Normal Mode Parameters Estimation by a VLA in Single-Shooting

In this article, we propose an orthogonality-constrained modal search (OCMS) method for estimating modal wavenumbers and modal depth functions using a vertical linear array (VLA). Under the assumption of a known sound speed profile, OCMS leverages the orthogonality of distinct modal depth functions to extract both the modal depth functions and their corresponding wavenumbers, even when the VLA and a monochromatic sound source remain stationary. The performance of OCMS is evaluated through numerical simulations under varying signal-to-noise ratios, different VLA apertures, varying numbers of VLA elements, VLA tilt and sound speed profile (SSP) uncertainty. The results demonstrate that OCMS is robust against noise, VLA aperture variations, and changes in the number of VLA elements, meanwhile, the algorithm maintains reliable performance when SSP uncertainty$< $1 m$/$s and VLA tilt angle$< 5^\circ$. Furthermore, the effectiveness of OCMS is validated using SwellEx96 experimental data. The relative error between the modal wavenumbers derived from experimental data and those computed via Kraken is on the order of$10^{-4}$.