*2.5. Dual-Polarized RVI Algorithm*

One of the conventional approaches for agricultural monitoring from quad-polarization SAR data is the calculation of the radar vegetation index (RVI) [40]. An adaptation for Sentinel-1 data assumes that σ0VV≈ σ0HH[78–80], such that

$$\text{RVI} = \frac{4 \ast \sigma\_{\text{VH}}^{0}}{\sigma\_{\text{VH}}^{0} + \sigma\_{\text{VV}}^{0}} \tag{1}$$

The radar backscatter coefficient, σ0, also known as the radar cross-section (RCS) per unit area, is the conventional measure of the intensity of the signal reflected by the surface. It is a normalized dimensionless number that varies significantly with the incidence angle, wavelength, and polarization, as well as with properties of the scattering surface itself [81].

Each RVI model in the present study was based only on one dataset with the maximum number of images acquired at an ascending orbit with the same incidence angle. These are considered to be the most favorable conditions for determining the RVI without incidence angle normalization. An RVI-based Kc model for cotton was not produced because of the lack of imagery acquired at the same incidence angle over the specific fields where the agro-meteorological measurements were performed. The models based on the suggested methods for local incidence angle normalization methods described in Sections 2.6 and 2.7 were compared to the models based on the RVI.

### *2.6. σ0-Based Local Incidence Angle Normalization*

The radar backscatter intensity depends on the incidence angle, with σ0 decreasing proportionally to the incidence angle increase in the intermediate range of incidence angles typical for Sentinel-1 and the majority of spaceborne SAR missions [43,82–84]. Based on this understanding, σ0 was normalized by multiplying it with the local incidence angle (θ) in the decimal degree scale:

$$
\sigma\_{\text{Norm}}^0 = \sigma^0 \ast \theta \tag{2}
$$

The normalization of σ0 is achieved by multiplying lower σ0 values obtained under shallower local incidence angles by higher θ values than the higher σ0 values acquired under steeper local incidence angles. In this study, different VV and VH polarization combinations of normalized σ0 values were used to model Kc, LAI, and crop height. The models described below (Equations (3)–(5)) were produced using polarization combinations that showed the best R<sup>2</sup> and RMSE values. The following polarization combination was used to model Kc and LAI in wheat, and LAI in processing tomatoes:

$$\text{V} = \sigma\_{\text{Norm, V-H}}^{0} + \sigma\_{\text{Norm, VV}}^{0} \tag{3}$$

where, and afterward, V is a vegetation variable being estimated.

The following polarization combination was used to model wheat and cotton height:

$$\text{V} = \sigma\_{\text{Norm, V-H}}^{0} - \sigma\_{\text{Norm, VV}}^{0} \tag{4}$$

Processing tomato and cotton Kc estimation models were based on

$$\mathbf{V} = \sigma\_{\text{Norm}\downarrow\text{VV}}^{0} \mathbf{V} \tag{5}$$

The descending winter wheat imagery showed a very low correlation with wheat variables and, therefore, was not used for the development of wheat models.

### *2.7. β0-Based Local Incidence Angle Normalization Method for Tomato and Cotton Height, LAI, and Kc Estimation*

A radar beam transmitted at a shallow angle travels longer distances through the vegetation canopy than a beam transmitted under a steep angle; thus, the attenuation of the former is typically higher than that of the latter. Apart from the beam two-way travel distance through the vegetation, the radar backscatter is affected by soil roughness, dielectric properties, and a combination of different types of scatterers that exist in each pixel [85,86]. The wheat fields in this study are flat, and the vegetation growth is uniform. Hence, scattering from the soil surface is mostly specular in the early part of the season, and the volume scattering component increases as the vegetation develops [87,88]. Unlike the wheat fields, the structure of processing tomatoes and cotton fields is more complex, with mounds and furrows. The distance between planted mound centers in all three processing tomato fields is two meters, and it is one meter in cotton. The difference between the elevation of the mounds is up to 15 cm in processing tomatoes and 12 cm in cotton. Consequently, the standard deviation of the surface height is up to 7.5 cm and 6 cm in processing tomatoes and cotton, respectively. According to the Peake and Oliver roughness criterion [89], the surface is considered rough if

$$\mathbf{h}\_{\text{rms}} > \frac{\lambda}{4\*\cos\,\delta} \tag{6}$$

where hrms is the standard deviation of the surface height variation; λ is the wavelength; and δ is the incidence angle. The incidence angle is slightly different from the local incident angle for slopeless surfaces, but this difference does not affect the calculation of the roughness criterion. Accordingly, in C-band SAR with an incidence angle range of 30◦–45◦, the roughness threshold is hrms > 1.5 cm for an incidence angle of 30◦, and hrms > 1.8 cm for an incidence angle of 45◦. Therefore, the processing tomato and cotton fields are rough, decreasing backscatter dependence on the incidence angles [85], and modifying the rate of backscatter change as the incidence angle increases [90]. Unlike the smooth wheat fields, every pixel in processing tomato and cotton fields contains multiple types of scattering: specular (plant-free furrows), double bounce (corners between furrows and mounds), and volume scattering in the canopy. Moreover, at some incidence angles, Bragg scattering caused by the row frequency might occur [91,92].

Owing to the complex surface structure in cotton and processing tomato fields, another transformation method specific to these fields was derived empirically in addition to the σ0 normalization method. This new method is based on the polynomial regression between plant variables multiplied by the newly derived attenuation coefficient sin(Radians(90 − θ) 3 ) and radar brightness (β0):

$$\text{V\*}\,\sin(\text{Radians}(\theta 0-\theta)^3) = \text{ A\*}\,(\mathfrak{k}^0)^2 + \text{B\*}(\mathfrak{k}^0) + \text{C} \tag{7}$$

where V is the plant variable (such as height, LAI, or Kc); θ is the local incidence angle in degrees; A, B, and C are the specific model coefficients; and β0 is the radar brightness coefficient [41] at either VV (processing tomatoes) or the sum of VV and VH polarizations (cotton). β0 is a dimensionless coefficient that corresponds to the reflectivity per unit area in the slant range. β0 is used because the radiometric correction of σ0 is based on a sea-level ellipsoid [41,93] and, therefore, less suitable for monitoring of rough surfaces and areas with a rugged topography [94]. Previous studies found β0 to be the best unencumbered estimate SAR measurement [41,95,96]. Using radar brightness is an established practice for research in space [97,98] and common practice in the analysis of RADARSAT-1 imagery [85,99], but not of Sentinel-1 data.

The attenuation coefficient sin(Radians(90 − θ)<sup>3</sup>) is used to account for the dependence of beam attenuation on the local incidence angle. The SAR beam interaction with objects on the ground can be described as a triangle in which the radar beam is the hypotenuse, and the local incidence angle θ is the angle between the hypotenuse and vertical cathetus normal to the surface (Figure 2). β0 values are reconstructed to a normalized value by applying a sine function to a cubed value of the (90 − θ) value in radians. The attenuation coefficient is linearly and inversely proportional to the local incidence angle, as shown in Figure 4. Therefore, by applying the suggested normalization, higher β0 values obtained under steeper (closer to vertical) local incidence angles are divided by higher coefficient values compared to β0 values acquired under shallower local incidence angles. The main difference between the σ0 and β0 methods is that the former applies a steeper increase to the radar backscatter (σ0) as the local incidence angle increases than the latter (β0). This difference in the behavior of the methods was created to take into account that as the incidence angle increases, the radar backscatter decreases more slowly for rough surfaces than for smooth surfaces [90,100].

**Figure 4.** The attenuation coefficient (sin(Radians(90 − θ)<sup>3</sup>)), as a function of θ (in degrees), and the local incidence angle. This attenuation coefficient was used to correct for the dependence of radar brightness β0 values on the local incidence angle.

Therefore, the normalized β0 value can be written as

$$\beta\_{\text{Norm}}^0 = \frac{\mathfrak{g}^0}{\sin(\text{Radians}(\mathfrak{g}0-\mathfrak{e})^3)}\tag{8}$$

where *β*<sup>0</sup>*Norm* is the normalized radar brightness in VV or VH polarization, β0 is the radar brightness in VV or VH polarization, and sin(Radians(90 − θ)<sup>3</sup>) is the attenuation coefficient. TheprocessingtomatoLAImodelutilizes thesumofnormalizedβ0 inbothpolarizations:

$$\text{LAI} = \frac{\text{\ $}\_{\text{VH}}^{0}}{\sin(\text{Radians}(\text{90}-\text{9})^{3})} + \frac{\text{\$ }\_{\text{VV}}^{0}}{\sin(\text{Radians}(\text{90}-\text{9})^{3})} \tag{9}$$

### *2.8. Calibration and Validation of Empirical Vegetation Variable Estimation Models*

The field-measured vegetation variables were used in regression models against the uncorrected radar backscatter parameters. Further, the proposed local incidence angle normalization methods were applied to the SAR images, and new empirical regression models were derived. Finally, the models based on the data prior to normalization and postnormalization and on the dual-polarized RVI were compared to assess the performance of the normalization process (Figure 5).

**Figure 5.** Derivation of the dual-polarized normalized (with σ0 or β0 normalization), non-normalized, and RVI models.

In every case, the same type of regression model (either linear or polynomial) and the same polarization combination were used in the comparison. Coefficients of determination in models based on non-normalized data were symbolized as R20, and their root-meansquare error was symbolized as RMSE0. The differences in the R<sup>2</sup> and RMSE following the normalization procedures were calculated. In addition, the Steiger variation [101] of the two-tailed Fisher Z-score tests [102] was performed to determine whether the difference in the models' R<sup>2</sup> is significant (α ≤ 0.05). The significance of the RMSE difference was calculated using the two-tailed Wilcoxon signed-rank test [103] to determine whether the difference in the models' RMSE was significant (α ≤ 0.05). According to the goals set in this study and due to a finite amount of available SAR imagery and ground truth data, all the available data were used to calibrate the empirical models to achieve the models' maximum reliability and estimation accuracy [104]. In order to additionally validate the models' estimation performance, the RMSE values of normalized models applied separately to each dataset (experiment) were also calculated and are presented in Tables S4–S8.
