OCMS-D: A Sparsity-Constrained Modal Orthogonality Method for Single-Frequency Source Depth Estimation with a Vertical Array

To address the challenge of estimating the depth of a single-frequency (tonal) source using a vertical line array (VLA) without prior knowledge of seabed parameters, a method called OCMS-D (orthogonality-constrained modal search-based depth estimation) is presented. The method exploits the orthogonality of mode depth functions as a physical constraint and leverages the sparsity of propagating normal modes in the received field. A convex optimization model is formulated to jointly estimate modal wavenumbers, depth functions, and complex mode amplitudes. Furthermore, a depth-sign search (DSS) is introduced to compensate for mode phase signs, which effectively suppresses sidelobes in the depth ambiguity function and enables high-accuracy depth estimation. Numerical simulations analyze the influence of signal-to-noise ratio, array parameters and sound speed profile (SSP) uncertainty on the algorithm’s performance. Comparisons with matched-mode processing (MMP) show that OCMS-D exhibits superior robustness to SSP uncertainty and requires a smaller array aperture. Experimental data from the SWellEx-96 campaign demonstrate that the method achieves depth estimation errors of less than 5.4 m for the shallows and less than 10.8 m for the deep spurce. Notably, it requires neither seabed prior information nor source motion, offering a robust array signal processing framework for tonal source localization.