Recovery of directional wave spectrum from sparse data with compressed sensing

Compressed sensing provides an efficient framework for reconstructing wave signals from reduced measurements. For multi-channel buoy data, the three displacement components exhibit intrinsic correlations, as wave motion contributes simultaneously to all directions according to linear wave theory. Meanwhile, conventional compressed sensing methods based on ℓ1-shrinkage tend to underestimate signal energy when sparsity is not strictly satisfied, leading to biased spectral estimation. This paper introduces a group sparsity constraint to promote joint sparse spectral representations across channels. An energy constraint is proposed in the form of a soft lower bound, enabling an isotropic rescaling of the recovered spectrum while preserving its sparse structure. The resulting model is solved using the alternating direction method of multipliers. Experiments with multi-channel buoy records indicate that the main spectral features can be recovered under the tested conditions, although weak spurious peaks may appear in the estimated directional spectra and reconstruction across long contiguous gaps remains challenging. These results suggest that the proposed approach can enable data compression by retaining a subset of the original measurements.