1 Introduction

Source localization in shallow water can be realized by using the acoustic propagation properties in the waveguide. The traditional method for the source range and bearing estimate is triangulation (Carter 1981; Huang and Barkat 1991), which utilizes the time delay between two or three horizontal subarrays and the bearing to measure the source range. Due to the frequent interactions of sound with the sea surface and sea bottom, the plane wave hypothesis of the triangulation method is not met, which, in turn, induces ambiguity in the degradation of the localization performance (Gong et al. 2013). Therefore, more accurate replicas that indicate the actual waveguide properties should be considered, especially in an under-ice shallow water environment, in which ice cover can introduce more multipath structures and energy attenuation (Yang 1989; Gavrilov and Mikhalevsky 2006). Model-based and environment-dependent methods, such as matched-field processing (MFP) (Baggeroer et al. 1993), modal dispersion method (Kuperman et al. 2001; Bonnel et al. 2014), array invariant (Lee and Makris 2006; Song and Cho 2017; Song and Byun 2021, 2022), and waveguide invariant (Cho et al. 2016) are usually employed in complex acoustic environments for source depth or range estimation. In this study, we apply MFP to localize the source in under-ice shallow water.

The MFP source localization is a typical model-based processing method that has been extensively applied to underwater localization. The first and most critical step is the generation of the replicas representing the receiving array, which considers the actual waveguide environment, such as sound speed profile and surface and bottom conditions. Subsequently, by correlating the replicas and the actual array measures, the source position can be induced from the ambiguity surface (Bucker 1976). To enhance the resolution, the minimum variance (MV) method optimizes the signal in the interest direction but suppresses the signals from other directions (Baggeroer et al. 1988; Schmidt et al. 1990). Jesus (1993) employed the subspace method to localize the transient signals in shallow water. Broadband processing can improve MFP robustness and resolution by using multiple frequency components (Brienzo and Hodgkiss 1993; Soares and Jesus 2003). The constraints and drawbacks of the MFP are sensitivity to the environment. To promote the MFP method’s robustness to the environmental mismatch, some modified approaches such as MV-EPC (Krolik 1992), MU-RSWP (Harrison et al. 1996), and maximum a posteriori probability methods (Gemba et al. 2017) have been applied. Currently, machine learning MFP methods are used for localization (Niu et al. 2017; Van Komen et al. 2021). Nevertheless, in the under-ice shallow water environment, the MFP has been barely investigated (Liu et al. 2019).

In the under-ice environment, the biggest challenge is the effect of ice on acoustic propagation properties. For example, for the first-year ice in the Arctic, the sea ice has an approximate thickness of 1–3 m with surface roughness and a random number of pressure ridges. Wolf et al. (1993) used the discrete ridge sea ice model to calculate the low frequency (less than 50 Hz) reflection coefficient and the propagation properties by the Burky-Twersky scattering theory in the Arctic (Diachok 1976; Liu et al. 2016). Kuperman and Schmidt (1986) applied the perturbation theory to calculate the ice reflection coefficients of the horizontal stratified rough ice. The module called OASR (Schmidt and Jensen 1985) refers to the implementation of the method proposed by Kuperman and Schmidt (1986). Hope et al. (2017) focused on the influence of sea-ice roughness on long-range acoustic propagation through the 900 Hz data from the ice-tethered buoys in a drifting network, and on how well the arrival structure could be predicted by the full wave integration model OASES. Worcester et al. (2023) used the data from 2016–2017 Canada Basin Acoustic Propagation Experiment to study the acoustic travel-time variability of the Beaufort Lens through parabolic equation model which was the latest widely used model in the under-ice environment (Duda et al. 2021; Barclay et al. 2023). In this work, we use the OASR to incorporate the sea ice reflection effect and produce replicas by the normal mode method.

This article is structured as follows. In Section 2, under-ice localization schemes are examined, including the theory of the ice reflection model, Bartlett, and the MV processor of the MFP. In Section 3, the Yellow Sea winter experiment is described, where a source of 650–750 Hz linear frequency modulated (LFM) signal at 5 m depth is collected by an 11 m-long VLA in a ca. 16.5 m depth sea with a ca. 5 cm-thick ice. The reflection of the thin ice and the propagation property are also investigated. At the end of Section 3, incoherent broadband Bartlett and MV methods are performed for real data to corroborate the localization performance. Lastly, conclusions are drawn.

2 MFP source localization in an under-ice environment

As introduced in the Section 1, there are numerous MFP source localization methods for various mismatch conditions. It is no need for validating all methods. The Bartlett method has the most robustness with low resolution while the MV method is a adaptive method which has the high resolution but is subject to environment mismatch. Consequently, in this work, we choose the Bartlett and MV method as the representatives for the under-ice validation.

2.1 Bartlett and MV processors of MFP

The basic ideas of the Bartlett methods are as follows. The array receiving data of the ith element in the frequency domain can be represented as

$${p}_{i }\left({f}_{k},r,z\right)={a}_{i}\left({f}_{k},r,z\right){\varvec{s}}\left({f}_{k}\right)+{\varvec{n}}\left({f}_{k}\right),$$
(1)

where r and z refer to the source range and depth, respectively. n(fk) is the white, Gaussian, zero-mean additive noise. s(fk) represents the source spectrum at a frequency of fk. The variable of \({a}_{i}\left({f}_{k},r,z\right)\) is the Green function. In normal mode theory, \({a}_{i}\left({f}_{k},r,z\right)\) can expressed as

$$\alpha_i\left(f_k,r,z\right)=\sqrt{\frac{2\uppi}r}\sum\limits_{n=1}^N\frac{\phi_n(z)\phi_n(z_i)}{\sqrt{k_n}}\mathrm e^{i\left(k_nr-\uppi/4\right)-\alpha_nr},$$
(2)

where \({\phi }_{n}({z}_{i})\) is the mode depth function of mode n. \({k}_{n}\) is the horizontal wavenumber, and \({\alpha}_{n}\) is the attenuation coefficient. In the under-ice environment, the MFP critical point is the generation of \({\alpha}_{i}\left({f}_{k},r,z\right)\) that considers the ice reflection effect.

For the array measurement, the cross-spectral density matrix can be attained by the spatial correlation of the ith and jth elements of the array.

$${R}_{ij}\left({f}_{k}\right)=\langle {p}_{i}\left({f}_{k},r,z\right){p}_{j}^{* }\left({f}_{k},r,z\right)\rangle,$$
(3)

where the asterisk indicates the conjugate of the complex receiving field. Then, the Bartlett processor of MFP can be given by

$${C}_\text{bart}\left({f}_{k},r,z\right)=\frac{\sum\limits_{i}\sum\limits_{j}{a}_{i}^{*}({f}_{k},r,z){R}_{ij}\left({f}_{k}\right){a}_{j}\left({f}_{k},r,z\right)}{{\sum\limits_{i}{R}_{ii}\left({f}_{k}\right)\sum\limits_{i}\left|{a}_{i}\left({f}_{k},r,z\right)\right|}^{2}},$$
(4)

where \({C}_\text{bart}\left({f}_{k},r,z\right)\) is the ambiguous surface of the MFP, in which the maximum corresponds to the true source depth and range. The Bartlett method is susceptible to the high sidelobes in the ambiguous surface, which will overwhelm the main lobe and result in an error estimation of the source position. The MV method can promote the resolution by constraining the sidelobes and keeping the main interesting position. Mathematically, the MV method is

$${C}_\text{MV}\left({f}_{k},r,z\right)=\sum_{i}\sum_{j}\frac{1}{{a}_{i}{\left({f}_{k},r,z\right)}^{*}{{R}_{ij}}^{-1}{a}_{j}({f}_{k},r,z)}.$$
(5)

To further promote the robustness and resolution of the MFP, broadband MFP exploits the gain in the frequency domain. The incoherent processing uses the frequency component of the complex acoustic pressure by accumulating the energy at the main lobes and suppressing the high sidelobes corresponding to a spurious position. The corresponding broadband processing is expressed as

$${{C}_\text{inc}(f}_{k},r,z)=\sum\limits_{k=1}^{K}{C}_\text{bart/MV}({f}_{k},r,z).$$
(6)

2.2 Ice reflection model

As stated in Section 1, the critical point for the MFP in under-ice sea is the measure of the ice effect on acoustic propagation. We use here the model proposed by Kuperman and Schmidt (1986), which considers the upper boundary as three horizontal stratified media of water, ice cover, and air half-space, as shown in Fig. 1. The upper and lower interface of the ice is rough, which includes the scattering loss. The acoustic field in each medium can be expressed as \(\chi\), which can be then broken down into two components, namely coherent (mean) \(\langle \chi \rangle\) or incoherent (scattered) components s. In Fig. 1, the subscript a, i, and w of \(\langle \chi \rangle\) represents air, ice, and water, respectively.

Fig. 1
figure 1

Schematic of the ice scattering model

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To calculate the reflection coefficient of the mean component \(\langle \chi \rangle\), we apply the boundary conditions in water-ice and ice-air interfaces. The condition in the interface can be represented as \({\text{B}}\left({\chi }_{j};{\chi }_{j+1}\right)=0\), where j and j + 1 refer to the medium above and below the interface, respectively. In the ice-water interface, for instance, \({\text{B}}\left({\chi }_{j};{\chi }_{j+1}\right)=0\) means the continuity of the normal displacements and normal stress. We conduct the Taylor expansion of the total field \(\chi\) and then use the property of the roughness γ to obtain the effective potential as follows:

$${\chi }^{*}=\langle\chi \rangle{|}_{z=0}+\frac{\langle{\gamma }^{2}\rangle}{2}\frac{{\partial }^{2}\langle\chi \rangle}{{\partial z}^{2}}{|}_{z=0}+\langle\frac{\gamma \partial s}{\partial z}\rangle{|}_{z=0}.$$
(7)

We can determine the incoherent interference term \(\langle \frac{\gamma \partial s}{\partial z}\rangle\) in the effective potential. To obtain the solution of the mean field \(\langle\chi \rangle\), we must solve the term \(\langle\frac{\gamma \partial s}{\partial z}\rangle\). Kuperman and Schmidt (1986) developed the Bessel transformed effective potential and used the perturbation method to solve the above problems. The detailed scheme is tedious and will not be elaborated on in this article. The mean field will be used here to calculate the reflection coefficients.

Afterward, we analyze the ice reflection property based on the experiment environment. The thickness of the ice plate is approximately 5 cm with high porosity and an rms roughness of 0.01 m during the experiment period. The ice is a little loose, which can be cracked easily by the fish vessel. The acoustic parameters of the ice are assumed by experience, and compressional and shear velocities are 2700 and 400 m/s, respectively. The corresponding compressional and shear attenuation coefficients are 0.3 and 1.0 dB/λ, respectively. The density of ice is 0.9 g/cm3. We obtain the reflection property of the 5 cm thin ice as a function of grazing angle and frequency (650–750 Hz with increments of 5 Hz) by the OASR model. In Fig. 2, the reflection loss is relatively small, with the most reflection loss less than 0.15 dB at a grazing angle of 56° resulting from the thin ice. The phase of the reflection coefficient is inversely proportional to the grazing angle and the frequency rather than 180°.

Fig. 2
figure 2

Reflection loss and phase of the ice reflection per bounce as a function of grazing angle

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In the following section, we will couple the reflection loss with the normal mode theory to produce the replicas for the MFP application in conjunction with the actual SSP in the real sea experiment. Then, we validate the localization performance of the incoherent Bartlett and MV processors using real data.

3 Real data processing

3.1 Setup of the experiment

The Yellow Sea winter under-ice shallow water experiment was done in January 2018. The receiving site is located at 39.354°N, 123.350°E, fluctuating as a consequence of the wind. Figure 3 illustrates the location of the receiving site and 3 km sending station. Figure 4 shows the ice state, and the ice cover is thin and fragile, with a 5 cm thickness. The sea bottom is relatively flat, with an approximate depth of 16.5 m in the experiment region. The wind is mild, and the air temperature is low, less than −18℃.

Fig. 3
figure 3

Location of the receiving site (red rectangular) and 3 km sending station (blue diamond)

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Fig. 4
figure 4

Picture of the sea ice

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The schematic of the experiment setup is displayed in Fig. 5. Both the receiving and sending sites are situated in a fish vessel that floats on the ice cover. In the receiving site, the VLA has 12 hydrophones with a spacing of 1 m and the uppermost element at a depth of 3 m. The signal-to-noise ratio (SNR) of the receiving data is low due to the absence of the forward amplifier module. The sending site transmits the signal at a designated distance in the range of 500 m to 10 km. The source is a UW350 transducer submersing at 5–10 m depth.

Fig. 5
figure 5

Schematic of the experiment setup and the sending and receiving sites

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The SSP in the experiment region forms a sound duct with a sound channel axis at 6 m depth. In the sending site, we use the AML 6000 sound velocity profiler to determine the SSP, which will then be used to make the environment file of KRAKENC. In the receiving site, only the RBR Temperature and Depth instrument is employed to determine the temperature profile and then convert it to SSP. In Fig. 6, the SSP is almost identical within the sending and receiving sites within the 3 km distance. There exists a weak sound channel, with the axis at a depth of 6 m.

Fig. 6
figure 6

SSP at the sending (solid blue line) and receiving (red dashed line) sites

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3.2 Acoustic propagation property in under-ice shallow water

We apply the normal mode theory to compute the replicas, which has the advantage of reducing the complexity of the calculation. Once the mode depth functions and the horizontal wavenumber are obtained, the replica in different source and receiver depths and ranges are readily built without running the model repetitively. In this section, we calculate the dispersion property of the normal mode within the source frequency domain of 650–750 Hz. There exists at most 10 modes in the 16.5 depth waveguide. The mode depth functions of modes 1, 4, and 9 are shown in subplots (a)–(c) in Figs. 7 and 8, which indicate that the mode depth function is approximately independent of the frequency.

Fig. 7
figure 7

Waterfall figure of the mode depth function of modes 1, 4, and 9 as a function of frequency

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Fig. 8
figure 8

Mode depth function of modes 1, 4, and 9 (the curves in each subplot consist of all frequencies with increments of 1 Hz)

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Figure 9 displays the transmission loss waterfall figure within 5 km. The modal interference is complex and faint, which results from the frequency interactions of the upper and lower boundaries. Almost all the modes interact with the upper and lower boundaries. The eigenray of the acoustic path is presented in Fig. 10, where we only show the rays within 5 bounces of the sea bottom for clarity. In Fig. 10, the red line means the direct path, the blue line means the path with only the bottom bounce, and the black line means the path with both surface and bottom interaction. The ice thickness is much less than the wavelength of sound (2 m), which renders little influence of the ice uncertainty on localization. Thus, it can be concluded in Fig. 10 that at a short distance of 3 km, the bottom modes or paths are the predominant contributors to the replica. In other words, the sea bottom constitution (sand, silt, etc.) and the corresponding acoustic parameter are the predominant factors of the localization errors.

Fig. 9
figure 9

Waterfall figure of transmission loss with the source at a depth of 5 m and 700 Hz

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3.3 Source localization

To process the array receiving data, we first bandpass the signal by a Finite Impulse Response with a passband of 600–800 Hz to highlight the LFM signal in the time domain. Afterward, by pulse compression, we determine the position of the LFM frame, which will be extracted in the receiving sequence with a length of 4 s, as shown in Fig. 11. We can deduce from Fig. 11 that a total of 4 successive snapshots (or frames) exist based on the predetermined sending schedule. Figure 12 provides the time-frequency spectrum in the 2.7 km source range. From Fig. 12, the LFM signal is faint, which indicates the low SNR due to the absence of the preamplifier module of the hydrophones.

Fig. 10
figure 10

Eigenray of the 5 m depth source and 10 m receiver

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Fig. 11
figure 11

Results of pulse compression

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Fig. 12
figure 12

Time-frequency spectrum of channel 1 and the first frame

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Using multiple snapshots and incoherent broadband MFP on real data, we obtain the ambiguity surface of the Bartlett and MV processors, as shown in Figs. 13a and b. We use 4 successive snapshots and 151 individual frequency components to attain the ambiguity surface. In other words, the final result of the Fig. 13 is the average and smoothness of total 604 individual ambiguity surfaces. As deduced from the GPS of the sending and receiving ship, the true distance of the 3 km station is 2.7 km, and the source depth is 5 m. Both the Bartlett and MV processors have the strongest highlight in the proximity of the true source position. The results of both processors are consistent, with the source being 2730 m in range and 5.5 m in depth. As expected, the MV processor has good performance in sidelobe suppression. For the MFP processor, the range and depth estimate errors are 30 and 0.5 m, respectively. It is worth noting that sidelobes in the ambiguity surface are high and are vulnerable to the wrong estimate. As stated in Section 3.2, the high sidelobes and the localization error are primarily from the uncertainty of the sea bottom.

Fig. 13
figure 13

Ambiguity surface at 2.7 km range: a Bartlett; b MV

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4 Conclusions

In this work, the MFP source localization of a 3 km range is implemented in an under-ice shallow water environment. We use the horizontal stratified elastic sea ice model and perturbation theory to calculate the sea ice effect and couple them with normal mode theory to produce the replicas. Afterward, the performance of incoherent Bartlett and MV methods is confirmed by real data, which is collected by a 12-element VLA with 1 m space for a source (650–750 Hz) at a 2.7 km range and 5 m depth. The experiment results show a range error within 2% and a depth error within 10%. The error primarily originates from the uncertainty of sea bottom parameters.