# On the Radiative Transfer Model for Soil Moisture across Space, Time and Hydro-Climates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{B}(H- and V-polarization) to physical variables (soil moisture, soil texture, surface roughness, surface temperature, and vegetation characteristics) is studied. Our results indicate that the sensitivity of brightness temperature (V- or H-polarization) is determined by the upscaling method and heterogeneity observed in the physical variables. Under higher heterogeneity, the T

_{B}sensitivity to vegetation and roughness followed a logarithmic function with an increasing support scale, while an exponential function is followed under lower heterogeneity. Surface temperature always followed an exponential function under all conditions. The sensitivity of T

_{B}at H- or V- polarization to soil and vegetation characteristics varied with the spatial scale (extent and support) and the amount of biomass observed. Thus, choosing an H- or V-polarization algorithm for soil moisture retrieval is a tradeoff between support scales, and land surface heterogeneity. For largely undisturbed natural environments such as SGP’97 and SMEX04, the sensitivity of T

_{B}to variables remains nearly uniform and is not influenced by extent, support scales, or an upscaling method. On the contrary, for anthropogenically-manipulated environments such as SMEX02 and SMAPVEX12, the sensitivity to variables is highly influenced by the distribution of land surface heterogeneity and upscaling methods.

## 1. Introduction

_{B}) followed by soil moisture retrieval [4,5,6,7]. Several scaling algorithms have been developed in the past using data fusion techniques integrating information from various scales (point/airborne/satellite), platforms (MODIS/LANDSAT/AVHRR etc.), sensors (active/passive), frequencies (P, L, C, and X), and land surface variables (surface temperature, NDVI, etc.) [8,9,10,11,12]. For most cases, coarse-scale knowledge is estimated from smaller spatial scale features in which upscaling has reduced to the problem of a change of support scale. Western and Bloschl (1999) [13] defined the scale triplet (support, spacing, and extent), where support is referred to the area (or volume) integrated by individual soil moisture measurements. The scaling methods incorporate their own conceptual model uncertainties apart from the uncertainties of products used for downscaling/upscaling [14,15,16]. Thus, for downscaling/upscaling of either soil moisture or T

_{B}, it first calls for understanding the propagation of errors/uncertainties through the scale and heterogeneity of land surface variables.

## 2. Materials and Method

#### 2.1. Heterogeneity Observed in Various Hydroclimates

**.**These field campaigns were conducted over approximately a 4–8 weeks’ window, primarily to validate soil moisture retrieval algorithms over a wide range of soil and vegetation conditions [31,32,33]. The airborne remote sensing data for these field campaigns were observed at either 800 m or 1.5 km resolution, which is the base resolution for our analysis. The 800 m pixels are upscaled to support scales of 1.6 km, 3.2 km, 6.4 km, and 12.8 km, and 1.5 km pixels are upscaled to 3 km and 9 km to coincide with two of the Soil Moisture Active Passive (SMAP) target products. The variables such as soil moisture, surface temperature, vegetation water content, and soil texture are obtained from various sources, which are discussed under each field campaign. Variables such as surface roughness, vegetation structure, and single scattering albedo, which are available only at sparse locations, are assumed to be scalars with a uniform probability distribution whose ranges are estimated from extensive field measurements.

#### 2.1.1. Southern Great Plains Experiment’1997 (SGP’97), Oklahoma

#### 2.1.2. Soil Moisture Experiment’2002 (SMEX02), Iowa

#### 2.1.3. Soil Moisture Experiment’2004 (SMEX04), Arizona

#### 2.1.4. Soil Moisture Active Passive Validation Experiment’2012 (SMAPVEX12), Winnipeg

#### 2.2. Soil Moisture Retrieval Algorithm

_{c}is the effective vegetation temperature [K]; ${e}_{p,\theta ,f}$ is the emissivity of the (rough) soil surface; ${r}_{p,f,\theta}$ is the rough surface reflectivity; ${\tau}_{p,f}$ is the nadir optical depth; ${\omega}_{p,f,\theta}$ is the single scattering albedo. The subscripts p, θ, and f denote the polarization, angle of incidence, and frequency of measurement, respectively. It is assumed that the canopy and soil temperature are equal during morning thermal crossover, which also corresponds to the planned time of observation for SMAP at around 6:00 h local sun time.

#### 2.3. Global Spatial Sensitivity Analysis: Sobol Method

**Y**) into partial variances caused due to each model input (

**X**) (either considered singularly or in combination), i.e.,

**E**and

**V**are the expectation and variance operators; refer to the Appendix A for more details. The so-called “Sobol” indices or “variance-based sensitivity indices” are obtained as follows:

**First Order Sensitivity Index**${\mathit{S}}_{\mathit{i}}=\frac{{V}_{i}\left(Y\right)}{V\left(Y\right)},$ the amount of variance of Y explained by input variable X

_{i};

**Second Order Sensitivity Index**${\mathit{S}}_{\mathit{i}\mathit{j}}=\frac{{V}_{ij}\left(Y\right)}{V\left(Y\right)},$ the amount of variance of Y explained by the interaction of the factors X

_{i}and X

_{j}(i.e., sensitivity to X

_{i}and X

_{j}not expressed in V

_{i}nor V

_{j});

**Total Sensitivity Index**${\mathit{S}}_{\mathit{T}\mathit{i}}={S}_{i}+{{\displaystyle \sum}}_{ij}{S}_{ij}+{{\displaystyle \sum}}_{ijl}{S}_{ijl}+\dots {S}_{1,2,\dots k}$, which accounts for all the contributions to the output variation due to factor X

_{i}(i.e., the first-order index plus all its interactions). In other words, the ${S}_{Ti}$ index is defined as a summation of the main, second, and higher order effects, which involves the evaluation over a full range of parameter space. If ${S}_{i}$ and ${S}_{Ti}$ are equal and the sum of ${S}_{i}$ (and thus ${S}_{Ti})$ is 1, then the model is additive (linear) in nature, otherwise if ${S}_{Ti}$ is greater than ${S}_{i}$ and ∑ S

_{i}< 1 or ∑ S

_{Ti}> 1, then the model exhibits non-linearity. Thus, linearity or non-linearity of model through time, scale, and hydroclimate can be determined from a summation of ${S}_{Ti}\u2019$s indices.

**Y**= f(

**X**) be a spatial model, then Y is the model output and

**X**are random independent model inputs, where

**X**is composed of

**U**and

**W**, with

**U**defined as random vectors and

**W**defined as two-dimensional spatial maps. Using the similar configuration,

**Y**is the simulated brightness temperature, f is the radiative transfer model, and

**X**is the input parameter vector with K = [X

_{K}_{1}, X

_{2}, X

_{3}, X

_{4}] as [Surface Roughness-RMS height (S), Surface Roughness-Correlation length (L), Vegetation Structure (B), Scattering Albedo (ω)] and

**W**as spatial maps [Soil Moisture (SM), Clay Fraction (CF), Surface Temperature (TSURF), Vegetation Water Content (VWC)]. The steps used to carry out the analysis are described in Figure 1. The variables such as RMS height (S) and correlation length (L) are sampled from a uniform distribution whose ranges are estimated from field measurements. The variables such as vegetation structure (B) and scattering albedo (ω) are also sampled from a uniform distribution whose ranges are defined based on vegetation type. However, experimental data for these variables are limited and are obtained from past literature [42,53]. The range of variability in surface roughness, vegetation parameter, and scattering albedo are assumed to remain constant at all scales, as there are no means to ascertain how those vary with scale. The mean values of land surface variables used in the analysis for various field campaigns are represented in Table 1. The spatial input variable at each support scale is defined as a real random variable with a uniform range for the number of pixels developed at that scale. That is, the numbers of pixels ‘n’ at each support scale are considered as equiprobable, with each pixel labeled with a unique integer in the set {1, …, n}. The number of pixels decreases with increasing the support scale, for example, the numbers of pixels at a 800 m support scale are more than 4000, and at a 12.6 km support scale, there are about 50 pixels. The first order and total Sobol sensitivity indices are estimated with confidence intervals using the bootstrap technique with resampling [54]. The number of samples N = 55,000 are used for model evaluations. Here, we discuss and present only total sensitivity indices.

#### 2.4. Upscaling Methods: Linear Upscaling vs. Inverse Distance Weighted (IDW) Upscaling

## 3. Results and Discussion

#### 3.1. Plant Structure

**E**is normal to the axis of the stalk at all incidence angles, and for a vertically polarized wave,

**E**is parallel to stalk, thereby coupling strongly [77]. This difference will be significant mainly over vegetation that exhibits some preferential orientation such as vertical stalks noticed for tall grasses, grains, wheat, corn, maize, forest, etc. For vegetation with near horizontally oriented primary branches or broad leaf plants (soybean, bean, canola, sunflower, etc.), there is stronger attenuation for a horizontally polarized wave than vertically polarized waves. It is reasonable to assume that the absorption/scattering loss factor will be approximately polarization independent for canopy components whose sizes are much smaller than the wavelength of observation.

^{2}. Due to the low vegetation and primarily random vegetation structure, the sensitivity to vegetation variables is insignificant for this region temporally, at all support scales and polarizations (Figure 3). Similarly, the SGP97 region (Oklahoma), which is dominantly rangeland and has a significant occupation of winter wheat crops with a total mean vegetation water content of 0.323 kg/m

^{2}, shows the insignificant contribution of vegetation variables towards brightness temperature (V- and H-polarization) at all spatial scales (Figure 4). SMEX02 (Iowa) has a total mean vegetation water content of 1.9 kg/m

^{2}. At the 800 m scale, H-polarization was observed to be (~1–4%) more sensitive than V-polarization. With an increasing support scale, V-polarization was observed to be ~2% (at 1.6 km) and ~15% (at 12.8 km) more sensitive than H-polarization for vegetation water content (VWC) and vegetation structure (B) (Figure 5). Under IDW analysis, H-polarization was always temporally more sensitive than V-polarization to vegetation water content by ~2–3% and vegetation structure (B) by ~3–9%, except on DOY 188, where V- and H-polarization showed the same sensitivities (Figure 5). That is, at lower vegetation water content (DOY 178, 182) and a support scale of 800 m under linear upscaling and at all support scales under IDW where the heterogeneity in VWC is maintained, a higher sensitivity to VWC and B at H-polarization is noticed. For DOY 188, with higher vegetation water content and higher support scales under linear upscaling where the heterogeneity disappeared with support scales (moving towards extent mean value), a higher sensitivity to vegetation structure (B) at V-polarization is observed. This confirmed that there is a stronger coupling with plants exhibiting definite vertical structures, and the coupling increases with increasing vegetation water content and decreasing heterogeneity. The SMAPVEX12 region shows a wide variety of agricultural crops such as cereals, soybeans, canola, corn, etc., with a total mean vegetation content water of 1.4 kg/m

^{2}. Unlike in SMEX02, the sensitivity to vegetation structure (B) was always observed to be higher for H-polarization at all scales. We hypothesize that the amount of vegetation water content determines the sensitivity of H- and V-polarization to the B variable. To test the hypothesis, the analysis was conducted including pixels with higher vegetation water content than those observed within the SMAPVEX12 extent, which resulted in higher sensitivity to vegetation structure (B) at V-polarization. This may occur because with increasing VWC, the dielectric constant of vegetation also increases, thereby increasing the sensitivity to V-polarization due to stronger coupling, as mentioned earlier. Thus, vegetation variable ‘B’ is a complex function of plant geometry and vegetation water content.

#### 3.2. Spatio-Temporal Scales in Different Hydroclimates

#### 3.2.1. Semi-Arid (SMEX04) Hydroclimate

^{2}= 0.99) decrease (increased) in sensitivity to surface temperature (RMS height, S) with support scale. A general decrease in R

^{2}is noticed for the for H-pol, and this is observed across all hydroclimates, as noted in Figure 3. Due to the rocky topography of SMEX04 region, the observed surface roughness is highly random, emphasizing the higher sensitivity to RMS height (S), which increased from ~11% (~63%) to ~38% (~87%) for V-polarization (H-polarization), and correlation length (L) increased from ~1% (~7%) to ~5% (~10%). This is in contrast to SMAPVEX12 and SMEX02, where the surface roughness is more correlated due to regular agricultural practices resulting in higher sensitivity to correlation length (L) as well. The sensitivity to vegetation variables (VWC, B, and ω) and clay fractions are negligible across both polarizations (H and V), upscaling methods, and spatio-temporal scales.

#### 3.2.2. Sub-Humid (SGP97) Hydroclimate

^{2}) for a smaller extent than observed at a larger extent (SM ~0.14 v/v and VWC ~0.32 kg/m

^{2}).

#### 3.2.3. Humid-Dfa (SMEX02) Hydroclimate

^{2}= ~0.95) and nearly disappeared at 1.6 km for H- and at 3.2 km for V-polarization. On the other hand, the sensitivity to surface roughness variables (S and L) followed a logarithmic function (R

^{2}= ~0.97) with an increasing support scale. Similarly, vegetation water content (VWC) and vegetation structure (B) followed a decreasing (R

^{2}= 0.96) and increasing logarithmic function. This is because of the greater variability in surface roughness and vegetation variables observed, particularly for the anthropogenically-modified region such as SMEX02. Thus, the impact of these variables is observed longer with a support scale following a logarithmic function, compared to the surface temperature that followed an exponential function. Under IDW upscaling, the sensitivity to land surface variables with support scales changed by ~1–2%. Sensitivity to variables such as soil moisture, surface temperature, and surface roughness remained uniform until 6.4 km, which later decreased by ~2% for soil moisture and surface temperature, and increased for surface roughness by ~2% at 12.8 km. The uniformity in sensitivity to these variables until 6.4 km under IDW is similar to the reasons explained above. However, for a selected extent, the variability in some land surface variables attains saturation at a certain scale (6.4 km), resulting in a decrease in sensitivity to those land surface variables upon further upscaling. The support scale at which this saturation is attained is dependent on the heterogeneity and extent of the analysis. The sensitivity to vegetation water content remained uniform until 3.2 km and decreased ~1% with an increasing support scale, and sensitivity to vegetation structure (B) increased linearly (R

^{2}=0.98) by ~6% with the scale. The sensitivity to clay fraction and albedo remained negligible under IDW and linear upscaling. Except for DOY 178, which was a relatively dry day, the sensitivity to clay fraction increased to ~3% at V-polarization and ~1% at H-polarization until 3.2 km.

#### 3.2.4. Humid-Dfb (SMAPVEX12) Hydroclimate

^{2}= 0.99) with scale. It is noteworthy that the sensitivity to VWC slightly increased with scale.

#### 3.3. Upscaling and Environmental Heterogeneity

#### 3.3.1. Homogeneous Environment (Sub-Humid and Semi-Arid)

#### 3.3.2. Heterogeneous Environment (Humid Dfa and Humid Dfb)

## 4. Conclusions

_{B}to variables remained nearly uniform or followed an exponential function with support scales. Under higher heterogeneity, the T

_{B}sensitivity to vegetation and roughness followed a logarithmic function with an increasing support scale. Surface temperature always followed an exponential function under all conditions. The upscaling methods play a significant role in representing non-linear effects of heterogeneous landscapes, as processes and variables change their significance with scale. It is recommended that the most sensitive variables to T

_{B}(H- or V-) should be upscaled using non-linear (e.g., IDW) methods to preserve spatial heterogeneity, and this is especially important under heterogeneous environments. The study emphasized that the observed heterogeneity and upscaling method will determine the sensitivity to land surface variables. This study is particularly relevant in order to establish the significant variables that can be used for downscaling and upscaling T

_{B}algorithms to various scales and under heterogeneous landscapes.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**X**), where Y is the output,

**X**= (${X}_{1},\dots {X}_{k})$ are k independent inputs, and f is the model function. If all the factors

**X**are allowed to vary over their range of uncertainty, then the corresponding uncertainty of the model output

**Y**is defined by its unconditional variance

**V**(

**Y**). The input factors are ranked based on the amount of variance removed from the output when we learn the true value of a given input factor X

_{i}. That is, ${V}_{{X}_{~i}}(Y|{X}_{i}={x}_{i}^{*}$) would be the resulting conditional variance as it is fixed to its true value ${x}_{i}^{*}$ taken over all ${X}_{~i}$ (all factors but ${X}_{i}$). However, the true value ${x}_{i}^{*}$ is unknown for each ${X}_{i}$. Hence, an average of ${V}_{{X}_{~i}}(Y|{X}_{i}={x}_{i}^{*}$) measured over all possible values ${x}_{i}^{*}$ will remove the dependence of ${x}_{i}^{*}$, thereby resulting in ${E}_{{X}_{i}}\left[{V}_{{X}_{~i}}\left(Y|{X}_{i}\right)\right].$ Using the following property (Mood et al., 1950) [80],

_{i}is an important factor. The conditional variance ${V}_{{X}_{i}}\left[{E}_{{X}_{~i}}\left(Y|{X}_{i}\right)\right]$ is the first-order (e.g., additive) effect of X

_{i}on Y and the associated first-order sensitivity index of X

_{i}on Y measure, which is written as ${\mathit{S}}_{\mathit{i}}=\frac{{\mathit{V}}_{{\mathit{X}}_{\mathit{i}}}\left[{\mathit{E}}_{{\mathit{X}}_{~\mathit{i}}}\left(\mathit{Y}|{\mathit{X}}_{\mathit{i}}\right)\right]}{\mathit{V}\left(\mathit{Y}\right)},$ while ${E}_{{X}_{i}}\left[{V}_{{X}_{~i}}\left(Y|{X}_{i}\right)\right]$ is customarily called the residual. As such, ${S}_{i}$ is a number always between 0 and 1. Sobol [48] introduced the decomposition of model function f into summands of increasing dimensionality:

**First Order Sensitivity Index**${\mathit{S}}_{\mathit{i}}=\frac{{\mathit{V}}_{\mathit{i}}\left(\mathit{Y}\right)}{\mathit{V}\left(\mathit{Y}\right)};$

**Second Order Sensitivity Index**${\mathit{S}}_{\mathit{i}\mathit{j}}=\frac{{V}_{ij}\left(Y\right)}{V\left(Y\right)};$ the amount of variance of

**Y**explained by the interaction of the factors X

_{i}and X

_{j}(i.e., sensitivity to X

_{i}and X

_{j}not expressed in individual X

_{i}and X

_{j});

**Total Sensitivity Index**${\mathit{S}}_{\mathit{T}\mathit{i}}={S}_{i}+{{\displaystyle \sum}}_{ij}{S}_{ij}+{{\displaystyle \sum}}_{ijl}{S}_{ijl}+\dots {S}_{1,2,\dots k}$; this accounts for all the contributions to the output variation due to factor X

_{i}(i.e., first-order index plus all its interactions). ${S}_{Ti}$ can also be defined by decomposing the output variance V(Y), in terms of main effect and residual, conditioning with respect to all factors but one, i.e., ${\mathit{X}}_{~\mathit{i}}$

**.**The unconditional variance can be rewritten as: $V\left(Y\right)-{V}_{{X}_{~i}}\left[{E}_{{X}_{i}}\left(Y|{X}_{~i}\right)\right]={E}_{{X}_{~i}}\left[{V}_{{X}_{i}}\left(Y|{X}_{~i}\right)\right]$, and dividing both sides by unconditional variance results in: ${\mathit{S}}_{\mathit{T}\mathit{i}}=\mathbf{1}-\frac{{\mathit{V}}_{{\mathit{X}}_{~\mathit{i}}}\left[{\mathit{E}}_{{\mathit{X}}_{\mathit{i}}}\left(\mathit{Y}|{\mathit{X}}_{~\mathit{i}}\right)\right]}{\mathit{V}\left(\mathit{Y}\right)}=\frac{{\mathit{E}}_{{\mathit{X}}_{~\mathit{i}}}\left[{\mathit{V}}_{{\mathit{X}}_{\mathit{i}}}\left(\mathit{Y}|{\mathit{X}}_{~\mathit{i}}\right)\right]}{\mathit{V}\left(\mathit{Y}\right)}$.

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**Figure 2.**Propagation of an unpolarized microwave radiation incident on a lossy dielectric structured vegetation (

**top**) and unstructured/bushy (

**bottom**) vegetation. The unpolarized incident wave propagates through structured vegetation emitting H-polarized wave and V-polarized radiation through bushy vegetation.

**Figure 3.**Soil Moisture Experiments 2004 (SMEX04), Total Sensitivity Index (TSI) for Brightness Temperature under V-polarization, and H-polarization using Inverse Distance Weighted (IDW) and Linear Upscaling. SM: Soil Moisture; CF: Clay Fraction; S: Root Mean Square Height; L: Correlation Length; TSURF: Surface Temperature; VWC: Vegetation Water Content; B: Vegetation Structure; ω: Single Scattering Albedo. The red and brown curves represent an exponential fit to TSURF (expo_TSURF, V-pol: R

^{2}= 0.997 & H-pol: R

^{2}= 0.99) and to S (expo_S, V-pol: R

^{2}= 0.97 & H-pol: R

^{2}= 0.89), respectively. The symbols represent the TSI values of the respective variables, while R

^{2}represents the goodness of fit.

**Figure 4.**Southern Great Plains 1997 (SGP’97), Total Sensitivity Index (TSI) for Brightness Temperature under V-polarization, and H-polarization using Inverse Distance Weighted (IDW) and Linear Upscaling. SM: Soil Moisture; CF: Clay Fraction; S: Root Mean Square Height; L: Correlation Length; TSURF: Surface Temperature; VWC: Vegetation Water Content; B: Vegetation Structure; ω: Single Scattering Albedo.

**Figure 5.**Soil Moisture Experiments 2002 (SMEX02), Total Sensitivity Index (TSI) for Brightness Temperature under V-polarization, and H-polarization using Inverse Distance Weighted (IDW) and Linear Upscaling. SM: Soil Moisture; CF: Clay Fraction; S: Root Mean Square Height; L: Correlation Length; TSURF: Surface Temperature; VWC: Vegetation Water Content; B: Vegetation Structure; ω: Single Scattering Albedo. The brown and green curves represent a logarithmic fit to S (log_S, V-pol: R

^{2}= 0.97 & H-pol: 0.87) and to VWC (log_VWC, V-pol: R

^{2}= 0.96 & H-pol: 0.93), while the red curve represents an exponential fit to TSURF (expo_TSURF, V-pol: R

^{2}= 0.95 & H-pol: 0.94). The symbols represent the TSI values of the respective variables, while R

^{2}represents the goodness of fit.

**Figure 6.**Soil Moisture Active Passive Experiments 2012 (SMAPVEX12), Total Sensitivity Index (TSI) for Brightness Temperature under V-polarization, and H-polarization using Inverse Distance Weighted (IDW). SM: Soil Moisture; CF: Clay Fraction; S: Root Mean Square Height; L: Correlation Length; TSURF: Surface Temperature; VWC: Vegetation Water Content; B: Vegetation Structure; ω: Single Scattering Albedo. The green and mint-green curves represent an exponential fit to VWC (expo_VWC, V-pol: R

^{2}= 0.998 & H-pol: 0.99) and to B (V-pol: R

^{2}= 0.98 & H-pol: 0.999). The symbols represent the TSI values of the respective variables, while R

^{2}represents the goodness of fit.

**Figure 7.**The conceptual model describes the classification of environments as homogenous and heterogeneous environments based on density and heterogeneity in biomass (green). Each of these environments can further be classified as water (blue) and energy (red) rich environments.

**Top-Left**quadrant represents a heterogeneous water rich environment, where brightness temperature is most sensitive to vegetation and soil moisture with higher order interactions of ~5–10%;

**Top-Right**quadrant represents a heterogeneous energy rich environment, where brightness temperature is most sensitive to vegetation and temperature with higher order interactions of >10–15%;

**Bottom-Left**quadrant represents a homogeneous water rich environment, where brightness temperature is most sensitive to soil moisture with no or very low higher order interactions;

**Bottom-Right**quadrant represents a homogeneous energy rich environment, where brightness temperature is most sensitive to soil moisture and temperature with higher order interactions of <5–10%. The transition from homogenous to heterogeneous, energy rich to water rich, and vice versa, can occur through spatio-temporal scales (spatial: extent and support; time: day, month, seasonality, climate change, etc.) following land-use/land-cover change and events such as precipitation, evapotranspiration, etc. The lasting transition from homogeneous to heterogeneous and vice versa can also occur at the climate change temporal scales.

**Table 1.**The mean values of land surface variables used in the analysis for various field campaigns.

Southern Great Plains 1997 (SGP97) | Soil Moisture Experiments 2002 (SMEX02) | Soil Moisture Experiments 2004 (SMEX04) | Soil Moisture Active Passive Validation Experiments 2012 (SMAPVEX12) |
---|---|---|---|

Oklahoma (Sub-Humid) | Iowa (Dfa-Humid | Arizona (Semi-Arid) | Winnipeg (Dfb-Humid) |

Mean Soil Moisture: 0.14 v/v | Mean Soil Moisture: 0.19 v/v | Mean Soil Moisture: 0.07 v/v | Mean Soil Moisture: 0.25 v/v |

Mean Soil Temperature: 298 K | Mean Soil Temperature: 315 K | Mean Soil Temperature: 319 K | Mean Soil Temperature: 290 K |

Mean Vegetation Water Content: 0.32 kg/m^{2} | Mean Vegetation Water Content: 1.9 kg/m^{2} | Mean Vegetation Water Content: 0.09 kg/m^{2} | Mean Vegetation Water Content: 1.4 kg/m^{2} |

Root Mean Square (RMS) height (cm): 0.27–1.73 | Root Mean Square (RMS) height (cm): 0.19–3.05 | Root Mean Square (RMS) height (cm): 0.71–23.28 | Root Mean Square (RMS) height (cm): 0.23–3.21 |

Correlation length (L) (cm): 3.4–32.18 | Correlation length (L) (cm): 0.43–26.95 | Correlation length (L) (cm): 8.7–119.5 | Correlation length (L) (cm): 2.5–24.5 |

Vegetation structure (B): 0–0.15 | Vegetation structure (B): 0–0.15 | Vegetation structure (B): 0–0.15 | Vegetation structure (B): 0–0.15 |

Scattering albedo (ω): 0–0.05 | Scattering albedo (ω): 0–0.05 | Scattering albedo (ω): 0–0.05 | Scattering albedo (ω): 0–0.05 |

ESTAR support scales: 0.8 km,1.6 km, 3.2 km, 6.4 km, 12.8 km | PSR/C support scales: 0.8 km,1.6 km, 3.2 km, 6.4 km, and 12.8 km | PALS support scales: 0.8 km,1.6 km, 3.2 km, 6.4 km, and 12.8 km | PALS support scales: 1.5 km,3 km, and 9 km |

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Neelam, M.; Mohanty, B.P.
On the Radiative Transfer Model for Soil Moisture across Space, Time and Hydro-Climates. *Remote Sens.* **2020**, *12*, 2645.
https://doi.org/10.3390/rs12162645

**AMA Style**

Neelam M, Mohanty BP.
On the Radiative Transfer Model for Soil Moisture across Space, Time and Hydro-Climates. *Remote Sensing*. 2020; 12(16):2645.
https://doi.org/10.3390/rs12162645

**Chicago/Turabian Style**

Neelam, Maheshwari, and Binayak P. Mohanty.
2020. "On the Radiative Transfer Model for Soil Moisture across Space, Time and Hydro-Climates" *Remote Sensing* 12, no. 16: 2645.
https://doi.org/10.3390/rs12162645