Horizontal-Transverse Coherence of Bottom-Received Acoustic Field in Deep Water with an Incomplete Sound Channel

Abstract

The horizontal-transverse coherence of low-frequency (300 Hz) and long-range (10–40 km) acoustic fields near the bottom in deep water is investigated based on experimental data obtained from the South China Sea. The results indicate that the horizontal-transverse coherence length exhibits a strong dependence on the source-receiver distance, with fluctuations consistent with sound intensity trends. In high-intensity regions, the horizontal-transverse coherence is relatively high, with a coherence length exceeding 600 λ, where λ is the acoustic wavelength, whereas in low-intensity regions, the horizontal-transverse coherence decreases significantly, with the coherence length shortening to 10–30 λ. The physical mechanisms underlying the horizontal-transverse coherence are analyzed using the ray theory. In high-intensity regions, the energy of the dominant ray (the ray with the highest energy) accounts for over 70% of the total energy of the rays, exerting a decisive influence on the coherence coefficient and leading to stable horizontal-transverse coherence in the received acoustic field. In contrast, in low-intensity regions, the energy distribution is dispersed, and when amplitude and phase disturbances due to spatial inhomogeneity are introduced, the horizontal coherence deteriorates significantly. The numerical simulations are also performed, and the results are consistent with the experimental observations.

1. Introduction

The coherence of the acoustic field serves as an important assessment of the temporal and spatial variations, exerting a profound influence on array gain [1]. In deep water with incomplete channels [2], due to the limitations imposed by water depth, the sound speed of the water near the seabed is lower than that at the surface. In such an oceanic environment, when the sound source is positioned near the sea surface, the target energy received by bottom-mounted hydrophones is dominated by those from direct paths, seabed-reflected paths, and surface-reflected paths. The relative proportions of these distinct acoustic components within the received energy undergo transformation as a function of propagation distance. Random environmental parameters, including seafloor topography [3], seabed sediment properties [4], sea surface roughness [4], and internal waves [5], emerge as significant factors influencing the sound propagation characteristics and coherence of the acoustic field. The inhomogeneity of the medium presents a challenge to the coherence of large-aperture receiving arrays, thereby exerting a substantial impact on the detection performance of such arrays. Consequently, an in-depth investigation of the patterns of acoustic field coherence is of great theoretical and practical importance for the design and performance enhancement of sonar systems in deep-water environments with incomplete sound channels.

In recent years, researchers have conducted extensive work on the horizontal-longitudinal and vertical coherence of the acoustic field [6,7,8,9]. Zhang et al. [9] found that for small longitudinal separations, the trend of horizontal-longitudinal coherence in shallow water is consistent with the variation of sound intensity with respect to transmission distance. Li et al. [8] analyzed spatial coherence in deep water using experimental data, revealing that when the reference sound signal is located in the convergence zone, the spatial distribution characteristics of both horizontal-longitudinal and vertical coherence in deep water closely resemble the spatial distribution structure of the sound transmission losses.

Investigations into horizontal-transverse coherence have primarily focused on shallow water areas [9,10,11,12,13,14,15]. Numerous numerical simulations and experiments have indicated that random environmental factors, such as sound-speed fluctuations and interface undulations, may reduce horizontal-transverse coherence [9,10,11,12]. Studies by Rouseff [12], Wan [13], and Carey [14] demonstrate that the dependence of horizontal-transverse coherence length on source-receiver distance in shallow water is relatively weak. In contrast, Zhang et al. [15] found that the horizontal-transverse coherence of the bottom-received acoustic field is highly sensitive to the source-receiver distance. There are few articles addressing horizontal-transverse coherence in deep water. Andrew [16] utilized a set of deep-water fixed sources and bottom-mounted horizontal arrays to calculate the horizontal-transverse coherence length at source-receiver distances of 200 km and 300 km, revealing that the coherence length at 300 km was greater than that at 200 km. This finding suggests that source-receiver distance is not the only factor influencing horizontal-transverse coherence length; rather, it is more dependent on the sound-speed inhomogeneity of the medium. In summary, research on horizontal-transverse coherence in deep water remains scarce, particularly regarding studies at large depths under incomplete channel conditions, which requires further investigation.

This study analyzes the horizontal-transverse coherence of the acoustic field by utilizing experimental data from a bottom-mounted horizontal array in deep water with an incomplete channel. It summarizes the distribution characteristics of horizontal-transverse coherence length as a function of distance. Section 2 provides a detailed introduction to the horizontal-transverse coherence experiments in an inhomogeneous ocean waveguide, discussing the results of the transmission loss and seabed horizontal-transverse coherence. In Section 3, numerical simulations are conducted using the parabolic equation model, and the simulation results of the variation of horizontal-transverse coherence with distance are found to be consistent with the experimental observations. Section 4 analyzes the physical mechanisms influencing the distribution pattern of horizontal-transverse coherence using the ray theory. Section 5 concludes the study.

2. Experiment

2.1. Experiment Description

In October 2022, a sound propagation experiment was conducted by the Institute of Acoustics of the Chinese Academy of Sciences in the South China Sea. The experiment utilized a combined operation of a ship and a bottom-mounted horizontal line array (HLA). The equipment configuration is shown in Figure 1, where the HLA consists of self-contained hydrophones and temperature-depth sensors. The total length of the HLA is approximately 3.1 km, with 95 hydrophones deployed at non-uniform intervals. The sampling frequency of the HLA was set to 24 kHz.

Figure 2 illustrates the seabed topography and experimental site layout, with O₁ and O₂ marking the positions of the starting and ending elements of the HLA, O as the central point, and T as the endpoint of the experimental track. During the experiment, the research vessel continuously deployed explosive sound sources along the O → T direction (perpendicular to the HLA) at a nominal depth of 200 m. The total length of the OT track is approximately 40 km. The received signals from the explosive sources within a 10.7 km range of the HLA exhibit clipping effects and were therefore excluded from the analysis.

On the explosive sound source trajectory, a multibeam echosounder system was used to conduct detailed measurements of the seafloor topography. Figure 3 provides a schematic of the seafloor topography along the OT survey line. From this figure, it is evident that the seabed depth gradually decreases as the distance extends from the HLA (r = 0) toward the far end. The height difference within the 40 km range is approximately 130 m, with an average slope of about 0.19°, indicating a relatively flat seafloor topography. The inset in Figure 3 shows the seafloor topography variation within the 29–31 km range along the propagation path, revealing a small-period and uneven undulation superimposed on the gentle slope.

By applying a two-dimensional (2D) Fourier transform to the measured seafloor topography, the 2D spatial spectrum of the topography can be obtained:

$$D(f_x,f_y)={\log }_{10}\left[{\displaystyle {\int }_{y_1}^{y_2}{\displaystyle {\int }_{x_1}^{x_2}d(x,y){\mathrm{e}}^{\mathrm{i}2\pi (f_{x}x+f_{y}y)} \mathrm{d}x\mathrm{d}y}}\right],$$

(1)

where i is the imaginary unit, $d$ represents the measured seafloor topography in the experimental area, and the 2D topography survey range is $x\in [x_1,x_2], y\in [y_1,y_2]$, with $(f_x,f_y)$ being the spatial frequency. The topography periods in the x and y directions are given by:

$$T_x=1/f_x,$$

(2)

$$T_y=1/f_y.$$

(3)

Figure 4a shows the 2D seafloor topography spatial spectrum of the experimental area. Figure 4b presents the one-dimensional (1D) seafloor topography spatial spectrum in the x-direction for $f_y={10}^{-4}{\mathrm{m}}^{-1}$, ${10}^{-3}{\mathrm{m}}^{-1}$, ${10}^{-2}{\mathrm{m}}^{-1}$, and ${10}^{-1}{\mathrm{m}}^{-1}$, respectively. Upon examining Figure 4, it can be observed that in addition to the large-period seafloor topography with the period $T\gg 40\mathrm{km}$, the experimental area also contains small-period uneven seafloor topography with periods ranging from 0.1 to 40 km.

Due to limitations in measurement conditions and computational models, it is impossible to use sea surface undulation and sediment conditions that fully match the actual environment in numerical simulations. Statistical models of boundary conditions are employed instead. The sea surface condition on the day of the experiment was classified as grade 2, with a relatively calm sea surface. Extensive numerical calculations [3] have shown that under such conditions, the impact of rough sea surfaces on the low-frequency, deep-water acoustic field is minimal. Additionally, according to reference [17], the surface sediment in the experimental region is primarily composed of calcareous biogenic clayey silt, with no significant variation in sediment type within the observed area. Therefore, in the simulations presented in this study, the effect of the rough sea surface is neglected, and a flat sea surface is assumed. Meanwhile, the thickness of the surface sediment layer and other key parameters, which will be further elaborated in Section 2.2, are assumed to remain horizontally constant within the experimental range.

On the day of the horizontal-transverse coherence experiment, the variations of seawater temperature and salinity with depth at point T were measured. Five days after the experiment, the same measurements were conducted at point O, where the sea conditions on the measurement day were classified as grade 5. According to reference [18], the sound speed at the experimental sites was calculated using the empirical formula:

$$c=1449.2+4.6T-0.055T^2+0.00029T^3+(1.34-0.01T)(S-35)+0.016z,$$

(4)

where T is temperature in degrees centigrade, S is salinity in parts per thousand, and z is depth in meters. The sound-speed profiles at both sites are shown in Figure 5a, and Figure 5b depicts the variation in the sound-speed difference between the two sites with depth. As shown in Figure 5b, the sound-speed difference between the two sites is within 1.2 m/s, except at approximately 60 m depth. Given the small impact of the sound-speed profile at 60 m depth on the deep-received acoustic field with a 200 m source, it is assumed that the sound-speed profile remains constant in both space and time for the simulation. Thus, the sound-speed profile at point T is used for subsequent analysis. Figure 5a indicates that the sound speed near the seabed is significantly lower than that near the sea surface, which is characteristic of a deep-water sound-speed profile with an incomplete channel.

2.2. Experimental Data Overview

During sound propagation, due to the dispersion of normal modes, the time-domain waveform of the signals received by the hydrophone can be divided into multiple signal clusters based on arrival time [11], as shown in Figure 6. With increasing reception distance, the number of signal clusters also increases. For example, at a distance of 11 km, the received waveform includes 2 signal clusters, while at 36 km, the waveform contains 4 clusters.

The target received signal, containing explosive sound signals from each cluster, is extracted with a duration of 4 s. Corresponding environmental noise signals of the same duration are also extracted, and the average signal-to-noise ratio (SNR) for a single hydrophone is calculated at different distances [18]:

$$SNR(r)=10{\log }_{10}\left[{\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N\frac{{{\displaystyle {\int }_{-\infty }^{\infty }\left|p_{\mathrm{s}}(t,r)\right|}}^2\mathrm{d}t}{{{\displaystyle {\int }_{-\infty }^{\infty }\left|p_n(t,r)\right|}}^2\mathrm{d}t}}\right],$$

(5)

where N is the number of hydrophones, $p_{\mathrm{s}}$ is the target signal, and $p_{\mathrm{n}}$ is the noise signal. The calculation results are shown in Figure 7. As illustrated in Figure 7, the SNR for each hydrophone at all received distances exceeded 30 dB. According to reference [9], the impact of noise on the calculation of the horizontal-transverse coherence coefficient is negligible in this case.

The Range-dependent Acoustic Model (RAM-PE) [19], based on the split-step and higher-order Padé approximation methods for the parabolic equation, exhibits superior computational efficiency and precision. This model is capable of addressing range-dependent sound propagation issues, such as those involving irregular seafloor topography, and has become a primary method for calculating acoustic fields in horizontally heterogeneous environments in underwater acoustics. Based on relevant literature regarding seabed sediment studies in the South China Sea continental slope region [20], a three-layer seabed model is adopted, consisting of a surface sediment layer with constant sound speed, a middle layer with a positive sound-speed gradient, and a semi-infinite basement layer with a constant sound speed. With the RAM-PE model, the sound speed and attenuation coefficients for each layer are inversely calculated [21], yielding the seabed parameters presented in

Table 1. Seabed geoacoustic model.

Layers	Sediment	Middle Layer	Basement
Sound speed (m/s)	1520	1700	1820
Sound speed gradient (s−1)	0	1.2	0
Density (g/cm3)	1.57	2.00	2.40
Density gradient (g/cm4)	0	0.004	0
Attenuation coefficient (dB/λ)	0.04	0.08	0.12
Attenuation gradient (dB)	0	0.002	0
Thickness (m)	40	100	Semi-infinite

. The seabed attenuation coefficient varies with frequency, and here, the coefficient corresponds to a central frequency of 300 Hz and a bandwidth of 20 Hz.

2.3. Analysis of Transmission Loss

The average energy of signals received by the HLA within the bandwidth $\left[f_0-{\displaystyle \frac{1}{2}}\Delta f,f_0+{\displaystyle \frac{1}{2}}\Delta f\right]$ is:

$$E_{\exp }(f_0,r)={\displaystyle \frac{1}{f_s^2}}{\displaystyle \frac{1}{M}}{{\displaystyle \sum _{m=1}^M\left|P(f_m,r)\right|}}^2,$$

(6)

where $f_{\mathrm{s}}$ represents the sampling rate, $M$ is the number of frequency points for calculation, $r$ is the horizontal transmission distance, and $P$ is the frequency spectrum of the received signal over a duration of 4 s. The average transmission loss within the bandwidth is calculated as:

$$T{L}_{\exp }(f_0,r)=SL(f_0)-(10{\log }_{10}[E_{\exp }(f_0,r)]-S_{\mathrm{htd}}),$$

(7)

where $SL$ is the source level and $S_{\mathrm{htd}}$ is the hydrophone sensitivity.

The transmission losses at various source-receiver distances are calculated at frequency 290–310 Hz. Four hydrophones (HTD1–HTD4) are selected, featuring relative transverse separations of 281 m (56 λ), 860 m (172 λ), and 1026 m (205 λ). The transmission losses of these four hydrophones are illustrated in Figure 8a, which shows that at long distances, the bottom-received acoustic field exhibits significant intensity variations: source-receiver distances of 11–28 km and 33–40 km correspond to low-intensity regions (regions with high transmission losses), while the range of 28–33 km corresponds to high-intensity regions (regions with low transmission losses). Figure 8b presents the seafloor topography along the paths from point T to the four hydrophones, revealing consistent seafloor slopes with uneven topographical undulation with small periods. Additionally, Figure 8a demonstrates that transmission losses vary for the same source-receiver distance but different transverse positions during sound propagation. In low-intensity regions, the difference in transmission losses between two hydrophones with a certain transverse separation can reach 5–10 dB, indicating significant fluctuations in the received acoustic field.

Figure 9 shows the standard deviation of transmission losses for the HLA at various distances. In low-intensity regions, the transmission losses’ standard deviation is relatively high, reaching 3–4 dB, while in high-intensity regions, it is around 1 dB, indicating strong stability of the acoustic field. This indicates that, in deep-water environments with incomplete channels, even minor factors, such as small-scale topographical undulations, rough surfaces, and sound-speed fluctuations, can significantly affect the horizontal-transverse characteristics of the acoustic field.

2.4. Analysis of Horizontal-Transverse Coherence

In the experiment, the acoustic propagation scenario of the HLA is shown in the 2D projection in Figure 10. The source is located at the coordinate origin, with receiving points designated as $(r,l)$ and $(r+\Delta r,l+\Delta l)$. $\Delta r$ represents the relative longitudinal separation in the direction of sound propagation, while $\Delta l$ indicates the relative transverse separation due to different sound propagation directions at the same distance.

To characterize the similarity between received signals at different positions, the delay coherence coefficient (also known as the time-domain coherence coefficient) is commonly used as an indicator. It is expressed as:

$$\Gamma [p(r,l),p(r+\Delta r,l+\Delta l)]=\underset{\tau }{\max }{\displaystyle \frac{{\displaystyle {\int }_{-\infty }^{+\infty }p(t;r,l)p(t-\tau ;r+\Delta r,l+\Delta l) \mathrm{d}t}}{{\displaystyle {\int }_{-\infty }^{+\infty }{\left|p(t;r,l)\right|}^2 \mathrm{d}t{\displaystyle {\int }_{-\infty }^{+\infty }{\left|p(t;r+\Delta r,l+\Delta l)\right|}^2\mathrm{d}t}}}},$$

(8)

where $p$ represents the time-domain sound pressure waveform of the signal and $\tau$ denotes the time delay. For the convenience of derivation and efficient calculation, the expression for the coherence coefficient in the frequency domain can be obtained through Fourier transform:

$$\begin{array}{l}\Gamma [P(r,l),P(r+\Delta r,l+\Delta l)]\\ =\underset{\tau }{\max }\left\{{\displaystyle \frac{\mathrm{Re}\left[{\displaystyle {\int }_{f_0-\Delta f/2}^{f_0+\Delta f/2}P(f;r,l)P_2^{*}(f;r+\Delta r,l+\Delta l){\mathrm{e}}^{\mathrm{i}2\pi f\tau } \mathrm{d}f}\right]}{\sqrt{{\displaystyle {\int }_{f_0-\Delta f/2}^{f_0+\Delta f/2}{\left|P(f;r,l)\right|}^2 \mathrm{d}f{\displaystyle {\int }_{f_0-\Delta f/2}^{f_0+\Delta f/2}{\left|P(f;r+\Delta r,l+\Delta l)\right|}^2 \mathrm{d}f}}}}}\right\},\end{array}$$

(9)

where $f_0$ represents the center frequency for calculation, $\Delta f$ denotes the bandwidth, * indicates the complex conjugate, and Re[•] represents the real part. If $\Delta f$ is sufficiently small, then $p(r,l)\approx p(r+\Delta r,l)$. When $\Delta l=0$, $\Gamma [P(r,l),P(r+\Delta r,l)]$ is referred to as the horizontal-longitudinal coherence coefficient, and when $\Delta r=0$, $\Gamma [P(r,l),P(r,l+\Delta l)]$ is referred to as the horizontal-transverse coherence coefficient. Generally, horizontal-longitudinal coherence is predominantly influenced by deterministic factors, such as source depth, receiver depth, and propagation distance. These factors give rise to predictable variations in coherence through the interference of normal modes. In contrast, horizontal-transverse coherence is primarily governed by stochastic environmental factors, such as irregular seafloor topography, sea surface roughness, and heterogeneous substrate conditions. These factors result in horizontal-transverse coherence displaying inherent unpredictability.

Experimental data is used to explore the statistical regularities of the horizontal-transverse coherence. The relative transverse separation between each pair of elements in the HLA and their corresponding signal coherence coefficient is calculated, with a frequency interval of 290–310 Hz and a frequency step of 0.25 Hz. Based on this, the coherence coefficients corresponding to array element pairs within a specific transverse separation range are averaged, yielding the average horizontal-transverse coherence coefficient for that separation range [11]. The average process can be expressed as:

$$\begin{array}{l}\rho (r,\Delta l)=\langle \Gamma [P(r,l),P(r,l+\Delta l)]\rangle \\ \hspace{1em}\hspace{1em}\hspace{1em}=\langle \underset{\tau }{\max }\left\{{\displaystyle \frac{\mathrm{Re}\left[{\displaystyle {\int }_{f_0-\Delta f/2}^{f_0+\Delta f/2}P(f;r,l)P^{*}(f;r,l+\Delta l){\mathrm{e}}^{\mathrm{i}2\pi ft} \mathrm{d}f}\right]}{\sqrt{{\displaystyle {\int }_{f_0-\Delta f/2}^{f_0+\Delta f/2}{\left|P(f;r,l)\right|}^2 \mathrm{d}\omega {\displaystyle {\int }_{{\omega }_1}^{{\omega }_2}{\left|P(f;r,l+\Delta l)\right|}^2 \mathrm{d}f}}}}}\right\}\rangle .\end{array}$$

(10)

Figure 11a illustrates the horizontal-transverse coherence coefficients of the experimental data at different distances, while Figure 11b shows the horizontal-transverse coherence coefficient curves at 31 km and 12.3 km reception distances. To quantify horizontal-transverse coherence and analyze the relationship between the coherence coefficient and array aperture, while also facilitating an intuitive observation of coherence variations across different frequencies, we define the horizontal-transverse coherence length of the received acoustic field as the relative transverse separation at which the horizontal-transverse coherence coefficient decays to 0.707. Figure 12 presents the curve of the horizontal-transverse coherence lengths of the received acoustic field as a function of the source-receiver distance.

By comparing Figure 11 and Figure 12, it is evident that the horizontal-transverse coherence of the received acoustic field exhibits a significant dependency on the source-receiver distance. Notably, the variation in horizontal-transverse coherence does not monotonically decrease with increasing distance; rather, it is consistent with the trend of transmission loss variations. Specifically, in high-intensity regions, the coherence coefficients remain at a relatively high level (greater than 0.707), with the coherence length extending beyond the detection range of the HLA. In contrast, in low-intensity regions, the coherence coefficients rapidly drop to 0.4–0.5, restricting the horizontal-transverse coherence length to a narrow range of approximately 10–30 λ (0.05–0.15 km).

3. Simulation Analysis

3.1. Simulation Analysis of Transmission Loss

With the experimental setup, numerical simulations of the acoustic field are conducted by the RAM-PE model. The HLA is positioned at the seabed. The geoacoustic model, as provided in Table 1, is used. In the simulations, all oceanic environmental parameters are kept constant except for the seabed attenuation coefficient, which varies with frequency. Simulations are carried out for the received acoustic fields at center frequencies of 300 Hz and 400 Hz, with a simulation bandwidth of 20 Hz and a frequency step of 1 Hz. The transmission losses of simulation and experiment are shown in Figure 13, which depicts that simulation results of transmission loss are consistent in the experimental data. Further analysis of the seabed-received acoustic field reveals that, with increasing distance, the transmission loss exhibits a distinct pattern of variation in intensity. Taking the seabed−received field at a center frequency of 300 Hz as an example, the source-receiver distance range of 28–33 km is categorized as a high-intensity region, while the ranges of 11–28 km and 33–40 km are classified as low-intensity regions.

3.2. Simulation Analysis of Horizontal-Transverse Coherence

Based on Equation (10), the horizontal-transverse coherence coefficients of the seabed-received acoustic field are calculated, with a frequency interval of 290–310 Hz. Figure 14 presents the horizontal-transverse coherence coefficients of the simulated-received acoustic field at different source-receiver distances. Figure 15 illustrates the variation of the horizontal-transverse coherence length of both experimental and simulated acoustic fields with source-receiver distance. By comparing Figure 11, Figure 14 and Figure 15, it is evident that the simulation results align with the experimental results, exhibiting the same trend. Specifically, the horizontal-transverse coherence of the received acoustic field is significantly dependent on the source-receiver distance, and its spatial distribution closely mirrors the fluctuations in transmission loss with increasing receiver distance. In particular, in high-intensity regions, the horizontal-transverse coherence coefficient remains above 0.707, and the coherence length extends beyond the detection range of the HLA. Conversely, in low-intensity regions, the horizontal-transverse coherence coefficient decreases rapidly, and the coherence length is confined to the order of several tens of wavelengths.

4. Ray-Based Interpretation of Horizontal-Transverse Coherence Spatial Distribution

4.1. Physical Mechanism of the Spatial Distribution Pattern of Horizontal-Transverse Coherence

The received acoustic field P at a distance r can be represented as a collection of sound rays:

$$P(r)={\displaystyle \sum _k{A}_k{\mathrm{e}}^{-\mathrm{i}2\pi f{t}_k}{\mathrm{e}}^{-{\delta }_k}{\mathrm{e}}^{\mathrm{i}{\xi }_k}},$$

(11)

where $k$ is the number of sound rays, $A$ is the sound pressure amplitude in the absence of disturbances, and $t$ is the time delay in the absence of disturbances. Then, $A{\mathrm{e}}^{-\mathrm{i}2\pi ft}$ represents the sound pressure in the undisturbed scenario. $\delta$ is the amplitude disturbance caused by inhomogeneous medium factors, such as seabed scattering and absorption, and $\xi$ is the random phase disturbance caused by inhomogeneous medium during propagation. The sound pressure at another receiver point, located at the same depth and source-receiver distance but with a transverse separation $\Delta l$ from the original receiver point, can then be expressed as:

$$P(r,\Delta l)={\displaystyle \sum _k{A}_k{\mathrm{e}}^{-\mathrm{i}2\pi f{t}_k}{\mathrm{e}}^{-{{\delta }_k}^{\prime }}}{\mathrm{e}}^{\mathrm{i}{{\xi }_k}^{\prime }},$$

(12)

where ${\delta }^{\prime }$ represents the accumulated amplitude disturbance along the given propagation path and ${\xi }^{\prime }$ represents the accumulated phase disturbance along that path. The received sound pressure at that receiver can then be expressed as the superposition of multiple clusters of sound rays:

$$P(r)={\displaystyle \sum _m{B}_m{\mathrm{e}}^{-\mathrm{i}{\varphi }_m}{\mathrm{e}}^{-{\alpha }_m}}{\mathrm{e}}^{-\mathrm{i}{\sigma }_m},$$

(13)

$$P(r,\Delta l)={\displaystyle \sum _m{B}_m{\mathrm{e}}^{-\mathrm{i}{\varphi }_m}{\mathrm{e}}^{-{{\alpha }_m}^{\prime }}{\mathrm{e}}^{-\mathrm{i}{{\sigma }_m}^{\prime }}},$$

(14)

where $m$ is the number of sound ray clusters, $B$ is the total sound pressure amplitude of the cluster in the absence of disturbances, and $\varphi$ is the total sound pressure phase of the cluster in the undisturbed scenario. $\alpha$ and ${\alpha }^{\prime }$ represent the amplitude disturbance terms caused by the inhomogeneous medium for the clusters along two different propagation paths. $\sigma$ and ${\sigma }^{\prime }$ represent the random phase disturbance terms introduced by the inhomogeneous medium along the same two paths. The cross−spectral matrix is then given by:

$$P(r)P^{*}(r,\Delta l)={\displaystyle \sum _m{\displaystyle \sum _n{B}_m{B}_n{\mathrm{e}}^{-({\alpha }_m-{{\alpha }_n}^{\prime })}{\mathrm{e}}^{-\mathrm{i}(\Delta {{\varphi }_{mn}}^{\prime }+\Delta {{\sigma }_{mn}}^{\prime })}}},$$

(15)

where $\Delta \varphi$ represents the phase difference between the sound ray clusters and can be expressed as:

$$\Delta {\varphi }_{mn}={\varphi }_m-{\varphi }_n.$$

(16)

For rays within the same cluster, we have:

$${\Delta {\varphi }_{mn}|}_{m=n}=0.$$

(17)

$\Delta {\sigma }^{\prime }$ represents the phase disturbance difference between two receiver points and within each ray cluster and can be expressed as:

$$\Delta {{\sigma }_{mn}}^{\prime }={\sigma }_m-{{\sigma }_n}^{\prime }.$$

(18)

When $\Delta f\to 0$, the numerator of Equation (10) can be expressed as:

$$\begin{array}{l}\underset{\tau }{\max }\left\{\mathrm{Re}\left[P(f,r)P^{*}(f,r,\Delta l){\mathrm{e}}^{\mathrm{i}2\pi f\tau }\right]\right\}\\ =\underset{\tau }{\max }\left\{\begin{array}{l}{\displaystyle \sum _m{\displaystyle \sum _n{B}_m{B}_n{\mathrm{e}}^{-({\alpha }_m-{{\alpha }_n}^{\prime })}}\times }\\ [\cos (\Delta {\varphi }_{mn}+\Delta {{\sigma }_{mn}}^{\prime })\cos (2\pi f\tau )+\sin (\Delta {\varphi }_{mn}+\Delta {{\sigma }_{mn}}^{\prime })\sin (2\pi f\tau )\end{array}\right\}.\end{array}$$

(19)

Thus, the horizontal-transverse coherence coefficient can be simplified as:

$$\begin{array}{l}\rho (r,\Delta l)\\ =\underset{\tau }{\underset{\tau }{\max }\left\{\begin{array}{l}{\displaystyle \sum _m{\displaystyle \sum _n{B}_m{B}_n{\mathrm{e}}^{-({\alpha }_m-{{\alpha }_n}^{\prime })}}}\cos (\Delta {\varphi }_{mn}+\Delta {{\sigma }_{mn}}^{\prime }-2\pi f\tau )/\\ \sqrt{{\displaystyle \sum _m{\displaystyle \sum _n{B}_m{B}_n{\mathrm{e}}^{-({\alpha }_m-{\alpha }_n)}\cos (\Delta {\varphi }_{mn}+\Delta {\sigma }_{mn})}}}/\\ \sqrt{{\displaystyle \sum _m{\displaystyle \sum _n{B}_m{B}_n{\mathrm{e}}^{-({{\alpha }_m}^{\prime }-{{\alpha }_n}^{\prime })}\cos (\Delta {\varphi }_{mn}+\Delta {{\sigma }_{mn}}^{″})}}}\end{array}\right\}},\end{array}$$

(20)

where $\Delta \sigma$ and $\Delta {\sigma }^{″}$ represent the phase disturbance differences between the sound ray clusters within the same receiver point and can be expressed as:

$$\Delta {\sigma }_{mn}={\sigma }_m-{\sigma }_n,$$

(21)

$$\Delta {{\sigma }_{mn}}^{″}={{\sigma }_m}^{″}-{{\sigma }_n}^{″}.$$

(22)

For rays within the same cluster, there is:

$${\Delta {\sigma }_{mn}|}_{m=n}=0,{\Delta {{\sigma }_{mn}}^{″}|}_{m=n}=0.$$

(23)

Generally, the coherence between rays belonging to different clusters is lower than that among rays within the same cluster. During long-distance propagation, random inhomogeneities in the medium, such as uneven seafloor topography, rough sea surfaces, varying seabed sediment, and sound-speed disturbances, can occur. These factors initially reduce the horizontal-transverse coherence among different signal clusters. Subsequently, as the effects of the inhomogeneous medium accumulate, the coherence among rays within the same cluster also gradually weakens. The combined impact of these factors leads to a decrease in the horizontal-transverse coherence of the received acoustic field.

In high-intensity regions, the energy distribution of signal clusters is highly concentrated, with the energy of the main signal cluster primarily coming from single-path rays. The in-phase superposition effect of rays causes the energy of the main signal cluster to significantly exceed that of other clusters, thus dominating the contribution to the horizontal-transverse coherence coefficients of the received field. At this point, Equation (20) can be further simplified as:

$$\begin{array}{c}\rho (r,\Delta l)\approx \underset{\tau }{\max }\left\{{\displaystyle \frac{B_1{B}_1{\mathrm{e}}^{-({\alpha }_1-{{\alpha }_1}^{\prime })}\cos (\Delta {\varphi }_{11}+\Delta {{\sigma }_{11}}^{\prime }-2\pi f\tau )}{\sqrt{B_1{B}_1\cos (\Delta {\varphi }_{11}+\Delta {\sigma }_{11})}\sqrt{B_1{B}_1\cos (\Delta {\varphi }_{11}+\Delta {{\sigma }_1}^{″})}}}\right\}\\ ={\displaystyle \frac{{\mathrm{e}}^{-({\alpha }_1-{{\alpha }_1}^{\prime })}}{\sqrt{\cos (\Delta {\varphi }_{11}+\Delta {\sigma }_{11})}\sqrt{\cos (\Delta {\varphi }_{11}+\Delta {{\sigma }_{11}}^{″})}}}{=\mathrm{e}}^{-({\alpha }_1-{{\alpha }_1}^{\prime })}.\end{array}$$

(24)

Under small-scale inhomogeneities, two receiving points at the same distance in the high-intensity regions, with a certain transverse separation, exhibit limited variation in the sound pressure amplitude of the main signal cluster, demonstrating strong stability, that is,

$${\alpha }_1\approx {{\alpha }_1}^{\prime } ,$$

(25)

such that $\rho (r,\Delta l)\to 1$. In contrast, in low-intensity regions, the energy of the received field is dispersed, with multiple signal clusters significantly impacting the horizontal-transverse coherence coefficient. Furthermore, in low-intensity regions, the inter-cluster phase difference is irregularly distributed within the range $[0,2\pi ]$, which makes the total sound pressure phase of the ray clusters highly sensitive to environmental disturbances. These two factors result in a sharp decrease in the horizontal-transverse coherence of the received acoustic field when there is a certain transverse separation between two receiver points, that is, $\rho (r,\Delta l)\ll 1$.

Reference [15] analyzes the horizontal-transverse coherence in shallow water, concluding that the coherence length in shallow water is highly dependent on the source-receiver distance and fluctuates synchronously with the sound field intensity as the distance varies. This spatial distribution pattern of horizontal-transverse coherence is consistent with the findings in the deep water presented in this study. The spatial distribution of horizontal-transverse correlation in both deep and shallow water is the result of the combined effects of normal mode (or multipath) interference and environmental inhomogeneities. However, due to the dissimilarities in the acoustic propagation structures between deep and shallow waters, the specific distance ranges corresponding to the strong and weak distributions of horizontal-transverse coherence vary between these two environments. Under the shallow water condition in reference [15], at frequency 508–640 Hz/80–101 Hz, the coherence length is greater than 170 λ/185 λ for most distances within 50 km. In the deep-water environment in this study, at frequency 290–310 Hz, the coherence length in high-intensity regions (28–33 km) is greater than 600 λ.

4.2. Simulation Analysis of Energy Proportion of Ray Clusters

The ray model BELLHOP [22] is recognized for its clear graphical outputs and explicit physical significance. Despite its diminished precision in complex environments, the model provides an intuitive analytical approach to ray propagation and demonstrates considerable adaptability and computational efficiency in the context of deep-water sound propagation. Consequently, BELLHOP has become a prevalent analytical tool for qualitative assessment of deep-water sound propagation problems. With the BELLHOP model, a numerical simulation of the received field in a flat seabed environment at 300 Hz was conducted. Figure 16 shows the signal delay structures at 24 km reception distance in low-intensity regions and 30 km reception distance in high-intensity regions. Orange markers indicate rays with one bottom reflection, yellow markers denote two bottom reflections, purple markers represent three bottom reflections, and green markers indicate four bottom reflections. The received signal forms multiple clusters based on arrival times [11]. Within the same cluster, relative delay differences are small, while those between different clusters are relatively larger. The dispersion both between and within clusters collectively determines the interference structure of the acoustic field [11]. From the delay structure at the high-intensity reception point, it is evident that sound energy concentrates in the main signal cluster, while the energy contributions from other clusters are relatively low. The energy within the main cluster primarily arises from a single propagation path. Conversely, in low-intensity regions, sound energy is dispersed across multiple clusters, with no individual cluster showing a significantly dominant energy proportion.

The proportion of single-path ray energy to the total energy of the signal rays can be expressed as:

$$R_i(r)={\displaystyle \frac{\left|A_i\right|}{{\displaystyle \sum _k{\left|A_k\right|}^2}}},$$

(26)

where R_i is the energy proportion of the i-th path signal ray and $A$ is the sound pressure amplitude. The path with the highest energy at the receiving point is identified as the main signal ray, with its energy proportion denoted as $R_{\mathrm{main}}$. Figure 17 illustrates how $R_{\mathrm{main}}$ varies with distance. It shows that at receiving points in high-intensity regions on the bottom, the acoustic field energy primarily derives from the main signal path, with $R_{\mathrm{main}}>70\%$. The phases of the rays within the main signal cluster are close together, and through in-phase superposition, the pressure amplitude of the main signal cluster significantly exceeds that of the other signal clusters, thereby exerting a decisive influence on the horizontal-transverse coherence coefficients of the received acoustic field. Consequently, the total sound pressure at the receiving points in high-intensity regions shows low sensitivity to both amplitude and phase disturbances, resulting in a stable pressure amplitude in the received acoustic field, which maintains a high horizontal-transverse coherence coefficient, that is, $\rho (r,\Delta l)\to 1$. In contrast, in low-intensity regions, $R_{\mathrm{main}}<40\%$, leading to a balanced sound pressure among clusters, with sound energy dispersed across multiple clusters and no dominant cluster. In this case, all signal clusters affect the horizontal-transverse coherence coefficients. Moreover, the sound pressure phase exhibits irregular distribution in the process of sound propagation, and the cumulative effect of random disturbances becomes increasingly significant, leading to considerable variations in the received acoustic field. Therefore, the high sensitivity to amplitude and phase disturbances results in a rapid decline in the horizontal-transverse coherence coefficients in low-intensity regions, that is $\rho (r,\Delta l)\ll 1$.

To analyze the impact of medium inhomogeneities on the horizontal-transverse coherence, a simulation topography is created by overlaying large-period and small-period uneven topography. Meanwhile, the distance from the source to each array element is set to the same wavefront distance. With the RAM-PE model, a simulation of the bottom-received field is conducted. The bottom-received field is simulated with the reception depth set at the seabed surface at frequency 290–310 Hz. Through a Butterworth filter, 2D spatial filtering is applied to the measured seafloor topography in Figure 1 to obtain large-period uneven topography (period greater than 40 km) and small-period uneven topography (period less than 5 km). As shown in Figure 18a, the large-period uneven topography from the source to each element is generally consistent, while the small-period uneven topography depicted in Figure 18b exhibits variations.

Figure 19a displays the spatial distribution of the horizontal-transverse coherence coefficients of the received field with only large-period uneven topography. The result shows that the coherence coefficients are all close to 1, indicating complete coherence in the transverse-received field. Subsequently, the effects of adding small-period uneven topography at amplitudes of 0.5, 1, and 2 times the original amplitude on the large-period uneven topography are calculated, as shown in Figure 19b–d. Figure 11 indicates that after introducing inhomogeneous media, the horizontal-transverse coherence in high-intensity regions is less sensitive to disturbances, maintaining a high coefficient with good stability. Conversely, in low-intensity regions, the coherence is more sensitive to disturbances. As the amplitude of small-period uneven topography increases, the coherence rapidly declines, exhibiting weaker stability in the coefficient. This is consistent with conclusions from ray theory analysis.

In actual marine environments, the presence of medium inhomogeneities leads to a degradation of horizontal-transverse coherence, which consequently reduces array gain and limits the underwater detection and communication capabilities of sonar systems. Given these acoustic field characteristics, sonar system design should be based on the level of the horizontal-transverse coherence to balance array aperture and system performance in order to achieve optimal performance in practical applications. Additionally, the deployment strategies for multi-array joint detection should be optimized based on coherence levels to enhance detection range and overall system efficacy.

5. Conclusions

This work utilizes experimental data from a large-aperture HLA in the South China Sea to investigate the variation of horizontal-transverse coherence in the bottom-received field under low-frequency (300 Hz) and long-range (10–40 km) conditions. The results indicate that the horizontal-transverse coherence exhibits a strong dependence on transmission distance, with trends consistent with the transmission loss. In high-intensity regions, the horizontal-transverse coherence is relatively high, with the horizontal-transverse coherence length exceeding 600 λ (3 km). In contrast, in low-intensity regions, the coherence length significantly decreases to a coherence length of 10–30 λ (0.05–0.15 km).

Using the ray theory, we analyze the physical mechanisms behind variations in the horizontal-transverse coherence coefficients. The acoustic field consists of multiple signal clusters with similar arrival times, and the dispersion of rays within and between clusters determines the interference structure, affecting the spatial distribution of the horizontal-transverse coherence. In high-intensity regions, the energy of the main signal rays exceeds 70% of the total energy, exerting a decisive influence on the horizontal-transverse coherence coefficients. The concentration of ray energy results in low sensitivity to phase disturbances caused by minor inhomogeneities, keeping the horizontal-transverse coherence coefficients close to 1. In contrast, in low-intensity regions, the energy of the main signal rays is less than 40% of the total energy, resulting in energy dispersion across multiple clusters. The phases of clusters are highly sensitive to phase disturbances from inhomogeneous media, significantly reducing the horizontal-transverse coherence coefficients.

Research on the horizontal-transverse coherence in deep water with incomplete channels has significant theoretical and practical implications. Uncertainty parameters during sound propagation, such as amplitude and phase disturbances from seafloor topography, sea surface roughness, and internal waves, all affect the coherence. Investigating the mechanisms of horizontal-transverse coherence under more variable environmental conditions, such as strong internal waves, heterogeneous bottom sediment types, and rough sea surfaces, and summarizing the patterns of influence that inhomogeneous media have on horizontal-transverse coherence will enhance the practical utility of coherence. This enhancement will enable more accurate predictions and applications of acoustic field coherence. This is also an important direction for future work.