ellipsoid of revolution.
If the heterogeneous material composing the earth were not arranged in this manner at the
outset, the stresses produced by gravity would tend to bring about such an arrangement; but
as the material is not a perfect fluid, since it possesses considerable viscosity, at least near the
surface, the rearrangement will be imperfect. In the partial rearrangement some stresses will
still remain, different portions of the same horizontal stratum may have somewhat different
densities, and the actual surface of the earth will be a slight departure from the ellipsoid of
revolution in the sense that above each region of deficient density there will be a bulge or bump
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 7
on the ellipsoid, and above each region of excessive density there will be a hollow, relatively
speaking. The bumps on this supposed earth will be the mountains, the plateaus, the conti-
nents; and the hollows will be the oceans. The excess of material represented by that portion
of the continent which is above sea level will be compensated for by a defect of density in the
underlying material. The continents will be floated, so to speak, because they are composed
of relatively light material; and, similarly, the floor of the ocean will, on this supposed earth,
be depressed because it is composed of unusually dense material. This particular condition of
approximate equilibrium has been given the name “ isostasy.”’
The adjustment of the material toward this condition, which is produced in nature by the
stresses due to gravity, may be called the “isostatic adjustment.”
The compensation of the excess of matter at the surface (continents) by the defect of
density below, and of surface defect of matter (oceans) by excess of density below, may be
called the “isostatic compensation.”
Let the depth below sea level within which the isostatic compensation is complete be called
the “depth of compensation.” At and below this depth the condition as to stress of any
element of mass is isostatic; that is, any element
of mass is subject to equal pressures from all di-
rections as if it were a portion of a perfect fluid.
Above this depth, on the other hand, each element
of mass is subject in general to different pressures
in different directions—to stresses which tend to
distort it and to move it.
Consider the relations of the masses, densi-
ties, and volumes, above the depth of compen-
sation, fixed by the preceding definition. The
mass in any prismatic column which has for its
base a unit area of the horizontal surface which
lies at the depth of compensation, for its edges
vertical lines (lines of gravity) and for its upper
limit the actual irregular surface of the earth (or
the sea surface if the area in question is beneath inland Column Sea Goast Column Ocean Column
the ocean) is the same as the mass in any other I:tustration No. 1.—Three unit columns showing ideal depth of
similar prismatic column having any other unit i a
area of the same surface for its base.* Tllustration No. 1 represents three such unit columns.
Let the depth of compensation be called h, and the mean surface density of the solid portion
of the earth be called 6. Then the mass of material in a column of unit area at the seacoast is
oh, + (density times volume). ,
Let the elevation above sea level of the irregular surface of the earth over the unit area of
an inland column be called H. Then the mass of material in the inland column above sea
level is OH. Also, let the density of that portion of the inland column between sea level and
the depth of compensation be called 6; Then the mass of material in the column is expressed
by the equation
Surface of ground
Mass in any land unit column =dH +0,h, (1)
By definition, at the depth of isostasy, any element of mass is subject to equal pressures
from all directions as if it were a portion of a perfect fluid. In order that this may be true,
the vertical pressures due to gravity on the various units of area at that depth must all be the
* It would be more accurate to use the words “inverted truncated pyramid” instead of “prismatic column.” The latter expression has been
selected because it is sufficiently exact for the purpose and corresponds to the allowable approximations actually made in the mathematical part
of the investigation.
+ For the purpose of this demonstration it is assumed that the average density of the earth’s crust below the seacoast between sea level and
the depth of compensation is equal to the average density of the solid portion of the earth’s surface (2.67). This assumption ignores the probability
that within a depth as great as 114 kilometers (the assumed depth of compensation) there is probably a slight increase in density with increase of
depth, due to increased pressure, the density being some unknown function of the depth. This neglect also appears in various other places in this
publication. It is shown later, under the heading “ Discussion of errors,”’ that this neglect introduces no appreciable errors into the computation.
Tt is justified, therefore, as a means of avoiding unnecessarily long and complicated statements.
8. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
same, and therefore masses of the various unit columns must all be the same. Therefore the
mass in a land unit column must be equal to the mass in a seacoast column, or
OH +6rh,=6h, (1a)
From equation (1a) it follows that
3,— Sa) (2)
Pod
The difference called 0, between 4 and 4; is expressed by the equation
3,-0- a (2a)
1
AH
ae 3
= 95, (3)
This difference between the normal density at the surface of the land and also throughout
a column at the seacoast on the one hand, and the density of an inland column below sea
level on the other hand, is the average compensating defect of density, and this difference
multiplied by the depth of compensation is the compensating defect of mass, 0,h,.
The total mass in the inland column may also be expressed by the equation (see illustra-
tion No. 1),
Mass in any land unit column =0H +0h,—6,h, (4)
As the mass in each unit column is the same, namely 6A,, it is obvious from equation (4)
that
OH=6,h, (4a)
This equation is a statement in mathematical symbols that in each unit column the com-
-pensating defects of mass below sea level must be exactly equal to the mass above sea level
which is considered to be the surface excess.
Equation (3) indicates that the compensating defect of density is proportional to the
elevation of the surface above the sea level as 0 and h, are assumed to be constant.
In an ocean unit column the top of the solid portion happens to be below sea level, being
a part of the bottom of the ocean. In the ocean column let the depth of the water be called D
and the density of the sea water 0. Then the depth of the solid portion of the column will be
h,—D. Let the density of this solid portion be called 6,. Then the mass of material in this
unit column will be expressed by the equation
Mass in any ocean unit column =6,D+0,(h, —D) (46)
By definition, this mass must equal the mass of the unit column at the seacoast, hence
byD + d4(h, — D) =sh, (4c)
From equation (4c) it follows that
_dhy—OyD
~ h,—D
The difference 0, between the density of the solid portion of the ocean column 6, and the
normal density 6 is expressed by the equation
_dhy—dyD
Oo (4d)
Ve ag (4e)
— h-D- od
The total mass in any ocean unit column may also be expressed by the equation (see illus-
tration No. 1),
Mass in any ocean unit column =6,,D + (0 +6,)(A,—D) (5)
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 9
As the mass in each unit column is the same, namely 6/,, it follows from equation (5) that
D(6—6y) =6,(h, —D) (5a)
That is, in the solid portion of each ocean unit column the compensating excess of mass
must be exactly equal to the defect of mass in the water portion of the column.
Equation (4f) indicates that the compensating excess of density is nearly proportional to
the depth of water, as 6 and 6, are assumed to be constant and (h,—D) is approximately
constant.
In this publication the mean surface density of the solid portion of the earth, 0, is assumed
to be 2.67. The density of sea water, 0,, is 1.027. With these values 0—6,,=0.6150. Hence,
for oceanic unit columns, equation (5a) becomes
(d—0,)D =0.615dD = 0,(h, —D) (5d)
and equation (4f) becomes
0.615D
b= 05 iy o
Note that equation (6) differs from equation (3) only by containing the factor 0.615; in
having D, a depth, in the place of H, an elevation; and in having (A,—D) as a denominator
instead of h,.
As a concrete illustration, consider three unit columns such as are indicated in illustration
No. 1, one beneath a mountain summit at an elevation of 3 kilometers, one underlying an
area which is at sea level, a portion of the seashore for example, and the third under the ocean
at a point where it is 5 kilometers deep. Let the depth of compensation be assumed to be
114 kilometers below sea level, and the mean surface density 6=2.67. In the first column the
ratio H to h, being op according to equation (3) the defect of density, 6,, is a of 2.67 or
0.07, and the density of the material below sea level is 2.67—0.07=2.60. In the second
column the density of the material is 2.67. In the third column the compensating excess of
density of the material underlying the ocean is, by equation (6), pO619)O) _ 93. =0.07
and the density of the material is therefore 2.67 + 0.07 =2.74.
Under such a mountain, therefore, if isostasy exists as defined by the stated assumptions,
the average density is about 3 per cent less than under the seacoast, and on the other hand,
under a portion of the ocean 5 kilometers deep the average density is about 3 per cent greater
than under the seacoast, down to the depth of compensation in each case.
As a rough approximation it may be stated, on the basis of the preceding paragraph,
that beneath areas which lie above sea level the density is defective by about 1 per cent for
each kilometer of elevation of the surface. Since much of the land portion of the earth’s
surface is at an elevation of less than 1 kilometer and very little of it above the elevation 3
kilometers, the compensating defects of density beneath most land areas are less than 1 per
cent of the mean density and exceed 3 per cent only under a few small areas on very high
mountains. Similarly, the compensating excesses of density under ocean areas seldom exceed
3 per cent as the depths exceed 5 kilometers (16 000 feet or 2700 fathoms) in but a small
portion of the ocean.
If the condition of equal pressures, that is of equal superimposed masses, is fully satisfied
at a given depth, the compensation is said to be complete at that depth. If there is a variation
from equality of superimposed masses, the differences may be taken as a measure of the
degree of incompleteness of the compensation.
In the above definitions it has been tacitly assumed that g, the intensity of gravity, is
everywhere the same at a given depth. Equal superincumbent masses would produce equal
pressures only in case the intensity of gravity is the same in the two cases. The intensity of
gravity varies with change of latitude and is subject also to anomalous variations which are
to some extent associated with the relation to continents and oceanic areas. But even the
10 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
extreme variations in the intensity of gravity are small in comparison with the variations in
density postulated. The extreme variation of the intensity of gravity at sea level on each
side of its mean value is only 1 part in 400. Even this small range of variation does not occur
except between points which are many thousands of kilometers apart. As will be shown later,
the postulated variations in mean densities are about 1 part in 30 on each side of an average
value. Hence, it is not advisable to complicate the conception of isostasy and introduce long
circumlocutions into its definition in order to introduce the refinement of considering the
variations in the intensity of gravity.
The variation of the intensity of gravity with change of depth below the surface need
not be considered, as its effect in the various columns of apaterial considered will be substantially
the same.
The idea implied in this definition of the phrase “depth of compensation,” that the isostatic
compensation is complete within some depth much less than the radius of the earth, is not
ordinarily expressed in the literature of the sub-
ject,* but it is an idea which it is difficult to
Sea level avoid if the subject is studied carefully from any
point of view. |
Surface of
ASSUMPTIONS AS TO ISOSTASY.
In the computations of the investigation here
published the depth of compensation is assumed
to be 113.7 kilometers under every separate por-
tion of the earth’s surface.
This is substantially the value given in The
Figure of the Earth and Isostasy, page 175. It
was the best value available at the time the com-
putation of the gravity reduction tables pub-
lished herein was commenced. A better value,
122 kilometers, became available while these com-
on putations were in progress, but too late to be used.
Cok
rc (See Supplementary Investigation in 1909 of The
ILLUSTRATION No. 2.—Three unit columns showing approximate
depth of isostatic compensation as used in computations. Figur e of the Earth, p- 77. )
The mean surface density of the earth—that
is, the mean density of the solid portion of the earth for the first few miles below the surface—
is assumed in this investigation to be 2.67. The phrase ‘of the solid portion of the earth” is
inserted in the preceding sentence to indicate that the ocean, with a density of only 1.027, is
excluded from this mean.
The computations concerned in this investigation were actually made on the assumption
indicated in illustration No. 2 instead of those indicated in illustration No. 1 and used on
pages 7-9. This slight change was made to simplify and facilitate computations and is justified
by the fact that the errors so introduced are negligible, as shown later under the heading
“Discussion of errors.’”’ In illustration No. 1 and in the corresponding text, the compensation
is assumed to extend everywhere to a depth of 113.7 kilometers below sea level. In illustration
No. 2 and in the actual computations, the compensation is assumed to extend everywhere to
a depth of 113.7 kilometers measured downward from the solid surface of the earth—that is,
from the land surface in land areas (above sea level) and from the ocean bottom in oceanic
areas (below sea level). For land areas, in computing the direct effect of the topography, the
portion above sea level was assumed to have the density 0 as indicated in illustration No. 2,
but in computing the effect of the isostatic compensation the density was assumed to be d—0,
* See, however, a reference to Pratt’s Hypothesis in Helmert’s Héhere Geodisie, II Theil, p. 367.
} For the data and considerations upon which this value is based, see The Solar Parallax and its Related Constants, by William Harkness,
Washington, Government Printing Office, 1891, pp. 91-92; see also The Figure of the Earth and Isostasy from Measurements in the United States,
p. 128.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 11
above sea level as well as below, 6, being computed from formula (3). The seacoast column
is the same in the two illustrations. Upon the assumption indicated in illustration No. 2 and
used in actual computations for oceanic compartments, formula (6) becomes
0.615 D
i (6a)
3,=0
In the computations of this investigation the compensation under each separate portion
of the earth’s surface is assumed to be uniformly distributed with respect to depth from the
surface down to the depth of compensation, 113.7 kilometers. In other words, the compen-
sating defect or excess of density under a given area is assumed to be, at all depths less than
the depth of compensation, exactly equal to the 6, of equations (3) and (6), which was defined
as being the average defect (or excess) of density.
Elsewhere * it has been assumed temporarily for investigation purposes that the compen-
sating defect (or excess) of density varies with respect to depth, being for example greatest
near the surface and diminishing uniformly to zero at the depth of compensation, its average
value being 06,.
In the pringival computations of this investigation the isostatic cotapenwation is assumed
to be complete under every separate portion of the earth’s surface, however small the area
considered. That is, equations (3) and (6) are assumed to be true los every separate unit of
area even though a very small unit be chosen, as for example, 1 square foot.
The authors do not believe that any one of these assumptions upon which the computa-
tions are based is absolutely accurate. The mean surface density is probably not exactly 2.67
and the actual surface density in any given area probably does not agree exactly with the mean.
The depth of compensation is probably not exactly 113.7 kilometers, and it possibly is some-
what different under different portions of the earth’s surface. The compensation is probably
not distributed uniformly with respect to depth. It is especially improbable that the com-
pensation is complete under each separate small area, under each hill, each narrow valley,
and each little depression in the sea bottom. It is exceedingly improbable, for example, that
as each ton of material is eroded from a land area, carried out of a river mouth, and deposited
on the ocean bottom, that corresponding changes of isostatic compensation occur at the same
time under the eroded area and under the area of deposition at just such a rate as to keep the
compensation complete under each.
The authors believe that the assumptions on which the computations are based are a
close approximation to the truth. They believe also that the quickest and most effective
way to ascertain the facts as to the distribution of density beneath the surface of the earth
is to make the assumptions stated, to base upon them careful computations for many observa-
tion stations scattered widely over the earth’s surface, and then to compare the computed
values with the observed values of the intensity of gravity in order to ascertain how much and
in what manner the facts differ from the assumptions.
In this investigation, accordingly, the intensity of gravity at many observation stations
has been computed on the assumptions stated. These computed values have been compared
with the observed values at these stations. The differences between the observed and the
computed values, the residuals, are due to two classes of errors. In the first class are errors in
the observations and in the computations. In the second class are errors in the assumptions.
The average and maximum magnitudes of the errors of the first class are fairly well known.
The magnitude and character of the residuals which may be produced by them are fairly well
known. It is shown in this publication that the residuals, differences between observed and
computed values of the intensity of gravity, are larger than may be accounted for by the first
class of errors. Therefore it is certain that the second class of errors are of appreciable size.
In other words, it is certain that the assumptions are appreciably in error. But, as the residuals
are but little larger than may be accounted for by the first class of errors, it is certain that the
assumptions are nearly correct.
* The Figure of the Earth and Isostasy from Measurements in the United States, pp. 156-163.
12 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
The residuals contain evidence not only as to the extent but also as to the manner in which
assumptions depart from the truth. To read and interpret this evidence precisely is exceedingly
difficult, because of the fact that the residuals are small. If the residuals were large, it would
be clear that the assumptions were far from the truth, and it would be easy to see in which
direction the truth lay. In the actual case it is difficult to ascertain in what way the assump-
tions should be changed to make them a closer approximation to the whole truth, while still
remaining a statement of general laws applicable to the whole United States.
FORMULE.
It was desired to compute the intensity of gravity at any selected station on the earth upon
the assumptions as to isostasy which have been stated. It was necessary to select the formule
and methods of computation.
The computations may be most conveniently made in two parts.
First, the intensity of gravity may be computed on an ideal earth having the same size
and shape as the ellipsoid of revolution which most nearly coincides with the sea-level surface
of the real earth, and having no topography and no variations in density at any given depth
below the surface. To convert the real earth into this ideal earth all material on the real earth
above sea level must be removed, the water of the ocean must be replaced by material of den-
sity equal to the mean surface density of the real earth, and all variations in density at any
given depth in the real earth must then be removed by taking out or injecting enough material
in each part to make the density conform accurately to the mean density in the real earth at that
depth. In this ideal earth the density will increase with increase of depth in the same manner
as it does upon an average in the real earth, but in the ideal earth all masses lying at the same
depth will have the same density, whereas in the real earth such masses have densities which
are known to differ slightly from each other.
This computation was made by using Helmert’s formula of 1901,* namely,
+ 7o=978.046(1 +0.005 302 sin 26—0.000 007 sin 724) (7)
The symbol ;, stands for the required value of gravity at a station on the ideal earth above
described in the latitude ¢. On such an ideal earth the value of gravity at the surface would
be a function of the latitude only, as expressed by this formula. The numerical value of 7, com-
puted from formula (7) is both the acceleration of gravity in centimeters and the attraction
of gravity in dynes on a unit mass (1 gram) at the station expressed in the centimeter-gram-
second system.
The form of this formula is fixed by theory. The three constants which it contains, namely,
978.046, 0.005 302, and 0.000 007, were computed from a large number of observations of gravity
at stations scattered widely over the earth’s surface. New and better values of these constants
may be obtained by further research and the use of more observations, but at the beginning of
this investigation the formula as written was believed to be the best representation available
s
* Der normale Theil der Schwerkraft im Meeresniveau, von F. R. Helmert, S. 328-336, Sitzungsberichte / der K6niglich Preussischen / Akademie
der Wissenschaften / zu Berlin, / Jahrgang 1901 / Erster Halbband, Januar bis Juni. See also Bericht tiber die relativen Messungen der Schwers
kraft mit Pendelapparaten fiir den Zeitraum von 1900 bis 1903, unter MAtwirkung von F. R. Helmert erstattet von E. Borrass, S. 133-136, Verhand-
lungen / der vom 4 bis 13 August 1903 in Kopenhagen abgehaltenen / Vierzehnten Allgemeinen Conferenz der / Internationalen Erdmessung / Redi-
girt vom stindigen Secretar H. G. van de Sande Bakhuyzen. / II. Theil: Spezialberichte. See also The Figure of the Earth and Isostasy from
Measurements in the United States, p. 172, for some comments upon this formula.
+ After the manuscript of this publication was completed, a letter addressed to the Superintendent, of which the following is a translation, was
received from Dr. Helmert:
: ) PoTsDAM, October 31, 1911.
es Bowie sent to me a small brochure for which I offer my best thanks to the sender and to you. Permit me to make a remark in regard to
my formula.
In 1901 I did indeed give:
-yo= 978.046 (1+-0.005302 sin 26—0.000007 sin 2 2 4)
This formula is based on the value of g in the Vienna system (Sterneck).
The American values of g are, however, referred to Potsdam. The constant 978.046 must, therefore, be modified by the application of —0.016
by which correction it is referred to Potsdam, as I have several times stated in my reports.
I therefore request that in your investigations in North America you will use the value
7o= 978.030 (1-++0.005302 sin 26—0.000007 sin 2 2 6)
as being my improved formula. f
I know that your scientists think that the value 978.038 is more suitable for the United States. That value, of course, may be used. I only
wanted to emphasize that, in so far as my work is concerned, the value of g in the United States is not 978.046, but 978.030.
Tt is clear that the values of gravity in the United States, used in this publication, are hased upon Potsdam, as shown on p. 73, and that, there-
fore, the position taken by Dr. Helmert in this letter is correct. The only manner in which this change ultimately affects the conclusions reached.
in this publication is shown on p. 75. .
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 13
of gravity at sea level on the ideal earth described in the preceding paragraph. During the
progress of this investigation a small correction to the constant 978.046 was derived, as shown
later in this publication. The formula, with this small correction applied, is believed by the
authors to be the best available at present for the purpose for which it is intended.
The Helmert formula of 1901 corresponds to a value of 298.3+0.7 for the reciprocal of the
flattening of the earth. This is in fair agreement with the best value now available for this
quantity as derived from observed deflections in the United States, namely, 297.0+0.5.*
The stations at which observations of gravity were made are situated on the real earth,
not the ideal earth, and are in general above sea level, not at sea level. The second portion of
the computation of the intensity of gravity at any observation station must therefore take ac-
count of the topography which exists upon the real earth, take account so far as is possible of
the variations in density beneath the surface of the real earth, and take account of the effect
of the elevation of the observation station above sea level.
The correction for elevation was computed by the formula
—0.000 308 6 H
in which H is the elevation of the station above sea level in meters. This correction of the
attraction upon a unit mass (1 gram) at the station is in dynes and reduces from sea level to
the actual station. It takes account of the increased distance of the station from the attract-
ing mass, the earth, as if the station were in the air at the stated elevation and there were no
topography on the earth. This is an old formula and needs no comment other than that it
has been adopted in this simple form by Dr. Helmert as being sufficiently accurate.
The real difficulty of the investigation was encountered when an attempt was made to
compute the effect, upon the attraction at a given station, of the topography which exists upon
the earth and of the isostatic compensation of that topography which is assumed to exist be-
neath the surface of the earth. For this purpose new formule and new methods of computa-
tion were found to be necessary.
It was desired to compute the effect upon the attraction at each station of all the topography
of the world and of the isostatic compensation of that topography. It was desired to do this
with sufficient accuracy to insure that all constant errors in the computed effects would cer-
tainly be less than 1 part in 200 and all accidental errors in the separate parts of the computa-
tion less than .0002 dyne. This, it was believed, would insure that the computed total correction
for any station would ordinarily be in error in so far as the computation alone is concerned by
less than 0.003 dyne. In order to make this computation with the specified degree of accuracy
with a minimum expenditure of time and energy the formule and methods of computation
about to be given were selected and used. This publication contains full information as to the
degree of success with which the computations were made, both as to accuracy and rapidity.
This degree of success is the proper measure of the excellence of the formule and methods of
computation selected.
The attraction of any elementary mass, dm, acting upon a mass of 1 gram at the station
of observation is, in dynes,
kdm
D (8)
in which k is the gravitation constant and D is the distance from the station to the elementary
mass. In order to get the result in dynes all quantities in this formula must be expressed in
the centimeter-gram-second system. :
The general expression for Newton’s law of gravitation is
MM.
F=k3/? Te z (8a)
* See Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, pp. 60, 77.
Tt See p. 651 of “ her die Reduction der auf der physichen Erdoberfliiche beobachteten Schwerebeschleunigungen auf ein gemeinsames Niveau
Von F. R. Helmert in Sitzungsberichte der K6niglich Preussischen Akademie der Wissenschaften 1903 Erster Halbband.
14 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
in which m, and m, are two masses each of dimensions infinitesimal in comparison with the
distance D between them and F is the attraction between the two masses. Newton’s law of
gravitation is frequently expressed merely in the form of a proportion, F being stated to be pro-
portional to oe The gravitation constant, k in formula (8a), is the factor by which the
product of two masses divided by the square of their distance asunder must be multiplied in
order to express the force exerted by those masses on one another. The gravitation constant
is not a mere numeral. Its dimensions are shown by the exponents in (L**M“T-*) if L, M, T
denote the units of length, mass, and time, respectively. That is, the gravitation constant is the
cube of a distance divided by the product of a mass and the square of a time.
Formula (8) is merely the special case of formula (8a) which is pertinent to the problem in
hand.
The value adopted in this investigation for k in the centimeter-gram-second system is
6673 (10). The basis of this adopted value is as follows, as stated by Dr. R. S. Woodward :*
In spite of the superb experimental investigations made particularly during the past quarter
of a century by Cornu and Baille (Comptes rendus, LX:XVI, 1873), Poynting (The Mean Density
of the Earth, by J. H. Poynting, London, Charles Griffin & Co., 1894), Boys (Philosophical
Transactions, No. 186, 1895), Richarz and Krigar-Menzel (Sitzungsberichte, Berlin Academy,
Band 2, 1896), and Braun (Denkschriften, Math. Natur. Classe, Vienna Academy, Bd. LXIV,
1897), it must be said that the gravitation constant is uncertain by some units in the fourth
significant figure, and possibly even by one or two units in the third figure.
The results of the investigators mentioned for the gravitation constant are, in C. G. S.
units, as follows, the first result having been computed from data given by MM. Cornu and
Baille in the publication referred to:
Cornu and Baille (1873) 6668 (10-1!)
Poynting (1894) 6698 (10711)
Boys (1894) 6657 (107!)
Richarz and Krigar-Menzel (1896) 6685 (10!
Braun (1897) 6658 co
Regarding these as of equal weight, their mean is 6673 (107) with a probable error of +5
units in the fourth place, or 1/1330th part. This is of about the same order of precision as that
deduced by Prof. Newcomb from astronomical data.
The uncertainty in the adopted value is, however, within allowable limits for the present,
investigation.
The vertical component at the observation station of the attraction expressed in formula
(8) is, in dynes,
sin f
kame (9)
in which f is the angle of depression, below the horizon of the station, of the straight line from
the station to the elementary mass.
This vertical component is all that is concerned in this investigation. The integral of all
such vertical components at the station, corresponding to all the elementary masses which
together constitute the earth, is the vertical force due to gravitation which acts on a mass of
one gram placed at the station. This vertical force expressed in dynes is necessarily numerically
equal to the acceleration (both being expressed in the centimeter-gram-second system) which
would be produced by gravitation acting upon any mass at the station left free to fall. They
are, of course, affected by the centrifugal force due to the earth’s rotation, but this effect need
not be considered in the discussion of these formule.
The term, gravity, is used in its generally accepted sense; that is, it is the resultant of the
earth’s gravitation and the centrifugal force due to the earth’s rotation.
* See p. 153 of an address entitled “‘The Century’s Progress in Applied Mathematics,” by R.S. Woodward, Bulletin of the American Mathe-
matical Society, 2d Series, Vol. VI, No. 4, pp. 133-163. In this address and in another by the same author entitled ‘‘ Measurement and Calcu-
lation,” published in Science, new series, Vol. XV, No. 390, pp. 961-971, June 20, 1902, are given excellent statements of the nature of the gravi-
tation constant and the importance of determining its value accurately.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 15
In general it will be found that throughout this publication the attraction (expressed in
dynes) is dealt with directly by preference rather than its numerical equivalent, the acceleration
(expressed in centimeters and seconds). This preference is due to the belief that thereby
circumlocutions are avoided and greater clearness secured in the conceptions.
If the station and the elementary mass, dm, are at the same elevation referred to sea level
0
b=5
and
_ O
D=2r sin 5
(see illustration No. 3), in which @ is the angle at the center of the earth subtended between the
station and the elementary mass and r is the radius of the earth.
If absolute accuracy were desired it would be necessary to use for r the average radius of
curvature, between the station and the mass considered, of the equipotential surface in which
they both lie. This average radius depends upon the elevation above sea level and also, since
the sea-level surface is an ellipscid of revolution (not a sphere), it depends upon the latitude
of the station and the azimuth of the line from the station to the mass under consideration.
But with sufficient accuracy for this investigation r is assumed to be constant with the value
637 000 000 centimeters in this and similar formule. This is equivalent to assuming, in
deriving these formule, that the station is on the surface of a spherical earth having the radius
stated. Under the heading “Discussion of errors”’ it will
be shown that this assumption is far within the allowable
limits of approximation.
By substituting these values of @ and D in (9) there
is obtained as the formula for the vertical component of
the attraction in dynes upon a unit mass at the station,
due to an elementary mass which is at the same elevation
as the station,
sin :
kdm——,
4r? sin? 5
=kdm E (10)
The single symbol F is used to represent that portion
of the formula
si g
im 5
4r? sin? 5
which depends simply upon the direction and distance of
the elementary mass from the station, because later it is
most convenient to deal with E separately from k and dm.
ILLUSTRATION No. 3.—Showing station and ele-
mentary mass at same elevation.
To divide both the numerator and denominator of (10) by sin 5 would simplify the expression,
but by so doing the close analogy between (10) and the more complicated expressions (15) and
(16) would become less obvious.
In each of the illustrations Nos. 3, 4, and 5, S represents the gravity station, and the circle
represents the intersection of the level surface which lies at the elevation of the station with a
plane defined by the station, the center of the earth (C), and the elementary mass considered.
16 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
B is the location of the elementary mass dm, £ is the angle between the horizon of the station
(SH) and the straight line from the station to B, and D is the distance from the station to B.
In illustrations Nos. 4 and 5, D, is the distance from the station to a point A at the same ele-
vation as the station and in the same vertical line as B, the location of the elementary mass.
Illustration No. 4 represents the case in which the elementary mass, dm, is higher than the
station, the difference of elevation being 4. In the triangle SAB, from the law of proportional
sines,
h cos :
: 2 (11)
ao
also, in this triangle, according to plane trigonometry,
D?=D2+h? +2D,h sin § (12)
From illustration No. 4 it appears that
a=3 (13)
and
B=Pa-Be (14)
By substituting from formule (11), (12), (13), and (14)
in (9) there is obtained as the formula for the vertical com-
ponent of the attraction in dynes upon a unit mass at the sta-
ticn, due to an elementary mass which is higher than the station,
6 h cos
sin{ 5— Se
Di +h+2D,4 sin 5 (15)
kdm =kdm E,
De +h? +2D,h sin :
Here again a single symbol, E,, is taken to represent that por-
tion of the formula which depends simply upon the direction
and distance of the elementary mass from the station.
Illustration No. 5 represents the case in which the elementary mass, dm, is lower than
the station, the difference of elevation being h. By the same process as that used above it
may be shown that the vertical component of the attraction in dynes upon a unit mass at the
station, due to an elemcntary mass which is lower than the station is
ILLUSTRATION No. 4.—Showing elementary
mass at greater elevation than station.
h cos g
hes ta 2
sin 5 +sin™ Sa
4/ D2 + —2D,hsin5 (16)
kdm 9 =kdmE,
D?2+h?—2D,A sin 3
in which £, is used to represent that part of the formula which depends simply upon the direc-
tion and distance of the elementary mass from the station.
It is important to note that the only approximation made in deriving formule (10), (15), and
(16) is that to which attention has already been called, namely that the radius of curvature
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 17
concerned at each station is 637 000 000 centimeters. In every other respect the derivation of
the formule is exact regardless of the distance of the attracting mass from the observation
station. The attracting mass may even be located at the antipodes of the station. These
formule were used in connection with all attracting masses which are so far from the station
that the curvature of the sea level surface must be taken into account in order to insure that
the errors of computation of the effects are less than 1 part in 200.
For masses near the station, the well-known formula for the attraction of a mass having
the form of a right cylinder upon a point outside the cylinder and lying in its axis produced
was utilized.* This formula, in a convenient form for the
present purpose, for the attraction in dynes upon a unit mass,
(1 gram) at the station, is
k2nd{ VP +h — Vet (h+t)?+¢} (17)
in which & is the gravitation constant, 0 is the density of the
material, c is the radius of the cylinder, ¢ is the length of an
element of the cylinder, and h is the distance from the attracted
point, the station, to the nearest end of the cylinder.
For a mass which has the form of a cylindrical shell, that
is, the difference of two concentric right cylinders of the same
length having different radii, c, and c, formula (17) becomes
hond{ Vee +h? — yep +h?— yee + h+t+ yer+ hte} (18)
This is the attraction in dynes upon a unit mass (1 gram) at
the station.
The formule (17) and (18) are exact if applied to cylinders
and cylindrical shells.
The justification of the radical departure from past prac-
tice represented by formule (10), (15), and (16), and by the ee ee
introduction of the gravitation constant into formule (17) and
(18) is the success attained thereby in securing quick and accurate computations. The reader is
therefore requested to suspend judgment until the remainder of this publication has been read
and the degree of success has been compared with that obtained by the use of any other
formule with which comparison is made.
DIVISION OF THE SURFACE OF THE EARTH INTO ZONES AND COMPARTMENTS.
In order to apply formule (10), (15), (16), (17), and (18) to the computation of the effect
of the topography and the isostatic compensation, the whole surface of the earth was divided
into zones by circles, each having the station at its center, and each zone was divided into equal
compartments by radial lines. The division adopted is shown in the following table. Tllus-
trations Nos. 10a@ and 10), page 48, show the shapes of certain compartments.
* For two statements of this formula see A Treatise on Attractions, Laplace’s Function and Figures of the Earth, by John TH. Pratt, third
edition, p. 46, and Traité de Mécanique Céleste, F. Tisserand, Tome II, pp. 71-72.
15593°—12——2
18 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
Deslenotion colnet of nee aes of Compartments
Meters Meters
A 0 2 1
B 2 68 4
C 68 230 4
D 230 590 6
: E 590 1 280 8
F 1 280 2 290 10
G 2 290 3 520 12
H 3 520 5 240 16
I 3 240 8 440 20
J 8 440 12 400 16
K 12 400 18 800 20
L 18 800 28 800 24
M 28 800 58 800 14
N 58 800 99 000 16
0 99 000 166 700 28
° 7 VW ° / ut
18 1 29 58 1 41 18 1
17 1 41 13 1 54 52 1
16 1 54 52 2 ll 53 1
16 2 11 53 2 33 46 1
14 2 33 46 3 03 05 1
13 3 03 05 4 19 138 16
12 4 19 13 5 46 34 10
ll 5 46 34 7 51 30 8
10 7 51 30 10 44 6
9 10 44 14 09 4
8 14 09 20 41 4
7 20 41 26 41 2
6 26 41 35 58 18
5 35 58 51 04 16
4 51 04 72 #13 12
3 72 13 105 48 10
2 105 48 150 56 6
1 150 56 180 1
For the numbered zones it was found to be more convenient to use the radii of the zone in
degrees and minutes of a great circle than in meters. The inner radius of zone 18 is the same
as the outer radius of zone O, that is, on a sphere of the adopted size, radius 637 000 000 centi-
meters, 1° 29’ 58’’ of a great circle (the inner radius of zone 18) has a length 166 700 meters (the
outer radius of zone O). Zone A commences at the station, and zone 1 ends at the antipodes of
the station. All the zones together cover the earth completely.
Zone A, with a single compartment, is a circle about the station with a radius of 2 meters.
Similarly, zone 1, with a single compartment, is a circle about the antipodes of the station with
a radius of 29° 04’ (3240 kilometers). Zones 18 to 14 each have a single compartment. All
other zones have from 2 to 28 compartments each, the number of compartments being even in
each case.
For each zone a special reduction table was prepared in the manner indicated hereafter
under the heading, ‘‘Computation of reduction tables.’”’ This table for each zone gives the
relation between the mean elevation of the surface of the ground in each compartment of that
zone and the effect of topography and the isostatic compensation in that compartment upon
the vertical component of the attraction at the station.
In making the arbitrary selection of radii of zones and of the number of compartments in
each zone, it was necessary to consider the effect of the size and shape of the compartment;
first, upon the time required to complete the computations; second, upon the accuracy of the
computations in so far as it depends upon the accuracy of the estimates made by the computer
of the mean elevation within each compartment; and, third, upon the accuracy of certain
necessary assumptions in the computation.
The larger the compartments are made, the smaller will be the number of compart-
ments, and therefore the smaller the number of estimates of mean elevation to be made, one for
each compartment. But as the compartments are made larger, the time required for each
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 19
estimate becomes greater. For with a large compartment it is necessary to estimate the mean
elevation more closely to secure a given degree of accuracy than with a small compartment; to
estimate to the nearest hundred feet, for example, instead of to the nearest thousand feet.
Also, the larger the compartment the greater the total range of elevations within the compart-
ment, and therefore the greater the time necessary to secure an estimate of the mean to a given
degree of accuracy. Hence the adoption of compartments either too large or too small would
have made the time required for the computation greater than would otherwise have been
necessary.
There are 317 compartments in all, 199 in the 15 lettered zones near the station, and 118
in the 18 numbered zones, all of which are more than 166 kilometers from the station.
It is believed that the size and shape of each compartment has been so fixed that the error
of computation for any compartment is ordinarily less than 0.0002 dyne, and is of the accidental
class. The basis of this belief will be indicated in connection with the topic, ‘Discussion of
errors.” Jt is known that notable success has been attained in securing rapid computation.
With the experience now available, a better selection of radii of zones, and of numbers of
compartments in each zone could be made. But such a new selection would make it necessary
to recompute the reduction tables. It is not probable that the improvement would be sufficient
to warrant this recomputation.
COMPUTATION OF REDUCTION TABLES FOR NEAR ZONES.
For zone A, comprising the surface of the earth in a circle around the station with a radius
of 2 meters, the reduction table was computed by formula (17).
The effect of the topography in this zone, if the station is on land, is the effect of a cylinder
of material having the density, 0, assumed to be the mean surface density of the earth, namely,
2.67, having a radius c=200 centimeters and a length, t, equal to the elevation of thestation. In
the formula h=o for this case, as the station is at the suid of the cylinder in question. The eleva-
tion of the surface of the ground in all parts of this small zone is assumed in the computation to
be the same as the elevation of the station.
In the computations it was necessary, of course, to express all distances in centimeters to
conform to the adopted value of k, which is expressed in the centimeter-gram-second system.
(See p. 13.)
The attraction computed is evidently a vertical force, as the station lies in the axis of the
cylinder, which is vertical.
The effect of the corresponding isostatic compensation was computed from the same
formula (17) with the same values of ¢ and A, but with ¢=11 370 000 centimeters, the assumed
depth of compensation (it should be remembered that compensation is assumed to begin at the
surface of the ground and at the bottom of the sea, see page 10), and with a value of 6, from
formula (3), page 8, substituted for 6, namely, 0, =n? 67-35 B00" in which Z is the
elevation of the surface above sea level (assumed to ie the same as the elevation of the station).
The isostatic compensation is thus treated as a cylinder of material of a negative density 0,,
or, in other words, as a negative mass just equal to the positive mass which would exist in this
zone above sea level if the actual density of all material in the zone above sea level were 2.67.
For a land compartment the computed effect of the topography is positive, an increase in
the downward attraction upon a unit mass (1 gram) at the station. The computed effect of
the isostatic compensation is negative, a decrease in the downward attraction upon a unit mass
at the station. The difference of the two is the resultant effect of the combined topography
and isostatic compensation. This resultant effect was computed for various assumed values
of the elevation of the station above sea level, and then the reduction table for zone A written
as shown on page 30. An inspection of the table will make it clear that as soon as a few of
the tabular values had been computed the remainder could be safely interpolated with the
required degree of accuracy.
20 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
To apply formula (17) to a station at sea, such as those occupied by Dr. Hecker on the
Atlantic and Pacific,* it is necessary in computing the effect of the topography to substitute
for 0 in formula (17) the value (0—6,) (see pp. 8-9, and illustration No. 2), the defect of
density of sea water in comparison with solid earth. The value of h is zero, the station being
assumed to be at sea level. The mass thus considered is a mass which is the difference between
that actually contained in the cylinder of radius 200 centimeters extending from the station at
sea level down to the bottom of the ocean, and the mass which would fill this same cylinder if
solid earth with a density of 2.67 were substituted for the sea water.
To apply formula (17) to a station at sea, in computing the effect of the isostatic com-
pensation, it is necessary to substitute for the 0 of formula (17) the
f iret, Gabe value of 6, computed by formula (6a), page 11, namely
0.615 D 0.615 D
Qo 7 2.8777 370 000
in which D is the depth of the water. In this case the h of formula
(17) is not zero but equal to D, as the upper limit of the compensa-
tion is at the ocean bottom at a distance D below the station.
a For an oceanic compartment the computed effect of the topog-
Sesundt Coad raphy (in this case submerged topography, or hydrography) is
negative, a decrease in the downward attraction upon a unit mass
at the station. That is, the attraction is less than it would be if
in the compartment from the ocean bottom to sea level material of
density 2.67 were substituted for the sea water which is actually in
this space. The computed effect of the isostatic compensation is
positive, an increase in the downward attraction upon a unit mass
at the station, for the compensation is in this case an excess of
density and of mass. The resultant effect is in this case again a
numerical difference.
Similarly formula (18). was used in computing the reduction
tables for zones B to O inclusive. It was used separately for the
topography and the isostatic compensation, and the results were
also combined. ‘The values for the 6 of formula (18) were the same
as have been stated already in connection with the application of
formula (17).
In using formula (18) to compute the effect of the topography
E at in land zones three cases arise.
: First, when the mean elevation of the surface of the ground in
the zone is the same as the elevation of the station, h is zero in
7 formula (18). (See illustration No. 6.) In this case the attracted
point, the station, is in the plane of the upper end of the cylindrical shell considered. This
cylindrical shell contains all the material in the zone, from the actual surface of the
ground down to sea level, the inner and outer radii of the shell being the same as the inner and
outer radii of the zone, and the length of an element of the cylindrical shell being the mean
elevation of the surface of the ground.
Second, when the station is above the mean elevation of the surface of the ground in the
zone, as indicated in the second case in illustration No. 6, h is the difference of elevation between
the station and the mean surface of the ground in the zone, and in other respects this case is
similar to the first one. For any land zone the computed effect of the topography in either
the first or the second case is always positive, an increase in the downward attraction at the
station.
ee
Third Case
a
t
t
—
ILLUSTRATION No. 6.—Showing topog-
raphy in land zones—three cases.
* Bestimmung der Schwerkraft / auf dem / Atlantischen Ozean / Sowie in / Rio de Janeiro, Lissabon und Madrid / Mit Neun Tafeln / von O.
Hecker; Berlin, 1903. Bestimmung der Schwerkraft / auf dem / Indischen und Groszen Ozean / und / An Deren Kiisten | Sowie Erdmagnetische
Messungen / Mit Zw6lf Tafeln. / von Prof. Dr. O. Hecker; Berlin, 1908.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 21
Third, when the station is below the mean elevation of the surface of the ground in the
zone, as indicated in the third case in illustration No. 6, the cylindrical shell containing the
topography is considered broken into two separate cylindrical shells, one above the other,
indicated as shell A and shell B in the illustration, and formula (18) is applied separately to the
two shells. Shell A extends from sea level to the elevation of the station, and its effect is com-
puted exactly as was that of the shell in the first case. Shell B contains the remainder of the
material in the zone above sea level. It extends from the level of the station up to the mean
elevation of the surface of the ground in the zone. In this shell c, and c, have the same values
as in shell A, h is zero, the station being in the plane of the lower end of the shell, and ¢ is the
difference between the elevation of the station and the mean elevation of the surface of the
ground in the zone. The effect of the material in shell B is an upward attraction at the station.
Hence the resultant effect at the station of the topography in this case is the difference of the
separate effects of shell A and shell B. This resultant effect will evidently be positive, a down-
ward attraction, if shell A is longer than shell B, and will be negative if shell B is the longer.
If the station is at an elevation exactly one-half of the mean elevation of the surface of the
ground in the zone, shell A and shell B are of equal lengths, and the resultant effect is zero.
For oceanic zones the first and second cases arise, but never the third case. Hence for
oceanic zones the computed effect is always negative, the downward attraction at the station
being always less than it would be if material of density 2.67 were substituted for sea water.
In applying formula (18) to the computation of the effect of the isostatic compensation
for land zones all three of the cases described above arise. Hence, in the third case, the effect
of the compensation was obtained by computing separately the effects of two shells correspond-
ing toshell A andshell B. In computing the effect of the compensation the length of an element
of the shell is 11 370 000 centimeters (the depth of compensation) in the first and second cases,
11 370 000 centimeters minus the difference between the elevation of the station and the mean
elevation of the ground in the zone in shell A of the third case, and simply the difference between
the elevation of the station and'the mean elevation of the surface of the ground in the zone in
shell B of the third case. In all these cases, including both shells in the third case, the value
to be used for 6 in formula (18) is that computed from formula (3), page 8, in which the
mean elevation of the surface of the zone is to be used for H and the assumed depth of com-
pensation for f,. As shell A is always much longer than shell B in connection with the com-
pensation, its effect always predominates, and the computed effect of the compensation for
these zones is always negative, a decrease of downward attraction at the station.
In applying formula (18) to the computation of the effect of the isostatic compensation
for oceanic zones the second case is the only one which arises, and the computed effect of the
compensation is always positive, an increase in the downward attraction at the station. The
value to be used for 6 in formula (18) is computed from formula (6a), page 11.
For zones B to O the combined effect of topography and compensation is not always a
numerical difference of the separate effects. In a few rare cases for land zones, namely, when
shell B of the third case happens to be longer than shell A, the effects at the station of the
topography and its compensation are both negative, and their combined effect is the numerical
sum.
To avoid circumlocutions a few paragraphs just preceding this have been worded as if
the mean elevation for the whole of each zone was dealt with in the computation. In zone F,
see page 18, which is divided into 10 equal compartments, the effect of the topography or of the
compensation in any one compartment upon the vertical component of the attraction at the
station is evidently exactly one-tenth of that computed for the whole zone from formula (18),
provided the elevation of the surface of the ground is the same throughout the zone. The
actual practice was to use formula (18) in computing the effects for a whole zone at once, then
to divide the result by the number of compartments in that zone (10 for zone F) to obtain
the effect of each compartment. These effects for separate compartments were then tabulated
in the reduction tables, and in using these tables the mean elevation for each compartment.
was used, not the mean elevation for the whole zone.
22 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
It was not found necessary to compute each separate value in the reduction tables for
zones B to O. For each of these tables a few scattered values, in each of several selected
columns, were computed. For each selected column the points so computed were plotted on
cross-section paper, using the assumed mean elevation of the compartments as abscisse and
the computed values as ordinates. When the number of plotted points was sufficient to
enable one to do so with the required degree of accuracy, a curve was drawn through these
points to represent all the required values corresponding to the column in question. The
intermediate values for the column were then scaled from the curve and entered in the table,
together with the computed values. After the values in a few columns of the table had been
so obtained, it obviously became possible to interpolate the values for the remaining columns
with the required degree of accuracy. The vertical differences in the columns, filled in from
the computations and curves, served as checks in making these interpolations.
Illustration No. 7 shows the curves used as indicated above in connection with the reduction
table for zone E. On each curve the computed points are indicated by small circles. The
curves were drawn by eye, using a draftsman’s flexible ruler. The shape of each curve and its
position relative to the other curves furnish a sensitive check for detecting errors in the plotted
values due to the computations or plotting.
As the computations of the reduction tables by formula (18) could be made much more
easily than by formale (10), (15), and (16), it was desired to extend the use of formula (18)
to as many zones as possible. It was found that out to zone L the errors secured by the use
of formula (18) in the manner already described were within allowable limits. It appeared
that when formula (18) was applied to zone O the principal error arose from the fact that a
point in the middle of this zone which is at the same elevation as the station lies 4500 feet below
the horizontal plane of the station on account of the curvature of the sea-level surface. It
appeared that possibly this particular error could be eliminated and a very close approximation
to the truth obtained by using for the A of formula (18) not the difference of elevation between
the station and the mean surface of the ground in the zone, but instead the difference of elevation
between the station and a point 4500 feet below the mean elevation of the zone. This would
have the effect of making the second correction in the table zero if the mean surface of the com-
partment lay in the horizontal plane of the station. Accordingly, the column headed “Station
above compartment, 800 feet,’ in the reduction table for zone O, was computed with a value
4500 +800 = 5300 feet for h, the next column with a value 4500 + 1600 =6100 feet for h, and so
on. Similarly the values in the column headed ‘‘Station below compartment, 800 feet,” were
computed with the value 4500 — 800 =3700 feet. The corrections in the column headed “Station
at same elevation as compartment”’ are applicable to compartments in which the mean elevation
of the surface of the ground is the same as that of the station. These values were computed
with h = 4500 feet in formula (18).
Similar modifications to take account of the curvature approximately were made in the
tables for zones M and N, but for zones nearer the station it appeared that such changes would
not amount to as much as 0.0001 dyne, and they were therefore not computed.
After computations for zone O were made by formula (18), using the modified method
indicated in the preceding two paragraphs, in which method the curvature of the sea-level
surface is taken into account in part, certain values of the table were also computed by formule
(10), (15), and (16), which are exact, the curvature being fully taken into account. This test
showed that the tabular values as computed by formula (18) by the method described are each
within 0.0002 dyne, and are in error by less than 1 part in 200 on an average. This made it
certain that the errors in zones M and N, and other zones nearer the station than zone O, are
well within the adopted limits. The test in zone O also indicated that for the next larger zone
the adopted limits of error might be exceeded if formula (18) were used, even with the modifi-
cation described. Therefore formule (10), (15), and (16) were used for all zones beyond O.
’
.
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EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 23
COMPUTATION OF REDUCTION TABLES FOR DISTANT ZONES.
To use these formule in computing the effect of the topography within a given zone for a
land area it is necessary to integrate the expression kdmE to include all elementary masses
within that zone between the surface of the ground and sea level. E is understood in this
statement to be the E of formula (10), the EF, of formula (15), or the E, of formula. (16) for each
elementary mass according to whether it is at the same level as the station, higher than the
station or lower than the station, respectively. Since kdmE is the vertical component of the
attraction in dynes upon a unit mass at the station due to an elementary mass, dm, the integral
stated is evidently the vertical component of the attraction due to all the elementary masses
which combined constitute the material lying above sea level in the zone in question. In the
integration k is a constant, and the sum.of all the elementary masses, dm, is the total mass m,
which is known in terms of the volume and density. No difficulty was encountered in dealing
with these quantities. But the expression for £ is a function of h and 6, which can not, so far
as the writers know, be directly integrated with respect to these quantities by calculus. There-
fore, an integration by numerical computation was made.
The vertical component of the attraction in dynes upon a unit mass at the station due to all
the topography within any zone lying entirely in a land area was therefore expressed as the
integral of kdmE or
km (average value of F for the zone) (19)
in which it is understood that the various values of EH, of which the average is taken, must
correspond to equal elementary masses, of which the sum is m, the total mass represented by
the topography in the zone.
Similarly the vertical component of the attraction in dynes upon a unit mass at the station
due to the isostatic compensation of the topography within any zone lying entirely in a land
area is also represented by formula (19). The negative mass involved is m, the values of E are
those fixed by the direction and distance of the compensation from the station, and h is made
to vary to cover the whole range occupied by the compensation, namely, from sea level down
to the depth 113.7 kilometers below that surface.
The effect of the topography and the effect of its compensation might have been computed
separately from formula (19), but it was believed that greater rapidity would be secured without
loss of accuracy by combining and dealing directly with the resultant difference of the effects
of the topography and its compensation.- Accordingly, the actual process followed is that
described in the following paragraphs.
The computation will be described first for land zones haying an elevation of 100 feet and
for the station assumed to be at sea level. The modifications introduced for other elevations,
for ocean zones, and for assumed positions of the station above sea level will be stated later.
For a selected value of 6, EH was computed by formule (10) and (16) for several equally
spaced values of h, varying from zero to the depth of compensation. Let the required mean
value of an infinite number of such equally spaced values, covering the depth of compensation,
be called E,. By successive trials with increasing numbers of equally spaced values of h it
was ascertained how many values were necessary in order to secure the required degree of
accuracy in the mean value, £,, corresponding to the selected value of 6. As E varies con-
tinuously according to a law which may be graphically expressed by a smooth and regular
curve, it was not difficult with the numerical values at hand to make certain that one had
secured the required degree of accuracy. [Illustration No. 8 is an example of such a curve,
which corresponds to @=1° 55’; that is, to compensation which lies in a part of zone 16. (See
page 18.) The values of h are plotted as abscissz and the corresponding values of EF as ordinates.
The small circles each represent a computed value of EH. A smooth curve has been drawn by
eye through these computed points. It is evident that as the curve is nearly a straight line
between successive computed points but little change would be secured in the mean by com-
puting more points. This is still more clearly and precisely shown in the following table,
corresponding to illustration No. 8, and showing the computed values of H and their first and
second differences.
24 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
Values of E for various depths, with 6=1° 58’.
Depth (10%) | Difference | ghecond |
Centimeters
3684
1/1611 370 000 11 026 +7342 — 60
2/1611 370 000 18 308 +7282 —133
3/1611 370 000 25 457 +7149 —209
4/16 X11 370 000 32 397 +6940 —255
5/1611 370 000 39 082: +6685 —332
6/16 X11 370 000 45 435 +6353 | —354
7/16 X11 370 000 51 434 +5999 —409
8/1611 370 000 57 024 +5590 —454
9/1611 370 000 62 160 +5136 —462
10/1611 370 000 66 834 +4674 —477
11/1611 370 000 71 031 +4197 —499
12/1611 370 000 74 729 +3698 —433
13/1611 370 000 77 994 +3265 —485
44/1611 370 000 80 774 +2780 —473
15/16 X11 370 000 83 081 +2307 —431
11 370 000 84 947 +1876
Mean 52 568
If an infinite number of points were computed on the curve shown in illustration No. 8
and the mean taken, instead of the mean of the finite number of points there shown, the change
in the computed mean would be represented by the average ordinate included between the
curve and the series of chords joining the computed points which are shown in the illustration.
As a convenient rough guide it was assumed that this average ordinate would usually be less
than one-eighth of the average second difference shown in the preceding table. That this
ratio, one-eighth, is a reasonably safe assumption in such a case may be verified either by trial
or by geometry, assuming the short portion of the curve between successive points to be an
arc of a circle.
A similar process of reasoning was followed to obtain the mean value of EF corresponding
to the topography for the same selected value of 6. Let Ey be the required mean value of an
infinite number of equally spaced values covering the range from zero to the arbitrarily selected
elevation, 100 feet. After E had been computed from formule (10) and (15) it usually appeared
that in order to secure a sufficiently exact value of Ey it was necessary to compute but two
values, one for h=o and one for h=100 feet, the mean of these two being sufficiently accurate.
It will be shown later how topography of a greater elevation than 100 feet was dealt with.
Keeping in mind that the negative mass, which is the isostatic compensation, is necessarily
exactly equal to the positive surface excess of mass, which is the topography, formula (19) as
applied to the topography and the compensation combined may be written
km(E7p— Eo) or km Ep (20)
Ex being written for H,— Eo.
As this process of computing Ez, corresponding to a selected value of 6, is slow it was impor-
tant to use good judgment in selecting the various values of @ for which the computation was
to be made. It was desired to obtain a sufficiently accurate value of Ep for every possible
value of 6 by computing a moderate number of values for selected values of 6. At first Ep
was computed for 6=180°, that is for the antipodes of the station, and for 0=90°, midway
between the station and its antipodes. Then the computation was made for a few more values
of 6 at large intervals. It soon became evident that Ep varies quite slowly and at nearly a
uniform rate if. is near 180°, but that for small values of 6, Ez varies at a large and rapidly
changing rate. Therefore, to secure a given degree of accuracy in interpolated values it was
evidently necessary to compute Ep for closely spaced small values of 0, but only for widely
separated large values of 0. The following table shows the various values of Hz actually
computed. The values of 0 shown in this table were selected by inspection by a step by step
process, computations being made first for two values only of @ as already stated, then for
values spaced at intervals of 30°, then at intervals of 10° for smaller values only, at intervals
of 5° for still smaller values, and so on. All values of Ep intermediate between those shown
in the table were obtained by interpolation. Illustration No. 9 shows a part of this table
expressed graphically.
ABAUNS TYOISOIOIS'S 0 FHL AB OBLNTHd
ol
"Y3 40 SANIVA JO NOILLVLINSSSHdSY IVOIHAVY—'6G “ON NOILVYLSNTH
6 f0 Sanjoy
8 256
06
001
orl
oOo o&
a
z
(pe) "7 40 Sampo
061
EF
FE
cT
OF
TOP
OGR.,
AP
HY
AN
D
Is
OSTATIC
ee
PE
NS.
ATI
ON
ON
GRA
VI
TY
25
8
° Ey
20
180 : )
160 00 461 Fg (10)
150 00 at a
0 1. 2 E
140 a + 62 8466 = Tp (10%)
130 0 bean = 62. 166 |
00 A 5.5 6 05 = 0
110 0 ae ~ Ga f 0
105 a A. 9805 = 64. ae - . 5552 E
100 a i ae 1425 = Pee ae bd 5591 ar 'p (1020)
9500| 4 75.2134 | — 7 1043) - Sa 1 ap z
ee ee | 2 i) ae ae
0 3. 7 . 05 | oe 30 1
80 00 a ao a i = 2 10 15 : a Bp (10
af + os: 196° — 84. 4071 e et aa + e782 = 88 :
65 HA 0h soo | 88. a4 | 305 8 0 08 - 50.8 =
07. = ae a: : = : =
ga) Hee “2s hn 3 0 Fis. ait = eS
00 4 3.2 —10 . 892 - 1 75 45 763. = il 8 = 362.
57 12 22 = =~ bale 8 3 +7 9 77 3 .6
00 a 5.1 1 68 ite 1 0 85 = 12 2 = 89
56 12 18 es . 476 8 + 8 2 32 4 25
00 at 7.0 12 72 = 15 07 ee .8 =
55 0 129, 150 ae rae 8. 00 t 831.2 rag 45e 4
a +131, 120 eo a = 7. 45 + aoe a =| 1000. b 2S
FO ag ae — 130. 0 : mae: 73 eG - 1432. = ee
00 a 3.4 13 88 ae 2 0 ae 83 2 15 2 = 29
52 0 135 710 Se arale 715 lL 4 pet B74.
2 +18 | ae a4. BGS — 2886 7 ong ral 599 2, ona
00 + 8.0: 13 05 = 3 00 41 = 16 9 = 24,
50 140 80 zs 6.8 . 005 6 4 +9 8 97 6 oil
00 + 5 13 48 = 3. 5 a 74. _ 18 «1 = 80,
49 14 45 eae 134 63 10 2 05 7 9
00 ae 3.1 14 84 os 3 0 a 08 19 3 = 43
48 14 i ee . 270 61 10 9 25 8 .6
00 ae 5.7 144 23 s, 3. 5 ae 46 = 20 9 = 14
47 14 84 a: 4 418 6 0 10: 61 8 Ll
00 ais 8.5 14 70 =< 3 0 ae 86 es 22 3 ss 93
46 151 70 es 7.2 574 . 54 1 13 9 9
00 + 4 15 32 = es 5 30 = ae .3 el vee
4 1 7 = 0. TA 5 + 9 ea
ae sr ze “iis | 3 05 5 8 ie | me | ae
43 0 160. 1 = . 278 = 1 30 1246 — 8 = 1345
iG Lee ie — 168. 08 eae 52 +128 — Foe 1693
40 0 171 6 i . 666 5 02 15 1304 3 6 i 1911
39 00 +175. 928 —Ne. 46 = Poe 5 10 +132 a - a
ll ae 92 ero ae 5 + 4 = 3 66
38 180. 6 1a 56 05 134 0 ee
00 + 13 178 3 3 5.6 5 + 4 = 388 60
37 0 184 - —18 717 6 70 00 1366 40 A = a
36 re +189. 569 = 172 = 014 4 55 +1388 = ae = 2466
35 to +194. 240 a1) 7. 877 -— 6 387 4 50 +1412 a es oy 2577
34 00 199, 876 ee ~ 13H 4 45 +143 - 4320 | - 269
00 + 37 198 1 < 7.2 4 + 6 Ss 482 6
33 204. 6 = .12 7 46 40 146 4 Se 282
00 ay a) 203 3 _ eo 4 41 0 a 658 1
32 0 210 —20 . 716 8 39 35 2 486 484: — 2954
ee 4216. 72 =o ats me 4 30 per = F058 = as
30 a 4993, 92 oe 00 _— 9. 883 4 25 ae - ae = 3946
29 5 +230. S 330. 76 — 10. oe ia Tie = 5503 = 8
28 0 4-238. 54 =e See 415 1598 = a = 359
ort de 04 237 ae 4 +1 eee ae
27 246 —2 - 76 12 10 628 6 =e
00 + . 06 46 Se 4 +1 ae ana
26 254. -2 -10 13 05 660 6 = ee
ae 66 55 ae 4 +1 = oa
25 0 263 —26 .10 14 00 ab 694 658 = 203
23 00 {84 65 a ~ 17:08 3 50 ae — 137 Se
59 00 396 82 ae = ist 3 45 1801 — T601 a
; Lo 3 +1 ee: 5
BL Ot +308. 02 ae 20 40 840 79 — 5236
0 + . 02 a 13. — . 72 3 +1 - 86
19 80 za 8 88 5% i = Ses = $88
a fou 08 ey = 3862 3 25 1868 = 0385 = eon
18 00 oe ae = 85 7 a0 = 108 = 1
17 oe ee =. 3 os aa = 1a = sn
16 3 404. 9 —45 . 60 = 54, 00 ate 26 _ 1268 - 557
. 98 oe =D 2 2 5 9
ne Han a = 8:8 4 ts = as = 88
15 0 442. 6 504, 47 = 68. 49 9 45 49 16 o 15462 _]j 570
aa a eee — 74.14 a 1356s — 16567 ee
14 2b 472. _5A . 04 - 80.4 35 ak 562 = 1782 a 231
0 +4 0 ae 74 8 9 23 264 1 9 a 131
13 3 88. 57 i 7. 0 ats 2 = 921. 76
0 +5) 2 ie 1.9 9 59 22 272, 9 4 = 142
13 0 05 59 ss 5.7 5 Be 6 =, 073 17
0 Le 6 — 598.7 10. 0 a 281 aa a
12 3 24.2 628 ae 2 : +2 : Soa 16 a2
12 00 ioe 860 6 aes 2 10 429) ay = 18174
toe ae 10.0 Ae tus - Bon a
; ao. —155. 0 -3 =
“ee “13.0 155 39m = 84855 = 2
. 198, 50 6 s 83) = 56
a 3 1 +3 70 28
aa iB] ie = a = ae
5.0 1 35 +4014 = Bune ~ BB oe
1 30 +4212 ae = ae
1 25 cr = bei = 43048
7 4 =
a ee vag | — ssa7
a= 96711 = aes
es eae
—10 ee
48
26 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
It was desired that each value of E, used in the computation, whether obtained directly
or by interpolation, must be correct within 1 part in 200.
Having sufficiently accurate values of Ez for each separate value of @ the next step is
essentially an integration with respect to @ as the variable.
The area of any zone lying between the limiting values of 0, 6,, and 6,, on the surface of
the sphere which is being considered, one having a radius of 637 000 000 centimeters (see
p. 15) is
2zxr* (cos 0,—cos 6,) , (21)
or (6.283186) (637000000)? (cos 6,—cos 6.)
Hence for this zone formula (20) becomes
kd [(6.283186) (637000000)? (cos 6,—cos 0,)] (mean value of Ez for the zone) (22)
in which for m there has been substituted its value in terms of density and volume, namely,
OH (area).
With the numerical values before one it is not difficult to determine that for zones of a
moderate width the average value of E, for the zone is with sufficient accuracy the mean of its
values at the two edges of the zone corresponding to 0, and 6,. Therefore, formula (22), which is
an expression for the required vertical component of the attraction in dynes upon a unit mass
at the station due to the combined effect of both the topography and its isostatic compensation
lying in a zone, may be evaluated by making separate numerical computations for separate
narrow zones and adding the values.
By examination of the table showing values of Ep it is evident that the separate zones
which may be used in this process are wide near the antipodes and decrease in width as 6
becomes smaller. The actual widths used did not exceed the following limits and were
occasionally less.
Limits of widths of subzones.
6g Limit of width
of subzone
o + o +7 ov
180 00 to 72 00 2 00
72 00 to 20 00 1 00
20 00 to 10 30 0 30
10 30 to 5 40 0 15
5 40 to 1 25 0 05
It was known from a reconnoissance of the problem that for all distant zones (beyond
6=1° 29’ 58’’) the value of the attraction computed from formula (22) would be nearly propor-
tional to H. Therefore, as a time-saving device, it was decided to determine such widths for
the selected zones and fix the number of compartments in each zone so that an attraction of
0.0001 dyne for any one compartment would correspond to a value for H of either 100, 1000,
or 10 000 feet in that compartment. In that case the computation would consist simply of
estimating the mean elevation within the compartment in feet and moving the decimal point
a certain number of places to the left to obtain the attraction in dynes.
The arbitrarily selected unit of elevation corresponding to 0.0001 dyne was 10 000 feet for
zones 1 to 6 (see tables on pp. 44-46), 1000 feet for zones 7 to 13, and 100 feet for zones 14 to 18.
The number of compartments in each zone was arbitrarily fixed as shown in the same tables.
By formula (22) the width of the zone was computed which would satisfy the condition that
the attraction in one compartment corresponding to a unit of elevation was exactly 0.0001
dyne. For example, for zone 3 having 10 compartments it must be 0.0010 dyne for the zone
if the mean elevation in the zone is 10 000 feet.
No difficulty was found in making this computation. An example of the actual arrange-
ment of the numerical work is shown below for zone 12 having 10 compartments.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 27
Computation of limit of zone 12.
[em. =kd H=0.00543061 for H=1000 feet.]
6 to Oe Cos 61—cos 6 a ie Ep (10m) | Area xX kmX Ep
o + Ws oO 4+ Vt
5 46 34 to 5 40 00 0. 0001905 0. 000004857 | 0. 00000002638 2198 0. 00005798
5 40 00 to 5 35 00 . 0001425 - 000003633 . 00000001973 2310 . 00004558
5 35 00 to 5 30 00 . 0001405 - 000003582 . 00000001945 2413 . 00004693
5 30 00 to5 25 00 - 0001383 © . 000003526 . 00000001915 2522 . 00004830
5 25 00 to 5 20 00 . 0001363 . 000003475 . 00000001887 2636 . 00004974
5 20 00 tod 15 00 . 0001341 . 000003419 . 00000001857 2758 . 00005122
5 15 00 to 5 10 00 . 0001321 . 000003368 . 00000001829 2888 . 00005282
5 10 00 to 5 05 00 - 0001299 . 000003312 . 00000001799 3024 . 00005440
5 05 00 to 5 00 00 . 0001278 . 000003258 . 00000001769 3170 . 00005608
5 00 00 to 4 55 00 . 0001257 . 000003205 . 00000001741 3330 . 00005798
4 55 00 to 4 50 00 . 0001236 . 000003151 . 00000001711 3504 . 00005995
4 50 00 to 4 45 00 . 0001215 . 000003098 . 00000001682 3690 . 00006207
4 45 00 to 4 40 00 . 0001194 . 000003044 . 00000001653 3888 . 00006427
4 40 00 to 4 35 00 . 0001173 . 000002991 . 00000001624 4097 . 00006654
4 35 00 to 4 30 00 . 0001151 . 000002934 . 00000001593 4321 . 00006883
4 30 00 to 4 25 00 . 0001131 . 000002884 . 00000001566 4564 . 00007147
4 25 00 to 4 20 00 . 0001109 . 000002827 . 00000001535 4822 . 00007401
4 20 00 to 4 19 12 . 0000176 . 000000449 . 00000000244 4987 . 00001217
; Sum=0,00100034
The change for 1’’ is about 0.000 000 25, therefore, the-inner limit of zone 12 is, to the-
nearest second, 4° 19’ 13’’.. With that limit the above sum becomes 0.001 000 34 — 0.000 000 25
=0.001 000 09. :
The basis for the arbitrary decisions as to unit elevations and number of compartments
in each zone will be indicated under the topic ‘‘Discussion of errors.’”’ It suffices to state here
that the selection was guided by the desirability of making the computations as rapidly as pos-
sible subject to the chosen standard of accuracy. Errors of judgment in one direction would
make the computation slow, and in the opposite direction would make the computation too
inaccurate.
The limits of zones 1 to 18 computed as indicated above are shown in the reduction tables
on pages 44-46, as well as on page 18.
To apply formula (22) to the computation for oceanic zones it was necessary merely to take
into account the fact that the defect of density represented by sea water is 0Q—d,=0.615 0.
(See p. 9.) Therefore, if the unit of elevation is 10 000 feet for a land compartment, correspond
ing to an attraction of 0.0001 dyne, it will be for an oceanic compartment to produce the same
effect
10 000 feet
0.615 =16 260 feet =2710 fathoms.
Hence the unit of depth shown for zones 1 to 18 in the reduction tables on pages 44-46.
The attraction computed from formula (22) for a given compartment is not strictly propor-
tional to H as assumed for a first close approximation. The limits of A used in computing Fp
must correspond to H. For a land compartment, as H is made greater EH, becomes smaller,
as it is an average value covering larger values of fh in formula (15). Also as H, the assumed
elevation, is made greater EH, tends to become smaller, for the isostatic compensation is assumed
to commence at the solid surface of the ground (above sea level) (see illustration No. 2, page
10), and to extend to a depth of 113.7 kilometers measured from that level. The limits of h
used in formule (15) and (16) must be fixed accordingly. Similar modifications must be
inserted for oceanic compartments, the compensation commencing in this case at the ocean
bottom, not at the sea level. As Hpand £,, and their difference Ep, vary slightly for different
values of H, the computed attractions in formula (22) are not strictly proportional to H as they
would be if Ey were independent of H. This departure from strict proportionality was found
28 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
upon investigation to be inappreciable for zones 1 to 13. For zones 14 to 18 a few special com-
putations were made to evaluate the corrections for departure from proportionality shown in
the reduction tables on page 44. These special computations were made by using the proper
limiting values of h as indicated above, comparing the computed values with the values based
on the assumption of proportionality and the original computations with 4 =100 feet, and tabu-
lating the differences as shown in the reduction tables. A few computations only were necessary
because the corrections were small, and regular in their variation.
It was also assumed in order to secure a first close approximation to the attraction required
that the station is at sea level. In general the station lies above sea level and, therefore, to
secure exact results the values of h, used in formule (15) and (16) in computing E; and E-, must
be differences of elevation between the station and the elementary mass, not merely the elevation
of the elementary mass as was assumed in the first approximation. To secure the corrections
for elevation shown in the reduction tables on page 44 a few special computations were made
on the exact basis, compared with the first approximation, and the differences tabulated as
corrections for elevation of the station. The corrections for elevation were found to be negligible
for zones 1 to 13, and to be small as shown in the reduction tables for zones 14 to 18. Because
the corrections are small and their variations regular but few special computations were necessary.
EXPLANATION OF REDUCTION TABLES.
The complete reduction tables for all the zones are given in the following pages. All tabular
values are the vertical components of the attraction upon a unit mass at the station expressed
in units of the fourth decimal place in dynes. It is equally true that these are corrections in
units of the fourth decimal place of centimeters, to the acceleration of gravity, expressed in the
centimeter-gram-second system.
These tables cover the whole of the earth’s surface, from the station of observation to its
antipodes. By their use one may quickly compute the effect upon the attraction of gravity, at
any station on the earth, of all the topography of the earth and of its isostatic compensation
assumed to be complete and uniformly distributed, with respect to depth, down to a limiting
depth of compensation of 113.7 kilometers.
The radii of the zones A to O are given in meters, while those for zones 18 to 1 are in degrees,
minutes, and seconds of an arc of a great circle.
The first column of each table from A to O contains values for the mean elevation of the
compartment as read from the maps. The second, third, and fourth columns contain the
corrections for the. topography, the compensation, and the algebraic sum of the corrections for
topography and the compensation respectively. These values are computed upon the assump-
tion that the station is at the same elevation as the compartment. For zone A the elevation of
the zone is necessarily that of the station, as its radius is only two meters. In the tables for zones
B to O corrections for the elevation of the stations above or below the compartments are shown,
The corrections for the topography and compensation, the station being at the same eleva-
tion as the compartments, are shown separately in columns 2 and 3 for the zones out to O, in order
that certain comparisons may be made between the effects of the assumption of complete local
isostatic compensation and of regional isostatic compensation complete within a stated distance
from the station. (See pp. 98-102.)
For the regular computations of the combined effect of topography and compensation, one
correction is taken from column 4 of each table from zone A to zone O. For zone A this is the
only correction. For zones B to L, inclusive, a second correction must be applied, as indicated.
to take account of the difference of elevation of the station and of the mean surfaces of the ground
in the compartment. To the correction based upon the assumption that the station is at the
same elevation as the compartment, taken from the fourth column of the table, is added alge-
braically the correction for elevation of station above or below the compartment in order to
obtain the total effect of topography and compensation. Thus, in zone E, if the mean elevation
of the surface of the ground in a compartment is 2000 feet, the first correction is +0.0016 dyne,
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 29
and if in this case the elevation of the station is 3000 feet, 1000 feet above the compartment, the
second correction is +0.0007 dyne, and the total effect of both the topography of this compart-
ment and its isostatic compensation is to increase the vertical component of the attraction on
a unit mass at the station by 0.0023 dyne.
It is understood that, for zones B to L, inclusive, the second correction, namely, for station
above or station below compartment, is zero if the elevation of the station is the same as the mean
elevation of the surface of the ground in the compartment. This fact is used in interpolation if
necessary. For example, in the case just cited in zone E, in which the mean elevation of the
surface of the ground in the compartment is 2000 feet, if the station happens to be 100 feet above
the compartment, the second correction would be + 0.0001 dyne, since the table indicates it to be
+0.0002 if the station is 200 feet above, and it is understood to be zero if the station is at the
same elevation as the compartment. Similarly, if in this case the station happens to be 100 feet
below the compartment, the second correction would be —0.0001 dyne since the table shows it
to be — 0.0002 dyne if the station is 200 feet below the compartment.
For zones C to O the first column of the tables contains elevations in both fathoms and in
feet. Those in fathoms are depths below sea level and are marked minus. The values in the
second, third, and fourth columns, corresponding to depths in fathoms, are computed on the
supposition that the station is at sea level and in the following columns, headed ‘‘Station above
compartment,” the station is assumed to be at the stated distances above sea level. Hence,
for all water compartments, there will be two corrections in the regular computations, one from
the fourth column and one from the proper column beyond the fourth. Thus, in zone E, if the
mean depth in the water compartment is 200 fathoms, and the elevation of the station above
sea level is 600 feet, the two corrections are —0.0004 dyne and —0.0002 dyne, and the total
effect of both topography in this compartment and its isostatic compensation is to decrease the
vertical component of the attraction on a unit mass at the station by 0.0006 dyne.
' For zones M, N, and O, as already explained in connection with the computation of the
tables (p. 22), the second correction does not necessarily become zero when: the station is at the
same elevation asthe compartment. Instead it has the value shown in the tables for these zones
in the extra column headed ‘‘Station at the same elevation as compartment.’’ In taking out
the second corrections for these three zones this extra column must be carefully noted, one must
take the second correction from it when the station and compartment happen to be at the same
elevation, and one must use the values in this column to control interpolations when the station
and compartment are nearly at the same elevation. Thus, if the mean elevation of the surface
of a compartment in zone M is 12 000 feet the second correction is +0.0001 if the station
is also at the elevation 12 000 feet, it is between + 0.0001 and +0.0002 if the station is less than
700 feet above the compartment, and it is between +0.0001 and —0.0002 if the station is less
than 700 feet below the compartment. :
For zones 18 to 14 three corrections are applied. The first is read directly from the map,
being 0.0001 dyne for each unit of elevation, the unit in each case being 100 feet, as indicated in
the heading of this table. The second is taken from the second column of the table, using the
first correction as an argument in entering the table. It takes account of the slight departure
of the actual correction from being strictly proportional to the elevation. The third correction
is taken from the last part of the table and takes account of the correction due to the elevation
of the station above sea level. Thus, in zone 17, if the correction as read from the map is
—0.0100 dyne, the elevation of the zone (the zone has but one compartment) being 10 000 feet,
then the correction for departure from proportionality is +0.0001, and if the elevation of the
station above sea level is also 10 000 feet the correction for its elevation is + 0.0003 and the total
effect of topography and compensation of this zone, upon the vertical component of the attrac-
tion upon a unit mass, at the station is — 0.0100 + 0.0001 + 0.0003 = — 0.0096 dyne. Similarly,
if zone 17 is all upon the ocean, and the average depth of the water is 2710 fathoms (or 100 of
the specified units of depth) the correction as read from the map is +0.0100 dyne, the correction
for departure from proportionality is +0.0001, and if the station is at the elevation of 5000
30 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
feet the correction for elevation is — 0.0001 and the total effect of the topography in this zone and
its isostatic compensation is + 0.0100 +0.0001—0.0001=0.0100 dyne.
The unit of elevation for zones 13 to 7 is 1000 feet, and for zones 6 to 1 is 10 000 feet.
These large units of elevation make it easy to estimate quickly the mean elevation within each
compartment with the required degree of accuracy.
Note that for zones 13-1 there are no corrections for elevation of station and for departure
from proportionality.
The reduction tables thus far described are believed to cover all cases which will arise when
the gravity station is on land. But in order to provide for the computation of the effects of
topography and isostatic compensation on the attraction at a gravity station on a vessel at
sea, such as those occupied by Dr. Hecker on the Atlantic and Pacific Oceans, the two supple-
mentary tables for use in connection with gravity stations at sea were prepared. These tables
are computed on the supposition that the observation station is at sea level, since the correction
for the small elevation above sea level to which the station is limited on board a ship would be
less than 0.0001 dyne in every case. But one correction is to be taken out from these tables
for each compartment. This correction is to be taken from the first table if that can be done
without using any of the values marked with an asterisk. Otherwise it is to be taken from the
second table in order to avoid large errors of interpolation which otherwise would occur on
account of the large second differences in the first table.
For the remaining zones 18 to 1 no such sea tables are necessary, as the regular tables pre-
pared for land stations cover all cases which will arise.
REDUCTION TABLES FOR LETTERED ZONES.
Zone A,
[Inner radius, zero; outer radius, 2 meters. One compartment.]
Correction for—
Elevation
of selon Topogra-
and com-
Topog- Compen- | phy and
partment | raphy sation compen-
sation
Feet
0 0 0
5 +1 0 +1
10 +2 0 +2
100 +2 0 +2
1 000 +2 0 +2
2 000 +2 0 +2
3 000 +2 0 +2
000 +2 0 +2
5 000 +2 0 +2
6 000 +2 0 +2
7 000 +2 0 +2
8 000 +2 0 +2
9 000 +2 0 +2
10 000 +2 0 +2
11 000 +2 0 +2
12 000 +2 0 +2
13 000 +2 0 +2
14 000 +2 0 +2
15 000 +2 0 +2
For zone A the correction to gravity is a function only of the elevation of the station, for
all land stations, as shown by the above table.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 31
Zone B,
{Inner radius, 2 meters; outer radius, 68 meters. Four compartments.}
Correction for elevation of station—
a ; Correction for—
: ation of Above compartment Below compartment
compart-
ment re Topogra-
Se oa Bompen- | 25feet | S0fect | 75feet | loo fect | 125 feet | 25 feet | 50 feet | 75 feet | 100 feet | 125 feet
sation
Feet
0 0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0
20 ed 0 +1 0 0 0 0 0
30 +2 0 +2 0 0 -1 -1 -1 —3
40 +3 0 +3 0 0 -1 -1 -1 —3
50 +3 0 +3 0 -1 -1 -1 oe —3 —6
60 +4 0 +4 0 -1 -1 -1 —2 -3 -6
70 +4 0 +4 0 -1 -1 —2 —2 —3 —6
80 +5 0 +5 0 -1 -1 —2 —2 -3 —6 -9
90 +5 0 +5 -1 —1 —2 —2 -3 —3 —6 -9
100 + 6 0 +6 —1 -l —2 —2 —3 —3 —6 -9 —12
150 +8 0 +8 -1 —2 -2 —3 —4 —2 —5 —8 -1l —14
200 +10 0 +10 -1 —2 —3 —4 -5 —2 -5 —7 —10 —12
300 +12 0 +12 -1 —2 —4 —5 —6 —2 —4 -6 —8 —10
400 +14 0 +14 -1 3 -—4 —5 —6 —2 —4 —6 -—8 -9
500 +14 0 +14 -1 —3 —4 —5 —7 —2 —3 —5 -—7 — 8
1 000 +16 0 +16 —2 —3 5 —6 —7 —2 -3 —5 —6 -7
2 000 +17 0 +17 —2 —3 —5 —6 —7 —2 —3 —5 —6 —-7
3 000 +17 0 +17 —2 3 —5 —6 —7 —2 -3 —5 —6 -7
4 000 +17 0 +17 —2 —3 —5 —6 —7 —2 —3 —5 —6 -7
5 000 +17 0 +17 —2 —3 —5 —6 —7 —2 = —5 —6 -7
6 000 +17 0 +17 —2 -3 —5 —6 —7 —2 —3 —5 -—6 —-7
7 000 +17 0 +17 —2 —3 —5 -6 -—7 —2 -3 —5 — 6 —7
8 000 +17 0 +17 —2 —3 —5 —6 —7 —2 -3 5 — 6 -7
9 000 +18 -1 +17 —2 —3 —5 —6 —7 —2 3 —5 —6 -7
10 000 +18 -1 +17 —2 -3 —5 —6 —7 —2 -3 —5 —6 -7
15 000 +19 -1 +18 —2 —3 —5 —6 —7 —2 —3 —5 —6 -7
It is assumed that the mean elevation for any compartment in this zone will never be
negative (below sea level) for any gravity station on land.
Lone C.
[Inner radius, 68 meters; outer radius, 230 meters. Four compartments.]
Correction for elevation of station—
Correction for—
Above compartment Below compartment
Mean
elevation
of com- Topog-
partment Goi raphy |
Topog- au and 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500 | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500
raphy | Pes? | com- | feet | feet | feet | feet | feet feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet
pensa-
tion -
Fathoms
— -9 0 -9
— 40 —.4 0 -4 0 0 0 0
Feet
0 0 0 0 0 0 0 0 0 0 0 0 0 0
100 +1 0 +1 0 0 0 0 0 0 0 0 0j— 1] -1] -—2
150 +2 0 +2 0 0 0 0 0 0 0 0/—1/—1] -1] -3 | -4
200 +4 0 +4 0 0 0 0 0 0 0/—1/—1]/—1| —2] —4| -6|-8
250 +6 0 + 6 0 0 0 0 0 0 0j/—1]—1)-— 2} —2); —4 | —7 |—10 |—12
300 8 0 +8 0 0 0 0 0 0 0j—1/—1j— 2} —2| —5 | —8 |—11 |—14 |—16
350 t10 0 +10 0 0 0 0 0 0} —-1)/— 1 |— 2 |— 2) —2| —5 | —8 |—12 |—15 |—18 |—20
400 +12 0 +12 0 0 0 0 0 0] —1/— 2|— 2j— 3 | —2] —5 | —8 |—12 |—16 |—19 |—21 |—24
450 +13 0 +13 0 0 0 0 0 0} —-1j—2/— 2)/-— 3] —2] —4] —7 |—11 |—15 |—19 ;—22 |—25 |—26
500 +15 0 +15 0 0 0 0 0 0] —1j/— 2|/— 3|— 4] —2|-—4 | —7 |—11 |-15 |—19 |—23 |—26 |—28 | —30
16 0 +16 0 0 0 0 0 o0| —-1)— 2|-— 3|— 4] —2] —4 | —6 |—10 |—14 |—18 |—22 |—26 |—28 | —31
m tis 0 +18 0 0 0 0 0] —1| —2/— 3/— 4 |— 5 | —2] —4 1 —6 |—10 |—14 |—18 |—22 |—26 |—29 | —32
700 +20 0 +20} 0 0 o| —1) -1] -2| —3 |— 4 j— 5 /— 6} —1] —3 | —6 |— 9 |—13 |—17 |~20 |—24 |—28 | —31
800 +22 0 +22 o| —1| —1| —2/ —2] -3] —4/-— 5 |— 6 |- 7 | —1] —3 | —5 j— 9 |—12 |—16 |—19 |—22 |—26 | —29
900 +24 0 +24 o| —1/ —2| —2) —2] —4] —5 |-— 6 |— 7 |- 8| —1] —3 | —5 |— 8 jJ—11 |—15 |—18 |—21 |—24 | —27
0 26 o| —-1} —2| —2) -—3| ~-4| -6/—7/— 8 |- 9| —1] —3] —5 |— 8 j—11 |—14 |-17 |—20 |—23 | —26
1 200 13 0 +28 —1|—2| —2] —3} —4] —5| -6|/— 7 |- 9 |—11 0} —2) —4 |— 7 |—10 |—13 |—15 |—18 |—20 | —23
1 400 +30 0 +30; —1] —2| —2| —3| —4] —5 | —6 |— 8 |—9 j-11 0| —2 | —4 |— 6 |— 9 J—11 | 14 |—16 |—18 | —21
1 600 +32 0 4+32|—-1] —2| —3| —3| —4] —6 | —7 |—_9 |-10 |—12 0 | —2 | —4 |— 6 |— 8 |—10 |—13 |—15 |—17 | —20
2 000 +34 0 +34] —1 | —2| —3| —4]} —5 | —6 | —8 |—10 |—11 |—13 0] —1] —3 |— 5 |— 7 |—10 |—12 |-14 |—16 | —18
0 36 | —1| —2| —3} —4} —5 | —7 |] —8 {—10 |—11 |—13 0| —1) —3 |— 5 |J— 7 J— 9 |—12 j—18 |-15 | —17
3 O00 tes 0 $38 —1|-—2/ —3] —4| —6) —7 | —9 |-10 |—12 |-14 0} —1,; —3 |— 5 |— 7 |— 9 |—12 |—14 |-15 | —16
5 000 +41 -1 +40] —1] —2| —3] —4| —6 | —7 | —9 |—10 |—12 |—14 0} —1; —2 |— 4 |— 6 jJ— 8 |—10 |—12 |-14 | —15
15 000 +44 —2 +42} —1/ —2] —3 | —4| -6] —7 | —9 |—11 |—12 |-14 0} —1 | —2 |— 4 |J— 6 |— 8 |-10 |—12 |-18 } —15
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
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g3— | se— | ec— | oc— | st— | o1— | t— | u—-|}6—]}2—]s—]e- |e- Ji— | st—| ot—| st—]er— | et-]or-|s—|e-|s-]e-]e-]e-]t Jo | ot |I- [Ost | o008
ee— | 92— | ¥— | Te— |-or— | oI— | et— | 1— J or— |2-—]¢—]e- Js— | r- | at—]|ot—] st fat} u-je—]|s—je-|s-je-]e- |e ]i jo feet JIT [set | Oe
te~ | ge— | se— | ee— | er— | 2t— | s1— | et— J or— | 2-]s—]e- | e— |1- | or—|st—|er—]a—|or—|s—]2—|e-|e—]e-]e-]IH] tH jo [set |I~ [ert | OOF
ze— | 6c— | 92— | e@— | oz— | zt— | 1— | ct— | or— }2-—]s—]e— J2— |i- | or—]u-]er—|m—-|o-|s—]2—]s—-]e—]e-]e-] I]t /o Jet [TH |e | ORE
ee— | og— | 2e— | e— | t2— | st— | st— | at— | o— | 8 —} 9 - | s— |2— J i— |st—]m—l|er-|m—-|6—-|s—|9-—|s-l]e-]e-fe-|I—|1t~ so [ert | I~ | br | 0098
ve- | 1e— | se— | se— | ze— | or— | or— | er— | m—|s—f9—]a— Je- |r |st—jer—ja-|u-j6é-js-]9-|s-]e- ie fI-]i jt jo jet [IK [er | OE
se— | ee— | 62— | 9e— | ee— | Oc— | AI— | I— | TI— | 6 —| 9 —] F— J e— | I~ | FI-]sI- | - | oI- | 6-]4—]9-|S-]e— |e] I-]/I-}I-]o fire JI c+ | 006 €
ze— | ¥8— | Og— | 22— | ¥e— | I2— | sI— | st— | et— J or— | 2 —] g— | e- |s- | e—Jst-]}m—-|6-|;8—|42—-|s—]s-]e- |e] I~ ]0 |O 10 [OF [0 Or+ | 000 €
ge— | SE~ | Te— ; 8— | H— | IZ— | sI— | SI- ] eI— J OI— 2 —] G— J e- | s-— | et—]u—]}o-|;6—-|8—/9—-|s—)F-]e-7pe-]I-]O |O 10 | set [0 se+ | 008 Z
OF— | 9e— | ce— | ez— | S2— | ee— | 1— | 9T— | eI— J OI—] 2—] 8— J e- |s- |a—|u-jor—|s—|z—/9—-|F—]e-]je-]I—}oO |0 |O JO | er JO ge+ | 009 @
we— | se— | be— | og— | 92— | e2— | or— | 9t— | et— ] or— | 8 — | s— | #- |e- ;1-jor-|6é—|s—|/9—-|s—]F—]e-]e—/I-]O [0 |0 [oO |r JO ret | 00F Z
H— | OF— | 9e— | ce— | 8a— | Fe— | Os— | AI— ] FI- ] TI-] 8— ] S— | F- J 2- Jor-|6—-—]|8—Jz—j|9-]F—-—]e—]se-]t—/O [I+] I+ |T+/ 1+) cer 10 ze+ | 008 Z
S— | a | Be— | Fe— | O€— | 92— | e— | SI— | SI- | I- | 8 —] 9- Je js- |e-—|s—|z—-j9-|F—]eF—-]|e@—]T-J]oO JO [It| T+ [ct] T+/ oF | 0 og+ | 0002
L— | eh— | ee— | se— | Te— | ge— | ec— | o— | 2I- | eI—] 6 —] 9- |#- js- |8—J2—-[9-J]s-—j|rF—-]e-—]e-]o JO [i+ }et)e+ [etl T+ | set | 0 8c+ | 008 T
9»— | er— | 6e— | Se— | ce— | 8e— | — | O2— | ZI— | eI— | 6 — | 9— |e Js-— JL-—]9-[S—]F—- |e —]e-—]/T—]o [I+] t+ et | st [et t+) vt fo 9+ | OUT
9¥— | eb— | OF— | 9e— | ze— | os— | Se— | e— | BI— } FI— J or— J 2- | S— Je— J4—-|s-—]|F—-]e-—j;s-}Ir—fo JO [I+ ) st fet) st fet | t+ | set | 0 Sot+ | 009 T
bh— | ch— | eg— | ye— | ee— | eo— | Se— | 12— |] SI— | FI- J Or— J A- | S— fe- J9-|s-—je—-]e-—je—|E—-|oO J|t+ [et jet | et] et ct | it) eet | 0 s+ | 008 T
WH | Tr | e€— | 9e— | ce— | es— | se— | te- | GI— | SI— | 1I- ] S— | S— Je- |S-—|F—-je-|s-JT—}oO [oO | t+ set |etler eter [Tt ar | 0 c+ | OOF T
OF— | Le— | ce— } ze— | ee— | Ss— | T2— |] 6I— | sI— ] 1I- S— _ }s— fe- JF—-fe-—]e-f;T-]/T—-joO |T+/etlsetjet jer ler ler [it | ost | 0 Oc+ | 008 T
ze— | e— | Te— | 82— | G2— | Z- | SI— | sI- | T1I- | 8— | s— fe- |F-je-—]se-|r—-jo jo J|r+jerljerlet)erjer ler | t+ err | 0 6I+ | ose T
ge— | es— | te— | se— | se— | te— | st1— | st—| 1—-|8- |s- Je-— Je-Je-|]e-J|r-jo jrt+ |otjerjet [et | ster jer | t+ | sry | 0 sI+ | 008 T
ge— | og— | 22— | — | — | 8I— | T- | 1- ; 8- | s-— Je-— fe-—|e-jI—]o |O |T+istletjet|er|rt pet pet lit yur jo Zit | OST T
zée— | 6o— | — | H— | — | 8I— ] I- | TI—]8- | s-— Je- fe-|rI-|I—]o j|T+ie+let [et] et | ot let let pet lit) ort | 0 9+ | OOLT
6c— | 9e— | &@— | O— | SI— | FI— | 1I— |] 8- |S— Je- |e-]1I—-jO Jo, |r t+tjetjet ler [er [e+ | et let [et | tt | sit | oO SI+ | 0S0 T
se— | so— | ee— | oe— | SI~ | FI— | TI—]} 8- |s- |e- | T-—]O Jo JrtlT+]etjpet ler e+ etl rt let yet | i+ |r | 0 FI+ | 000 T
s— | — | et— | zi— | at—for—}z— Je— fe-— |r —-fo Jrtl[r+[otle+l|et st] et let [et | ot] et] t+ | ert [oO I+ | 096
w— | te— | 6r— | zI— | wI— |or—|z- |}s—- Je- Jo Jo JrI+[et|etfet|et | et | Fe [et | et] et pet i+ )art | 0 I+ | 006
w— | st—}9t—]et—jor—Jz- ]s- Je— Jo JI tli +i[ot+i[et et] tlre ot |e | ot [e+ pet i+ |) Tt 10 Ti+ | 098 -
os— | st— | 91— | et— Jor— J z— | s— Jem JT + [T+ [eti[etjetljetl| e+ )et [et | ot) e+ Ft | et | t+ [ork | 0 oI+ | 008
4i—|st—}ai—} 6 —j2- | s— fe Jr tl[otletyetl[etlet+ l/r + let yet rt et et let jit je + | 0 6+ | OcL
wi~|ot—]et—}6e—|z- |s- Je- Jetfetrietiets/etle+] r+) et) et set e+ ot let jit] s+ | 0 s+ | 002
si—|a—|6—]z—- |s- Je- |atietjerletle+/et+] r+] et [rt et | bt] pt pet i+} et [0 s+ | os9
a—|tm—|6—]2- |s- le- Jatletletjet|r+)et]e +] et jot et et] ot pet i+ et oy L+ | 009~
t—|6—/9- Je- le- Jotletle+letla+ [e+ let] stlrt jot lot | ot let jit jot | 0 9+ | og
o—|s—fo- ji— Je-— Jotietl[etset+|r+]e +] e+] ot [et] et let] et [etl t+ ys + jo s+ | oos
L—-|s- Je- J[e- Jetletletlet|e rye +] e+ et [or] tt [ot /et [et [i+] r+ | 0 b+ | 09%
g—|e- Je- fa- Jetietl{[etset|r+] r+] e+] Ft ]et] ot ppt [et fet lit je+ jo s+ | 006
s— Je- Je- Jetletletfet]rt+ pt] rt+ let let fet [et pet per jit set [0 e+ | ose
e— |e- Je- Jetfetl[etl/etl/etrletjer rt jeter jer ser yet [it (et | 0 e+. | 008
o |0 Jo Jo |o |o |o0 |o0 Jo |o }6 }O |0 Jo jo 0 0 0
V2
Z- j@- |e- |e- |’ | s- | s- |e ]%—-]z-/e-]e@—/T-|I-|r—- jo Ba< |e
e— |&- [%s- |e—-]|%@-]e-|e@—-]%—-|e—-]1-|F- |0 b— | 00r—
z— |%—-|e—-|@—- | s—-|%—-]1I-|2—- |o dee) 0ST
—|%-]s-|I-]|m- |o u— | 00¢—
SULO"IDT
gaag | aap | aaer | gear | a2oy | a9ay | aay | gear | a0ay | aay | gaay | aoap | 4004 | aay | gaan | 9905 | ga0r | a00p | 9005 | aay | g20y | 900y |a00p | 3a0p | 409 | g00x | goer | 920F | eStrod
OorE | Oost | Cozt | Cort | oot | 006 | O08 | OOL | 009.| OF | OOF | ODE | O0Z | OOF | OOFT | OOET | OOZT | OOTT | OOOT | 008 | OO | ODL | O09 | OOS | OOF | OOF | OZ | OOT | -m0os | NOM. | Aades
: pue ae -sodoL| anew
sydea 3 ~yredur0a
queuyiedu10) MOTEg quawy1edai0a oA0ogy -80do.1, JO WOT}CA
-ofa Uva
[‘syuemyredu0o xIg ‘siajeUT NGG ‘SNIPs JeyNO ‘srajyauT Oge*SNIpPwI 1oUTT)
‘q 2u0Z
33
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
ee— | Sc— | 8I— | 9I— | SI— | OI— | 8 — | 9 —|F—)}e—-]IT- 0 eo— | 6I— | FI— | GI— | OI— | B— _| 9- i Aad — ad 0 0 elt+ an LL+ 000 ST
66 || S6—-") OL [20 | He | | 8 pe | eS ea la «10 te— | 6I— | FI— | GI— | OI— | B—_— | 9— i s— I- 0 0 @Lt+ $— OL+ 000 #1
se— | 96— | 6E— | LI— | FI— | TI | 6 — | 9 — | FP - |e] i fo S— | 8I— | 8I— | TI— | 6 — | 8— | 9- t—- Cr ic 0 0 Th+ &— Lt 000 €1
$8— | 42— | OC— | LI— | FI— | T1I- | 6 — | 9 -—|E¢—-—)}E—]e- | 0 &o— | 8I— |-8I-— | T1I—- | 6 — | 2— | S— &- ot i Ss 0 0 69+ e—- GL+ 000 21
9s— | 82— | Te— | 8I— | SI— | 7I— | OT—- |} 2 -— | $ —| § —]fVe- | 0 co— | 4I— | sI— | OI— | 6 — | L—- =| a s— o— = 0 0 89+ &—- be 000 IT
8&— | O&— | G2— | GI— | 9I— | SI-— | OT-— | 8 —| 9 —|; F—]fse- | 0 @o— | LI— | @I— | OT— | 8 — | 9— | F— &— oT t— 0 0 99+ &— 69+ 000 OT
OF— | G— | &e— | OC— | LAI— |] FI- | TI- | 8 —| 9 -—|F—-—]s- | 0 Té— | 9I-— | 1I-— | 6 — | 8 — | 9- =| F—- g— G— oe 0 0 go+ os cot 000 6
Gh— | &&— | F— | Te— | SI-— | SI— | GI-— | 6 — | 9 -—| F-—]s- | 0 O@—- | 9I- |} Ti-— | 6 —|L—]9- | F o— [= [= 0 0 zo+ C= pot 00 8
Sh— | b— | SS— | Ge— | SI— | SI— | AI—-— | 6 — | 9 —| F—J]s- | 0 0¢- | SI- |} 11-6 —| 2—] s- | t— co rs 1 0 0 09+ o— Zot 000 8
St— | 98— | 9— | GB | BI— | SI-— | AI— | 6 —| L -— | F-—] S— | 0 6i—' | FI— | 0l— |} 8 -— 19 — ps | 8 o— I- 0 I+ 0 8g+ o— 09+ 00g 2
Le— | L8— | Le— | Se— | 6I— | 9I— | EI— | OI— | 2 — | $ — | S— | I- | 8I-— | FI-}]6-—|s—}9-—]r ]E- [= 0 0 I+ 0 gst oT so+ 000 2
Le—- | L8— | Le— | Fe— | OC— | 9I— | SI— | OI— | 2 — |} $ — |] E— J I- | kI-] &l-—]6é—j}2—-—]9-—]Fr |] o- I- 0 0 I+ 0 got o— zg+ 008 9
8h— | 88— | 82— | He— | OC— | 9I— | SI— | OI— | 8 — | S$ — | O— _ J I- | ZI- |] &I-]6—-—}]2—-—]s—]r—- | e- i 0 0 I+ a G+ o— g¢+ 009-9
Ge [GE 86 | S6= | Te 20 | SS | OR ee Pe eS Ee | ce | Se | ae he SS re Pe t= 0 0 I+ T+ est 6— gc+ 00F 9
Os— | OF— | 62—- | SZ— | To— | LI— | FI— | OI- | 8B — | $— |] E— | I- | o9—-]etI-|;s—-—]9—-—|s—]e- J se- i= 0 0 I+ Le ost o— $+ 002 9
Ts— | Te— ; O&— | 9G— | Ge— | SI— | FI— | TI- | 8 — |; F —| O- | I- | 9I—-]} @-|s—-1|9-—|s—]e— | s- 0 0 I+ Aa I+ Igt+ o— est 000 9
CS | ERS | OS | Ze ee |) ST | SE | ES | 8 ee eh Se a | ee 1 oe eS |e 0 0 I+ It It os+ o— oo+ 008 ¢
FS— | G— | Te— | Le— | &S— | GT— | OT— | @I— | 6 — | 9 — | B— Y= |S BS | he es ee SH eS 0 0 I+ I+ Lae 6r+ I= os+ 009 ¢
gG— | &— | c&— | 8o— | Fe— | OZ— | OI— | VI-— | 6 — } 9— | E- | I- | FI—-];OI-|9—-—;F-—]e-—|Fe jI- T+ T+ or G+ I+ 8h+ I= 6r+ OOF ¢
9S— | Fr— | E8— | 62— | Fe— | OC— | 9TI— | SI-— | 6 — | 9 —| F— | I- | I-] OI-]|9-]|F—|E—-—]e 10 i I+ or ot+ I+ Le+ Ee 8b+ 00z ¢
£g= | Sh= | SE—" || Go=— | ‘SG— | OS= | 2h | St |: Ol | ee |S | SH 6 eee eS eH Pe 0 i I+ G+ ot I+ cpt I- 9+ 000 $
sc— | 9F— | be— | OF— | 9E— | Te— | LI— | SI— | OI— | 2 — | ¥— | T- | VI-|s8—j|;s—-—|/F&—-—16-—]|I- 10 Le T+ ot ot At tet t= cet 008 +
6S LP SS TE | Oe To ZI SE OFS 2 Se ES EE Se ee ee SS TS «10 ot ot o+ ét+ T+ a+ T= e+ 009 +
OO= | SF= | 9S— | C8— | Zo= |e | SES | BIS Ol | 2H EO Ole | te lk eH eS Ee T+ ot ot o+ ot I+ Tet+ i ch+ 00 +
O9—- | 8h— | 9E— | GE— | LE— | @— | SI— | F1— | TI-— | 2—]} F—- | I- | Ol-—]9-j;e-J;e-|t-jJo T+ ot G+ ot+ a+ I+ 6g+ t= Ort 002 +
O9— | Gh— | LE— | GE— | 8Z— | ES— | BI— T= | 2=(%> | I> | 6— |S —te=—)}1—|90 It | t+ Gar e+ e+ é+ T+ Le+ eo get 000 +
I9— | OS— | 8E— | ES— | 8E— | FE— | BI— cL= | S=— | s= | E= | s— | PH | Pe | 0 I+ | 3+ e+ e+ e+ e+ T+ 98+ tT 28+ 008 &
0o9— os— | 68— | F&— | 6B— | Fo— | OZ— éI— | 8 — | $— o— £2 ES EO T+] e+ €+ ot e+ Sar e+ [+ ret = oe+ 009 €
6S— | OS— | 6e— | SE— | 6Z— | SV— | O2— GE GE | Gee Gf Quer g at T+]/e+ ]e+ | + yt e+ e+ e+ E+ ze+ = Se-F OOF €
ss— | OS— | 6E— | SE— | O8— | S2— | O6— GI |G — | S— |S | SESS | Lae Pe Se | Se ee e+ a+ oe e+ ot og+ t= Eee 002 €
gS— | 6h— | 6E— | SE— | O&— | 9Z— | IZ— I> |} 6—|s=— | oS | RSL | Lae oa ee | ee Pe g+ e+ e+ e+ ot 8o+ = 6+ 000 €
Le— | 8&— | FE— | O&— | 9S— | T2—- &I-|6—-|]§- |@- | 8-10 @t+)/e+)]/et] e+ [ot g+ yt a+ e+ e+ 9+ i= l¢@+ 008 z
Gr— | LE— | FE— | 63— | S2— | Te— = |e— | 9=— |e | Se— | 0 e+/e+)/e+)e+ | e+ 9+ G+ gt e+ ot + i Fo+ 009 Z
98— | €&— | GE— | SS— | le— sI-—}6—|]9- J@- JT -;t +l] +)/e +] 9+ ]e+ | 9+ 9+ o+ ¢+ $+ e+ 1+ [= oot OOF Z
FE— | SE— G— | $S— | Te— sI— | OI— | 9- o—- 0 @+tirt+]e+}] e+ )ot 9+ 9+ g+ g+ e+ a+ 6I+ t= 0+ 002 &
cE— | O&— | LO— | EZ— | O3— eI-— |6—}9- |@- | T+/€+}/¢4+/9+)/9+]9+ | 9+ it S+ gt e+ e+ oT+ [= Li+ 000 Z
86— | SG— | Ge— | EI— @I— |6 —|}$—- 1e- J7tlr+]/9t+}9+)]/94+)] 9+ 9+ i+ c+ G+ b+ a+ FI+ 0 FI+ 008 T
* Go— | OG— | LI— ti | 8 | s= o- $+ )r+]/9+}9t+]}L2+] lt b+ i+ c+ g+ $+ o+ Ti+ 0 II+ 009 T
8I— | 9I- Ol— |S=— | S=— | o— (Rr SE 2+ 2a) ea] eb | l+ c+ ot t+ a+ 6+ 0 6+ OOF T
WM | te (6K | 2H ps (ee Re se Le he ea ce oP L+ ot e+ e+ ot Loe 0 L+ 006 T
Ol-18-—|9-|]F—- |e- |G+t}|Stlo+tl}/a+i19t+] 9+ [9+ 9+ g+ e+ e+ ot gt 0 Gb 000 T
9-|o¢-—[e—- |T- |St+t}G¢+{/9+]}/9+]/9+1/9+ | 9+ 9+ c+ e+ or T+ e+ 0 er 008
Be fe P+tig¢til/G¢t+y;o+]}/o4+)] ot c+ e+ e+ e+ ot I+ ot 0 ot oo9
o— I- [e+] +]e +] +]e +] e+ | et e+ e+ ot ot I+ T+ 0 Le 00F
0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0
pad
Leo |e oH Pe | t= [tk t= = I- I- 0 Di 0 t= os —
CH SS eS eS (eS te o— o- on I- I= L= 0 b= oor —
PRS |e] ee be t &- o— o— Ls @—- 0 6 osI—
=F —|F — | P=" | F— r= = o- o- I- tO 0° - 00¢—
= Bry ee | r= ¥— = &- oc [= a> 0 0 = ose—
B= | = —— §= = o- 1 So 6 0 6 — o0g—
— &— &—- o— {= H= 0 i- ose—
c= o— I= = 0 eI- 00F—
o— ‘= sT— T+ 9I-— Osh—
la aA I+ 8I- oos—
SULOy DT
qoay | Jook | Joog | Joos | Jooy | Joos | Jaoy | Joo | Joos | JooF | Joo} | Joos | Joos | Joos | Jooy | Joo} | Jooy | aay | yooy 902} qo} qaoy 40a} 429} uns
000 | 00S% | 000Z | OOST | ODOT | OOFT | OOZT | OOOT | 008 | 009 | OOF | 002% | 0D0E | ODSZ | 000% | OOST | OOOT | OOFT | OOGT O00T 008 009 00F 002 | _gadiuoo woryes sud
T Aer
pue Ayd | -usdur0g | -Sodoy, queued
queuyredm00 Mojeg quewyieduros saoqy ~eiBodoy, -ui00 JO uoT}
BAaT? UBIAL
—U01}84S JO WOT}BAI]O JOJ WOT}DILI0D —J0} WOT}daLI0D
[-squourqseduroo yy 4t97
‘s1012U OSZI ‘SNIPBI 19}NO {s1ajaw YEG ‘SNTpPeI JoUUT]
“ql au0g
3
15593°—12
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
34
—UO0T}21S JO WOTJBAV]a 1OJ WOTJIeII09
Io} WoOT}oeLI09
Ie— | 6I— | 24I— | SI— | gI— | 11- | 8 —} 9 — | e— &—- |3- I- L= 9- c= | t= o— Ls 0 0 I+ Tt I+ 0 got y- oL+ 000 ST
Te— 6I— | ZI— | SI— | &I- | TI—- | 8 —} 9 — | g— 8— | 3- r= §= | $= |= &— o> I= 0 0 I+ I+ I+ 0 got - 69+ 000 #1
&— | O— | 8I— | 9I—- | fI-— | 11- ] 6 — |} 2 — |} s— _ | e- J z- I= |S= |= (8 se fe |-0 I+ Tt T+ I+ It I+ 9+ b—- L9+ ~| 000 &T
ve— =| Se— | GI- | I= | SI— | ZI— | OI | 2 | ee Lee | ee eS ea Wig) TS '10 0 T+ I+ I+ It ir I+ 09+ y- ihe 000 I
9— | &— | 12— | 4I— | FI- | I— | OI— | 4 — | ¢— J e— Je- |I-— J e-— Jz—- Je- J1- Jo I+ | 8+ G+ ot T+ I+ T+ 6o+ s— 9+ 00 IT
2e— | — | @W— | SI— | St— | zI— | oI— | 8 — | g— _ | F— J s- | I- |e-— |e- |e- |t— Jo T+ | ot ot ot ot ot+ I+ Lg+ o= 09+ 000 IT
82— | S2— | e%— | GI— | 9I— | SI— | TI-— | 8 — | s— | r— Je- J I- }e- fe- rs 1/0) I+ {@+ | ot ot ot ot e+ I+ oo+ §= sc+ 00S OT
6Z— | 98— | &— | OS— | AI— | FI- | sI— |} 6 — | 9—-—« FH | Z—- I- |@- |@- |I- |0 T+ | 2+ | d+ ot a+ a+ e+ I+ wet s— g+ 000 OT
o&— | Zo— | H— | OS— | LZI— | SI— | eI— | OI— | 2— | F— | Z2-— |I- | I-— |1I- Jo Tse: TEE | egcbe | BE t+ a+ a+ ot I+ oot 2 got 00S 6
O&— | 42— | %— | O2— | AI— | SI— | EI— | oOI— | 2— | F- | 2—- | I- |I- Jo 0 I+ | 8+ | ot | &+ a+ e+ t+ a+ I+ 6h+ C= t+ 000 6
Te | 8e— | — | We | 2I— | et | er OI- |4— |%- |@- | I- JO It | T+ | Ut | Ut | ot | + e+ e+ ot+ ot T+ Le+ ci og+ 009 8
‘TE— | 8— | FB— | I2— | 8I— | 9I— | €I— | OT— | z— | S— | z—- I- |0 I+ | 3+ [t+ | ot | e+ | E+ e+ e+ e+ o+ I+ Spt o— Lb+ 00Z 8
ee— | 6@— | S2— | @— | SI— | 9I— | EI— | OI— | Z—- | S— | s— | I— | it | t+ Jet fat fet jet | rt e+ SF ot ot Tt eet o— Sb+ 008 ZL
ee— | 6@— | S2— | @— | BI— | LI— | FI- | TI— J z— | e— Je— J i— Jit fet fot fat fet fat fet e+ e+ ot ot T+ Tet o- ee+ 00F Z
Fe— | O&— | 98— | €S— | O— | ZI— | FI- | TI— | 8— | s— J e— | I— Jot fet fat jet |e+ | et | et o+ e+ e+ e+ I+ 6E+ o— Tr+ 000 2
ve— | TE— | 9B— | Es— | OS— | AI- | SI— | gI— | S—_ J 9-— | e— | I— | st fet fet Jet [ot | e+ [ot o+ e+ e+ or T+ gé+ o— or+ 008 9
gg— Té— | 246— | be— | OZ— | SI— | SI— | 7aI— | 6— | 9-— | e— | I— Jet fet Jet | e+ | ot Jot fot c+ e+ e+ or I+ Le+ o— 6g+ 009 9
gg— Té— | 22— | Fe— | IZ— | SI— | SI— | maI— | 6—_ | 9— J e— | I— Jet | e+ [H+ | o+ | Ft [ot | ot c+ ot s+ e+ I+ 96+ a 8e+ OOF 9
gs— Té— | 240— | be— | TZ— | 8I— | SI— | Z7I— | 6—_- | 9— ~ | e— | I-— Jet | e+ Jot | ot [ot | ot Jot c+ 7+ e+ e+ T+ e+ C= 98+ 002 9
se— | T€— | 246—- | e— | T2— | SI— | SI— | I— | 6— -9— fe-— | I— | ot pot Jet fot | ot jot fet G+ e+ e+ é+ Tt eet o— sgt 000 9
se— | 1€— | 4e— | — | IZ— | 8I-— | sI— | gI— | 6—_ | 9— J e— | I— | et | e+ [ot Jot |ot+ fot fot c+ au $s e+ T+ eet o—- bet 008 ¢
se— | Té— | 2e— | — | IZ— | 8I-— | SI— | zI— | 6—_ | 9- | e- | I— | et | et [9+ | ot |ot+ Jot Jot o+ ot b+ e+ I+ og+ oF cet 009 ¢
se— Té— | L6— | ¥o— | — | SI— | 9I— | ZSI— | 6B—_—-« | 9— S| E-— «|S CL GH+ 1 GH+ [St CJ ot J OF Jot | o+ g+ ot e+ e+ I+ 6+ @- Té+ 00 ¢
ge Té— | 20— | H— | I2— | SI— | SI— | ZI— | 6B—_- | 9— | E-— | s— | 9+ | 9+ | Ot | Ot JOH | OF | 9+ c+ ot b+ e+ I+ 8o+ e- og+ 006 ¢
be— Té— | L0— | #— | — | SI— | SI-— | AI— | 6—_- | 9— S| -— «| S— | OF Lot J Ot Jot Jot [ot | 9+ 9+ c+ tt e+ I+ 9¢+ oT 82+ 000 ¢
Fe— T&— | 26— | %— | l— | SI— | SI— | AI— | 6—_- || 9— «| E— | s— | 9+ | 9+ J Ot Jot Jct fst | at 9+ c+ ot SF I+ Sor a 98+ 008 +
yE— | €— | 8B— | H— | Z— | 8I- | STI— | SI-— | 6@— _- | 9- J e— fo- Jzt fet fzt [Lt Jat fat fat 9+ o+ ot e+ ot Ft iL Sot 009 >
g&— | O&— | Zo— | — | T7— | 8I— | SI—-} @I— | 6—_- || 9— FW LB Cf At fet fet fet fet fat fat 9+ o+ e+ e+ ot c+ t= e+ OOF *
s&— | O&— | La— | H— | IZ— | 8I-— | STI— | ZI-— | 6— | 9— «| B— YU lt fet fet fet fat fat fat 9+ gt+ ot e+ o+ Tot [= cot 006 +
c&— | O&— | Z2— | €Z— | OZ— | ZAI— | FI— | ZI-— | 6— | 9- J e-— | s— | st | st | st fst fst fat fat 9+ ¢+ ot e+ ot 61+ | 0+ 000 +
Té— | 6Z— | Ze— | €@— | OZ— | ZI- | FI— | TI— | 6—_ | 9— | e— | s— {st | st |st [st fst fst fat 9+ G+ bt e+ ot git I- 6I+ 008 €
O&— | 82— | 9Z— | SZ— | BI— | 9I— | FI— ] TI— | 8B— |} 9- J e— | I— | st | st | st | st | st Jet fat 9+ c+ $+ e+ or 9I+ I- Zit 009 €
62— | 4o— | SZ— | e2— | 6I— | 9I— | EI- ] TI— | 8B—_ | 9— | e— J I— | 6+ | 8+ | st | st | st | st fat 9+ c+ e+ e+ ot Sit I- 9I+ O0F €
83— 93— | F— | I2— | 6I— | 9I— | EI— | OT— | 8B— _ | 2z—- c= I- |6+ | 8+ 8+ | 8+ |8t+ | 8+ | 2+ 9+ G+ $+ e+ ot FI+ T= ST+ 00Z €
9— | 86— | $— | OS | SI— | ST = | BI | Ol | a= | 9m |e | I 16+ Le | ee et let ol et | et 9+ c+ tt e+ a+ Zi+ fC eit 000 €
we | Ge | GI | LI | FI | ZI—- | Bm 1 em Le Le Le 1) | ot pet lst ee pee | et 9+ c+ e+ e+ ot Ti+ T= ZIt+ 008 Z
T@— | 8I— | 9I— | FI— | BI— | 6 — | L— | S— «YW LL T— | 6+ [8+ [8+ [8+ | 8t | st | 2+ 9+ G+ ot e+ ot OI+ t= Ti+ 009 @
Ob | SE | 8S | HS | 8 = ee SS = PS Per 1S PS |) B+ SAR cede tzoe 9+ c+ e+ SH ot+ 8+ load 6+ OOF @
P= "| SI | OLS 18 — |e GS Em | TO 1 6 OL BH PSH OL 8H Of ete tks | 9+ ot+ F+ im I+ Boe T= St 00% @
I= | t= 1 Ol FS — 19 |G | SS | b= es | Bae PSHE | ete ese ae ee 9+ c+ a+ e+ I+ 9+ Ls bec 000 &
S— I= Ee | F— be= | T— LB | ab | 2 | 2+ Lor [ot | o+ G+ b+ S$ ot I+ y+ 0 Pate 009 T
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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35
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
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ee os 2 or or a = 6—- L- c= §= ik I+ ot ot+ e4- e+ e+ Ser e+ T+ 8h+ LE aot 00S #1
I€— | La— | &— | OZ— | 9I— | FI— eI— G= = c= io t= ie ot o+ Soe e+ St e+ ot+ I+ 9F+ F- og+ 000 #1
= = = = = = L— 6 — i= s= o> o- ot s+ Se b+ et e+ §+ e+ ot+ ce+ t= 6F+ oos et *
a os iz ie ot rH ot oI— l= c= g= co o+ SHE e+ ot e+ e+ e+ a e+ eet i= Let 000 &T
Z&— | 8Z— | S2— | Te— | ZI— | STI— GI- oI— wire C= s= o— e+ s+ e+ bt e+ e+ $+ e+ ot oh+ = 9b+ 00 ZT
ZE— | 6— | SZ— | TZ— | 8I— | SI— sI— oI— L= c= 8— o— e+ s+ ot e+ e+ e+ t+ e+ ot OFT i a ppt 000 @T
Z&— | 6Z— | 9Z— | G— | BI— | SI- gI— or— L= 9- F- o- e+ e+ g+ g+ cr t+ tt e+ a+ set 8= Te+ 00g IT
= — | 9—- — ae — Si oI— 8- 9- = a b+ e+ g+ c+ o+ e+ a+ €+ o+ LE+ = 0F+ 000 IT
ie Oe - ee co or sI— OT 8—- 9—- y— o~ o+ o+ 9+ 9+ G+ gt+ t+ §+ ot set e-— ge+ 00S OT
$e— Tgé— | 42— | €2— | 6I— | 9I— FI— TI- 6—- 9- —— o— o+ St+ 9+ 9+ 9+ gt+ gt e+ ot e+ $= | oer 000 OT
Fe— Té— | ZE— | &]— | OS— | LZI— FI— TI- 6— 9- a o- 9+ 9+ 9+ 9+ 9+ 9+ c+ §+ ot Té+ &— bet 00¢ 6
¥e— Te— | La— | Fe— | OZ— | LZT— FI— II- 6— 9— t= o— 9+ 9+ Lt+ L+ 9+ 9+ c+ §+ a+ 6o+ 8 eet 000 6
- = = = = A bI— TI- 6— 9- = c= Lt Loe Lt Lt 9+ 9+ c+ e+ ot Lo+ c= og+ 009 8
i i ie ie oe i bI- II- | 6— 9- 7 Z— L+ L+ 2+ 2+ 9+ 9+ G+ e+ ot 9e+ a 8o+ 002 8
ee— | og— | 2e—| ee— | 0¢— | 2zI-— | FI- | aI- | 6— 9- — o- 2+ L+ 2+ L+ 9+ 9+ St e+ ot P+ o 9¢+ 008 2
ge— | o&— | 92— | &@— | OZ— | ZI— FI— oI— 6—- 9= t= o— s+ 8+ 8+ 8+ oe 9+ gt e+ ot oot c= tot O0F Z
ZE— | 6Z— | 9Z— | EV— | OS— | 2I— sI- 6I— 6—- 9- F— o—- 8+r 8+ 8+ s+ i+ 9+ g+ e+ ot Te+ eo e+ 000 2
— < = = =| i= FI— gI— 6- 9- F—- o- 8+ st 8+ 8+ Lt+ 9+ gt a+ ot 0+ oT cot 008 9
ie = oe eo a it FI— si= 6= = : oo st gt oe st i+ 9+ + e+ of 6t+ a lotr 009 9
(— | te— | Se— | we | GI | BI i See aI— 6— 9- r= o— 6+ 6+ st st it 9+ G+ e+ ot+ sIt+ ae Go+ 00F 9
6z— | 92- | #— | 12— | st—|9t- |FI- |ezi- | 6- 9- — o= 6+ 6+ s+ s+ L+ 9+ c+ $+ et 21+ o— 61+ 002 9
6s— | 9e— | &—- | 12— |} sI—|9I- |rI- | II- | 6- 9- a a 6+ 6+ 8+ s+ L+ 9+ c+ t+ e+ 9t+ a en 000 9
—_ ss = — Fa oe ra 1 Sd 6—- 9—- i oa co 6+ 6+ st gt L+ 9+ c+ $+ e+ cT+ o— 2T+ 008 ¢
se ao = oe ar or i i 6— 9- ¥—- o— 6+ 6+ 8+ st Lt 9+ c+ a+ e+ ci+ o— Ze 009 ¢
L@— | HE— | G— | OS— | BI— | 9I—- FI— cL= 6— 9- a o— 6+ 6+ 8+ 8+ Le 9+ G+ e+ e+ FI+ C= ot OOF ¢
9%— | — | G— | 6T— | LI— | SI—- &I— TI- 6—- o= t—- o= 6+ 6+ 8+ 8+ bb 9+ ¢t+ ot ot e+ o- GI+ 00¢ ¢
Sc— | 86— | I6— | ST— | 9T— | FI éI— oI- 8- 9—- t= a- 6+ 6+ 8+ st 2+ 9+ g+ e+ e+ aI+ o- I+ 000 ¢
ze— | et— | 21— | gt— | et- - - - |e- Je- |e- fer fot fet fst fat fot [ot fet fat fort I- ut 00s +
e i ot i a1 _ {TH b a g= .= o— 6+ 6+ 8+ 8+ lt 9+ c+ e+ ot 8+ I- 6+ 000
aiI-|1- |6- |s— |9- - |e |t- Jé6t jet jst jst Jet jot fst Jet fet fot I- L+ 008 €
oI- 8—- LCS o= p= o- 1 8+ 8+ tr a 9+ c+ e+ e+ et c+ I- 9+ 000 €
1 Go c= c= o— | 8+ + as 9+ o+ b+ ot ot+ T+ Sar I- e+ 00S Z
GI t= c— 67 a L+ 2h 9+ 9+ c+ t+ a ot I+ ot+ i © ab 000 @
I- I- I- c+ ot+ pt t+ t+ or e+ e+ it EF 0 Te 00¢ T
0 0 F+ Pt e+ Cr e+ e+ ot T+ I+ 0 0 0 000 T
0 e+ e+ ot a+ ot Bate I+ 0 0 0 0 0 00s
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36
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
— W01}¥}S JO MO}VAV[d JO} MOT}0eII09
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9E~ | Be— | 8o— | Go~ | Te— | SI- | sT— | eI— | or— J 2- FF JI- Jo+]o+fot fat [zt fat fat fot fot Jet fer frt | ect tS get 00g &T
9— | BE | BS— | Go— | Te— | BI— | SI— | T— J or— J 2— | H— JT— Jatfatfat fet fae fat fat fot fot [ae fet fre | eet ae et 000 &T
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E- | Te— | 8e— | Fe— | Te— | SI— | sI— | I— | or— | 2— | F— Jt- |st+][et ist fat fat [at [r+ fot for fat fet frit | ect a Le+ 00¢ II
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8e— | O&— | 2e— | H— | TE— | SI— | SI— | sI— J or— | Z— J F— JI- Jot+let lst jst (st fzt+ fzt fot fot fot fet fat | ost t= e+ 008 OL
Be | O— | 2e~ | FE— | OS | SI~ | SI | GI- J or— | A~ | F— | I- Jo+]ot]st |st jst jst fet fot fot fot fet Jet fort s— aot 000 OL
Te | 6— | 92— | Se— | OS | BI— | SI— | GI— | or— | z— | F- | I- |6+]oet]ot |st |st+ |st+ |st fot fot fet fet fat fart Sa 06+ 00S 6
OF | 8B— | Se— | EZ— | OS | SI— | SI~ | @I— J or— | z— JF ]I- |o+]ot et Jot 8+ |st [st fst Jot [ot fet fet fort c= 6I+ 000 6
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quouljiedui0s Moleg quouljziedui0s saoqy -sodoy, Jo HOWeA
: -9]9 UE
['squeujiedur0s u997x1g
“S19}9UL OFSE ‘SNIPS Jo}NO ‘s19}eUN OZEE ‘SNIPB1 JaUuU]]
"H auoz
37
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
og— 9F— | GF— | 8E— | EE— | 8S— | FE— | OS— | ST— | TI— | 2—- = sit | ett | et+ | e+ | st+ | att | I+ Ti+ 6+ gt 9+ e+ wot L= Teé+ 000 ST
6F— ch— | Ih— | LE— | GE— | 8Z— | F— | OS— | ST— | TI— | 2—- e- FIt | FL+ | E+ | PEt |-FT+ | SI+ | ott Ti+ oT+ 8+ 9+ e+ got Li og+ 00¢ #1
8h— vr— | Te— | 9E— | GE— | 8Z— | FO— | GI— | ST— | IT— | 2—- o= FI+ | FL+ | PI+ | FT+ | PI+ | €T+ | 2t+ TI+ oI+ gt 9+ €+ oot 9—- 8+ 000 #1
Le sh— | OF— | 9E— | TE— | 4O— | SS— | GI— } ST— | TT— | £—- s= I+ | H1+ | FI+ | FE+ | FTF | SIt+ | a+ Ti+ oI+ st 9+ e+ 0¢+ 9- 9o+ 00¢ &T
9F— Gh— | 6E— | SE— | TE— | 4e— | ES— | BI— | ST— | TI— | £Z- = Pit | I+ | PL+ | PT+ | PTF |-8E+ | I+ Ti+ oI+ st 9+ e+ 6r+ 9—- Sot 000 &L
Sr TF— | 88— | FE— | OF— | 9Z— | GS— | BT— | SI— | TI— | L—- e— | e+ | e+ | tt | Ft | T+ | Ott | att Ti+ oI+ 8+ 9+ e+ Sit 9—- oot 00¢ GT
oe OF— | LE— | EE— | OF— | 9Z— |} GB— | BI— | ST— | TI— + 2— &— Git | bI+ | FI+ | FT+ | FEF | SI+ | att Ti+ oI+ gt 9+ e+ 2T+ g— oot 000 oI
6E— | SE— | G&— | 62— | GS— | To— | 8I— | FI— | TI— | Z— &- GI+ | FI+ | FI+ | FT+ | FT+ | SIt+ | at Ti+ 6+ Lt+ c+ e+ sIt+ ¢— 0¢+ 00S TT
LE— | FE— | TE— | 8S— | GZ— | Te— | SI— | FI— | OT— | 2—- 8 Sit | FI+ | FI+ | HI+ | FT+ | ST+ | ait Ti+ 6+ Le G+ e+ FI+ G— 6I+ 000 TL
FE— | TE— | 83— | FE— | T]— | LT— | FI— | OT— | LZ—- = Git | I+ | PI+ | HE+ | FE+ | ST+ | att Ti+ 6+ Lt G+ €+ vit c= 61+ 008 OT
€E— | O€&— | LE— | FB— | OS— | LT— | ET— | OT— | Z— o> St+ | FL+ | bt+ | F1+ | FI+ | ett | ZI+° Ti+ 6+ L+ s+ e+ eit+ g— sI+ 009 OT
€&— | O€— | L2— | &U— | OS— | 9T— | ET— | OT— | 2Z— c= SI+ | PI+ | 1+ | T+ | FT+ | Ott | 1+ oI+ 6+ Lt G+ e+ sI+ Gc 8I+ OOF OT
GE— | 6G— | 9Z— | EV— | OS— | 9T— | ET— | OT— | 2Z— ¢= GI+ | PL+ | FE+ | PL+ | I+ | OEt | IT+ oI+ 6+ Lt+ G+ e+ a+ c= 1+ 002 OT
ZE— | 6G— | 9V— | G— | 6I— | 9T— | SI— | OT— | L— c~ Git | PE+ | PT+ | FT+ | I+ | VE+ | 1+ oI+ 6+ Lt ct ot+ a+ ; om 9T+ 000 OT
8Z— | SS— | G— | GI— | 9I— | SI— | OT— | L— &- GI+ | FI+ | PE+ | FI | PTF | Ut+ | TT+ oI+ 6+ Lt G+ ot Ti+ - gt+ 008 6
8G— | SZ— | G— | 6I— | 9T— | ST— | OT— | L— 2 SI+ | HI+ | bI+ | FT+ | SI+ | B+ | I+ or+ 6+ l+ S+ ot Ti+ y= 7 009 6
L6— | ¥O— | GB— | 6I— | 9T— | ET— | OT— | L— e= PIt | FL+ | PI+ | FT+ | St | r+ | Ti+ oI+ 6+ Lt c+ ot or+ ~— I+ 00F 6
L6— | ¥6— | T@— | 6I— | 9T— | SI— | OLT— | LZ— S= Pit | PT+ | I+ | E+ | Ett | 2+ | 11+ oI+ 6+ Lt G+ ot oT+ p— PI+ 00é 6
9%— | ES— | TZ— | 8SI— | SI— | GI— | 6 — | 9— C= I+ | I+ | FI+ | ST+ | Ett | Or+ | I+ or+ 6+ L+ ct ot 6+ $- gI+ 000 6
€o— | O@— | 8SI— | SI— | @I— | 6 — | 9— = It | PI+ | PI+ | St | I+ | a+ | t+ or+ 6+ L+ G+ a+ 6+ ¥- ei+ 008 8
€c— | OC— | 8I— | SI— | @I— | 6 — | 9—- e— | bi+ | w+ | br+ | ett | Sit | a+ | 11+ oI+ 8+ l+ c+ ot 6+ ~—- ett+ 009 8
co— | 6I— |.4I— | FI= | 1l— | 8 — | se c= PI+ | I+ | St+ | &T+ | I+ | at+ | t+ 6+ 8+ 9+ gt G+ 8st r= ort 007 8
eo—.| 6I— | ZI— | FI— | TI-— |] 8 — | S— €— | bit | bI+ | et+ | et+ | Ett | ott | oft 6+ 8+ 9+ G+ ot gst F= ait 002 8
T@—- | 8I— | 9I— | €I— | II-— |} 8 — | s— = I+ | I+ | et+ | t+ | eit | I1+ | of+ 6+ 8+ 9+ e+ o+ L+ = Ti+ 000 8
A= | A= | l= | Ok— | 8S— | = o- PI+ | I+ | ert | a+ | att | Ti+ | or+ 6+ 8+ 9+ a+ e+ 9+ = 64+ 00S 2
9I— | FI— | at— | OI-— | 8 — | S— o— ert | st+ | a+ | at+ | at+ | 1i+ | or+ 6+ 9) ae 9+ y+ o+ G+ a= 8+ 000 2
FI— | ZI— | oI— | 2 — | S—- o— ett | ot+ | a+ | 11+ | 11+ | Or+ | 6 + gst Lee 9+ ian e+ q+ s- 8+ 00g 9
GH T= | 6S | 2— 7] se oc it | 2+ | 1+ | 11+ | 11+ | O1+ | 6 + 8+ Lt+ G+ e+ o+ y+ c= 9+ 000 9
6-— (219 — 1 2= o- @i+ | Ti+ | 11+ | Or+ | Or+ |} 64+) 8+ Lt+ Oot G+ e+ o+ e+ o— c+ 00s ¢
Ree | ores Qa o— Ti+ | 11+ | or+ | OT+ | OT+ |6+)8 + Lt 9+ e+ e+ ot ot+ os e+ 000 ¢
2S | eS o— OI+ | Olt }oT+ | 6+; ;6e4+);8t 2+ 9+ 9+ e+ e+ ot a+ o> y+ 00S F
QS eS eS I- Ort | ort }e+l]oe+/8t{/styat 9+ gt+ e+ €+ o+ toe Z- e+ 000 +
Ber ee; i 6+/6t+) Sti stlz+tl{[z2+)/9+ g¢+ G+ e+ e+ T+ Tt oT e+ 00¢ &
P| oF dos St stlzaty~2ti ~atloa+ty et ge y+ g+ a+ I+ 0 = T+ 000 €
oT pie L+/L2+,}/9+/9+/9+]/¢+/¢4+ y+ et €+ ot I+ 0 ie T+ 00S Z
o- | oe 9+/9+]/o+]/o+);o+}]ot+]/Ft+ e+ et+ G+ ot I+ 0 LE T+ 000 3
iL P+l[rtl/r+i r+ {rt+jetrlyet e+ ot+ a+ T+ I+ 0 i= Le 00s T
er etietiletiletj et etlor ot ot+ It T+ 0 0 0 0 000 T
GHC lS ITE | Le | te I+ Ee 0 0 0 0 0 0 oo¢
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 0d
Go| Se — | oH | oe | r= L= Le l= [= 0 0 0 0 002 —
Pe) | eS a [8 eo G6 67 o— i i 0 I+ LS oor —
GS | RS e=— |e = f= == o— i = £= I+ (a 009 —
. 9—-—|]9-|\S- P= ¥- s- oT i oe L= It C= 008 —
99> c= Ea c= o—- = g7 ot = 000 T—
9- Ques = o—- = oo ot GS 006 T—
= r—- = tT a ot LA OOF T—
= ¢= = f= a+ 6—- 009 IT—
&—- oc 6 — a+ II—- 008 T—
o— I e+ FI 000 2—
SUOUDT
4a0y | rey | yaex | qo0y | goog | aoaz | goax | ao0z | qo0y | ye0y | goog | yey | goog | y003 | yo0p | a00p | yoy | yoay | aooy | soar | gooy | gooy | gaoy | yoop | uoMes
000ZT |O00TT | COoOT | 0006 | 0008 | 0002 | C009 | CODE | 000% | CODE | 000% | OOOT | OONZT | OOOTT | OODOT | 0006 | 0008 | 0004 | 0009 000¢ 0007 0008 0002 O00T — mores Ayer
-ued m0: -sodo, \ueul
Ayder 9 ue ~yxed 0100
quaurjredur0s Mopoyt quourjredui00 eaoqy “Bodo, JO NONBA
-919 Weep,
—U01}8]s Jo WOTLBANTA Joy UOT}DEII0;)
—JoJ WOT}Oe1I09
[‘s}aeujiedui00 £400M LT,
"SIO{OUL OFFS ‘SNIPVI Jono ‘sIo}our OFZG ‘sNIpeI souUT]
"I 90g
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
38
6e— 93— | Z— | 6I— | ST— | TI—- 9- o— Te+ | 0+ | 6T+ | SI+ | ZT+ | ¢T+ I+ oI+ 8+ 9+ e+ FIt OI— Ft 000 ST
88— 9B— | Z@— | SI— | SI— | II—- 9- | 3- Tot | OS+ | GI+ | ZI+ | OT+ | FI+ TI+ OI+ st gt+ e+ sI+ oI— e+ 00¢ FT
Le— St— | T— | 8SI— | FI— | TI—- 9- | 3- To+ | OS+ | 6T+ | LI+ | 9T+ | FI+ Ti+ 6+ Lt Gt et ait G= Tot 000 #1
9g— Fo— | T2— | LI— | FI- | TI 9— | 2— | 02+ | 6I+ | SI+ | ZI+ | OT+ | FI+ Ti+ 6+ Lt G+ et Ti+ 6 — oo+ 00S eT
se— €6— | OG— | LI— | FI— | IT— G— |e— | Oct | 6T+ | SIt+ | OT+ | ST+ | FI+ Ti+ 6+ L+ q+ a+ oI+ 6—- 61+ 000 &T
ee— @3— | OC— | LI— | FI— | OT— G-— | e-— | oo+ | 6T+ | ZT+ | 9T+ | ST+ | I+ Ti+ 6+ 1+ o+ c+ 6+ 3 = LIF 00S 21
se— @o— | 6I— | 9I— | I— | OT— G— |%—- | 03+ | 8T+ | ZI+ ] 9T+ | ST+ | F1+ Ti+ 6+ L+ e+ ot 8+ = 9I+ 000 ZT
T@— | 8I— | SI— | €I— | OI- S— |%— | 61+ | 8tt+ | 2ZI+ | 9I+ | I+ | eT+ oI+ s+ L+ e+ ot L+ 8 sit 00S TT
Ié@— | 8I— | SI— | @I— | OI— G— | o— | 61+ | 8It+ | 9I+ | ST+ | FI+ | sI+ oI+ st 9+ ot o+ be b= I+ 000 IT
O@— | LI— | #I— | @I— | OT— g— |e— | 81+ | 2zt+ | 91+ | Stt+ | 1+ | ett or+ gt 9+ e+ ot 9+ Lo sit 00¢ OT
6I— | 9I— | FI— | TI— | 6 — g— ot SI+ | LI+ | ST+ | FI+ | ST+ | I+ 6+ st 9+ e+ a+ gt+ (ed git 000 OT
8I— | 9I— | €I-— | TI-— | 8 — $— | 2— | 241+ | 91+ | ott | bI+ | STt+ | ZI+ 6+ Lt+ 9+ pt ot+ a+ g9- OI+ 00¢ 6
LI— | SI— | 8I— | OI- 18 — F-— |t- 20+ | OT+ | GSI+ | I+ | I+ | I+ 6+ Lok: 9+ p+ ot e+ 9 64+ 000 6
9I— | FI- | aI- | OI-— J 2 — b— | @— | Ott | STt+ | HI+ | Ott | att | 11+ 6+ J2+ [ot b+ ot e+ o= 8+ 00g 8
SL" | STS 6: = 2 = bcs o- QT+ | SI+ | FI+ | I+ | att | Ti+ 8+ bok G+ e+ ot ot oo sob 000 8
TI-— | OI-— | 8 —|9—- &— oe I+ | bI+ | Sit | SI+ | 11+ | OT+ gt+ Dee c+ e+ ot Ls oe 9+ 000 2
eh Sh = i eI+ | et+ | ci+ | 01+ |}6+)8 4+ Lor gt e+ e+ I+ 0 P= r+ 000 9
== pS i= a Ti+ | Ot+ | oT+ |]6+/}/8+)2+ 9+ G+ e+ et T+ 0 oo Ce 000 ¢
g= I= |0 6 FE Se Se Be Oe bp oe G+ pt e+ ot Tt y= or o+ 000 F
I- |0 L+]/9+}/9+]9+/¢9+ )/¢9+ r+ e+ a+ ot+ It 1 Si o- T+ 000 €
I> 10 G+t)/p+l]p+ipti p+ryet+ e+ e+ a+ ot I+ t= LS 0 000 &
It [@+]e+r [et ;otlyeti tt I+ Tt T+ I+ 0 LS PS 0 000 T
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
198
€-|}@-|/¢-|¢- > LS I= t= Tt Esk T+ 0 00F —
y- |b ‘aa c= o—- I- t= 0 a+ og 008
r- S= o—- I- bate 6: E R= 006 T—
smoyyog
aay yooy | ya0q | gaaq | yooy | yooy aay | yaay | ya0z | qooy | yoy | goog | yooy | yooy qa0y 4oay qoay 40a} 400} woryes
o00zT 0008 | 0002 | ooo9 | ooo | coor 0002 | ooot | co0zt | ooott | cooot! o006 | 0008 | o002 coos | coor | aooe | ooo | oot | -wodtt0>| goes | xqdter
soae -uedwm0g | -80doz, jueul
Yer 4redui0o
queurredui0s Mofeg yuewyredu100 eaoqy -dodoL Jo WOTyeA
-9[9 URAyy
—U01}81S JO WOLYBANTA JOJ WOTJDeLION
—J0} WOTJOeIIOD
[syueujsedu0s ueeyxIg *s10}9TT OOF ZI ‘SNIPeI J0jNO ‘s19J9UI OFFS ‘SNIP JouNT]
fee
39
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
tI— | F1—|et— | 2t—|or—|6— |2— | 9- | ¢—- | e—- | s- | I- | ett | 1+ | ort | 6+ st | 2+ | 9+ g+ pt e+ ot T+ Sok @I— SI+ 000 ST
tI— | et—|zi— | 1—|or—|]s— | 2—- |9- | s-— | e- | e- | I- | ett | T+ | ort | 6+ 8+ | 2+ | 9+ gt ot e+ ot Ta ot sI— PI+ 00S FT
t= eI— | 2I— | T1— | O1— | 8— L= 9- c= ; oe o— i It+ | O1+ | OT+ | 6+ 8+ Lt+ 9+ gt e+ e+ a+ I+ e+ i= sit 000 FT
ei— |zti-|}t1—]}]or-}6 —|s— |2— |9- | ¢- | #- | @- | I- [1+ |] ott oIt |} 6+ | Bt | 2+ | 9+ g+ b+ e+ ot Tt I+ Ii- It 00¢ &T
eI- Cl} TE= | 01> | 6 =| 8— L a= c= c= o— i= Ti+ | oOl+ | 6+ | 8+ 8+ Lt+ 9+ ct a+ e+ a+ It I+ T= Zit 000 €T
oI— II-— | 0I- }6 — | 8 —]2- 9= 9—- c= 2 o— 0 Ti+ |ot+ |6 +] 8+ i+ 9+ 9+ c+ e+ e+ ot T+ 0 oI— OIt+ 00¢ ZT
ZI— | t— 1 0t- |e — fem | 2— | 9 |e [8 |e Te ot+}/6et+ioa+]st |zt | ot | ot e+ e+ e+ a+ T+ 0 oI— oI+ 000 21
6I— l—|0i— |@ — 1s —|i- | 9- |S Fee |e Pe 10 ort |/6+]/6+]st | zt | ot | ot a+ P+ e+ o+ I+ 0 6- 6+ 00S IT
i- |oi-}6—]8s—|24—-—]9- |s—- |¢s- | F#- | &- | I- 10 ort |e+{st}]zt jzt | 9t | ot e+ P+ e+ e+ T+ t 6— gt 000 TT
i |e 14-1 S— Lee fs |e i= fem tie ie 6t+}/sti stl] z+ |9t jot | ot e+ e+ ot I+ I+ tS 6= gt 00S OT
Gi 1c |6—1e—|2—19- |. |r pe | em I-18 6+ti/stist}]z+ jot [ot | St ot e+ ot I+ t+ T= = Le 000 OT
oI- 6-18 — | L=|9—|e= | eo leo [eo |e LT 10 6+ istist]zt+ [ot [9+ | Gt au e+ ot T+ It o— 8 — 9+ 00S 6
= 6—le-lb—|a—-|@—- |e le | t— 1e- TT | 0 gst{/L2+]L2+]9+ [ot | ot | o+ e+ e+ a+ T+ T+ o— ia g¢+ 000 6
G— 1F=—(R=—|2=—10—|S— | F—- |e 1a | a= TT 18 Stlz+]2+}]9o+ [ot [St | t+ et e+ ot Tt I+ o— £5 G+ 00¢ 8
gs—- |8—l|/2—/9=—19=—/|8=— | F- | 8- Le | oo I- |0 L+i/L+]9+]9+ | G+ [St | + e+ e+ e+ T+ I+ o— Le Ge Hk 000 8
L—-{]9—-/]9-—|]¢¢-—]*%- |#— | €- | e- | s- I- |0 L+i/9+/9+]¢+ |o+ |F+ | Ot ct o+ et T+ It o— 9- a+ 000 2
QS = S| RS e o. €— o— T- |.0 9+/G$+/E+]%4+ $+ e+ e+ e+ ot+ a+ T+ I+ o— go iS) te 000 9
= | ES e- oe eS oT b= 0 G¢+/G+]e +) Ft e+ e+ a+ ot T+ T+ I+ 0 o— i= c+ 000 ¢
a o— o— a Lt 0 Pr+yjrt+]e+)et+ e+ ot @t+ a+ T+ I+ I+ 0 o— aa L+ 000 F
i Se = LL I- 0 et+ijetryaor|ot ot ot e+ a+ I+ I+ T+ 0 o— G— 0 000 €
1 I-— |0 ZeiS+iscw ler | et | ee Pre T+ I+ T+ I+ 0 i Le 0 000 @
0 TEP eh tee Pe Lie I+ 0 0 0 0 LS Le 0 000 T
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
at
PH |b tH t= (he ie t= [= tT L I- 0 I+ tt+ 0 oor —
ee i I- i E= 1I- 0 T+ et i= 008 —
t= = i I+ Pa oe 006 T—
SULOYIOT
yooy | yo0y | y00y | yo0y | yooy | 4003 | Joos | Joor | ooy | Joay | JOOE | JOOF | JooT | Jooy | JOO] | 400} |, 790y | Jooy | Joos 409} 00} 402} 490} Joo} wOT}es
00zz | 0099 | G009 | OOFS | OOSF | OOS | CODE | ODE | OOFS | OORT | OOST | 009 | 00ZL | 0099 0009 | OOPS | OOSF | OOZF | 0098 0008 00S 008T 008T 009 -uedu0o mores Ayder
Codex [wedurog | Bodog, qyueur
eed ~yredu100
qyuewji1edur0s MoTeg queuyiedui0s sa0qgy -sodoy, Jo WOTywA
-9[9 Wee,
—U01}246 JO WOLZVASTS IO} WOTJOeI1I09,
[sqyueu7zredui0s A,U0MT, “SIOJOTI QOS ST ‘SNIPeI JoyNO ‘sI0JeU OOF ZT ‘snipes Jouuy]
BA
10} WOTJD91I0Q
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
40.
OI— | 6— |8— J4~ | 9— | 9~ |s— |¥- |e- |e- Js- | t- Jot |et [et Jat Jor fot Jot Jot Jot fet fat fit for sI— 6+ 000 $T
67 |8— |4= |9~ [s= [s— ]F— |e | e- Js- JH JIi- [st fet [et Jot fot fot Jat fet fet Jet [re fre foe I- st 000 1
6 |8~ |4= Jom [e— | s~ ]F— Je— | e- [s- |r JI- [st fet fat jot fot [ot fat fet fet fat fit jt foe eI— Lt 000 81
6— |8— Je4— | 9- | S— | S—- J- [e- | e-— |e- | I— |i [st fst fet Jot fet Jot Jat e+ e+ e+ Tt te 9- gI—- 9+ 00$ eT
8— |LZ- |]9- ]9- [G- |e- |F— Je- Je-— Je- |I- ]I— Jat |zt+ fot Jot Jot | ot | ete e+ e+ e+ I+ It 9- eI— or 000 @t
B— [4~ | 9- | S- |e |e | F- [e- Je- |e |I- | t— |2t+ fot Jot [ot fot fat |at e+ e+ + I+ I+ 9- I— Sr 00¢ TT
L- jL~ |9- |G- |%- ]F- J e- Je- [z- |e- JI- JiI— [z+ }ot Jot Jot [et | ot e+ e+ a+ ot iF T+ 9- II- ot 000 IT
L- |t- [9- |G—- |$- |e- Je- Je- [e- |e- |I— | i— Jzt ot Jot [ot Jet b+ | e+ et ot o+ I+ I+ 9- oI— e+ 00S OT
4- |9- 19- |S- |%—- |#- Je- [e- |e- Je- | I— | I— Jot Jot [ot [ot fot ot | et e+ o+ o+ It It 9- oI— b+ 000 OT
9— |9- |S—- |G— |%— |F- |e- J e- | e- |e- |I- Io 9+ | 9+ | G+ | Gt | e+ | et Jet e+ ot+ ot I+ 0 o> 6 — e+ 00S 6
9— |s- |S- |F- | F- J e- | e- Je- |e- |I- |I- Jo 9+ | St | ot | e+ | o+ | et | e+ o+ o+ T+ I+ 0 = 6 — e+ 000 6
Ss |e | e= Les (ea [ee ie | |e. Io Gt | G+ (G+ | b+ |F+ | et | e+ o+ ot T+ I+ 0 = 6e== e+ 00S 8
ge (b= |e | 8= [ee 1 e= bee | re it [0 Gt |} c+ [e+ | b+ | P+ | et Jet e+ ot T+ It 0 c= 8 —- et 000 8
P= | F=— 1S PS [oS Pee be I- |T- |0 Gt |}G+ [$+ | b+ Jet | et Jat ot Tt I+ T+ 0 cf 1 a+ 000 2
e-— |e- |@- | e- i = I- |0 + | Ft | F+ J Ot fet | ot [z+ ot+ T+ I+ T+ 0 i an os I+ 000 9
o-— |%@- | B- i to I-— |1- |0 et |e+ fet | et fet | z+ {ot a+ T+ I+ I+ 0 Ee. C= T+ 000 ¢
I- i oe I- |1T- |0 St [et 1 e+ (ee | ee 1 i+ I+ I+ I+ T+ Tet 0 ¢= = I+ 000 *
ke [0 0 0 c+ | 2+ [ot | 3+ [2+ | t+ I+ T+ 0 0 0 0 s— 2 0 000 €
0 0 T+ Te |e re T+ T+ T+ 0 0 0 0 o— Oo 0 000 @
0 iP Pie | re [ede ie ie 0 0 0 0 0 0 L= i= 0 000 T-
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
19g
= {t= re IS = Te T= I- 0 0 0 0 I+ Lt 0 o0g —
I- |T- | I- I- Li I- f— 0 at ct 0 009 —
I- I- LS T= os i 0 ot ag i 006 —
t= te 0 e+ y+ i 00 I—
0 e+ 9+ o— 00S I—
SULOYIOT
yoo | 409 | ya0q | eax | go0q | Joos | gooq | 00x | 400g | 4095 | goa | goog | ga0q | goaz | go0q | goa.) yoo | yoay | gooy | oor | yoor | yooy | yooy | aoa | tones
OOFS8 | OOLL | OOOL | OOED | 0O9S | DOG | OOS | OOSE | 008% | OOTZ | OOFT | OOL | 00FB | OOLL | COOL | ODES | ODDS | OD6F | OOF oose 008¢ 001% OOFT 00L -uedui0. mores Aqder
pue i Tau
-uedurog | -Zodoy, z
Aydei 4redui0o
queuyreduioo Mopeg yuounredur0s evoqy -Bodoy, Jo mora
-9[9 Ueayl
—01} 81S JO MOLVA IO} WOT}IILION —I0} W0{}001109
[‘syueurjzedur0o inoj-AyUOM J, *SI9}OUL ON 8z ‘SNIPeI 12]NO ‘s19}9UT OO ST ‘sNIpeI JaUUy]
"T au0g
41
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
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EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
42
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43
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44 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
REDUCTION TABLES FOR NUMBERED ZONES.
Zone 18.
[Unit of elevation 100 feet (27.1 fathoms for depths). 9=1° 41’ 13” to 1° 29” 58”. One compartment. ]
Correction | Correction for elevation of station at— Correction {| Correction for elevation of station at—
ae as Moe departure! ones as Ae departure
- ropor-
see Somali” 5 000 feet 10 000 feet 15 000 feet siete “tionality 5000 feet ‘| 10000 feet 15 000 feet
+150 +1 —2 —5 —7 0 0 0 0 0
+125 +1 —2 —4 —6 — 25 0 0 +1 +1
+100 +1 —2 —3 —5 — 50 0 +1 +2 +2
+ 75 0 —1 —2 —3 — 75 0 +1 +2 +3
+ 50 0 -1 —2 ~2 —100 0 +2 +3 +5
+ 25 0 0 —1 -1
Zone 17.
[Unit of elevation 100 feet (27.1 fathoms for depths). 9@=1° 54’ 52” to 1° 41’ 13. One compartment.]
+150 +3 —2 —4 —6 0 0 0 0 0
+125 +2 —2 -3 —5 — 25 0 +1 +1 +1
+100 +1 -l —3 —4 — 50 0 +1 +1 +2
+ 75 +1 —l —2 -3 — 75 +1 +1 +2 +38
+ 50 0 -1 —l —2 —100 +1 +1 +3 +4
+ 25 0 -—1 -l -l
Zone 16.
[Unit of elevation 100 feet (27.1 fathoms for depths). 0@=2° 11’ 53” to 1° 54” 52”. One compartment.]
+150 +4 —2 -—3 —5 0 0 0 0 0
+125 +3 -1 —3 —4 — 25 0 0 +1 +1
+100 +42 —1 —2 -—3 — 50 0 +1 +1 +2
+ 75 +1 -1 -1 —2 — 75 +1 +1 +1 +2
+ 50 0 -l1 -1 —2 —100 +1 +1 +2 +3
+ 25 0 0 -1 al
Zone 165.
[Unit of elevation 100 feet (27.1 fathoms for depths). @=2° 33/ 46” to 2° 11’ 53’. One compartment.]
+150 +5 =i = =4 0 0 0 0 0
+125 4d i =2 =3 — 25 0 0 0 bel
+100 +2 ==if af i ag — 50 0 0 1 +1
+ 75 41 =i -1 2 — 75 Ay ey. +1 +2
+ 50 +1 0 -1 -1 —100 +1 +1 +2 +3
+ 25 0 0 0 =i
Zone 14.
[Unit of elevation 100 feet (27.1 fathoms for depths). @=3° 03/ 05’ to 2° 33’ 46”. One compartment.]
+150 +6 -1 —2 —3 0 0 0 0 0
+125 +4 —1 2 —3 — 0 0 0 0
+100 +3 —l —l —2 — 50 0 0 +1 +1
+ 50 +1 0 —l —1 —100 +2 +1 +1 +2
+ 25 0 0 0 0
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 45
Zone 13.
{Unit of elevation 1000 feet (271 fathoms for depths). @=4° 19’ 13’ to 3° 03’ 05’. Sixteen compartments. ]
No correction for elevation of station. No correction for departure from proportionality.
Zone 12.
[Unit of elevation 1000 feet (271 fathoms for depths). @=5° 46’ 34” to 4° 19/13’. Ten compartments. ]
No correction for elevation of station. No correction for departure from proportionality.
Zone 11.
[Unit of elevation 1000 feet (271 fathoms for depths). 97° 51’ 30” to 5° 46 34”. Hight compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 10.
[Unit of elevation 1000 feet (271 fathoms for depths). 9=10° 44’ to 7° 51’ 30’. Six compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 9.
[Unit of elevation 1000 feet (271 fathoms for depths). 0@=14° 09’ to 10° 44’ Four compartments.]
i
No correction for elevation of station. No correction for departure from proportionality.
Zone 8.
[Unit of elevation 1000 feet (271 fathoms for depths). 9=20° 41’ to 14° 09’. Four compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 7.
[Unit of elevation 1000 feet (271 fathoms for depths). 6=26° 4/’ to 20° 41’. Two compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 6.
[Unit of elevation 10 000 feet (2710 fathoms for depths). 6=35° 58 to 26° 41’. Eighteen compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 8.
[Unit of elevation 10 000 feet (2710 fathoms for depths). @=51° 04’ to 35° 58’. Sixteen compartments. ]
No correction for elevation of station. No correction for departure from proportionality.
Zone 4.
[Unit of elevation 10 000 feet (2710 fathoms for depths). @=72° 13’ to 51° 04’. Twelve compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 8.
[Unit of elevation 10 000 feet (2710 fathoms for depths). 9=105° 48’ to 72° 13’. Ten compartments. ]
No correction for elevation of station. No correction for departure from proportionality.
46 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
Zone 2.
[Unit of elevation 10 000 feet (2710 fathoms for depths). @=150° 56’ to 105° 48’. Six compartments.]
No correction for elevation of station. No correction for departure from proportionality.
Zone 1.
[Unit of elevation 10 000 feet (2710 fathoms for depths). 6=180° to 150° 56’. One compartment only.]
No correction for elevation of station. No correction for departure from proportionality.
SPECIAL REDUCTION TABLES FOR SEA STATIONS.
[Corrections in dynes in units of the fourth decimal place. Station at sea level.]
Depth Zones |
Fathoms A B Cc D E F G H I J K L M N Oo }
5 000 —1 -1ll -—27 -39 -—53 ~-55 -—48 -40 -—40 -—34 -18 -3 +441 +51 +32 |
4 800 —1 -1l1 -27 -39 -—53 -55 -—47 -—39 -—38 -~32 -17 -—2 +441 +449 +831
4 600 -1 -ll -27 -~39 -—53 -—54 -—46 -37 -—36 -29 -15 —-1 +440 +47 +429
4 400 —-1 -ll —27 -38 -53 -54 -45 -36 -—34 -27 -—14 -—1 +40 +45 +28
4 200 —1 -ll -—27 -—38 -—53 -53 -—44 -34 -—32 -—25 —12 0 +389 +44 +27 |
4 000 —1 —ll —27 —-38 -52 -—52 -—43 -—33 -30 —23 —11 +1 +39 +42 +26 |
I
3 800 —-1 -ll -27 ~—38 ~—51 —51 —42 —31 —28 -—21 —9 +41 +38 +440 +24 |
3 600 -1 -ll -27 -38 -—51 -50 —41 —29 -—26 —19 — 8 +42 +37 +38 +23 |
3 400 —1 —-ll -—26 -37 -50 -49 -—39 -—28 -24 -17 —7 +42 +36 +36 +22 |
3 200 -l -ll -—26 -37 -—49 -48 -—38 -—26 -—22 -15 —5 +43 +35 +34 +421
3 000 —-l1 -ll -—26 -37 -48 -46 -36 -—24 -20 -13 —4 +3 +433 +32 +19
2 800 —l -ll —26 -37 -—47 -44 -34 -—23 -18 -ll —3 +44 +32 +30 +418 |
2 600 -1 -ll -26 —36 -—46 -—43 -—32 -21 -16 -10 — 2 +4 +30 +28 +417 |}
2 400 —1 -ll -26 -36 -—44 -—41 -30 -20 -14 —8 —1 +4 +28 +26 +415 |
2 200 -1 —ll —26 -35 ~—43 -39 -27 -17 -12 —6 —1 44 426 +24 414 |;
2 0N0 —1 -10 -—26 -35 -42 -37 -25 -15 -ll —5 0 +4 +25 +422 +412
1 800 —1 -10 -—26 -—34 -—41 -34 -22 -13 -—9 -—4 0 +4 +23 +20 +412 |
1 600 -1 -—10 —25 -—34 -—40 -32 -—19 —-10 —7 —2 41 44 +420 +418 +10 |
1 400 —1 -10 -25 -34 -—38 -—28 -16 —9 —5 —2 +1 +44 419 416 +9 }:
1 200 -—l1 -10 -25 -33 -35 -25 -13 —7 —3 —1 +1 +483 417 414 48
1 000 —-1 -10 -—25 -—31 -—31 -—20 -10 —5 —- 2 0 +1 +42 +14 412 +6 |
800 —-1 -10 -25 -30 -27 -15 -—7 -—3 1 0 +1 42 411 +9 +5 |:
600 —1 —10 -—23 -—26*-21 -10 —4 -1 1 +141 4249 47 44
400 —1 -—10 -—22 -—21*-13*-5 —2 —-1 0+14+1 41 464542
200 —1*-10 *-17*-l1l*-4*-1 0 0 O +1 41 41 43 42 41
0 0 *0 *0 *0 *0 *0 0 0 0 0 0 0 0 0 0
\
* Use table following for these values on account of large second difference.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 47
Supplementary table for use in connection with gravity stations at sea.
[Correction in dynes in units of the fourth decimal place. Station at sea level.]
Depth Zone
Fathoms B Cc D E F
800 —25 —27
750 —24 —26
700 —24 —24
650 —24 —22
600 —23 -—26 -—21 —10
550 —23 -—26 -19 — 9
500 —23 -25 -17 —7
450 —23 -—23 -15 — 6
400 —22 -21 -13 — 5
350 —21 -19 -ll —4
300 —20 -17 -—9 —38
250 -19 -14 —6 —2
200 —-10 ~—17 -ll —4 —-1
150 -—10 -15 —7 —2 —-1
100 —-9 -ll —4 —-I1 0
75 -—-8 —9 —2 —1 0
50 —-7 —-5 -1 -1 0
25 -5 -1 -1 0 0
10 — 2 0 0 0 0
0 0 0 0 0 0
USE OF TEMPLATES.
For each scale of map or chart to be used in the computations there was prepared a sheet
of transparent celluloid with the circles and radial lines which define the limits of the zones
and compartments drawn to the same scale.
Such a template is shown in illustration No. 10a as used for maps on a scale of 1/10000.
The zones are marked with their designating letters, and the scale of the template is ordinarily
marked on each. No attempt has been made to reproduce the illustration to the proper scale.
Each template consists of a sheet similar to that indicated in illustration No. 10a carrying
lines bounding the compartments which lie on one side of the reference line. By turning the
template 180° in azimuth on a map it serves also to fix the position of the remaining compart-
ments. While in use the template is placed on a map with the center of the circles at the
station and with the reference line lying in the meridian. As a convenient designation the
compartments in any zone are numbered in the clockwise direction commencing with the first
which is to the eastward of north from the station.
Illustration No. 106 shows a template such as was used on maps on a scale of 1/6013500.
This necessarily shows more distant zones than illustration No. 10a. The dotted radial lines
in zones 14 to 18 are not compartment boundaries. Each of these zones has one compartment
only. They are lines dividing each of the zones into ten equal parts, as it was found convenient
in estimating the mean elevation for such large zones to make separate estimates for each part
rather than to make an estimate at once for the whole compartment or zone. For the same
purpose dotted lines are shown in zone 7 separating each of its two compartments into five
equal parts. ; : ates
By the use of these transparent (celluloid) templates the many circles and radial lines
fixing the limits of the zones and compartments on a given map for any station were super-
posed on the map by the mere process of laying the template on the map in the proper position.
The use of the templates saved a very large amount of labor which would otherwise have been
necessary in drawing the many zones and compartments on several hundred maps. It also
left the maps without damage or defacement.
48 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
In computing the correction for topography and isostatic compensation for a given station
the computer places the appropriate template in the proper position on the best contour map
available. He then estimates the mean elevation of the surface in each compartment from
the contour lines on the map, seen through the template, and at once takes out from the reduc-
tion tables the two corrections for that compartment and records them in the proper places
on the computation forms. As he has the reduction tables constantly before him he is con-
tinually guided as to the accuracy with which the estimate of mean elevations must be made
in order to secure the corrections with the required degree of accuracy. As a rule this estimate
Reference Line
Reduced trom template used
on maps of 1/60/3500 scale
Reauced from template used
on maps of Y10000 scole
Ls
ILLusTRATION No. 10 (a).—Template for maps of ILLUSTRATION No. 10 (b).—Template for maps of scale
scale 1/10000 (reduced). 1/6013500 (reduced).
may be made very quickly, for as indicated in the reduction tables, an approximate elevation
of a compartment is sufficient. This is especially true in the numbered zones 13 to 1, for which
the unit elevations are either 1000 or 10 000 feet.
EXAMPLES OF COMPUTATIONS OF CORRECTIONS.
The following table is a sample of the computations, and in it are given the values (in
units of the fourth decimal place in dynes) of the correction for topography and isostatic
compensation for each compartment of zones A to 1 at the San Francisco gravity station.
This station is near the open coast, is 85 miles from the 1000-fathom line, and is only 375 feet
above sea level.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 49
San Francisco, Cal., gravity station No. 54.
[p=87° 47’ 22/7. 4=122° 25’ 40’. Elevation=375 feet.]
A B c D E F G H I J k L M N Oo
+2 0/4+13 —1 +3 0) lL 61 0 0 0 0 00 00 00; 00 0 0 0 0 00;—2 0; -10
+13 -1 +9 0; +1 +1] +1 «0 0 0 0 0 0 0 0 0 0 0 0:0 0 0 0 0 0 0 0 0
+13 -1 49 0] +2 +1 0 0 0 0 0 0 00 0 0 0 0 0 a 0 0/— 4 0 0 0 0 0
tig +9 0/42 +1 0 0 0 0 0 0 0 0 0 0 0 0 0 0/—1 0}/-— 40 0 0 0 0
+2 +1 0 0 0 0 0 0 00 0 0 0 0 0 0/—1 0/— 1 0/-2 0 00
aL bl f 4 : . ; 0 0 0 0 ‘a 10 { 00;/—3 0)/ -—10
00 00 0 0 00;/—3 0} -—10
00 0 0 0 0 0 0 00 0 0 0 Olf—1 O0/— 8 O;f—1 0 00
0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 { 00 00
0 0 00 0 0 0 0 0 0 0 0 0 0 0 0;/+2 0) —2 9
0 0 0 0 0 0 0 0] -1 0 0 0 0 0 00;/+2 0} —20
0 0. 0 0 0 0 0 0 0 0 ‘0 0 0 0/ +10 0 00
{ 00 00 0 0 0 0 0 0 0 0 0 0; +10 0 00
0 0 0 0 0 0 0 0 0 0 0 0 0 0;+4 0 00
00 0 0 0 0 0 0 0 0 00 0 0;+1 0 00
0 0 0 0 00 0 O/—1 O/f—2 Olf—-1 0/1 +2 0
00 0 O;/-1 0 00 0 0 0 0 GO| + 8 0
0 0 0 0 0 0 0 0 00 OO) = 2:0) fl 0
0 Oo}; 00 0 0 0 0 0 0 00 +12 0
00 0 0 00 0 0 0 0 +13 0
0 0 00 0 0 0 0 +13 0
0 0 00 0 0 +13 0
0 0 00 0 0 +13 0
0 0 00 0 0 +12 0
9 0 —1 0jf-1 0 +9 0
00 0 0 00 +60
0 0;-—1 0 eG
0 0 00 —20
= 00 00 -—30
0 0 —-20
+2 04/452 -—4}] +35 0/+9 +6] +1 0 0 0 0 0 0 0 0 0; —2 0] —1 0} —7 0] —14 0] +15 0] +99 9
+2 +48 +35 +15 +1 0 0 0 0 —2 -1 -7 — 14 +15 +99
18 17 16 15 14 13 12 Bl 10 9 8 7 6 5 4 3 2 1
—4 0 Te Oo Kaee @aas OSE b —- 56 —6; —5] —5] —7] -2 0 0 ol 0 0 0 0
+28 0 (+28 0 [\+32 0 1\+35 0 [\+38 0 0 =a mes Stheals a <4 . c a e : +1
Sel este es ey ey th a at ae
-— 7 +5 +5] +8
= 3 +9] +8 tos 749 0 0 +1 oi +1
{ 0 +11] +10 { 0 . 0 ad 0; +1
— 2} +n] +9/\+4 0 0 o} +1] 41
+ 5] +7 0 0 oO} +1] 41 { 0
+ slp -1N4+4 0 0 +1] +41 0
+ 9 { -ol’ Oo} +1 41] 41
+ 10 Oo} +1) 41] 41
+ 10 0 +1 +1 0
Sg oO} 41 oli 0
+ 8 +1 +1 0
4 4 +1] 44 0
= ff +1 ek 0
{ 0 +1] 41
+1 0
+1 0
+1
+1
+24 0] +21 0] +20 0} 419 0} +17 Of +25] +23) +21) +14) +10 +15}; +10| +9/ +9 +8) +5] +4] +1!
+24 +21 +20 +19 +17
Sum of all zones = +446.
At the top of the table are given the latitude, longitude, and elevation of the station. In
actual practice the zones may be arranged in any convenient manner on a single sheet. Here
they are placed in such a way as to show them in as compact a form as possible.
The headings of the several columns indicate the zones by letter or number, it being under-
stood that the zones are in the order of their distances from the station, namely, A to O, and
18 to 1, the zone A being at the station with its inner radius zero.
In a zone having more than one compartment, the compartments are numbered clock-
wise, the first one being to the north of the station and just to the east ot the meridian passing
through the station. Having this arrangement of compartments in mind, one can readily see
15593°—12 4
50 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
in the table for any station the effect of the different prominent topographic features. This
is noticeable in zone O for both the San Francisco and Pikes Peak gravity stations.
At San Francisco for the first 13 compartments of zone O the corrections are all zero or
negative. These are all land compartments. In compartment 14, nearly due south from the
station, a positive correction, due to the ocean, first appears. In compartments 15 to 25, all
to the westward of the station, the corrections are all positive, showing the influence of the
deep waters of the Pacific. Compartments 26 to 28 are land compartments, showing the influ-
ence of the Coast Range to the northwestward of San Francisco. The similar influence of the
part of the Coast Range to the southeastward of San Francisco lying within this zone is shown
in compartments 10 and 11.
It will be noticed that there is a double column for each of the zones B to O. The first
column gives the effect on the intensity of gravity at the station, due to the topography
and the isostatic compensation of the several compartments based upon the assumption that
the station is, in each case, at the same elevation as the compartment. The mean elevation
of the compartment is obtained from the map or maps used. Entering the table for the par-
ticular zone with this elevation, this correction is obtained from the fourth column, which is
headed ‘‘Correction for topography and compensation.” In the second column for zones B to
O is given the effect of the intensity of gravity due to the elevation of the station above
or below the average elevation of each compartment. These quantities are given in the tables
under the headings ‘‘Correction for elevation of station above compartment” and ‘‘Correction
for elevation of station below compartment.”
In taking out the second correction it must be kept in mind, as already noted on pages
22 and 29, that it does not become zero in zones M, N, and O when the station is at the same
elevation as the compartment, but, instead, has the values shown in the special column in the
reduction tables for these zones. For zones B to L the second correction is zero when the
station is at the same elevation as the compartment.
Two columns are given for each zone 18 to 14, the first one showing the correction as
read from the map and given in the first columns of the reduction tables for those zones, while
the second column contains the algebraic sum of the corrections for the departure from pro-
portionality and for the elevation of the station above sea level.
For each of the zones 13 to 1, there is only one column of figures, which are the corrections
for the compartments as read from the map, each compartment of zones 13 to 7 having a cor-
rection of 0.0001 dyne for each 1000 feet in elevation (271 fathoms for depth), and zones 6 to 1
having a correction of 0.0001 dyne for each 10 000 feet of elevation (2710 fathoms for depth).
The algebraic sums for each column is given at the foot of the column and immediately
below these separate sums is given the algebraic sum for the zone. The sum-for all zones is
+446 in the units used in the computation or +0.0446 dynes. This is the correction at San
Francisco for the topography of the entire earth and its compensation.
It was found at times to be desirable to treat in two parts the corrections for a compart-
ment which contained both land and water areas. The corrections for land and water for the
compartments treated in this way are connected in the table by brackets, the first number
being for the land portion and the second for the water portion of the compartment in question.
In determining the correction for any portion of a compartment the table is entered with the
elevation of that portion as the argument as if it were the elevation of the whole compartment,
but the correction entered in the computations is only that proportion of the total correction
which the area of the portion of the compartment bears to its total area.
The elevations close to the gravity station at San Francisco are low and in no case inside
of zone I is the height of a single compartment more than 700 feet above sea level. In zone L
one compartment to the eastward of San Francisco, in the Coast Range, has an average elevation
of about 800 feet. Zone I. is just beyond the change of sign due to distance (see p. 65), and
therefore the correction for that compartment is not over 0.0001 dyne. In zone M there are
several compartments near the compartments of zone L in the Coast Range, already referred to,
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 51
with elevations of about 1000 feet, each of which causes corrections of — 0.0004 dyne (see third
and fourth numbers in the column for zone M). In zone N there are two compartments having
depths of about 900 fathoms, which cause corrections of +0.0010 dyne. The land compart-
ments in this zone do not have elevations above 1000 feet. A portion of zone O extends well
beyond the 1000-fathom line, which causes corrections as large as +0.0013 dyne for several
water compartments.
The corrections for the land and water portions of each of the zones 18 to 14 are given
separately, the correction being minus for the land and plus for the water. Most of the water
sections of these zones are far out in the Pacific Ocean. Each of these zones has only one
compartment, but for convenience in reading elevations and depths from the maps, each zone
is divided into 10 parts and for each part the correction is taken from the reduction tables as
one-tenth of the value given for the whole zone for an elevation equal to that of the part in
question. The table was entered only once to obtain for the zone the correction for the
elevation of the station above sea level. For each of the zones 18 to 14 at San Francisco the
algebraic sum of the corrections for departure from proportionality and for elevation of station
is zero.
Each of the zones 13 to 1 has only one column of figures in the table, as there are no cor-
rections for elevation of station nor for departure from proportionality. The total correction
for each of these zones is plus, showing that the effect of the water compartments predominates.
There was no interpolation of values in any of the zones for the gravity station at San Fran-
cisco. All values were computed directly from the maps and charts.
The following table gives in detail the computation of the effect of topography and its
isostatic compensation at the gravity station Pikes Peak, which is a mountain station far from
the ocean. The station is much above the general elevation of the surrounding country.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
52
‘Ajeatoadsar “ojog ‘uostuuny pure ‘Iesueq ‘ssulidg opes0jog 48 ‘Gp pue ‘FF ‘Zh “SON SUOT}EIS APIAVIS WO UOTejodsajuy Aq PoUre}qo aoa SOl[e}] UT WMoYs san[eA—ALON
"TLSI-+ = seuoz [Ye Jo wing
I+ + 9+ s+ 6+ 6+ + + - = st— s8— 6o— 19— s9- 3S 3
53 5 - = i It 0o9—- | 2+ 99- | e+ L- | 8+ - {et L-
seats ; ze ane ma ee: wee tS ven - . so Tt oo fat go- | et u- Jet uw }et
T ra g ¥ ¢ 9 L 8 6 or IL ar &T tI ST oT 21 81
6z8— FeE— o6e—- —«. oe 6L+ Tos+ olb+ Shot cist co9+ ozs + oTe+ LST+ al+ G+
Set Lse— | e+ o9e— "| 06+ Ose— | L6+ G2I— | LIT+ se— | 9I+ GOt | OLT+ BPEt | G+ Tet | 09+ Wt | &— GOO+ | GF—- GOot | FE- E+ | TI SoTt | O e+ 0 Vt
1 We
I+ 2I-
I+ 8I-
a
It LI- y+ 9-
I+ 6I- b+ 9-
te ti rb o~
ee ae t+ 9-
Te 2 F+ 9- 9+ B- 6+ I+
I 2I= b+ 9— 9+ B-— 6+ @I+
I~ $I- Br -9=' G+ 3- 8+ Ft
I+ FI- t+ 9- 9+ b— 8+ eit
IF $I- St S— rm. G= Ge ge 8+ Ft 6+ oIt 9+ Tot
I+ FI= gt 8— tr g= 9+ 3- 8+ + 8+ eit G+ e+
I+ @I- Gt 0&— 9+ 08— gs 9= 9+ 3- 8+ t+ St Sit §+ 96+
I+ 0OIT— e+ 08— L+ $8 $+ 9- 9+ o- 6+ €+ 8+ e+ Ft 96+
I+ 6—- G+ Lo— Sa $8 Beg ot 1- St Ft Lt I+ Gt Set G+ get e
Te 3 = gt &e~- 9+ ge— $+ 9- 9+ B- St St Loe Stet Gt &%+ e+ Trt
It g-= e+ ig- o> t= e+ G— oF 2= s+ 9+ 9+ SIt+ Gt + F+ OF+ 0 T9t+
Tse 80 St 9B— 9+ 22— Tr G= 9+ 3g- 8+ St 8+ FIt+ b+ GB+ e+ 66+ T+ 9+
re 6-= G+ 8I- L+ €4 b+ F-— Oa SE L+ 9+ 9+ ST+ 9+ Gt G+ set I+ 29+ 8 = 61+
I+ 6- G+ 9I- 9+ Te- BP it 9+ L+ 6+ @It+ 9+ f+ y+ OF+ 0 19+ 0 oL+
I+ 0I- G+ 8I- L+ To- St b- 9+ B- 6+ F+ OI+ OT+ L+ 0¢+ G+ e+ I+ 19+ eS Tet P= 19+
Te Or c+ 8I- 9+ 0¢- G+ 9+ @- Go OIt 6+ L+ 61+ Gt e+ It 19+ - t+ g— sot
IF 6—- G+ 0@- Lt+ t-— ce 3 9+ 3- OI+ 3+ OI+ 6+ L+ 61+ 9+ set I- 09+ PS TE €— got — r+ 10 8It
Ts 6° E+ 02— be Lo e+ G— 9+ 6- OI+ 3+ Olt 6+ Z+ 61+ g+ get Z— 8o+ II- 0L+ Bo it 0 a+ | 0 Sit
Tt 6-— t+ 0— L+ lg- J+ 9- ch 3= OIt &+ TIt 64+ 8t sit g+ get e— 6o+ 6— OL+ i 294 f&- G+ |0 8It
te a> Gt 8I- Lt 8%— &+ 9—- oF 2= oI+ €&+ 6+ I+ L+ 61+ 9+ 98+ Z— got 2° 0S GIS $- Gt |0 8It+ 0 e+
oO N n 1 x f I H 9 az a a 0 a v
("30037 ¢80 PI=UOTCACT A
‘00 460 oSOI=Y
“st oN uoung hnavipn “ojo ‘yDrq $2%%d
‘BI AG 88=F]
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 538
The arrangement of this table is the same as that for the station at San Francisco, which
was discussed in detail. The corrections for the zones A to 14, at Pikes Peak, were computed
from the elevations read from maps. For zones 13 to 1 they were interpolated (in the manner
explained later under the heading ‘‘Saving of time by interpolation’’). from stations Nos. 42, 44,
and 45, which are at Colorado Springs, Denver, and Gunnison, Colo., respectively. The total
value of the effect of the topography and its isostatic compensation as obtained by the inter-
polation is given in the table for each of the zones 13 to 1. The leaders shown in the columns
for these zones indicate the number of compartments in each zone.
As Pikes Peak is an inland station, there are no water compartments within the computed
' zones A to 14,
As was the case in the table showing the corrections for the different zones at the San
Francisco gravity station, zones B to 14 at Pikes Peak have two columns of figures each. In
each zone the first column shows the effect of the topography and compensation with the station
at the same elevation as the several compartments, while the second column of figures shows
the corrections due to the elevation of the station above or below the compartment.
It is interesting to notice the change of sign at zone F of the correction for elevation of
station (see p. 52), the change of sign due to distance between zones J and K, in the first column
for these zones, also the change in the sign of the total correction between zones K and L.
Pikes Peak is a conical-shaped mountain, which accounts for the corrections for the several
compartments of each of the near-by zones being of about the same size. The effect of the
mountains to the westward is clearly shown in zones M, N, and O, but especially in zone O, the
corrections being larger in the lower half of each column corresponding to compartments west
of the station than in the upper half of the column in each of these zones.
CORRECTIONS FOR TOPOGRAPHY AND ISOSTATIC COMPENSATION, SEPARATE ZONES.
In the following table are given the total corrections for each zone, for topography and its
isostatic compensation, to the intensity of gravity at each of the 89 gravity stations used in this
investigation. There is also given the total correction for each station, this necessarily being
the sum of the corrections for the separate zones. The values are given in units of the fourth
decimal place in dynes.
The names and numbers of the stations are given in the headings of the table, while the
letters or numbers of the zones are shown in the first column. The value for each zone at a
station was obtained from the computations of the corrections for the separate compartments
of the zones. Samples of such computations made at a station are given in tables on pages 49
and 52 for the gravity stations at San Francisco and on Pikes Peak.
The figures in italics represent the accepted interpolated values for the correction for
topography and its compensation as explained on pages 58-60. The other figures are the values
for the zones for which the corrections were obtained directly from maps and the reduction
tables.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
54
Correction for topography and isostatic compensation, separate zones
sig!| MRESESOS ORR SE RSNA Hoh RsREHMessaehy Ss) i) | CBee eRaa Reise sasha Sees h ema
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EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 55
Correction for topography and isostatic compensation, separate zones—Continued.
ee sagitngton, i i ; Cam-
Zone IS 8. thsonian| Baltimore, | Philadelphia, Princeton, Hoboken, New York,| Worcester, | Boston, bridge
Office, Institution, No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. ae
No. 21 No. 22 ;
A +2 +2 +2 +2 + 3 +2] + 2) + 2) 4+ 2 Ae
B +12 + 8 +24 +12 + 40 + 8 + 27 + 56 + 16 + 12
C + 2 0 + 4 +4 + 16 0 + 7 + 64 + 4 0
D 0 0 + 6 0 + 6 0 + 2 + 31 + 1 0
E 0 0 0 0 0 0 0 + 11 0 0
F 0 0 0 0 0 0 0 + 7 0 0
G 0 0 0 0 0 0 0 0 0 0
H 0 0 0 0 0 0 0 0 0 0
I 0 0 0 0 0 0 0 0 0 0
J 0 0 -1 0 0 0 0 — 10 0 = ¢
ee ee es ee er
0 0 — 3 0 0 - -
M —12 —12 —20 — 6 —- ll —12 — 12. — 27 —- 4 — 9
N -17 —17 —16 —10 — 16 —18 — 18 — 25 — 12 — 15
O —23 —23 —20 —19 — 22 —25 — 26 — 28 — 10 — 14
18 =5 — 65 — 6 -3 — 4 = 6) == 6) = BP 278 — 2
17 —8 — 8 Sait — 6 -— 6 -§) — 5] — 8] - 1 anf
16 -—9 -—9 a) —6 -— 6 —s| -— 8] - 4] -1 — 2
15 — 8 —8 — 8 — 3 ae +i} +°1/ -— 2g] - 2 — 2
14 —4 — 4 = 0 + 1 +3/ + 3} -— 2] ~ 2 -— 2
12 +38 +38 +7 +12 + 18 +4) +44) 4+ 9) 411 + 10
12 +13 +13 +14 +19 + 20 +22} +22} + 24] + 26 + 25
1 +18 +18 +19 +21 + 21 +21{ +21] +23} + 25 + 25
10 417 +17 +17 +16 + 16 4+16| +176) +17]; +18 + 18
9 +11 +11 +10 +11 + 11 +11 + 11 + 12 + 12 + 12
8 +12 +12 +13 +14 + 14 +15) +15) +127) +17 + 17
7 +6 +6 +6 +6 + 6 +6|/ + 6} + 6) +6 + 6
6 +6 +6 +6 +6 + 6 +6| + 6| + 6) +6 + 6
5 +7 +7 +7 +6! + 6 +6| + 6} + 6] + 6) 4+ 6
4 +6 +6 +6 +6 + 6 +6) + °6) + 6] +6 + 6
3 +6 +6 +6 +6) + 6 +6) + 6] + 6] + 6| +6
2 +4 +4 +4 +4| + 4 +4] + 4/ + 4) + 4] + 4
1 +1 ed by +1 + 7 +1 + 7 eed |! Pepe + 1
Total +40 +34 +57 +93 +130 +79 +106 +178 +133 +101
i Ithae: Cleveland, | Cincinnati, | Terre Haute,| Chicago Madison, | St. Louis Kansas Ellsworth,
Zone No. ai No. 33 No.33 | No.34’| No.85 || No.36) | No.37' | No.38 | %, | “No. 40
2 2 2 + 2 + 2 + 2 + 2 + 2
7 i a a pe Lr ‘ee +56 +62 +56 +64 + 68
C + 4 +88 +78 +84 +60 +72 +95 nee Be oc
D + 4 +59 +48 +57 +28 +42 +70 + + + a
E 0 +27 +20 +22 +12 +16 +30 +13 +3 ee
F 0 + 6 +10 0 +2 +4 +10 0 oe + a
G 0 0 0 0 0 0 ‘ ; a + ;
= i a ; ‘ 0 0 0 0
} 5 —16 —16 —11 — 8 —7 —16 — 3 —16 — 16
K ) —20 —20 -17 —10 -— 9 a — ae - rs
Ll 0 —32 —24 —20 —12 -l1 - _ ; = _ a
M — 6 —50 —42 —42 —30 —22 —57 = ae _ a
N — 4 —56 —41 —50 —37 —26 —48 = =a =
O — 15 —58 —45 —48 —35 Be = = : =e ca ve
ay SS =e Zi = as Gil vee)! asad) ete) ae
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EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 85
Note that for the ocean stations, Nos. 1 to 7, there is in each case but one change of sign
as successive zones are considered, namely, the change of sign due to distance. The water
compartments predominate in their effects in every zone.
At station No. 1, over a deep part of the ocean where the bottom is nearly level to a great
distance from the station, the positive corrections beyond the change of sign due to distance
have nearly the same aggregate as the negative corrections before the change of sign, and,
therefore, the total correction for topography and compensation is small. At stations Nos. 2
and 3, near the Tonga Deep, this balance of positive and negative corrections is slightly dis-
turbed in one sense—the positive correction predominates; and at stations Nos. 4 and 5, over
the Tonga Deep, the balance is greatly disturbed in the opposite sense, the negative corrections
being largely in excess: These are probably typical cases.
Note that stations Nos. 6 to 9 constitute a progressive series of four in relation to topogra-
phy. No. 6 is over deep water near an oceanic island, No. 7 over water of moderate depth
nearer to the oceanic island, No. 8 near sea level on the coast of a high oceanic island, and No. 9
on a high summit of such an island. Note that the corrections for topography and compensa-
tion stand in order, namely, +0.019, +0.078, +0.162, and +0.469. A comparison of values
for corresponding zones in the preceding table for these four stations will indicate the manner
in which the positive corrections gradually gain predominance as the station is made-to approach
from deep water to the summit of an oceanic island. While making this comparison it will be
well to consult pages 65-71 in regard to the change of sign due to distance.
Stations Nos. 11 and 12 are like station No. 8 in being near sea level on the shore of an
oceanic island surrounded by deep water. Note the resemblance between these three stations
as to the correction for separate zones. In each case the sum of the corrections out to zone L
is small, but beyond that large positive corrections appear and the total correction for each
station is positive and large, corresponding to the known fact that large values of gravity are
ordinarily observed in such a location. .
Station No. 13 is remarkable for having unusually small corrections in every zone—all
positive.
Station No. 14 shows a succession of values characteristic of stations on a high plateau
far from any ocean. The large positive corrections for near zones are more than offset by still
larger and more numerous negative corrections beyond the change of sign due to distance,
which occurs at zone J, and the total correction is, therefore, large and negative. The very
large negative values in zones K to O are due to the fact that the high plateau extends far
enough from the station to fill these zones. The negative corrections are numerous because,
the station being far from the nearest ocean, the water effects do not predominate and positive
corrections do not appear again until a very large zone is reached, namely, No. 6, of which the
inner radius is 2900 kilometers.
A comparison in detail of the corrections for separate zones at stations Nos. 15 and 16.
will show why the corrections for topography and compensation tend to be large and positive
for a station above the general level in a mountainous country and negative for a station far
below the general level in the same region. Note that the positive corrections for small zones
are much smaller at station No. 16 at the bottom of one of the deep valleys than at station
No. 15 on a high summit of the Alps, and that the change of sign due to distance occurs before
zone G at No. 16 and after zone J at No. 15. These two differences between the two stations
are due largely to the effect of corrections due to the differences of elevation of the station and
the zone (‘station below compartment” and ‘‘station above compartment”) shown in the
reduction tables on pages 30-43. Consult especially the reduction table for zone G on page 35 in
connection with the correction for zone G at these two stations. It will also be noted that for
the same reason the negative corrections, beyond the change of sign due to distance and before
the water effects begin to predominate; are larger for corresponding zones at station No. 16 as
arule. This is especially noticeable for zones K, L, M, and N.
86 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
DISCUSSION OF ERRORS.
As the methods of computation used in this investigation are novel in many respects, it
is important to consider the accuracy of each part of the process. As it has been stated that
the desirability of selecting such methods as would give the required results with the minimum
expenditure of time has been continually kept in mind, it may seem probable that this close
attention to the economics of the problem has diverted attention from the requirements of the
problem as to accuracy.
Throughout the investigation very close attention has been paid at every step to insuring
the maintenance of the required degree of accuracy. It is not feasible within the allowed
limit of length of this publication, and without printing all the details of the computation, to put
before the reader all the evidence which has been considered by the writers in estimatmg the
magnitude of errors from various sources. The discussion of errors which follows serves,
however, to show in a general way the methods by which the estimates of error were made
and to put the estimates on record for future reference and for reexamination by others.
Let it be assumed for the moment that the purpose of the present investigation is to
compute the value of gravity at each observation station by taking adequately into account
the effect of every portion of the earth’s mass in producing an attraction at the station. In
order to accomplish this the computation must take into account adequately all the facts as to
the shape of the earth’s surface (its topography) and all the facts as to density at all points
within the earth. These two sets of facts serve to locate with reference to the station every
portion of the attracting mass.
If this be considered the true purpose of the investigation, the real measures of the total
errors made in the attempt are the residuals of the attempt, namely, the apparent anomalies
by the new method shown in the table on page 74. Each anomaly is the difference between
the computed value of the attraction upon a unit mass (1 gram) at the station and the directly
observed value of that attraction. The degree of accuracy attained may be expressed by
saying that the largest anomaly is — 0.095 dyne (at stations Nos. 53 and 56, Seattle, Wash.),
that the mean anomaly without regard to sign is 0.017 (p. 76), and that as computed from
these anomalies considered as errors the probable error of the result at a single station is +0.014
(p. 75).*
The total error, as defined above, the apparent anomaly at each station, is the aggregate of
errors of three different classes. The first class comprises the errors in the observed value of
the attraction at the station. The second class includes all errors in the computed values of
the attraction at the station. Among these are errors due to numerical inaccuracy in the
computations, due to errors of approximation in the formule used, and errors due to the faults
and incompleteness of the maps which were used. The third class includes such errors as are
due to the difference between the actual arrangement of density in the earth and the arrange-
ment which has been assumed. The assumed distribution of densities is that fixed by the
statement that under every part of the earth’s surface the isostatic compensation is complete
and uniformly distributed with respect to depth down to a limiting depth of 113.7 kilometers
(p. 10).
The purpose of this discussion is to give the reader an estimate of the probable average
magnitude of the errors of the first and second classes and to compare this with the total error
as expressed by the anomalies, thereby securing an estimate of the magnitude of the errors of
the third class. From this point of view the errors of the third class are the portions of the
apparent anomalies which may not be accounted for as due to errors of the first or second
class. The smaller the errors of this third class are found to be the more nearly the assumed
distribution of densities agrees with the actual. The errors of this class furnish a good basis
for further investigation as to the actual distribution of densities in the earth.
* This mean and probable error are based upon the anomalies at 87 stations in the United States, the two stations Nos. 53 and 56, at Seattle
Wash., being rejected. 7 :
1
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 87
ERRORS OF OBSERVATION.
The half-second pendulums, described in Appendix 15, Coast and Geodetic Survey Report
for 1891, were used in the relative determination of gravity at each of the 89 stations in the
United States used in this investigation. The observations were made during seasons of less
than 6 months each, and the pendulums were standardized at the base station (in the basement
of the office of the Coast and Geodetic Survey at Washington) both before and after each season.
Three pendulums constituted a set, each pendulum being swung through at least two periods
of approximately eight hours each in determining the intensity of gravity at a station or
while obtaining the periods of the pendulums at the base stations. The necessary time observa-
tions were made with a portable astronomical transit set up in the vicinity of the gravity station.
The apparatus was used during standardizations in the same manner as in the field.
The following table shows the magnitude of the probable errors of the relative intensity of
gravity at 85 of the stations in the United States used in this investigation. The stations for
which no probable errors were computed are the base stations, the Smithsonian Institution,
Washington, D. C., Baltimore, and Seattle University.
Stations Probable error,
in dynes
8 +0. 003
14 + .002
58 + .001
5 . 000
Average + .0013
The probable errors shown above are those due to the accidental errors made at the stations
in the field. Let it be assumed that the accidental errors in obtaining the mean periods at Wash-
ington from the standardizations of the pendulums are approximately equal to the probable
errors in the field means. Then the total probable error for a station may be considered as a
combination of the probable error of the standardization and the probable error of the field
station. On this assumption the maximum probable error is +0.004, and the average probable.
error is +0.0018 for the mean result at any station. The actual error is probably at no station
more than four times the average probable error, or 0.0072 dyne, and the average actual error is
much lower than that. It is believed that the assumption stated above tends to give estimates
which are too large rather than too small. .
The following special statement is necessary for the seven stations, Ely, Pembina, Mitchell
Lake Placid, Potsdam, Wilson, and Alpena. Upon the return of the gravity party to the base
station, in November, 1909, after having observed at these stations, it was found that the period
of each of the three pendulums used during the season had considerably shortened. After having
made two complete determinations of the periods a very thin film of foreign substance was dis-
covered on the supporting plane of each of the three pendulums. Upon the removal of this
substance the pendulums resumed their former periods. In addition to the stations mentioned
above, North Hero and Iron River were occupied while the pendulums were probably affected
by the foreign substance on the planes. These two stations were reoccupied during a subsequent
season, and the values obtained for the intensity of gravity agreed closely with those obtained
during the first occupation of those stations, provided it was assumed. that the foreign substance
affected the periods of the pendulums to the same extent at those stations as during the first
determination of the periods at Washington in N ovember. North Hero and Tron River were
considered as base stations in determining the value of the intensity of gravity at Lake Placid,
Potsdam, Wilson, and Alpena, which stations had been occupied after North Hero and before
Iron River. Iron River and Washington were considered as base stations for Ely, Pembina, and
Mitchell, these three stations having been occupied after Tron River and just before the return to
Washington after the close of the season. The intensity of gravity used for North Hero and Iron
88 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
River was that determined during the reoccupation of those stations in 1910, and the value of the
period at the base station was that determined by the first standardization after the close of the
season in 1909.
The periods for the first and second occupation of North Hero differed by 0.0000057 second,
while at Iron River they differed by 0.0000042 second, and at the base station the difference
between the period given by the first standardization in November, 1909, and the mean period
of the two standardizations in May and October, 1910, was 0.0000033 second. This seems
to indicate that the effect of the foreign substance on the periods at the stations between North
Hero and the base station gradually decreased during the season between July and November,
1909. If the error in the adopted mean period at any of these stations is as much as 0.000 002 5
second, then the error in the value of the intensity of gravity at the stations from this cause
is 0.010 dyne. If a similar error was made at one of the base stations (North Hero, Iron River,
or Washington), the error due to this cause is 0.005 dyne. Hence, it is possible that there may
be errors as great as 0.015 dyne in the adopted values of the intensity of gravity at the
stations Lake Placid, Potsdam, Wilson, Alpena, Ely, Pembina, and Mitchell. It is believed,
however, that the actual error for each of those stations from all causes is less than 0.010 dyne.
In general the pendulums show approximately the same period at the base station in Wash-
ington during successive standardizations. There is given below a table showing the mean
period of the three pendulums forming the “A” set for the base station:
Date of stand- Period in
ardization seconds
Jan., 1909 0.500 707 5
June, 1909 .500 707 7
Dec., 1909 - 500 706 4
May, 1910 .500 705 7
Oct., 1910 .500 707 0
Mean . 500 706 9
It was assumed in each case that the pendulums were in normal condition. The values
obtained at the base stations in November, 1909, were not included in this table, on account of
the presence of foreign substance on the planes in the heads of the pendulums during those
standardizations. For the gravity work done during the years 1909 and 1910, the period
adopted for the base station in reducing a season’s work (except the season between July and
November, 1909) was the mean of the periods obtained at the beginning and at the end of the
season.
ERRORS OF COMPUTATION.
The first step in computing the attraction at a station was to compute by the Helmert
formula of 1901 the attraction 7., at a point on an ideal earth at sea level in the same latitude as
the actual station. The ideal earth referred to is one having the same size and shape as the
ellipsoid of revolution which most nearly coincides with the sea-level surface of the real earth
and having no topography and no variations in density at any given depth below the suresee,
(See p. 12.)
The Helmert formula of 1901 is based upon many gravity determinations widely distributed
over the earth’s surface, and in consequence probably gives a close approximation to the desired
values. The available indirect evidence gives strong support to the belief that this formula, in
which the constants are computed from gravity observations, is of a very high degree of accuracy:
For example, the values of the flattening of the earth, as computed by this formula and as com-
puted from geodetic observations in the United States, are of about the same degree of accuracy
and agree closely. The value of the reciprocal of the flattening derived from the Helmert
formula of 1901 is 298.3 +0.7, and from geodetic observations in the United States is 297.0 0.5.*
* Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, p. 60
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION-ON GRAVITY. 89
This is a confirmation, by independent observations of a different kind from those on which the
formula is based, of the accuracy of the second constant in the Helmert formula.
But, on the other hand, the Helmert formula of 1901 is based upon selected coast and
inland stations. The present investigation indicates that even at these carefully selected
stations there is probably small systematic error due to the failure, by the methods of reduction
used in connection with the derivation of the Helmert formula, to take account properly of
the effects of the topography and its isostatic compensation. A correction (0.007) serving to
eliminate this systematic error as completely as is possible at present, has been derived from
the observations in the United States and applied to the first constant in the Helmert formula
of 1901. (See p. 75.) It is believed that the Helmert formula of 1901 so corrected is a true
representation within less than 0.003 dyne on an average of the attraction at sea level on the
ideal earth, if the formula is limited in application to the range of latitudes occurring in the
United States. .
The correction for elevation (p. 13), the next step in the computation, is of such a nature
that it is reasonably certain that the errors made in computing it are very small, usually not
more than 0.001 dyne. An error of 3 meters in the elevation makes but 0.001 dyne error in
the computed correction. For the gravity stations in the United States the elevations are
known as a rule within 3 meters and at very few if any of the stations is the error in elevation
more than 15 meters. .
The value of the gravitation constant (k) adopted in this investigation is 6673 (10-4),
and it is estimated that the probable error of this adopted value is one part in 1330. (See
p. 14.) This constant enters directly as a factor into each formula for computing the correction
for topography and isostatic compensation. (See formule (10), (15), (16), (17), and (18),
pp- 15-17.) Hence, the probable error of one part in 1330 in the gravitation constant produces an
error of the same proportional part in each computed correction for topography and compensation.
The largest of these corrections (see p. 74) is only 0.187 for station No. 43, Pikes Peak. Even
for this case the probable error in the correction due to error in the gravitation constant is only
0.0001 dyne (0.187/1330), and is therefore negligible in connection with the present investigation.
Similarly, any error in the assumed mean surface density of the earth will produce an
error of the same proportional part in the computed correction for topography and compensa-
tion corresponding to each land compartment. The mean surface density has been assumed
to be 2.67 in this investigation. It is reasonably certain that the mean density of the whole
of that portion of the earth which lies above sea level does not differ from this by as much as
one-twentieth part.* At Pikes Peak, station No. 43, the sum of the corrections for all land
compartments is probably greater than for any other one of the 89 stations in the United States
used in this investigation. At this station this sum is about +0.180 dyne.+ An error of one-
twentieth part in this would be only 0.009 dyne. An inspection of the tables on pages 54-58
indicates that as a rule the sum of the corrections for land compartments for stations in the
United States is less than 0.020 and an error of one-twentieth part would, therefore, ordinarily
be less than 0.001 dyne. .
In general the density of sedimentary rocks tends to be less than 2.67, not unfrequently
as much as one-tenth part less. On the other hand, igneous rocks and rocks which have been
buried to a great depth tend to be of density greater than 2.67. These local departures of the
densities from the assumed mean, 2.67, produce errors of the third class, which have been
defined as errors due to the difference between the actual arrangement of densities in the earth
and the assumed arrangement. These effects of local departures of density from the mean
are a part of the anomaly at the station rather than errors in determining the anomaly, Hence,
the discussion of them will be taken up later as a part of the discussion of the meaning of the
anomalies. .
sity of the earth, 2.67, and this estimate of its uncertainty are based largely upon the information
: : oe Ee Sent, fae ue, by William Harkness, Washington, Government Printing Office, 1891, pp. 91-92,
Be te ane a ones A to 10 at this station. (See p. 56.) Zone 9 is the nearest zone containing any oceanic compartments.
eee ait the estimates of density of rocks in the vicinity of 10 of the gravity stations here treated as given on p. 530f Appendix T
se hs on ouctettn Survey Report for 1894, ‘Relative determinations of gravity with half-second pendulums and other pendulum observa-
of the Co:
tions,” by G. R. Putnam and G. K. Gilbert.
90 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
On page 15 attention is called to the fact that in deriving the formule by which all compu-
tations for distant zones have been made the earth is treated as a sphere with a radius of
637 000 000 centimeters, although it is actually a spheroid.
In zones M, N, and O errors due to this approximation are evidently negligible for, as
shown on page 22 and in the reduction tables for those zones, pages 41-43 (consult the column
headed ‘‘Station at same elevation as compartment”), if all of the curvature were neglected
and the earth’s surface treated as a plane, the error introduced would be only 0.0001 dyne in
any one compartment of these zones. The curvature of the actual spheroid in any azimuth
within the limits of the United States differs by less than one five-hundredth part from that of
the assumed sphere and, therefore, the error for any compartment due to the cause under
discussion must necessarily be much less than 1/500 of 0.0001 dyne in zones M, N, and O. The
errors must be still smaller for near zones.
For more distant zones a general consideration of the geometric relations, shown in illus-
tration No. 15, page 67, indicates that the error is probably considerably greater. Without a
detailed investigation the three following considerations seem to the writers sufficient to assure
one that the total error due to this cause is probably less than 0.001 dyne at every station.
First, the total correction for topography and isostatic compensation beyond zone O is less
than —0.060 dyne at every one of the 89 stations. Second, the actual radius of the earth
varies from 6357 kilometers at the pole to 6378 kilometers at the equator; that is, from 13 kilo-
meters less (1/490 part) to 8 kilometers greater (1/800 part) than the assumed radius. These
differences may be considered as maximum vertical displacements of material in very distant
zones from its assumed position. The displacements are small in comparison with the distance
to the zone in these cases. Third, on the actual spheroid the radii in various azimuths from
the station are different. For example, for a station in the central portion of the United States
in latitude 39° the radius of curvature in the meridian is 6361 kilometers, 9 kilometers less (one
part in 710) than the assumed value, 6370 kilometers, and in the prime vertical at this same
station the radius of curvature is 6387 kilometers, 17 kilometers greater (one part in 370) than
the assumed value. Hence, in each zone the errors of the kind under consideration tend to be
compensating to a considerable extent, some parts of the zone lying farther from the center of
the earth than the assumed curvature places them and other parts of the same zone, lying in
different azimuths, being nearer to the center than the assumed curvature would place them.
Assuming for the moment that the elevations and depths shown on the maps and charts
used are correct, the errors made by the computer in estimating the mean elevation or mean
depth within each compartment did not, as a rule, produce any error even in the fourth decimal
place in dynes. In zone A an error of at least 5 feet in estimated elevation is necessary in order
to make an error of 0.0001 dyne in the computed correction even if the elevation of the station
is less than 10 feet. In this zone if the station has an elevation greater than 10 feet, the cor-
rection is 0.0002 dyne in every case. In zone F it takes an error of 200 feet or more in the
estimated elevation to produce an error of 0.0001 dyne in the computed correction; in zone M
500 feet or more; in zones 18 to 14, 100 feet; and in zones beyond 14, 1000 feet or more. (Consult
the reduction tables, pp. 30-47.) In many cases the total range of elevation within a compart-
ment, as shown by the map, is less than that necessary to produce a change of 0.0001 dyne in the
correction taken from the reduction table. In these cases no error in the correction arose from
the estimation of the mean elevation. Still more frequently the range of elevation within the
compartment is not more than three or four times that necessary to produce a change of 0.0001
dyne. It is probable that in such cases the estimate of mean elevation was rarely in error by
more than the quantity corresponding to 0.0001 dyne. For perhaps one-tenth of all the com-
partments the computer found so large a range of elevations shown on the map that his estimate
of mean elevations was necessarily made with considerable care and attention to the details of
the contour lines, and even then the correction taken from the reduction table may be in error
by two or more units in the fourth decimal place. It is believed that the aggregate of such
errors for a station is seldom greater than 0.001 dyne. For, as indicated above, difficulties were
encountered in making the estimate of the mean elevation with sufficient accuracy at only a
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 91
small percentage of the compartments; the errors so made tend to be in the accidental class;
the difficulties were obvious to the computer and he, therefore, exercised unusual care in the
extraordinary cases, even to the extent of subdividing the compartments and making a separate
estimate for each subcompartment; and finally for each station a second computer inspected
the computation made by the first and made a second estimate covering some of the compart-
ments in which there were obvious difficulties in making a sufficiently accurate estimate. For
compartments for which two estimates were made the mean of the two was used, unless they
differed so much as to lead to detection of an error in one or the other.
In making the computations for topography and isostatic compensation, the elevations of
compartments were read from the maps without making any allowance for the fact that glaciers
have a much lower density than the land. A computation was made to show the effect upon
the intensity of gravity at station Gornergrat of the defect of density of the glacial ice in its
vicinity in comparison with solid earth of the assumed density, 2.67. An inspection of the
maps of this region showed that 37 of the 102 compartments in the zones E to K were over
ice, and the shapes of the clear portions of the valleys indicated that the average thickness of
this ice in the several compartments varies from a few feet to more than 600 feet. The presence
of ice in the zones closer to the station than zone E and farther from the station than zone K
was believed not to affect the intensity of gravity at the station.
An average density of unity was assumed for the glacial material in making this computa-
tion. This is believed to be near the truth, for the heavy material carried by the glacier (sand
and gravel) is probably approximately balanced by cavities and the lightness of the clear ice
in comparison with water. This makes a defect of density of approximately 1.67 in portions
of the topography of certain compartments. This should make a minus correction to the
computed effect of the topography and a plus correction to the effect of the isostatic compen-
sation. The largest correction found for any one compartment due to this lack of density was
0.0004 dyne, while the average correction for a compartment was less than 0.0001 dyne. In
the near zones the effect of ignoring the lack of density in the glacier made the computed value
of gravity too great, while, owing to the change of sign with distance from the station (see pp.
65-70), the effect of such neglect in the more distant zones was to decrease the computed value
of gravity. The total result for station Gornergrat was to make the computed value of gravity
too great by 0.0006 dyne, a negligible quantity. It is probable that the effect on the intensity
of gravity of assuming glacial ice to have a density of 2.67 in the computations of the effect
of topography and isostatic compensation upon the intensity of gravity has not caused an
error of more than 0.0010 dyne at any one of the stations treated in this investigation.
In using the mean elevation within a compartment as the argument in entering the reduc-
tion tables on pages 30-47, it is tacitly assumed that the influence of a unit of area of a given
elevation is the same wherever it is located in the compartment. This is only approximately
true. For example, in zone 13 (limiting radii 3° 03’ 05’’ and 4° 19’ 13’ ’) Ep is 5000 at the
outer edge of the zone and 13 600 at the inner edge. (See p- 25.) The influence of a unit of
area of a given elevation on the outer edge of the zone is, therefore, 5000/ 13600 = 0.37 as great
as on the inner edge. If, therefore, in this zone the elevations nearer the outer edge in one com-
partment happen to be much greater than elevations nearer the inner edge, the correction
taken from the table by using the mean elevation as an argument will be too large. Similarly,
if the slope in the compartment happens to be downward from the inner edge toward the outer
edge the correction taken from the table will be too small. a
When the arbitrary selection of radii of zones and of number of compartments in each
zone was being made the danger of errors from this source was kept constantly in mind (see Pp.
18), and each compartment was made so small that the estimated errors due to this cause in
any compartment would ordinarily be less than 0.0002 dyne. The details of the manner in
which this estimate was made can not be conveniently shown here. Evidently the narrower
the zone is made the smaller the error from this cause, both because Ep will be more nearly
the same on the two edges of the zone and because the difference between the average elevation
of the near topography and of the distant topography in each of the compartments of a zone
92 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
tends to be small. Similarly, after the width of the zone is fixed, the smaller the compartments
are made within the zone the smaller will be the error, for the less will be the range of elevations
included within the compartment and the larger will be the change of elevation corresponding
to an effect of 0.0001 dyne. The values of E, were before the investigator for all outer zones
at the time each decision was made. For inner zones for which formula 18, page 17, was used,
an indirect method of obtaining the equivalent of the change in Ep was utilized. The investi
gator also had before him the experience and data obtained in connection with the previous
investigation of the figure of the earth * which enabled him to estimate the maximum difference
of elevation between the inner and outer edges of any given compartment which would probably
be found at any station.
Errors due to this cause will evidently be of the accidental class, since in some, zones a
downward slope toward the station will produce an error of one sign and in others the reverse slope
will produce an error of the opposite sign. In the 317 compartments concerned in the compu-
tation at a given station there will be but few compartments, sometimes none, in which this
error is as great as 0.0001 dyne, and errors of both signs will probably occur among these few.
It is believed that the aggregate error due to this cause at a station seldom exceeds 0.0005 dyne.
The errors due to the faults and incompleteness of the maps and charts used are believed
to be very small as a rule. The aggregate error for all numbered zones is probably seldom,
if ever, greater than 0.002 dyne. For the lettered zones, zones which lie near the station, the
aggregate error in some cases may be two or three times this limit. The reduction tables (pp.
30-47) show that for the nearer lettered zones the elevations must be known with greater accuracy
in.general than for the more distant numbered zones, and since the compartments are small
in the lettered zones it is necessary to know the details of the topography. The magnitude of
the aggregate error at a given station, due to faults and incompleteness of maps and charts,
therefore, depends principally upon the accuracy of the maps and charts covering the region
close to the station rather than that of those covering distant regions. f
Some errors are made in locating the compartment boundaries on the maps, due to
unavoidable inaccuracy in constructing the templates, to inaccuracy in placing the templates
on the maps, to special difficulties encountered in connection with the distortion of scale on
Mercator ‘charts, and to shrinkage and, therefore, error of scale of the maps and charts. With
the templates and maps before one it is evident that the aggregate effect of these errors at a
station is ordinarily negligible. In general the effect of an error in locating a compartment
boundary is simply to throw a small part of the area which belongs in one compartment into
an adjoining compartment, where its influence on the computed correction is nearly the same
as if it had been placed in its proper compartment..
The methods followed in computing the reduction tables have been stated on pages 19-28.
The precautions taken were such as to insure that no tabular value is in error by more than 0.0002
dyne, and that in general the tabular values are correct to within 0.0001 dyne. The intervals
between tabulated values have been so selected, with due regard to second differences, as to
insure that the errors made in interpolating between them, using first differences only, shall
ordinarily be less than 0.0001 dyne.
How large are the errors introduced into the computed topographic effect on: the intensity
of gravity by the interpolation of values corresponding to outer zones? The complete com-
putation was made for only six stations. Each new station to be computed was so chosen,
if possible, as to lie within the triangle defined by the nearest three stations for which the
computation had already been made, and near the center of said triangle. From these three
surrounding stations the interpolation, if any, was made.
The computation was commenced with the inner smaller zones and proceeded outward.
The two rules used by the computers in deciding at what zone it was allowable to begin to
accept the interpolated values and to accept them for all larger zones were, as stated on page
63, as follows:
* The Figure of the Earth and Isostasy, etc., pp. 125-127.
+ For a more detailed statement of the considerations upon which the judgment expressed in this paragraph is founded, see The Figure of the
Earth and Isostasy, etc., p. 124.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 93
Rule 1—Commence to accept the interpolated values as final with the first zone for which
such interpolation is allowable under rule 2, provided that it is beyond the zone containing
the nearest of the three stations from which the interpolation is made.
Rule 2.—Let 0.0005 dyne be the interpolation limit for any zone. Subject to rule 1, .
acceptance of the interpolation may begin with a given zone if each of the three zones next
within it shows an agreement between the interpolated and computed values which is within
the interpolation limit. _
Under rule 2, at any station the maximum error made by accepting interpolated values
would be, in dynes, 0.0005 times the number of zones interpolated, if the error of interpolation
I—C (interpolated minus computed) always had the same sign. It was believed, however,
that the agreement between the interpolated and computed values (commencing with zones
not smaller than those contemplated under rule 1) would tend strongly to be closer and closer
for successive zones proceeding outward. It was also believed that there would be a strong
tendency for the various differences between interpolated and computed values for several
zones such as are interpolated under the rules to include values having both the plus and minus
signs, and, therefore, for the errors in the accepted interpolation to tend to be eliminated from
the final result for the station.
The correctness of these beliefs is established by the results secured during the progress
of the computations. From the results of the computations of 48 stations a comparison between
the computed and interpolated values was secured at each station on from 2 to 10 zones. In
81 per cent of the cases the average value, without regard to the sign of I—C (nterpolated
minus computed) was less for the outer one-half of the zones on which both interpolation and
computation was made at that station than for the inner half of such zones. Also in 56 per
cent of the cases there were found to be both plus and minus signs of the values of I—C at
the station.
These tests confirm the theory to such an extent that it is believed that the total error
introduced into the computed effect of topography and compensation at a station by the
acceptance of interpolated values is seldom greater than 0.0022 dyne and is, as a rule, not
more than one-half that amount. In addition to the evidence stated in the paragraph above,
this estimate of 0.0022 dyne is based upon the fact that the average difference between the
‘computed and the interpolated values for the three zones (see rule 2) next within the one for
which the interpolation is accepted, at any station, is in general 0.0002 dyne or less. The
average number of zones per station for which interpolated values were accepted is 11. If
the error for each interpolated zone were 0.0002 and all were of the same sign, the error would
be 0.0022 on an average. However, as the outer zones have more overlapping of areas, the
interpolated and computed values for those zones should agree on an average more closely
than these values for the three zones next preceding the zone at which interpolation begins,
and as these errors are of the accidental class and not all of the same sign, there is a tendency
for the errors of interpolation to be eliminated from the final result for the station. One may,
therefore, conclude that the total error caused by accepting the interpolated values is so small
negligible.
7 sate oh to which the isostatic compensation extends has been assumed to be fixed by
a surface which lies 113.7 kilometers below sea level, but, as noted on page 10, in order to sim-
plify and to facilitate the computations the depth of compensation has in the computations
really been reckoned from the solid surface of the earth, not from sea level. This computing
device has, therefore, virtually displaced the isostatic compensation upward on land areas by
a distance equal to the elevation of the surface of the area above sea level, and downward for
ocean areas by a distance equal to the depth of the particular part of the ocean considered.
For near zones this displacement of the compensation produces negligible effects because the
total effect of the compensation is small (consult the reduction tables for zones A to I, pp. 30-37).
For the very distant zones, 13 to 1, this displacement of the compensation produces effects which
are certainly negligible, since the reduction tables, pages 45 a 46, sea ba is no
appreciable correction for elevation in these zones. For intermediate zones J to 14 small appre-
94 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
ciable effects are probably produced in some cases by the virtual displacement of the isostatic
compensation introduced as a computing device. Though no special investigation of the
aggregate of effects has been made it is believed to be small. In other words, the actual
computation made on the supposition that the depth of compensation is 113.7 kilometers
measured from the solid surface, is believed to be practically in agreement as to numerical
results with the computation which theoretically should have been made on the supposition
that the depth of compensation is 113.7 kilometers measured from sea level.
Within the great depth, 113.7 kilometers, to which isostatic compensation extends there
is probably a slight increase of density with increase of depth, due to increased pressure. No
account has been taken of this in the process of computation, as already noted on page 7. It
may appear at first sight that this neglect introduces some error into the computed results,
but it does not. The isostatic compensation as used in the computation is essentially an excess
or defect of density referred to the normal density for each level concerned within the depth
of compensation. It matters not in the computation of the effects of topography and iso-
static compensation whether the normal relation of density to depth is such that there is no
appreciable increase of density within the depth of compensation or whether there is consid-
erable increase within that depth, for the excesses and defects of density constituting the
isostatic compensation are referred to this normal law, not to a constant density for all depths.
The point at which the relation of density to depth enters this investigation, though not explic-
itly, is in the derivation of the Helmert formula of 1901. Any actual change in the distribu-
tion of density with respect to depth would in general change the observed value of the intensity
of gravity and would cause one or more of the constants of this formula to change. There-
fore, the constants in this formula as derived from observations correspond to the actual relation’
between depth and density, though that relation is not known.
NATURE OF APPARENT ANOMALIES.
There have been discussed on the preceding pages the principal possible sources of error of
the first and second classes, defined on page 86. Among these sources are the errors in the in-
strumental determinations of gravity at each station, errors in the corrected Helmert formula
of 1901, errors in the corrections for elevation, errors in the adopted values of the gravitation
constant and the mean surface density, the erroneous assumption in certain parts of the compu-
tation that the sea-level surface is a sphere rather than a spheroid, errors in the estimated mean
elevations in the different compartments, errors due to variations of elevation within each com-
partment, errors in the maps and charts used, errors in locating compartment boundaries, errors
of interpolation for outer zones, and errors in computing the reduction tables. The errors of each
of these kinds are nearly or quite independent of the others, and follow different laws of distribu-
tion. In estimating the effects of all these errors at a station one must therefore consider them
as accidental errors and that their combined effect is the square root of the sum of their squares
rather than merely their sum. On this basis the writers estimate that the probable error of the
computed anomaly at a station by the new method is about +0.003 dyne on an average. In
other words, the chances are even for and against the proposition that the actual error in the
computed anomaly at a station is greater than 0.003.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 95
The basis for this estimate is in part indicated in the following table:
Estimate of errors of the first and second classes.
: Maximum Average prob-
Source of error probable error | able error of
ofany station | any station
Observations of gravity +0. 004 +0. 002
Helmert formula of 1901, corrected +0. 003 +0. 002
Correction for elevation +0. 003 +0. 001
Gravitation constant _ +0. 000 +0. 000
Mean surface density +0. 005 +0. 001
Defects and incompleteness of maps +0. 004 +0. 001
Acceptance of interpolation +0. 001 +0. 000
From all other causes +0. 001 +0. 001
The square root of the sum of the +0. 009 +0. 003
squares, or the probable error of the
the final result subject to all the sepa-
rate errors enumerated
If the whole anomaly be considered as an error, then the probable error for all stations due
to all causes is +0.014 (see p. 75), this probable error being computed from the 87 apparent
anomalies available in the United States after rejecting the two Seattle stations. It should be
noted that this computation includes the third class of errors defined on page 86, those due to
the departures of the actual arrangement of densities beneath the surface from the arrangement
which has been assumed. The magnitude of the errors of this third class, the real anomalies
sought, may be estimated as that part of the total error computed, as indicated above, from the
apparent anomalies, which is not accounted for by errors of the first and second classes, namely,
+ (0.014)? — (0.003)? = +0.0137.
These two values, +0.003 and +0.0137, may be interpreted as follows: The second being
about five times the first, the apparent anomalies shown on page 76 under the designation
‘‘ Anomalies, new method,” are upon an average composed of one part errors of observation and
computation to five parts actual anomaly at the station, due to the departure of the actual
arrangement of densities from the assumed arrangement. The quantities labeled ‘‘ Anomalies,
new method,’’ are therefore a close approximation to the real anomalies sought. They are a
possible basis for further investigation as to the actual distribution of density within the earth. —
THE METHOD NOT SUBJECT TO HIDDEN ERRORS.
This discussion of errors would be seriously incomplete if it were closed without calling at-
tention to certain characteristics of the computations on which this investigation is based which
insure safety against certain classes of obscure but serious errors.
The process of integration by the method of computing a large number of separate values of
the function (see pp. 23-27), which has been used in this investigation, is very clumsy and inele-
gant, as seen from the mathematical point of view, but from the practical point of view of one
who desires to solve the problem of computing the effect of all the topography ‘of the world and
of its isostatic compensation upon the intensity of gravity at a given station, it has a very differ-
ent aspect. From the latter point of view it appears that the method is sufficiently rapid to
make its use permissible and that it is clearly safe against errors, whereas the alternative mathe-
matically elegant method is unsafe.
As to the rapidity of the method, it was found in practice that the necessary reduction tables
for zones covering the whole earth were computed in the equivalent of about 800 hours of time
for one computer. This seems to be a reasonable time when one considers the importance
and difficulty of the problem solved. Moreover, these tables made it possible to make the
remaining portions of the computation very rapidly. They enabled the computer in 17 hours
to compute the effect of all the topography of the world and its isostatic compensation upon
the intensity of gravity at any given station on the earth’s surface, and to be certain that the
96 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
errors of the computed result are confined within the very narrow limits indicated by the pre-
ceding discussion of errors. This in turn furnishes a safe basis, and in the opinion of the writers
the only safe basis yet available, for an accurate determination of the flattening of the earth
from gravity observations; for any effective investigation of the theory of isostasy by means
of gravity observations; for any investigation of the real meaning of the apparent anomalies
of gravity, such, for example, as those on small oceanic islands; and in fact for any safe general
conclusions from observations of the intensity of gravity on the earth’s surface.
The method used in this investigation of obtaining the integrals of the expressions (9), (15),
and (16), pages 14-16, by computing many numerical values, is safe against excessive or unseen
errors because of the fact that the computer has before him in these many numerical values a
clear and definite means of knowing how large are his errors of approximation. For example,
when facing the actual problem of determining the mean value £, (see p. 23) with various com-
puted values of E before him, there is little difficulty in deciding safely how many values of E
to compute in order to be certain of a given degree of accuracy in the mean value. Various
similar examples from this investigation might be cited.
On the other hand, if the computer resorts to the more elegant method, from the mathe-
matical point of view, and first transforms formule (9), (15), and (16) by simplification into
forms which can be integrated by calculus, he is, while making the simplification, in grave danger
of introducing errors of approximation which he believes to be small, but which are in reality
large. The writers believe that in this particular problem this danger has not been escaped in
the past. For example, the conclusion that it is not necessary to take distant topography into
account, a conclusion which has been acted upon in many previous investigations, and which
this investigation shows to be erroneous, has apparently been reached in the past by dealing
with unsafe approximations in the literal or symbolic form. So, too, it seems to the writers that
one can not overlook the necessity of taking curvature very fully into account if one has the
numerical values before him, but may easily overlook it if he is dealing with symbols and
formule only.
Another characteristic of the method of computation used in this investigation, which is
very important as a means of securing safety against unseen errors, is the fact that it deals with
the actual irregular surface of the earth rather than with a geometrical surface which is assumed
to fit the earth’s surface in the vicinity of the station. It is true that the irregular surtace
actually used in the computation is made up of 317 level surfaces, one for each compartment
of each zone, the mean elevation in each compartment being the argument with which the
reduction tables are entered. But the compartments near the station are so small that the
surface upon which the computation is based is, in these zones, a very close approximation to
the actual irregular surface. The one compartment of zone A is a circle with a 2-meter radius.
Each of the four compartments of zone B has an area of less than 4000 square meters. The
agreement between the assumed surface and the actual irregular surface of the earth is less
close for the more distant topography, but there is still, even for the most distant zones, an
approximation to the actual irregular surface. The precautions taken in fixing the size and
shape of the separate compartments insure, in fact, that even for these distant zones the approx-
imation to the actual irregular surface is sufficiently exact to keep the errors in the computed,
effects of topography and compensation well within the allowable limits.
In any computation of the effects of topography and compensation in which any part of
the earth’s surface is assumed to conform to the geometrical surface, in which, for example, a
mountain or an oceanic island is assumed to have a conical shape, or the distant topography
is assumed to ke a plain of indefinite extent, it is desirable to consider with extreme care how
much error may be introduced into the computations by such assumptions, to consider care-
fully what evidence the computer has that these errors are small in each separate case. Such
errors once introduced into an investigation remain there regardless ot the degree of mathe-
matical elegance and precision which may be maintained thereafter. The writers believe that
the more carefully this point is examined the more fully the advantages of the methods of com-
putation used in the present investigation will be appreciated.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 97
EFFECTS OF TOPOGRAPHY AND COMPENSATION—WHY COMBINED.
In the investigation of the figure of the earth and isostasy by means of observed deflections
of the vertical, the whole effect of the topography was first computed and later the effect of the
isostatic compensation was combined with it.* In the present investigation, based on gravity
determinations, the effects of topography and of compensation have been combined as early as
was feasible in the processes of deriving formule and of computing. Thus, as indicated on
pages 23 and 24, instead of computing the two effects separately for distant zones they were
combined in formula (20) and the resultant effect computed at once and tabulated in the reduc-
tion tables on pages 44-47. So, too, for near zones the principal part of the reduction tables
(pp. 30-43) refers to resultant effects, not to separate effects. The only columns in these tables
showing separate effects are columns 2 and 3, and these were not used in the regular com-
putations.
Why was this departure made from the methods of the earlier investigation ?
This departure was decided upon immediately after a preliminary reconnoissance of the
problem. It then appeared probable that, for all zones except for those very near the station,
the two opposing effects of topography and compensation would be nearly equal, and their dif-
ference, therefore, much smaller than either one. Under these circumstances it appeared that
to compute each of the opposing effects with sufficient accuracy to secure the required degree
of accuracy in their difference it would be necessary to secure several significant figures in the
computation. If this supposition were true, it would be necessary in making the separate
computations, either to make the compartments of the separate zones very small and numerous,
and hence the computation very slow, or, otherwise, if large compartments were used, it would
be necessary to make the estimate of mean elevation in each compartment with such a high
degree of accuracy as to be both slow and difficult. On the other hand, it appeared that in
the direct computation of the resultant difference of effects, it would be necessary to use but
two or three significant figures in the computation, that the compartments could be made large
and therefore not very numerous, and that only an approximate estimate of the mean elevation
in each compartment would be required and could, therefore, be made quickly and easily.
It seemed, therefore, that so much would be gained in rapidity and ease of computation by
the proposed departure from the earlier practice that these gains should outweigh all other
considerations.
Now, this investigation being complete, the writers have an opportunity to review the deci-
sion in the light of accumulated facts and greater experience. In that light it appears that
the decision was wise for zones which are more than 26° from the station—zones 6 to 1 of the
present investigation. For these zones the difference of the effect of the topography and the
effect of the compensation is less than one-tenth of either; that is, Zz is less than one-tenth of
either Ep or Ec (p. 25). For nearer zones the difference, as a rule, is a much greater propor-
tional part. Hence, for these nearer zones the gain in rapidity and ease made by dealing
directly with the difference of effects rather than with the separate effects was not great, and
therefore the decision was not wise. Moreover, it appears now that if the separate effects had
been computed for these nearer zones it would have given the investigator a clearer and more
precise insight into the problems involved. It would also have facilitated studies of the rela-
tion of the computed results to the assumption as to the depth of compensation and possibly
to some other assumptions.
If, therefore, an entire new investigation were being made the writers believe it would be
wise to compute the two effects separately for zones A to O and 18 to 7, but the gain to be
secured does not seem to be sufficiently great to warrant the revision of the present investiga-
tion and the remodeling of the reduction tables here printed.
* The Figure of the Earth and Isostasy from Measurements in the United States, pp. 68-73.
15593°—12——7
98 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
REGIONAL VERSUS LOCAL DISTRIBUTION OF COMPENSATION.*
The question whether each topographic feature is completely compensated for by a defect
or excess of mass exactly equal in amount directly under it, or whether the topographic feature
is compensated for by a defect or excess of mass distributed through a more extensive portion
of the earth’s crust than that which lies directly beneath it, is a very important one. The
theory of local compensation postulates that the defect or excess of mass under any topographic
feature is uniformly distributed in a column extending from the topographic feature to a depth
of 113.7 kilometers below sea level. The theory of regional compensation postulates, on the
other hand, that the individual topographic features are not compensated for locally, but that
compensation does exist for regions of considerable area considered as a whole.
In order to have local compensation there must be a lower effective rigidity in the earth’s
crust than under the theory of regional compensation only. In the latter case there must be
sufficient rigidity in the earth’s crust to support individual features, such as Pikes Peak, for
instance, but not rigidity enough to support the topography covering large areas.
Certain computations have been made to ascertain which is more nearly correct, the
assumption of local compensation or the assumption of regional compensation only. In making
such computations it is necessary to adopt limits for the areas within which compensation is
to be considered complete. A reconnoissance showed that the distant topography and com-
pensation need not be considered, for their effect would be practically the same for both kinds
of distribution. As a result of this reconnoissance it was decided to make the test for three
areas, the first extending from the station to the outer limit of zone K (18.8 kilometers), the
second from the station to the outer limit of zone M (58.8 kilometers), and the third, to the
outer limit of zone O (166.7 kilometers).
The computed effect of the topography in each compartment and zone is the same under
the two methods. The effect of compensation is assumed to be the same for each compartment
and zone which is beyond the limit of the area adopted for the test. The effect of compensation
within that limit is computed for each compartment in the case of the theory of complete local
compensation, while in the case of regional compensation only, it is obtained from one operation
after the average elevation within the area considered is known.
The regular computations of the effect of topography and compensation had been completed
at 56 stations in'the United States, Nos. 1 to 56, inclusive, and at all of the stations not in the
United States, used in this investigation, before it was planned to make computations based on
the theory of regional compensation within limited areas. In the regular computations for
these stations the effect of topography and compensation for zones A to O was taken from the
fourth column of the reduction tables (see pp. 30-43), and no record was made of the elevations
of the several compartments as read from the maps. In making the supplemental computations
these tables were entered with the previously computed values of the combined effect of the
topography and compensation as arguments, and the approximate values of the elevations of
the several compartments of zones A to O were taken from column 1 of the reduction tables,
and the values of the effect of compensation taken from column 3. The supplementary com-
putations were not made for all of the stations between Nos. 1 and 56 on account of the large
amount of work involved.
While making the computations of the effect of tépography and compensation for stations
Nos. 57 to 89 (except station No. 84), a table was made for each station, giving the elevation
of each compartment out to zone O as read from the map. With these elevations the reduction
tables were entered and the effect of compensation was taken out separately from column 3.
The total effect of compensation under the theory of local distribution was obtained for each
of the areas considered by adding the values of the effect of compensation for the several com-
partments of each of the zones. The mean value of the elevation of each zone was obtained
by taking the mean of the elevations of its several compartments, and the mean elevation of
* The investigation under this heading was made at the suggestion of Mr. G. R. Putnam, of the Coast and Geodetic Survey
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 99
each of the three areas considered (limited by zones K, M, and O) was obtained by combining
the elevations of the various zones, the elevation of each zone being given a weight equal to its
percentage of the total area under consideration.
The regional compensation for the total amount of topography in the area considered was
assumed to be uniformly distributed both vertically and horizontally throughout a column of
depth 113.7 kilometers and of a cross section equal to the area of the topography—that is,
successively from the station to the outer limits of zones K, M, and O. The effect of the com-
pensation upon the intensity of gravity at the station was computed by formula (17), in which
the several terms have the same significance as stated on page 17.
The table following shows the comparison of the effects of local compensation and regional
compensation for 41 stations in the United States and 4 stations not in the United States.
It also shows the anomalies by the first method and for 3 cases at each station by the second
method. The first column gives the number and name of the station. The second column
gives the total correction for topography and compensation by the method of local compensa-
tion. In the third column.are shown the values of the compensation for the topography
included in the area extending from the station out to zone K, the compensation being assumed
to be complete and local, In the fourth column are given the values of the compensation for
the topography within the same area, but with regional compensation only, which is assumed
to be uniformly distributed and complete, within the area limited by the outer circumference
of zone K.
Columns 5 and 6 are similar to 3 and 4, except that the area considered extends from the
station to the outer limit of zone M. The same statement applies to columns 7 and 8, except
that the area considered extends from the station to the outer limits of zone O. The ninth
column contains the new-method anomalies, based upon complete local compensation, and the
last three columns show the anomalies for the three cases under the theory of regional compen-
sation only.
100
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
The mean, without. regard to sign, of the anomalies by the new method for the 41 stations
in the United States shown in the above table is 0.017 dyne. For the regional compensation
Effect of compensation within outer limit of— Anomaly, Anomaly with regional com-
Effect of eh ensation within outer
topog- method 1 imit of —
Number and name of raphy Zone K Zone M Zone O =
station and +0.007)
compen- (local
sai ‘ c en-
ac Local | Regional} Local | Regional} Local | Regional entioa) Zone K | Zone M | Zone O
Stations in United States
42. Colorado Springs, :
Colo. —0. 007 —0: 036 —0. 036 —0. 094 —0. 093 —0. 165 —0. 164 —0. 009 —0. 009 —0.010 —0.010
43. Pikes Peak, Colo. + .187 | — .052 | — .044 | — .118 | — .100 | — .189 | — .172 | + .019 | + .011 | + .006 | + .002
44, Denver, Colo. — .015 | — .026 | — .028 | — .076 | — .085 | — .152 | — .169 | — .018 | — .016 | — .009 | — .001
45. Gunnison, Colo. — .001 | — .041 | — .044 | — .120 | — .128 | —.212 | —.210 | + .018 | + .021 | + .026 | + .016
46. Grand Junction,
Colo. — .051 | — .026 | — .028 | — .082 | — .089 | — .156 | — .170 | + .022 | + .024 | + .029 | + .036
48, Pleasant Valley
. Junction, Utah + .024 | — .040 | — .041 | — .103 | — .100 | — .171 | — .159 | + .002 | + .003 | — .001 — .010
49. Salt Lake City,
Utah — .041 | — .026 | — .028 | — .075 | — .078 | — .137 | — .143 | + .008 | + .010 | + .o11 | + .014
54, San Francisco, Cal. | + .045 - 000 +000 | — .002 | — .003 | + .009 | + .033 | — .025 | — .025 | — .024 | — .049
55. Mt. Hamilton, Cal. | + .120 — .012 — .012 | — .017 — .009 — .018 — .003 — .005 — .005 — .013 — .020
57. Iron River, Mich. + .014 | — .007 | — .008 | — .020 | — .020 | — .031 | — .024 | + .036 | + .037 | + .036 | + .029
58. Ely, Minn. + .008 | — .006 | — .008 | — .018 | — .021 | — .031 | — .029 | + .021 | + .023 | + .024 | + .019
59. Pembina, N. Dak. | — .009 | — .004 | — .004 | — .o11 | —~.012 | — .023 | — .025 + .017 |} + .017 | 4+ .018 | + .019
60. Mitchell, S. Dak. — .006 | — .006 | — .007 | — .016 | — .019 | — .033 | — .037 | — .001 -000 | + .002 | + .003
61. Sweetwater, Tex. + .009 | — .O11 | — .012 | — .028 | — .029 | — .049 | — .049 | — .031 | — .030 | — .030 | — .031
62. Kerrville, Tex. + .013 — .009 | — .010 | — .024 — .025 — .038 — .0382 | + .029 + .030 + .030 + .023
63. El Paso, Tex. + .001 — .020 | — .021 — .054 — .055 — .098 — .104 + .005 + .006 + .006 + .011
64. Nogales, Ariz. + .038 | — .020 | — .020 | — .046 | — .041 | — .076 | — .069 | — .052 | — .052 | — .057 | — .059
65. Yuma, Ariz. — -010 | — .001 | — .001 | — .004 | — .006 | — .012 | — .018 | + .007 | + .007 | + .009 | + .013
66. oes Cal. - 000 +000 | — .00L | — .002 | — .004 | — .011 | — .024 | — .052 | — .051 | — .050 | — .039
67. Goldfield, Nev. + .027 -030 | — .030 | — .077 | — .078 | — .137 | — .141 | — .015 | — .015 | — .014 | — .O11
68. Yavapai, Ariz. + .034 - 030 — .030 — .080 — .080 — .137 — .129 — .001 — .001 — .001 — .009
69. Grand Canyon,
Ariz. — .096 | — .028 | — .029 | — .079 | — .080 | — .136 | — .127 | — .o12 | —.o11 | — 011 | — .02
70. pallup N. Mex. + .014 | — .036 | — .036 | — .095 | — .095 | — .163 | — .156 | — .015 | — .915 | — .015 | — :022
71. Las Vegas, N. Mex.; + .017 | — .036 | — .035 | — .094 | — .094 | — 160 | — .150 + .001 -000 | + .001 | — .009
72, Shamrock, Tex. + .007 — .013 — .012 | — .031 — .031 — .055 — .056 + .030 + .029 + .030 + .081
73. Denison, Tex. _ — .001 | — .004 | — .004 | — 010 | — .009 | — .018 | — .017 | + .003 | + .003 + .002 | + .002
74, Minneapolis, Minn. | — .005 | — .004 | — .005 | — .012 | — .013 | — .022 | — . 024 + .057 | + .058 | + .058 | + .059
75, Lead, 8. Dak. + .044 | — .026 | — .027 | — .064 | — .061 | — .102 | — .089 | + .050 | + .051 | + .047 + .037
76. Bismarck, N. Dak. | — .005 — .008 — .009 — .024 — .026 — .044 — .047 - 000 + .001 + .002 | + .003
77. Hinsdale, Mont. — .017 | — .010 | — .012 | — .030 | — .034 | — .058 | — .067 | + .027 | + .029 + .0381 | + .036
78. Sandpoint, Idaho — .044 | — .014 | — .014 | — .045 | — .049 | — .086 | — .095 - 000 -000 | + .004 | + .009
79. Boise, Idaho — .042 | — .016 | — .018 | — .047 | — .051 | — .094 | — .108 | + .006 | + .008 + .010 | + .020
80. Astoria, Oreg. + .008 000 000 | — .002 | — .005 000 | + .008 | — .015 | — .015 | — .012 | — .023
81. Sisson, Cal. + .015 | — .022 | — .026 | — .058 | — .059 | — .096 | — .0s8 | — .012 | ~-008 | — lon | — - 020
82. Rock Springs, Wyo.| — .001 | — .036 | — .034 | — .093 | — .093 | — 169 | — .177 + .011 | + .009 | + .011 | + .019
83. Paxton, Nebr. + .002 014 | — .016 | — .041 — -043 | — .073 | — .077 | — .008 | — .006 | — .006 | — .004
85. North Hero, Vt. — .009 000 | — .001 | — .003 | — .007 | — .012 | — .016 | — .001 -000 | + .003 | + .003
86. Lake Placid, N. Y. | + .032 | — .011 | — .012 | — .024 | — .021 | — .033 | — .020 + .004 | + .005 ! + .001 | — .009
87. Potsdam, N. Y. — .004 | — .002 | — .003 | — .008 | — .010 | — .017 | — .017 | 4 -o19 + .020 | + .021 | + .019
88. Wilson, N. Y — .002 -000 | — .002 | — .003 | — .004 | — .011 — .017 | — .012 | — .010 | — .011 | — .008
89. Alpena, Mich. -000 | — .004 | — .003 | — .010 | — .008 | — .016 | — .016 | — .022 | — 2093 | — -024 | — .022
Mean with regard
to sign - 002 < z
Mean without re- af + .003 | + .003 | + .001
gard to sign -017 017 -017 -019
Stations not in United
States
15. Gormarsrat, Switz- ‘ee sis i
erlan +. — .04 — .04 — .099 — .081 — .140 — .0938 |24. @
16. St, Maurice, Swite- | : se eee | ae ake
erlan . = -. — .024 | — .064 | — .069 | — .103 | — .086 e - 005 _
6 Honolua, Hawai ne - ne + .003 | + .005 | + .008 -014
ian Islands +. -. -. + .011 }/ + 019 | + .072 137 - 0! . =
9. Menne es ae = PME oe Oe ot Set ats
ian Islands + -469 | — .089 | — 036 | — .070 | — .036 | -- .020 | + .108 | + .183 | + .170 | + 149 | + .055
Mean with regard
to sign 5
Mean without re- ; + ora + .069 | + .058 | + .007
gard to sign .072 - 069 - 058 021
1 See p. 74. 2 See p. 81.
the means, without regard to sign, for the anomalies of the same stations are 0.017 dyne, 0.017
dyne, and 0.019 dyne, respectively, for the three cases of areas limited by zones K, M, and O.
The mean anomaly, without regard to sign, for these 41 stations in the United States is
practically the same for the two methods of distribution of compensation.
regard to sign, for the regional compensation only, with zones
the same as for the local compensation—that is, 0.017 d
sign, for the regional compensation is 0.019 dyne for zone O.
The mean, without
K and M limiting the area, is
yne—while the mean, without regard to
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 101
The means, without regard to sign, of the anomalies for the six stations, Nos. 54, 62, 65,
66, 80, 81, on or near the coast, are as follows: Local compensation, 0.023 dyne; regional com-
pensation to zones K, M, and O, 0.023, 0.023, and 0.028 dyne, respectively.
The means, without regard to sign, of the anomalies for the 14 stations, Nos. 57, 58, 59, 60,
61, 72, 73, 74, 76, 77, 83, 87, 88, 89, which are in the interior of the United States and not in
mountainous regions, are: Local compensation, 0.020 dyne; regional compensation to zones K,
M, and O, 0.020, 0.021, and 0.020 dyne, respectively.
The means, without regard to sign, of the anomalies for the 21 stations, Nos. 42, 43, 44,
45, 46, 48, 49, 55, 63, 64, 67, 68, 69, 70, 71, 75, 78, 79, 82, 85, 86, in the above table, which are
in the mountainous regions, are: Local compensation, 0.013 dyne; regional compensation to
zones K, M, and O, 0.013, 0.014, and 0.017 dyne, respectively.
The means for the stations in the interior not in mountainous regions show that there are
no differences of importance in the four mean anomalies. This is what one would expect with
no prominent topographic features near a station, the effect of the compensation being prac-
tically the same whether the compensation is local or distributed uniformly over an area of
greater extent.
The results for the stations at or near the coast and those in mountainous regions show
that the mean, without regard to sign, is practically the same for the method of local distribu-
tion and for regional distribution with zones K and M limiting the area considered. The mean
anomaly for the method of regional distribution, with zone O limiting the area in the case of
stations on or near the coast, is 22 per cent larger than the anomaly of the method of local com-
pensation. The mean anomaly for the mountain stations in the case of regional distribution
to zone O is 31 per cent greater than the anomaly for the local compensation.
If the separate anomalies in the United States be compared, it is found that in 16 cases
out of 41 the anomaly with local compensation assumed is smaller than with regional compen-
sation assumed uniformly distributed to zone K (18.8 kilometers), and only 13 cases in which
it is larger. Similarly, there are 20 cases out of 41 in which the anomaly with local compensa-
tion is smaller than with regional compensation extending to zone M (58.8 kilometers), and only
15 cases in which it is larger. There are 26 cases out of 41 in which the anomaly with local
compensation assumed is smaller than with regional compensation assumed to extend to zone
O (166.7 kilometers), and only 12 cases in which it is larger. In all other cases the two anomalies
compared are identical to the last decimal place used, the third.
The evidence either for or against local compensation in comparison with such regional
compensation distributed uniformly over these moderate distances is necessarily slight and
possibly inconclusive. For, as shown in the table, the difference between computed effects of
compensation in the two cases compared is very small upon an average. The whole evidence
is furnished by these very small differences, which are frequently less than the errors of obser-
vation and computation. As shown by the table, there is but one station among the 41—
namely, No. 43, Pikes Peak—at which the difference between the computed effect of local com-
pensation and the computed effect of regional compensation uniformly distributed to zone K
exceeds 0.004. Such a difference tends to become greater as the distance over which the
regional compensation is supposed to be uniformly distributed is increased, but columns 7 and 8
of the table show that even when the regional compensation 1s assumed to extend to zone O,
a distance of 166.7 kilometers from the station, there is only one station among the 41—namely,
station No. 54, San Francisco—at which the difference between the computed effect of local
compensation and the computed effect of regional compensation exceeds 0.017 dyne.
Nevertheless the evidence, slight as it necessarily is, indicates that the assumption of local
compensation is nearer the truth than the assumption of regional compensation uniformly
distributed to zone K (18.8 kilometers). The evidence is still stronger in the same direction
when the comparison is made between local compensation and regional compensation extending
uniformly to the greater distances, 58.8 and 1 66.7 kilometers, represented by zones M and O,
It is possible that the assumption of regional compensation only, extending uniformly to
some distance from the station less than 18.8 kilometers, may be nearer the truth than the
102 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
assumption of local compensation. But it is evident that it would be exceedingly difficult to
test this supposition effectively by gravity observations, for the evidence available would neces-
sarily consist in general of still smaller differences than the very small differences dealt with
above in connection with the comparison of local compensation and regional compensation
extending to zone K. It appears to the writers, therefore, that the large amount of labor
necessary to extend this investigation to the remaining 48 stations in the United States, or to
smaller assumed distances as limits for the assumed regional compensation, would not be justi-
fied at this time by the results, as the evidence secured would probably be inconclusive. At
some future time, when more evidence is available from additional gravity stations, an exten-
sion of the investigation may be advisable.
The evidence shown at the bottom of the table from four stations not in the United States
is conflicting and inconclusive. In this connection one should consider the peculiar conditions
at the two stations on the Hawaiian Islands. These are islands which are evidently of volcanic
origin and where the processes of vulcanism are still apparently active.
It is stated above, in substance, to be the belief of the writers that the evidence indicates,
though it does not prove, that the assumption of local compensation is nearer the truth than
the assumption of regional compensation only, distributed uniformly to a distance of 166.7 kilo-
meters, or 58.8 kilometers, or even to the small distance 18.8 kilometers from the station. It
is also admitted as a possibility that an assumption of regional compensation only, distributed
to some still smaller distance from the station, may be nearer the truth than the assumption
of local compensation. If the writers stopped their statement of the case here their real views
might be misunderstood. It is hoped, therefore, that the following quotations from page 11 of
this publication will prevent misunderstanding:
“The authors do not believe that any one of these assumptions upon which the computations are based is absolutely
accurate.’’
“It is especially improbable that the compensation is complete under each separate small area, under each hill
p D ,
each narrow valley, and each little depression in the sea bottom. It is exceedingly improbable, for example, that
as each ton of material is eroded from a land area, carried out of a river mouth, and deposited on the ocean bottom,
the corresponding changes of isostatic compensation occur at the same time under the eroded area and under the area
of deposition at just such a rate as to keep the compensation complete under each. The authors believe that the
assumptions upon which the computations are based are a close approximation to the truth.”
The following paragraph,* written before the investigation of this particular question by
means of gravity observations was commenced, expresses the belief of the writers of the
present publication:
“In the above statement that the separate topographic features of the continent are compensated, it is not intended
to assert that every minute topographic feature, such, for example, as a hill covering a single square mile, is separately
compensated. It is believed that the larger topographic features are compensated. It is an interesting and impor-
tant problem for future study to determine the maximum size, in the horizontal sense, which a topographic feature
may have and still not have beneath it an approximation to complete isostatic compensation. It is certain from the
results of this investigation that the continent as a whole is closely compensated and that areas as large as States are
also closely compensated. It is the writer’s belief that each area as large as one degree square is generally largely
compensated. The writer predicts that future investigations will show that the maximum horizontal extent which
a topographic feature may have and still escape compensation is between one square mile and one square, degree.
This prediction is based, in part, upon a consideration of the mechanics of the problem.”
It seems’ clear to the writers that if the area taken be sufficiently small immediately sur-
rounding a station, the assumption of regional compensation only, uniformly distributed over
this area will be nearer the truth than local compensation distributed strictly in accordance
with the elevations within an area. It appears, however, from the inconclusive evidence fur-
nished by the gravity observations that the radius of this area is probably less than 18.8 kilo-
meters, which radius is within the outer limit indicated in the preceding paragraph. It also
appears that the gravity observations will probably not yield conclusive evidence as to which
hypothesis is nearer the truth for still smaller areas since the differences between the effects
according to the two hypotheses applied to these very small areas are so minute as to be very
difficult to observe.
* From p. 169 of The Figure of the Earth and Isostasy from Measurements in the United States.
EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 103
TEST OF DEPTH OF COMPENSATION.
In this investigation, as stated on page 10, the isostatic compensation has been assumed.
to be complete and uniformly distributed to the depth of 113.7 kilometers. This was the most
probable value of the depth of compensation available at the time the investigation was com-
menced. This depth had been obtained from investigations based entirely upon observed
deflections of the vertical in the United States. Later portions of those investigations have
shown that the most probable value now available for the depth of compensation is 122 kilo-
meters.*
It is evidently desirable, before concluding the present investigation, to ascertain whether
it is possible to determine the depth of compensation from the gravity observations with as
great accuracy as it has already been determined from the observed deflections of the vertical,
and whether numerical corrections of importance would result from changing the assumed
depth. from 113.7 to 122 kilometers. Accordingly, the approximate test here reported upon
was made to settle these two questions.
For the assumed depth of compensation, 85.3 kilometers, the values of Ez were computed
for a few values of 6 (0 being the distance from the station expressed in angular measure) by
the methods and formule set forth on pages 23 and 24. Each of these values was compared with
the corresponding values, as shown on page 25, computed for the assumed depth of compensation,
113.7 kilometers. The comparisons indicated that the reduction in Ey caused by changing
the assumed depth from 113.7 to 85.3 kilometers, if expressed as a percentage, varied but little
from zone to zone among the numbered zones. Accordingly, a few computations only, made
it possible to construct the part of the table shown below which refers to numbered zones.
Similarly, the effect of compensation alone was computed for some of the lettered zones
on the assumption that the depth of compensation is 85.3. It appeared that the change of the
assumed depth from 113.7 to 85.3 reduced the computed effect of compensation by amounts
which, expressed as a percentage, were practically constant (at 33 per cent) from zones A to
zone F, and beyond that point changed in a regular manner, as shown in the first part of the
table printed below.
Percentage of change in compensation and in Ep when the assumed depth of compensation is
: dane from 118.7 to 85.3 kilometers.
Zone i ag Zone E,
A +33 18 —-17
B +33 17 —18
Cc +33 16 —-19
D +33 15 —21
E +33 14 —22
F +33 13 —23
G +32 12 —24
H +32 11 —24
I +31 10 —24
J +29 9 —24
K 427 8 —25
G 123 7 —25
M +14 6 —25
N +03 5 —25
O -ll 4 —25
3 —25
2 —25
1 —25
By use of this table the changes shown in the following table for 10 stations in the United
States and 1 in the Hawaiian Islands were computed. In making the special investigations
stated under the heading, “Regional versus local distribution of compensation,” the effect of
* Supplementary Investigation of the Figure of the Earth, p. 77.
104 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY.
compensation alone for each lettered zone had already been computed for certain stations,
including the 11 used in the present test. Hence, for these zones the required change, as shown
below, was obtained at once by multiplying the effect of compensation for a given zone by the
percentage shown in the preceding table for that zone. For each numbered zone at a station
the total correction for that zone, as shown in the tables on pages 54-58 and 84, was multiplied
by the percentage of reduction in Ez for the zone, as shown in the above table, the total
correction for the zone being sensibly proportional to Ep.
Changes in computed correction for topography and compensation produced by changing the
assumed depth of compensation from 113.7 to 85.3 kilometers.
[All tabular values are in units of the fourth decimal place in dynes.]
é i Mauna ount |Salt Lake} Lake Tron
Zone | Fembina, Caren, | ‘Peak himseken, Be eek ee Junction, [Hamilton,| City, | Placid, | River,
: No. 69 | No. 43 | No. 54 Tehanas.. o: 0. No. 55 | No. 49 | No.86 | No. 57
A 0 0 0 0 0 0 0 0 0 0 0
B 0 0 — 2 0 -—1 0 0 0 0 0 0
Cc 0 0 — 2 0 — 3 -— 1 - 1 0 —1 0 0
D 0 —2 -— 5 0 — 6 — 2 — 2 —2 — 2 0 0
E 0 — 3 — 8 0 — 8 — 5 — 3 -3 -— 3 —1 0
F 0 — 3 —12 0 —13 —7 — 3 —3 — 3 — 2 0
G 0 — 5 —13 0 —16 — 8 — 5 —4 — 5 —4 0
H 0 — 8 —18 0 —18 —10 -—7 —5 —7 —2 0
I 0 —15 —28 0 —30 —19 -13 —6 —13 — 6 — 6
J =p —18 —28 0 —23 —23 -15 —6 -17 —7 —5
K ay —29 —38 0 —30 —32 —26 —8 —25 -11 —10.
L —5 —34 —42 —-2 —26 —39 —36 —4 —32 —12 —10
M -—7 —50 —60 3 -—15 —58 —56 —3 —49 —10 —12
N 0 0 -1 0 0 — 1 -—i] 0 -—i1 0 0
oO +7 +29 +40 —il —37 +35 +38 0 +33 +5 + 6
18 +2 +10 +12 -—4 -—12 +11 +13 0 +11 + 2 +1
17 +2 +10 +12 —4 —14 +11 +14 0 +12 + 2 +1
16 +2 +9 +13 —4 —16 +12 +14 0 +12 + 2 + 1
15 +3 +10 +13 -—4 -17 +14 +14 0 +14 +2. +1
14 +3 +11 +13 —4 —-18 +14 +14 —2 +14 “+ 2 + 2
13 +6 +19 +19 — 6 —36 +21 +23 —5 +25 + 3 + 4
12 +3 +13 +12 — 6 —24 +11 +13 —5 +15 +1 + 3
11 +3 + 6 +7 —5 —20 + 8 + 8 —5 +9 -— 1 + 2
10 +3 +1 + 4 - 3 -—15 +2 + 4 —4 + 3 — 3 + 2
9 +2 — i] 0 — 2 -—9 -—1 -—1 —2 -— 1 —2 0
8 +1 — 3 — 2 —4 —10 — 3 — 3 —4 — 3 — 3 0
7 -l1 — 2 — 2 —2 — 5 — 2 — 2 —2 — 2 —2 -i1
6 —2 — 2 — 2 — 2 -— 5 — 2 —2 -2 — 2 — 2 —2
5 -3 — 2 —2 — 2 —2 —2 — 2 —2 — 2 — 2 —2
4 —2 — 2 — 2 — 2 -1 — 2 —2 —2 — 2 — 2 —il
3 —-l1 — 2 -1 -1 -i1 -— 1 —]) —-1 -1 -1 - 1
2 —l -—i1 -1 -1 -—1 -i1 -1 —1 -— 1 -—1 -i1
1 0 0 0 0 0 0 0 0 0 0 0
Total.| -5 —§4 | —124 | —72 | —432 | =80 | —a7 | 2s poe a 4 \
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O.H.Tittmann, Superintendent %
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ILLUSTRATION No. 13.—MAp SHOWING LOCATION OF GRAVITY STATIONS USED IN THE INVESTIGATION.
PRINTED BY THE U.S.GEGLOGICAL SURVEY
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ILLUSTRATION No. 16.—LINES OF EQUAL ANOMALY FOR NEW METHOD OF REDUCTION.
PRINTED BY THE U.S.GEOLOGICAL SURVEY
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ILLUSTRATION No. 17.—LINES OF EQUAL ANOMALY FOR BOUGUER METHOD OF REDUCTION.
PRINTED BY THE U.S.GEOLOGICAL SURVEY
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ILLUSTRATION No. 19.—ILLUSTRATION FROM SUPPLEMENTARY INVESTIGATION IN 1909 OF THE FIGURE OF THE EARTH AND ISOSTASY, SHOWING RESIDUALS OF SOLUTION H, ALL STATIONS, WITH AREAS OF EXCESSIVE AND DEFECTIVE DENSITY, AND SHOWING ALSO ALL GRAVITY STATIONS WITH NEW-METHOD ANOMALIES.
Cornell University Library
QB 331.U58
TTT
1924 006 300 820 oi aincves
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