untitled Near-infrared digital photography to estimate snow correlation length for microwave emission modeling Ally Mounirou Toure,1,* Kalifa Goïta,1 Alain Royer,1 Christian Mätzler,2 and Martin Schneebeli3 1CARTEL, Département de Géomatique Appliquée, Université de Sherbrooke, 2500 Blvd Université, Sherbrooke, Québec J1K 2R1, Canada 2Institute of Applied Physics (IAP), University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland 3WSL Institute for Snow and Avalanche Research (SLF), Flüelastrasse 11, 7260 Davos-Dorf, Switzerland *Corresponding author: ally.toure@usherbrooke.ca Received 9 May 2008; revised 3 November 2008; accepted 22 October 2008; posted 14 November 2008 (Doc. ID 95774); published 12 December 2008 The study is based on experimental work conducted in alpine snow. We made microwave radiometric and near-infrared reflectance measurements of snow slabs under different experimental conditions. We used an empirical relation to link near-infrared reflectance of snow to the specific surface area (SSA), and converted the SSA into the correlation length. From the measurements of snow radiances at 21 and 35 GHz, we derived the microwave scattering coefficient by inverting two coupled radiative transfer models (the sandwich and six-flux model). The correlation lengths found are in the same range as those determined in the literature using cold laboratory work. The technique shows great potential in the determination of the snow correlation length under field conditions. © 2008 Optical Society of America OCIS codes: 280.0280, 280.4991, 120.5630, 120.5700. 1. Introduction There are various radiative transfer models designed to simulate the snow emission in the microwave range, e.g., the Helsinki University of Technology snow emission model (HUT, [1]), the microwave emission model of layered snowpacks (MEMLS, [2]), the dense medium radiative transfer model (DMRT, [3]), and the strong fluctuation theory (SFT, [4,5]). HUT and DMRT on the one hand use the snow grain size as a structure parameter; on the other hand, MEMLS and SFT use the correlation length to quantify the snow microwave scattering. For practical purposes, snow hydrologists have defined snow grain size as the greatest diameter of prevailing grains in the snow layer [6,7]. However, it has been shown that using the maximum extent of characteristic grains can lead to a strong overesti- mation of the effective grain size applicable in scat- tering models [8]. This optical grain size corresponds to the diameter of noncontacting spheres with the same surface area and the same ice volume as the snowpack under consideration, and thus with the same specific surface area (SSA) [9]. For example, the effective size of long needles is quite small, be- cause the value is close to the diameter of the nee- dles; for disks of large diameter, the optical grain size could also be small, depending on their thickness [10]. Stellar snow crystals can have a maximum ex- tent of 1 cm, but their thickness is only 20 to 40 μm. The optical grain size and the correlation length are the appropriate parameters to describe electromag- netic wave scattering [8]. The correlation length can be understood as a measure of the average 0003-6935/08/366723-11$15.00/0 © 2008 Optical Society of America 20 December 2008 / Vol. 47, No. 36 / APPLIED OPTICS 6723 s o u r c e : h t t p s : / / d o i . o r g / 1 0 . 7 8 9 2 / b o r i s . 3 7 6 2 4 | d o w n l o a d e d : 1 6 . 5 . 2 0 1 6 distance beyond which variations of the dielectric constant in one region of space become uncorrelated with those in another region [11]. At a distance greater than the correlation length, the values can be considered random. The correlation length pc is proportional to the optical grain size Do, and both are inversely proportional to SSA [12]. Also in the computation of the microwave scatter- ing of a granular medium such as snow, the physi- cally meaningful structure parameters are pc and Do [8,10,13,14]. One way to determine these quanti- ties is through the measurement of SSA. Until re- cently, there was no easy way for a practical determination of the SSA under field conditions. Because of that, there have been some attempts to establish empirical relationships between the correlation length and the snow grain size. The ap- proaches are approximations based on observations made at one site. The empirical formulas cannot be used in all circumstances. The concept of using the scattering of light to determine SSA and thus Do was first speculated by Giddings and Lachapelle [15] and Warren and Wiscombe [16] found first con- firmation by using the Mie theory to describe the op- tical properties of snow in the solar spectrum. The dependence of snow reflectance on grain size was used in optical remote sensing to map grain size in the surface snow layer [17,18]. The fact that SSA can be directly converted to optical grain size was shown by Mitchell [19]. The traditional method of de- termining SSA involves cold-laboratory work, which is time consuming and laborious [20,21]. Therefore, an important model-input parameter has so far been unavailable in the field. Advances in digital camera technology have made it possible to measure the snow reflectance using a simple digital camera fitted with an appropriate fil- ter. An exponential relationship was found in [22] be- tween the SSA and the reflectance of snow at the near-infrared (NIR) spectrum. Matzl and Schneebeli used the relationship to determine the SSA of snow profile with an uncertainty of less than 15%. A coupled contact illumination probe and field spectro- radiometer was used by Painter et al. [23] to deter- mine the optical grain size under field conditions with greater precision than the NIR technique. But the NIR technique has much higher spatial re- solution imagery of SSA (a 1 mm thick layer can be clearly identified) than the spectrometer measure- ments technique (2 cm spatial resolution). This char- acteristic makes the NIR method interesting because it can also be used to accurately determine the snow stratigraphy, and locate ice lenses, hard crust, and thin layers in the profile [22]. All these parameters greatly influence the snow microwave emission. Furthermore, NIR photography is simpler and more convenient for field work. The equipment is basic; only a digital camera and the Spectralon for calibra- tion are needed. No source of artificial light is needed. In this study, we considered the NIR technique for determination of the snow SSA and its correlation length. To check whether the NIR reflectance can be used to determine the correlation length for use in snow microwave emission modeling, we carried out experimental work at the snow station of the Swiss Federal Institute for Avalanche Research at Weissfluhjoch, Davos. We performed two types of experiment: 1. The first was to measure the snow slab bright- ness temperature at 21 and 35 GHz in different experimental conditions, and from that we retrieved the absorption and scattering coefficients through the inversion of the six-flux radiative transfer equations [24]. 2. The second experiment was to determine the SSA of the snow sample using NIR digital photogra- phy, and from that information, to compute the cor- relation length of the samples. 2. Experiments A. Geographical Location of the Site The site of the measurements is Weissfluhjoch, Davos, Switzerland, in the Dorftälli, a locally flat area at 2540 m above sea level. This is the location of the snow station for the Swiss Federal Institute for Snow and Avalanche Research. At this station, regular weather and snow parameter measurements are made. The measurements were conducted from 12 to 22 April 2006, i.e., just before the spring melt began. B. Material We used two portable, linearly polarized radiometers operating at 21 and 35 GHz (Table 1) built at the In- stitute of Applied Physics, University of Bern. A blackbody box kept at ambient temperature was used to calibrate the radiometers. An aluminum plate was used to serve as the perfect reflector, a 5 cm thick blackbody plate was used to serve as the perfect absorber of microwave radiations, and a 3 cm thick styrofoam plate was used to thermally insulate and to help support the slab. The output vol- tage of the radiometers was recorded by a data log- ger. The voltage is linearly related to the brightness temperature (Tb). The digital camera used for the NIR photography was a Sony Cybershot DSC- P200. The charge-coupled device (CCD) used in the camera is sensitive to NIR light. A filter was placed Table 1. Properties of the Portable Dicke Radiometers Properties Radiometer 1 Radiometer 2 Center frequency f ½GHz� 21 35 Bandwidth Δf ½GHz� 0.8 0.8 Integration time t ½S� 1–8 1–8 Sensibility [K] 0.1 0.1 3 dB beam width φ 9° 9° Horn antenna Pyramidal Pyramidal 6724 APPLIED OPTICS / Vol. 47, No. 36 / 20 December 2008 over the CCDs to block the visible light. The wave- length of the detected light is in the range of 840 and 940 nm. The photos were taken under diffuse light with the camera placed on a tripod at approxi- mately 1 m from the sample. When there was direct radiation, a white curtain was used to create a diffuse light. C. Tb Measurement Procedures The principal setup of the experiment is shown in Fig. 1. The radiometers were mounted on a frame that allows easy variation of incidence angle and change of polarization. The snow sample is placed on the styrofoam plate on a metal table. Between the snow and the sample, an absorber or metal plate was inserted. The samples were placed near the radiometer antennas to ensure that all the measured radiation came from the sample surface and volume. A box at the rear contains the device for adjusting the voltage of the power supply, as well as a car battery as a spare power supply in case of failure of the cen- tral power generator of the snow station. The output voltages of the radiometers were recorded by a data- logger [polycorder (PC)]. a. Removal of the slab. The preparation of the snow samples was one of the most delicate tasks of the measurement procedure. This method requires homogeneous samples of dry snow to study their mi- crowave properties as a function of their structure. To measure the brightness temperature (Tb) of the snow, we first had to mechanically remove 70 × 60 cm blocks of snow with thicknesses varying between 9 and 20 cm. Snow was excavated at a pre- viously undisturbed site to ensure that the layers were as homogeneous as possible. The snowpack was examined to find layers of sufficient thickness. Once a layer was chosen, a sample was prepared by cutting the layer with metal plates, taking great care to keep the slab undisturbed. The sample was then placed on a 3 cm thick styrofoam plate to help support the sample. Tests done by Wiesmann et al. [24] showed that the transmissivity of the styrofoam is better than 0.99 in all conditions for the frequency range used here. The styrofoam had the advantage of thermally insulating the samples from the various underlying bases during handling and measure- ments. In the case of new snow, the handling was very difficult, even on the styrofoam plate, because those samples were likely to break apart or to show fissures, requiring the preparation of new samples. The surface of the samples was then smoothed by a straight metal edge, a process that had to be per- formed very accurately. The slabs were then placed on a platform in front of the radiometers for measure- ments (Fig. 1). Thirty-three samples were used for the experimental measurements. The radiometric measurements were taken immediately after the sample was extracted. After the radiometric mea- surements were finished, the snow temperature was determined at different points near the center and the edges of the samples. The maximum tem- perature difference did not exceed 2 K. The snow density was also measured. The density ranged from 69 to 400 kg=m3. b. Measurements. The brightness temperatures of the snow slabs were measured in two different si- tuations to determine the reflectivity and the trans- missivity of the samples: (a) the snow slab on the aluminum plate at 45° angle of incidence, at both ver- tical and horizontal polarization; (b) the snow slab on the absorber at 45° angle of incidence, at both hori- zontal and vertical polarization. For calibration pur- poses, the sky brightness temperature and that of the blackbody box at ambient temperature were also measured. The depth at which the sample was taken was also recorded. D. Near-Infrared Photography Before taking the NIR photos, the side of the slab was carefully smoothed with a blade. The calibration tar- gets were then placed on the sample (Fig. 2). The tar- gets were manufactured from Spectralon standards with NIR reflectances of 50% and 99%. The distance between the camera and the profile wall was approxi- mately 1 m. To compensate for tilting or turning of the camera with respect to the surface of the slab, a geometric correction of the digital image was needed. The correction required at least three tar- gets to be inserted on the profile with a known dis- tance from each other and known geometry (Fig. 2). The nearest neighbor method was used to transform the target coordinates in the digital image congru- ently with their original geometry and distance [22]. The NIR reflectance (nir) of the snow was calibrated with regard to the pixel intensity of the targets: nir ¼ a þ bi; ð1Þ Fig. 1. (Color online) Experimental setup: the radiometer is mounted on a frame directed at the snow sample (9 to 20 cm) which is placed on a 3 cm thick styrofoam plate on a metal table. Between the snow and the sample an absorber or metal plate was inserted. The polycorder (PC) and the battery box are also shown. 20 December 2008 / Vol. 47, No. 36 / APPLIED OPTICS 6725 where i is the intensity of each pixel and a and b are determined by a linear regression on the pixel intensities of gray-scale standards. E. Correlation of Reflectance and SSA It was found by Matzl and Schneebeli [22] that the correlation between the NIR reflectance of the snow samples calculated from the calibrated digital image and their SSA is about 90%. The spatial resolution of the imagery is very high. In images covering between 0.5 and 1 m2, even layers of 1 mm thickness can be documented and measured (Fig. 3). The error in- creases with increasing SSA from 3% for SSA below 5 mm−1 to about 15% for SSA values above 25 mm−1: SSA ¼ A⋅enir=b; ð2Þ where nir is in %, A ¼ 0:017 � 0:009 mm−1, and b ¼ 12:222 � 0:842 [22]. 3. Radiometric Properties of the Snow Slab The aim of the radiometric property studies is to un- derstand how various Tb are related to the emissiv- ity (e), reflectivity (r) and transmissivity (t) of a snow slab, and eventually to the snow scattering and absorption coefficients. To connect the measured Tb to the snow internal scattering and the reflections at the interfaces, we used the simple sandwich model based on the radia- tive transfer proposed and developed by Wiesmann et al. [24]. We made two types of measurement: a. Brightness temperature of snow on metal plate (Tbmet). Figure 4(a) presents the situation of a snow slab with thickness d on a metal plate. The upwelling brightness temperature Tbmet is composed of the emitted snow radiation, caused by the physical snow temperature Tsnow of the sky radiation Tbsky that is reflected from the slab (total reflectivity rmet), and of Tbsky that is transmitted through the snow slab and reflected by the metal plate. The up- welling brightness temperature of the snow on the metal plate Tbmet is Tbmet ¼ ð1 − rmetÞTsnow þ rmetTsky; ð3Þ where rmet is the total reflectivity of the snow on the metal plate, which comprises the internal reflectivity of the slab and that of the interface and can be expressed as follows: rmet ¼ ri þ ð1 þ riÞ2Rmet; ð4Þ where Rmet is a function of internal transmissivity (t) and reflectivity (r) and of the interface reflectivity (ri): Rmet ¼ r þ t2ð1 − rÞ−1 1 − rri − rit 2ð1 − rÞ−1 : ð5Þ Fig. 2. (Color online) Preparation of the snow slab for the NIR photography. Four gray-white targets are placed on a smoothed snow side. Fig. 3. (Color online) NIR photo of a 13 cm thick inhomogeneous slab laying on a blackbody and styrofoam slabs: the top layer is newly fallen snow with high NIR (SSA ¼ 38:5=mm), the middle layer is refrozen snow with very low NIR (SSA ¼ 7:7=mm), and the bottom layer consists of rounded snow (SSA ¼ 17:5=mm). 6726 APPLIED OPTICS / Vol. 47, No. 36 / 20 December 2008 b. Brightness temperature of snow on absorber (Tbabs). Similar to the case of the snow on metal plate, the upwelling brightness temperature of the snow on absorber Tbabs is [Fig. 4(b)] Tbabs ¼ ð1 − rabsÞTsnow þ rabsTsky; ð6Þ where it is assumed that the snow and absorber temperature are the same and that rabs is the total reflectivity of the snow on the absorber, including the internal reflectivity of the slab (r) and that of the interface air/snow (ri): rabs ¼ ri þ ð1 − riÞ2Rabs; ð7Þ where Rabs similarly to Rmet is expressed as follows: Rabs ¼ r þ rit2ð1 − rriÞ−1 1 − rri − ðritÞ2ð1 − rriÞ−1 : ð8Þ The internal reflectivities (r) and transmissivities (t) can be computed from the measured Tbmet and Tbabs if ri (air–snow interface) is known. If the snow inter- face is considered smooth, ri can be computed using the Fresnel reflectivity formula at the given inci- dence angle, polarization, and dielectric slab permit- tivity ε. To obtain the values of r and t, we just need to solve the system of Eqs. (4), (5), (7), and (8). To link the slab internal reflectivity r and trans- missivity t of our samples on the one hand, to the in- ternal scattering and absorption coefficients on the other hand, we used the six-flux model proposed by Wiesmann et al. [24]. The six-flux model is a sim- plified radiative transfer model that helps to account for all the radiation that is propagating in the snow slab. It is a model that reduces the radiation at a given polarization to six-flux streams along and op- posed to the three principal axes of the slab. The hor- izontal fluxes represent the trapped radiation whose internal incidence angle θ is greater than the critical angle for total reflection θc: θ > θc ¼ arcsin ffiffiffiffiffiffiffiffi 1=ε p ; ð9Þ where ε is the relative permittivity of the slab. The vertical fluxes represent the radiation which does not undergo total reflection; this is the radiation that interacts with the space above the snowpack. In the case of plane-parallel and isotropic slabs, the six- flux model is reduced to the well-known two-flux model, and the two-flux absorption coefficient γ0a and scattering coefficient γ0b can be written in terms of six-flux parameters: γ0a ¼ γað1 þ 4γcðγa þ 2γcÞ−1Þ; ð10Þ γ0b ¼ γb þ 4γ2c ðγa þ 2γcÞ−1; ð11Þ where γa is the six-flux absorption coefficient, γb is the six-flux backscattering coefficient, and γc is the corner scattering coefficient (the coefficient for cou- pling between the vertical and the horizontal fluxes). Wiesmann et al. [24] also established that, for a snow layer of thickness d, the internal reflectivity (r) and transmissivity (t) can be expressed as r ¼ r0ð1 − t20Þð1 − r20t20Þ−1; ð12Þ t ¼ t0ð1 − r20Þð1 − r20t20Þ−1; ð13Þ where the one-way transmissivity t0 of the slab is t0 ¼ expð−γd=ðcos θÞÞ; ð14Þ and the reflectivities r0 of infinite slab thickness is r0 ¼ γb0ðγa0 þ γb0 þ γÞ−1 ð15Þ and a function of the two-flux absorption, scattering coefficients (γ0a and γ0b), and the damping coefficient γ, which can also be expressed as a function of γ0a and γ0b: γ ¼ ðγa0ðγa0 þ 2γb0ÞÞ1=2: ð16Þ Fig. 4. (Color online) Principles of snow sample measurements: (a) brightness temperature Tbmet of snow on metal plate and (b) brightness temperature Tbabs of snow on absorber. Correspond- ing values of blackbody radiation Tbbb were also measured. 20 December 2008 / Vol. 47, No. 36 / APPLIED OPTICS 6727 For an isotropic scattering, the total scattering coef- ficient γs is given by the sum γs ¼ 2γb þ 4γc; ð17Þ and the ratio 2γc=γb can be expressed as follows: 2γc=γb ¼ x=ð1 − xÞ; ð18Þ where x is x ¼ ððε − 1Þ=εÞ1=2: ð19Þ From Eqs. (10)–(19) we have the full picture of how the internal reflectivity r and transmissivity t are linked to the six-flux scattering coefficients γb, γc, and γs. We solved the system of Eqs. (10)–(19) to de- rive the scattering coefficients. 4. Results and Discussion The scatterplot in Fig. 5 shows a trend for SSA to in- crease with decreasing density. The plot shows char- acteristic SSA-density clusters for newly fallen snow (density, ρ < 200 kg⋅m−3 shown in triangles), rounded grain (200 < ρ < 300 kg⋅m−3 shown in filled circles) and for compacted rounded grain snow (ρ > 300 kg⋅m−3 shown in diamonds). The same clus- ters were also found by Matzl and Schneebeli [22], and it is assumed that this loose relationship is due to the fact that metamorphism and sintering of- ten have an impact on both SSA and density at the same time. In fact, Kerbrat et al. [21] have shown that SSA is not correlated to snow density. The loose relation that seems to suggest a correlation between SSA and density is due to a side effect that, on aver- age, when the snowpack increases in density, its SSA decreases. That is not always the case. Rosenfeld and Grody [25] also demonstrated that from midseason, an internal restructuring of the snow occurs that can dramatically affect SSA even though the total mass of individual particles, as well as the total density of the snowpack, remain constant. The representation of the scattering coefficient at 21 and 35 GHz at vertical polarization versus SSA [Figs. 6(a) and 6(b)] shows an exponential decrease of the scattering coefficient with the snow specific surface. The correlation coefficients between the scattering coefficient at 21 and 35 GHz at vertical po- larization and the SSA are −0:57 and −0:64, respec- tively. New snow with high SSA has a low scattering coefficient, and settled snow with rounded grains has low SSA and a high scattering coefficient. Samples 15 and 19, taken at the bottom of the snow- pack and made up of coarse grains, have very low SSA and a very high scattering coefficient. The same trend can be seen at horizontal polarization [Figs. 6(c) and 6(d)]. The correlation coefficients be- tween the scattering coefficient at 21 and 35 GHz at horizontal polarization and the SSA are −0:64 and −0:77, respectively. Figure 7 is the representation of the scattering coefficient at 21 and 35 GHz at vertical polarization [Figs. 7(a) and 7(b)] and at 21 and 35 GHz at horizon- tal polarization [Figs. 7(c) and 7(d)] versus snow den- sity. As expected, there is no correlation between the scattering coefficients and the snow density. Denser samples are not always characterized by larger scat- tering coefficients. These results show that the scat- tering is mainly dependent on SSA. Foster et al. [26] also demonstrated that the SSA is so dominant in snow microwave scattering that the cumulative contribution of other structural features is over- whelmed. According to Mätzler [8] and Debye et al. [12], the relation between the correlation length (pc) and the SSA is as follows: pc ¼ 4 ð1 − vÞ SSA ; ð20Þ where v ¼ ρ=ρice is the volume fraction of the ice (ρ is the density of the snow and ρice ¼ 917 kg=m3 is the density of ice). We used the formula to convert the SSA derived from the NIR photography into the correlation length. The representations of the pc versus the scat- tering coefficient at 21 and 35 GHz at vertical polar- ization [Figs. 8(a) and 8(b)] show increasing scattering coefficient with pc. Coarse snow grains have high correlation lengths, which translate into high scattering coefficients. The same trend can also be seen for the scattering coefficients at 21 and 35 GHz at horizontal polarization [Figs. 8(c) and 8(d)]. Similar results were found by Wiesmann et al. [24] on samples collected during the winters of 1994/1995 and 1995/1996 using the same methodology to ex- tract the scattering coefficient and cold laboratory measurements to determine the pc. The plots of Fig. 5. (Color online) Scatterplot of SSA and snow mean density. Three different types of snow are distinguished: snow with density ρ < 200 kg⋅m−3 (shown as triangles), density 200 < ρ < 300 kg⋅m−3 (shown as filled circles), and snow with density ρ > 300 kg⋅m−3 (shown as diamonds). 6728 APPLIED OPTICS / Vol. 47, No. 36 / 20 December 2008 correlation length versus scattering coefficients at vertical and horizontal polarization of the samples with densities above 200 kg⋅m−3 are fitted with a power law fit: γsp ¼ d2p:ðpcÞc3p: ð21Þ The fits are shown by the solid lines. The parameters d2p and c3p, (p represents the vertical or horizontal polarization) and their equivalent found by Wiesmann et al. [24] (d2wie, c2wie) are provided in Table 2. The pc we found are in the same range as those of Wiesmann et al. [24], but our scattering coef- ficients are generally higher than those found by them. The power law fits (Fig. 8) in [24] and are stee- per compared to those of our data. The first possible reason for these discrepancies between our data and that of [24], especially at 21 GHz and horizontal po- larization could be due to mechanical damage of the radiometer antenna at 21 GHz. The antenna tended to shake slightly when a change in polarization from vertical to horizontal was made. The discrepancies could also arise from the fact that there was a differ- ence between our experimental setup and that of [24]. They used a metal frame of the same size as the samples to hold together the snow slabs. Using the frame reduces the edge effects because the frame acts as a mirror for radiation hitting the edge, and thus simulates a larger sample. More importantly, most of our samples were made up of very hard den- sified snow and the values of pc are mostly concen- trated between 0.1 and 0:2 mm. The 2005/2006 winter season was exceptional. The snow depth was about 2 m, while the average snow depth usually recorded in the area is 1:5 m [27,28]. This means that the metamorphism yielded different types of snow grain. For snow density exceeding approximately 350 kg⋅m−3, which is the case for most of our samples, Fig. 6. (Color online) Representation of the scattering coefficient at (a) 21 GHz vertical polarization, (b) 35 GHz vertical polarization, (c) 21 GHz horizontal polarization, and (d) 35 GHz horizontal polarization versus the SSA. Samples with density ρ < 200 kg⋅m−3 are shown as triangles. Samples with density ρ > 200 kg⋅m−3 are shown as filled circles. The numbers on the right of the symbols indicate the sam- ple’s number. The solid curve is the exponential fit of the dense samples. 20 December 2008 / Vol. 47, No. 36 / APPLIED OPTICS 6729 pore spaces are very small (due to compaction) and the kinetic growth process is limited. “Hard” depth hoar develops [29,30] under these conditions. Hard depth hoar crystals are composed of sharp angular crystals, but are relatively smaller and stronger (due to higher degree of bonding) than classic depth hoar made up of larger faceted cup shaped grains with thinner walls (smaller pc). These larger hollow crystals tend to grow in low density snow with large pore spaces [29]. The hard depth hoar tends to have a higher pc than the classic one. The near-infrared reflectance depends on the opti- cal grain size, which is inversely related to SSA but is independent of snow density. As already mentioned at the beginning of this section, SSA and snow den- sity are not correlated. Snow is a random granular medium in which grains cannot be considered as in- dependent scatterers [10]. The snow emission model MEMLS is based on this principle. This is why it uses the correlation length, a parameter that takes into account both the SSA and the snow density [8,10,12]. The estimated correlation lengths from the NIR technique and measured snow parameters, including density, temperature, and sample thickness were used as input parameters in MEMLS to simulate the brightness temperatures at 21 and 35 GHz. Figure 9 shows the comparison between the mea- sured brightness temperature of a snow sample placed on a metal plate and snow emission model (MEMLS) predictions. Figures. 9(a) and 9(b) are the results for 21 GHz at vertical and horizontal polarizations. Figures 9(c) and 9(d) represent the re- sults for 35 GHz at vertical and hroizontal polariza- tions. The correlation coefficient ranges from 0.55 at 21 GHz to 0.75 at 35 GHz. There can be several rea- sons for the scatter, especially at 21 GHz, including the fact that volume scattering is stronger at 35 GHz than at 21 GHz. Furthermore, a systematic error was added to the measurements at 21 GHz be- cause, during the measurements, the 21 GHz radio- meter was less stable than the 35 GHz instrument. Fig. 7. (Color online) Representation of the scattering coefficient at (a) 21 GHz vertical polarization, (b) 35 GHz vertical polarization, (c) 21 GHz horizontal polarization, and (d) 35 GHz horizontal polarization versus snow density. The numbers on the right of the symbols indicate the sample’s number. 6730 APPLIED OPTICS / Vol. 47, No. 36 / 20 December 2008 5. Conclusion We used the relations established by Debye et al. [12] and by Matzl and Schneebeli [22], respectively, to link infrared reflectances of snow to the SSA, then the SSA to the correlation length, knowing the den- sity of snow. From the measurements of brightness temperatures at 21 and 35 GHz, we derived the scat- tering coefficient of the samples by inverting the sandwich and six-flux radiative transfer models. We showed that the scattering coefficient increases with the correlation length. The relationship can be defined by a power law. The pc we found are in the same range as those de- termined by Wiesmann et al. [24]. New snow has a pc range from 0.06 to 0:08 mm, settled densified snow is between 0.08 and 0:2 mm, and the depth hoar range is Fig. 8. (Color online) Double logarithmic representation of the scattering coefficients at (a) 21 GHz vertical polarization, (b) 35 GHz ver- tical polarization, (c) 21 GHz horizontal polarization, and (d) 35 GHz horizontal polarization versus the correlation length. Samples with a density ρ < 200 kg⋅m−3 are shown as triangles; samples with a density ρ > 200 kg⋅m−3 are shown as filled circles. The numbers on the right of the symbols indicate the sample’s number. The solid lines are the power law fit of the dense samples; the broken lines represent the power law fit found by Wiesmann et al. [24]. Table 2. Fit Parameters for the Six-Flux Scattering Coefficient Versus the Correlation Length and the Correlation Coefficientsa Frequency (GHz) Polarization d2, ½m−1� c3 R d2wie, ½m−1� c3wie Rwie 21 Vertical 2.967 0.975 0.83 68.3 2.95 058 Horizontal 2.514 0.342 0.44 – – – 35 Vertical 3.514 0.843 0.73 260.4 3.13 0.90 Horizontal 5.781 0.876 0.73 – – – ad2, c3, R) compared to those determined by [24] (d2wie, c3wie, Rwie). 20 December 2008 / Vol. 47, No. 36 / APPLIED OPTICS 6731 between 0.2 and 0:4 mm. This confirms the theoretical relationship [Eq. (20)] established by Debye et al. [12] thatconnectsthesnowcorrelationlengthtoitsspecific surface.Thetechniqueshowsgreatpotentialinthede- terminationofsnowcorrelationlengthunderfieldcon- ditions, even though it still needs some improvements (i.e., elements such as uniform illumination of the snow-pitwall).Theimprovementofthedetermination of the correlation length is critical in the solving of the many-to-one relationship between the snow water equivalent and the measured brightness temperature through the data assimilation scheme. This work was supported by the National Science and Engineering Research Council of Canada and by Environment Canada (CRYSYS Program). We are grateful to the Institute of Applied Physics (IAP), University of Bern, for the invaluable logistical sup- port they provided us with for the field work, and to the Centre de Calcul Scientifique (CCS) of Université de Sherbrooke and especially to HuiZhong Lu who has been useful in helping to optimize the inversion program. References 1. J. T. 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