1. Introduction
The direction of arrival (DOA) of a signal propagated over an ocean waveguide is a primary factor in various applications and procedures, such as estimating of the location of the source and restoring the transmitted signal in a passive environment [
1,
2,
3]. DOA estimation has recently been applied in Green’s function estimation of unknown sources (e.g., ships) [
4,
5,
6,
7]. The frequency–wavenumber (
f–
k) analysis, as well as reliable and robust delay-and-sum (DAS) beamforming, can estimate the DOA of a uniform linear array with
d-m spacing [
8,
9,
10]. In the
f–
k analysis, the DOA is calculated by using the ratio of the wavenumber, determined by the frequency of the signal, with the wavenumber derived by spatial sampling. If the frequency of the signal exceeds the design frequency (=
, where
c is the nominal sound speed in water of 1500 m/s), the angle would be inaccurately estimated due to spatial aliasing. A statistical approach can be used to rectify the DOA of a broadband signal in a single-path environment [
9]. However, angle correction is difficult when multipaths exist, and the frequency band of the signal is significantly higher than the design frequency (i.e., sparse).
For a sparse vertical array configuration, DOA estimation by using DAS beamforming is highly likely to fail, and recent efforts have been made to overcome this problem. Abadi et al. proposed a beamforming approach based on the concept of frequency difference, known as frequency-difference (FD) beamforming [
11]. FD beamforming is a method of beamforming a difference-frequency component that is equal to the difference between two frequencies extracted by the product of two relatively high-frequency components [
11,
12,
13]. Xenaki et al. applied the concept of compression sensing (CS) to beamforming in 2014 [
14,
15,
16,
17,
18]. CS-based beamforming is a method used for estimating the DOA by using a convex optimization solution under sparse vertical array configurations.
In this paper, we propose an extended f–k analysis for a sparse vertical array configuration by utilizing the FD concept of and explaining the relationship with the existing FD beamforming. Hereafter, the algorithm proposed in this paper is referred to as the frequency difference–wavenumber (–k) analysis.
The remainder of this paper is organized as follows: in
Section 2, we review the frequency–wavenumber analysis and present the mathematical formulation of the
–
k analysis proposed herein. In
Section 3, we describe the shallow-water acoustic variability experiment (SAVEX15) conducted in the northeastern East China Sea (ECS).
Section 4 compares the
f–
k analysis,
–
k analysis and FD beamforming via simulation for the SAVEX15 environment, demonstrating the feasibility of DOA estimation by using the
–
k analysis. In
Section 5, we describe the estimation of the DOAs of cracking sounds produced by snapping shrimps, which were recorded by using a sparse vertical array during the SAVEX15 experiment, using the
–
k analysis and FD beamforming. Finally, the concluding remarks are summarized in
Section 6.
4. Numerical Simulation
The ray-tracing code BELLHOP was used to generate the received signals for the simulation [
26,
27]. The Green function between the source and receiver can be calculated by using the following equation:
where
is the arrival amplitude, including any phase shift from boundary bounces, and
is the arrival time. These two variables are the outputs of BELLHOP. The received signals were generated by multiplying the Green function with the frequency-domain transmitted signal and performing an inverse Fourier transform.
As previously mentioned, we used two ray-path arrivals (i.e.,
) to mimic the scenario of snapping shrimp, discussed in
Section 5. The source depth and the range between the source and the VLA were set to 100 m and 210 m, respectively [
24].
Figure 2a shows the
f–
k analysis of the simulation data. The slopes appear to be visible; however, overall, it is featureless to the extent that the angle cannot be estimated, although there were only two ray paths. This is because of the near field of the array as well as striation interference. Considering that the far field of the array is reached when
is less than unity [
28], the source will be in the near field because this parameter at 11 kHz (the lower frequency bound) is greater than 27 (i.e.,
). If the source is in the near field, the slope related to the DOA in the
f–
k analysis inevitably spreads, and the influence of the spread is increased in the case of a sparse vertical array configuration. With
, the x-axis was converted from a wavenumber to a physically detectable angle, as shown in
Figure 2b. Recalling that the angle range that the array can detect by using Equation (
8) decreases when the frequency of the signal is higher than the design frequency, the angle range becomes extremely narrow for an extremely sparse vertical array configuration, as discussed in
Section 2. The gray-shaded region represents the regions that are not physically detectable, and, when the design frequency is 200 Hz, the detectable angle range of the array at 24 kHz is within
.
Figure 2b, similar to
Figure 2a, is featureless, as if the pattern of striations is random. Hence, the DOA cannot be corrected by using periodicity.
First, FD beamforming was applied to the same simulation (see
Figure 3). To minimize the number of cross terms generated due to the multipath, the output of FD beamforming was incoherently averaged over 11 kHz
22.6 kHz with 10-Hz intervals. The y-axis in
Figure 3 indicates the difference frequency. To confirm the trend of an increasing difference frequency, the difference frequencies, which are user-chosen parameters, were set from 0 Hz to 1400 Hz with 1-Hz intervals. This is comparable to simulating a signal with a frequency band of 0–1400 Hz. The arrows in
Figure 3 represent the DOAs calculated by using the image method based on the center of the VLA as the reference angles. The red and blue arrows correspond to the direct (12.6
) and surface-reflected (−36.1
) paths, respectively.
Two vertical lines (mainlobes) are observed in the FD beamforming output, and the angles corresponding to two vertical lines are in good agreement with the DOAs. Several curves in
Figure 3 are grating lobes caused by spatial aliasing and are a mix of grating lobes produced by each path. In
Figure 3, the white dotted line represents the maximum limit frequency (i.e., 400 Hz) that satisfies the far field of the array for the geometry considered here. A frequency higher than 400 Hz (lower part of the white dotted line in
Figure 3) satisfies the near field of the array, resulting in sidelobes emerging around the main or grating lobes due to the angle spread.
The results of the proposed algorithm for simulation data are displayed in
Figure 4a. The result of
–
k analysis is incoherently averaged over 11 kHz
22.6 kHz with 10-Hz intervals, similar to that of FD beamforming. The difference between
Figure 2a and
Figure 4a is remarkable. In contrast to the featureless
f–
k analysis (see
Figure 2a), the output of the
–
k analysis shows that the two main gradients are clearly separated because multipath interference is mitigated.
Although spatial aliasing exists in the difference-frequency band of 0–1400 Hz, clear separation allows for the DOA estimation through periodicity. However, beyond the maximum limit frequency (white dotted line), other minor slopes are formed in addition to the major slopes, which shows the angle spread due to the near-field effect explained in FD beamforming. This angle spread can be relaxed if the range is increased, whereas other conditions remain fixed.
Figure 4b shows the result of wavenumber-to-angle conversion, using
, as shown in
Figure 2. The gray-shaded region represents the angle range that the array cannot detect physically. When the difference frequency is lower than the design frequency, an output at all angles was achieved, similar to that in FD beamforming. However, as the difference frequency is increased, the angle range decreases. Nevertheless, the
–
k analysis can detect a wider angle compared with the
f–
k analysis.
Except for the gray-shaded region,
Figure 3 and
Figure 4b are identical. This relationship is consistent with that described in Sec.
Section 2 between the
f–
k analysis and DAS beamforming and is corroborated by
Figure 5. Additional averaging (i.e., double averaging) in the range of difference frequencies can improve the robustness of DOA estimation [
11,
12,
19,
20].
Figure 5 shows a comparison between DAS beamforming, the results of double averaging from the
–
k analysis and FD beamforming. Averaging for
was performed with 1-Hz intervals, excluding the gray-shaded regions. The green dashed line represents DAS beamforming, whereas the black solid line and the red dotted line represent the double averaging outputs of the
–
k analysis and FD beamforming, respectively. All the results are normalized to the peak value. First,
Figure 5a shows the results of double averaging within the difference frequency band lower than the design frequency as well as the result of DAS beamforming.. Two large pulses are detected in DAS beamforming. However, the exact angle could not be calculated because of oscillations within each pulse due to the sparseness of the array. By contrast, double averaging of the
–
k analysis and FD beamforming provided consistent results at all angles, with two distinct angles (direct: 13.8
, surface-reflected: −36
), demonstrating consistency between
Figure 3 and
Figure 4b. Compared with the reference angles, the maximum error of the angle estimated in
Figure 5a is approximately 1
, indicating that the angle is successfully estimated.
Figure 5b shows the results of double averaging over all difference frequencies (i.e., 0–1400 Hz) used in this paper. The result of DAS beamforming is identical to that shown in
Figure 5a. The two mainlobes of double averaging remained similar and distinguishable. For simulation data, the angles estimated from the
–
k analysis and FD beamforming are as follows: (1)
–
k analysis, 16.4
(direct) and −36.5
(surface-reflected); and (2) FD beamforming, 16.4
(direct) and −35.0
(surface-reflected). Compared with the reference angles, the maximum error in the estimation of all angles using the two approaches is 4
(i.e., the surface-reflected path in FD beamforming). This error is believed to be caused by the angle spread due to the near-field effect on difference frequencies above 400 Hz, and the error decreased as the range increased. In addition, there is a noticeable difference in the sidelobes between
Figure 5a,b. The difference between the
–
k analysis and FD beamforming is based on whether the gray-shaded regions are included when double-averaged over all difference frequencies. This is the region where the grating lobes of FD beamforming are formed, and all grating lobes are included in the case of double averaging in FD beamforming. This yields the same effect as adding background noise. As a result, the sidelobe of FD beamforming is larger than that of the
–
k analysis. The simulation confirmed that, although both approaches can estimate DOAs, there may be a difference in the sidelobe depending on the difference-frequency band for double averaging.
5. Experimental Results
To verify the proposed algorithm by using experimental data, we analyzed a set of direct and surface-reflected noises collected on JD 141 (JD141 06:55:30). Cracking sounds along the VLA are shown in
Figure 6a. The direct and surface-reflected paths, which were separated at around 30.7 s, are denoted by D and S, respectively. The snapping shrimp was in the near field of the array, as evidenced by the direct path with the shape of a spherical wavefront. The surface-reflected noise was dispersive due to the rough sea surface. As an example of a cracking sound, which has a higher spectral energy at frequencies above 10 kHz [
24], the spectrogram of the received signal at the middle hydrophone (i.e., 51.25 m; eighth channel) are displayed in
Figure 6b. The dominant frequency band of cracking sounds was found to be 11–24 kHz (frequency band between the white dotted lines in
Figure 6b) and was used to estimate the DOAs by using FD beamforming and the proposed algorithm.
Through simulations, we found that the
f–
k analysis for the sparse vertical array configuration is featureless. As this featurelessness appears similarly in the data of FD beamforming, the
f–
k analysis for the snaps measured from the experiment is not displayed.
Figure 7,
Figure 8 and
Figure 9 illustrate the results along with the experimental data in the same manner as the simulation results (see
Figure 3,
Figure 4 and
Figure 5). Recall that averaging for
was performed with 10-Hz intervals between 11 kHz and 22.6 kHz.
Figure 7 shows the FD beamforming output as the experimental result counterpart of
Figure 3. Two vertical lines appear between −60
and 30
in the frequency band below 400 Hz, as in the simulation; however, afterward, the two lines spread and disappeared in the frequency band. This phenomenon is believed to be caused by the array shape. In contrast to the simulation, which assumes that the array is a straight line, the array during the experiment was not straight and might have had a curvature because of various factors such as current. This curvature can potentially cause an angle spread similar to that caused due to the near-field effect, and a direct arrival through a relatively closer path will be more sensitive. Nevertheless, except for the difference in the background noise, the patterns of the main and grating lobes in
Figure 3 and
Figure 7 are similar, suggesting that the
–
k analysis of the experimental data is similar to that of the simulation data.
Figure 8 shows the output of the
–
k analysis as the counterpart of the experimental results in
Figure 4. The experimental data were comparable to the simulation results, as expected from the FD beamforming output. Furthermore, the angle spread and decay caused by the abovementioned array shape were more clearly highlighted by the angle-related slope in the
–
k domain (see
Figure 8a). Despite the addition of background noise and the influence of the array shape compared with the simulation, the two main lines can still be clearly identified in
Figure 8.
Figure 9 depicts the results of the experimental data obtained by employing double averaging, which can improve the robustness and the output of DAS beamforming. Double averaging is performed with an interval of 1 Hz starting at 0 Hz, and the upper bounds of double averaging in
Figure 9a,b are 200 Hz (design frequency) and 1400 Hz, respectively. As all angles can be detected by using the
–
k analysis in the difference-frequency band within the design frequency, the results of FD beamforming and
–
k analysis double-averaged from 0–200 Hz coincide, as illustrated in
Figure 9a. This is the expected result, which is the same as the simulation result. By contrast, data in
Figure 9b, in which the upper bound of double averaging is 1400 Hz, differ from the simulation data (see
Figure 5b). The intensity of the direct path is greater than that of the surface-reflected path in the simulation, as shown in
Figure 5b, whereas the intensity of the surface-reflected path is greater in the
–
k analysis in
Figure 9b. Although not shown here, the intensity of the direct path steadily decreased when the upper bound of double averaging is gradually increased from 200–1400 Hz. This drop in the direct path intensity is expected owing to the angle spread. In contrast to the surface-reflected path, wherein the angle spread is not apparent due to detectable angle restriction, the intensity of the direct path decreased because of destructive interference caused by the angle spread. Nonetheless, we confirmed that the two mainlobes can be clearly identified and that the sidelobes, after double averaging the
f–
k analysis, are lower than those after FD beamforming, as in the simulation. The two peaks in
Figure 9a, which are not affected by angle spread, are 10.5
(direct) and −38
(surface-reflected) shifted by −2
from the angles (see
Figure 5a) estimated in the simulation. Because there was an array tilt during SAVEX15 [
7,
24], which caused a shift in the angular axis, the angle shift between the experimental and simulation data is reasonable.