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Ocean reverberation is an important issue in underwater acoustics due to the significant influence on working performance of the active sonars. In this paper, a uniform bottom-reverberation model is proposed based on ray theory, which can calculate monostatic and bistatic reverberation intensity and explain the generation process of deep-water reverberation. The mesh meth-od is firstly used in this model by dividing bottom scatterers into a number of grids. Then reverberation is calculated based on the exact time of scattering signal generated on each grid. Due to exact arrival time, the presented model can provide more accurate result than classical models, in which scatterers are usually treated as circular rings or elliptical rings. Numerical results are compared with reverberation data collected from the South China Sea deep-water experiment with different receiving distances and depths. The simulated and experimental results agree well overall.

Reverberation is an important subject in the field of ocean acoustics. Ocean reverberation is usually the main background interference of active sonars, and it is also widely concerned because it carries rich information that can be used in the inversion of environmental parameters.

Considerable work was done on shallow-water reverberation in the past several decades. In shallow water, the reverberation models based on normal mode^{ }[

Based on the ray theory, a uniform bottom reverberation model is proposed, which can calculate monostatic and bistatic reverberation intensity, and explain the generation process of deep-water reverberation. The mesh method is firstly used in this model by dividing bottom scatterers into a number of grids. The propagation time of different acoustic paths can be obtained more accurately, so the reverberation intensity is calculated accurately and efficiently.

In this section, a deep-sea bottom reverberation model is established based on ray theory. The reverberation signal can be expressed as the process of wave propagating from source to bottom scatterer and then spreading back to the receiver after scattering. Assuming the source is located at ( 0 , z 0 ) , according to ray theory, sound pressure at the scattering element ( r , z b ) can be expressed as:

p ( r , z b ) = ∑ n = 1 N p n = ∑ n = 1 N A n ( s ) ϕ n ( s ) e i ω τ n ( s ) ω , (1)

where r is the horizontal position of the scatterer, z_{b} is the water depth, N is the total number of sound ray, A_{n}(s) is the amplitude of sound ray, ϕ n ( s ) is the function of beam amplitude, ω is the angular frequency of the sound source,

τ n ( s ) = ∫ 0 s 1 c ( s ' ) d s ' is the delay time of sound ray from the source to the scatterer, s is the path of the ray beam, and c ( s ' ) is the sound speed of sea water

on the sound ray. When the source is far enough away from the scatterer, the sound ray can be approximated as a plane wave. Therefore, the incident sound field excited by a point source at the scattering element can be approximately expressed as the superposition of N plane waves.

According to the reciprocity principle, the scattering pressure from the scatterer at unit area is expressed as

p ( r ) = ∑ i = 1 N ∑ j = 1 M p i n c , i ( r i ) × p s c a t t , j ( r j ) × g ( θ i n c , i , θ s c a t t , j , φ ) , (2)

where p i n c , i is the incident wave transfer function, p s c a t t , j is the scattering wave transfer function, r i represents the distance between the source and the scatterer of the ith incident sound ray, and r j represents the distance from scatterer to the receiver of the jth scattered sound ray. N and M represent the total number of incident sound ray and scattering sound ray, respectively. g ( θ i n c , i , θ s c a t t , j , φ ) is a three-dimensional scattering function, that is the plane wave amplitude scattering from per unit area. In the monostatic reverberation calculation, the incident and scattered sound rays are always in a vertical plane. Therefore, the scattering coefficient is only related to the two variables of incident grazing angle θ i n c , i and scattering grazing angle θ s c a t t , j . For bistatic reverberation, the incident sound ray and the scattered sound ray are not in the same vertical plane, we also need to introduce the scattering azimuth angle φ to describe the scattering function as shown in

Based on the Lambert backscattering model, Ellis and Haller established a three-dimensional scattering model suitable for long-range reverberation. The scattering coefficient is expressed as [

g ( θ i n c , i , θ s c a t t , j , φ ) = μ sin θ i n c , i sin θ s c a t t , j + υ ( 1 + Δ Ω ) 2 e − Δ Ω 2 σ 2 . (3)

The first term in formula (3) is the Lambert scattering coefficient, caused by backscattering. The second term caused by side scattering is presented based on the Kirchhoff approximation and the Helmholtz equation under the assumption that bottom is isotropy and the roughness of the interface is in accordance with the Gaussian distribution. μ is the backscatter intensity, σ is the lateral scattering intensity, Δ Ω is a measure of the deflection of the scattered ray from the specularly reflected ray given by

Δ Ω = cos 2 θ i n c . i + cos 2 θ s c a t t , j − 2 cos θ i n c . i cos θ s c a t t , j cos φ ( sin θ i n c . i + sin θ s c a t t , j ) 2 . (4)

According to Equation (2), the bottom reverberation intensity received from scatterer of unit area can be expressed as:

I s c a t t = ∑ i = 1 N ∑ j = 1 M | p i n c , i ( r i ) | 2 | p s c a t t , j ( r j ) | 2 | g ( θ i n c , i , θ s c a t t , j , φ ) | 2 . (5)

Supposing the pulse intensity emitted from the source is I 0 ( τ ) , and the pulse duration is τ 0 , the reverberation intensity at time t can be expressed as the superposition of the scattered signals received at that time,

R ( t ) = ∫ 0 τ 0 I 0 ( τ ) I s c a t t d A ( t , τ ) , (6)

where arrival time t = τ i n c , i + τ s c a t t , j , τ i n c , i and τ s c a t t , j are the time of ray from source to scatterer and from scatterer to receiver, respectively, d A ( t , τ ) is the scattering area. The scattering area is divided into k scattering elements, and the area of the kth element is represented by Δ s k . The reverberation intensity received by the hydrophone can be expressed as discrete form:

R ( t ) = I 0 ∑ k = 1 K I s c a t t Δ s k . (7)

In early literatures, when calculating monostatic bottom reverberation, i.e., source and receiver are in the same horizontal position, the scattering region contributing to reverberation at a certain time is usually regarded as a ring. Similarly, for the calculation of bistatic bottom reverberation, the scattered sound waves received by the receiver at the same time are considered to be from the scattering region of an elliptical ring with source and receiver respectively located at the two focal points of the elliptical ring. This kind of method assumes that rays of different propagation paths reach the same scatterer at the same time when calculating reverberation. However, in the deep-water environment, the ray paths from source to scatterer and from scatterer to receiver are diverse. Meanwhile the propagation time is obviously different. Therefore the area contributing to the reverberation at any given moment is not a ring or an elliptical ring. In order to obtain accurate results, we do not use ring or elliptical ring to divide the scatterer. Instead, bottom is divided into grids firstly, i.e. dividing the seabed into a large number of rectangular scatterers as shown in

In

Scatterers are symmetrical in four quadrants of the coordinate system including source and receiver. However, the source depth and receiver depth are usually different. In other words, the influence of scatterers on the reverberation result is not completely symmetric with the y coordinate axis, whereas it is symmetric with the x coordinate axis. Therefore, only the scatterers in first and second quadrant are considered in reverberation calculation, and then the calculated results are doubled to obtain the final reverberation intensity.

For the environment of the experimental area in the South China Sea,

here are: water depth is 3472 m, source frequency is 500 Hz, source depth is 200 m, receiver depth is 3021 m, and the distance between source and receiver is 5 km. The bottom acoustic parameters and scattering coefficient are the same as those given in next section. For the reverberation intensity at 9-s time (see

Using the deep-water bottom reverberation model proposed in Section 2, we calculate reverberation intensity for different conditions. In numerical simulations,

we take 3472-m water depth and use the sound velocity profile measured in the experiment conducted in the South China Sea. A single-layer liquid bottom model with semi-infinite space is used. Acoustic parameters in the bottom are as follows: sound velocity is 1580 m/s, density is 1.6 g/cm^{3}, absorption coefficient is 0 .3dB/λ . The bottom characteristics represented by these parameters are consistent with those measured through core sampling during the experiment. The scattering coefficients given in Equation (3) are as follows: 10 lg ν = − 10 , 10 lg μ = − 32 , and ( 180 ∘ / π ) σ = 10 ∘ .

We present a deep-water bottom reverberation model based on ray theory,

which can calculate both monostatic and bistatic reverberation intensity. In the conventional methods, scattering region contributing to reverberation signal at a certain time is usually treated as a circular ring or an elliptical ring. Different from them, bottom is firstly divided into a number of grids, and then reverberation is calculated on the basis of exact time from each grid to the source and receivers. As a result, this model can calculate deep-water reverberation more accurately.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11874061, 11434012, 11474302).

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, L.H., Qin, J.X., Li, Z.L. and Liu, J.J. (2018) A Deep-Water Bottom Reverberation Model Based on Ray Theory. Journal of Applied Mathematics and Physics, 6, 2445-2452. https://doi.org/10.4236/jamp.2018.611205