# A Method for Estimating Source Depth Based on the Adjacent Mode Group Acoustic Pressure Field
## 1 Introduction  
The research on source depth estimation in this study is based on passive detection technology, and passive depth localization of an acoustic source is an important research aspect. Specific applications include underwater source detection, marine search and rescue, navigation and tracking and so on. At present, the field of passive detection direction source depth estimation can be divided into two categories: preliminary estimation of source depths and accurate estimation of source depths. In terms of preliminary estimation, only the approximate depth of the source in the waveguide needs to be estimated. [PERSON] et al. [1] proposed a passive source depth classification method using a horizontal linear array. [PERSON] et al. [2] studied depth classification of a source as a binary classification problem and proposed a decision index based on classical mode filters. Accurate estimation requires accurate acquisition of source depth [3; 4; 5]; thus, this study focused on the accurate estimation of source depth, and the singular points of the acoustic field were used as a bridge between the adjacent mode group (AMG) acoustic pressure field and the depth of the source. The AMG is the combination of modes of different orders, for example, (6,7)-AMG represents the combination of the sixth and seventh modes. [PERSON] et al. [6] drew a streamline of underwater energy flow in 1985, tracked the energy flow of an acoustic field and determined that there are singular points near the energy streamline, which provided a new research direction for underwater acoustics. Later, [PERSON] and [PERSON] [7] carried out an experiment on the plane wave incident to an elastic spherical shell in water, which proved that the acoustic pressure at the vortex point was close to 0 Pa and the average particle velocity at the saddle point was close to 0 m/s. [PERSON] [8] analyzed the streamline behavior near singular points in a two-dimensional acoustic field by using differential equations to describe the intensity of the streamline. On the basis of the assumption that the average particle velocity at the saddle point is close to 0 m/s during this period, it was determined that saddle points also occur at positions where the phase difference between the average particle velocity and acoustic pressure is an odd multiple of \(\pi/2\). In 2001, [PERSON] and [PERSON] [9] studied the singular points and acoustic power flow generated by the interference of two modes in an ideal waveguide in shallow water. The characteristics of acoustic power flow were more intuitively explored through acoustic power flow figures and grazing angle figures. In 2002, [PERSON] [10] proposed a mode extraction method based on singular value decomposition, providing technical support for obtaining important mode distributions and conducting singular points analysis in the field of acoustics. In 2008, underwater singularities were truly detected [11]. [PERSON] [12] demonstrated in 2019 that the distribution of singular points is related to the signal-to-noise ratio through experiments to detect them and that vortex points will shift over time, which could become a source of information for future changes in the marine environment. In 2023, [PERSON] et al. [13] conducted simulation verification on singular points and inverted the depth of the source through the depth of the singular points. In summary, AMG acoustic field singular points are generated by mode interference, with a periodic distribution in the distance direction, and these singular points contain rich environmental information, including waveguide information and source information. In addition, the distribution of singular points in the acoustic field corresponds to the distribution of acoustic pressure and particle velocity fields. Therefore, the AMG acoustic pressure field also contains rich source information. For the study of source depth estimation, the traditional method is matched field processing (MFP), but MFP has many drawbacks [14].