Transverse horizontal spatial coherence in shallow water

Since the advent of large-aperture array processing, more and more attention has been paid to the sound field correlation, which has fundamental limit to the array gain of spatial coherent signal processing. The two dominant mechanisms that degrade the spatial coherence are normal modes (or multi-paths) interference and the environmental variability caused by several relevant oceanographic processes. In the present study, the transverse horizontal spatial coherence of explosive signals has been studied experimentally by a bottom-mounted array in the Northern South China Sea. And the effects of normal mode interference on the transverse horizontal spatial coherence have been analyzed numerically. Expressed in terms of wavelengths, the coherence length is shown to be larger than 170λ/185λ at acoustic frequency 508-640Hz/80-101Hz in shallow water. It is much greater than Carey’s shallow-water result 30λ estimated from array signal gain after assuming a specific functional form for the coherence (The Journal of the Acoustical Society of America 104, 831 (1998)). It, however, is consistent with Rouseff’s modelling result of a coherence length larger than 100λ (The Journal of the Acoustical Society of America 138, 2256 (2015)). Both Carey and Rouseff argue that the transverse horizontal spatial coherence length depends only weakly on range, in direct. In the present study, however, the coherence length is shown to depend highly on source-receiver range, and it fluctuates synchronously with the sound-field intensity while range varies.


Introduction
The spatial correlation of acoustic field is an important characteristic in underwater acoustics. The resolution of a horizontal array can be improved by increasing its aperture. However, the spatial coherence of acoustic field has physical limitations on the performance of array beamforming spatial gain. Generally, an array's coherence length gives the maximum separation between two points for which coherent processing produces useful gain when a distant source is at broadside [1].
Many works have been done on the horizontallongitudinal correlations in the past years [2][3][4][5][6][7]. However, there are relatively rare results of the transverse horizontal correlation because it is difficult to deploy a large-scale horizontal array. Carey [8] investigated shallow-water experimental data taken from around the world, and estimated transvers horizontal coherence length L coh from array signal gain after assuming a specific functional form for the coherence. Expressed in terms of wavelengths, the average coherence length L coh of his shallow-water results was 30λ. Rouseff [1] developed a statistical model that quantifies how linear internal waves affects the transverse horizontal L coh , neglecting all other forms of environmental variability. Rouseff concluded that L coh is strongly dependent on where the source and array are positioned in the water column. As for a range of 20km, internal wave energy of 400 J/m 2 and a water column of 80m, the modeling result of L coh is between 70λ and 180λ at acoustic frequency higher than 200 Hz but below 800Hz, while L coh is between 220λ and 330λ below 100 Hz. Both Carey and Rouseff argue that the transverse horizontal spatial coherence length depends only weakly on sourcereceiver range. Zhang [9] investigated the experimental data in the Yellow Sea, and he found that L coh at the same range fluctuated synchronously with the soundfield intensity while frequency varies.
In this paper, we present the experimental result about the transvers horizontal spatial correlations of the sound field in shallow water by using the experimental data obtained in the Northen South China Sea. And we investigate the transvers coherence in shallow water by combining the normal modes theory with geometry analysis. Fig.1 sketches the scenario of interest. An acoustic source at position (0, 0, z s ) is oriented broadside to a distant horizontal line array (HLA) at range r and depth z r . The blobs in the figure represent environmental factors that can affect array coherence. These include linear and non-linear internal waves, sea-surface waves, neutrally buoyant intrusions, bottom roughness, and properties of the seabed [1]. Spatial coherence, which describes the similarity between the acoustic fields at two spatially separated points, has no widely accepted definition, but for present purposes, the horizontal correlation coefficient of the acoustic fields measured along the HLA is defined as 2 2 ( , , ) ( , ; ) (  ,  ;  )  max  ( , ; )  (  ,  ; ) rrl p r l t p r r l l t dt p r l t dt p r r l l t dt

Backgrounds
where p(r,l ;t) is the sound pressure at (r,l) in time t domain,  is the time delay, ∆r and ∆l is the longitudinal and transverse separation respectively. Eq.
(1) will be used to calculate the correlation coefficient in the following experiment data analysis. The acoustic fields with high correlation ρ=1 are said to be coherent and ρ=0 are referred to as completely incoherent.
On condition that ∆r is small enough, we have where ρ(r,0,∆l) is defined the transverse horizontal correlation coefficient. The horizontal separation ∆l =L coh such that ρ(r,0, l+L coh )= ρ(r,0, l)e -1 (4) defines the transverse horizontal coherence length L coh as used in this paper.

Experiment setup
In June of 2017, an experiment was conducted in the Northern South China Sea. As shown in Fig.2

Data processing
The sound speed profiles along the propagation track are shown in Fig.4, which are calculated based on the XBT data. The mean temperature profile measured by the vertical TD array near the HLA is shown in Fig.5. We see that there is obvious negative gradient in the speed/ temperature profiles all through the water column along the propagation track during the experiment. The time-variation of the temperature profile near the HLA is shown in Fig.6, measured by the vertical TD array in 65 hours. It shows that there were linear internal waves around the HLA location during the experiment. But we could not find obvious non-linear internal waves in Fig.6.
Four received explosive signals are shown in Fig.7, of which the source-receiver distance are 57 km, 44km, 33km and 26km respectively.
The transmission loss (TL) of the received signals between 508Hz and 640Hz are calculated, as shown in Fig.8. The red dot-dash line in Fig.8, display name TL 1 , is the first order curve fitting result of TL. In this paper, the relative local energy intensity is calculated by    The peaks on the curve of δE versus source-receiver distance indicate local maximums of explosive soundfield intensity, as shown in Fig.9(b). By Eq. (1), the experimental spatial correlation coefficients at different ranges are shown in Fig.9(a), while the explosive signals are processed in 508-640 Hz frequency band. As we can see in Fig.9, the transverse horizontal spatial correlation coefficient depends highly on the source-recerver range, and it fluctuates synchronously with the sound-field intensity while range varies. For example, the sound-field intensity has a local minimum at range of 31.4 km and a local maximum at range of 43.7km (marked by red circle in Fig.9(b)). Correspondingly, the coefficient coefficient has a local minimum and maximum at 31.4km and 43.7km relatively. Fig.11 shows the curve of the correlation coefficient and its standard deviation with transverse separation varying at 31.4km and 43.7km.   The same phenomenon alse occures in other frequency bands, as indicated in Fig.10. This phenomenon is inconsistent with what Carey and Rouseff described in papers. Both of them argue that the transverse horizontal spatial coherence length depends only weakly on range. As for the 508 to 640 Hz frequency band, the transverse coherence length is shown to be larger than 170λ (expressed in terms of certer-frequency wavelength) at most ranges less than 50km in shallow water. It is much greater than Carey's shallow-water result 30λ estimated from array signal gain after assuming a specific functional form for the coherence, but consistent with Rouseff's modeling result. Fig. 11. Correlation coefficient and its error range at 31.4km (a) and 43.7km (b), 508Hz-640Hz. A similar experiment has been conducted in the same area of South China Sea in November 2016 (Exp2016). The bottom-mounted HLA with unequal spacing between adjacent elements had an aperture of about 3km. The spatial correlation coefficient and its standard deviation in 80-101 Hz frequency band are shown in Fig.12. Obviously, the transverse coherence length is larger than 185λ at range of 10.9 km. Only one explosive source was casted braodside to the large array, so the spatial correlation data at other ranges could not be obtained.

Theoretical analysis
In this section, a theoretical analysis will be made for the phenomenon of transverse horizontal correlation coefficient fluctuating with range.
where P(r,l ;ω) is the spectrum of the sound pressure p(r,l ;t), * represents the complex conjugation, ω denotes the angular frequency, ω 1 and ω 2 are the lower and upper angular frequencies.
For simplicity, let ∆r=0 and ω 1 =ω 2 , so we have * ( , ; ) ( , ; ) ( , ) ( , ; ) ( , ; ) Re P r l P r l l rl P r l P r l l On condition that sound speed and density depend only on depth z, the underwater acoustic pressure field generated by a point source can be expressed by normal modes theory. The sound pressure can be written as where Ψ m (z s ) and Ψ m (z r ) are the eigenfunctions at the source and the receiver, respectively. k m is the horizontal wavenumber of the mth mode. δθ m (r, ∆l) is the mode phase transverse variation (MPTV).
According to Eq. (7)-(11), we have 4 interference between different modes. At some points in the sound field, the modes which make a major contribution to the total energy are almost in phase, so the sound-field intensity will have local peaks at these points in the rang-depth pattern. Similarly, the intensity will have local valleys at the points where the major modes are out of phase.

Situation of modes in phase
At the points where the major modes are in phase, Besides, we can do the following analysis. The correlation coefficient of P(r,l) and P(r, ∆l) can be written as where Φ(r,l) and Φ(r, l+∆l) is the phase angle of P(r,l) and P(r, l+∆l) respectively. In other words, the correlation coefficient is equal to the cosine of the angle between P(r,l) and P(r, l+∆l) on the complex plane. At the points where the major modes are in phase, the phases of the major modes are approximately equal, so small variation of mode phases, δθ m (r, ∆l), has little influence on the phase angle of the total sound field which is approximately the sum of the major modes, as shown in Fig.13. As a result, . 13. Phase angle geometry. The major modes (mode 1, 2 and m) have similar phases. Small variation of mode phases has little influence on the phase angle of their sum.
So ρ(r,∆l) is approximately equal to 1 at these points.

Situation of modes out of phase
At the points where the major modes are out of phase, the phases of the major modes are dispersed from 0 to 2π, so the cosine and sine of   mm r  are random variables in the range of -1 to 1, and Eq. (16) is no longer valid. In this case, it is difficult to simplify Eq. (12) as we do in Section 4.1. However, we can still analyse the correlation coefficient from the perspective of geometry. As shown in Fig.14, the phases of the major modes are dispersed from 0 to 2π, so small variation of mode phases has great influence on the phase angle of their sum. In a statistical sense, the value of Φ(r,l)-Φ(r, l+∆l) is distributed randomly in range of 0 to 2π. So we can deduce directly

Simulation
We use the normal mode program KrakenC to calculate the acoustic field at 100 Hz. In the simulation, the depth of water column is 100m, the sound speed profile is calculated according to the temperature in Fig.5 and thought to be range-independent. The source/receiver depth is 50m/100m. A two-layer liquid bottom model is used and the thickness of the sediment layer is 10 m. The sound speed is 1550 m/s, the density is 1.6 g/cm 3 , and the attenuation coefficient is 0.09 dB/λ inside the sediment layer. The infinite basement has a sound speed of 1650 m/s, a density of 2.0 g/cm 3 , and an attenuation coefficient of 0.18 dB/λ. Fig.15(a) shows the simulated TLs at ranges from 50km to 53km.
For simplicity, the transverse variation, δθ m (r, ∆l), of each mode phase caused by the variation of ocean environment is assumed to be subjected to uniform distribution and to be independent and identical distribution. On these assumptions, the average correlation coefficient of 200 Monte-Carlo simulation results is shown in Fig.15(b), of which the y axis is the standard deviation of MPTV, δθ m (r,∆l). Compared with Fig.15(a), the correlation coefficient is shown to depend highly on range, and it fluctuates synchronously with the sound-field intensity while range varies. In Fig.15, WAPD has local minimum at range of 50km and 52km, where the sound-field intensity and the standard deviation of MPTV have local maximums, as analysed in Section 4.1; WAPD has local maximums at range of 50.9km and 52.7km, where the sound-field intensity and the standard deviation of MPTV have local minimums, as analysed in Section 4.2.
It needs to be pointed out that the fluctuation of correlation coefficient with range is not due to the fluctuation of MPTV, δθ m (r,∆l). Generally speaking, δθ m (r,∆l) changes slowly with range r. Even if the standard deviation of MPTV keeps constant with range r, the average correlation coefficient still fluctuates with r, as shown in Fig.15(b). The experimental result and theoretical analysis indicate that the interference between normal modes is the fundamental cause of the fluctuation of correlation coefficient and acoustic intensity with range. Rouseff's modelling result of the transverse horizontal coherence length depends weakly on range, because of "neglecting the m≠m' cross terms" (See Eq. (19) in Ref. [1]).

Conclusions and perspectives
The transverse horizontal spatial coherence in shallow water is investigated by using the experimental data obtained in the Northern South China Sea and by combining the normal modes theory with geometry analysis. It is shown that the transverse horizontal spatial coherence depends highly on the source-receiver range, which fluctuates synchronously with the sound-field intensity while range varies, and that the fundamental cause of this phenomenon is the interference between normal modes. At the ranges where the major modes are in phase, the transverse coherence is strong locally as well as the acoustic intensity, and the converse is also true.
What's more, the experimental results showed that the transverse coherence length was larger than 170λ at most ranges less than 50km in shallow water as for the 508 to 640 Hz frequency band, and was larger than 185λ at a range of 10.9km in shallow water as for the 80 to 101 Hz frequency band. These results of coherence length indicate that it is possible to utilize large aperture (e.g. 200λ) horizontal line array for coherent processing in shallow water. This is of great significance for long distance acoustic detection and communication in shallow water.
The obvious degradation of the correlation coefficient when the source is dropped in the area with strong slope sea-floor, as shown in Fig.3, Fig.9(a) and Fig.10, and the major environmental reasons leading to coherence degradation with transverse separation in shallow water will be investigated in the future.