Striation-based source depth estimation with a vertical line array in the deep ocean

Abstract

A striation-based method with a vertical line array is proposed for source depth estimation. Broadband striation structures of direct and surface-reflected arrivals after propagating to receivers near the ocean bottom are applied. A tracking algorithm for the striation structures is proposed based on the extended Kalman filter. A cost function for source depth estimation is presented by matching the traces of the measured striations with a library of model-based traces under different source depths. The method is demonstrated on array data collected during an acoustic research experiment in the South China Sea in 2016.

1. Introduction

In deep ocean, depth insensitivity is a disadvantage for source depth estimation. The detection of sources by deploying bottom-moored hydrophones has been analyzed in recent years.^1–6 The interference feature of direct (D) and surface-reflected propagation (so-called Lloyd mirror)⁷ has been studied in source depth estimation. Light (constructive interference) and dark beams (destructive interference) are apparent in the sound fields of deep oceans. A simple image theory expression^3–5 for the interference pattern has been derived from either modal or ray propagation expression. The passive depth separation^3–5 of sources has been studied using vertical line arrays (VLAs) deployed below the critical depth in a deep ocean with modified Fourier transform. This method presents a power output striation with a vertical beamformer along direct arrival angles. This striation fluctuates at the structure frequency for different source depths. The performance metrics,⁶ such as the resolution and ambiguity in transformed-based depth estimation, are analyzed in detail. However, the signal-to-noise ratio (SNR) is insufficient for the detection of sources at a single frequency, especially for quiet submerged sources.

This study analyzes a depth estimation method based on a broadband interference structure that can distinguish near-surface and submerged targets. The interference structure is obtained from the broadband intensity output of bottom-moored VLA steered in the direction of arrival (DOA) of the downward propagation from a near ocean-surface source. The intensity plot versus frequency and source range shows striations of high and low intensity, which are related to the interference of the Lloyd mirror.⁸ The number of striations and their positions in a certain frequency bandwidth are sensitive to the source locations, especially for the source depth. Therefore, when the source ranges of a moving source are known, the source depth can be estimated from the features of these striations. When the source range is unknown, the intensity plot versus the frequency and DOA of the downward propagation shows a similar interference structure and can be used for source depth estimation. Given that the striations gradually form with moving source, a tracking algorithm based on the extended Kalman filter (EKF) model⁹ is presented to track the positions of the striations changing with the time. Finally, the position and number of striations are used to obtain the source depth by matching the estimates with the model-based values at different source depths. A cost function is presented to quantify the difference between the experimental estimates and model-based values.

2. Method

Sections 2.1–2.3 present the striation structure in simulation, EKF for tracking striations, and source depth estimation. The simulations are presented to illustrate the steps of source depth estimation.

2.1. Striation structure in simulation

The simulation environment is consistent with the measured environment of an experiment, which is illustrated in Sec. 3. Thus, the conclusions from the simulation are useful for understanding the observations in the experiment. Figure 1(a) shows the sound speed profile (SSP). The sound channel is an incomplete deep-water channel and does not have critical depth because the sound speed at the sea bottom is less than the sound speed on the sea surface. The mixed-layer depth is approximately 40 m, whereas the sea depth is approximately 3904 m. The seabed is relatively flat. The receiver array moored at the bottom comprises 16 equally spaced hydrophones with 4 m spacing. The center depth of the array is approximately 3700 m. Sediment sound speed, density, and attenuation coefficient are approximately 1560 m/s, 1.6 g/cm³, and 0.2 dB/λ, respectively. The Bellhop¹⁰ model is used to demonstrate the feasibility of the proposed method. The intensities of array outputs of the conventional beamformer (CBF) at the DOA of the direct arrival (hereafter called D-DOA) along the frequency are calculated. The intensity plot versus the frequency and D-DOA, which changes with the source range, exhibits striation structures.

Fig. 1.

(Color online) (a) SSP of the experiment. Striation structures of the source depths in simulation: (b) 3, (c) 10, and (d) 50 m. Illustrations of the simulated source at source depth of 10 m, (e) striation tracking by EKF, and (f) estimation result with the cost function.

Figures 1(b)–1(d) show the simulated interference structures at different source depths. No interference is observed incident from the grating lobe range of the array.¹¹ Therefore, the beam intensity spectra at the steering angle covers the frequency from 50 to 1000 Hz. Continuous striations of high or low intensities are observed. Meanwhile, the number of striations increases with increasing source depth. These results indicate that the position and the number of striations are closely related to the source depth in deep oceans. In this paper, the striation with low intensities is referred to as dark striation and used as the observations in the following model of filtering. A single striation with high intensities (light striation) covers wider frequency ranges than dark striation as shown in Figs. 1(b)–1(d). Thus, tracking the local minimum intensities in the dark striations could result in a smaller uncertainty than tracking the local maximum intensities in the light striations.

2.2. EKF for tracking striations

The EKF is used for striation tracking. The state vector is

$$D_{k+1}=\left[\begin{array}{c}{f}_{k+1}\\ B_{k+1}\\ {\dot{f}}_{k+1}\\ {\dot{B}}_{k+1}\end{array}\right]=\left[\begin{array}{c}{f}_k+\Delta d{\dot{f}}_k\\ B_k+\Delta d{\dot{B}}_k\\ {\dot{f}}_k\\ {\dot{B}}_k\end{array}\right]+w_k,$$ (1)

where $(f_k,B_k)$ is the frequency and beamforming intensity of striations at ping $k$ of the D-DOA sequence, $({\dot{f}}_k,{\dot{B}}_k)$ represents the corresponding instantaneous tracking velocity of $f$ and $B$, $\Delta d$ is the difference of the D-DOA between pings in the simulated striation, and $w_k$ is the process noise, which is assumed to be a white Gaussian noise. For transition tracking, the transition is based on the reasonable assumption that the velocities at ping $k+1$ are the same as those at ping $k$, and the variations are included in the state noise. Therefore, ${\dot{f}}_{k+1}={\dot{f}}_k$ and ${\dot{B}}_{k+1}={\dot{B}}_k$. The observation vector is

$$Z_k=h(D_k)+v_k,$$ (2)

where $h(\cdot )$ is the nonlinear observation function and $v_k$ is a white Gaussian noise process that accounts for errors by the measurement system. The function $h(\cdot )$ represents that it searches the spectrogram to follow the minimum intensity of striation within a small frequency region centered at the frequency according to the previous ping.¹² $Z_k$ is given by a combination of the dynamic frequency and minimum intensity of striation at ping $k$. The estimation of $D_{k+1}$ is based on $Z_{k+1}$ and $D_k$. The update of the state vector is as follows:

$${\widehat{D}}_{k+1|k+1}={\widehat{D}}_{k+1|k}+K_{k+1}{e}_{k+1},$$ (3)

where ${\widehat{D}}_{k+1|k}$ is the a priori estimate of ${\widehat{D}}_{k+1}$ given the previous state $D_k$, $K_{k+1}$ is the Kalman gain, and the new information is $e_{k+1}=Z_{k+1}-h({\widehat{D}}_{k+1|k})$.

The striation traces by the EKF tracking algorithm are demonstrated in Fig. 1(e). The lines are the tracing results, and the background is the striation structure of the interference pattern in simulation in Fig. 1(c). The striations are selected by searching for the local minimum intensity, along with the D-DOA, by a given appropriate initial frequency.

2.3. Source depth estimation

The cost function for the source depth estimation is defined as

$$\begin{array}{c}C\left(z_s^a\right)=({\left(1-\frac{1}{N_1}\sum _{n=n_{\min }}^{n_{\max }}\sum _{\left(\theta ,f\right)}{P}_{\hspace{0.17em}\text{exp}\hspace{0.17em}}\left(\theta ,f,z_s|\left(\theta ,f\right)\in T_{\text{model}}^n\left(z_s^a\right)\right)\right)}^{-2}\\ { +{\left(\frac{1}{N_{\theta }}\sum _{m={\theta }_{\min }}^{{\theta }_{\max }}\left|\text{sign}\left(N{f}_{\hspace{0.17em}\text{exp}\hspace{0.17em}}\left(\theta ,f,z_s\right)-N{f}_{\text{model}}\left(\theta ,f,z_s^a\right)\right)\right|\right)}^{-2})}^{-1/2},\end{array}$$ (4)

where the first term is related to the position of striations, and the second term is related to the number of striations. The variables $z_s^a$ and $T_{\text{model}}^n$ are the assumed source depth and trace of the nth striation in the simulation, respectively. First, a mesh is created between the frequency and D-DOA domain, and $(\theta ,f)$ represents the center point of one mesh. When the experimental striation trace passes through the mesh point of $T_{\text{model}}^n$, we denote $(\theta ,f)\in T_{\text{model}}(z_s^a)$, and the corresponding position $P$ has a value of 1. Otherwise, $P$ has a value of 0. Meanwhile, $\text{sign}(\cdot )$ is the sign function, $\mathit{N}\mathit{f}$ is the number of traces at each D-DOA, and $N_1$ and $N_{\theta }$ are the numbers of trace mesh points and D-DOA of the experimental data, respectively. The magnitude of each term must be balanced through normalization in which 1 represents a complete mismatch and 0 represents a perfect match. In this method, the maximum value of the cost function is $1/\sqrt{2}$, and the ideal minimum is 0. By minimizing the cost function, the optimal value of $z_s^a$ equals the real source depth when no environment mismatch is assumed in the simulation.

An interpolation scheme is selected to prevent the estimation error caused by the overall shift and missing of the striations. The first term is disabled if the position of striations has an overall shift. In this way, the second term allows the number of striations to contribute to the cost function. By contrast, the second term is ineffective if striations are missing because of tracker error. Subsequently, the first term enables the position of striations to contribute to the cost function. The interpolation between these two terms is defined so that one of them can take precedence. To verify our depth estimation method, Fig. 1(f) shows the curve of the cost function in Eq. (4). The estimated result of simulated source with the depth of 10 m is the true value.

3. Experimental verification and discussion

The experimental data obtained from the ship is processed using uniform 3 s snapshots. The sample rate of the hydrophones is 10 kHz. First, the broadband CBF is applied to compute the bearing time recordings (BTRs), as shown in Fig. 2(a). The upward and downward propagations are noticeable and can be separated by DOA. The D-DOA of the downward propagation with time (the dotted line) is then obtained. The symmetry in the D-DOA at about 2400 s is an indication that the source reached its closest point of approach (CPA) at that time. Fourier transform is used for data processing, and the beamformer is reapplied at each frequency with the corresponding D-DOA at each time ping. The beam intensity output is recorded in the time-frequency plane in Fig. 2(b). The time sequence is denoted in terms of the snapshots, which are dependent on the D-DOA.

Fig. 2.

(Color online) (a) Wideband BTR data, (b) spectrogram produced by the experimental data of Fourier decomposition, which is the CBF intensity output with the frequency from 50 to 1000 Hz and the corresponding D-DOA at each time ping, (c) comparison between the striation structures of the estimated (2.6 m) and experimental source depth, and (d) values of the cost function.

Figure 2(c) compares the striation structures of the estimated (2.6 m) with those of the experimental source depth. The dotted lines are extracted by the EKF from the striation structure when the time sequence is converted to the D-DOA in Fig. 2(b), and the background is the striation structure in simulation. Figure 2(d) shows that the cost function is sensitive to source depth, thereby resulting in high resolution. The result of this method is in accordance with the depth range of the ship propeller.

A flow chart of the source depth estimation scheme is displayed in Fig. 3. The method includes two branches: the simulation and experimental data processing. The steps of the simulation are as follows. First, the striation structure simulation for different source depths with the experimental environment is carried out. Second, the striations are tracked by EKF, and the relationship between frequency and D-DOA is recorded. The process of dealing with experimental data comprises three steps. First, the D-DOA estimation of the experimental data is performed by the broadband CBF. Second, the intensity of array output at the D-DOA and each frequency is obtained by Fourier transform, and the plot of intensities versus D-DOA and frequency shows the striation structure of the experiment. Finally, the EKF is used for striation tracking. With the striation traces of simulation and experiment, the source depth is estimated by minimizing the cost function, which is related to the position and number of striations between the simulation and experimental data.

Fig. 3.

Flow chart of source depth estimation based on striation structure.

4. Conclusions

This paper has described a method of source depth estimation based on a broad band interference structure and presented the results of its application to experimental data. The method is straightforward and easily implemented. The DOA obtained by the wideband CBF is recorded for the striation structure so that the information such as the range and time at CPA and source speed is not required in this method. The striation structure with a frequency band has an advantage in the application of experimental data. An interpolation scheme of the cost function is used for robustness. This scheme consists of matching measured and simulated striation structures with the experimental environment to obtain the source depth. In practical applications, the broadband over which the striation structure is performed may be chosen to improve SNR.