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

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.

💡 Research Summary

The paper introduces a novel method, Orthogonality Constrained Modal Search‑based Depth estimation (OCMS‑D), for estimating the depth of a monoharmonic (single‑frequency) underwater source when the source–array range is fixed and seabed environmental parameters are unknown. Traditional depth‑estimation techniques fall into two categories: interference‑based methods that require broadband signals and short ranges, and matched‑field processing (MFP) that relies heavily on accurate ocean‑bottom models. Both approaches become unreliable when seabed properties are uncertain or unavailable.

OCMS‑D overcomes these limitations by exploiting two physical properties of normal‑mode propagation: (1) sparsity of propagating modes at a given frequency and range, and (2) orthogonality of mode depth functions within the water column. First, the Orthogonality Constrained Modal Search (OCMS) algorithm estimates the normal‑mode wave numbers (kₘ), the corresponding depth functions ψₘ(z), and complex modal amplitudes aₘ directly from the pressure data recorded by a vertical linear array (VLA). This is formulated as an ℓ₁‑norm minimization with a residual constraint (Eq. 5) and solved using the CVX convex‑optimization toolbox. Crucially, the algorithm does not require any prior knowledge of seabed sound speed, density, or attenuation, and the VLA need not span the entire water depth.

After obtaining the modal amplitudes, the sign of each mode remains ambiguous. Incorrect sign assignment leads to large sidelobes in the depth‑ambiguity function D(z), degrading depth resolution. To resolve this, the authors propose a Depth‑Sign Search (DSS) procedure. For each possible combination of mode signs, a normalized depth‑ambiguity function is constructed and compared with a theoretical Dirichlet‑kernel‑shaped function using the Kullback‑Leibler (KL) divergence. The sign combination that minimizes the KL divergence is selected, and the final source depth is taken as the location of the maximum of D(z) under this optimal sign set.

Simulation studies employ a realistic sound‑speed profile (SSP) measured in the Yellow Sea, a VLA with 30 hydrophones spaced 1 m apart (first element at 1 m depth), a source at 20 m depth radiating a 596 Hz tone, and a horizontal range of 5 km. The simulated field, generated with the KRAKEN normal‑mode model and corrupted by 30 dB Gaussian noise, contains about ten propagating modes. OCMS accurately recovers the modal wave numbers, depth functions, and amplitudes; higher‑order modes show larger wave‑number errors due to reduced orthogonality caused by seabed interaction, yet the overall depth‑estimation error remains below 2 m.

Experimental validation is performed with two real data sets. In the Yellow Sea experiment, using a 4‑second analysis window, the absolute depth error never exceeds 2.4 m. In the SWellEx‑96 experiment, with a 10‑second window, the shallow source (≈20 m) is estimated within 5.4 m and the deep source (≈70 m) within 10.8 m. These results compare favorably with conventional CMFP and the previously proposed Seabed‑Independent Depth Estimation (SIDE) method, especially considering that OCMS‑D requires neither seabed parameters nor source motion.

Key advantages of OCMS‑D are: (1) complete independence from seabed environmental knowledge, (2) ability to operate with a VLA that does not cover the full water column, (3) applicability to single‑frequency signals, enabling real‑time depth tracking of stationary sources, and (4) robust resolution of modal sign ambiguity via the DSS cost function. Limitations include reliance on the orthogonality assumption, which may be weakened for high‑order modes that couple strongly to the bottom, and potential sign‑combination ambiguities when only a few modes are present.

Overall, OCMS‑D offers a practical, physics‑driven solution for underwater source depth estimation in environments where traditional model‑based methods fail, and it holds promise for integration into real‑time sonar and acoustic surveillance systems.