**4. Discussion**

Accurate estimation of ϑ*<sup>v</sup>* was envisaged using a linear equation of σ<sup>0</sup> *VV* and <sup>σ</sup><sup>0</sup> *VH* radar cross section from bare agricultural soils. A thorough data collection campaign was undertaken during 2017 and 2018, synchronizing with the pass of the satellite. Bare soil areas were mostly post-harvest cropped areas with little or no crop residue, depending on the crop sown. In the study area, 50% of the agricultural land comprises rice cropped and irrigated from a seasonal stream. Sentinel-1 SAR, dual polarized imagery was used to estimate soil moisture over bare soils using a semi-empirical model. Model parameters were estimated using linear and multi-linear regression. Performance evaluation was conducted based on a 70:30 ratio of sampled points and low RMSE was found between the observed and estimated soil moisture, when a linear relationship between σ<sup>0</sup> *VV* and <sup>σ</sup><sup>0</sup> *VH* was combined for 2017 and 2018.

**Table 5.** Empirical constants (A, B, and T) of the localized model. The total number of samples for each date was 62. The linear equation was derived using 70% of the total population.


**Table 6.** Empirical constants (A, B, and T) of the generalized model. The total number of samples for 2017 was 368, for 2018 it was 427, and for a combination of 2017 and 2018, it was 795. The linear equations were derived using 70% of the total sample of each year and combined year.


#### *4.1. Relationship between* σ<sup>0</sup> *VV and* <sup>σ</sup><sup>0</sup> *VH with Observed Data*

Soil moisture data were collected over two years (2017 to 2018) during the dry summer season from March to May from the 62 plots on the dates of the satellite passes. The relationship between σ<sup>0</sup> *VV*, <sup>σ</sup><sup>0</sup> *VH*, and observed soil moisture by individual dates of the satellite pass (13 images) showed that in both years, the backscatter and observed soil moisture had a significant positive correlation [2,10,27,28]. In both years, VV polarization had a higher backscatter dB value than VH polarization. In cross-polarization (VH), signal attenuation occurs due to volumetric scattering [29]. In 2017, soil moisture constantly increased from 4 March to 27 April. The R<sup>2</sup> between radar backscattering coefficient and in situ measurements of soil moisture is reported in Table 3. A sudden increase in R<sup>2</sup> (VV) can be observed on 15 May, corresponding to the consecutive rainfall events that occurred during the three days before the date of the satellite pass (Figure 8). This means that there is a better correlation for high values of soil moisture, probably because under this condition, the radar backscattering coefficient's dependence on soil moisture is more important than it is on surface roughness.

Similarly, an unexpected increase in σ<sup>0</sup> *VH* was observed (Figure 5). 27 May 2017 (Table 3) had a low R<sup>2</sup> value from σ<sup>0</sup> *VH* compared to the rest of the dates due to the rainfall event (Figure 8), weeds or crop residue moisture [24]. In 2018, R2 for the relationship between σ<sup>0</sup> *VV* and observed soil moisture was significant during March because of residual soil moisture (i.e., the crop residual moisture influenced the radar backscattering coefficient, Table 3). Residual soil moisture was low on 4 April and 22 May due to evaporative demand and higher between 16 April and 10 May due to consecutive rainfall events (Figure 8). R2 did not decrease from March to May, probably due to irregular changes in crop residue moisture, since σ<sup>0</sup> *VH* is sensitive to it [24]. The R2 values from <sup>σ</sup><sup>0</sup> *VH* during March were relatively low despite no rainfall in the month because of residual soil moisture from the previous crop. The cumulative moisture due to rainfall during April is reflected in the low R2 of 16 April and 28 April (Figure 8). During May, high R<sup>2</sup> values were due to bare soils. A linear combination of σ<sup>0</sup> *VV* and <sup>σ</sup><sup>0</sup> *VH* during each date in 2017 produced higher R2 compared to R<sup>2</sup> from individual polarization. This shows that the addition of σ<sup>0</sup> *VV* and <sup>σ</sup><sup>0</sup> *VH* or vice versa improves the backscatter and soil moisture relationship rather than a single relationship with different polarization (Table 3). A similar relationship existed during 2018 from a linear combination of σ<sup>0</sup> *VV* and, which improved R<sup>2</sup> significantly (Table 3).

**Figure 8.** Temporal soil moisture, backscatter, and rainfall, 2017–2018 (March to May).

#### *4.2. Localized and Generalized Relationships*

To operationalize the accurate estimation of soil moisture for decision making, a global relationship was envisaged considering all dates during the dry season. The R2 value of global relationship from VV polarization during 2017 was 0.68, which was higher than the mean of the local relationships. The generalized relationship was found to be more useful for an accurate soil moisture estimate. In addition, R<sup>2</sup> for the generalized relationship performed better than the mean of the localized relationship (0.67) with VH polarization. The scenario during 2018 from VV polarization was more influenced by rainfall events in the dry season. The R2 values ranged from 0.56 to 0.69 with a mean of 0.62 from localized relationships and 0.66 from the generalized relationship, which was more than the local mean. R<sup>2</sup> was very low from VH polarization due to cumulative moisture from rainfall events. However, the generalized relationship produced a lower R2 than the mean localized relationship for VH in 2018 (see Tables 3 and 4). The usefulness of a generalized relationship was exhibited with a consistent increase in the accuracy of the soil moisture estimates over two years. The relationship from VV and VH polarization during 2017 showed significantly lower R<sup>2</sup> than the linear combination of VV and VH during the same year. Similarly, also during 2018, R<sup>2</sup> was significantly higher than the individual polarization. Finally, the best relationship was obtained when the linear combination of two polarizations was combined (appended) for the two years 2017 and 2018, than from single polarizations combined for the two years. It was inferred that generalized relationships are more promising in terms of building a model compared to localized relationships, which may not relate to the entire population.

## *4.3. Modeling the Relationships*

The relationships of localized and generalized modeling were explored and tested for multicollinearity, especially linear combination models σ<sup>0</sup> *VV* <sup>+</sup> <sup>σ</sup><sup>0</sup> *VH*. Multicollinearity is a statistical phenomenon in which two or more predictor variables in a multiple regression model are highly correlated [30]. To detect multicollinearity, we used an indicator called variance inflation factor (VIF), which is a tool to measure and quantify how much the variance is inflated [30]. If any of the model's VIF values exceed 5 or 10, it is an indication that the associated regression coefficient is poorly estimated because of multicollinearity [31]. The P value indicates statistical significance for independent variable contribution in the model, which is explained in Section 2.4.3.

For generalized models, nine different types of linear relationships were explored with σ<sup>0</sup> *VV* and <sup>σ</sup><sup>0</sup> *VH* data (Table 6) during 2017 and 2018. In 2017, among the three possible models σ<sup>0</sup> *VV*, <sup>σ</sup><sup>0</sup> *VH* and, <sup>σ</sup><sup>0</sup> *VV* <sup>+</sup> <sup>σ</sup><sup>0</sup> *VH*, individual models σ<sup>0</sup> *VV* and <sup>σ</sup><sup>0</sup> *VH* showed a low RSE and the *p* value was statistically significant. When σ0 *VH* was added to <sup>σ</sup><sup>0</sup> *VV*, the RSE was lower than in the individual models. This indicates a very good relationship with a low VIF as well as *p* value for both backscatter coefficients. Similar results were observed in 2018. Generalized models derived by combining two years (2017 and 2018) of data showed similar results as individual year models. This study also showed that the linear combination equations from the localized models also performed well with low VIF (<2) and a *p* value statistically significant for both backscatter coefficients (Tables 7 and 8).

A collinearity test on the generalized and localized models showed that the VIF for a linear combination of both backscatter coefficients (VV + VH) was <3. Hence, these models are non-collinear. All models showed low *p* value, indicating that both backscatter coefficients made meaningful addition to the models. During modeling relationships with a linear combination of individual backscatter coefficient, it was inferred that the individual backscatter coefficients were non-collinear, contributing to R<sup>2</sup> independently. It was found that the localized models from individual dates varied over time, and any one equation with a low RSE and VIF may not represent the whole season. In addition, the generalized models produced lower RSE representing the whole season, and were hence better than each localized model.


**Table 7.** Analysis of variance in the generalized model.

\*\*\* *p* value < 0.001.

**Table 8.** Analysis of variance in the localized model.


\*\*\* *p* value < 0.001.

#### *4.4. Validation of Models*

Models were validated using 30% of the sampled points. Results for the localized models are summarized in Table 5. In 2017, the lowest RMSE (0.01) was found on 21 April. Figure 8 shows that no rainfall or very weak rainfall was observed on this day. An increase in RMSE was observed on 15 May. Similarly, in 2018, the lowest RMSE was observed on 23 March and the highest (0.03) on 16 April 2018, probably due to the increase in rainfall. The results seem to show that the RMSE of the models is related to the amount of rainfall. Localized models performed better in drier soils.

As far as the generalized models are concerned, the validation results showed that generalized models obtained using co-polar σ<sup>0</sup> *VV* data provided a lower RMSE than those based on cross-polar σ0 *VH* data for both 2017 and 2018 and taking all data acquired from 2017 to 2018. We also found that the linear combination of both co-polar and cross-polar backscattering coefficients always provided a lower RMSE than the models using only one polarization. The best results came when using the linear combination of polarizations and all the data acquired along the two years, resulting in an RMSE of 0.02 (Table 6). This globalized model was used to produce maps of soil moisture and its spatial variability (Figures 9–11). This is probably the most important result, as a simple multi-linear model using both co-polar and cross-polar Sentinel-1 data acquired over long time periods can reproduce the spatial variability of soil moisture.

**Figure 9.** Spatial variability in the soil moisture in Siruguppa *taluk* during 2017.

**Figure 10.** Spatial variability in the soil moisture in Siruguppa *taluk* during 2018.

**Figure 11.** Spatial variability in the estimated soil moisture from combining all the dates in 2017 and 2018.

#### **5. Conclusions**

This study aimed to accurately estimate the soil moisture of bare, post-harvest agricultural areas collected from Siruguppa *taluk* (sub-district) in the Karnataka state of India. Fifty percent of this agricultural area is grown with rice that is irrigated by seasonal canal irrigation. An accurate estimate of volumetric soil moisture (ϑ*v*) was envisaged using a semi-empirical model based on a linear equation of co-polarized and cross-polarized radar cross section obtained by Sentinel-1 images. A thorough data collection campaign was undertaken during 2017 and 2018 during the pass of the satellite.

Both localized and generalized models were developed using Sentinel-1 image independently and all images together, respectively. Results indicate that the accuracy of the soil moisture estimates increased when using both co-polar and cross-polar images instead of only σ<sup>0</sup> *VV* or <sup>σ</sup><sup>0</sup> *VH*, independently.

The use of localized models revealed that the RMSE of soil moisture estimates decreased corresponding to dry periods, with little or no rainfall. This indicates that better estimates of soil moisture can be obtained for drier soils. Coming to globalized models, soil moisture estimates with lower RMSE were observed when merging all data acquired in 2017 and 2018, and co-polar and cross-polar images, with a R2 of 0.7 and RMSE of 0.02. The availability of a large amount of in situ data collected over a large area demonstrated that a generalized linear model based on the joint use of co-polar and cross-polar C-band SAR images acquired for a long time period, with a short revisiting time of twelve days, could capture spatial variability in soil moisture. This is an important result as the availability of Sentinel-1 data can provide farmers with timely and accurate estimates of soil moisture and enable the mapping of its spatial variability by using simple semi-empirical models. This information, when provided in the immediate weeks and months preceding the cropping season, could be very crucial in determining planting dates and assessing early season plant growth, thereby playing a key role in influencing productivity.

**Author Contributions:** Conceptualization, methodology, and software: A.K.H.; Investigation: M.I.A., A.K.H., G.N., and A.W.; Original draft preparation: M.I.A., A.K.H., and G.N.; Review and editing: M.I.A., A.K.H., A.W., and G.N.; Funding acquisition: A.W. All authors have read and agree to the published version of the manuscript. **Funding:** The authors gratefully acknowledge the financial support given by the Earth System Science Organization, Ministry of Earth Sciences, Government of India (IITM/MM-II/ICRISAT/2018/IND-11 and IITM/MM-II/CRIDA-ICRISAT-IIPR /2018/IND-9) to conduct this research under Monsoon Mission. This research was supported by the CGIAR Research Program on Climate Change, Agriculture and Food Security (CCAFS) carried out with support from the CGIAR Trust Fund and through bilateral funding agreements. For details, visit https://ccafs.cgiar.org/donors.

**Acknowledgments:** The authors are thankful to Prasad S. Thenkabail, Western Geographic Center, USGS, for his valuable insights into data analysis, and João Catalão, Department of Geographic Engineering, Geophysics and Energy, Faculty of Sciences, University of Lisbon, for engaging and fruitful discussions. The authors are also thankful to Smitha Sitaraman for English language editing.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study, in the collection, analyses, or interpretation of data, and in the writing of the manuscript, or in the decision to publish the results.
