Source Range Estimation Using Linear Frequency-Difference Matched Field Processing in a Shallow Water Waveguide

Abstract

Matched field processing (MFP) is an established technique for source localization in known multipath acoustic environments. Unfortunately, in many situations, imperfect knowledge of the actual propagation environment and sidelobes due to modal interference prevent accurate propagation modeling and source localization via MFP. To suppress the sidelobes and improve the method’s robustness, a linear frequency-difference matched field processing (LFDMFP) method for estimating the source range is proposed. A two-neighbor-frequency high-order cross-spectrum between the measurement and the replica of each hydrophone of the vertical line array is first computed. The cost function can then be derived from the dual summation or double integral of the high-order cross-spectrum with respect to the depth of the hydrophones and the candidate sources of the replicas, where the range that corresponds to the minimum is the optimal estimation. Because of the larger modal interference distances, LFDMFP can efficiently provide only one optimal range within the same range search interval rather than some conventional matched field processing. The efficiency of the presented method was verified using simulations and experiments. The LFDMFP unambiguously estimated the source range in two experimental datasets with average relative errors of 2.2 and 1.9%.

1. Introduction

Matched field processing (MFP) was developed in the mid-1970s [1] and remained a popular source localization technique through to the 1990s [2]. MFP can be successful under low signal-to-noise ratios (SNRs) when the actual propagation environment is well known. Conventional matched field processing (CMFP) is the simplest scheme. It is a cross-correlation technique developed for matching the values of the replica field computed with the measured values at the output of a vertical linear array. Different from beamforming technology, MFP incorporates the physical multipath effects through the simulated replica field to localize the source.

However, the primary limitation of MFP arises from two aspects: the actual-to-model mismatch and sidelobes due to modal interference. The former usually leads to a lack of robustness in the MFP source localization estimation and increases in severity with an increasing frequency and array-to-source range. Even for a low frequency, the environmental mismatch may induce phase fluctuations and limit the utility of MFP. Prior research [3] indicated that 1 kHz is an approximate nominal upper frequency limit for the successful application of CMFP in shallow water.

Adaptive techniques can provide a higher resolution, such as the minimum variance distortionless response (MVDR) [4,5], multiple constraint method (MCM) [6], and multiple signal classification (MUSIC) [7], which all offer higher source localization resolution than CMFP; however, these techniques based on the direct use of the complex field are sensitive to environmental mismatch, especially for a high frequency, where incomplete knowledge of the environment can lead to a serious decline in localization performance.

To improve the tolerance of MFP for environmental mismatch, some robust techniques or algorithms based on CMFP were proposed. These include broadband matched field processing [8], matched mode processing (MMP) [9,10], matched beam processing (MBP) [11,12], compressed matched field processing [13,14,15], frequency-difference matched field processing (FDMFP) [16,17,18,19], machine learning techniques [20], multiple constraints methods [21], and sector-focused stability [22]. MMP and MBP specifically transform measurement data from the element domain to the modal and beam domains, respectively, before conducting matching procedures. Compared with CMFP, MMP exhibits reduced susceptibility to environmental mismatch, while MBP mitigates the seabed’s impact. FDMFP, which was proposed by Worhtmann in 2015, expanded upon frequency-difference beamforming techniques [23] to adapt MFP for high-frequency source localization in shallow water. The FDMFP technique is realized by matching the autoproduct, which is a quadratic product of the measured field at two nearby frequencies (interval is $\Delta \omega$), with the frequency-difference replicas based on the low-frequency propagation model. It was shown by both simulation and experimental results that FDMFP can be more robust against environmental mismatch than CMFP.

The abovementioned improves the robustness, but to some extent, the resolution of the results is sacrificed. More importantly, the problem of periodic sidelobes caused by modal interference still exists. To suppress the periodic sidelobes and improve the robustness, a linear formulation of FDMFP is proposed in this paper, which differs from the physical mechanism of FDMFP proposed by Worthmann [16]. The formulation is based on normal mode theory, in which the replicas of two nearby frequencies are calculated in the same way as CMFP without recourse to the frequency difference replicas. The distinct advantage of LFDMFP is that we can provide a clear explanation that the mode pair interference distances of LFDMFP are far larger than those of CMFP (the ratio of the modal interference distances of LFDMFP to those of CMFP is approximately $\beta \omega /\Delta \omega$). Therefore, LFDMFP can efficiently provide an optimal range estimation within the distance search interval rather than some optimal ranges of the CMFP. Although the range resolution is degraded, the robustness of LFDMFP to the environmental fluctuation is enhanced. Furthermore, the depth estimation of the source could be evaluated by utilizing the completeness of the normal modes at the optimal range. The formulation was confirmed by numerical simulations and experimental results.

The main contributions of this study are summarized as follows:

(1) Linear frequency-difference matched field processing for estimating the source range is proposed. The LFDMFP utilizes larger modal interference distances to effectively solve the periodic sidelobes, thus greatly reducing the occurrence of “false targets”.

(2) The difference processing enhances the tolerance of LFDMFP to environmental mismatches of fluctuations and improves the robustness.

(3) The performance of the LFDMFP was analyzed using simulations, and the effectiveness of the LFDMFP was verified using data from two offshore experiments.

The remainder of this article is organized into four sections. The mathematical derivation and theoretical underpinnings of the proposed method are presented in Section 2, which also includes the specifics for LFDMFP. Section 3 presents the method performance analysis. Section 4 documents the results of this method based on experimental data recorded during the Sound Propagation Experiment in Laoshan Bay in 2005 and the Sound Propagation Experiment in the South China Sea in 2019. Section 5 provides the conclusions of this study.

2. Theory and Algorithms

2.1. The Conventional MFP Method

For simplicity, a range-independent stratified ocean environment was considered in this study. An N-element vertical line array (VLA) of hydrophones is disposed at a distance of $r_0$ from a source at the depth of $z_0$. The complex sound pressure of the frequency $\omega$ received by the ith hydrophone is given by

$$P(\omega ,z_i,r_0,z_0)=S\left(\omega \right)G(\omega ,z_i,r_0,z_0)+N(\omega ,z_i)$$

(1)

where $S\left(\omega \right)$ is the spectrum of the source, $N(\omega ,z_i)$ is the ambient noise considered as the zero-mean Gaussian noise, and $G(\omega ,z_i,r_0,z_0)$ denotes the Green’s function between the source and the ith hydrophone. The normal mode representation of the replicas (Green’s function) in the far field of the source in cylindrical coordinates can be expressed by

$$G(\omega ,z_i,r_s,z_s)={\displaystyle \frac{i\rho \left(z_s\right)}{\sqrt{8\pi r_s}}}{e}^{i{\displaystyle \frac{\pi }{4}}}\sum _n{\displaystyle \frac{{\Phi }_n\left(z_s\right){\Phi }_n\left(z_i\right)}{\sqrt{k_n\left(\omega \right)}}}{e}^{i[k_n\left(\omega \right)+i{\alpha }_n\left(\omega \right)]r_s}$$

(2)

where $k_n\left(\omega \right)+i{\alpha }_n\left(\omega \right)$ and ${\Phi }_n\left(z\right)$ are the complex eigenvalue and eigenfunction of the nth normal mode, respectively; $z_s$ is the depth of the candidate source and $r_s$ is the distance to the VLA.

CMFP for localizing a source is based on matching the acoustic pressure field measured at the array of sensors with the modeled replica fields $G(\omega ,z_i,r_s,z_s)$ computed for the acoustic waveguide via a numerical propagation model over a grid of possible source positions $(r_s,z_s)$, with the position estimate taken to be the grid point of the maximum match. The normalized ambiguity function of CMFP is defined as

$$E(r_s,z_s;\omega )={\displaystyle \frac{{\displaystyle \sum _{i=1}^N}{G}^{\ast }(\omega ,z_i,r_s{z}_s)P(\omega ,z_i,r_0,z_0)}{\sqrt{{\displaystyle \sum _{i=1}^N}|G(\omega ,z_i,r_s,z_s){|}^2{\displaystyle \sum _{i=1}^N}|P(\omega ,z_i,r_0,z_0){|}^2}}}$$

(3)

where the range and depth corresponding to the maximum of $E(r_s,z_s;\omega )$ are the optimal estimations of the source location.

In the far field of the point source, the low-frequency acoustic field comprises modal components that propagate long distances at shallow grazing angles and vertical components that rapidly decay because of the strong interaction with the bottom. It is recognized that the MFP is a generalized beamforming process, where there exists periodic ambiguity (sidelobes) due to modal interferences [24]:

$${\widehat{r}}_s=r_0-l{\lambda }_{mn}(l=0,\pm 1,\pm 2,\dots \dots )$$

(4)

where ${\lambda }_{mn}=2\pi /|k_n-k_m|$ is the modal interference distance between the mth and nth modes.

2.2. The LFDMFP Method

Broadband-matched field processing, while capable of partially suppressing sidelobes, requires substantial computational resources due to the need for multiple calculations of replica fields across the band. Instead of broadband-matched field processing, we reformulated LFDMFP to resolve the problem of sidelobes. Initially, the cross-spectrum of the ambiguity functions of two neighbor frequencies is computed for each candidate location of the source. The cross-spectrum process is called the high-order cross-spectrum. The ambiguity function $\Gamma (r_s;\omega ,\Delta \omega )$ of LFDMFP is defined by the integral along the candidate source depth. The process of integration is called dual summation or a double integral:

$$\begin{array}{cc}\hfill & \Gamma (r_s;\omega ,\Delta \omega )=\left|{\int }_0^{H}E(r_s,z_s;\omega +\Delta \omega )E^{\ast }(r_s,z_s;\omega )d{z}_s\right|\hfill \\ \hfill & \approx A^2\left|\sum _n{\displaystyle \frac{{\Phi }_n^2\left(z_0\right)e^{-[a_n(\omega +\Delta \omega )+a_n\left(\omega \right)](r_0+r_S)}}{k_n\left(\omega \right)k_n(\omega +\Delta \omega )}}{e}^{i[k_n(\omega +\Delta \omega )-k_n\left(\omega \right)](r_0-r_S)}\right|\hfill \end{array}$$

(5)

where $A=1/\sqrt{{\sum }_l{\varphi }_l^2\left(zs\right)/k_l{\sum }_m{\varphi }_m^2\left(zs\right)/k_m}$ is the normalization coefficient. In the derivation above, the orthogonality of the eigenfunctions of the normal modes is used.

$${\int }_0^H{\Phi }_n(\omega ,z_s){\Phi }_m(\omega +\Delta \omega ,z_s)d{z}_s\approx {\delta }_{nm}$$

(6)

Provided that the eigenfunctions are weakly dependent on the frequency or $\Delta \omega$ is small $(\omega /\Delta \omega \gg 1)$, Equation (6) is a good approximation. The eigenfunctions can then be approximated by neglecting the high-order terms of the Taylor expansion and represented as

$$k_n(\omega +\Delta \omega )\approx k_n\left(\omega \right)+{\displaystyle \frac{\Delta \omega }{v_{gn}\left(\omega \right)}}\cdots$$

(7)

where $v_{gn}=\partial \omega /\partial k_n$ denotes the group velocity of the nth mode. These properties are well satisfied if $\omega$ is significantly larger than the cutoff frequency of the nth mode. Substituting Equation (7) into Equation (5) gives the approximation result of the following ambiguity function of LFDMFP:

$$\Gamma (r_s;\omega ,\Delta \omega )\approx A^2\left|\sum _n{\displaystyle \frac{{\Phi }_n^2\left(z_0\right)e^{-[{\alpha }_n(\omega +\Delta \omega )+{\alpha }_n\left(\omega \right)](r_0+r_S)}}{k_n\left(\omega \right)k_n(\omega +\Delta \omega )}}{e}^{i{\displaystyle \frac{\Delta \omega (r_0-r_S)}{{\nu }_{gn}}}}\right|$$

(8)

The key difference between LDFMFP and CMFP lies in the encoding of the source range information within exponential terms. The source range information in LDFMFP is encoded in the exponential terms of $e^{i\Delta \omega (r_0-r_s)/v_{gn}}$ rather than $e^{i{k}_n(r_0-r_s)}$; then, the modal interference cycles are given by

$${\Lambda }_{mn}={\displaystyle \frac{2\pi }{\Delta \omega \left(\frac{1}{v_{gm}}-\frac{1}{v_{gn}}\right)}}$$

(9)

Furthermore, for a shallow water waveguide, there is the following approximate estimation:

$${\displaystyle \frac{{\Lambda }_{mn}}{{\lambda }_{mn}}}=\left|{\displaystyle \frac{k_m-k_n}{\Delta \omega \left(\frac{1}{v_{gm}}-\frac{1}{v_{gn}}\right)}}\right|=\left|{\displaystyle \frac{\omega \left(\frac{1}{v_{pm}}-\frac{1}{v_{pn}}\right)}{\Delta \omega \left(\frac{1}{v_{gm}}-\frac{1}{v_{gn}}\right)}}\right|\approx \beta {\displaystyle \frac{\omega }{\Delta \omega }}$$

(10)

The notion of the waveguide invariant is used in Equation (10). In a typical shallow water waveguide, $\beta \approx 1$. Provided that $\Delta \omega \ll \omega$, it is clear that the modal interference distances in the LFDMFP processor are much larger than those of the CMFP. This is the inherent physical principle that causes LFDMFP to be robust against environment fluctuation because ${\Lambda }_{mn}$ may be larger than the spatial correlation radius of environment fluctuations in comparison with ${\lambda }_{mn}$. As a penalty, the range resolution of LFDMFP is degraded due to ${\Lambda }_{mn}\gg {\lambda }_{mn}$. The influence of the frequency difference on the range estimation is analyzed in Section 3.

2.3. Cost Function

2.3.1. The Cost Function of the Range Estimation

The feasibility of the reformulated LFDMFP was first tested using a numerical simulation in the shallow water waveguide with a negative thermocline shown in Figure 1a. The depth of the water was 60 m, and the sound speed, density, and attenuation of the semi-infinite fluid bottom were set to 1800 m/s, 1.8 g/cm³, and 1 dB/$\lambda$, respectively. The source was arranged at a depth of 15 m and a 60-receiver 1 m interval VLA was deployed at 10 km away from the source. The sound fields at two frequencies of 300 and 310 Hz were computed using the software KRAKENC with the former eight normal modes. The computational grid of the replicas was set to 1 m in depth and 100 m in range.

In this numerical example, the modal interference distances of the first two normal modes were ${\lambda }_{12}\approx 0.75$ km and ${\Lambda }_{12}\approx 22.2$ km. Theoretically, the modal interference distance ratio of LFDMFP and CMFP was about 30. In the simulation environment, the modal interference distance ratio of the first two normal modes was about 29.6, which was consistent. Figure 1b shows that in $\Gamma (r_s;\omega ,\Delta \omega )$, there were some “spurs” because the orthogonality of normal modes is destroyed by only concerning the sound field in the water. To cancel the “spurs”, the cost function to estimate the source range was redefined as

$$Cost\left(r_s\right)=|2\ast \Gamma (r_s,\omega ,\Delta \omega )-{\Gamma }_{\omega }\left(r_s\right)-{\Gamma }_{\omega +\Delta \omega }\left(r_s\right)|$$

(11)

where ${\Gamma }_{\Omega }\left(r_s\right)=\left|{\int }_0^{H}E(r_s,z_s;\Omega )E^{\ast }(r_s,z_s;\Omega )d{z}_s\right|\dots (\Omega =\omega ,\omega +\Delta \omega )$ are called the auto-spectra of the frequencies of $\omega$ and $\omega +\Delta \omega$, respectively. $\Gamma (r_s;\omega ,\Delta \omega )$ is called the cross-spectrum between $\omega$ and $\omega +\Delta \omega$.

Figure 1c shows how the cost function in Equation (11) changed with the range. Because the “spurs” inherent in the auto-spectra ${\Gamma }_{\omega }\left(r_s\right)$ and ${\Gamma }_{\omega +\Delta \omega }\left(r_s\right)$ and the cross-spectrum $\Gamma (r_s;\omega ,\Delta \omega )$ similarly manifested themselves, $Cost\left(r_s\right)$ could cancel most of the “spurs” when $r_s$ was close to the real range of the source (Figure 1c). However, the main lobe of the cost function became so fat that the range resolution was degraded. Inspired by Figure 1b, when the estimated range was closer to the real range of the source, the coherent components of the auto-spectra and cross-spectrum decreased, and the difference between them became smaller. When $r_s$ equals $r_0$, $Cost\left(r_s\right)$ reaches the minimum and the estimation of ${\widehat{r}}_s$ is

$${\widehat{r}}_s=min\left\{Cost\left(r_s\right)\right\}$$

(12)

When the cost function of LFDMFP is similar to that of CMFP, the best source range estimation corresponds to the maximum value of the ambiguity function for observation. The cost function in Equation (12) is normalized by Equation (13), and the final cost function is obtained. To distinguish Equation (12) from Equation (14), the cost function only refers to Equation (14) in the subsequent simulation and experimental verification part:

$$cost\left(r_s\right)=min\left(Cost\left(r_s\right)\right)/Cost\left(r_s\right)$$

(13)

$${\widehat{r}}_s=\underset{r_s}{max}\{cost\left(r_s\right)\}$$

(14)

Figure 1d illustrates that the performance of the reformulated cost function in Equation (14) changed with the range, where the black line represents the LFDMFP and the red line represents the slice of ambiguity function at the true depth of the CMFP. The resolution of the CMFP was better than the LFDMFP. However, the strong sidelobes certainly disturbed the best range estimation, especially in the condition of low SNRs. Based on the simulation results depicted in Figure 1, the newly formulated LFDMFP effectively mitigated the issue of multiple solutions by ensuring that only one prominent peak was observed within the range search interval, which enhanced the accuracy of the range estimation.

2.3.2. The Cost Function of the Depth Estimation

It is worth mentioning that the depth of the source can be evaluated using the completeness of the normal modes ${\displaystyle \sum _n}{\Phi }_n\left(z_s\right){\Phi }_n\left(z_0\right)\approx \delta (z_s-z_0)$ at the optimal range in the case where the environment is known. The cost function of the depth estimation is defined as

$$\Xi (z_s;\omega )=\left|E({\widehat{r}}_s,z_s;\omega )\right|\propto \left|\sum _n{\displaystyle \frac{{\Phi }_n\left(z_s\right){\Phi }_n\left(z_0\right)}{k_n\left(\omega \right)}}\right|\propto {\displaystyle \frac{1}{k\left(\omega \right)}}\delta (z_s-z_0)$$

(15)

According to Equation (15), the depth estimation in the simulation environment of Figure 1a is shown in Figure 2. Although LFDMFP cannot directly obtain the source depth estimation, after obtaining the range estimation, information on the source depth estimation can be obtained using Equation (15). The focus of this study was to explore the advantages of LFDMFP in source range estimation. The depth estimation is just additional information derived from Equation (15).

2.4. Algorithm

The pseudocode implementation of the LFDMFP is presented in Algorithm 1. In the following sections, the results of the LFDMFP simulation performance analysis are given in Section 3 and subsequently validated with experimental data in Section 4.

Algorithm 1 LFDMFP

Input: Time domain pressure $p\left(t\right)$ recorded by VLA and sound speed profiles Output: Source range ${\widehat{r}}_s$ and source depth ${\widehat{z}}_s$  1: Compute the power spectrum using the time domain pressure recorded by the VLA and select the two adjacent frequencies ${\omega }_1$ and ${\omega }_2$ and their spectra $P(\omega ,z_i,r_0,z_0)\dots \omega ={\omega }_1,{\omega }_2$.  2: Divide the search grid and compute the replica field $G(\omega ,z_i,r_s,z_s)\dots \omega ={\omega }_1,{\omega }_2$ using the propagation model.  3: Compute the high-order cross-spectra using Equations (3) and (5):   $E(\omega ,z_i,r_s,z_s)=\int G\ast (\omega ,z_i,r_s,z_s)P(\omega ,z_i,r_0,z_0)d{z}_i\dots \omega ={\omega }_1,{\omega }_2$.   $\Gamma (r_s;\omega ,\Delta \omega )=\int E(\omega ,z_i,r_s,z_s)d{z}_s\dots \omega ={\omega }_1,{\omega }_2;\Delta \omega ={\omega }_2-{\omega }_1$.  4: Compute the cost function of the range estimation using Equations (11) and (13):   $Cost\left(r_s\right)=|2\ast \Gamma (r_s,\omega ,\Delta \omega )-{\Gamma }_{\omega }\left(r_s\right)-{\Gamma }_{\omega +\Delta \omega }\left(r_s\right)|$.   $cost\left(r_s\right)=min\left(Cost\left(r_s\right)\right)/Cost\left(r_s\right)$.  5: Compute the source range: ${\widehat{r}}_s=\underset{r_s}{max}\{cost\left(r_s\right)\}$.  6: Compute the source depth: ${\widehat{z}}_s=\left|E(\omega ,z_i,{\widehat{r}}_s,z_s)\right|$.

3. Simulation and Performance Analysis

LFDMFP is constrained by various factors, such as the signal-to-noise ratio (SNR), aperture of the linear array, and frequency difference ($\Delta \omega$). Among these factors, some exhibit similar performance to LFDMFP compared with CMFP, such as the SNR and aperture of the linear array, while the frequency difference is unique to LFDMFP. The previous study [25] analyzed the influence of the aperture of a VLA (including the array length and the number of elements) on CMFP. To adequately sample the pressure field, a fully spanning aperture is necessary. Furthermore, the aperture of the VLA affects the orthogonality of the normal mode shown in Equation (6). Once the array satisfies the minimum length requirement, the performance is largely determined by the number of array elements. To provide a unique representation of each mode, the number of elements should be at least equal to the number of modes. Based on this, a VLA with a fully spanning aperture and a number of elements that far exceeded the number of modes was set up in this simulation experiment. We discuss the influence of the SNR and frequency difference on the method proposed in this paper.

3.1. The Influence of the SNR

In this simulation, the ocean sound channel environment and geometry were consistent with the waveguide environment shown in Figure 1a. The source was arranged at a depth of 15 m and the 60-receiver 1 m interval VLA was deployed 10 km away from the source. The sound fields at two frequencies of 300 and 310 Hz were computed using the software KRAKENC with the former eight normal modes. Figure 3 illustrates the influence of the SNR on the performance of the method, with 1000 Monte Carlo experiments conducted for each SNR. The SNR was set to be uniformly distributed between −20 and 20 dB; the noise type was complex Gaussian white noise and the power signal-to-value ratio was applied, as represented by Equation (17). Figure 3 demonstrates the error of range estimation of the LFDMFP and CMFP under different SNR conditions, with the horizontal axis representing the SNR and the vertical axis representing the RMSE of the range estimation.

$$noise=\sigma A{e}^{2\pi iB}$$

(16)

where $\sigma$ is the standard deviation of the noise, A satisfies the standard normal distribution, and B is a uniformly distributed random number between 0 and 1. The SNR is defined as

$$SNR=10lg[{\displaystyle \frac{1}{N{\sigma }^2}}\sum |p{|}^2]$$

(17)

where N is the number of arrays, and p denotes the complex sound pressure, as calculated by Equation (2). The error was computed using Equation (18):

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{({\widehat{r}}_s-r_0)}^2}{n}}}$$

(18)

According to Figure 3, both the LFDMFP and CMFP produced unsatisfactory range estimations at low SNRs. This was because the MFP necessitated precise spectral matching, which demands that the signal’s distinct spectral features remain minimally influenced by noise. Consequently, at low SNRs, the introduction of random spectral variations can compromise the signal’s spectral integrity, thereby complicating the matching process. Under low-SNR conditions, the method proposed in this paper exhibited stronger robustness compared with the CMFP; furthermore, the performance was enhanced, which was primarily due to LFDMFP’s cross-spectral processing of adjacent frequencies’ ambiguity functions, which ensured that noise from different frequencies remained non-interfering, and thus, contributed to its superior robustness against the Gaussian white noise.

3.2. The Influence of the Frequency Difference

The incorporation of the frequency difference imposes constraints on the performance of LFDMFP. Both excessively large and small frequency differences can adversely affect the accuracy of the range estimation. Due to the coupling relationship between the frequency differences, frequency, and waveguide environment, it is difficult to quantitatively determine the applicable range of the frequency difference. Hence, this section discusses the qualitative analysis of the impact of the frequency difference selection on LFDMFP. The frequency difference alters the orthogonality of the normal modes and the phase difference. In this simulation, the SNR was 10 dB, while the sound source and VLA positions, as well as the ocean sound channel environment and geometry, were consistent with the waveguide environment shown in Figure 1a. The lower frequency of the two neighbor frequencies was set to 300 Hz.

Figure 3. Influence of SNR on the two methods.

Figure 3. Influence of SNR on the two methods.

The frequency difference influences the phase difference of the normal modes, which plays a dominant role in the localization performance of LFDMFP. The Taylor expansion in Equation (7) requires that the frequency difference should not be too large. As the frequency difference increases, the modal interference in LFDMFP decreases, which weakens its ability to suppress periodic ambiguities. Conversely, when the frequency difference is too small, the approximations in Equation (19) apply, which leads to Equation (8) transforming into Equation (20), which represents a power spectral function at frequency $\omega$. Consequently, the cost function of LFDMFP approaches a constant, thus rendering the method ineffective.

$$k_n(\omega +\Delta \omega )\approx k_n\left(\omega \right)+{\displaystyle \frac{\Delta \omega }{v_{gn}\left(\omega \right)}}\cdots \Rightarrow k_n(\omega +\Delta \omega )\approx k_n\left(\omega \right)$$

(19)

$$\Gamma (r_s;\omega ,\Delta \omega )\approx A^2\left|{\sum }_n{\Phi }_n^2\left(z_0\right)/k_n\left(\omega \right)k_n(\omega +\Delta \omega )\right|$$

(20)

On the other hand, the impact of the frequency difference on LFDMFP is its effect on the orthogonality of normal modes. The approximation in Equation (6) relies on the premise that the frequency difference should not be too large. Figure 4 illustrates the influence of six frequency differences (0.01, 1, 10, 50, 100, and 300 Hz) on the orthogonality with the eight normal modes. It is evident that as the frequency difference increased, the values of the off-diagonal elements grew, which compromised the orthogonality of the modes. Figure 5 shows the range estimation of the LFDMFP corresponding to the six frequency differences. Obviously, when the frequency difference was 0.01 or 1 Hz, the results in Figure 5a,b were similar to the theoretical analysis, which were manifested as the cost function changing irregularly with range. As the frequency difference increased, the LFDMFP accurately estimated the range of the sound source. When the frequency difference exceeded 100 Hz, sidelobes appeared, which affected the range estimation at low SNRs.

By fixing the lower frequency at 300 Hz and adjusting the frequency difference, the curve of the LFDMFP distance estimation error as a function of the frequency difference was obtained, as shown in Figure 6, where the horizontal axis represents the magnitude of the frequency difference and the vertical axis represents the error of the range estimation.

In this simulation, the effective range of the frequency difference was 5–100 Hz. When the frequency difference was small (less than 1 Hz), the cost function of the LFDMFP exhibited irregular variations with range, which rendered the range estimation of the source ineffective. As the frequency difference increased, the range estimation performance of the LFDMFP improved. When only considering the first two normal modes, 12 Hz was the upper limit frequency difference with only one modal interference distance within the range search interval. In this simulation, the effective mode of the normal mode was eight. As the number of normal modes increased, the interference span of the LFDMFP increased; therefore, the upper limit of the frequency difference also increased. Under high SNRs, the results shown in Figure 5 and Figure 6 appeared. However, in actual waveguide environments, the SNR was low, and the large frequency difference led to high sidelobes, which could obscure the main lobe and cause false alarms. It is important to note that the range of frequency difference was dependent on the reference frequency and the waveguide environment. As the reference frequency increased, both the upper and lower bounds of the frequency difference range increased.

4. Experimental Results

The method presented in this paper was applied in two offshore experiments to show its effectiveness. The two experiments were the Qingdao Laoshan Bay Sound Propagation experiment conducted by the Hydroacoustic Laboratory of Ocean University of China and the Sound Propagation experiment in the South China Sea.

4.1. Sound Propagation Experiment in Laoshan Bay

The Hydroacoustic Laboratory at the Ocean University of China conducted a sound propagation experiment in August 2009 in Laoshan Bay of Qingdao. The experiment employed a dual-vessel operation mode, with the receiving ship anchored at station F to receive the transmitted signals. The receiving array was a VLA, while the launching ship emitted signals from various stations (H1–H6). Figure 7 illustrates the distribution of the experimental stations.

The experimental environment could be fairly approximated to an 18 m water depth Pekeris waveguide with the sound speed profiles shown in Figure 8a. The acoustic parameters of the half-space sediment ($c_b$ = 1596 m/s, ${\rho }_b$ = 1.62 g/cm³, ${\alpha }_b$ = 0.28 dB/$\lambda$ ) used here were obtained through the MFP inversion of the same data. The VLA had N = 15 sensors with a sampling rate of 12 kHz, which were uniformly spaced between 2.2 and 16.2 m. The source transmitted an M-sequence pseudorandom signal with a central frequency of 750 Hz and a bandwidth of 200 Hz. During this experiment, the depth of the source was maintained at 11 m. Figure 8b is a snapshot of the VLA-received signal and the transmitted M-sequence signal after matching filtering. Only two normal modes were observed.

This section verifies the effectiveness of the presented method by presenting the results from analyzing the H3 station data. At this station, the real distance between the source and the VLA was 4.62 km. The selected frequency band ranged from 780 to 804 Hz with frequency intervals of 3 Hz. The grid used to calculate the replica field was set to a depth of 0.2 m and a distance of 10 m. The replica field was calculated every 3 Hz using the KRAKENC. Figure 9 shows the results of the two methods for the H3 station. At this station, the target source range was 4.62 km according to the GPS. Figure 9a shows the CMFP incoherent averaging results at the six frequencies, where the black box represents the estimated range of the source. High sidelobes appeared within the range search interval, which hindered the true range estimation. The range estimation of the CMFP was 5.11 km, with an estimation error of 10.84%, which was a large deviation from the actual range. The range estimation of the LFDMFP is depicted in Figure 9b. The frequencies used for the LFDMFP were 780 and 786 Hz, and thus, a frequency difference of 6 Hz. The range estimation of the LFDMFP was 4.64 km, with an estimation error of 0.4%.

Table 1. The dependence of ${\Lambda }_{12}$ on the frequency difference.

$\mathit{\omega },\mathit{\omega }+\mathbf{\Delta }\mathit{\omega }$ (Hz)	${\mathbf{\Lambda }}_{12}$ (km)	$\mathit{\omega }/\mathbf{\Delta }\mathit{\omega }$	${\mathbf{\Lambda }}_{12}/{\mathit{\lambda }}_{12}$	$\mathit{\beta }$
780, 783	157	260	300	1.154
780, 786	77.6	130	150	1.154
780, 789	51.9	87	100	1.154
780, 792	39.3	65	75	1.154
780, 795	31.5	52	60	1.154

lists the dependence of the interference distance (${\Lambda }_{12}$) on the frequency difference ($\Delta \omega$), which was computed using Equation (9) in the actual experiment environment. In this experiment, the interference distance ${\lambda }_{12}$ of the first two normal modes at the frequency 780 Hz was approximated to 0.52 km (Figure 9a). Table 1 shows that ${\Lambda }_{12}/{\lambda }_{12}$ was proportional to $\omega /\Delta \omega$ and the coefficient of the proportionality was the waveguide invariant $\beta \approx 1.154$. When adjusting the frequency difference within the frequency range of the sound source signal and selecting multiple sets of data for the LFDMFP, the obtained range estimation results of the LFDMFP varied with the frequency difference, as shown in Figure 10, where the bar represents the error size corresponding to the frequency difference. According to the results in Figure 9 and Figure 10, despite the modest resolution of the range estimation, the result was accurate and the error was less than that for the CMFP.

${\Phi }_n\left(z_s\right)$ was calculated by using the sound speed profile displayed in Figure 8a, and the depth estimation was approximated to 12 m (Figure 9c).

4.2. Sound Propagation Experiment in the South China Sea

On 15 September 2019, an experiment that involved a towed sound source and a VLA was conducted in the South China Sea at a depth of 82 m. The sound speed of the water body in the experimental sea area is shown in Figure 11a, where the average sound speed was 1534 m/s. The sound speed, density, and attenuation of the sediment were 1573 m/s, 1.97 g/cm³, and 0.75 dB/$\lambda$, respectively. The receiving ship was equipped with a 32-element VLA with non-uniform spacing. In the shallow water area, the array elements were spaced 2 m apart; in the bottom water area, they were 1 m apart; in the intermediate water area, they were 1.5 m apart. The specific depth of the VLA deployment is shown in Figure 11b. The launch ship towed a sound source with a central frequency of 300 Hz, which emitted a linear frequency modulation signal that ranged from 280 to 350 Hz. Throughout the towing, the sound source maintained a depth of approximately 31 m.

This section verifies the effectiveness of the LFDMFP method through experimental data from two stations with distances of 8.47 km and 13.71 km. Significant internal wave activity was observed in the latter. Within the selected frequency band (280–340 Hz), 21 frequencies with an interval of 3 Hz were selected, and the candidate source grid for replica calculation was configured in an area with a depth of 2 m and a distance of 10 m.

Figure 12 shows the comparison of the range estimation between the two methods when the sound source distance was 8.47 km. Figure 12a shows the incoherent averaged CMFP results at the band. The black circle marks the estimated position of the CMFP, with a range estimation and error of the CMFP of 9.16 km and 8.14%, respectively. Figure 12b shows the range estimation of the LFDMFP when the frequencies 297 and 301 Hz were used. The range estimation of the LFDMFP was 8.52 km, with an estimation error of 0.5%. By using the sound speed profile displayed in Figure 11a, the depth estimation was approximated to 34 m (Figure 12c). Figure 13 shows the range estimation of the two methods when the sound source distance was 13.71 km. The range estimations of the CMFP and LFDMFP were 12.98 and 13.60 km, respectively, with range estimation errors of 5.32 and 0.8%, respectively. The depth estimation was approximated to 24 m (Figure 13c). Unlike the processing results of the other stations, in this station, the optimal range estimation of the CMFP was not within the periodic sidelobes, which indicates that the internal wave activity significantly affected the amplitude and phase of the sound field, thereby resulting in significant differences between the estimated and actual distances of the CMFP. However, we found that the LFDMFP demonstrated strong robustness in the face of mismatch (Figure 13b). The preliminary analysis suggested that this may have stemmed from the mechanism of the LFDMFP when handling the frequency difference, which converted the phase part to $k_n(\omega +\Delta \omega )-k_n\left(\omega \right)$. Compared with the phase part $k_n\left(\omega \right)$ of the CMFP, this change significantly improved the adaptability to the horizontal wavenumber changes. In addition, the LFDMFP also alleviated the influence of the internal waves on the eigenfunctions to a certain extent during its depth integration process. This also had a positive effect on improving the robustness of the LFDMFP in the face of a mismatch. Through these mechanisms, the LFDMFP more stably handled the complex signal changes and thereby demonstrated more reliable performance in practical applications.

Figure 14 depicts the range estimation results of the CMFP and LFDMFP over time against the actual source distance and entire range estimation error. The size of the error bar represents the relative error, which was calculated using Equation (21). Within the experimental trajectory range, the average relative error of the LFDMFP was 1.9%, which demonstrated that the LFDMFP had greater potential compared with the CMFP regarding the range estimation. Furthermore, the data from a single station position illustrated that the LFDMFP proposed in this paper exhibited robustness against environmental mismatches caused by internal waves.

$$error=|{\displaystyle \frac{{\widehat{r}}_s-r_0}{r_0}}|\times 100\%$$

(21)

5. Conclusions

A novel linear array-signal-processing technique based on the normal mode theory and CMFP is presented to estimate the range of a source with only two neighbor frequency components of the radiated sound field. Its performance was investigated for source localization in simulations and experimental data. Four conclusions were drawn from this research effort: (1) The LFDMFP was successful at estimating the range in shallow water. Using measurements from the Qingdao Laoshan Bay Sound Propagation Experiment and Sound Propagation Experiment in the South China Sea, the technique’s source range and depth estimations were unambiguous. The average absolute errors of the range estimation in the two experiments were 2.2 and 1.9%, respectively. (2) Due to the larger modal interference, only one peak appeared within the range search interval. Therefore, the LFDMFP could efficiently control the defect that CMFP provides multiple optimal estimations of the source range estimation. (3) The degraded resolution, which could be improved by combining the CMFP in the case where the knowledge of the actual propagation environment is known perfectly, may allow us to mesh grids with a larger range step size to reduce the computational cost of the replicas. (4) Preliminary analysis indicated that the LFDMFP had a higher tolerance for the environmental mismatch of fluctuations, which was verified by the data-processing results of the Sound Propagation Experiment in the South China Sea.

The above conclusions are positive since the simulation and experimental results validated the effectiveness of the LFDMFP. However, users need to make certain adjustments to the method based on the experimental environment and signal parameters, such as the frequency difference. Furthermore, it is hard to give a quantitative evaluation of how the frequency difference affects the range estimation. Further analysis of other factors that affect the performance of LFDMFP and a mathematical explanation of why LFDMFP is robust to environmental mismatch will be elaborated in future publications.