The primary objective of separating normal-mode interference spectra (NMISs) is to extract more fine-grained coherent structures between different pairs of modes in a shallow water sound field. However, NMISs generally exhibit nonlinear phase frequency and amplitude frequency relationships in one-dimensional broadband sound intensity measurements, which poses a challenge for effective separation. To address the difficulty arising from the nonlinearity of NMISs, this study presents a variational dispersion mode decomposition (VDMD) method. An innovation of VDMD is to incorporate the physical model of NMISs into the framework of Variational Nonlinear Chirp Mode Decomposition (VNCMD). First, it utilizes the waveguide invariant and the interference frequency factor to characterize the phase function of NMISs. Second, this parameterization of the nonlinear phase function provides an analytical expression for the gradient of the objective function in the variational optimization processing of VNCMD, thereby improving the efficiency of its iterative search and enabling accurate extraction of NMISs. Finally, experimental results from a Yellow Sea trial demonstrate that the separated NMISs agree well with the reference modes, validating the effectiveness of our proposed method.

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.