Depth estimation of a monoharmonic source using a vertical linear array at fixed distance

TL;DR

The paper introduces OCMS-D, a method to estimate the depth of a monoharmonic underwater source at fixed range using a vertical linear array without prior seabed parameters. By combining orthogonality-constrained modal search with a depth-sign search, it jointly estimates normal-mode wavenumbers, mode depth functions, and complex amplitudes from a single-frequency signal, and then computes a depth ambiguity function to localize the source depth $z_s$ with sign compensation. The approach is validated numerically and via Yellow Sea and SWellEx-96 experiments, achieving sub-meter depth accuracy in favorable conditions and robust performance across SNRs, array configurations, and mode counts. This seabed-parameter-agnostic, real-time compatible method eliminates reliance on detailed environmental models and demonstrates practical potential for underwater target localization and tracking. The results also indicate how VLA design and processing window length influence depth estimation accuracy, guiding future deployments.

Abstract

Estimating the depth of a monoharmonic sound source at a fixed range using a vertical linear array (VLA) is challenging in the absence of seabed environmental parameters, and relevant research remains scarce. The orthogonality constrained modal search based depth estimation (OCMS-D) method is proposed in this paper, which enables the estimation of the depth of a monoharmonic source at a fixed range using a VLA under unknown seabed parameters. Using the sparsity of propagating normal modes and the orthogonality of mode depth functions, OCMS-D estimates the normal mode parameters under a fixed source-array distance at first. The estimated normal mode parameters are then used to estimate the source depth. To ensure the precision of the source depth estimation, the method utilizes information on both the amplitude distribution and the sign (positive/negative) patterns of the estimated mode depth functions at the inferred source depth. Numerical simulations evaluate the performance of OCMS-D under different conditions. The effectiveness of OCMS-D is also verified by the Yellow Sea experiment and the SWellEx-96 experiment. In the Yellow Sea experiment, the depth estimation absolute errors by OCMS-D with a 4-second time window are less than 2.4 m. And the depth estimation absolute errors in the SWellEx-96 experiment with a 10-second time window are less than 5.4 m for the shallow source and less than 10.8 m for the deep source.

Figures (14)

• Figure 1: The parameters of a range-independent waveguide, along with a VLA and a monoharmonic source within the waveguide.
• Figure 2: Comparison of estimated horizontal wavenumbers, mode amplitudes (magnitude), and mode depth functions with Kraken-calculate reference values, respectively. Red markers indicate reference values, while blue markers represent estimated values.
• Figure 3: Depth ambiguity function $D(z,q_0)$ (blue line) obtained using OCMS-D and the reference depth (green line). The result of depth estimation ${\hat{z}_s} = 20$m.
• Figure 4: Estimation results of source depth and modal parameters, along with their standard deviations, obtained by OCMS-D under different SNRs. (a) Estimation results of source depth. (b) Estimation results of wavenumbers. The red dashed lines indicates the Kraken-calculate reference value. (c) Estimation results of the mode depth functions. The red dashed lines indicates the zero error lines. (d) Estimation results of modal amplitudes. The blue lines and blue shaded regions in figures represent the mean and standard deviation intervals of the estimated results, respectively.
• Figure 5: The AE of depth estimation results under different array parameters and SNR. (a) and (b) are the MAE for the SNR-Aperture plane and SNR-Number plane, respectively. The blue lines in (c) and (d) are the results for SNR=30 dB in (a) and (b), respectively. The red lines in (c) is the results for a bottom VLA when the SNR = 30 dB.