Novel Clustering Schemes for Full and Compact Polarimetric SAR Data: An Application for Rice Phenology Characterization

Information on rice phenological stages from Synthetic Aperture Radar (SAR) images is of prime interest for in-season monitoring. Often, prior in-situ measurements of phenology are not available. In such situations, unsupervised clustering of SAR images might help in discriminating phenological stages of a crop throughout its growing period. Among the existing unsupervised clustering techniques using full-polarimetric (FP) SAR images, the eigenvalue-eigenvector based roll-invariant scattering-type parameter, and the scattering entropy parameter are widely used in the literature. In this study, we utilize a unique target scattering-type parameter, which jointly uses the Barakat degree of polarization and the elements of the polarimetric coherency matrix. Likewise, we also utilize an equivalent parameter proposed for compact-polarimetric (CP) SAR data. These scattering-type parameters are analogous to the Cloude-Pottier's parameter for FP SAR data and the ellipticity parameter for CP SAR data. Besides this, we also introduce new clustering schemes for both FP and CP SAR data for segmenting diverse scattering mechanisms across the phenological stages of rice. In this study, we use the RADARSAT-2 FP and simulated CP SAR data acquired over the Indian test site of Vijayawada under the Joint Experiment for Crop Assessment and Monitoring (JECAM) initiative. The temporal analysis of the scattering-type parameters and the new clustering schemes help us to investigate detailed scattering characteristics from rice across its phenologicalstages.


Introduction
study. We measured soil moisture at each field in two sampling locations, 155 arranged in two parallel transects along the row direction. The separation 156 between each transect was ≈ 40 m. We measured the pointwise soil moisture 157 using theta-probe. Nevertheless, the soil underlying the rice crops was satu-158 rated during the majority of the growth stages due to irrigation and rainfall 159 events. We measured vegetation samples at two points of each field due to 160 the spatial heterogeneity within the field, which is due to the irregular growth statistics of bio-physical and soil parameters are given in Table 1. 3. Satellite data pre-processing   To reduce the speckle effect in S, the multi-looked Hermitian positive semi-definite 3×3 coherency matrix T is obtained from the averaged outer product of the target vector k P (derived using the Pauli basis matrix, Ψ P ) with its conjugate (Lee and Pottier, 2009).
where N denotes the square window size for spatial averaging and Tr is the 206 sum of the diagonal elements of the matrix. two extreme cases, the EM wave is said to be partially polarized, 0 < m < 1. 215 Barakat (Barakat, 1977) provided an expression of m for the N × N 216 coherency matrix. This expression is used in this study to obtain the degree 217 of polarization m FP from the 3 × 3 coherency matrix T for FP SAR data as, where | · | is the determinant of a matrix.
Here, η 1 and η 2 are two auxiliary variables representing the tangent of the 231 ratios between the diagonal elements (T 11 and T 22 + T 33 ) of the coherency 232 matrix, T, and the total polarized scattering power (m FP Span).

233
We define: where γ FP can be related to the average scattering-type parameter, Cloude

244
The eigen-decomposition of T can be expressed as, where Σ is the 3 × 3 diagonal matrix with non-negative elements, λ 1 ≥ λ 2 ≥ 246 λ 3 ≥ 0, which are the eigenvalues of T. The pseudo probabilities, p i obtained 247 from the eigenvalues are defined as,   plot is represented by two bounding curves, Curve I and Curve II in figure 2.
The CP mode measures a projection of the 2 × 2 complex scattering 255 matrix S as, where the subscript C can be either the left-hand circular (L) transmit with 257 a + sign or the right-hand circular (R) transmit with a − sign. The 2 × 2 258 covariance matrix is then obtained from the elements of the scattering vector 259 as, For CP-SAR data, the 4 × 1 Stokes vector g can be written in terms of 261 the elements of the 2 × 2 covariance matrix C 2 as, where ± corresponds to left and right circular polarization respectively.

263
From the elements of g, the backscatter power in the same sense (SC = 264 g 0 − g 3 2 ) and opposite sense (OC = g 0 + g 3 2 ) to the transmitted circular po-265 larization is utilized to derive the scattering-type parameter for the compact- where the total power Span is defined as, Here, ζ 1 and ζ 2 are two auxiliary variables representing the tangent of the 273 ratios between the opposite and same sense circular polarized backscatter 274 powers (OC and SC) and the total polarized scattering power (m CP Span).

275
Similar to FP, we define: where γ CP can be analogously related to the polarization ellipticity parameter However, in order to compare, the two parameters within 278 the same range, they are suitably scaled as, χ = −2χ and θ CP = 2γ CP which 279 is a roll-invariant parameter (detailed in Appendix A.2) is given as, Similar to θ FP , it can be noticed from (18) that for a pure dihedral scat-

286
The expression for the Barakat degree of polarization for the compact-287 polarimetric case is given as, The eigen-decomposition of C 2 can be expressed as, where Σ is a 2 × 2 diagonal matrix with non-negetive elements, λ 1 ≥ λ 2 ≥ 0, 290 which are the eigenvalues of C 2 . The pseudo probabilities, p i obtained from 291 the eigenvalues are defined as, which are then used to define the scattering entropy (H CP ) for CP-SAR data 293 as, As mentioned earlier for the FP case, we use the quantity in the H CP /θ CP polar plot as shown in figure 3. Similar to FP, the feasible regions in the H CP /θ CP polar plot is represented by two bounding curves,

297
Curve I and Curve II in figure 3.

299
In this study, we propose clustering schemes equivalently for both FP

326
In this section, we analyze the proposed clustering framework using the 327 C-band San-Francisco RADARSAT-2 SAR data. Following this, we perform 328 a detailed case study for the unsupervised clustering of rice phenology over 329 Vijayawada, India.

405
The density of the data points in Z6 and Z9 zones has also increased 406 on 29 Jul, while rice transplantation was undergoing in some other fields.

407
Therefore, a moderately high accumulation of data points can also be seen 408 in Z3 ( figure 10a and figure 10b). Moreover, the previously sown rice fields 409 had achieved a higher vegetative stage due to which the areal coverage by the 410 crop canopy had increased, thereby slightly decreasing the scattering entropy. for different dates are related to rice phenological changes. In this regard, the 481 null hypothesis states that there exists no relationship between the changes in the clusters and rice phenology (i.e., the change is due to randomness). The p-values (95 % confidence level) as shown in Table 3 indicates that we can 484 reject the null hypothesis, and therefore, there is evidence that the changes 485 in the unsupervised clusters are due to rice phenology. It is noteworthy that the differences in the characterization capability be-487 tween FP and CP SAR data depends on the type and geometry of the targets. The elements of the C 2 (Ψ) matrix are:

554
Alongside this, note that |C 2 | and Tr(C 2 ) are roll-invariant, where | · | is the 555 determinant and Tr(·) is the trace of a matrix. Therefore, the 2D Barakat 556 degree of polarization, m CP = 1 − 4|C 2 | Tr(C 2 ) 2 is also roll-invariant. Hence, 557 we conclude that the proposed scattering-type parameter for CP SAR, is independent of Ψ, i.e., it is a roll-invariant parameter.
The scattering matrix S for the FP SAR data can be written as, Toolbox). We have used this toolbox for simulating CP data from FP SAR 577 data.

578
Appendix C. Software/Codes to extract FP and CP parameters 579 We obtain the 3×3 coherency matrix, T from the full-polarimetric SAR