## 2 Doppler Anomaly Extraction
We therefore distinguished \(f_{D\text{C0}\text{r}}\) between a term relative to the land \(f_{D\text{C0}\text{r}_{L}}\) and the term relative to the sea \(f_{D\text{C0}\text{r}_{S}}\):  
\[f_{D\text{C0}\text{r}_{L}}(r_{L})=\sum_{n=0}^{N}a_{\text{L}_{n}}\,r_{L}^{n} \tag{7}\]  
\[f_{D\text{C0}\text{r}_{S}}(r_{S})=\sum_{n=0}^{N}a_{S_{n}}\,r_{S}^{n} \tag{8}\]  
where we used the subscript \(L\) and \(R\) to distinguish the terms related to the land and to the sea, respectively; and \(N\) is the polynomial order. Our tests also show that a range expansion to a fourth-order, i.e., \(N=4\), provides a satisfactory trade-off between the removal of the regular pattern related to the background variation of the Doppler Centroid and the preservation of components associated to the sea surface velocity. The polynomial coefficients are estimated through a classical Least Square approach, and the resulting coefficients are combined afterward as explained in the following. More specifically, focusing for example on the land component, we assume to have P measurement points on the land:  
\[\left\{ \begin{array}{c}f_{D\text{C0}\text{r}_{L}}(r_{L_{1}})=f_{1}\\.\\.\\.\\.\\ f_{D\text{C0}\text{r}_{L}}(r_{l_{p}})=f_{P}\end{array} \right. \tag{9}\]  
thus allowing the following matrix problem formulation:  
\[\left[ \begin{array}{cccc}1&r_{L_{1}}&\dots&r_{L_{1}}^{N}\\.&.&\dots&.\\.&.&\dots&.\\.&.&\dots&.\\ 1&r_{L_{p}}&\dots&r_{L_{p}}^{N}\end{array} \right]\left[ \begin{array}{c}a_{L_{ 0}}\\.\\.\\.\\ a_{L_{N}}\end{array} \right]=\left[ \begin{array}{c}f_{1}\\.\\.\\.\\ f_{P}\end{array} \right] \tag{10}\]that is:  
\[\mathbf{R}_{L}\mathbf{a}_{L}=\mathbf{f}_{L} \tag{11}\]  
where \(\mathbf{R}_{L}\) is the \(P\)x (\(N+1\)) range matrix expressed in Equation (10) and \(\mathbf{a}_{L}\) is the (\(N\)+1) x 1 vector of coefficients on the land area that we need to find. Hence, now, it is possible to estimate the coefficients \(\mathbf{a}_{L}\) as follows:  
\[\mathbf{a}_{L}=\left(\mathbf{R}_{L}\mathbf{R}_{L}^{\mathrm{T}}\right)^{-1}\mathbf{R}_{L}^{ \mathrm{T}}\mathbf{f}_{L} \tag{12}\]  
In the same way, it is possible the estimate the coefficients relative to the say \(Q\) points located in the sea region:  
\[\mathbf{a}_{S}=\left(\mathbf{R}_{S}\mathbf{R}_{S}^{\mathrm{T}}\right)^{-1}\mathbf{R}_{S}^{ \mathrm{T}}\mathbf{f}_{S} \tag{13}\]  
The estimated coefficients relevant to the sea and land areas are then combined to achieve the final estimation of the range component: in this process, only the coefficients from order 1 to N are averaged in order to decouple the zero order contribution on the two different regions. Assuming, as an approximation, that the variance of the estimated coefficients is proportional to the inverse of the number of measurements in two areas, i.e., \(P\) and \(Q\), we applied the following combination to find t:  
\[\left[\mathbf{a}_{1}.....\mathbf{a}_{N}\right]^{\mathrm{T}}=\left[\mathbf{a}_{L_{1}}..... \mathbf{a}_{L_{N}}\right]^{\mathrm{T}}\frac{P}{P+Q}+\left[\mathbf{a}_{S_{1}}.....\mathbf{ a}_{S_{N}}\right]^{\mathrm{T}}\frac{Q}{P+Q} \tag{14}\]  
The range component to be subtracted from the data at each azimuth line, i.e., \(f_{DC0_{r}}(r)\), is achieved by substituting \(\mathbf{a}_{0}=\mathbf{a}_{L_{0}}\), i.e., the land offset, and \(\mathbf{a}_{1}.....\mathbf{a}_{N}\) resulting from Equation (14) in Equation (6). The final DCA estimation, is evaluated by subtracting \(f_{DC0_{r}}(r)\) from \(f_{DC}(x,r)\) estimated over the patches with the Madsen approach described in [29], and then applying the previously mentioned azimuth notch filter. The final DCA is geocoded to provide a product loadable in GIS environments. During the geocoding stage, the line of sight (LOS) director cosine is evaluated and provided as an output together with the sea mask.