# Vertical Correlation and Array Gain Analysis for Vertical Line Array in Deep Water
## 1 Introduction  
Array gain (AG) is a crucial factor for the detection sensor system, which describes the improvement of the signal-to-noise ratio (SNR) between the sensor array and the single element. It can also be expressed as the difference between the signal gain (SG) and the noise gain (NG), which are related to the spatial correlation of the signal and noise, respectively [1]. Spatial correlation of sound field characterizes the similarity of the received signal at different spatial locations. Because of the high correlation of signal received for the horizontal line array, the AG is easy to obtain by using the beamforming technology. However, for the vertical line array (VLA), the received signals of each element contain a complex multipath structure and the signal correlation at different depths is usually not high enough. Thus, and it leads to rapid performance degradation in the beamforming processing. Previous researches of the vertical correlation in the acoustic field are focused on the shallow water environment. [PERSON] et al. pointed out that when the source and the sensor array are below the thermocline, the vertical correlation length as a function of element interval in units of wavelength increases with frequency and range [2]. In the multipath channel, a method has been proposed to estimate the vertical correlation of the sound field averaged over depths and ranges. Correlation peak appears at the same reception depth as the source depth while ignoring the surface loss andvolume absorption [3]. An experiment was conducted in the shadow water, and conclusion reveals that vertical correlation depends on the horizontal range rather than element depth when the received signal at low frequency is in the iso-velocity shallow water with a silt-sand bottom [4]. [PERSON] et al. compared influence of different carrier frequencies on correlation characteristics when considering irregular acoustic channel boundary. The result indicates that the vertical correlation decreases as the central frequency increases [5]. In comparison with the shallow water, the multipath effect for the long-range sound waves is distinctly weaker in the deep water. Ray theory is applied to estimate the vertical correlation in the multipath channel [6]. Reference [7] infers that the vertical correlation in the deep water is closely related to the hydrophone spacing, signal frequency, and signal waveform distortion caused by multipath interference in the sound field. Then, the coupled-mode theory is applied to analyze the influence of the signal frequency and receiving range on the vertical correlation [8]. [PERSON] et al. conducted a deep-water experiment in the South China Sea, and they discussed the vertical correlation of the acoustic field in large depths. Results reveal that in the convergence zone, vertical correlation coefficient periodically oscillates as the receiving depth increases [9]. Noise correlation is another vital factor that makes a great influence on the AG. [PERSON] and [PERSON] proposed the typical isotropic noise model and the surface noise model and discussed their spatial correlation [10]. Following consideration of the seawater attenuation, the spatial correlation of the wind-driven noise in deep water is analyzed [11; 12]. In addition to the classical noise models above, scholars have explored the spatial correlation of multiple noise models. [PERSON] et al. modelled the low-frequency marine environmental noise field in the direct-arrival zone. They discussed the vertical directivity and spatial correlation under three wind speeds by using the ray method and parabolic equation [13]. The characteristic of shipping noise is studied, and spatial correlation functions are also derived from the directionality density function [14]. Based on the development of the correlation theory for the acoustic field, AG research of the VLA is also carried out.