A simple method to reconstruct coastal surface currents from single SAR Doppler observations using the continuity equation

Doppler-derived surface currents from Synthetic Aperture Radar (SAR) provide only a single velocity component, the radial (range-directed) velocity $u_r$, which limits their direct use in oceanographic applications. Here we present a simple and physically constrained method to reconstruct the orthogonal azimuthal velocity component $v_{\varphi}$, yielding a two-dimensional estimate of the surface current field. The approach exploits the horizontal continuity equation together with solid boundary constraints, making it well suited for coastal and semi-enclosed regions. The method is applied to Sentinel-1 Doppler observations over the Skagerrak Sea and evaluated against two high-resolution ocean circulation models, Norkyst-800 and the data-assimilative Norkyst-DA. Despite relying on a single, noisy Doppler-derived velocity component, the reconstruction recovers coherent and dynamically consistent circulation patterns. Comparisons with the model fields yield correlation coefficients of about 0.9 for the observed radial component and 0.7--0.75 for the reconstructed azimuthal component, with root-mean-square errors on the order of 0.2~m~s$^{-1}$. The azimuthal velocity exhibits larger biases than the radial component, consistent with the predominantly ageostrophic nature of the transverse circulation in the Skagerrak and its sensitivity to wind forcing and model formulation. The reduced bias relative to the data-assimilative model, together with consistency with inter-model differences, indicates that SAR observations can be used to diagnose model biases not only in the directly observed radial component $u_r$ but also in the reconstructed azimuthal component $v_{\varphi}$. These results demonstrate that single-look SAR Doppler measurements contain sufficient dynamical information to recover two-dimensional surface current structure when combined with physically based constraints, providing a valuable observational resource for coastal circulation studies and model evaluation.