Whittle–Matérn Roughness Covariance for C-Band SAR Inversion over Bare Agricultural Soil

This study evaluates whether the Whittle–Matérn (WM) covariance representation improves pixel-wise SAR inversion over bare agricultural soil which can exhibit scale-dependent roughness. WM includes a smoothness parameter that controls covariance and roughness spectrum shape and has a Gaussian–Gamma scale-mixture interpretation. It was compared with exponential, Gaussian, and powered exponential (PE) covariance models within the same modified extended advanced integral equation model. Co-registered RADARSAT-2 and Sentinel-1 observations provided five channels for inversion. All experiments used identical observations, ancillary inputs, parameter bounds, objective function, and multi-start optimization settings. The analysis focused on the three co-polarized channels, for which covariance structure enters the scattering model directly. At a co-polarized residual tolerance of 2dB, WM produced model-consistent solutions for 425 of 439 retained pixels, compared with 384, 387, and 423 pixels for the exponential, Gaussian, and PE models, respectively. Its blind point ratio was 3.19%, versus 12.53%, 11.85%, and 3.64%, respectively. On the 342 pixels retrievable by all four models, WM gave the lowest mean absolute error for all channels except Sentinel-1 VH. The largest improvement was obtained for Sentinel-1 VV, where WM reduced RMSE to 0.501dB, compared with 0.820dB, 0.786dB, and 0.606dB for the exponential, Gaussian, and PE models. WM also yielded the most complete retrieval maps and scene mean soil moisture estimates closest to the available SMAP and station reference means. These results show that covariance representation is a material source of structural difference in unregularized multi-sensor SAR inversion.