For an acoustic receiver deployed at the bottom of the direct arrival zone of a submerged source at short horizontal ranges in deep ocean, the interference pattern of the direct and surface-reflected acoustic arrivals shows periodic modulation, which is directly related to the source depth, source frequency, and vertical arrival angle. In this work, the interference cycle presented in the frequency domain is used to extract the broadband source depth, with the vertical arrival angle obtained from the ratio of vertical acoustic intensity and horizontal acoustic intensity from the signal recorded by a single vector sensor. Experimental results demonstrate the source depth estimation without requiring knowledge of the ocean environment.

## 1. Introduction

In the deep ocean, for an acoustic receiver deployed at the bottom of the direct arrival zone, where the direct ray from a shallow source arrives without boundary reflection, the direct (D) and surface-reflected (SR) arrivals usually dominate the acoustic field as the predicted transmission loss of the D and SR arrivals is much lower compared with other bottom-reflected arrivals. According to the image source theory, the interference pattern of the D and SR acoustic arrivals shows periodic modulation, which is directly related to the source depth, source frequency, and vertical arrival angle (VAA).^{1,2} Source depth estimation using this interference structure with a vertical line array (VLA) has been analyzed in recent years.^{1–5} McCargar and Zurk first introduced a robust source depth estimation method in the reliable acoustic path (RAP)^{6} zone without requiring knowledge of the ocean environment using a VLA, which utilizes the depth-dependent modulation changes of the interference structure along the arrival angle trace at a certain frequency.^{1} However, a big extent of the target's track in arrival angle is needed to get enough resolution in depth estimate to compensate for the lack of broadband signals. Similar to the interference cycle, the time delay of the D and SR arrivals is also related to the source range and depth. By tracking the time delays between the D and SR arrivals, a moving source localization with a single hydrophone was achieved.^{7} However, several measurements coupled with a motion hypothesis are needed. Weng proposed another source range and depth estimation method with a single hydrophone based on two cycles of the acoustic intensity periodic structure among the bottom-reflected, bottom-surface-reflected, surface-bottom-surface-reflected, and bottom-surface-reflected arrivals in the shadow zone.^{8} In this method, two cycles of the acoustic intensity periodic structure have been used to compensate for the lack of array aperture.

As the interference pattern between the D and SR acoustic arrivals is related to the source depth, source frequency, and arrival angle, when one parameter of the source frequency or the arrival angle is known, the source depth can be directly extracted from the interference cycle of the other. Instead of utilizing the interference cycle along the arrival angle at a single frequency in Refs. 1 and 2, the depth-dependent modulation changes of the interference structure along the frequency domain at a certain arrival angle is used in this work.

## 2. Theory and technique

### 2.1 The interference pattern

The interference pattern of the D and SR arrivals is illustrated in this subsection following Ref. 2. Ignoring the stratification of the ocean water, ray bending and the contribution of the bottom-reflected arrivals, the acoustic field from a submerged source with depth $zs$ is a combination of the direct and surface-reflected arrivals as shown in Fig. 1. By image source theory, the pressure field can be described as a pair of out of phase point sources, and the received pressure at point $(r,z)$ is given by

where $P1(r,z,\omega )$ and $P2(r,z,\omega )$ are the pressure of the D and SR arrivals, respectively, the wavenumber in the environment with sound speed $c$ is given by $k=\omega /c$, $S(\omega )$ is the source complex spectral amplitude, $\Gamma $ is the surface reflection coefficient, and $R\u2212$ and $R+$ are the slant ranges of the direct and surface-reflected paths, respectively, $R\xb1=r2+(z\xb1zs)2$. Due to its high acoustic impedance, the air-sea interface can be approximated as a pressure release boundary. In addition, at low frequencies, rough surface scattering is negligible for the steep reflection angles. Thus, the surface reflection coefficient $\Gamma \u2248\u22121$.

For an acoustic source and receiver close to the upper and bottom region of the deep ocean, respectively (i.e., the depth separation between source and receiver is large, $z\u226bzs$), only considering the phase difference of the direct and surface-reflected arrivals yields the following expression:^{2}

where $R=r2+z2$, $sin\u2009\theta s=z/R$, and $\theta s$ is the arrival angle (in practice, $\theta s$ is a weighted average arrival angle of the D and SR arrivals). According to Eq. (2), the acoustic intensity at the receiver can be approximated as

where $\rho $ is the water density. As shown in Eqs. (2) and (3), the interference pattern of the D and SR acoustic arrivals shows periodic modulation in source depth, wavenumber (or source frequency), and vertical arrival angle. To achieve source depth estimation, one can use the interference cycle in arrival angle domain at a certain frequency through tracking the source for a long range with a VLA.^{1} In contrast, one can use the interference cycle in frequency domain at a certain VAA for broadband sources. The value of VAA can be obtained from the ratio of vertical acoustic intensity and horizontal acoustic intensity with a single vector sensor. This source depth estimation method will be illuminated in the following.

### 2.2 Arrival angle estimation

When the bottom-reflected paths are neglected, the particle horizontal velocity received by a vector sensor deployed at the deep water can be expressed as

where $\theta 1$ and $\theta 2$ are the vertical arrival angle of the D and SR arrivals, respectively. As the depth separation between source and receiver is large, both $\theta 1$ and $\theta 2$ are close to $\theta s$, Eq. (4) herein can be approximated as

Similarly, the particle vertical velocity can be approximated as $Vz(r,z,\omega )\u2248P(r,z,\omega )\u2009sin\u2009\theta s/\rho c$. Thus, the horizontal and vertical intensity flux can be written as

respectively, where the superscript asterisk indicates the complex conjugate operator. Thus, the vertical arrival angle $\theta s$ can be estimated from the ratio of the vertical and horizontal intensity flux,^{9}

where $\omega 1$ and $\omega 2$ are the lower and upper bounds of the available frequency band, respectively.

### 2.3 Source depth estimation

The sinusoidal form of the depth-dependent modulation in Eq. (3) suggests that the source depth estimation could be achieved for broadband sources once the vertical arrival angle $\theta s$ is obtained. This modulation is contained in the cosine term, which shows the modulation is periodic in frequency $f$ with period^{2}

The modulation period can be estimated using a Fourier analysis of the spectrum of the acoustic intensity. When the value of the frequency modulation period $fperiod$ and vertical arrival angle $\theta s$ are obtained, the source depth can be estimated by^{2}

In this paper, the multiple signal classification (MUSIC) algorithm is used to analyse the frequency cycle of the spectrum of the acoustic intensity, as it is more effective than fast Fourier transform for short analysis sequences, i.e., more effective when the available frequency bandwidth of the acoustic intensity is not broad enough to achieve high resolution of the frequency cycle and source depth.

## 3. Experimental results

The approach described in Sec. 2 has been used to analyse experimental data. The experiment was conducted on April 20, 2018 in the South China Sea. In the experimental area, the ocean bottom is almost flat and the mean water depth is about 3472 m. Explosive sources with nominal explosive depth equal to 200 m were used. The receiver was a four-component ocean-bottom seismometer (OBS) deployed at the ocean bottom as shown in Fig. 2(a), with a sampling rate equal to 250 Hz, which provides three components of the particle velocity and the pressure field. During the experiment, 14 sources were dropped from the rear deck of the ship and were distributed along a line from 0.5 to 5.5 km distance from the receiver.

An example of depth estimation using the signal recorded by the OBS for the source exploding at 4.21 km is presented in Fig. 2, where in panel (a) the sound speed profile (SSP) along depth is shown as a solid line. We show the signals representing the sound pressure and the three components of the particle velocity in Fig. 2(b). There are two distinct peaks in the waveform envelope corresponding to the D and SR acoustic paths, the arrival time difference of these two paths is about 0.16 s. Figure 2(c) presents the spectrum of the acoustic intensity of the signal from 59.8 to 60.8 s reported in Fig. 2(b), showing obvious interference structure due to the coherent summation of the D and SR sound. Only the frequency band from 20 to 100 Hz is used as the energy of the seismic acoustic below 20 Hz is non-negligible and the energy higher than 100 Hz is filtered out by the internal circuit of the OBS.

The spectrum of Fig. 2(c) could be considered as an input parameter to be treated by MUSIC (the function pmusic of matlab^{10} was implemented). Results are shown in Fig. 2(e). Based on the fact that $fperiod=1/tdiff$ and Eq. (9), the X axes are transformed into frequency period and depth in Figs. 2(d) and 2(f), respectively. Due to the interference of the D and SR acoustic paths, the frequency period value of the highest peak in Fig. 2(d) is 6.07 Hz, corresponding to the highest peak in Fig. 2(e) with the time difference equal to 0.16 s. The frequency period value of the second highest peak in Fig. 2(d) is 56.7 Hz, with the arrival time difference equal to 0.018 s in Fig. 2(e). This peak comes from the coherent summation of the shock wave and the first bubble pulse of the source, as its time difference value is close to that of the shock wave and the first bubble pulse which is about 0.023 s. The second highest peaks of the other 13 sources also show close time difference values around 0.023 s. The other three lower peaks in Figs. 2(d)–2(f) are false peaks since the signal subspace dimension in the function pmusic is set to 10, indicating the existence of five peaks in the half space of the output of the pmusic function.

The particle horizontal velocity $Vr(r,z,\omega )$ is calculated by $Vr(r,z,\omega )=Vx(r,z,\omega )\u2009cos\u2009(\varphi s)+Vy(r,z,\omega )\u2009sin\u2009(\varphi s)$, where $\varphi s$ is the source azimuth angle obtained from $\varphi s=arctan(\u2211\omega 1\omega 2Iy(r,z,\omega )/\u2211\omega 1\omega 2Ix(r,z,\omega ))$. Following the procedures in Sec. 2.2, the value of the VAA is estimated to be 39°. The depth estimation result using this value with the reference sound speed $c$ taken as the speed at the receiver (equal to 1520 m/s), is presented in Fig. 2(f), where the real source depth is plotted as dashed line. As shown in Fig. 2(f), the highest peak appears around the true depth of the source, in excellent agreement with the image source theory prediction. The second highest peak at 21 m is a false peak due to the bubble pulse of the source.

The VAA and source depth estimation results of all 14 sources are given in Fig. 3. Consistent with the VAA of D and SR paths calculated using BELLHOP,^{11} the estimated VAA generally decreases with the source-receiver range. As illustrated in Fig. 3(b), all 14 estimated source depths are around the nominal source depth. Generally, when the estimated VAA is smaller than the average value of the model calculated VAA of the D and SR paths, shown as a solid line in Fig. 3(a), the estimated depth is bigger than the nominal source depth and vice versa.

## 4. Conclusion and discussion

This paper described a source depth estimation method using a single vector sensor near the ocean bottom without environmental knowledge except the reference sound speed. The approach is based on using simple image-source theory expressions to describe the interference structure of the direct and surface-reflected arrivals from a submerged source at short horizontal ranges. This interference structure shows periodic modulation in the source depth, source frequency, and vertical arrival angle. With the vertical arrival angle obtained from the ratio of the vertical and horizontal intensity flux, the source depth is estimated from the frequency cycle of the spectrum of the acoustic intensity. This approach is plausible due to the practically ubiquitous strong reflection from the sea surface, and the insensitivity of the arrival time difference of the D and SR paths with steep arrival angles to the fluctuation of the water sound speed profile. The proposed approach has been proven to be efficient in source depth detection for a set of experimental data.

According to Eq. (9), the performance of this source depth estimation method is affected by the following three aspects: available frequency bandwidth, reference sound speed, and accuracy of the arrival angle. The available frequency bandwidth determines the resolution of the source depth, with wider bandwidth resulting in higher depth resolution. The value of the reference sound speed is expected to have little influence on the processing performance as it is a kind of mean sound speed from the source to sea surface, resulting in a small relative error even if the SSP is unknown. The accuracy of the arrival angle is crucial as even a small angle error will result in a large depth estimation error; thus, high signal-to-noise ratio is usually needed.

In contrast to the isovelocity model in Fig. 1, the sound velocity and the angle of the ray change with depth in the real environment. For sources at relatively short ranges, variation of the sound speed with depth has little effect on the frequency interference cycle as the arrival angle is steep, while for remote sources, ray bending, i.e., the difference of the arrival angle at the receiver and upper region, is non-negligible, and Eq. (3) is no longer accurate to describe the interference structure. Thus, more work, such as frequency interference cycle modification considering the water stratification, is needed in the future to improve the accuracy of the source depth estimation for remote sources.

## Acknowledgments

This work was supported by the Frontier Science Research Project of the Chinese Academy of Sciences (Grant No. QYZDY-SSW-SLH005) and the National Natural Science Foundation of China (Grant Nos. 11804364, 11804362, and 11874061).