Single Hydrophone Passive Source Range Estimation Using Phase-Matched Filter

Abstract

Algorithms working in mode space instead of directly matching the received complex sound pressure were developed to improve computational efficiency and robustness, but these algorithms may be inconvenient to apply in practice because manual operations are often inevitable when performing modal filtering. Based on a phase-matched filter, an imperfect matching scheme named the modal phase based matched impulse response (MP-MIR) is proposed to estimate the source range rapidly and conveniently with a single hydrophone. The field to be matched is still the received complex sound pressure. The replica field is a sum of several “phase” modes, which can be efficiently and conveniently synthesized merely with the horizontal wavenumbers of normal modes and the source–receiver range. The effectiveness of the proposed MP-MIR was demonstrated in localizing 84 emissions along a weakly range-dependent track at ranges of 2.54–20 km in the South China Sea. Although it was found, from cross-correlation coefficients, that the received signals showed strong variation even between adjacent emissions, MP-MIR outperformed the classical matched impulse response (MIR) with a lower standard deviation in most cases, demonstrating good robustness and potential for practical applications.

1. Introduction

Parameter estimation using a single hydrophone has been persistently sought over the past decades in the underwater acoustic community because it does not involve the use of synchronized arrays, and it is very suitable for small robotic platforms or monitoring that needs to cover a large area. In the problem considered here of ranging with broadband emissions, the early algorithms were implemented directly based on the received complex sound pressure [1,2]; then some advanced feature-based algorithms (mainly multipath [3,4] and modal dispersion [5,6,7,8,9]) were explored. However, most of these algorithms have many or few shortcomings in terms of computational efficiency or robustness, or especially in the operational complexity of feature extraction, which, to some extent, limit their application in practice.

Limited to a single receiver, relevant techniques localizing a source or characterizing the propagation environment rely upon the frequency diversity of the channel’s response instead of upon the spatial diversity as in conventional matched field processing (MFP) [10] does. Clay [1] suggested a time reversal of the channel impulse response to localize a source in simulation. Then, Li and Clay [11] verified this method in a laboratory waveguide using air as the medium of propagation. Hermand [2] used a so-called model-based matched filter to inverse geoacoustic properties with data from Yellow Shark’94 experiments. In essence, Clay and Hermand use the same method (matched filter [12]) but performed it in the time domain and the frequency domain, respectively. This classical method proposed for working with broadband emissions is referred to as matched impulse response (MIR), below. MIR does not require feature extraction and thus is convenient for applications. However, the calculation of replica fields for MIR (i.e., complex sound pressure) involves all of the source/receiver geometry and geoacoustic parameters, resulting in an extremely high computational burden.

For low-frequency emissions (a few hundred Hertz) in shallow water, modal dispersion is a remarkable phenomenon and dominates the received signal structures. Since it has been demonstrated that, by working in mode space, it is easy to deal with environmental mismatch [13] and this way of working is able to achieve economic processing time [14] compared to a conventional MFP processor [10] in array signal processing, mode-based single-hydrophone algorithms are also continuing to attract attention [5,6,7,15]. A main representative of such algorithms is the method based on the dispersion curve (the position of the modes in the time–frequency domain) [9]. Since the dispersion curve is related only to the modal phase, the method allows the estimation of the source–receiver range without searching for source depth. A performance of taking into account both modal phase and modal amplitude was recently reported by Le Gall [8]. It was found that this processing is quite robust to mismatch, but the performance depends strongly on the source/receiver depth due to the sensitivity to depth of the modal amplitude. The problem of implementing mode-based algorithms mainly occurs at modal feature extraction (often referred to as modal filtering), where the difficulty of the step and the investment in time on it may be underestimated because manual operations are inevitable [16].

All of ranging algorithms mentioned above try to perform “perfect” matching between the replica field and the measured field, where “perfect” means the two fields using exactly the same signal features to complete the match, e.g., the two fields both using complex sound pressure in MIR. However, imperfect matching in which the replica field uses only some of the signal features included in the measured field to perform the matching is sometimes also feasible. A phase-matched filter is a typical representative of imperfect matching. It is known in signal processing that a matched filter [12] can be obtained by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. In contrast, a phase-matched filter does the same thing, but uses only the phase information of the template signal. In seismic signal processing, a phase-matched filter is often used to detect weak surface waves [17,18] because the phase of the dispersion of a given path is easier to be predicted than the amplitude of the dispersion. Recently, Geroski and Dowling [19] reported an MFP-style localization algorithm termed phase-only matched autoproduct processing (POMAP) for passive source localization in the deep ocean using array signals. Similar to the phase-matched filter, POMAP ignores the amplitude weight used in its previously reported version [20] to improve its robustness. Phase-matched filter and POMAP seem to support the contention that an imperfect matching scheme has the potential to outperform a conventional “perfect” matching scheme.

This study explores the application of a phase-matched filter [17] on single-hydrophone ranging and aims to provide an imperfect matching scheme that gives full consideration to robustness, computational efficiency, and also convenience for applications. The pursuit of the conventional ranging scheme MIR, which is based on a matched filter, is a search for (the most similar) replica of the received channel impulse response in both amplitude terms and phase terms. Alternatively, a ranging scheme based on a phase-matched filter will remove the search on amplitude terms. The proposed modal phase based MIR (MP-MIR) uses the complex sound pressure as the measured fields and “phase” modes (modal amplitudes at all frequencies are defaulted to the constant 1, and thus only the phase terms of each modes is computed and utilized) as the replica field. In this way, MP-MIR not only avoids the step of modal feature extraction (modal filtering) to improve implementation convenience, but also retains the robustness and computational efficiency of using the modal phase.

This paper is organized as follows. Section 2 outlines the details of the proposed ranging approach. Section 3 tests the feasibility of MP-MIR in the case of environmental mismatch through a numerical simulation. Section 4 presents the results of experimental signal processing. Finally, Section 5 draws some conclusions.

2. Background Information

2.1. Phase-Matched Filter

A matched filter is a very important detector in signal processing. Here, a simple review of matched filters is described to understand the difference between a matched filter and its variations, i.e., the phase-matched filter which is the basis of this paper. Readers who want to learn more about matched filters are referred to an introduction published by Turin in 1960 [12].

Let $H\left(f\right)$ and $s\left(f\right)$ be the Fourier transform of the impulse response of a filter and the input signal, respectively. Then the instantaneous signal power of the filter output at time $t_0$ is

$$P_{\mathrm{signal}}\left(t_0\right)={\left|\int H\left(f\right)s\left(f\right)e^{j2\pi f{t}_0}\mathrm{d}f\right|}^2.$$

(1)

Let $N_0/2$ be the power spectral density of the input noise; then the average noise power of the filter output is

$$P_{\mathrm{noise}}=\frac{N_0}{2}\int {\left|H\left(f\right)\right|}^2\mathrm{d}f.$$

(2)

The output signal-to-noise ratio (SNR) of the filter can be given by the ratio of Equations (1) and (2):

$${\mathrm{SNR}}_{\mathrm{o}}=\frac{{\left|\int H\left(f\right)s\left(f\right)e^{j2\pi f{t}_0}\mathrm{d}f\right|}^2}{\frac{N_0}{2}\int {\left|H\left(f\right)\right|}^2\mathrm{d}f}.$$

(3)

Schwarz’s inequality can be stated as

$${\left|\int f_1\left(x\right)f_2\left(x\right)\mathrm{d}x\right|}^2\le \int {\left|f_1\left(x\right)\right|}^2\mathrm{d}x\int {\left|f_2\left(x\right)\right|}^2\mathrm{d}x.$$

(4)

The equality holds if $f_1\left(x\right)=k{f}_2^{\ast }\left(x\right)$, where k is an arbitrary constant and ∗ indicates the complex conjugate. If we identify $H\left(f\right)$ with $f_1\left(x\right)$ and $s\left(f\right)e^{j2\pi f{t}_0}$ with $f_2\left(x\right)$, we can write

$${\mathrm{SNR}}_{\mathrm{o}}=\frac{{\left|\int H\left(f\right)s\left(f\right)e^{j2\pi f{t}_0}\mathrm{d}f\right|}^2}{\frac{N_0}{2}\int {\left|H\left(f\right)\right|}^2\mathrm{d}f}\le \frac{2}{N_0}\int {\left|s\left(f\right)\right|}^2\mathrm{d}f.$$

(5)

As a matched filter should provide the maximum SNR at its output for a given signal, we obtain

$$H\left(f\right)=k{s}^{\ast }\left(f\right)e^{-j2\pi f{t}_0}.$$

(6)

A phase-matched filter, like the one Herrin and Goforth used to detect Rayleigh waves [17], ignores the amplitude term of $H\left(f\right)$:

$$H_{\mathrm{p}}\left(f\right)=\frac{H\left(f\right)}{\left|H\left(f\right)\right|}=\frac{s{\left(f\right)}^{\ast }}{\left|s\left(f\right)\right|}{e}^{-j2\pi f{t}_0}.$$

(7)

Thus, the output SNR of a phase-matched filter is

$${\mathrm{SNR}}_{\mathrm{p}}=\frac{{\left|\int H_{\mathrm{p}}\left(f\right)s\left(f\right)e^{j2\pi f{t}_0}\mathrm{d}f\right|}^2}{\frac{N_0}{2}\int {\left|H_{\mathrm{p}}\left(f\right)\right|}^2\mathrm{d}f}=\frac{{\left|\int \left|s\left(f\right)\right|\mathrm{d}f\right|}^2}{\frac{N_0}{2}\int 1\mathrm{d}f}.$$

(8)

A special case of Equation (4) is formed as

$$\frac{2}{N_0}\int {\left|s\left(f\right)\right|}^2\mathrm{d}f\ge \frac{{\left|\int \left|s\left(f\right)\right|\mathrm{d}f\right|}^2}{\frac{N_0}{2}\int 1\mathrm{d}f}$$

(9)

if we identify $s\left(f\right)$ with $f_1\left(x\right)$ and constant 1 with $f_2\left(x\right)$, which indicates that the output SNR of phase-matched filter will be not greater than that of the matched filter.

Obviously, the phase-matched filter suffers some performance loss in a noisy environment because it ignores the amplitude information of the input signal $s\left(f\right)$. However, deducing the sensitivity of the power detector in this way may help to improve the robustness in the case of amplitude mismatch and to save the computation related to the amplitude.

2.2. MIR

An underwater acoustic waveguide shows sophisticated spatiotemporal variation, which often goes beyond the processing ability of plane-wave based methods, such as bottom interaction [21]. In 1976, Bucker [10] introduced an important concept, i.e., the ambiguity surface, by formulating a conventional MFP processor, where a realistic environment is used to predict the wave field. MFP matches the spatial structure of the complex pressure field (at one or more discrete frequencies), so it requires horizontal or vertical arrays. In 1999, Hermand [2] demonstrated that a single hydrophone is able to determine geoacoustic properties based on broadband waterborne measurements. Hermand’s proposed method (i.e., MIR) is a classical technique to exploit the frequency diversity of the channel’s impulse response (also be known as Green’s function in the time domain and the transfer function in the frequency domain).

The normalized ambiguous surface of MIR is defined as

$$A\left(\mathbf{p}\right)=\frac{\left|{\sum }_{n=1}^N{g}_{\mathrm{pre}}^{\ast }\left(f_n;\mathbf{p}\right)g_{\mathrm{mea}}\left(f_n\right)\right|}{\sqrt{{\sum }_{n=1}^N{\left|g_{\mathrm{pre}}\left(f_n;\mathbf{p}\right)\right|}^2}\sqrt{{\sum }_{n=1}^N{\left|g_{\mathrm{mea}}\left(f_n\right)\right|}^2}},$$

(10)

where $\mathbf{p}$ denotes the parameters to be estimated, and $g_{\mathrm{pre}}\left(f\right)$ and $g_{\mathrm{mea}}\left(f\right)$ denote the predicted and measured transfer function at frequency f, respectively. $g_{\mathrm{pre}}\left(f\right)$ is calculated with acoustic models. $g_{\mathrm{mea}}\left(f\right)$ can be obtained by performing source deconvolution [22] on the measured signal $p_{\mathrm{mea}}\left(f\right)=g_{\mathrm{mea}}\left(f\right)s\left(f\right)$ of the received hydrophone:

$$g_{\mathrm{mea}}\left(f\right)=\frac{p_{\mathrm{mea}}\left(f\right)s^{\ast }\left(f\right)}{max\left\{{\left|s\left(f\right)\right|}^2,ϵ\right\}},$$

(11)

where $s\left(f\right)$ is the transmitted signal. Parameter $ϵ$ is introduced as a minimum allowable source spectral amplitude to avoid dividing by a very small number or by zero and is usually chosen as a small percentage of $max{\left|s\left(f\right)\right|}^2$. The optimal estimates of the desired parameters are determined by

$$\widehat{\mathbf{p}}=maxA\left(\mathbf{p}\right).$$

(12)

The MIR processor of Equation (10) is a pure power MF detector without any a priori hypothesis about sophisticated spatiotemporal variation in acoustic waveguides. Its performance depends greatly on the degree of fineness of the modeled environment compared to the physical environment. A fine model requires complex calculation. More importantly, it is less possible to measure (and model) detailed environmental information in practical applications.

2.3. Modal Theory

According to normal mode theory [23], the transfer function of an ocean waveguide can be interpreted as a sum of several modal components:

$$g\left(f;z_s,z_r,r\right)=\sum _{m=1}^M{a}_m\left(f\right)exp\left(j{\varphi }_m\left(f\right)\right),$$

(13)

where $g\left(f;z_s,z_r,r\right)$ denotes the transfer function for the source and receiver located at depths $z_r$ and $z_s$, respectively, with horizontal range r; M is the number of propagating modes; and $a_m\left(f\right)$ and ${\varphi }_m\left(f\right)$ are the amplitude and phase of the mth mode, respectively.

In a range-dependent waveguide, the adiabatic normal mode theory formulates the amplitude term $a_m\left(f;z_s,z_r,r\right)$ and the phase term ${\varphi }_m\left(f;r\right)$ as

$$\begin{array}{cc}\hfill a_m\left(f;z_s,z_r,r\right)=& Q\frac{{\psi }_m\left(f;0,z_s\right){\psi }_m\left(f;r,z_r\right)}{\sqrt{k_m\left(f;r\right)r}}\hfill \\ \hfill & \times exp\left(-{\int }_0^r{\beta }_m\left(f;r^{\prime }\right)\mathrm{d}{r}^{\prime }\right),\hfill \\ \hfill {\varphi }_m\left(f;r\right)=& {\int }_0^r{k}_m\left(f;r^{\prime }\right)\mathrm{d}{r}^{\prime },\hfill \end{array}$$

(14)

where Q is a constant, ${\psi }_m\left(f;z\right)$ is the modal depth function, and $k_m\left(f\right)$ and ${\beta }_m\left(f\right)$ are the real part and the imaginary part of the horizontal wavenumber, respectively. In normal mode theory, ${\psi }_m\left(f;z\right)$ relies on both the waveguide environment (water depth, sound speed profile, seabed geoacoustic properties) and the depth, while $k_m\left(f\right)$ solely depends on the former and does not depend on the depth; ${\beta }_m\left(f\right)$ represents the attenuation of the waveguide environment.

With Equation (14), it is clear that the phase term does not depend on the source/receiver depth or on the bottom absorption, but is determined by the rest of the parameters (including the desired source-receiver range), unlike the amplitude term, which relies on all of parameters of the source/receiver geometry and the waveguide environment. This property is exploited to improve the computational efficiency for ranging in the coming section.

2.4. The Proposed Approach

MP-MIR is achieved by applying a phase-matched filter instead of a matched filter to the MIR processor Equation (10) in mode space. The normalized ambiguous surface of the proposed MP-MIR is defined as

$$A\left(\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)=\frac{\left|{\sum }_{n=1}^N{g}_{\mathrm{mp}}^{\ast }\left(f_n;\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)g_{\mathrm{mea}}\left(f_n\right)\right|}{\sqrt{{\sum }_{n=1}^N{\left|g_{\mathrm{mp}}\left(f_n;\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)\right|}^2}\sqrt{{\sum }_{n=1}^N{\left|g_{\mathrm{mea}}\left(f_n\right)\right|}^2}},$$

(15)

where the sum of “phase” modes (modal amplitudes at all frequencies are defaulted to the constant 1) gives the replica field

$$g_{\mathrm{mp}}\left(f;\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)=\sum _{m\in {\mathbb{M}}_{\mathrm{r}}}exp\left(j{\varphi }_m\left(f\right)\right)$$

(16)

and ${\mathbb{M}}_{\mathrm{r}}$ denotes the set of modes included in the replica field. Since it is possible for the modal depth function to take a zero (or nearly zero) value, not all theoretically existing modes are expected to be included in the measured transfer function $g_{\mathrm{mea}}\left(f\right)$. Correspondingly, the number of modes included in the replica set ${\mathbb{M}}_{\mathrm{r}}$ should, in general, be less than the number of theoretically existing modes M.

If we let ${\mathbb{M}}_{\mathrm{g}}$ denote the set of modes included in the measured field, we obtain

$$g_{\mathrm{mea}}\left(f\right)=\sum _{i\in {\mathbb{M}}_{\mathrm{g}}}{a}_i\left(f\right)exp\left(j{\varphi }_i\left(f\right)\right).$$

(17)

Thus, the numerator of Equation (15) can be rewritten as

$$\begin{array}{cc}\hfill \sum _{n=1}^N{g}_{\mathrm{mp}}^{\ast }\left(f_n;\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)g_{\mathrm{mea}}\left(f_n\right)=& \sum _{n=1}^N\sum _{m\in {\mathbb{M}}_{\mathrm{r}}}exp\left(-j{\varphi }_m\left(f_n\right)\right)\sum _{i\in {\mathbb{M}}_{\mathrm{g}}}{a}_i\left(f_n\right)exp\left(j{\varphi }_i\left(f_n\right)\right)\hfill \\ \hfill =& \sum _{n=1}^N\sum _{i\in {\mathbb{M}}_{\mathrm{r}}\cap {\mathbb{M}}_{\mathrm{g}}}{a}_i\left(f_n\right)+\hfill \\ \hfill & \sum _{n=1}^N\underset{m\notin {\mathbb{M}}_{\mathrm{g}}}{\sum _{m\in {\mathbb{M}}_{\mathrm{r}}}}\underset{i\notin {\mathbb{M}}_{\mathrm{r}}}{\sum _{i\in {\mathbb{M}}_{\mathrm{g}}}}{a}_i\left(f_n\right)exp\left(j{\varphi }_i\left(f_n\right)-j{\varphi }_m\left(f_n\right)\right)\hfill \end{array}$$

(18)

The sum over frequencies is separated into diagonal ($m=i$) and off-diagonal ($m\ne i$) terms. As the value of ${\varphi }_i\left(f_n\right)-{\varphi }_m\left(f_n\right)$ varies with subscripts n, m, i, off-diagonal terms in general sum in an incoherent way. Thus, the value of the numerator is dominated by diagonal terms. Assume that mode m is added to ${\mathbb{M}}_{\mathrm{r}}$. If $m\in {\mathbb{M}}_{\mathrm{g}}$, the addition tends to increase the sum of diagonal terms and further increase the value of $A\left(\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)$, whereas, if $m\notin {\mathbb{M}}_{\mathrm{g}}$, it simply tends to increase the denominator and thus decrease the value of $A\left(\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)$. Therefore, Equation (15) will be a maximum when set ${\mathbb{M}}_{\mathrm{r}}$ equals to set ${\mathbb{M}}_{\mathrm{g}}$. The optimal parameters can thus be determined from

$$\widehat{\mathbf{p}}=maxA\left(\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right),$$

(19)

where ${\mathbb{M}}_{\mathrm{r}}$ serves as an unknown parameter and can be adaptively determined by iterating over all possible combinations of modes to maximize $A\left(\mathbf{p},{\mathbb{M}}_{\mathrm{r}}\right)$.

The aim of Equation (15) is to attempt to decrease the influence of environmental mismatch by requiring fewer parameters to predict the main characteristics of received signals that are directly related to the desired parameter (source–receiver range). This trick also helps to increase the ranging efficiency by saving the computation related to amplitudes.

3. Simulations

Due to the simplification of practical waveguides by an acoustic model and the measurement uncertainty of environmental parameters, waveguide environment mismatch is a problem that must be considered in practice. In this section, the performance of MP-MIR in a case of environmental mismatch is tested under a “realistic synthetic” scenario with bathymetry and sound speed profile (SSP) mismatch used as in the case in the experiment signal processing in Section 4. Classical MIR is employed to provide a reference for the robustness of the MP-MIR ranging results.

The nominal waveguide environment between the receiver site and the 5th source site in the experiment is utilized to produce replica fields. As shown in Figure 1, the depths of the source and the receiver are 11 m and 52 m, respectively; the horizontal range between the two depths is 14.7 km. The bathymetry extracted from the GEBCO2020 global bathymetric product [24] shows a weakly range-dependent bottom. Because bottom properties were not measured during the experiment, a half-space bottom with compressional speed $c_b=$ 1600 m/s, density ${\rho }_b=$ 1.69 g/cm${}^3$ and absorption ${\alpha }_b=$ 0.2 dB/$\lambda$ is experimentally determined as the true bottom. The SSP utilized in the replica field computation is depicted in Figure 1b and is consistent with that observed in the experiment.

The range-dependent acoustic model (RAM) [25] and the normal mode model KRAKEN [26] are employed to compute replica fields of MIR and MP-MIR in [100, 200] Hz, respectively. During the computation, unknown parameter set $\mathbf{p}$ involves only the source–receiver range and bottom sound speed. The bottom density ${\rho }_{\mathrm{b}}$ is determined by the bottom sound speed $c_{\mathrm{b}}$ with the speed–density Hamilton sediment empirical relationship [27]:

$$c_{\mathrm{b}}=2330.4-1257.0{\rho }_{\mathrm{b}}+487.7{\rho }_{\mathrm{b}}^2,$$

(20)

where 1.25 $\mathrm{g}/{\mathrm{cm}}^3<{\rho }_{\mathrm{b}}<2.10$$\mathrm{g}/{\mathrm{cm}}^3$. Other necessary parameters of the acoustic model are assumed to be known.

A total of 12 “measured” signals with source depths uniformly increasing from 5 m to 60 m were simulated with a plane bottom and a negative gradient SSP, as shown in Figure 1 (dashed lines). For the waveguide environment, the difference in bathymetry and SSP between the measured field and the replica field introduced a certain amount of mismatch between the two fields, as expected, for the purpose of testing the robustness of MP-MIR for the waveguide environment mismatch.

The normalized short-time Fourier transform (STFT) spectrogram of the simulated received signal in the nominal waveguide environment, as well as the dispersion curves of the first four modes that predicted by adiabatic normal mode theory, are depicted in Figure 2. Since dispersion curves are well fitted to modes, it can be seen from the figure that there are only three (the 1st, 3rd, and 4th) modes included in this signal, and their energy distributions are very different, though dispersion curves indicate that there may be up to four modes in [100, 200] Hz. The quantitative difference between actually included modes and possibly existing modes shows the necessity of determining the optimal modal combination ${\mathbb{M}}_{\mathrm{r}}$ in MP-MIR. MIR does not involve this issue due to its search for source depths, placing a cost on ranging efficiency.

Figure 3a,b gives the ambiguous surfaces of MP-MIR (${\mathbb{M}}_{\mathrm{r}}=3,4$) and MIR for the “measured” signal at a source depth of 10 m. Circles indicate the maximum on ambiguous surfaces, and squares indicate the real source–receiver range and bottom sound speed of the “measured” signal. It can be seen from the two figures that both MP-MIR and MIR give peaks of less than 1 and the estimates deviated from the true value (the circles do not coincide with the squares) when waveguide environment mismatch occurs. However, the distance between the circle and the square for MP-MIR is smaller than that for MIR, which demonstrates that MP-MIR shows better adaptability to environmental mismatch. In addition, it should be noted that ${\mathbb{M}}_{\mathrm{r}}=3,4$ gives a maximum over all possible choices of ${\mathbb{M}}_{\mathrm{r}}$, thus its ambiguous surface is shown in Figure 3a instead of the ambiguous surface of the set ${\mathbb{M}}_{\mathrm{r}}=1,3,4$, which indicate the real included modes in the signal. As is explained in Section 2.1, a phase-matched filter compresses the energy of a time-dependent modulation (i.e., mode components) to an instantaneous output. Usually, the compression gain of a wide modulation is bigger than a narrow modulation. In the form of Equation (15), where the output is normalized among all modes included in ${\mathbb{M}}_{\mathrm{r}}$, adding a narrow modulation (mode 1) tends to increase the value of the denominator more than the value of the numerator, leading to an reduction in the ratio. Therefore, mode 1 was excluded under the maximization principle [Equation (19)].

Figure 4a,b shows the ranging results of MP-MIR and MIR for the 12 “measured” signals. It is apparent that MP-MIR gives similar ranging results to MIR. Both the two methods give precise source–receiver ranges with a relative deviation of less than 10%, except for the "measured" signals at source depths of 25 m and 45 m, where the estimates show significant errors because the sidelobes give the maximum peak.

Figure 5a,b shows the bottom speed estimates from MP-MIR and MIR for the 12 “measured” signals. It can be seen from the two figures that MP-MIR gives more concentrated and precise estimates than MIR, demonstrating a significant improvement in the precision of estimating bottom speed in mismatched environments by avoiding the use of modal amplitude information, which is sensitive to mismatch, and using only information from the modal phase.

The investigation above demonstrates that the MP-MIR approach can achieve satisfactory accuracy and robustness and may be able to provide a competitive improvement compared to MIR without increasing the preprocessing step as feature-based methods do. In addition, the method was found to achieve an improvement in computational efficiency compared to MIR due to avoiding of the search for source depth.

4. Experimental Results

In this section the performance of the MP-MIR approach was evaluated with experimental signals collected in July 2021 by a bottom-moored hydrophone at a depth of 52 m at about (E$109.1^{°}$, N$17.8^{°}$) in the South China Sea.

Figure 6 depicts the experimental environments where depth data have been extracted from the GEBCO2020 global bathymetric product [24] because no measured bathymetries are available in this experiment. As Figure 6a shows, the bathymetry around the experimental area gradually shoals from south to north and thus forms a weakly range-dependent environment for sound propagation. The receiver site (white circle) and the 7 source sites (white stars) are located in an approximately straight line in the south-north direction and are between 2.54 km (nearest) and 20 km (farthest) apart. The SSP used in the subsequent ranging process is shown in Figure 6b, which was measured at the 7th (farthest) source site. Since the temperature and depth logger did not reach the bottom, probably due to the high current velocity during the SSP measurement, the rest of this SSP near the ocean bottom is patched according to other SSP records around the experimental area at other times.

Seven group signals were transmitted at a nominal source depth of 11 m while sailing along the source sites from south to north. Each group includes 12 upsweep signals, with pulse duration of 5 s and frequency band 20–200 Hz. Figure 7a gives the spectrogram of the 5th group of signals after source deconvolution performed with Equation (11). It is obvious that signals below 100 Hz are seriously contaminated by noise; thus only the 100–200 Hz components are employed in the subsequent ranging process. Figure 7b–m show spectrograms of all 12 signals in group 5. The energy distributions of those spectrograms are obviously similar but without the same details. Figure 8 gives the correlation coefficient matrix of these 12 signals. It is found that the cross-correlation coefficients between any two signals varies from 0.29 to 0.69 with an average value for the cross-correlation coefficients of only 0.46. Such a low averaged cross-correlation coefficient indicates that the received signals varied strongly, even over several seconds. In addition, the energy distributions of the 12 spectrograms are also a little different from that of the simulation example depicted in Figure 2, which may be a result of the imprecise waveguide environment information (SSP, bathymetry, or other parameters). Source ranging was performed with a search over source–receiver ranges and bottom sound speeds, the same as we performed it in Section 3.

Figure 9 shows all the ranging results of $7\times 12=84$ signals of MP-MIR and MIR. Red circles and blue circles represent each of the range estimates; black horizontal lines represent the true ranges. It is apparent that most estimates show good consistency with the true source–receiver ranges. However, MP-MIR provides more concentrated ranging results than MIR, especially for signals in the 5th group, which show a strong temporal variability. As the standard deviations (STD) of 12 signals of each group shows in Figure 10, MP-MIR gives a smaller STD for the middle five groups and a larger STD for only the 1st and the 7th groups. In addition, the fluctuation of the STD curve of MP-MIR is much smoother than that for MIR. The statistical results demonstrate that the MP-MIR approach is robust in practical application.

The environmental mismatch problem was encountered in the processing where (1) SSP and especially bathymetry are not precisely known and (2) the received signals are found to vary rapidly even over several seconds. However, the results demonstrate that the MP-MIR approach is robust in this practical application and shows less fluctuation among the same group estimates than the classical MIR in most cases.

5. Conclusions

In this paper, we introduced an efficient and robust approach termed the modal phase based matched impulse response (MP-MIR) on a single-hydrophone source ranging on a weakly range-dependent shallow-water environment. The same framework of a classical MIR was adopted in MP-MIR with an adjustment, according to the phase-matched filter mechanism, to produce replica fields where the sum of several “phase” modes (amplitudes of a mode at any frequency are defaulted to the constant 1 to retain only phase information) that are synthesized by the horizontal wavenumber taking the place of the original complex sound pressure. Since the modal phase does not depend on the source/receiver depth or the bottom absorption and is determined mainly by the rest of the parameters of a waveguide, fewer parameters makes the computation of ranging more efficient than that based on complex sound pressure replica fields. The computation can also be easily extended to the range-dependent case based on adiabatic normal mode theory. In addition, MP-MIR is convenient to apply in practice because the received complex sound pressure is directly adopted as measured fields without the processing step of feature extraction.

The performance of MP-MIR was demonstrated by both a numerical simulation and 84 experimental signals. The simulation was performed under a “realistic synthetic” waveguide environment for an experiment in the South China Sea with intentional bathymetry and SSP mismatch. MP-MIR gave satisfactory and comparable ranging results as well as more concentrated and precise bottom speed estimates among the 12 “measured” signals (whose source depths increased evenly from 5 m to 60 m) compared with MIR. In the experiment, received signals were found that strongly varied even over several seconds, leading to some differences in modal energy distributions on the time–frequency plane between the adjacent emissions. However, MP-MIR outperforms MIR in most cases with less fluctuation among same-group estimates, even when SSP and bathymetry in particular are not precisely known. The results of the simulation and the experiment seem to suggest that MP-MIR is robust in application.

As an unconventional idea, an imperfect matching scheme (e.g., MP-MIR using only phase information in the replica field but both phase and amplitude information in the measured field) has not been extensively studied for parameter estimation using a single hydrophone. This paper demonstrates its success both in simulation and experimentally with a designed algorithm, MP-MIR, that gives full consideration to robustness and efficiency as well as convenience for applications. Ongoing work on MP-MIR includes (1) a comparison of the influence of mismatch and signal-to-noise ratio, (2) the feasibility of application to detailed geoacoustic inversion, and (3) more experimental evaluations for practicability.