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Passive target detection under low signal-to-noise ratio is essential in various underwater circumstances, but difficult as lacking of targets’ prior information, special propagation characteristics and effect of background noise. Based on the time–frequency representation, we develop a narrow band time–frequency space matched method. The time–frequency matrix is derived based on the ray theory, the warping expression, and the pooling function. Then the sliding cumulative convolution of the matrix obtains a higher signal to noise ratio output. The likelihood ratio test detector is derived for passive target detection. Analysis and the ocean experiment results show that the proposed approach outperforms other passive detectors.

The principle of underwater acoustic signal propagation is of vital importance to realize the “digital ocean”. However, underwater circumstances are becoming more complex and multi-factorial because of raising human activities, changing climate, to name a few. For this study, we formulate a mathematical model to describe the complex variation of underwater propagating acoustic signals, and the solving method are presented. Firstly, the perturb-coefficient nonlinear propagation equation is derived based on hydrodynamics and the adiabatic relation between pressure and density. Secondly, physical elements are divided into two types, intrinsic and extrinsic. The expression of the two types are combined with the perturb-coefficient nonlinear propagation equation by location and stochastic parameters to obtain the stochastic nonlinear differential propagation model. Thirdly, initial and boundary conditions are analyzed. The existence theorem for solutions is proved. Finally, the operator splitting procedure is proposed to obtain the solution of the model. Two simulations demonstrate that this model is effective and can be used in multiple circumstances.