# Advancing NASA’s AirMOSS P-Band Radar Root Zone Soil Moisture Retrieval Algorithm via Incorporation of Richards’ Equation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{2}with FLUXNET tower sites in regions ranging from boreal forests in Saskatchewan, Canada, to tropical forests in La Selva, Costa Rica. The radar snapshots are used to generate estimates of the SMPs via inversion of scattering models of vegetation overlying soils with variable soil moisture distributions. The retrieved SMPs are in turn assimilated or otherwise applied by hydrologists to estimate land surface model hydrological parameters over the nine biomes, generating a high-resolution time record of root zone soil moisture evolution. These hydrological parameters are ultimately integrated with an ecosystem demography model to predict the respiration and photosynthesis carbon fluxes [14].

## 2. Mathematical Derivations

#### 2.1. Richards’ Equation

^{−1}] is the water flux density, θ [L

^{3}L

^{−3}] is the soil water content, h [L] is the pressure head (absolute values are considered here for convenience), K [LT

^{−1}] is the unsaturated hydraulic conductivity, t [T] is time, and z [L] is soil depth assumed positive downward from the soil surface.

#### 2.2. New Solution to Richards’ Equation

_{s}and θ

_{r}are the saturated and residual volumetric water contents, respectively, K

_{s}is the saturated hydraulic conductivity, P is an empirical parameter related to the soil pore size distribution, and h

_{cM}is the effective capillary drive introduced by Morel-Seytoux and Khanji [35]. As shown below in Equation (31), P is related to the van Genuchten parameter m. Assuming P = 1, the RE is reduced to a linearized form for which analytical solutions to various flow processes exist [16,17,18]. However, the assumption P = 1 is rarely met under realistic conditions; most soils exhibit a P significantly larger than 1 [33]. Thus, P is treated as a soil parameter in the following derivations, where the resulting nonlinear RE is solved analytically. Note that the soil hydraulic parameters θ

_{s}, θ

_{r}, P, K

_{s}, and h

_{cM}are constant for a given soil and do not change with time. Therefore, they should be discerned from the so-called “free parameters”, which vary temporally to fit the SMP at any given time.

_{1}, a

_{2}, and a

_{3}are the integral constants. Combining Equations (10), (11), (14), (18) and (19), and denoting the constants b

_{1}, b

_{2}, and b

_{3}, a simple closed-form solution is obtained:

_{r}is negligibly small, allows transformation of Equation (20) to the final SMP solution:

_{1}, c

_{2}, and c

_{3}are the combined constants or the final free parameters of the RE-based SMP model.

_{1}, c

_{2}, and c

_{3}(i.e., ∂θ/∂c

_{1}, ∂θ/∂c

_{2}, and ∂θ/∂c

_{3}) is proportional to θ

^{1−P}that yields very large numbers for most cases. Therefore, small changes in c

_{1}, c

_{2}, and c

_{3}can lead to a significant change of the SMP shape. This means that direct derivation of c

_{1}, c

_{2}, and c

_{3}through inversion is not as straightforward as proposed in [14], for example, to find the optimum polynomial parameters. We resolve this problem by replacing the free model parameters (c

_{1}, c

_{2}, and c

_{3}) with soil moisture values at three arbitrary depths, θ

_{1}(z

_{1}), θ

_{2}(z

_{2}), and θ

_{3}(z

_{3}). Thereby we capitalize on both the physical significance of the free model parameters and the desirable sensitivity of the SMP, Equation (21), to variations of θ

_{1}, θ

_{2}, and θ

_{3}. Assuming that Equation (21) coincides with these three points, then c

_{1}, c

_{2}, and c

_{3}can be calculated directly from θ

_{1}, θ

_{2}, and θ

_{3}as follows:

#### 2.3. Second Order Polynomial Approximation

_{cM}larger than the maximum depth of interest (i.e., z/h

_{cM}< 1), all the terms with orders higher than 2 can be neglected. Hence Equation (26) can be reduced to the second order polynomial presumed in [14]:

## 3. Validation of the Proposed SMP Model

#### 3.1. Numerical Data

^{−1}and atmospheric pressure head of −1000 m were assumed. To simulate the drying process after a wetting event, the initial soil profile was assumed to be saturated from z = 0 to 30 cm and air-dry from z = 30 to 50 cm. Hydraulic properties of three vastly different soil textures including sand, loam and clay were used to parameterize the HYDRUS simulations. The van Genuchten (VG) [30] soil hydraulic functions were applied with HYDRUS default soil hydraulic parameters listed in Table 1:

_{cM}for each soil (presented in each plot), first an initial guess was made. Then three arbitrary data points, θ

_{1}(z

_{1}), θ

_{2}(z

_{2}), and θ

_{3}(z

_{3}), from the Hydrus simulations were used to calculate c

_{1}, c

_{2}, and c

_{3}with Equation (22) to Equation (25). Finally, P and h

_{cM}were optimized to visually best fit the simulation results.

_{cM}) requires measurement of the soil hydraulic functions, which is impractical for large-scale applications. To avoid this complication, the following approximations for P and h

_{cM}from VG parameters is proposed:

_{cM}and the same K at θ* = 0.5. These equations are introduced here because average VG parameters for various textures are well documented in the literature [40] (Table 1) and they can be approximated with pedotransfer functions from easy-to-measure textural properties (sand, silt and clay percentages) [41]. The calculated parameters P and h

_{cM}based on Equations (31) and (32) with VG parameters provided in [40] for the 12 soil textural classes of the United States Department of Agriculture (USDA) classification scheme are listed in Table 1.

_{cM}= 350 cm (i.e., the optimum values obtained from HYDRUS-1D simulations shown in Figure 1) instead of the values listed in Table 1.

_{cM}>> 50 cm. Based on the values shown in Table 1, it is unlikely to find a natural soil which meets this condition, since P = 1 corresponds to an extremely coarse-textured soil and large h

_{cM}corresponds to a fine-textured soil. This mismatch highlights the advantage of applying the general model, Equation (21), rather than its reduced form, Equation (28), in the AirMOSS algorithm.

_{cM}, as applied for the evaporation process shown in Figure 1 (top row). The small discrepancies are due to the nature of the analytical solution. According to Equation (20), an initial soil moisture profile dries with time if b

_{1}, b

_{2}and b

_{3}remain constant. This means that the constants of Equation (20) vary with time in the SMPs depicted in Figure 3. This time-dependency of constants contradicts the underlying assumption of the separation of variables method. Hence, while application of Equation (21) for a wetting process is possible in practice, it is theoretically not justifiable.

#### 3.2. Measured Data

_{cM}= 350 cm were used for Equation (21). For both Equations (21) and (28), θ values measured at 5, 20 and 50 cm depths were applied for direct calculation of the three free model parameters. It was assumed that below the dynamic zone the water content is uniform and equal to that of the wetting front. It is well documented that at later evaporation stages, a dry zone develops close to the soil surface and the so-called drying front (or vaporization plane) recedes below the surface. In this case, the Buckingham–Darcy law is not applicable for modeling soil water content above the drying front without accounting for the vapor flow contribution, because the pressure head gradient approaches infinity at the drying front [44]. Therefore, it was assumed for Equation (21) that soil water content above the drying front (i.e., where Equation (21) intersects θ = 0) is zero.

_{cM}= 87 cm. Equation (21) was visually fitted to measured data via varying θ

_{1}(z

_{1}), θ

_{2}(z

_{2}), and θ

_{3}(z

_{3}) in Equations (22)–(24).

## 4. AirMOSS Retrieval Algorithm

#### 4.1. Background

#### 4.2. Considerations Regarding the New Model Application

_{1}, θ

_{2}and θ

_{3}at three arbitrary depths (z

_{1}, z

_{2}and z

_{3}) are optimized instead of a, b and c. Therefore, the problem of finding the upper and lower bounds is more straightforward because of the physical meaning of θ

_{1}, θ

_{2}and θ

_{3}from which free parameters of Equation (21) can be directly calculated with Equations (22)–(24).

_{1}, θ

_{2}and θ

_{3}shown in Figure 7 are discussed below.

_{2}> θ

_{1}and θ

_{3}. Equation (21) is always valid for this case. Case B is similar to later drying times for which θ

_{3}> θ

_{2}> θ

_{1}. Equation (21) is valid for this case, unless θ

_{3}is larger than a critical value θ

_{c}. The critical value can be approximated with Equation (33) that ensures validity of Equation (21) in most cases:

_{3}< θ

_{c}is necessary for inversion from a mathematical point of view, although the condition θ

_{3}> θ

_{c}is rarely observed in natural settings. Therefore, the new solution can be easily applied to the profiles of cases A and B. These two cases can be merged into a single case when the two constraints of θ

_{1}< θ

_{2}and θ

_{3}< θ

_{c}are satisfied.

_{2}< θ

_{1}and θ

_{3}, however, cannot be predicted with Equation (21), as θ is undefined within part of the profile where c

_{1}z + c

_{2}exp (z/h

_{cM}) + c

_{3}is negative. For such a case, it is suggested to release the two constraints for cases A and B and rather assume P = 1 in order to replace Equation (21) with Equation (26), which is approximately the same as the polynomial, Equation (28), as indicated in Figure 7.

#### 4.3. Preliminary Inversion Results

_{cM}, did not require extra information as they were directly estimated from soil texture (Table 1). The Metolius pixel exhibits a loamy sand soil with P = 5.52 and h

_{cM}= 2.94 cm, and the BERMS pixel a sandy loam soil with P = 6.73 and h

_{cM}= 5.70 cm.

_{1}, z

_{2}, and z

_{3}have been chosen to coincide with 0, 20, and 45 cm. The applied inversion parameters are identical to the ones that had been previously used for these sites with the polynomial function assumption. The lower bounds for the unknowns, namely θ

_{1}, θ

_{2}and θ

_{3}, are 0.01, 0.05, and 0.01 m

^{3}/m

^{3}, respectively, for both sites. Upper bounds for the unknowns are 0.10, 0.15, and 0.1 m

^{3}/m

^{3}, respectively, for both sites. These bounds were chosen based on in situ soil moisture data.

_{1}, θ

_{2}and θ

_{3}is lower than the sensitivity of the polynomial to variations of a, b and c, which are in fact a special case of c

_{1}, c

_{2}and c

_{3}. Secondly, the unknowns of the new solution are physical parameters. Therefore, the corresponding lower and upper bounds could be deduced with more confidence from in situ data when compared to the second order polynomial. Note that large dynamic ranges of the unknowns would result in inaccurate SMP retrievals, due to the limited length of the Markov chain in the simulated annealing method, discussed in [14], and the ill-posedness of the problem.

_{1}< θ

_{2}and θ

_{3}< θ

_{c}). For the case that the error between measured and calculated backscattering coefficients cannot be satisfactorily minimized with these constrains, case C can then be assumed (i.e., P = 1).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The best fit of Equation (21) (solid lines) to HYDRUS-1D simulation results (circles) for simultaneous evaporation and drainage in three different soils. The soil parameters in Equation (21), P and h

_{cM}, were treated as fitting parameters (top row), taken from Table 1 (middle row), or were set to P = 1 and h

_{cM}>> 50 cm leading to the second order polynomial (bottom row). Data below the wetting front were not considered for least square fitting. The fitting accuracy is quantified with the mean absolute error, ε (cm

^{3}cm

^{−3}).

**Figure 2.**The best fit of Equation (21) (solid lines) to HYDRUS-1D simulation results (circles) for evaporation in the presence of a shallow water table at z = 1 m for three different soil textures.

**Figure 3.**The best fit of Equation (21) (lines) to HYDRUS-1D simulation results (circles) for a constant-flux (= 1 cm h

^{−1}) infiltration process into a loam soil with parameters given in Table 1.

**Figure 4.**Comparison of Equations (21) (new solution) and Equation (28) (second order polynomial) with measured soil moisture data from Soil Climate Analysis Network (SCAN) site number 2078 (Madison, Alabama) at Julian days (JD) 31 and 40, 2013. The clay soil parameters, P = 15.9 and h

_{cM}= 350 cm, were used in Equation (21). PWP: permanent wilting point; SMP: soil moisture profile.

**Figure 5.**Comparison of Equations (21), new solution, and Equation (28), second order polynomial, with measured soil moisture data from SCAN site number 2078 (Madison, Alabama) at Julian days (JD) 87 and 96, 2012. The clay soil parameters, P = 15.9 and h

_{cM}= 350 cm, were used in Equation (21).

**Figure 6.**The best fit of Equation (21) (lines) to measured soil moisture data (dots) from SCAN site number 2026 (Walnut Gulch, Arizona) at various dates during 2010. The loam soil parameters, P = 6.6 and h

_{cM}= 87 cm, were used for fitting Equation (21). The fitting accuracy is quantified with the mean absolute error, ε (cm

^{3}cm

^{−3}).

**Figure 7.**Three possible arrangements of θ

_{1}, θ

_{2}and θ

_{3}(model parameters) throughout the drying process. Data were extracted from Figure 13a in Tabatabaeenejad et al. [14]; profile 2 (

**case A**), profile 6 (

**case B**) and profile 4 (

**case C**). Values of P = 6.6 and h

_{cM}= 87 cm were assumed.

**Figure 8.**AirMOSS retrieved soil moisture profiles using the new solution (RE) and second order polynomial for two flights over Metolius on 21 August 2015 and over BERMS on 28 September 2015. The inversion accuracy is quantified with the mean absolute error, ε (cm

^{3}cm

^{−3}). BERMS: Boreal Ecosystem Research and Monitoring Site; RE: Richards’ equation.

**Table 1.**Average van Genuchten model parameters for the 12 soil textural classes of the United States Department of Agriculture classification scheme [40] as well as parameters for Equations (2) and (3) calculated with Equations (31) and (32).

Soil Texture | θ_{r} | θ_{s} | α (cm^{–1}) | n | K_{s} (cm/Day) | P | h_{cM} (cm) |
---|---|---|---|---|---|---|---|

Sand | 0.045 | 0.43 | 0.145 | 2.68 | 712.80 | 4.83 | 2.38 |

Loamy sand | 0.057 | 0.41 | 0.124 | 2.28 | 350.20 | 5.52 | 2.94 |

Sandy loam | 0.065 | 0.41 | 0.075 | 1.89 | 106.10 | 6.73 | 5.70 |

Loam | 0.078 | 0.43 | 0.036 | 1.56 | 24.96 | 8.89 | 17.90 |

Silt | 0.034 | 0.46 | 0.016 | 1.37 | 6.00 | 11.60 | 78.94 |

Silt loam | 0.067 | 0.45 | 0.020 | 1.41 | 10.80 | 10.84 | 51.64 |

Sandy clay loam | 0.100 | 0.39 | 0.059 | 1.48 | 31.44 | 9.79 | 13.46 |

Clay loam | 0.095 | 0.41 | 0.019 | 1.31 | 6.24 | 13.05 | 100.39 |

Silty clay loam | 0.089 | 0.43 | 0.010 | 1.23 | 1.68 | 16.00 | 481.18 |

Sandy clay | 0.100 | 0.38 | 0.027 | 1.23 | 2.88 | 16.00 | 178.22 |

Silty clay | 0.070 | 0.36 | 0.005 | 1.09 | 0.48 | 31.92 | 4.19 × 10^{5} |

Clay | 0.068 | 0.38 | 0.008 | 1.09 | 4.80 | 31.92 | 2.62 × 10^{5} |

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Sadeghi, M.; Tabatabaeenejad, A.; Tuller, M.; Moghaddam, M.; Jones, S.B. Advancing NASA’s AirMOSS P-Band Radar Root Zone Soil Moisture Retrieval Algorithm via Incorporation of Richards’ Equation. *Remote Sens.* **2017**, *9*, 17.
https://doi.org/10.3390/rs9010017

**AMA Style**

Sadeghi M, Tabatabaeenejad A, Tuller M, Moghaddam M, Jones SB. Advancing NASA’s AirMOSS P-Band Radar Root Zone Soil Moisture Retrieval Algorithm via Incorporation of Richards’ Equation. *Remote Sensing*. 2017; 9(1):17.
https://doi.org/10.3390/rs9010017

**Chicago/Turabian Style**

Sadeghi, Morteza, Alireza Tabatabaeenejad, Markus Tuller, Mahta Moghaddam, and Scott B. Jones. 2017. "Advancing NASA’s AirMOSS P-Band Radar Root Zone Soil Moisture Retrieval Algorithm via Incorporation of Richards’ Equation" *Remote Sensing* 9, no. 1: 17.
https://doi.org/10.3390/rs9010017