2.1. Synthetic dataset

2.1.3. Synthetic dataset generation

Once the synthetic inputs are generated, we run the calibrated IEM to generate the backscatter coefficients for the grid level using the grid soil moisture values (${\sigma }_{V{V}_g}$, ${\sigma }_{V{H}_g}$) or for the plot level using the soil moisture at plot scale (${\sigma }_{V{V}_p}$, ${\sigma }_{V{H}_p}$). Then an absolute error corresponding to the SAR observation accuracy was added to the simulated backscattering coefficients to obtain a more realistic synthetic dataset. For Sentinel-1, this error is defined by the absolute radiometric accuracy, which is equal to 0.70 dB and 1.0 dB for VV and VH polarizations respectively [10]. Accordingly, for each element of our dataset, 5 noise samples were randomly selected from the zero-mean Gaussian noise distribution with a standard deviation of 0.7 and 1.0 dB, respectively for VV and VH. The randomly selected noise values were then added to the IEM’s simulated $\sigma$ at both scales (plot and grid). Finally, our noisy synthetic dataset, in VV and VH polarizations is composed of about 8 million elements.

Table 1 shows an example of the possible combinations of our input parameters in an 8-column data format. Each row represents a unique data combination defined by a given radar incidence angle ($\theta$), surface roughness, and soil moisture at plot and grid scales. ${\sigma }_{V{V}_p}$ and ${\sigma }_{V{H}_p}$ were simulated using ($\theta$, $HRMS$, $MVp$) while ${\sigma }_{V{V}_g}$ and ${\sigma }_{V{H}_g}$ were simulated using ($\theta$, $HRMS$, $MVg$).

Table 1. Example of synthetic data generated by the calibrated IEM using Sentinel-1 wavelength.

Table 1. Example of synthetic data generated by the calibrated IEM using Sentinel-1 wavelength.

IEM inputs				IEM outputs
$\mathsf{\theta }$	HRMS	MVg	MVp	${\mathsf{\sigma }}_{{\mathbf{VV}}_{\mathbf{p}}}$	${\mathsf{\sigma }}_{{\mathbf{VH}}_{\mathbf{p}}}$	${\mathsf{\sigma }}_{{\mathbf{VV}}_{\mathbf{g}}}$	${\mathsf{\sigma }}_{{\mathbf{VH}}_{\mathbf{g}}}$
20.0	0.5	4.0	6.56	-12.16	-27.58	-9.30	-21.21
20.0	0.5	4.0	6.45	-10.72	-24.56	-9.06	-19.27
20.0	0.5	4.0	10.76	-10.21	-24.82	-8.77	-20.32
...	...	...	...	...	...	...	...
45.0	3.8	40.0	36.98	-6.04	-15.49	-7.67	-17.01
45.0	3.8	40.0	32.38	-6.36	-17.66	-8.60	-17.55
45.0	3.8	40.0	38.40	-4.78	-15.53	-6.40	-17.39

In this study half of the synthetic dataset is used for training the neural networks and the other half for their evaluation, the evaluation half is referred to as the validation dataset.

2.2. Real dataset

3.1. Inversion algorithm

3.2. Evaluated Sentinel-1 configurations

• Configuration 1: Analyze the effect of Sentinel-1 polarizations
  Three inversion Sentinel-1 configurations were tested: (1) VV polarization alone; (2) VH polarization alone; and (3) both VV and VH polarizations. In this configuration the soil roughness parameter HRMS is ignored. They can be formulated as:

  $$\begin{array}{c}Config\_1\_VV:MVp=\mathcal{N}({\sigma }_{V{V}_p},\theta ;\mathit{w})+ϵ\\ Config\_1\_VH:MVp=\mathcal{N}({\sigma }_{V{H}_p},\theta ;\mathit{w})+ϵ\\ Config\_1\_VV\_VH:MVp=\mathcal{N}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},\theta ;\mathit{w})+ϵ\end{array}$$

  (9)
• Configuration 2: Separate the MVp estimation domain into two separate domains one for dry to slightly wet and one for very wet
  Using the same polarizations as in the previous configuration, we separate our MVp solution search domain into two domains: one with a search for a solution for dry to slightly wet soil conditions and one for a solution for wet to very wet soil conditions. This configuration needs a priori information on MVp (a priori dry to slightly wet or very wet). Partitioning the estimation domain into distinct dry and wet conditions and training dedicated neural networks for each domain may significantly enhance soil moisture estimation accuracy [16]. By focusing on domain-specific patterns and relationships, the specialized neural networks can capture the complexities associated with soil moisture variations more effectively. In the case of a priori dry to slightly wet soil, $\mathcal{N}$ will be built up using the synthetic training dataset elements with MVp between 4 and 30 vol.%. Contrarily, in the case of a priori very wet soil conditions, $\mathcal{N}$ will be developed using the synthetic training dataset elements with MVp between 20 and 40 vol.%. An overlap of 10 vol.% on MVp is considered between the dry to slightly wet and the very wet training datasets of $\mathcal{N}$. During the evaluation, the dry $\mathcal{N}$ is applied on attributes with MVp < 25, while the wet $\mathcal{N}$ is applied on attributes with MVp ≥ 25. In an operational context, the choice between ${\mathcal{N}}_{dry}$ or ${\mathcal{N}}_{wet}$ is determined by meteorological data, primarily focusing on precipitation. For example, if there has been significant rainfall one or two days before the S1 acquisition, the ${\mathcal{N}}_{wet}$ would be used; otherwise, the ${\mathcal{N}}_{dry}$ is applied. In this configuration the soil roughness parameter HRMS is ignored.
  This is formulated as:

  $$\begin{array}{c}Config\_2\_VV:MV{p}_{dry}={\mathcal{N}}_{dry}({\sigma }_{V{V}_p},\theta ;\mathit{w})+{ϵ}_{dry}\\ MV{p}_{wet}={\mathcal{N}}_{wet}({\sigma }_{V{V}_p},\theta ;\mathit{w})+{ϵ}_{wet}\\ \\ Config\_2\_VH:MV{p}_{dry}={\mathcal{N}}_{dry}({\sigma }_{V{H}_p},\theta ;\mathit{w})+{ϵ}_{dry}\\ MV{p}_{wet}={\mathcal{N}}_{wet}({\sigma }_{V{H}_p},\theta ;\mathit{w})+{ϵ}_{wet}\\ \\ Config\_2\_VV\_VH:MV{p}_{dry}={\mathcal{N}}_{dry}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},\theta ;\mathit{w})+{ϵ}_{dry}\\ MV{p}_{wet}={\mathcal{N}}_{wet}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},\theta ;\mathit{w})+{ϵ}_{wet}\end{array}$$

  (10)
• Configuration 3: Assess the added value of using the grid information in addition to plot scale
  In this configuration, we hypothesize that incorporating backscatter coefficients at the grid scale into the soil moisture estimation process, in addition to backscatter coefficients at the plot scale, can improve the accuracy of MVp estimation, potentially offering an alternative to the domain-separated approach which necessitates weather data for selecting the appropriate neural network (dry to slightly wet or very wet). This hypothesis assumes that integrating grid coefficients can inform $\mathcal{N}$ about the soil moisture status in the study area, enabling the inversion model to adapt to both dry and wet soil characteristics. We also chose to use both polarizations as its the most precise configurations. There are two subcases of configuration 3, formulated as:

  $$\begin{array}{c}Config\_3\_grid:MVp={\mathcal{N}}_{MVp}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{MVp}\\ \\ Config\_3\_grid\_MVg:M{V}_g={\mathcal{N}}_{MVg}({\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{MVg}\\ MV{p}_{dry}={\mathcal{N}}_{dry}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{dry}\\ MV{p}_{wet}={\mathcal{N}}_{wet}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{wet}\end{array}$$

  (11)

  Equations (11) give a detailed presentation of the neural networks and their inputs used in configuration 3. In $Config\_3\_grid$ only ${\mathcal{N}}_{MVp}$ was used, this network was trained using the backscatter coefficients for VV and VH polarizations on the grid and plot scales. In $Config\_3\_grid\_MVg$ three $\mathcal{N}$ were used, ${\mathcal{N}}_{MVg}$ is trained to estimate $MVg$ using the backscatter coefficients for VV and VH on the grid scale, the $MVg$ estimated from ${\mathcal{N}}_{MVg}$ will serve as dry/wet domain separator if estimated $MVg<25$ vol.% the config will use ${\mathcal{N}}_{dry}$ in order to estimate $MVp$ otherwise it will use ${\mathcal{N}}_{wet}$. The second and third $\mathcal{N}$ were trained using the backscatter coefficients for VV and VH polarizations on the grid and plot scales, ${\mathcal{N}}_{dry}$ uses backscatter coefficients on $MVp<30$ vol.% and ${\mathcal{N}}_{wet}$ uses backscatter coefficients on $MVp>20$ vol.%. In this configuration the soil roughness parameter HRMS is ignored.
• Configuration 4: Analyze the added value of using soil roughness estimates
  In this last configuration, we hypothesize that training neural networks with soil roughness estimates, in conjunction with incorporating grid backscatter coefficients and partitioning estimation domains, can potentially improve soil moisture estimation accuracy. This hypothesis suggests that by accounting for the complex relationships between soil moisture, surface roughness, and backscatter signals, neural networks can better capture the intricacies of soil moisture variations across various surface conditions. There are two subcases of configuration 4, formulated as:

  $$\begin{array}{c}Config\_4\_grid:HRMS={\mathcal{N}}_{HRMS}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{HRMS}\\ MVp={\mathcal{N}}_{MVp}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},HRMS,\theta ;\mathit{w})+{ϵ}_{MVp}\\ \\ Config\_4\_grid\_MVg:M{V}_g={\mathcal{N}}_{MVg}({\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{MVg}\\ HRM{S}_{dry}={\mathcal{N}}_{HRM{S}_{dry}}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{HRM{S}_{dry}}\\ HRM{S}_{wet}={\mathcal{N}}_{HRM{S}_{wet}}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},\theta ;\mathit{w})+{ϵ}_{HRM{S}_{wet}}\\ MV{p}_{dry}={\mathcal{N}}_{dry}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},HRM{S}_{dry},\theta ;\mathit{w})+{ϵ}_{dry}\\ MV{p}_{wet}={\mathcal{N}}_{wet}({\sigma }_{V{V}_p},{\sigma }_{V{H}_p},{\sigma }_{V{V}_g},{\sigma }_{V{H}_g},HRM{S}_{wet},\theta ;\mathit{w})+{ϵ}_{wet}\\ \end{array}$$

  (12)

  In $Config\_4\_grid$ two $\mathcal{N}$ were used. ${\mathcal{N}}_{HRMS}$ was trained to estimate the soil roughness using the backscatter coefficients for VV and VH polarizations on the grid and plot scales, while ${\mathcal{N}}_{MVp}$ was trained to estimate the soil moisture using the backscatter coefficients for VV and VH polarizations on the grid and plot scales in addition to soil roughness. Further, to account for the uncertainties on HRMS estimated from SAR images [7] in the training phase, a zero-mean Gaussian noise was added to HRMS with a standard deviation of 0.5. Estimated $HRMS$ from ${\mathcal{N}}_{HRMS}$ is used to estimates $MVp$.
  In $Config\_4\_grid\_MVg$ five $\mathcal{N}$ were used, like in $Config\_3\_grid\_MVg$, ${\mathcal{N}}_{MVg}$ is used to separate the estimation domain of $HRMS$ and $MVp$ into two domains dry-wet/very wet using the same thresholds on $MVg$ as $Config\_3$. Then, $HRMS$ is estimated according to dry or wet conditions (${\mathcal{N}}_{HRM{S}_{dry}}$ or ${\mathcal{N}}_{HRM{S}_{wet}}$) and then this estimate of HRMS is used in the network to estimate MVp (${\mathcal{N}}_{dry}$ or ${\mathcal{N}}_{wet}$).

3.3. Evaluation metrics

4.1. Using synthetic validation set

4.2. Using real Dataset

Appendix