Published online 16 November 2005
Published in Vadose Zone J 4:1152-1160 (2005)
DOI: 10.2136/vzj2004.0109
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Effect of Water Saturation on Radiative Transfer
Dominik Bänningera,*,
Peter Lehmanna,
Hannes Flühlera and
Jonas Tölkeb
a Soil Physic, Institute of Terrestrial Ecology ETH Zürich, Switzerland
b Institut für Computeranwendungen im Bauingenieurwesen, TU Braunschweig
* Corresponding author (baenni{at}env.ethz.ch)
Received 14 July 2004.
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ABSTRACT
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Porous media such as tissues, soil materials, porous rocks, and concrete become darker with increasing water content. Most methods used to quantify this behavior are based on statistical models. Physical models have rarely been used for this purpose. In this study, we implemented a radiative transfer model to analyze the measured reflectance and transmittance of prepared moist soil slabs, accounting for the physical processes of light reflection, refraction, and absorption. We calculated the trace of numerous beams in the image of vertical slab cross sections. This image represents the spatial distribution of the solid, water, and air phase in the soil slab. We implemented this model to test how the arrangement and optical properties of the water phase influences the reflectance. To validate the model assumptions, we qualitatively compared the calculated reflectance and transmittance with measurements. We concluded that reflectance and transmittance depend not only on the amount of water but also on the spatial distribution of water within the pore space of the porous medium.
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INTRODUCTION
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POROUS MEDIA are known to become darker when they are wetted. In many fields of research, such as precision farming, moisture control in greenhouses, arid land irrigation, and tracer mapping at soil surfaces, the relationship between soil reflectance and soil water content is of interest. In this study we investigated reflectance in the visible and near-infrared range of moist and structured surfaces. For visible and near-infrared light the penetration depth of light into soil is on the order of a few millimeters (Bänninger, 2004). This implies that soil reflectance is dominated by the uppermost soil layer. Due to an extremely steep gradient in water content at the soil–atmosphere interface (Hirasaki and Yang, 2002), the degree of saturation of this uppermost layer may differ from the average water saturation in the topsoil. Here we discuss the influence of near surface water content on radiative transfer as an index of the variable surface water content.
Several models have been proposed for describing the relationship between soil reflectance and soil water content. Most of these models are based on simple mathematical or statistical relations (Bowers and Hanks, 1965; Bowers and Smith, 1972; Muller and Decamps, 2000; Weidong et al., 2002, 2003). The relationship between light absorption and water saturation is most often described with a curvilinear function. Other models use the reflected intensity in the wavelength range of the water absorption bands to relate reflectance to water saturation (Harbert et al., 1974; Finney and Norris, 1978; Kano et al., 1985). An absorption band is a narrow wavelength range with significantly higher light absorption.
We studied the potential of the beam tracing model presented by Bänninger (2004) to describe radiative transfer in moist soil samples while taking into account the size of the particles and the arrangement of the fluid phases. This model describes mechanistically the physical processes of light scattering in porous samples. With respect to the variable water content, a realistic description of the water phase geometry is crucial. Hoa (1981) reported, for example, that light transmittance is a function of the size distribution of the water-filled pores. The beam tracing model was originally designed to calculate radiative transfer through dry soil samples, represented by cross-sectional images. Cross-sectional images show the spatial distribution of the solid phase and the pore space in a vertical cross section through the sample. To adapt the model for moist soil samples, we distribute the water and air phase within the pore space of the cross-sectional images and assign a respective refractive index to each of the phases. We first discuss different methods to calculate the water distribution in an arbitrary cross-sectional image. Next, we calculate the radiative transfer through the images representing moist soil samples. To test the validity of the model, results are compared with measured data. Finally, we discuss an application of the model in which the degree of water saturation of a soil sample is determined from the measured reflectance.
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THEORY
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The Radiative Transfer Model
An analysis of the reflectance of light from a wet soil requires an understanding of the influence of water on the light scattering processes. To model the effect of water on light scattering, we use a beam tracing model (Bänninger, 2004) designed to describe the light scattering process in complex scattering structures. Beam tracing is calculated in two-dimensional space. Reducing a three-dimensional medium to a two-dimensional cross section provides a first-order approximation. As shown in the study of Vokov and Remizovich (1998), this reduction in dimensions has a fairly small effect on the resulting reflectance and transmittance. In the two-dimensional form, the modeling task is less demanding and can be run on a desktop computer. The light scattering medium is then represented by a pixel image with a given width and depth, representing the vertical cross section through the particulate medium. Different gray values are assigned to the air, water, and solid phase. To model and measure the transmittance, the sample must be thick enough to avoid direct light rays between the source and detector, but thin enough to avoid complete absorption. Due to this restriction, the thickness of a soil slab can be varied only in a relatively narrow range because of the high absorption properties of soil material (Tidewell and Glass, 1994; Niemet and Selker, 2001; Niemet et al., 2002).
Reflectance and transmittance can be estimated by tracking the optical paths of many beams in the medium. The optical path of each beam starts at the light source. The light source is defined by the sample width L and the distance yshift from the sample (Fig. 1)
. The parameter 
defines whether the light is collimate (parallel), partially diffuse, or diffuse. The optical paths of light beams in the medium are determined by the processes of propagation, absorption, and scattering (Fig. 2) . These processes are calculated by solving the linear equation given by the position and propagation direction of the beam. Light absorption is calculated using Lambert-Beer's Law.
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Fig. 1. Definition of the light source used in the beam tracing model. L, width of the light source; , half opening angle of light source; yshift, distance between sample and light source; D, edge length of the sample.
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Fig. 2. Schematic of the beam tracing approach for a four particle medium. The incident beam I hits the surface of a solid particle and is forked into a reflected and transmitted beam. Thus, the number of light beams increases quickly. The letter s denotes the scattered light beams leaving the medium, and the letter a the absorbed light beams. The gray intensities of the lines characterize the number of preceding scattering events.
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Scattering occurs when a light beam hits an air–water, air–solid, or solid–water interface. Scattering (i.e., the directions and intensities of the reflected and transmitted beams after hitting an interface) may be calculated using Snell's Law and the Fresnel equations. These two laws consider only geometric optics, while other effects such as diffraction are neglected. The application of geometric optics is restricted to wavelengths that are shorter than the size of the particles. To solve Snell's Law and the Fresnel equation, one has to know the refractive indices of each phase. The refractive index 
is a complex number. Its real part defines the direction of the refracted beam, while the imaginary part describes the absorption of the light by the medium. Each material has a specific relationship between wavelength and the refractive index. Figures 3 and 4
depict the spectrum of the complex refractive indices of water and ice. The wavelength dependence of the refractive index is responsible for the existence of absorption bands.
Wet porous media frequently contain bound water (Bordillo and Marti, 2000), especially fine textured sedimentary rocks (Fischer et al., 1998) and rocks of magmatic origin (Schneebeli et al., 1995). This fact has to be accounted for because of the different molecular arrangement of the water molecules in bound water. Analogous to the refractive indices of water and ice, the optical properties of free and bound water are different. In our model, we use the refractive index of ice to describe bound water (Foster and Ewing, 1999). This assumption is based on findings of Bruni and Ricci (1998), who found that interfacial water undergoes a transition to a cubic ice-like metastable phase.
Calculating the radiative transfer of moist soil samples using the beam tracing model requires detailed information about the spatial arrangement of water in the pore space. We generated partially water-filled cross sections using four different methods to distribute the water and air phase in the pore space. For the first two methods, disc-shaped particles were distributed randomly to represent a cross section with a given porosity. In a first attempt we added a water film around each particle. This method is referred to as the "added-water-film method." An increasing film thickness simulates increasing water saturation. This method cannot realistically reproduce the curvature of the air–water interface. In a second approach, we simulated the water distribution using a pore network model. The saturated pore network is drained according to the size of the voids between the particles. The pore size classes are drained in decreasing order. This procedure corresponds to the insertion of air-filled discs of decreasing size into the water phase. The air–water interface curvatures of these discs correspond to the respective water potentials.
The added-water-film and the pore network models both neglect the continuity in the air and/or water phase. For the third and fourth methods we calculated the water distribution in a tomographed sand cube to obtain more realistic solid and water phase distributions. With the third method, we calculated the drainage of the cube using a three-dimensional pore network model. This approach is similar to the two-dimensional pore network model, except that drainage depends on the size as well as the continuity of the pores. The fourth method simulates the drainage of the cube with a Lattice–Boltzmann approach (Tölke et al., 2002; Lehmann, 2002). The water distribution at consecutive steps of the drainage yields cross-sectional images with the same arrangement of the solid phase at different degrees of water saturation.
Simulation Designs
The main objective of this study was to show how the water content, the distribution of water, and its optical properties influence the reflectance of wet soil. For this purpose we performed two sets of simulations. We also conducted a third set of simulations in a first attempt at solving the inverse problem, that is, estimating the water content from soil reflectance measurements.
In the first series of simulations, we focused on the influence of the water content on radiative transfer. We applied the four methods to generate cross sections of different water contents. The added-water-film method and the two-dimensional pore network model were used for cross sections containing randomly distributed disc-shaped particles with diameters of 127 µm. The size of the image was 6 by 1.8 mm. We illuminated the sample with diffuse light from a light source with a width of 3 mm. The wavelength was set to 
= 900 nm, the refractive index was w = 1.32 + i4.87 x 10–6 for water and s = 1.8 + i1 x 10–4 for soil. The three-dimensional pore network model was applied to a cube with dimensions of 0.7 by 0.7 by 0.63 mm. The sand-filled cube was tomographed with synchrotron X-ray at the Swiss Light Source at PSI, Switzerland, with a resulting voxel size of 3.5 µm. The diameter of the sand particles ranged from 100 to 200 µm. We used the Lattice–Boltzmann method to calculate the water and air dynamics in a cut-out (200 x 200 x 180 voxels) of the tomographed sand-filled cube. Figure 5
depicts the resulting cross-sectional images for each of the four methods.
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Fig. 5. Examples of the four methods used for arranging the water phase in the two-dimensional particulate structure: (a) added-water-film, (b) two-dimensional pore network, (c) three-dimensional pore network, (d) Lattice–Boltzmann.
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In the second series of simulations we analyzed the significance of bound water on the reflectance of moist soil. For that purpose, each particle was covered with bound water. In the first model run we used the same optical properties for free and bound water, while in the second run the optical properties of ice were assigned to the bound water. In reality, the thickness of such water films is on the order of a few molecular layers. For modeling purposes, the film must be at least four pixels in thickness to avoid numerical artifacts. Multiplied with the pixel size of the images, the resulting film thickness is larger than in reality. This approach hence leads to an overestimation of the influence of bound water
The beam tracing model describes the influence of water on radiative transfer quite well. Estimating the water content from radiative transfer should therefore, in principle, be possible. The feasibility of solving the inverse problem is tested in the third series of simulations. Kano et al. (1985) proposed using the relative changes in the reflectance between dry and wet spectra to estimate intermediate water saturations. Based on this idea, we relate the depth of the absorption band 
a in the reflectance spectrum to the water saturation 
of the soil (Fig. 6)
. The value of a is determined by comparing the measured reflectance at a wavelength outside and within the absorption band.
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Fig. 6. Spectral characteristic of a dry and wet soil sample in the vicinity of an absorption band of water. Symbols are defined in the text.
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Using the variables defined in Fig. 6, we approximated a with
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where the superscripts "d" and "w" refer to the spectra of the dry and wet sample respectively, and the subscripts 1 and 2 to the wavelengths 1 and 2, respectively. We assume that R1d – R1w 
R2d – R2w. The degree of water saturation, , for a measured a is then calculated as
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where b and c are fitting parameters. Equation [2] must include the origin of the coordinate system, since a must be zero at = 0. To determine the degree of water saturation for a given soil, Eq. [2] must be calibrated to obtain the parameters b and c. This requires at least two measurements of the reflectance and the corresponding saturation . To test the feasibility of this method, we estimated the degree of water saturation of samples for which we calculated the reflectance with the beam tracing model.
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MATERIALS AND METHODS
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We measured the reflectance R and the transmittance T of soil materials. Soil materials of different origin and texture were packed in thin sample holders of 0.5-, 1-, and 2-mm thickness. Each soil material was fractionated into seven particle size classes using sieves with mesh sizes of 63, 71, 125, 142, 224, 250, 450, and 500 µm. We chose different combinations of size fractions and sample thickness because these properties are relevant to the reflectance. The soil materials were taken from a spodosol at Guberwald (Richard et al., 1978) and a gley soil at Alpthal (Disserens, 1992). The spodosol samples were taken from the eluvial (Ae) and underlying Fe-enriched (Bs) horizon. The gley samples were taken from the clay-rich, reduced gley horizon (Gr). The Munsell soil color of the dry soil fractions was in the range of 10YR8/1 to 10YR7/1 (Ae), 10YR8/3 to 10YR7/3 (Bs), and 2.5Y5/2 to 2.5Y4/2 (Gr).
To pack the sample holders with the size fraction of a soil material, the samples were continuously vibrated to achieve dense and homogeneous packing. The light source was a quartz lamp mounted to an Ulbricht sphere. Light leaving the circular opening of the Ulbricht sphere is nearly 100% diffuse. The samples were illuminated with the entire wavelength spectrum of the light source. The reflectance was scanned with the ASD Field Spec (Analytical Spectral Devices, 2003) in wavelength increments of 1.4 nm with a spectral resolution of 3 nm in the visible and near-infrared band. In the short-wave infrared band, the increments were 2 nm and the spectral resolution was 10 nm. To measure the reflectance, the samples were placed on a white target (Spectralon) to have a standardized condition at the backside of the sample. The experimental setup for measuring R and T is depicted in Fig. 7
. To obtain different water saturations, the samples were irrigated with a certain amount of water. To avoid heterogeneities induced by the wetting procedure, we placed the sample holder in a horizontal position and removed the upper glass (Fig. 8)
. In this arrangement, the wetted surface (63 by 63 mm) is large compared with the infiltration depth (sample depth is 0.5–2 mm). To reach a homogeneous moisture distribution, the samples were stored for several hours in a chamber with a relative humidity close to saturation before the measurements were performed. To avoid condensation of water between the glass and the soil—an obstruction for measuring the radiative transfer—the glass surfaces were cleaned with acetone before packing (Niemet and Selker, 2001). Cleaning the glass surface removes ultrafine particles that could promote nucleation.
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Fig. 7. Setup for measuring reflectance and transmittance of wet soil samples. The right-hand side depicts a cross- section through the plane of the optical path. Due to the distance of the sample from the opening of the light source, the incident light on the sample is partially diffuse, as indicated by the dashed lines.
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RESULTS
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From the experiments we found that water saturation has a strong effect on radiative transfer. The reflectance decreases and the transmittance increases with increasing water saturation (Fig. 9)
. The decrease in reflectance is smaller for darker than for brighter soils (Fig. 10)
. Figure 11
illustrates the reflectance and transmittance as a function of relative saturation. The measured reflectances were very high, caused by the white target used as the background for the reflectance measurement. Especially in the case of thin samples, a large portion of light is reflected by the white background. To test our measurements, we compared them with published data from Weidong et al. (2002) and Whiting et al. (2003). The Munsell soil colors of the dry samples used in those studies were 2.5Y5/4A and 10YR5/3, respectively. Figure 12
reveals that the shape of the relationship between reflectance, R, and water saturation measured in our study is similar to those found in the literature. The absolute values of R, however, differed from the literature data because of the white target used as the background in our study.
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Fig. 9. Measured spectra of the ferric soil material (Bs horizon) illustrating the influence of water saturation on reflectance and transmittance. The particle size was 63 to 71 µm; the sample thickness was 0.5 mm. Reflectance was measured by placing the samples on a white support.
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Fig. 10. Dependence of reflectance on water content for the particle size fraction 142 to 224 µm of different soil materials. The reflectance was measured at 1000 nm. Gr refers to the strongly reduced clayey material, Bs to the yellow-reddish sand coated by oxidized Fe hydroxides, and Ae to the bleached quartz sand.
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Fig. 11. Measured dependence of reflectance and transmittance on water saturation degree. Data are taken from Fig. 9. The particle size fraction was 224 to 250 µm and the wavelength 900 nm.
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For the first series of simulations we used four different methods to investigate the effect of water content on light scattering. Comparing the first two methods, the added-water-film and the two-dimensional pore network model, the results are different at the higher water contents (Fig. 13 and 14)
. This indicates that not only the amount of water in the pore space influences radiative transfer, but also the spatial distribution of the water phase. The results obtained with the pore network model (Fig. 14) seem to be more plausible since the slope of the calculated reflectance spectra decreases continuously with increasing water saturation. This qualitative trend coincides with the measured results (Fig. 11). The radiative transfer calculated with the third method (three-dimensional pore network) is shown in Fig. 15 . The resulting reflectance curve is similar to the results obtained with the simpler second method (Fig. 14). In the case of the fourth method, we calculated the radiative transfer in seven cross sections of the scanned sand cube as a function of the water distribution (Fig. 16)
. The water distribution in the sand cube was calculated with the Lattice–Boltzmann approach. The calculated curves did not show the expected gradual change in the water content. This is caused by the small size of the analyzed images (200 x 200 pixels). The small ensemble of particles is statistically not representative for the entire cross section of the soil slab.
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Fig. 13. Computed dependence of reflectance and transmittance on water saturation. The water distribution was calculated with the added water film method.
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Fig. 14. Computed dependence of reflectance and transmittance on water saturation. The water distribution was calculated with the pore network model.
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Fig. 16. Reflectance and transmittance computed for seven cross sections of the tomographed sand cube. The cross sections were taken at depths of 140, 210, 280, 350, 420, 490, and 560 µm, respectively.
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With the second series of simulations we examined the significance of bound water on radiative transfer. The values of reflectance modeled with and without bound water were found to be almost the same (Fig. 17)
. Even unrealistically large differences between the refractive indices of free and bound water did not yield larger differences in reflectance.
The third series of simulations was performed to explore the possibilities of extracting soil moisture estimates from measured reflectance. For this purpose we calculated the reflectance at 1 = 500 nm and 2 = 1400 nm and determined the spectral depression a. Figure 18
shows that the regression model, Eq. [2], closely described the observations.
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DISCUSSION
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From the first series of simulations and the experimental data, we conclude that reflectance decreases with increasing water content. With increasing water content, less light is scattered and a higher portion of light is propagated in the forward direction. For wavelengths in the range of an absorption band (e.g., = 1450 nm), water absorbs the light, leading to drastically reduced reflectance and transmittance. The modeled results (Fig. 14) and measured data (Fig. 11) both indicate that transmittance values are minimal at non-zero water saturation degree. Thus, for very low water saturations, the light absorption is more significant than the scattering behavior. We conclude that removing the very last layer of water increases the proportion of forward propagating light. The water and air distributions were modeled with four different methods. Results show that the spatial arrangement of the water is relevant for radiative transfer. Especially for small samples, the arrangement is more dominant than the amount of water. Thus, cross-sectional images used for radiative transfer modeling must contain a large number of particles to represent the scattering medium.
The second series of simulations showed that the influence of bound water on radiative transfer was relatively small. Since the model overestimates the film thickness of bound water, one may expect that bound water has a relatively minor impact on radiative transfer. We analyzed the influence of bound water at = 700 nm. Similar results are expected for wavelengths ranging from 350 to 2500 nm since the difference between the refractive indices of water and ice (as a model for bound water) change only moderately.
The third series of simulations were used to test whether water contents can be estimated from measured reflectance spectra. The estimation errors in were on the order of 15% of the actual water content value. In our example the calibration curve (a) was based on 11 data points. In practice it is very likely that fewer data points are available to calibrate the relationship between absorption and water content. The estimation error would then increase even more. The relatively large estimation errors in are caused by the regression model given by Eq. [2], which does not capture the uneven shape of the relationship of vs. a. Future research should address the physical reasons for these deviations. However, for many applications the precision obtained in our study may be well sufficient, such as for mapping the spatial distribution of the water content on exposed soil surfaces.
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SUMMARY
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The objective of this study was to simulate light reflectance and transmittance of a variable wetted soil. To quantify the effect of water saturation on light transfer, we used the beam tracing model. This model explicitly accounts for the geometry of the solid, liquid, and gas phases. The model therefore requires detailed spatial distributions of the solid, water, and air phase along cross-sectional images of soil samples. We showed that both the amount and the spatial arrangement of the water phase affect radiative transfer. The modeled relationship between water content and radiative transfer properties was in agreement with the trends measured with soil slabs. We also briefly discussed the possibility of solving the inverse problem, that is, whether the beam tracing model can be used for estimating water saturation from reflectance data. The accuracy of the estimation is limited due to a rough regression model used for the inversion.
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TOP
ABSTRACT
INTRODUCTION
