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.