Dynamic mode decomposition by tensor evolution for source depth.

Depth estimation of shallow acoustic sources using a deep-ocean near-bottom vertical line array relies on depth-sensitive interference, which appears as an oscillatory structure in the frequency-angle domain in broadband matched beam-intensity processing (MBIP). For low signal-to-noise ratio, the nonlinear feature extraction in MBIP transforms array-domain noise into feature-domain distortions, making depth estimation fragile. Addressing this issue, we propose a tensor evolution-based depth estimation method inspired by dynamic mode decomposition, which interprets a sequence of broadband MBIP surfaces as the output of a dynamic system from a time-varying autoregressive operator. The operator is represented by a low-rank Tucker decomposition, which separates coherent feature modes and their temporal modes from incoherent noise. The target depth is inferred via Fourier summation applied to a dominant feature mode. Deep-ocean numerical experiments demonstrate that tensor evolution-based depth estimation improves robustness at low signal-to-noise ratio, consistently outperforming matched field processing and broadband MBIP.