Effects of particle size on bidirectional reflectance factor measurements from particulate surfaces

Zhongqiu Sun; Yunfeng Lv; Zhijun Tong

doi:10.1364/OE.24.00A612

1. Introduction

As solar light penetrates into particulate surfaces, it is partially reflected back because of its interactions with the constituents and structures of those surfaces. The angular distribution of the reflected signal and its variation with the wavelength of the light provide essential information about the physical and compositional properties of a particulate surface [1]. Using common multi-angle spectroscopic techniques, we can simulate the interaction of light with particulate surfaces, which is an essential tool for characterizing their properties [2]. For certain particulate surfaces, such as desert sand and cultivated soil, the spectral reflectance is a somewhat sensitive indicator of the physical characteristics of these surfaces, e.g., particle size, which is a significant geological property whose spatial variation controls many important ecological, soil-related and geomorphic processes in arid and semiarid regions [3,4]. Moreover, in the context of remote sensing applications, the bidirectional reflectance distribution function holds great promise for the extraction of structural details about particulate surfaces compared with traditional unidirectional measurements. For these crucial scientific tasks, the understanding, identification and modeling of the reflectance of particulate surfaces with different particle sizes is necessary to allow data relevant to the characterization of particulate surfaces to be derived from measurements and models.

Optical remote sensing theory has established the existence of a relationship between the spectral reflectance of particulate surfaces and their particle size distributions [5,6]. Because of its relevance to planetary spectroscopy, the effect of particle size on particulate reflectance has been studied extensively [7–11]. In general, reflectance increases with diminishing particle size, although the effect at specific absorption wavelengths is more complex. Theoretical simulations based on Mie scattering models and field verification suggest that the effect mentioned above is also observable for natural particulate surfaces on Earth in hyperspectral data collected in the field [4]. In several previous studies, a variety of modeling approaches and remote sensing or spectroscopic data have been used to derive the particle sizes of minerals, mineral mixtures, sands and sediments. For example, using the semi-empirical bidirectional reflectance model developed by Hapke [12], it may be possible to characterize solid surfaces in terms of their particle sizes. Johnson et al. [13] created a simple algorithm relating particle size and spectral reflectance, which can be used to retrieve mineral abundances and particle sizes from spectral reflectance data. In addition, image data, ground-based spectral data and laboratory measurements have been used to study particle size variations and transport processes in sand seas and dune fields [14,15]. Ghrefat et al. [16] also used AVIRIS data, along with spectral laboratory measurements, for the particle size characterization of sand surfaces. However, that study was based on the linear regression relationship between the absorption band depth and the gypsum size fraction.

Because of the anisotropic reflectance of particulate surfaces, the reflectance characteristics of such a surface can best be analyzed and modeled using both ground- and satellite-based sensors, which acquire both spectral and angular reflectance information at small and large scales, respectively [17]. Several earlier works have identified the potential of anisotropic reflectance data to provide information regarding soil surface roughness [18–20]. Currently, researchers are becoming increasingly interested in combined instrument approaches, in which multi-angular instruments with spectrometers are used to better address the structural complexity of particulate surfaces [21–26]. This concept has been applied to assess changes in the structure of soil surfaces on Earth [27–29]. However, none of the cited publications provides reflectance measurements of natural particulate surfaces with different particle sizes over a wide viewing range on the hemisphere above a sample, although measurements of this type have proven valuable for particle size characterization [30].

The objectives of this study are to quantitatively analyze the effect of particle size on the reflectance of particulate surfaces and to combine the measured results with photometric model parameters to characterize the physical properties of particulate surfaces. To achieve these goals, we should (1) outline the set-up of the measurement system and scheme, (2) present the anisotropic reflectance factors of the investigated particulate samples, (3) understand how particle size influences the samples’ reflectance, and (4) analyze the relationship between the particle size and the model parameters. The remainder of this paper is organized as follows. A brief introduction to the photometric model is provided in the next section. In Section 3, the details of the investigated samples and the measurement procedures are outlined. The measured results and their analysis are presented in Section 4. The relationship between the model parameters and the particle size is discussed in Section 5. In Section 6, we present the conclusions of this study.

2. Model description

Among the various photometric models used in geophysics and planetology, one of the most frequently cited models, developed by Hapke [12,31–34], may make it possible to characterize a material based on reflectance data are acquired under various angular and illumination conditions by adjusting different free parameters. Pinty et al. [35] extended this work to describe soil surfaces on Earth in which the individual particles have a non-uniform angular distribution. Then, Jacquemoud et al. [36] presented a simplified formulation that required six parameters and attempted an explanation of both the backward and forward scattering of light by soil surfaces. Thus, this model and its parameters are likely to be useful for characterizing soil surfaces [2,37]. We selected this model rather than the other form of the Hapke model, which has been critically assessed by Shkuratov et al. [38], because (1) it has previously been applied in investigations of terrestrial particulate surfaces [2,36,37,39,40] and (2) its simulation results are close to the original angular reflectance distribution of a particulate surface, especially with regard to the “hot spot” in the backward scattering direction [24]. The model is given as follows:

R = \frac{ω}{4} \frac{μ_{0}}{μ_{0} + μ} {[1 + B (α, h)] P (α, α') + H (μ_{0}) H (μ) - 1}

where,

\cos α' = \cos i \cos e - \sin i \sin e \cos φ

\cos α = \cos i \cos e + \sin i \sin e \cos φ

B (α, h) = \frac{1}{1 + (1 / h) \tan (α / 2)}

P (α, α') = 1 + b \cos α + c \frac{3 \cos^{2} α - 1}{2} + b' \cos α' + c' \frac{3 \cos^{2} α' - 1}{2}

H (x) = \frac{1 + 2 x}{1 + 2 \sqrt{x (1 - ω)}}

In these equations, ω is the single-scattering albedo; μ₀ and μ are the effective cos(i) and cos(e), respectively; i is the incident zenith angle; e is the emergent angle (viewing zenith angle); φ is the azimuthal angle (between the planes of incidence and emergence); α is the phase angle (between the incoming and outgoing light directions); and α’ is the angle between the specular and outgoing light directions. The function B₍_α_,_h₎ represents the backscattering of the light as a function of α and a roughness parameter h, which increases in value as the surface becomes smoother [2]. The phase function P(α, α’) is proposed for the interpretation of both backward and forward scattering from a particulate surface; b, c, b’, and c’ are the coefficients of the scattering phase function. The term H(μ₀)H(μ)-1 approximates the contribution from multiple scattering [35], and x is used to indicate the substitution of μ₀ or μ into the equation. In Eq. (1), the reflectance “R” refers to the ratio of the reflected brightness of the sample to that of a surface that exhibits Lambertian reflectance. Although several researchers have demonstrated the shortcomings of this model [41–44], we do not intend to provide a full accounting of the long, complex history of the model, which has previously been provided by other publications [38,45,46] and references within.

3. Samples and measurements

3.1 Samples properties

The samples were acquired from a desert pathway (located at 45°02′50″N, 121°49′12″E) and from Songnen Plain in Northeast China (located at 47°08′33″N, 130°42′05″E). All samples were acquired from the surface layers at the sampling sites, which are no deeper than 2 cm. The desert sample consists of quartz, feldspar and impurities. The grains are nearly translucent, and the sample is yellow in color. Because the cultivated soil was acquired from the humus layer, it contains a larger proportion of organic matter and is dark in color. The choices of desert and cultivated soil were dictated by the requirement that the samples not only be representative of arid (or semiarid) and cultivated conditions but also consist of groundcover components from terrestrial ecosystems that are widely distributed on land surfaces [3,47]. The soil properties of the samples are listed in Table 1. Our motivation is that we wish to study the structure rather than the composition of the particulate surfaces. All other relevant descriptions of these samples, such as samples pictures and SEM (scanning electron microscopy) images, have been provided in a previous publication [48]; we also show photomicrographs of a part of our samples in Fig. 1.

Table 1. The average value from 10 samples was used to characteristic each soil property considered in this study. N indicates that we did not measure this parameter. C-soil stands for cultivated soil.

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Fig. 1 The photomicrographs of our samples, A corresponding to 0.375 mm desert sand, B corresponding to 0.375 mm cultivated soil, and the small-scale surface structure of A and B are designated as C and D, respectively.

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3.2 Particle size characterization

The size distributions of all of the samples measured in the laboratory were measured using sieving techniques. Before measuring the reflectance of these particulate samples, we separated them by passing them through a series of sieves after they had been allowed to dry undisturbed for a week at room temperature (we assumed that the soil surface moisture was uniform and stable during the measurement process). We sieved the samples using two 0.9 mm, 0.45 mm and 0.3 mm sieves, producing six different size distributions: 0.9 mm, 0.45-0.9 mm, 0.45 mm, 0.3-0.45, 0.3 mm and smaller than 0.3 mm. An effective particle size, R_eff, was calculated for each sample as follows [4]:

R_{e f f} = \frac{\sum_{i}^{n} R_{i}^{3} {(\frac{Δ N}{Δ R})}_{i}}{\sum_{i}^{n} R_{i}^{2} {(\frac{Δ N}{Δ R})}_{i}}

where R_i is the mean of the opening size of sieve i and that of the next larger sieve. (△N/△R)_i was calculated as follows:

(\frac{Δ N}{Δ R}) i = \frac{3}{4 π ρ} R_{i}^{- 3} \frac{Δ M_{i}}{Δ R_{i}}

where ρ is the density of the soil particles, △M_i (in units of g) is the mass on sieve i and △R_i (in units of mm) is the difference between the opening size of sieve i and that of the next larger sieve. Based on Eq. (7), the R_eff values for both types of particulate surfaces were 0.9 mm, 0.675 mm, 0.45 mm, 0.375 mm, 0.3 mm and 0.15 mm.

Subsequently, each sample was poured into a columnar container with a diameter of 15 cm and a height of 10 cm. The containers were lightly shaken to allow for deposition; each surface was flattened, and all significant macroscopic topographical effects were eliminated by scraping a rule across the top edge of the columnar container. Then, we used a horizontal rule and a level bubble to keep the container horizontal on the objective stage. To mimic the natural land surface conditions, the surfaces were not pressed and the samples were not packed; photographs of the sample surfaces have been presented in our previous publications [24,48] (in Fig. 1 in each case).

3.3 Reflectance measurements

In this study, we used the Northeast Normal University Laboratory Goniospectrometer System (NENULGS) to perform the reflectance measurements of the particulate surfaces. NENULGS has been used to detect phase and spectral reflectance curves and the linear polarization degree of snow, soil and ice when the incident light is unpolarized [26,48–51]. The basic NENULGS configuration consists of a goniometer, an artificial illumination source and an Analytical Spectral Devices FieldSpec 3 (ASD FS3) spectroradiometer.

With its oblique and vertical axes, the goniometer can place the fiber-optic cable of the ASD spectrometer at any point on the hemisphere above a sample. In the measurement process, because of the limitations of the structure of our apparatus, 8° is the smallest phase-angle measurement we can perform when rotating the arm in the viewing direction, and thus, we did not conduct measurements at phase angles smaller than 8° for any incident angle in the principal plane. Therefore, measurements in the viewing angle range (37-53°) were absent in the backward direction when the incident angle was 45°. Each reflectance curve measurement began at the nadir direction. The increments in the viewing zenith angle were 2° and 3° in the principal plane, e.g., [0°, 2°, 5°,…70°]. The increment in the viewing zenith angle was 5° at other azimuth angles, e.g., [0°, 5°, 10°,…65°, 70°]. The measurement interval in the azimuth direction was 10°.

The illumination zenith angles were 45° and 60° during the measurement process in this study. The distance from the sensor to the sample surface was 0.15 m; the size of the field of view was 8° for the reflected measurements of the soil and sand samples. Thus, the sensor, whose viewing zenith angle varied from 0° to 70°, collected data from a circular footprint of 2.1 cm in diameter to an elliptical footprint with a 6.7 cm major axis. Because the diameter of the illuminated area, which was 10 cm, was larger than the viewing footprint area at all times, we did not need to consider the effect of the surroundings outside the container on the reflectance.

The bidirectional reflectance factor (BRF) of each sample, which is defined as the ratio of the reflected radiant flux dL’ from the sample surface area (dA) to the reflected radiant flux dL from an ideal and diffuse surface of the same area (dA) in the identical viewing geometry under single-direction illumination [52], was measured in the laboratory. This factor is calculated as follows:

B R F = \frac{d L' (i, φ_{i}; e, φ_{e})}{d L (i, φ_{i}; e, φ_{e})}

where i and φ_i are the zenith and azimuth angles, respectively, of the incident direction and e and φ_e are the zenith and azimuth angles respectively, of the viewing direction. The Spectralon plane provided by the manufacture was used to represent a perfect Lambertian panel in this study. Strictly speaking, the reflectance factor measured in this study should be referred to as the bi-conical reflectance factor (BCRF) [52]. However, to maintain a definition of the reflectance that is identical to that used in previous studies [21,22,53], we assume here that the BRF is approximately equal to the BCRF.

We also calculated the anisotropy reflectance factor (ARF) of each soil surface to enable a comparison of the anisotropic characteristics of samples with different particle sizes. The ARF can be defined as the ratio of the bidirectional reflectance factor in a specific viewing direction to the nadir reflectance factor under illumination in the same direction [54], and it is given as follows:

A R F = \frac{B R F (λ, i, φ_{i}; e, φ_{e})}{B R F_{n a d i r} (λ, i, φ_{i})}

where BRF is the bidirectional reflectance factor, BRF_nadir is the nadir reflectance factor and λ is the wavelength.

4. Results and analysis

4.1. The reflectances of the particulate surfaces

In this section, we will verify that our measured reflectance results for the particulate surfaces are consistent with previous studies. The spectral reflectance curves of the sand and cultivated soil surfaces with a particle size of 0.9 mm measured at different viewing zenith angles in the principal plane are shown in Fig. 2; for these measurements, the incident zenith angle was 60°. Figure 2 shows that the maximum reflectance of both types of particulate surfaces was observed in the backward scattering direction when the viewing zenith angle was 52° (corresponding to an 8° phase angle) and that the minimum reflectance was observed in the forward scattering direction when the viewing zenith angle was −60° (corresponding to a 120° phase angle). It is also clear from Fig. 2 that the reflectance increased with a decreasing phase angle in the principal plane. This is because the particles on the soil and sand surfaces created shadows, leading to a reduction in the BRF in the forward scattering direction, where shadows are most visible compared with other viewing directions. In the backward scattering direction toward the Sun’s position, each particle hides its own shadow, which results in a clear backscattering component [19]. These results, which are expected for reflection from particulate surfaces, are consistent with those of previous studies in which particulate surfaces similar to ours have been measured [21,22,28].

Fig. 2 The reflectance curves of particulate surfaces with a particle size of 0.9 mm at different viewing zenith angles in the principal plane, where the incident zenith angle was 60°. For both types of particulate surfaces, the minimum reflectance was observed in the forward scattering direction (−60°) and the maximum reflectance was observed in the backward scattering direction (52°).

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To further illustrate the reliability of our measurements of these samples, we also studied how the reflectances of the particulate surfaces depended on the incident angle. Figure 3 shows the reflectance curves of the particulate surfaces with a particle size of 0.45 mm in the nadir direction, for incident zenith angles of 45° and 60°. An increase in the reflectance with an increase in the incident zenith angle, which is a well-known phenomenon in soil remote sensing.

Fig. 3 The reflectance curves of particulate surfaces with a particle size of 0.45 mm measured in the nadir direction, for incident zenith angles of 45° and 60°, respectively. The upper two curves correspond to the desert sand surface, and the lower two curves correspond to the cultivated soil surface.

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4.2. The reflectances of particulate surfaces with different particle sizes

In the present study, we limit our attention to the observed effects of particle size, and we assume that the chemical composition of soil of the same type is identical for each particle size to allow us to relate any spectral variations to particle size effects. Figure 4 shows the spectral reflectances of particulate surfaces with different particle sizes measured in the nadir direction for an incident zenith angle of 45°. An obvious increasing trend in the reflectances of the desert sand and cultivated soil is evident as the particle size decreases from 0.9 mm to 0.15 mm. The average increases in the reflectance of the desert sand at visible wavelengths (350-740 nm) are 0.7%, 0.68%, 0.08%, 1.17% and 0.71% for decreases in the particle size from 0.9 mm to 0.675 mm, from 0.675 mm to 0.45 mm, from 0.45 mm to 0.375 mm, from 0.375 mm to 0.3 mm and from 0.3 mm to 0.15 mm, respectively; the average increases in the reflectance of the desert sand at near-infrared wavelengths (740-2500 nm) are 1.15%, 2.08%, 1.08%, 2.01% and 1.42% for the same decrements in particle size from 0.9 mm to 0.15 mm. By contrast, for the cultivated soil, with the exception of an average reflectance difference of 2% between surfaces with particle sizes of 0.15 mm and 0.3 mm at near-infrared wavelengths, the average increases in reflectance observed with decreasing particle size from 0.9 mm to 0.3 mm are small compared with those for the desert sand: the average increases in the reflectance at visible wavelengths are 0.5%, 0.03%, 0.02%, 0.01% and 0.01% for the consecutive investigated decrements in particle size from 0.9 mm to 0.15 mm, whereas the average increases in the reflectance at near-infrared wavelengths are 0.76%, 0.46%, 0.89% and 0.92% for these same particle size decrements. These trends are consistent with the results of measurements of sand and soil surfaces reported in previous studies [4,30,55,56], although the authors of these studies did not quantitatively analyze the reflectance differences. As seen from the reflectance spectra in Fig. 4, the absorption characteristics suggest the presence of water in both types of samples, despite the fact that they were dried for a week at room temperature before the measurement process.

Fig. 4 The reflectance curves of particulate surfaces with different particle sizes measured in the nadir direction for an incident zenith angle of 45°

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4.3. ARF and BRF distributions of particulate surfaces with different particle sizes

In addition to knowledge regarding spectral behavior, improvements to the quantitative retrieval of physical sand and soil properties via remote sensing also require information regarding the angular structure of the bidirectional reflectance values [57]. First, as shown in Fig. 5, we compared the ARFs of desert sand surfaces with different particle sizes with the results measured by Zhang et al. [30] to demonstrate that the effects of particle size on the ARF were identical for both studies, because both studies were performed in the red band and with an incident zenith angle of 45°. Based on the ARF values of the sand surfaces shown in Fig. 5, we found that a larger particle size corresponded to more obvious anisotropic characteristics of the sample. The most obvious ARF differences between samples with different particle sizes were observed in the backward and forward scattering directions. These reflectance characterizations in our study were similar to those in a previous study [30]. However, we focused here on the BRF and ARF at 1589 nm because the multi-angular reflectances of terrestrial surfaces in this band can be obtained from the Research Scanning Polarimeter (RSP) [58], the reflectance at 1589 nm has been found to be sensitive to changes in particle size [4,6,56], and researchers have previously noted that it would be worthwhile to extend the exploration of photometric models toward infrared wavelengths [59].

Fig. 5 The ARFs of desert sand surfaces with different particle sizes measured at different viewing zenith angles in the principal plane; the incident angle was 45°, and the wavelength was 670 nm.

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This non-Lambertian behavior mentioned above is a well-known phenomenon in the remote sensing of particulate surfaces. From the perspective of geometrical optics, for an opaque, rough soil surface with irregularities in the soil texture caused by the presence of particles and aggregates that are large compared to the wavelength of the incident light, it is impossible to illuminate the entire surface directly; these elements produce shadows that becomes an important factor influencing the shape of the reflectance pattern of the soil surface [18]. The reflectance variations caused by these shadows when viewed by a sensor are the fundamental reason for the low-reflectance behavior observed in the forward scattering direction, because the wave energy leaving the shaded areas is many orders of magnitude smaller than the energy reflected from the sunlit soil fragments [19]. When measurements are performed in the backward scattering direction, each particle scatters toward the position of the Sun and hides its own shadow, which results in a backscattering reflectance peak (referred to as a hot spot in this paper). Furthermore, small-scale roughness of the host particles that is of the same order as the incident wavelength also influences the reflectance pattern of a particulate surface [60–63]. However, this study does not focus on this topic.

Figure 6 shows the ARFs of particulate surfaces with different particle sizes measured at different viewing zenith angles; in these measurements, the incident zenith was 45° and the wavelength was 1589 nm. It was found that at larger particle sizes, the difference between the minimum and maximum ARFs was greater. The dependence of the ARF on particle size at 1589 nm is similar to the results discussed above for a wavelength of 670 nm. The polar plots of the BRFs of the soil samples (in Figs. 7 and 8) and the polar plots of the ARFs of the soil samples (in Figs. 9 and 10) clearly show the variations in angular reflectance observed when interpreting the soil surface radiometry, such as low reflectance in the forward scattering direction and relatively high reflectance in the backward scattering direction, lower BRFs at larger particle sizes for both types of samples at all zenith viewing angles, and more obvious anisotropic reflectance characteristics of the samples with larger particle sizes. We selected the reflectances of the particulate surfaces with particle sizes of 0.9 mm, 0.45 mm and 0.3 mm as the data on which to base Figs. 7-10 because these three particle size distributions were the most accurate of the six considered. The model presented by Cierniewski and Karnieli [18], which indicates that higher roughness of a soil surface generally tends to result in a higher variation in reflectance, may explain the results of our study. Larger particles correspond to higher roughness of the soil surface, resulting in the production of more shadowed areas in this study compared with the behavior of surfaces with smaller particles. Such shadowed areas are visible at viewing directions near the nadir and at more oblique angles in the forward scattering direction; thus, the reflectances of the soil surfaces with larger particles decreased more rapidly than those of the soil surfaces with smaller particles, especially in the forward scattering direction, as shown in Figs. 7 and 8. Furthermore, compared with a smoother surface (with a smaller particle size), a rougher soil surface will hide more multiple scattering of light between particles in the backward scattering direction in our measurements directions. Overall, these effects result in lower reflectances for surfaces with larger particles in our range of viewing azimuth angles.

Fig. 6 The ARFs of particulate surfaces with different particle sizes measured at different viewing zenith angles; the incident zenith angle was 45°, and the wavelength was 1589 nm. The negative viewing angles correspond to the forward scattering direction.

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Fig. 7 Polar plots of the BRFs of cultivated soil surfaces with different particle sizes for all viewing zenith angles at 1589 nm; the incident angle was 60°. The radial distance from the center of each plot represents the viewing zenith angle, with a maximum value of 70°. Rotation about the center represents a change in azimuth. An azimuthal angle of 0° corresponds to backward reflectance in half of the illumination principal plane. Near the hot spot (from 52° to 68°), the absent values were replaced with the values corresponding to a viewing zenith angle of 68°.

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Fig. 8 Polar plots of the BRFs of desert sand surfaces with different particle sizes for all viewing zenith angles at 1589 nm; the incident angle was 60°.

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Fig. 9 The polar plots of ARFs of cultivated soil with different particle size for all viewing zenith angles at 1589 nm; the incident angle was 60°.

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Fig. 10 The polar plots of ARFs of desert sand with different particle size for all viewing zenith angles at 1589 nm; the incident angle was 60°.

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The left-right symmetry of the BRFs and ARFs with respect to the light-source principal plane of the sand surface can be clearly observed in Figs. 7 and 8 and Figs. 9 and 10. This is because we selected the percentile of the reflectance value, and the difference between each side of the principal plane is close to zero. Hence, a high degree of symmetry was demonstrated in this study. We also found that the left-right reflectance symmetry was common to both the desert sand and cultivated soil surfaces (with particle sizes of 0.9 mm, 0.675 mm and 0.45 mm in diameter) investigated in our study. For this reason, we assumed reflectance symmetry for all samples and performed half-range azimuth measurements (in 330 viewing directions) for the remaining samples.

4.4. Quantitative characterization of the effects of particle size on the reflectance of particulate surfaces

We calculated the ratio between the BRF difference (BRF_0.15mm-BRF_0.9mm) and the BRF for the particulate surface with a particle size of 0.9 mm to illustrate the effects of particle size on reflectance. Figures 11 and 12 show that the maximum difference in BRF between samples with particle sizes of 0.15 mm and 0.9 mm is located in the forward scattering direction, whereas the minimum difference in BRF is located in the backward scattering direction at 1589 nm. When the incident zenith angle was 45°, the maximum difference reached 59% and the minimum difference was equal to approximately 0.5% for the cultivated soil, whereas for the desert sand, the maximum difference was 28% and the minimum difference was equal to approximately 3%; when the incident zenith angle was 60°, the maximum difference was 51% and the minimum difference was 0.7% for the cultivated soil, whereas for the desert sand, the maximum difference was 28% and the minimum difference was equal to approximately 3%.

Fig. 11 The change in the sample BRF from a particle size of 0.9 mm to a particle size of 0.15 mm ((BRF_0.15mm-BRF_0.9mm)/BRF_0.9mm) for an incident zenith angle of 45° at 1589 nm; the cultivated soil results are shown on the left, and the desert sand results are shown on the right.

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Fig. 12 The change in the sample BRF from a particle size of 0.9 mm to a particle size of 0.15 mm ((BRF_0.15mm-BRF_0.9mm)/BRF_0.9mm) for an incident zenith angle of 60°; the cultivated soil results are shown on the left, and the desert sand results are shown on the right.

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It is clear that the low-reflectance sample exhibited the maximal reflectance difference when the particle size changed from 0.15 to 0.9 mm in this study. This is because the desert sand (the relatively high-reflectance samples) generated more multiple scattering, whereas the cultivated soil (the relatively low-reflectance samples) generated less multiple scattering. This means that the shadows produced by the desert sand particles appeared “lighter” than the shadows produced by the cultivated soil particles for similar measurement conditions and particle sizes. In brief, the greater intensity of multiple scattering reduced the BRF difference for the desert sand samples in the considered range of viewing zenith angles. In addition, the roughness (which was of the same order as the incident wavelength) of the individual aggregates in the cultivated soil samples was higher than the roughness of the individual desert sand particles with the same diameter, as indicated by the scanning electron microscopy images of these particles shown in [48] and Fig. 1. These small-scale surface roughnesses will also influence the scattering properties of the particles [64,65], but we do not give this effect much consideration in this context.

5. The relationship between the model parameters and the particle size

In our previous study [24], we noted that the values of the parameters b, c, b’ and c’ in the model discussed in this paper cannot be used to describe the angular distributions of the light scattered by these particulate surfaces because the scattering properties of sand and soil surfaces are based primarily on backward scattering [1]. For the reason noted above, in this study, we focus on the relationship between the model parameters h and ωand the particle size. The model used in this paper is highly non-linear and contains several parameters; these characteristics make it difficult for the inversion procedure to find unique useful solutions. Without a priori information on the parameter values, the inverse problem typically consists of determining the optimal set of variables that minimizes the difference between the modeled and measured values through iterative numerical calculations. Following a method used in previous studies [2,35,36], a non-linear least-squares fitting procedure was applied to solve the inverse problem. The optimization performance was assessed based on the square root of the mean squared error (RMSE):

R M S E = \sqrt{\frac{\sum_{k = 1}^{n} {(R_{m} - R)}^{2}}{N_{f}}}

where R_m and R are the measured and modeled bidirectional reflectance values, respectively, of a sample for the same illumination and observation conditions; n is the total number of observations; N_f is the number of degrees of freedom, which is equal to the number of independently measured data points minus the number of parameters estimated in the procedure. This inverse modeling problem was coded in MATLAB using the “Isqnonlin” function.

5.1. Comparisons with measured data

Figure 13 shows a comparison between the measured and modeled BRFs of cultivated soil surfaces with particle sizes of 0.9 mm, 0.45 mm and 0.3 mm at a wavelength of 1589 nm for an incident zenith angle of 60°. From this figure, we can observe that the modeled results are in good agreement with the measured results. The absolute values of the average difference between the measured and modeled BRFs for all particulate surface samples at arbitrary incident zenith angles are less than 2%; this difference value is equal to (R-R_m)/R_m, where R is the modeled BRF and R_m is the measured BRF. We use Fig. 14, corresponding to an incident zenith angle of 45° and desert sand samples, to illustrate this small difference. We find that the model underestimates the BRF in the backward scattering direction and overestimates the BRF in the forward scattering direction. Based on the results presented in Figs. 13 and 14, it is clear that the model is very effective for computing the BRFs of particulate surfaces, as shown in previous studies [2,24,35,36].

Fig. 13 Comparison between the measured and modeled BRFs of cultivated soil surfaces with particle sizes of 0.9 mm, 0.45 mm and 0.3 mm at 1589 nm for an incident zenith angle of 60°.

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Fig. 14 The values of the difference (R-R_m)/R_m between the measured and modeled BRFs of desert sand surfaces with particle sizes of 0.9 mm, 0.45 mm and 0.3 mm at 1589 nm for an incident zenith angle of 45°.

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5.2. Assessment of the relationship between the model parameters and particle size

We have previously reported a monotonic relationship between the roughness parameter h and the particle size at wavelengths of 560 nm and 670 nm [24]; in other words, as the surface became smoother, h increased. However, the inverted values of the parameter h that are shown in Tables 2-5 do not support the results of our previous study. Instead, it is found that h varies irregularly with a varying particle size at a wavelength of 1589 nm; see the values of h given in the fourth column vs. the particle sizes in the second column in these tables. To further illustrate the relationship between the parameter h and the particle size, we present only the best-fit parameters of the model for 0.9 mm, 0.45 mm and 0.3 mm particulate samples at a wavelength of 865 nm in Tables 6-9. We find that the phenomenon in which an increasingly smooth surface results in an increasing value of h was not observed for the samples considered in this study; thus, we must clarify that our previous result reported in [24] was a special case. Based on these results, it is clear that the roughness parameter h cannot be directly related to the particle size, at least for the particulate samples investigated here. By comparing all of the parameters obtained through inversion for the same sample at different incident zenith angles, we find that the best-fit parameters do not depend on the incident angle. According to the results presented in this paper, we suggest that the roughness parameter h cannot be used as an indicator to characterize the particle size, although several studies have presented certain special cases in which such characterization is possible [2,24,36].

Table 2. The best-fit parameters of the model for the cultivated soil at 1589 nm with an incident zenith angle of 45°.

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Table 3. The best-fit parameters of the model for the desert sand at 1589 nm with an incident zenith angle of 45°.

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Table 4. The best-fit parameters of the model for the cultivated soil at 1589 nm with an incident zenith angle of 60°.

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Table 5. The best-fit parameters of the model for the desert sand at 1589 nm with an incident zenith angle of 60°.

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Table 6. The best-fit parameters of the model for the cultivated soil at 865 nm with an incident zenith angle of 45°.

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Table 7. The best fitted parameters of model for desert sand at 865 nm, the incident zenith angle was 45°.

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Table 8. The best fitted parameters of model for cultivated soil at 865 nm, the incident zenith angle was 60°.

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Table 9. The best fitted parameters of model for desert sand at 865 nm, the incident zenith angle was 60°.

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The single-scattering albedo ω, which may be regarded as as the most robust parameter [66], is the primary focus of our study. This parameter exhibited a clear negative correlation with the change in the particle size at wavelengths of 865 nm and 1589 nm for incident zenith angles of 45° and 60°; see the values of ω given in the third column vs. the particle sizes in the second column in Tables 2-9. Without a doubt, the single-scattering albedo ω increased with decreasing particle size at these selected wavelengths, which are feasible for use in remote sensing applications for terrestrial surfaces, such as bare soil [67,68]. A previously published study has demonstrated that even with a limited set of angular configurations, the photometric model can be constrained and the accuracy of each parameter can be estimated [59]. Our previous study [24] has proven that using parameters inverted from measurements made only in the principal plane may lead to an increase in the difference between the measured and modeled results, but in that study, no evaluation was made to determine whether using measurements performed in the principal plane to invert the parameters would influence the relationship between the single-scattering albedo and the particle size. To address this issue, in Tables 10 and 11, we present the best-fit parameters of the model for measurements of all samples at a wavelength of 1589 nm in the principal plane. We find that the single-scattering albedo did not increase with decreasing particle size for the special cases of 0.45 mm cultivated soil and 0.3 mm desert sand.

Table 10. The best fitted parameters of model for cultivated soil at 1589 nm in the principal plane, the incident zenith angle was 45°.

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Table 11. The best fitted parameters of model for dessert sand at 1589 nm in the principal plane, the incident zenith angle was 45°.

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To avoid drawing erroneous conclusions from our study, we present the best-fit parameters of the model for measurements of 0.9 mm, 0.45 mm and 0.3 mm particulate samples at a wavelength of 865 nm in the principal plane for an incident zenith angle of 45°; see Tables 12 and 13 (for cultivated soil, ω = 0.4436, 0.4266 and 0.3678 for particle size = 0.9 mm, 0.45 mm and 0.3 mm, respectively; for desert sand, ω = 0.8264, 0.7895 and 0.6865 for particle size = 0.9 mm, 0.45 mm and 0.3 mm, respectively). Remarkably, the single-scattering albedo shows a positive correlation with the particle size at 865 nm, which is an unreasonable phenomenon according to previous studies [26,59], although these studies did not use the same expression of the model. A similar issue is also evident for the measurements of the desert sand samples at a wavelength of 1589 nm in the principal plane for an incident zenith angle of 60°; see Table 14 (ω = 0.8708, 0.7821, 0.6814, 0.7958, 0.8716 and 0.8668 for particle size = 0.9 mm, 0.675 mm, 0.45 mm, 0.375 mm, 0.3 mm and 0.15 mm, respectively). Moreover, an unexpected result in which the relationship between the single-scattering albedo and the particle size appeared to be random was observed during the inversion process for the measurements of cultivated soil samples at a wavelength of 1589 nm with an incident zenith angle of 60°, although these values are not listed in the table.

Table 12. The best fitted parameters of model for cultivated soil at 865 nm in the principal plane, the incident zenith angle was 45°.

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Table 13. The best fitted parameters of model for desert sand at 865 nm in the principal plane, the incident zenith angle was 45°.

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Table 14. The best fitted parameters of model for dessert sand at 1589 nm in the principal plane, the incident zenith angle was 60°.

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We surmise there are two reasons for the irregular relation observed between the particle size and the single-scattering albedo that was inverted from measurements made in the principal plane, by contrast to the regular relation observed between the particle size and the single-scattering albedo that was inverted from measurements made over a wide viewing zenith angle. First, the model parameters may be non-unique and the fitting procedure may be finding local minima, meaning there are many different combinations of parameters that will yield a good fit to the same measured data; thus, the parameters should be interpreted only as one possible set that fits the measurements. It should be noted that in the absence of a priori information on the parameter values, the starting values used in the iterative fitting procedure were identical for all particulate surface samples considered in this study. Second, previous researchers [35,59] have shown that the accuracy of the inversion procedure increases with finer-resolution sampling in the solar and viewing zenith angles, which means that using BRFs measured in the principal plane to invert the parameters will increase the difference between the measured and modeled reflectance of a particulate surface [24]; in other words, the parameters are essentially dominated by the measurements over solid angles which are present in this paper. In Fig. 15, we show polar plots of R² that represent the regression relationship between the BRF and the particle size at a wavelength of 1589 nm when the incident zenith angle is 45°. Interestingly, a strong regression relationship between the particle size and the BRF is evident in all viewing directions; this relationship between the particle size and BRF of desert sand in the nadir direction at wavelengths of 1723 nm and 2169 nm has been presented in [4]. We also show the relationship between the particle size and the reflectance at 1589 nm to illustrate that the result reported in [4] is similar to ours shown in Fig. 16, although we did not evaluate the BRFs at 1723 nm and 2169 nm in this study. It is also found that the relationships between the BRF and the particle size are different for the two types of samples. We suspect that this is because their soil properties are different; in particular, the roughness of the individual aggregates in the cultivated soil samples is generally higher than the roughness of the individual desert soil particles of a similar diameter. However, we cannot determine which of the factors mentioned above is responsible for the different relationships between particle size and reflectance for the two types of samples investigated in this study. Nevertheless, based on these results, the existence of a stable negative correlation between the particle size and the single-scattering albedo that is inverted from the BRF measured over a wide range of viewing angles for all samples at arbitrary wavelengths and incident angles is incontrovertible, because the BRFs of particulate surfaces with different particle sizes at different viewing angles will contribute to the inversion procedure in a regular manner, especially for the single-scattering albedo.

Fig. 15 Polar plots of R² that represent the logarithmic regression relationship between the BRF of a cultivated soil surface and the particle size at 1589 nm (left) and the linear regression relationship between the BRF of a desert sand surface and the particle size at 1589 nm (right), for an incident zenith angle of 45°.

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Fig. 16 The logarithmic relationship between the BRF of a cultivated soil surface and the particle size at 1589 nm and the linear relationship between the BRF of a desert sand surface and the particle size at 1589 nm; the incident zenith angle was 45°, and the BRF was measured in the nadir direction.

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Furthermore, the regression correlations between the particle size and the single-scattering albedo inverted based on all viewing directions for the two types of samples at different incident zenith angles are shown in Figs. 17 and 18. From these figures, it is obvious that the single-scattering albedo, ω, is the only parameter that effectively represents the properties of a particulate surface, such as its particle size. However, if a researcher should wish to relate the single-scattering albedo of a particulate surface such as those considered here to its particle size using the photometric model, the measurements should be performed at a wide range of viewing zenith angles for each different incident zenith angle; at least, the inverted single-scattering albedo determined based only on measurements in the principal plane cannot be used to establish a relationship with the particle size.

Fig. 17 The regression relationship between the single-scattering albedo (SSA) inverted based on all viewing directions and the particle size for both types of samples, for an incident zenith angle of 45°.

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Fig. 18 The regression relationship between the single-scattering albedo (SSA) inverted based on all viewing directions and the particle size for both types of samples, for an incident zenith angle of 60°.

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6. Conclusion

The bidirectional reflectance factors (BRFs) of two types of soil samples with six identical particle size distributions were measured at different viewing zenith angles and two incident zenith angles over a wide viewing range on the hemisphere above samples. The anisotropy reflectance factors (ARFs) were also calculated to characterize the anisotropy of the soil surfaces. The findings indicated that in general, a larger particle size corresponds to a lower BRF in all viewing directions, as well as a greater difference in ARF between the forward and backward scattering directions [30]. A quantitative analysis of the BRF differences between surfaces with particle sizes of 0.15 mm to 0.9 mm for the two types of samples revealed that the minimum difference was observed the backward scattering direction, whereas the maximum difference was observed the forward scattering direction, and this difference was more distinct for the low-reflectance (cultivated soil) samples than for the relatively high-reflectance (desert sand) samples of the same particle size. Then, these results were used to invert the model parameters. We found that by using a photometric model [2,36], we were able to retrieve the best fits to the measured reflectance; we prove the robustness of the model for computing the reflectances of particulate surfaces and also once again demonstrated the shortcomings of this photometric model.

This study also demonstrated that the parameters of the photometric model, such as the single-scattering albedo ω, are effective for empirically characterizing particulate surfaces in terms of their particle size, but measurements must be made in a sufficient number of directions over a wide viewing range on the hemisphere above samples as showing in this paper. Moreover, the results presented in this paper not only can be used in the study of the particle sizes of soil surfaces in the laboratory but also offer a potential means of estimating the particle sizes of particulate surfaces using multi-angular remote sensing data. The validity of our conclusion, which empirically relates the single-scattering albedo of a particulate surface to its particle size, can be verified by investigating different types of particulate media found on Earth. However, for the extension of the results to many of the surfaces found on other bodies in the Solar System, much smaller particles would be a better choice for future experiments.

Acknowledgments

We would like to thank the two anonymous reviewers for the insightful reviews. This work was supported by the National Natural Science Foundation of China (Grant No. 41401379 and 41571343).

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Samples	Desert sand (%)	Silt (%)	Clay (%)	Coarse Desert sand (%)	SOC (%)	Color (Munsell)
Desert	9.5%	69.9%	13.1%	7.5%	N	10YR 5/3
C-Soil	19.4%	59.1%	21.5%	N	3.7%	2.5Y 3/1

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Cultivated Soil	0.9 mm	0.8061	0.0919	0.0753	0.1678	−0.4942	−0.0473	0.005
	0.675 mm	0.8172	0.1033	0.0353	0.1151	−0.5061	−0.0273	0.004
	0.45 mm	0.8244	0.1111	0.0056	0.0706	−0.5178	−0.207	0.008
	0.375 mm	0.8353	0.1049	−0.0064	0.0493	−0.4941	−0.0085	0.005
	0.3 mm	0.842	0.0977	−0.007	0.0339	−0.4709	−0.0119	0.007
	0.15 mm	0.865	0.1016	−0.0707	−0.0526	−0.4637	0.0113	0.008

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.945	0.0705	−0.241	0.1055	−0.615	−0.1056	0.005
	0.675 mm	0.95	0.0675	−0.307	0.095	−0.602	−0.135	0.006
	0.45 mm	0.954	0.0686	−0.216	0.012	−0.573	−0.0629	0.007
	0.375 mm	0.959	0.0765	−0.386	0.0336	−0.54	−0.182	0.006
	0.3 mm	0.961	0.065	−0.252	−0.0211	−0479	−0.149	0.004
	0.15 mm	0.968	0.0825	−0.479	0.0741	−0.5597	−0.17	0.005

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Cultivated Soil	0.9 mm	0.835	0.0107	0.08	0.0975	−0.6401	−0.5171	0.008
	0.675 mm	0.8438	0.0105	0.0756	0.0851	−0.6358	−0.1191	0.007
	0.45 mm	0.8534	0.0131	0.0567	0.0512	−0.6205	−0.0745	0.009
	0.375 mm	0.8606	0.0144	0.0378	0.0241	−0.6328	−0.0454	0.008
	0.3 mm	0.8665	0.0187	0.0261	0.002	−0.6028	−0.0013	0.009
	0.15 mm	0.8977	0.0236	−0.1091	−0.0379	−0.5679	0.2257	0.012

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.8865	0.0371	0.6185	−0.0096	0.3026	−0.1942	0.012
	0.675 mm	0.8976	0.0556	0.5164	−0.0621	0.1888	−0.1325	0.01
	0.45 mm	0.9065	0.1483	0.3559	−0.1394	0.0853	−0.0791	0.01
	0.375 mm	0.9147	0.1475	0.3415	−0.1613	0.0738	−0.0749	0.009
	0.3 mm	0.9214	0.1437	0.322	−0.184	0.0609	−0.0644	0.009
	0.15 mm	0.942	0.1204	0.1183	−0.2301	−0.1476	0.024	0.013

Effects of particle size on bidirectional reflectance factor measurements from particulate surfaces

Abstract

1. Introduction

2. Model description

3. Samples and measurements

3.1 Samples properties

3.2 Particle size characterization

3.3 Reflectance measurements

4. Results and analysis

4.1. The reflectances of the particulate surfaces

4.2. The reflectances of particulate surfaces with different particle sizes

4.3. ARF and BRF distributions of particulate surfaces with different particle sizes

4.4. Quantitative characterization of the effects of particle size on the reflectance of particulate surfaces

5. The relationship between the model parameters and the particle size

5.1. Comparisons with measured data

5.2. Assessment of the relationship between the model parameters and particle size

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (18)

Tables (14)

Equations (11)

Optics Express

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Cultivated Soil	0.9 mm	0.5761	0.0738	0.0592	0.1537	−0.3591	−0.0301	0.005
	0.45 mm	0.6072	0.0805	0.0005	0.1357	−0.4398	0.0176	0.005
	0.3 mm	0.6832	0.0378	−0.1369	0.1987	−0.694	0.1286	0.004

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.891	0.0797	−0.0959	0.121	−0.5233	−0.0725	0.006
	0.45 mm	0.9017	0.0879	−0.1164	0.0655	−0.4922	−0.0269	0.007
	0.3 mm	0.913	0.0933	−0.1002	0.0269	−0.4519	−0.0143	0.007

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Cultivated Soil	0.9 mm	0.6207	0.0054	0.0524	0.1345	−0.4515	−0.1299	0.006
	0.45 mm	0.6224	0.0048	0.0953	0.1633	−0.3045	−0.105	0.006
	0.3 mm	0.6391	0.0162	0.0594	0.0324	−0.4774	0.023	0.005

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.895	0.0073	0.1641	0.0935	−0.4926	−0.0758	0.008
	0.45 mm	0.91	0.0109	0.0489	0.0198	−0.5681	−0.0154	0.009
	0.3 mm	0.9117	0.0119	0.1873	0.0075	−0.3583	−0.0172	0.008

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Cultivated Soil	0.9 mm	0.79	0.089	0.0991	0.3033	−0.4742	0.0304	0.006
	0.675 mm	0.7932	0.0861	0.0824	0.3622	−0.5089	0.1779	0.005
	0.45 mm	0.8082	0.0898	0.0556	0.2787	−0.554	0.1642	0.006
	0.375 mm	0.8054	0.0852	0.0652	0.369	−0.5116	0.3098	0.006
	0.3 mm	0.8198	0.0845	0.0322	0.309	−0.486	0.2525	0.007
	0.15 mm	0.8379	0.0816	−0.0107	0.3222	−0.4951	0.4166	0.009

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.9089	0.0588	0.0651	0.5184	−0.4347	0.394	0.011
	0.675 mm	0.9191	0.0558	−0.0737	0.5661	−0.4519	0.3682	0.011
	0.45 mm	0.923	0.062	0.021	0.4329	−0.3649	0.3273	0.009
	0.375 mm	0.9397	0.0573	−0.18	0.4886	−0.4485	0.3193	0.007
	0.3 mm	0.9288	0.0534	0.0475	0.5268	−0.2798	0.4678	0.007
	0.15 mm	0.9463	0.0352	−0.1494	0.6228	−0.3975	0.425	0.006

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Cultivated Soil	0.9 mm	0.4436	0.0652	0.2483	1.0289	−0.3451	0.7851	0.007
	0.45 mm	0.4266	0.069	0.2534	1.2424	−0.3386	1.034	0.007
	0.3 mm	0.3678	0.0671	0.3696	1.8414	−0.3165	1.688	0.007

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.8264	0.0584	0.1448	0.7758	−0.3575	0.588	0.01
	0.45 mm	0.7895	0.0566	0.314	1.1175	−0.2485	1.1176	0.009
	0.3 mm	0.6865	0.0459	0.7055	2.1016	−0.006	2.285	0.009

Samples	Particle size	ω	h	b	c	b’	c’	RMSE
Desert Sand	0.9 mm	0.8708	0.1781	0.5409	−0.2158	0.1697	−0.2014	0.009
	0.675 mm	0.7821	0.1498	1.3324	−0.1208	0.9048	0.0309	0.008
	0.45 mm	0.6814	0.1953	1.9813	−0.0107	1.5899	0.2683	0.008
	0.375 mm	0.7958	0.1648	0.3129	−0.0826	1.0145	0.0981	0.007
	0.3 mm	0.8716	0.1271	0.869	−0.1448	0.5844	−0.0237	0.008
	0.15 mm	0.8668	0.1135	1.112	−0.1799	0.7906	0.1047	0.01