Time arrival structures of the empirical Green’s function extracted fromambient noise in shallow water
Xishan Yang(1)(2) , Fenghua Li(1)(1) State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China.

oucyxs@163.com,(2) University of Chinese Academy of Sciences, Beijing, 100049, China. lfh@mails.ioa.ac.cn

Abstract
Theoretical and experimental results show that the empirical Green’s function (EGF) can be extracted by correlating the ambient noise between two sensors. Ambient noise was recorded continuously on two horizontalline arrays (HLAs) deployed in shallow water with horizontal separations of approximately 0.5 km. Stable EGFs
(20–400 Hz) were extracted from ambient noise correlations between the HLAs using array beamforming which
can accelerate the convergence rate of EFGs significantly. The EGFs extracted from the short-distance arrays
have three distinct envelopes which correspond the head wave, the direct wave and the surface-reflected wave.
The fluctuations of travel times of EGFs are shown to be in agreement with the variations of sound speed files.
It is helpful in developing passive acoustic tomography systems. .
Keywords: Noise correlations, Arrival structures, Green’s function

1INTRODUCTION

It has been demonstrated that the cross-correlation function of the Brownian noise at two positions yields thearrival-time peaks as though there were an acoustic source at one position and a receiver at the other position
within a diffuse ultrasonic field.[10] The same theory of correlating the random signals to extract their coherent
aspects has been applied theoretically and experimentally to both land seismic[7] and ocean environments[8].
The time derivative of the correlations leads to an approximation of the empirical Green’s function(EFG) between the sensor points, which provides an alternative possibility for monitoring the ocean environment. The
surface reflected ray paths have been obtained the local Green’s function on a horizontal array with a lengthof about 120m.[4] The critical angle can be extracted from the strength of the surface reflected ray paths. The
deterministic normal modes have also been retrieved from cross-correlations of acoustic noise on three moorednear-bottom instruments with horizontal separations of approximately 5km, 10km. [1]In the ocean, though, the constantly changing environment means that the propagation path could change before the correlation reaches stochastic convergence. To enhance the emergence rate of EGFs, researchers have
implemented the beam-forming technique on a vertical array[9] and on two separated arrays[9, 5]. Our study
will show the improvement on the extraction rate of the EFGs from two horizontal arrays by beamforming
technique. The time arrival structures of the EFGs are analyzed and the different ray paths of the EFGs areexpected to be used in the passive acoustic tomography.
This paper is organized as follows. Section 2 presents a generalized theoretical formulation for the Green’sfunction extraction with conventional beam-forming. Section 3 investigates the time arrival structures of theEFGs extracted from ocean ambient noise in shallow water. The conclusions are summarized in Section 4.

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2THEORY

The cross-correlation of the signals received at two separate sensors (the mth hydrophone of array A and thenth hydrophone of array B, as shown in Fig. 1), is given byZ Tr /2

Cm,n (τ) =−Tr /2p(rA,m ,t + τ)p(rB,n ,t)dt,(1)where Tr is the recording duration, rA,m and rB,n are the locations of the two sensors, and p(r;t) is the pressure
field recorded at each sensor. In the frequency domain, Eq. (1) becomes
Cm,n (ω) =1
p(rA,m , ω)p∗ (rB,n , ω),Tr(2)where p∗ (rB,n , ω) is the complex conjugate of p(rB,n , ω), and ω is the angular frequency. The time derivativeof the NCF has been proved to be approximately the sum of time-domain EGFs[11]:dCm,n (τ)≈ −G(rA,m , rB,n ; τ) + G(rB,m , rA,n ; −τ),dτ(3)where G(rA,m , rB,n ; τ) is the causal EGF located at rA,m transmitting to rB,n , G(rB,n , rA,m ; −τ) is the acausal EGF
transmitting from rB,n to rA,m , and τ is the time delay between the two receivers.

Figure 1. (Color online) Definition of the geometric variables for waves that travel from different directions
to two receiver arrays. The regions of constructive interference are indicated by gray, and the blue regions
correspond to the noise at the sides where τ < 0.
To improve the convergence of the cross-correlation function, the signals from two arrays of hydrophones canbe beam-formed and used in place of the signals from the two individual sensors. Now, it is assumed thatθ = θ0 is one of the end-fire directions of the mth hydrophone of array A and the nth hydrophone of array B,
as shown in Fig. 1. When the array is not very long, the plane wave beam on array A in the end-fire directionof θ = θ0 can be expressed asbA (θ0 ) = wHA (θ0 )pA ,4466(4)where wA (θ0 ) = [wA1 (θ0 ), wA1 (θ0 ), . . . , wAM (θ0 )]T is the weight vector of array A, with wAm (θ0 ) = exp (−iωτm (θ0 )).
For simplicity and clarity, the regular frequency ω has been omitted. Here, (•)H denotes a complex transposeoperation, (•)T is a transpose operation, pA = [p(rA,1 ), p(rA,2 ), . . . , p(rA,M )]T is the frequency-domain signal received by array A, and M is the number of array elements. Similarly, the plane wave beam on array B isdenoted by bB (θ0 ). The cross-correlation function between the beams of array A and array B in the direction
θ0 isHH
CAB (θ0 ) = bA (θ0 )bH
B (θ0 ) = wA (θ0 )pA pB wB (θ0 ).

(5)With the beam-forming technique, the end-fire direction signals beneficial for extraction
are not affected, but√
the signals from non-end-fire directions are filtered out. One can obtain 20 log10 MN array gain in ideal
conditions,[2] which means the emergence rate of extraction will be enhanced significantly.

3EXPERIMENT

3.1 Overview of experiment

Figure 2. Schematic of the layout of the experiment site and the two bottom horizontal arrays. Arrays A andB are approximately parallel to each other. A vertical thermistor chain was placed between arrays A and B toacquire sound speed profiles.

Figure 3. (Color online) Spectrogram of the ambient noise recorded over 24 h in the frequency band
[10–1000 Hz]. The maximum value was set to unity (i.e., 0 dB).
The experiment was carried out in the South China Sea in 2018. Two self-recorded separated horizontal arrays,arrays A and B positioned as shown in Fig. 2, were deployed on the bottom to collect noise signals in shallow

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water. The distance between arrays A and B is about 0.5 km. The two arrays are approximately parallel toeach other. A thermistor chain was used to achieve water column sound speed profiles (SSPs) during theexperiment for some time periods. Other technical features of the environment parameters have been described
previously.[3]A typical normalized noise spectrum is shown in Fig. 3. The noise spectrum changes considerably and it
indicates that the ocean noise is highly non-stationary. The ambient noise level appears to be significantly
higher in the low-frequency band (f < 200 Hz) for most of the 24 h. At low frequencies, the ambient noise
field is primarily dominated by shipping noise.
3.2 Experimental results

Figure 4. Envelope of EGFs extracted from different element numbers for an accumulation time of 24 h.
Here, arrays A and B were used to extract the EFGs between them. 24 hours of ambient noise was segmented
into consecutive short intervals, each of duration 10 s. Prior to computing the cross-correlation functions, each
10 s of recorded data was filtered in the dominant frequency band (20–400 Hz).
Figure 4 compares the EGFs obtained by two hydrophones and two arrays for different element numbers. It
can be seen from the figure that the beamforming technique significantly improve the rate of EFGs extraction
from ocean ambient noise.
The time-domain of the EFG between the two short-distance is shown as the black line in Fig. 5a. From
the figure, it can be seen that the signal contains three envelopes. At the same time, the ray-based Bellhop
simulation was calculated for 210 Hz, which is the center frequency of 20–400 Hz. The arrival-time structureof the ray-based simulation is also shown by solid line in Fig. 5b. It can be found that the second peakcorresponds to the direct path. The third peak corresponds to the eigenray striking the sea surface once.
From the simulation, it can be found that the first peak of the extracted signal does not relate to any eigenray in water column. From its arrival time, it can be obtained that the sound speed of the first envelope isabout 1610 m/s, which is almost equal to the sediment sound speed in the experimental area. It is known thatoceanic sources can cause head waves at the water-sediment interface. Consider a point source that produces
a spherically spreading wave in a simple waveguide. Inside the critical range, where the incident ray is above
the critical angle, the direct wave splits into reflected and transmitted waves.[6] Beyond the critical range, thedirect wave is totally reflected. However, at the critical range, where the incident ray is exactly at the criticalangle, the transmitted wave travels parallel to the interface. This part of the transmitted wave is the so-called
lateral wave.
In the EFGs extraction, one correlates noise signals between two receivers by averaging over a sufficiently longtime, and then only the noise sources propagating through both receivers contribute to the EGFs. Thus, the4468

Figure 5. (Color online) (a) Comparison of ray-based time arrivals (colorful stem pattern) of Green’s function
with the EFG extracted from experiments (blue line). (b) Ray-based simulated eigenrays corresponding to theray paths in (a), and the black line represents the head wave.

Figure 6. (Color online) Time-domain EGFs extracted from ambient noise between arrays A and B by beamforming every 24 h for 30 days. The maximum amplitude in each signal was set to 1.

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first envelope comes from lateral waves transmitting through two arrays in sediments. After correlating thenoise, these lateral waves become a head wave in the extracted EFG. The sediment sound speed, which plays asignificant role in the sound propagation, can be inverted directly.
Figure 6 shows the experimental EFGs every 24 h for 30 days. From the figure, it can be seen that the arrival
times of head waves are constant over time. However, the arrival times of the second and third envelopes
fluctuate with different time due to the change of the SSPs in water column.

4CONCLUSIONSThe use of beamforming technique dramatically improves the strength of the correlated signal and reduces thetime needed to build up the signal peak when correlating ambient noise recorded at horizontally separated sensors. Stable EGFs (20–400 Hz) were extracted from the ambient noise correlations from three arrays for about
five moths certificate the robustness of this methods. Head waves, which can be used to invert the sedimentsound speed, were obtained from the EFGs between two short-distance arrays. The arrival time fluctuations ofthe EFGs extracted from two long-distance arrays can be used to conduct the passive tomography.

ACKNOWLEDGEMENTS
This research was supported by the National Natural Science Foundation of China under Grant Nos.11874061
and 11434012.

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