The received sound intensity of bottom-mounted line array varies as the submerged sound source moves in the direct arrival region, which resulting from interference between the direct and surface-reflected propagation paths, modulates with the target depth. In this work, the Fourier integral method from McCargar and Zurk [J. Acoust. Soc. Am. 133, EL320–EL325 (2013)] has been improved for depth estimation with a horizontal line array, and the matched sound intensity structure method from Zheng, Yang, Ma, and Du [J. Acoust. Soc. Am. 148, 347–358 (2020)] is introduced as a comparison. The two methods are verified in a deep ocean experiment.

In the deep ocean, a bottom mounted array forms a direct arrival region (DAZ), in which sound signals arrive at the hydrophone mainly through the direct and surface-reflected ray paths.1 The propagation properties of the DAZ were presented in Refs. 1 and 2 but have not been applied to correlation or gain analysis. In the DAZ, the acoustic properties were researched and applied for source localization,2–4 while the strong coupling of sound fields between range and depth makes it becomes a challenge for depth estimation.

In general, three kinds of methods are often used for target depth estimation in deep ocean. The first one is matched field processing,5 which obtains the range-depth ambiguity surface by matching the measured field with a replica field. The depth corresponding to the surface peak is the optimal estimation result. However, the strategy of matching the sound pressure amplitude and phase simultaneously makes the matched field processing method very sensitive to model mismatch.

Multipath time delay difference is another method used to estimate target depth, which has been widely studied and applied in the passive monitoring of marine mammals.6,7 The calls of marine cetaceans are a natural strong pulse sound source. Nosal et al.8 achieved the three-dimensional positioning of a sperm whale with at least four widely spaced, bottom mounted hydrophones, by using the delay difference of direct and surface-reflected acoustic arrivals. The time delay difference between direct and surface-reflected acoustic arrivals is the key to depth estimation. However, it is difficult to extract the multipath delay difference directly for continuous weak signals due to the low signal-to-noise ratio and the large array aperture requirement in passive detection.

The third type of method utilizes the interference structure of the sound field to estimate the target depth. This method is based on the physical characteristics of the interference structure resulting from the interference between direct and surface-reflected acoustic arrivals in receive range, depth, and frequency domains, which modulate with the source depth. McCargar2 and Zurk et al.3 constructed a modulation relationship between the sound field interference structure and source depth based on the image source theory and achieved depth separation of surface and submerged targets in simulation with a vertical line array. Zheng et al.4 achieved a better depth resolution and better ability to estimate the depth of a very shallow source with the matched beam-intensity method in simulation. This type of method is suitable for moving sound sources with constant depth and shows the advantage of narrowband frequency requirement.

Many of the methods mentioned previously are based on the vertical receive line array. Two depth estimation methods with a bottom mounted horizontal line array (HLA) will be introduced in this paper, and their effectiveness will be tested in a deep ocean experiment.

Figure 1 illustrates the image source theory geometry. A submerged point source moves with a constant depth z s and radial velocity v s at the endfire direction of a bottom mounted HLA, emitting a tone with spectral amplitude, S ( ω ). From image source theory, the received pressure on the HLA at mth snapshot is approximately given by
p ( r m , ω ) S ( ω ) ( e i k R m R m e i k R m + R m + ) ,
(1)
where m = 1 , 2 , 3 , , M, R m and R m + are the slant ranges of direct and surface-reflected paths, respectively, r m is the horizontal range from the source to the HLA midpoint at mth snapshot, and wavenumber k = ω / c. In the deep ocean, when the source depth satisfies z s R m, the expression of Eq. (1) can be further approximated as
P ( r m , ω ) 2 i S ( ω ) e i k R m R m sin ( k z s sin θ m ) ,
(2)
where R m = H 2 + r m 2, H is the water depth, and the arrival angle θ m satisfy tan θ m = ( H / r m ). The received sound intensity can be expressed as
I ( r m , ω ) = | P ( r m , ω ) | 2 ρ c = 2 | S ( ω ) | 2 ρ c R m 2 ( 1 cos ( 2 k z s sin θ m ) ) .
(3)
Obviously, the factor cos ( 2 k z s sin θ m ) in Eq. (3) shows the periodic oscillation of sound intensity, and the oscillation frequency is proportional to the source depth.

According to the reciprocity theorem, the transmission loss (TL) as a function of range and depth at 100 Hz for a point source put at the sea bottom generated by KRAKEN9 is shown in Fig. 2(b), where the water depth is 900 m and a measured sound speed profile (SSP) shown in Fig. 2(a) has been used in the simulation. The two black lines in Fig. 2(b) correspond to the acoustic paths of emergent angles 25° and 60° from the source, respectively. The evenly resampled TL in the sine of the emergent angle is shown in Fig. 2(c), which shows that the interference structure varies with the receive depth. Thus, the source depth can be estimated based on the modulation relationship between the received sound intensity structure and source depth, or by matching the received sound intensity structure with the replica intensity structure directly. The two methods will be introduced in the following section.

Multiplying R m 2 at each side of Eq. (3), the range compensated sound intensity I ( r m , ω ) becomes
I ( r m , ω ) = I ( r m , ω ) R m 2 = 2 | S ( ω ) | 2 ρ c ( 1 cos ( 2 k z s sin θ m ) ) ,
(4)
where I ( r m , ω ) oscillates periodically with a frequency of k z s / π as sin θ m changes. Supposing τ m = sin θ m, multiplying the depth modulation factor e i 2 k z τ m and integrating along the source arrival angle path, Eq. (4) becomes
τ 1 τ M I ( r m , ω ) e i 2 k z τ m d τ m = 2 | S ( ω ) | 2 ρ c τ 1 τ M ( 1 cos ( 2 k z s τ m ) ) e i 2 k z τ m d τ m ,
(5)
where z is the unknown depth. Here, τ 1 = sin θ 1, τ M = sin θ M, it is assumed that the horizontal range between the target and receiver array varies monotonically as the target moves, and τ m = sin θ m varies monotonically correspondingly. Defining the depth weighting function,
Q F ( z , ω ) = | τ 1 τ M I ( r m , ω ) e i 2 k z τ m d τ m | = 2 | S ( ω ) | 2 ρ c | τ 1 τ M ( 1 cos ( 2 k z s τ m ) ) e i 2 k z τ m d τ m | .
(6)
Expanding the factor e i 2 k z τ m in Eq. (6) yields
Q F ( z , ω ) = 2 | S ( ω ) | 2 ρ c | τ 1 τ M e i 2 k z τ m d τ m τ 1 τ M ( ( cos ( 2 k z s τ m ) cos ( 2 k z τ m ) i cos ( 2 k z s τ m ) sin ( 2 k z τ m ) ) ) d τ m | .
(7)
Applying the transformation law of trigonometric functions, Eq. (7) can be further expressed as
Q F ( z , ω ) = 2 | S ( ω ) | 2 ρ c | τ 1 τ M e i 2 k z τ m d τ m 1 2 τ 1 τ M e i 2 k ( z z s ) τ m d τ m 1 2 τ 1 τ M e i 2 k ( z + z s ) τ m d τ m | .
(8)
It has to be noticed that the factors e i 2 k z τ m and e i 2 k ( z + z s ) τ m in Eq. (8) periodically oscillate as τ m changes, thus
τ 1 τ M e i 2 k z τ m d τ m 0 ,
(9)
when the sine of the arrival angle extent is large enough for 2 k z | τ 1 τ M | 2 π, and
τ 1 τ M e i 2 k ( z + z s ) τ m d τ m 0 ,
(10)
when the source depth is deep enough (under the condition of z s R m) for 2 k ( z + z s ) | τ 1 τ M | 2 π. So, the depth weighting function can be expressed approximately as
Q F ( z , ω ) | S ( ω ) | 2 ρ c | τ 1 τ M e i 2 k ( z z s ) τ m d τ m | .
(11)

According to the stationary phase method, Q F ( z , ω ) obtains the maximum value only when z = z s. It should be emphasized that the range compensation in Eq. (4) is necessary when the changes of range along the integration path are comparable to the ocean depth, and the de-meaned of I ( r m , ω ) is necessary to make sure Q F ( z , ω ) = 0 when z = 0.

As shown in Fig. 2(b), the TL interference structure at a fixed emergent angle extent varies with the receive depth. Matching the beamformer output along the target trajectory with the replica sound intensity structure, the depth weighting function can be expressed as
Q M ( z , ω ) = m = 1 M I t r ( sin θ m ) I r e ( z , sin θ m ) m = 1 M I t r 2 ( sin θ m ) m = 1 M I r e 2 ( z , sin θ m ) ,
(12)
where I t r ( sin θ m ) is the beamformer output along the source trace, and I r e ( z , sin θ m ) is the replica sound intensity at search depth z and arrival angle θ m. The measured water depth and SSP are used for the generation of the replica field, which will be helpful in improving the depth estimation resolution.

A deep ocean experiment was conducted at the north of the South China Sea, where the water depth is about 900 m, and an HLA was put at the sea bottom. During the experiment, a towed sound source moved at the endfire direction of the HLA, emitting a 100 Hz cw signal continuously at a depth of about 55 m. A commercial ship with Automatic Identification System (AIS) passed away at the endfire direction of the HLA, which will be viewed as a surface target.

The HLA beamformed output power as a function of time and of the sine of arrival angles is shown for a commercial ship in Fig. 3(a) and for a towed submerged sound source in Fig. 3(b) at a frequency of 100 Hz. When the sine of the arrival angle varies from 0.35 to 0.75, the horizontal range varies from 2.4 to 0.8 km, respectively, which shows three oscillation cycles as shown for the submerged sound source in Fig. 3(b), while the periodic oscillation phenomena is not obvious for the commercial ship, as shown in Fig. 3(a).

The range compensated power output along the sound source trace was de-meaned before being put into Eq. (6). The results of the Fourier integral method are shown in Fig. 4(a). A peak at depth 18 m is seen in the depth weighting output for the commercial ship as shown in Fig. 4(a), while the commercial ship nominal draft is 7 m, and the estimated depth is about 11 m deeper than the nominal draft. For the submerged sound source, the peak of depth weighting output at a depth of 62 m, about 7 m deeper than that of the measured depth.

The requirement of 2 k z | τ 1 τ M | 2 π and 2 k ( z + z s ) | τ 1 τ M | 2 π in Eqs. (9) and (10) will not be satisfied when the sound source is shallow, which will deviate the estimated result from the true depth. The constant SSP approximation will bring some errors for the Fourier integral method as well. Although the estimated depths show some errors, the Fourier integral method still seems reasonable for distinguishing the submerged sound source from the surface source with a depth discrimination boundary of 20 m, for example.

Substituting the de-meaned beam output along the target in Fig. 3 into Eq. (12), the results of the matched sound intensity structure method are shown in Fig. 4(b). The peak of the depth weighting output for the commercial ship at a depth of 8 m with a weighting of 0.87, and at 54 m for the submerged sound source with a weighting of 0.88, agrees quite well with the true depth.

This paper has introduced two methods for depth estimation in a deep ocean direct arrival zone with an HLA; the improved Fourier integral method based on the depth modulation arises from the interference between direct and surface-reflected arrivals, which is robust to environmental mismatch. This method shows some system errors when the sound source is shallow, and shows errors for constant sound speed approximation. The matched sound intensity structure method matched the beamformer output intensity structure with the replica intensity structure directly and shows a more accurate depth estimation for both the surface and submerged sound source when the model mismatch is small.

The arrival angle extent and the HLA aperture may have a significant effect on the depth estimation performance, which needs further research in the future.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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