Analysis of the Adaptability and Application of Matched-Field Processors for Stationary and Maneuvering Targets in Shallow Water

Abstract

Passive acoustic localization in complex shallow waters requires algorithms tailored to specific operational constraints. This paper investigates the adaptability, computational efficiency, and statistical performance boundaries of five matched-field processing (MFP) methods—Bartlett, Minimum Variance Distortionless Response (MVDR), Multiple Signal Classification (MUSIC), Reduced Covariance Matrix (RCM), and Rank and Trace Minimization (RTM)—using the Elba-93 sea trial dataset. Error metrics and processing complexities are systematically evaluated across stationary and maneuvering target scenarios. Rigorous non-parametric statistical tests reveal distinct operational boundaries: under stationary conditions dominated by systemic environmental mismatch, energy-based processors guarantee reliable baseline stability. Conversely, under snapshot-deficient dynamic conditions tracking a receding target, standard high-resolution subspace methods become highly vulnerable to trajectory jumps. In such highly dynamic scenarios, adaptive energy-based processors (specifically MVDR) exhibit the most stable tracking continuity and lowest numerical peak errors. Simultaneously, the operational adaptability of subspace methods is improved via covariance matrix reconstruction (CMR). Specifically, the RCM technique effectively decouples unstructured sensor noise, mitigating maximum trajectory deviations and providing a balanced trade-off between computational efficiency and robustness. Statistical evaluations confirm the fundamental performance boundaries in static environments, while highlighting sample-size limitations in highly dynamic scenarios, thereby establishing a realistic, evidence-based benchmark for marine engineering applications.

1. Introduction

Underwater acoustic target localization (UATL) is a fundamental capability in ocean science and marine sensing. Early passive localization techniques often relied on simple array signal processing using time or phase differences, but their performance was limited by the complex multipath interference inherent in ocean waveguides [1]. To exploit this multipath propagation constructively, physics-based methods were developed, with matched-field processing (MFP) emerging as a cornerstone technique in the 1970s [2,3,4]. MFP estimates the source location by correlating the measured acoustic field across a hydrophone array with a dictionary of pre-computed replica fields generated by acoustic propagation models. The effectiveness of MFP has been demonstrated in numerous at-sea experiments [5,6].

Beyond deep-ocean exploration, reliable underwater acoustic target localization is increasingly critical for near-shore applications, particularly concerning usual activities in ports and harbors [7,8]. Modern port environments face dense maritime traffic and require continuous monitoring to ensure navigational safety. In these complex shallow-water settings, passive acoustic surveillance systems are essential for tracking autonomous underwater vehicles (AUVs), monitoring unauthorized underwater intrusions, and managing routine harbor traffic [9]. However, localizing fast-moving targets in such confined coastal environments directly exacerbates the physical challenges of environmental mismatch and snapshot deficiency [10]. Therefore, establishing a robust and adaptable localization framework holds direct practical value for coastal security and daily port management, broadening the relevance of matched-field methodologies to a wider marine engineering community.

Despite its theoretical elegance, the practical deployment of MFP is fundamentally hindered by the environmental “mismatch” problem [11,12]. As extensively discussed in contemporary marine engineering and acoustics literature, the intricate multipath effects and dynamic boundary conditions in shallow waters can severely degrade the spatial correlation of acoustic fields [13]. The accurate construction of regional sound speed profiles (SSPs) is critical for building reliable underwater acoustic systems, as it greatly affects signal propagation modes and localization accuracy [14]. Consequently, a fundamental trade-off exists between spatial resolution and environmental robustness. The conventional Bartlett processor provides interference-tolerant spatial estimation but suffers from wide mainlobes and high ambiguous sidelobes. High-resolution adaptive methods, such as the Minimum Variance Distortionless Response (MVDR) and subspace-based Multiple Signal Classification (MUSIC), offer pinpoint accuracy [15]. However, these high-resolution estimators are highly sensitive to operational constraints, particularly when tracking a fast-maneuvering target. To prevent the spatial smearing of the acoustic signature induced by continuous target motion, the integration time must be shortened [16]. This stringent kinematic constraint depletes the available snapshots, leading to an ill-conditioned or rank-deficient sample cross-spectral density matrix (CSDM). Under such snapshot-deficient dynamic conditions, standard subspace methods frequently fail, manifesting as severe subspace leakage, signal self-nulling, and unacceptable localization jumps.

To address these kinematic constraints and restore the operational adaptability of subspace methods under snapshot deficiency and environmental noise, covariance matrix reconstruction (CMR) techniques have been extensively investigated in robust Direction-of-Arrival (DOA) estimation [17]. Similar matrix reconstruction and sparse recovery strategies have been actively explored to guarantee high-resolution performance under challenging ocean conditions. The performance of subspace methods deteriorates when the CSDM is contaminated by unstructured sensor noise and colored interference. The Reduced Covariance Matrix (RCM) technique addresses spatially uncorrelated sensor self-noise, which strictly concentrates on the main diagonal of the CSDM. By explicitly removing the diagonal elements, RCM suppresses the noise floor and restores the orthogonality of the signal subspace under limited snapshots [18]. Furthermore, to address more complex colored ambient noise, Rank and Trace Minimization (RTM) frames the CSDM estimation as a convex optimization problem. Leveraging the low-rank property of the signal subspace and the sparse trace constraints of the noise, RTM adaptively separates and subtracts the noise covariance [19]. Advanced techniques such as Gaussian process regression with normal-mode-based kernels have also been proposed to enhance MFP performance by exploiting the correlation of acoustic fields at different receiving depths [20]. This guarantees a mathematically elegant and robust reconstruction of the acoustic scene.

Building upon these robust physical and statistical reconstructions, contemporary MFP research continues to grapple with the profound impacts of environmental variability on acoustic propagation. For instance, recent studies have highlighted how complex oceanographic features, such as the structural characteristics of mesoscale eddies, introduce significant localization uncertainty into matched-field processors [21]. To combat such environmental mismatches and stabilize source range estimation in shallow water waveguides, alternative physical formulations, such as linear frequency-difference MFP, have been actively proposed [22]. Concurrently, researchers are seeking to bridge the gap between rigorous acoustic propagation models and advanced computational frameworks. This has led to the emergence of physics-informed machine learning approaches, where physics-informed neural networks (PINNs) and their correction-aided variants are integrated directly into the MFP pipeline to enhance passive source range estimation while adhering to physical principles [23,24].

In recent years, the rapid advancement of machine learning (ML) and deep learning (DL) has offered a data-driven paradigm to bypass the explicit physical modeling constraints of MFP. Groundbreaking studies have demonstrated the considerable potential of feed-forward neural networks and convolutional architectures in mitigating environmental mismatches for shallow-water localization [25,26]. Recent advances in deep learning have shown remarkable capability in underwater sound source depth estimation and localization, particularly when combined with vector acoustic features and advanced network architectures [27]. While these data-driven models achieve remarkable accuracy by implicitly learning non-linear inverse mappings, physics-based array signal processing remains indispensable in practical marine applications. The “black-box” nature of neural networks lacks the strict mathematical interpretability required for critical decision-making, and their deployment typically relies on massive site-specific datasets that are difficult to obtain in complex operational waters [28]. Therefore, enhancing the robustness of physics-based spatial processors remains a critical research imperative.

It is essential to clarify that the primary scientific contribution of this study is not the derivation of novel mathematical algorithms, but rather to serve as a comprehensive engineering benchmark and physical evaluation of well-established matched-field methodologies. Rather than pursuing a singular optimal localization algorithm in idealized simulations, this paper systematically investigates the adaptability and application boundaries of five representative processors—Bartlett, MVDR, MUSIC, RCM, and RTM—utilizing the empirical Elba-93 sea trial dataset [29,30]. By employing high-fidelity 3D ambiguity surfaces and multi-dimensional error metrics, this study establishes a reliable, evidence-based practical framework for algorithm selection in dynamic marine operations.

The remainder of this paper is organized as follows. Section 2 introduces the environmental parameters of the Elba-93 sea trial dataset and details the simulation data setup for both stationary and maneuvering sources. Section 3 formulates the mathematical models of the five matched-field processing algorithms and provides a comparative analysis of their computational complexities. Section 4 provides a comprehensive performance evaluation, including high-fidelity spatial resolution comparisons, multi-dimensional error metrics, and rigorous statistical significance testing across different operational scenarios. Finally, Section 5 concludes the study, discussing its practical limitations and potential directions for future research.

2. Experimental Setup and the Elba-93 Sea Trial Dataset

To evaluate the practical adaptability and boundary performance of the aforementioned matched-field processors, empirical data collected during the Elba-93 sea trial is employed in this study.

2.1. Environment of the Elba-93 Sea Trial

The experimental site presents a range-independent shallow-water waveguide with a total depth of 127 m, located off the northern coast of Elba Island. As depicted by historical hydrological measurements, the aqueous layer is dominated by a downward-refracting sound speed profile (SSP). Specifically, the sound celerity decreases across a prominent thermocline spanning depths of 60 m to 77 m, scaling down from $1526\mathrm{m}/\mathrm{s}$ near the sea surface to $1508\mathrm{m}/\mathrm{s}$ at the bottom boundary.

Regarding the acoustic properties, the seabed is modeled as a two-layer structure: a 2.5-m thick sediment stratum overlying a semi-infinite basement. The sedimentary layer exhibits a sound speed gradient from $1520\mathrm{m}/\mathrm{s}$ to $1580\mathrm{m}/\mathrm{s}$, with a constant density of $1.75\mathrm{g}/{\mathrm{cm}}^3$ and an attenuation coefficient of $0.13\mathrm{dB}/\lambda$. The underlying rigid basement features a sound speed of $1600\mathrm{m}/\mathrm{s}$, a density of $1.8\mathrm{g}/{\mathrm{cm}}^3$, and an attenuation of $0.15\mathrm{dB}/\lambda$. Acoustic receptions were collected by a 48-element vertical line array (VLA). The hydrophone sensors were uniformly distributed with a 2 m spacing, comprehensively covering the water column from $18.7\mathrm{m}$ down to $112.7\mathrm{m}$.

2.2. Stationary Source Simulation Data Setup

In the stationary evaluation scenario, a physical acoustic source was tethered at a fixed spatial coordinate, approximately $79\mathrm{m}$ in depth and $56\mathrm{km}$ in horizontal range relative to the VLA. The corresponding spectrum is illustrated in Figure 1.

To perform spatial-spectral and subspace matched-field localization, a comprehensive dictionary of theoretical replica fields must be generated to scan the region of interest. Utilizing the KRAKEN normal-mode acoustic propagation program, the spatial search domain was gridded with high granularity. The range domain spanned from 4.0 km to 7.0 km with a fine spatial increment of 10 m, while the depth domain covered 70 m to 90 m with a 0.1 m resolution (Figure 2).

For the empirical data processing, a prolonged integration window of 20 s was utilized to calculate the sample cross-spectral density matrix (CSDM). This extended observation period guarantees a sufficient accumulation of temporal snapshots, ensuring the sample CSDM achieves full rank. Consequently, performance degradation in this static scenario is primarily isolated to systemic environmental mismatches rather than snapshot deficiency.

2.3. Maneuvering Source Simulation Data Setup

The dynamic tracking scenario involves an underwater target radially receding from the receiving array at a nominal depth of 70 m. While the complete maneuvering event lasted for 10 min, the acoustic source exhibited intermittent transmission characteristics. To ensure the integrity of the spatial-spectral estimation and avoid contamination from signal dropouts, the empirical data utilized for this analysis is strictly isolated to the initial continuous active transmission burst. This specific temporal window spans from 30 s to 60 s of the trial recording. Over this 30 s duration, the target’s radial distance advanced from approximately 5.945 km to 5.990 km. The corresponding spectrum is illustrated in Figure 3.

To facilitate continuous spatial localization for this moving target, a dynamic replica scanning grid was configured. The simulated search boundary was constrained between 5.8 km and 6.2 km in range, and 60 m to 85 m in depth (Figure 4). The theoretical copy fields were similarly generated via KRAKEN at the identical 170 Hz.

In stark contrast to the stationary case, the CSDM integration time was restricted to 5 s. This short temporal window is mechanically required to prevent severe spatial smearing of the acoustic signature induced by the target continuous displacement. However, this kinematic constraint reduces the available snapshot count, intentionally creating a rank-deficient matrix environment. This setup rigorously tests the operational boundaries of conventional high-resolution methods and validates the necessity of covariance reconstruction techniques.

3. Matched Field Processing Algorithm

Matched-Field Processing (MFP) formulates underwater acoustic localization as a generalized spatial filtering problem in complex ocean waveguides. Let $\mathbf{x}\left(t_k\right)\in {\mathbb{C}}^{N\times 1}$ denote the complex pressure vector received by an N-element vertical line array at the k-th snapshot for a specific focal frequency. The sample cross-spectral density matrix (CSDM), denoted as $\mathbf{R}\in {\mathbb{C}}^{N\times N}$, is estimated by temporally averaging over K consecutive snapshots within a predefined integration window:

$$\mathbf{R}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K}\mathbf{x}\left(t_k\right){\mathbf{x}}^H\left(t_k\right)$$

(1)

where ${(·)}^H$ represents the Hermitian transpose. The fundamental objective of MFP is to match this measured CSDM with a theoretical replica vector $\mathbf{w}(r,z)\in {\mathbb{C}}^{N\times 1}$, which models the acoustic propagation from a candidate source position at range r and depth z. The replica vectors are normalized such that $\parallel \mathbf{w}(r,z)\parallel =1$.

3.1. Energy-Based Spatial Processors

3.1.1. The Bartlett Processor

The Bartlett processor, also known as Conventional Beamforming (CBF), is the most fundamental MFP algorithm. It estimates the source location by maximizing the spatial correlation power between the measured CSDM and the theoretical replica vectors. The spatial spectrum of the Bartlett processor is defined as:

$${\mathbf{P}}_{CBF}(r,z)={\mathbf{w}}^H(r,z)\mathbf{Rw}(r,z)$$

(2)

While highly robust to environmental uncertainties, the Bartlett processor is intrinsically limited by the Rayleigh resolution criterion, suffering from wide mainlobes and high ambiguous sidelobes.

3.1.2. The Minimum Variance Distortionless Response

To mitigate the severe sidelobe interference inherent in CBF, the MVDR processor (or Capon beamformer) is employed. It adaptively minimizes the total output noise variance while maintaining a distortionless unity gain in the hypothetical look direction. The optimization problem is formulated as ${\min }_{\mathbf{h}}{\mathbf{h}}^H\mathbf{R}\mathbf{h}$ subject to ${\mathbf{h}}^H\mathbf{w}=1$. The resulting spatial spectrum is given by:

$${\mathbf{P}}_{MVDR}(r,z)={\displaystyle \frac{1}{{\mathbf{w}}^H(r,z){\mathbf{R}}^{-1}\mathbf{w}(r,z)}}$$

(3)

In practical dynamic scenarios, the limited number of snapshots K often causes $\mathbf{R}$ to be ill-conditioned. To ensure stable matrix inversion, diagonal loading is applied by replacing $\mathbf{R}$ with $\mathbf{R}+\mu \mathbf{I}$, where $\mathbf{I}$ is the identity matrix and $\mu$ is a small regularization parameter typically proportional to the noise variance.

3.2. Subspace-Based High-Resolution Processor

The Multiple Signal Classification (MUSIC) algorithm transcends energy-based filtering by exploiting the orthogonality between the signal and noise subspaces. Performing eigenvalue decomposition on the CSDM $\mathbf{R}$ yields:

$$\mathbf{R}={\mathbf{U}}_s{\mathbf{\Lambda }}_s{\mathbf{U}}_s^H+{\mathbf{U}}_n{\mathbf{\Lambda }}_n{\mathbf{U}}_n^H$$

(4)

where ${\mathbf{\Lambda }}_s$ and ${\mathbf{\Lambda }}_n$ are diagonal matrices containing the L dominant eigenvalues associated with the effective signal components and the remaining $N-L$ smaller eigenvalues corresponding to the noise, respectively. The matrices ${\mathbf{U}}_s$ and ${\mathbf{U}}_n$ consist of the corresponding eigenvectors that span the signal subspace and the noise subspace.

Theoretically, the true replica vector $\mathbf{w}$ is effectively orthogonal to the noise subspace ($\parallel {\mathbf{U}}_n^H\mathbf{w}\parallel =0$). However, in dynamic scenarios bounded by severe snapshot constraints, the noise subspace is highly susceptible to rank-deficiency and eigenvalue perturbation. To enhance robustness and account for multipath energy spreading, the spectrum is equivalently formulated using the projection matrix of the expanded signal subspace ${\mathbf{P}}_s={\mathbf{U}}_s{\mathbf{U}}_s^H$:

$${\mathbf{P}}_{MUSIC}(r,z)={\displaystyle \frac{1}{1-\left|{\mathbf{w}}^H(r,z){\mathbf{P}}_s\mathbf{w}(r,z)\right|}}$$

(5)

This projection-based formulation achieves distinct, “needle-like” mainlobes but remains highly vulnerable to spatial aliasing induced by environmental mismatch.

3.3. Covariance Matrix Reconstruction Techniques

To restore the operational adaptability of subspace methods under snapshot deficiency and complex ambient noise, CMR techniques are introduced as a preprocessing step before subspace extraction.

3.3.1. Reduced Covariance Matrix

In practical sensing systems, spatially uncorrelated sensor self-noise predominantly concentrates on the main diagonal of the CSDM. The RCM technique provides a straightforward yet highly effective reconstruction by explicitly annihilating these diagonal elements:

$${\mathbf{R}}_{RCM}=\mathbf{R}-\mathrm{diag}\left\{\mathrm{diag}\left(\mathbf{R}\right)\right\}$$

(6)

where the inner operator $\mathrm{diag}\left(\mathbf{R}\right)$ extracts the main diagonal elements of the matrix $\mathbf{R}$ into a column vector, and the outer operator $\mathrm{diag}(·)$ maps this vector back onto the main diagonal of a new matrix with all off-diagonal elements set to zero. This subtracted term mathematically represents the uncorrelated white noise component.

By performing eigenvalue decomposition on ${\mathbf{R}}_{RCM}$ to extract the purified signal subspace, this diagonal-deletion approach notably suppresses the noise floor and prevents severe subspace leakage under limited snapshots.

3.3.2. Rank and Trace Minimization

To address more complex, colored environmental interference, the RTM technique frames CSDM denoising as an iterative optimization process. It models the measured matrix as:

$$\mathbf{R}={\mathbf{R}}_{sig}+\mathbf{Q}$$

(7)

where ${\mathbf{R}}_{\mathrm{sig}}$ denotes the pure, noise-free signal covariance matrix containing the target’s spatial information, which is theoretically low-rank due to the limited number of dominant propagation modes. The term $\mathbf{Q}$ represents the unknown, full-rank ambient noise covariance matrix.

In theoretical frameworks, leveraging the low-rank property of ${\mathbf{R}}_{\mathrm{sig}}$ and the sparsity constraints of the noise trace, RTM aims to decouple the low-rank signal subspace from the noise by solving a strict semi-definite programming (SDP) convex optimization problem. However, utilizing standard interior-point solvers for this optimization incurs an prohibitive computational burden for dynamic real-time localization. Therefore, to ensure both computational efficiency and tracking robustness in this study, the RTM algorithm is implemented via an iterative eigen-truncation approximation.

The core objective of this practical implementation is to iteratively estimate ${\mathbf{Q}}_{\mathrm{est}}$ to minimize the residual noise power while preserving the positive semi-definiteness of the signal components. Given the initial CSDM $\mathbf{R}$, the noise covariance matrix is initialized as ${\mathbf{Q}}^{\left(0\right)}=0.1·\mathrm{mean}\left(\mathrm{diag}\left(\mathbf{R}\right)\right)\mathbf{I}$, where $\mathbf{I}$ is an $L\times L$ identity matrix. The algorithm proceeds iteratively through the following steps:

Signal Covariance Update: In the k-th iteration, the pure signal covariance matrix is approximated by subtracting the noise estimate:

$${\mathbf{R}}_{\mathrm{sig}}^{\left(k\right)}=\mathbf{R}-{\mathbf{Q}}^{(k-1)}$$

(8)

Eigenvalue Decomposition (EVD): The EVD is performed on the updated signal matrix:

$${\mathbf{R}}_{\mathrm{sig}}^{\left(k\right)}={\mathbf{V}}^{\left(k\right)}{\mathbf{D}}^{\left(k\right)}{\left[{\mathbf{V}}^{\left(k\right)}\right]}^H$$

(9)

where the eigenvalues in the diagonal matrix ${\mathbf{D}}^{\left(k\right)}$ are sorted in descending order ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_L$.

Noise Power Re-estimation: Assuming the number of dominant sources is $N_{\mathrm{sub}}$, the ambient noise power ${\sigma }_{\mathrm{noise}}^2$ is re-estimated by averaging the noise-subspace eigenvalues:

$${\sigma }_{\mathrm{noise}}^2={\displaystyle \frac{1}{L-N_{\mathrm{sub}}}}{\displaystyle \sum _{i=N_{\mathrm{sub}}+1}^L}\max ({\lambda }_i,0)$$

(10)

Noise Matrix Update: The noise constraint matrix is updated subject to a strictly positive lower bound ($ϵ={10}^{-10}$) to ensure numerical stability:

$${\mathbf{Q}}^{\left(k\right)}=\max ({\sigma }_{\mathrm{noise}}^2,ϵ)\mathbf{I}$$

(11)

Based on extensive empirical tuning in the Elba-93 dataset, this iterative process rapidly converges. The convergence criterion is practically set to a fixed termination of 5 iterations ($k_{\max }=5$), which optimally balances noise suppression and computational cost. After the final iteration, the refined signal covariance matrix is obtained as ${\mathbf{R}}_{\mathrm{RTM}}={\mathbf{R}}_{\mathrm{sig}}^{\left(k_{\max }\right)}$. The signal subspace eigenvectors ${\mathbf{E}}_s^{\mathrm{RTM}}$ are extracted from ${\mathbf{V}}^{\left(k_{\max }\right)}$, and the corresponding RTM ambiguity output is formulated identically to the MUSIC projector:

$$P_{\mathrm{RTM}}(r,z)={\displaystyle \frac{1}{\max (1-|{\mathbf{w}}^H(r,z)\left({\mathbf{E}}_s^{\mathrm{RTM}}{\left[{\mathbf{E}}_s^{\mathrm{RTM}}\right]}^H\right)\mathbf{w}(r,z)|,{10}^{-5})}}$$

(12)

By dynamically decoupling unstructured noise prior to subspace projection using this reproducible iterative strategy, RTM bridges the gap between the high-precision resolution of MUSIC and the steady-state tracking robustness of the Bartlett processor, bypassing the need for complex external optimization solvers.

3.4. Computational Complexity Analysis

For practical underwater surveillance systems, the computational complexity and data processing time are critical parameters that govern real-time deployment. The computational burden of matched-field processors generally comprises two stages: cross-spectral density matrix (CSDM) processing and ambiguity surface searching. Let L denote the number of hydrophone sensors (in this study, $L=48$) and $N_s$ denote the total number of spatial search grid points.

The conventional Bartlett processor is computationally the most efficient, requiring only matrix-vector multiplications with a complexity of $\mathcal{O}\left(N_s{L}^2\right)$ for spatial matching. The MVDR processor requires an additional matrix inversion (or pseudo-inversion) of the CSDM, resulting in a complexity of $\mathcal{O}(L^3+N_s{L}^2)$.

Subspace-based methods, including MUSIC and RCM, necessitate the eigenvalue decomposition (EVD) of the CSDM to extract the signal and noise subspaces. The EVD operation intrinsically incurs a complexity of $\mathcal{O}\left(L^3\right)$. Therefore, their overall computational cost is roughly on par with MVDR, scaling as $\mathcal{O}(L^3+N_s{L}^2)$, making them highly suitable for practical applications.

In theoretical formulations, the RTM algorithm frames the covariance matrix reconstruction as a strict trace-minimization convex optimization problem. Solving this semi-definite programming (SDP) problem using standard interior-point solvers generally incurs an exorbitant computational burden ranging from $\mathcal{O}\left(L^{3.5}\right)$ to $\mathcal{O}\left(L^4\right)$ per iteration, which is prohibitive for real-time tracking. However, to fulfill the engineering mandate of this study, the RTM implemented herein adopts a computationally efficient iterative eigen-truncation approach. By performing EVD iteratively to update the noise variance estimation (typically converging within $K=5$ iterations), the practical complexity of the implemented RTM is notably reduced to $\mathcal{O}(K{L}^3+N_s{L}^2)$.

In summary, while all evaluated methods possess manageable complexities for modern processing architectures, Bartlett remains the most lightweight. Subspace methods (MUSIC, RCM) introduce moderate but acceptable EVD overheads. The accelerated RTM provides advanced noise separation but remains the most computationally demanding among the five due to its iterative nature. Consequently, when balancing dynamic localization accuracy and real-time computing resource requirements, RCM emerges as a highly balanced and viable solution.

3.5. Error Assessment Metrics

To systematically quantify the localization precision and steady-state robustness of the five matched-field processors across different operational scenarios, multi-dimensional error metrics are established. Let $N_t$ denote the total number of temporal snapshots (or processing frames). For the i-th snapshot, let ${\widehat{y}}_i$ represent the estimated spatial coordinate (either range ${\widehat{r}}_i$ or depth ${\widehat{z}}_i$), let $y_i$ denote the corresponding truth coordinate ($r_i$ or $z_i$). The absolute error (AE) at snapshot i is defined as $A{E}_i=|{\widehat{y}}_i-y_i|$. The performance metrics—Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Maximum Absolute Error (MaxAE), and Mean Absolute Percentage Error (MAPE)—are formulated as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N_t}}\sum _{i=1}^{N_t}A{E}_i^2}$$

(13)

$$\mathrm{MAE}={\displaystyle \frac{1}{N_t}}{\displaystyle \sum _{i=1}^{N_t}}A{E}_i$$

(14)

$$\mathrm{MaxAE}=\underset{1\le i\le N_t}{\max }\left(A{E}_i\right)$$

(15)

$$\mathrm{MAPE}={\displaystyle \frac{1}{N_t}}{\displaystyle \sum _{i=1}^{N_t}}\left({\displaystyle \frac{A{E}_i}{y_i}}\right)\times 100\%$$

(16)

where RMSE serves as a comprehensive metric that penalizes large localization deviations, MAE reflects the average steady-state precision, MaxAE indicates the worst-case scenario (crucial for evaluating severe localization jumps), and MAPE standardizes the relative error across different spatial dimensions by normalizing the absolute error against the true target coordinate $y_i$.

4. Results and Discussion: Adaptability and Boundaries

Building upon the mathematical formulations and evaluation metrics established in Section 3, this section systematically evaluates the practical localization precision and steady-state robustness of the five matched-field processors. To rigorously delineate their operational boundaries, the evaluation is conducted across two distinct real-sea scenarios extracted from the Elba-93 dataset: a stationary target subject to systemic environmental mismatch, and a fast-maneuvering target characterized by severe snapshot deficiency. The comparative analysis employs high-fidelity 3D ambiguity surfaces to visualize spatial resolution, followed by multi-dimensional error metrics to quantitatively assess tracking adaptability under dynamic kinematic constraints.

4.1. Performance Analysis of Stationary Source

4.1.1. Spatial Resolution and Ambiguity Surfaces

The visual representation of spatial resolution is demonstrated via the 2D and 3D ambiguity surfaces in Figure 5. For the Bartlett processor, the spatial spectrum exhibits a broad mainlobe and elevated background noise floors due to the Rayleigh resolution limit. Consequently, this widespread energy distribution challenges the separation of closely spaced targets. In contrast, the adaptive MVDR and subspace-based processors (MUSIC, RCM, RTM) yield sharper spatial peaks with effective background noise suppression. Since the 20-s integration time ensures an accurately estimated full-rank CSDM, the signal and noise subspaces are well-defined, allowing MUSIC, RCM, and RTM to achieve their theoretical super-resolution capabilities without suffering from rank-deficiency-induced spatial aliasing.

This phenomenon corroborates the fundamental physical trade-off in array signal processing: subspace methods trade environmental robustness for spatial resolution. Because the subspace projection yields extremely sharp spatial peaks, even a slight mismatch in the replica field vector drastically offsets the peak from the true location. In contrast, the broad, “fat” mainlobe of the Bartlett processor acts as a spatial buffer, maintaining the peak energy relatively closer to the true target center despite environmental perturbations. Interestingly, the RCM technique, by explicitly eliminating the diagonal sensor self-noise, slightly mitigated this mismatch sensitivity. Ultimately, for static targets operating under severe environmental mismatch with adequate snapshot accumulation, energy-based processors guarantee a stable, albeit spatially broadened, baseline detection. Within the specific confines of the current shallow-water evaluation, they serve as a highly practical baseline choice for comparable underwater monitoring systems.

4.1.2. Analysis of Error Metrics

While high-resolution processors excel in spatial sharpness, their operational reliability under environmental uncertainties tells a different story. As recorded in the localization metrics, a systemic localization bias exists across all methods due to the unavoidable discrepancies between the assumed Kraken acoustic model and the true ocean parameters.

Remarkably, the empirical data reveals that the energy-based processors exhibit stable mismatch resilience compared to the subspace methodologies. The Bartlett and MVDR processors yielded a steady range RMSE and MAE of 503.77 m, with a tightly bounded MaxAE of 510 m. Conversely, the standard MUSIC and the sophisticated RTM methods, despite their ultra-narrow mainlobes, reported a higher range RMSE and MAE of 521.27 m, peaking at a MaxAE of 530 m. The error metrics are summarized in

Table 1. Quantitative comparison of range localization performance metrics for the stationary source.

Method	RMSE (m)	MAE (m)	MaxAE (m)	MAPE (%)
Bartlett	503.77	503.75	510.00	9.00
MVDR	503.77	503.75	510.00	9.00
MUSIC	521.27	521.25	530.00	9.30
RCM	510.00	510.00	510.00	9.10
RTM	521.27	521.25	530.00	9.30

and

Table 2. Quantitative comparison of depth localization performance metrics for the stationary source.

Method	RMSE (m)	MAE (m)	MaxAE (m)	MAPE (%)
Bartlett	4.98	4.98	5.00	6.30
MVDR	5.02	5.02	5.10	6.35
MUSIC	5.19	5.19	5.30	6.57
RCM	4.98	4.98	5.00	6.31
RTM	5.19	5.19	5.30	6.57

.

It is noteworthy that under the stationary target scenario, the performance of Bartlett and MVDR estimators are almost identical (Range RMSE of 503.77 m) (Figure 6).This convergence is attributed to the high signal-to-noise ratio (SNR) and the long integration time applied, which ensures a high-fidelity estimation of the CSDM. Under such sufficient snapshot conditions, the MVDR processor tends to degrade towards the performance of the conventional Bartlett beamformer.

Furthermore, the standard MUSIC and the RTM-enhanced MUSIC yielded identical localization results. This is because the iterative noise estimation in the applied RTM implementation simplifies the ambient noise field as isotropic white noise. In this specific case, subtracting a scaled identity matrix from the CSDM only shifts the eigenvalues but does not alter the eigenvectors. Since the MUSIC algorithm’s performance is solely dependent on the geometry of the signal and noise subspaces, the resulting spatial spectra are mathematically equivalent. This observation highlights that the true advantage of RTM would be more pronounced in scenarios dominated by strong, spatially correlated interference.

4.2. Performance Analysis of Maneuvering Source

4.2.1. Spatial Resolution and Ambiguity Surfaces

In this section, the algorithms are evaluated under the dynamic tracking scenario. Although the complete sea trial recording spans a 10 min duration, the analysis is strictly localized to a continuous active acoustic emission segment occurring between the 30th and 60th seconds. Within this specific 30 s window, the target source moves radially from approximately 6.75 km to 6.71 km. To prevent severe spatial smearing of the acoustic signature induced by continuous target displacement, the integration time for the sample covariance matrix is drastically constrained to 5 s. Accordingly, the matched-field localization is executed at 5-s consecutive intervals, commencing at 32.5 s, capturing processing frames at 32.5 s, 37.5 s, 42.5 s, 47.5 s, and 52.5 s. This stringent kinematic constraint inherently depletes the available snapshots, forcing the algorithms to operate under a rank-deficient matrix heavily perturbed by target motion. To visually assess the spatial resolution without redundancy, the ambiguity surfaces at a representative temporal frame at 37.5 s, are extracted for detailed discussion as shown in Figure 7.

At this specific juncture, the impact of spatial smearing begins to manifest. The Bartlett processor’s ambiguity surface, exhibits a severely smeared, continuous mainlobe that comfortably engulfs the true localization coordinate, acting as a spatial buffer against kinematic perturbations. The MVDR processor suppresses the background noise and focuses the energy, placing the estimated peak in close proximity to the reference target. This visual stability is a precursor to its excellent statistical performance. In stark contrast, the standard MUSIC algorithm demonstrates significant vulnerability. Due to the limited snapshots and continuous target displacement, the signal energy fractures into multiple spatial cells. This phenomenon, known as subspace leakage, distorts the orthogonality between the replica vectors and the noise subspace. Consequently, the MUSIC ambiguity surface exhibits a highly multi-modal energy distribution, where the global maximum erroneously jumps to a prominent false sidelobe. As anticipated, the RTM algorithm yields a spatial spectrum virtually identical to MUSIC. Remarkably, the RCM technique restores operational adaptability. By explicitly annihilating the uncorrelated sensor self-noise, RCM purifies the signal subspace, rebalancing the energy distribution and pulling the estimated peak back toward the true target location.

4.2.2. Analysis of Error Metrics

Figure 8 presents the time-varying localization results for the maneuvering source over five discrete temporal frames, specifically focusing on the short-term segment. As illustrated in Figure 8a, all five processors capture the general receding (outward-moving) trend of the source. However, visible fluctuations are observed in the high-resolution methods. Notably, the MUSIC and RTM processors exhibit a measurable range deviation, which can be attributed to the instability of the signal subspace estimation under snapshot-deficient conditions during source maneuvering. In contrast, the MVDR method demonstrates greater stability in range tracking, achieving a closer alignment with the true trajectory.

Regarding depth estimation, as shown in Figure 8b, the target’s nominal depth remains constant at 70 m. However, the estimated depths across the five frames display varying degrees of variation. The energy-based Bartlett processor consistently exhibits larger depth offsets. Meanwhile, while the subspace methods (MUSIC and RTM) match the true depth at certain snapshots, they exhibit trajectory deviations at other frames. This phenomenon underscores the performance dichotomy mentioned previously: while subspace-based methods offer higher resolution in static cases, their sensitivity to the time-varying CSDM in maneuvering scenarios leads to reduced tracking robustness.

The operational reliability of these algorithms under kinematic constraints is quantitatively revealed by the tracking errors at discrete processing frames and overall statistical metrics, as shown in Figure 9. It should be noted that the horizontal axis in Figure 9a,b represents the specific time instances of the five discrete processing frames across the observation period. The numerical error metrics are further summarized in

Table 3. Quantitative comparison of range localization performance metrics for the maneuvering source.

Method	RMSE (m)	MAE (m)	MaxAE (m)	MAPE (%)
Bartlett	140.96	140.25	163.75	2.35
MVDR	88.92	85.25	128.75	1.43
MUSIC	110.55	103.25	178.75	1.73
RCM	133.30	132.25	163.75	2.22
RTM	110.55	103.25	178.75	1.73

and

Table 4. Quantitative comparison of depth localization performance metrics for the maneuvering source.

Method	RMSE (m)	MAE (m)	MaxAE (m)	MAPE (%)
Bartlett	6.79	6.70	8.00	9.57
MVDR	2.77	2.70	3.50	3.86
MUSIC	4.54	3.60	8.00	5.14
RCM	5.84	5.50	8.00	7.86
RTM	4.54	3.60	8.00	5.14

.

The statistical data in Table 3 and Table 4 are consistent with the visual trajectory observations. For range localization, the MVDR processor yields the lowest error values among the evaluated methods, producing an RMSE of 88.92 m and a MaxAE of 128.75 m. In contrast, standard subspace methods MUSIC and RTM show higher peak errors (MaxAE = 178.75 m). The RCM technique, despite yielding a higher range RMSE (133.30 m) than MUSIC under these specific snapshot conditions, limits the MaxAE to 163.75 m, thereby mitigating the maximum trajectory deviation observed in standard MUSIC.

For depth localization in Table 4, the MVDR processor maintains a lower error profile, with a depth RMSE of 2.77 m and a maximum absolute error of 3.50 m. The conventional Bartlett processor exhibits an RMSE of 6.79 m in depth tracking. Furthermore, the subspace-based processors (MUSIC, RCM, and RTM) all record maximum peak errors of 8.00 m (MaxAE = 8.00 m), reflecting their sensitivity to the snapshot deficiency induced by the receding target kinematics.

These comparative metrics illustrate an operational characteristic of the evaluated algorithms: while subspace methods provide higher spatial resolution in stationary environments, MVDR exhibit more stable tracking continuity when evaluating fast-maneuvering targets under snapshot-deficient conditions.

4.3. Statistical Significance Analysis

To rigorously ascertain whether the numerical discrepancies observed among the five processors are statistically meaningful or merely a product of natural data variability, a comprehensive non-parametric statistical evaluation was conducted. Because the error metrics evaluated across the discrete temporal frames constitute repeated measures, a global Friedman test was performed on the absolute error series. Subsequently, a post-hoc multiple comparison test (Tukey–Kramer method) was systematically applied to evaluate all 10 possible pairwise combinations among the algorithms. The statistical confidence intervals and mean column ranks are visualized in Figure 10 for the stationary target and Figure 11 for the maneuvering target, where overlapping horizontal lines indicate a lack of statistical significance ($p\ge 0.05$).

For the stationary source scenario evaluated under severe environmental mismatch, the global Friedman tests yielded extremely significant variance across the five processors ($p=7.33\times {10}^{-12}$ for range and $p=5.13\times {10}^{-12}$ for depth). As visually corroborated by Figure 10a,b, the subsequent post-hoc pairwise comparisons revealed three critical statistical insights:

There is no statistically significant difference between Bartlett and MVDR ($p=0.999$ for range; $p=0.612$ for depth), formally proving that energy-based methods share statistically equivalent baseline robustness against systemic environmental uncertainties.

The pairwise comparisons between standard MUSIC and the RTM processor yielded identically non-significant results ($p=1.000$ for both range and depth). This empirically verifies the linear algebraic mechanism discussed in Section 4.2 that uniform trace minimization mathematically degenerates to standard subspace projection under the implemented assumptions.

Most importantly, statistically significant performance degradation was confirmed when comparing standard subspace methods against baseline processors. For instance, the differences between Bartlett and MUSIC are highly significant ($p=1.26\times {10}^{-6}$ for range; $p=7.18\times {10}^{-7}$ for depth). Conversely, applying the RCM technique successfully mitigated this degradation, resulting in a statistically significant error reduction compared to MUSIC ($p=5.26\times {10}^{-3}$ for range; $p=1.98\times {10}^{-6}$ for depth).

These robust statistical findings systematically validate that in static, mismatch-dominated environments, the degradation of standard subspace methods, the comparative resilience of Bartlett and MVDR, and the decoupling efficacy of RCM are not artifacts of random sample fluctuations, but fundamental, statistically significant algorithmic characteristics.

Conversely, for the maneuvering target scenario evaluated under snapshot-deficient dynamic conditions, the statistical evaluation yields a completely different trend compared to the stationary case. As depicted by the overlapping confidence intervals in Figure 11a,b, the global Friedman tests yielded $p=0.106$ for range and $p=0.190$ for depth. Correspondingly, all post-hoc pairwise comparisons across the five methods resulted in $p>0.05$ (Not Significant). This indicates that, mathematically, there is no statistically significant difference in performance among the five algorithms within the confines of this specific dynamic evaluation.

This lack of statistical significance corroborates the methodological limitation regarding sample size. It is fundamentally attributed to the highly restricted number of samples ($N=5$ temporal evaluation frames) available in the intermittent dynamic tracking sequence. This limited sample size provides insufficient statistical power (sensitivity) to overcome the stringent $p<0.05$ confidence threshold. Therefore, although adaptive energy-based processors (MVDR) and CMR techniques (RCM) exhibited measurable numerical reductions in overall errors and peak tracking deviations (MaxAE) under snapshot deficiency, these numerical variations currently remain mathematically within the inherent margin of natural data variability.

In summary, the comprehensive statistical evaluation solidifies the operational boundaries of these algorithms. In static environments with adequate temporal samples, the degradation of standard subspace methods and the decoupling efficacy of RCM are proven to be fundamental, statistically significant algorithmic characteristics. However, under highly dynamic conditions with strictly constrained sample sizes, the observed tracking improvements serve as empirical engineering observations rather than strict statistical certainties. This formally dictates that all practical recommendations formulated in this study must be cautiously interpreted within these specific snapshot limitations, emphasizing the critical need for future cross-validation across larger-scale, highly dynamic experimental databases.

5. Conclusions

The quantitative conclusions drawn herein are inherently restricted to the specific hydroacoustic conditions, receiving system configurations, and frequency bands of the Elba-93 empirical dataset. In diverse practical surveillance operations, absolute localization accuracy will inevitably fluctuate. Specifically, lower signal-to-noise ratio (SNR) levels will further diminish subspace orthogonality, while different sound speed profiles (SSPs) and seabed compositions will significantly alter the severity of the environmental mismatch. Second, due to the highly challenging intermittently-emitting nature of the maneuvering source in this trial, the dynamic tracking evaluation was constrained to limited temporal segments, which impedes a full-sequence continuous trajectory assessment with high statistical power. It must be noted that higher maneuvering speeds inherently dictate shorter integration times, which will directly exacerbate the snapshot deficiency phenomenon analyzed in this study.

Nevertheless, while the absolute error metrics are site-specific, the relative operational boundaries, statistical characteristics, and computational efficiencies revealed are fundamentally governed by the mathematical formulations of the processors. Based on rigorous statistical significance testing and computational complexity analysis, practical recommendations can be carefully formulated. For underwater monitoring systems operating under severe environmental uncertainties with stationary or slow-moving targets, the Bartlett and MVDR processors are suggested as highly practical baseline choices due to their statistically proven resilience to environmental mismatch and low computational overhead. Conversely, for tracking fast-maneuvering targets where integration times must be strictly reduced, standard subspace methods suffer from severe subspace leakage and trajectory jumps. In such snapshot-deficient dynamic tracking operations, the adaptive MVDR processor empirically demonstrates the most stable tracking continuity and yields the lowest peak errors. If subspace-based high-resolution capabilities are strictly required by system design, deploying CMR-enhanced methods like RCM serves as an effective approach; it actively suppresses unstructured noise to mitigate severe target loss while maintaining a manageable computational complexity suitable for real-time deployment.

Future research efforts will pivot towards addressing these current limitations. The primary focus will be validating these algorithmic boundaries across diverse, large-scale experimental databases encompassing varied SSPs, extreme SNR conditions, and continuous high-speed kinematic trajectories to establish robust temporal stability indicators. Furthermore, future work will explore data-driven paradigms, specifically employing deep learning frameworks capable of learning complex non-linear waveguide mappings, aiming to fundamentally overcome environmental mismatch and elevate tracking precision in dynamic marine environments.