Spectral element solution of the depth-dependent kernel functions in wavenumber integration theory of underwater acoustic propagation

Variational calculation of underwater acoustic propagation delays in non-uniform sound-speed fields with field data validation

• Comparative analysis of Finite Difference, Finite Volume, and Spectral Elements methods for full 3D time-domain acoustic wave propagation modelling. • Benchmarks ranging from simplified geometries that enable analytical comparison, to complex heterogeneous domains. • Implementation of a dedicated Finite volume-based acoustic solver in OpenFOAM with absorbing boundaries. • Comparison of omnidirectional and directional sources to analyze directivity effects on the resulting acoustic wave filed. • Best applicability range of each numerical method for near- and far- field acoustic prediction. A comparative study of three numerical methods - Finite Difference (FD), Finite Volume (FV), and Spectral Element Method (SEM) - for modeling underwater acoustic propagation is presented. The time-domain acoustic wave equation is solved using an in-house FD code, the open-source SPECFEM3D software for SEM, and a newly developed FV-based acoustic solver implemented and released within the OpenFOAM framework, extending a software environment traditionally used for computational fluid dynamics to underwater acoustics applications. The methods are systematically assessed through benchmark problems, ranging from homogeneous unbounded and semi-infinite domains to the Pekeris waveguide and a Gaussian canyon. Comparisons with analytical solutions demonstrate that all solvers accurately reproduce monopole and dipole radiation in simplified configurations. However, the analysis reveals that directional sources introduce non-trivial numerical sensitivities, even in simple environments. These effects manifest as spurious reflections and dispersion-related distortions, whose severity depends on the source implementation and the numerical scheme. The results show that SPECFEM3D generally provides the highest accuracy and robustness in heterogeneous and geometrically complex environments, while the in-house FD code and FV-based solver are more sensitive to dispersion but can recover accuracy through increased spatial resolution. Strategies to mitigate source-related artifacts, such as non-reflective hard sources and reduced source regions, are discussed. A preliminary investigation of moving sources highlights their straightforward implementation in FD and FV solvers, while requiring additional care within the SPECFEM3D framework. Overall, this work provides practical guidance on the accuracy, robustness, and applicability of different solvers for simulating underwater noise in near- and far-field conditions, while laying the ground for future source–propagation coupling within acoustic analogy frameworks in OpenFOAM.

Underwater sound field prediction conventionally relies on numerically solving the Helmholtz equation, which is computationally expensive and struggles to meet real-time requirements. This study presents the shallow water Fourier neural operator (SWFNO), an end-to-end model based on the Fourier neural operator, for the rapid and accurate prediction of the amplitude and phase of two-dimensional underwater sound fields in shallow water environments with variable sound speed profiles (SSPs). To enhance the model's generalization capability, we design a hybrid sample generation scheme combining Gaussian random fields with simulated shallow water SSPs, constructing a training dataset that balances randomness and physical plausibility. Simulation results demonstrate that SWFNO achieves high sound field prediction accuracy on both test sets and generalization datasets containing actual ocean SSPs, demonstrating its robustness to out-of-distribution samples. Furthermore, the model exhibits super-resolution prediction capability, enabling high-resolution sound field prediction without retraining, and achieves a speedup of over an order of magnitude compared to the traditional spectral method for large-scale samples. These results indicate that SWFNO is an efficient and accurate surrogate model for underwater acoustic propagation, offering a promising route toward real-time, high-fidelity, and intelligent sound field prediction in complex marine environments.

Ray-based models have been extensively applied in underwater acoustic propagation modeling because of their favorable physical interpretability and engineering practicality. Nevertheless, in complex ocean environments, conventional acoustic propagation models still face several limitations, including low computational efficiency, empirically determined grid settings, and inadequate local refinement capability, which restrict their application in high-accuracy and high-efficiency simulations. To address these limitations, an adaptive receiver-grid construction method for the BELLHOP model is proposed in this study. The method adaptively adjusts receiver-grid spacings by using seafloor bathymetric features and sound-speed-profile gradient characteristics as the primary driving factors. Specifically, local grid refinement is introduced in the receiver-grid region of critical acoustic propagation areas, whereas relatively coarse grids are employed in non-critical regions, thereby improving acoustic-field resolution while reducing the overall computational cost. Simulation results show that the proposed method effectively improves the transmission-loss computation efficiency and spatial resolution of the BELLHOP model in complex ocean environments, thus providing a practical approach for rapid and high-precision underwater acoustic propagation modeling.

This study focuses on how mesoscale eddies at the Kuroshio boundary in the East China Sea modulate underwater acoustic propagation. Using high-resolution reanalysis data from the Hybrid Coordinate Ocean Model (HYCOM) and validated acoustic ray-tracing simulations, the OW + SLA method is employed for eddy identification and classification. Statistical analysis of 120 eddy events from 2015 to 2020 clarifies their seasonal variation characteristics. Warm eddies shift the convergence zone 15–30 km away from the sound source and broaden it by 20–40%, while cold eddies shift it 10–25 km toward the source and narrow it by 15–35%. A linear relationship exists between eddy amplitude and acoustic transmission loss (TL = 72.4 + 0.42 h, R2 = 0.61), where TL is the transmission loss in decibels (dB) and h is the eddy amplitude in meters (m), and there are depth-dependent transmission loss modulation effects. These results provide practical guidance not only for sonar system design and acoustic communication optimization but also for error correction in underwater acoustic navigation systems operating in eddy-prone environments.

Normal mode simulations of underwater acoustic propagation can be computationally intensive, particularly for broadband signals or iterative applications like inversion. An approach using neural network (NN) is introduced to approximate and accelerate these simulations. The NN predicts modal parameters, such as the horizontal wavenumbers and modal depth functions. Modal parameters are predicted and can subsequently be used to compute propagation for arbitrary source-receiver configurations. To address the challenge of dynamic ocean environments or unknown seabeds, the model is trained across different range-independent environments. Training data were generated using the Kraken normal mode code for shallow oceanic waveguides with variable environmental parameters and frequencies within 50-500 Hz. The proposed NN is conditioned on mode and frequency, enabling efficient broadband predictions. Evaluated on environments unseen during training, the NN accurately approximates modal parameters. Once trained, the proposed approach can reduce computation time for modal parameters by an order of magnitude compared to conventional codes, such as Kraken. This efficiency could support demanding applications, like geoacoustic inversion or simulations on computationally constrained platforms.

Abstract The calculation of long-distance underwater acoustic propagation loss in the deep ocean is of significant importance to ocean acoustics research. Traditional numerical methods typically require solving complex partial differential equations, which are time-consuming and computationally expensive. Data-driven methods, which rely on large datasets for modelling and prediction, possess strong nonlinear fitting capabilities. In this study, we propose a data-driven method for predicting acoustic transmission loss in long-distance underwater acoustic propagation in the deep ocean. The method first utilizes a numerical model to generate a large-scale sample dataset, providing a solid data foundation for model training. Subsequently, a deep learning model based on convolutional neural networks is constructed to automatically learn key features of acoustic transmission loss. In the experiments, the model was trained on acoustic transmission loss data under varying source depths, frequencies, and sound speed profile conditions, enabling it to rapidly predict the distribution of long-distance acoustic transmission loss in deep-ocean environments. Compared with traditional numerical methods, the proposed method not only significantly reduces computation time but also achieves comparable accuracy to numerical models.

Automatic classification of marine vessels using radiated acoustic noise is a key requirement for naval operations andunderwater surveillance. However, classifier performance degrades under varying environmental and operationalconditions such as sea state, sound speed profile (SSP), and target range. This study examines the impact of sea state variability, SSP conditions, and target range on the performance andgeneralization capability of convolutional neural networks (CNN) based marine vessel classifiers. Comprehensive mathematical models were developed to simulate marine vessel noise, sea state–induced ambientnoise, and underwater acoustic propagation. Vessel noise was generated for multiple operational speeds, whilepropagation effects were modeled using the Bellhop ray tracing model with Gaussian beam tracing, incorporating SSP(Sound Speed Profile) variability and range-dependent propagation loss. Three CNN models were trained using datasetscorresponding to sea states one, four, and seven, and evaluated across all environmental and operational scenarios. The classification performance varied significantly with environmental and operational conditions. The CNN trained onsea state seven exhibited superior generalization across other sea states. A performance drop of approximately 30% wasobserved when the target range reached 25 km. SSP dependent analysis revealed strong sensitivity in convergence andshadow zones, where correct classification depended critically on precise range information. ROC analysis showed thatmodels trained under rough sea conditions achieved about 20% better generalization compared to those trained on lowersea states. The results demonstrate that environmental and operational variability strongly influences acoustic classificationperformance. Training with data representing rough sea conditions improves model robustness and generalization. Thestudy highlights the necessity of accurately modeled synthetic datasets as a practical alternative to limited and costly real-world acoustic recordings with incomplete environmental characterization.

Abstract Understanding both present and evolving ocean conditions is essential for accurate modeling of underwater acoustic propagation and its applications in science, engineering, and environmental monitoring. The four-dimensional temperature and salinity structure from the surface to seabed critically influences acoustic transmission. Modern ocean modeling systems combine three main components: observations, a numerical forecast model, and a data assimilation system. Of the tens of millions of global ocean observations assimilated daily, most are satellite-based surface measurements, while subsurface water column measurements number only in the thousands. This imbalance underscores the severe undersampling of the ocean interior and calls into question the utility of even high-resolution ocean forecasts for submesoscale acoustic propagation modeling. This paper presents a quantitative, glider measurement–based approach for determining the spatial and temporal resolution requirements for ocean observations. Building on the authors’ earlier work using an ocean model–based framework, this glider-based analysis reinforces prior findings: The resolution needed to constrain ocean models effectively, thereby improving acoustic propagation modeling, can far exceed current observational capabilities, particularly in challenging signal-to-noise environments. This analysis employs glider data extracted from the National Oceanic and Atmospheric Administration (NOAA) World Ocean Database. Significance Statement This study assesses the degree to which regional ocean models, data assimilation systems, and ocean observation data can resolve submesoscale variability at levels needed for accurate underwater acoustic propagation modeling. Applications include prediction of sound transmission in dynamic shelf environments and assessment of environmental variability impacts on acoustic performance. Under certain conditions, submesoscale operations can be shown to rely on accurate ocean predictions resolved to several hours temporally and several kilometers spatially. Ocean observation data, needed to constrain ocean models consistent with these resolution levels, typically fall substantially short of these requirements. A methodology is presented here for the quantitative determination of ocean monitoring resolution requirements leveraging ocean glider water column measurement data.

The normal mode model is widely used to solve underwater acoustic propagation problems in horizontally layered waveguides. However, ocean elastic bottoms are often modeled as fluids or replaced by an equivalent reflection coefficient in most current models; for this reason, the solution of elastic modes and seismic wavefields cannot be solved. To overcome this constraint, a solving method utilizing the Chebyshev collocation method (CCM) for the seismo-acoustic normal mode model within arbitrarily layered inhomogeneous fluid-solid media is proposed. The CCM is applied to discretize the modal equations of compressional wave (P-wave) and shear wave (S-wave), along with the boundary and interface conditions. A complex matrix eigenvalue system is established by restructuring the global discrete modal equation matrix, on which proper interface conditions (fluid-solid and solid-solid) as well as boundary conditions have been enforced. Solving this eigenvalue problem yields the horizontal wavenumbers and the full-depth modal functions; the full-wavefield is then constructed by summing up contributions of each of the modes. Numerical experiments demonstrate that the proposed method accurately computes both the acoustic and seismic wavefields. Moreover, detailed analyses reveal that its accuracy is comparable to that of rpress and significantly surpasses that of kraken and krakenc.

Forecasting underwater acoustic propagation in oceanic frontal areas is a difficult task due to their unstable dynamics. In this work, we propose to fit a Gaussian Process model, with a kernel derived from a structure model, to infer the position of the front from profiler data. Samples from the Gaussian Process can be used to generate sound-speed fields. Parabolic equation simulations on those samples show a good agreement with experimental acoustic data in propagation parallel to and across the front. As it can be intuitively expected, the discrepancy is a bit higher for across-front propagation due to strong range-dependence. However, these discrepancies are statistically due to Gaussian Process samples which proportion do not exceed 10% of the simulated data.

Accurate and energy-efficient localization of autonomous underwater vehicles (AUVs) remains a fundamental challenge due to the complex, bandwidth-limited, and highly dynamic nature of underwater acoustic environments. This paper proposes a fully adaptive deep reinforcement learning (DRL)-driven localization framework for AUVs operating in Underwater Acoustic Sensor Networks (UAWSNs). The localization problem is formulated as a Markov Decision Process (MDP) in which an intelligent agent jointly optimizes beacon selection and transmit power allocation to minimize long-term localization error and energy consumption. A hierarchical learning architecture is developed by integrating four actor–critic algorithms, which are (i) Twin Delayed Deep Deterministic Policy Gradient (TD3), (ii) Soft Actor–Critic (SAC), (iii) Multi-Agent Deep Deterministic Policy Gradient (MADDPG), and (iv) Distributed DDPG (D2DPG), enabling robust learning under non-stationary channels, cooperative multi-AUV scenarios, and large-scale deployments. A round-trip time (RTT)-based geometric localization model incorporating a depth-dependent sound speed gradient is employed to accurately capture realistic underwater acoustic propagation effects. A multi-objective reward function jointly balances localization accuracy, energy efficiency, and ranging reliability through a risk-aware metric. Furthermore, the Cramér–Rao Lower Bound (CRLB) is derived to characterize the theoretical performance limits, and a comprehensive complexity analysis is performed to demonstrate the scalability of the proposed framework. Extensive Monte Carlo simulations show that the proposed DRL-based methods achieve significantly lower localization error, lower energy consumption, faster convergence, and higher overall system utility than classical TD3. These results confirm the effectiveness and robustness of DRL for next-generation adaptive underwater localization systems.

Physics-informed neural networks (PINNs) have recently attracted considerable attention as a framework for solving partial differential equations. Underwater sound-field prediction fundamentally relies on solving acoustic wave equations, making PINNs a natural candidate for this application. This paper reviews recent developments in PINN-based modeling of underwater acoustic propagation, which we group into two main lines of research. The first introduces mathematically motivated simplifications of the governing equations and then employs PINNs as efficient solvers; examples include ray-based PINNs and PINN estimators of modal wavenumbers. The second focuses on improving computational performance by tailoring network architectures and hyperparameters, such as spatial domain-decomposition strategies. While PINNs demonstrate significant potential, challenges persist regarding computational efficiency and convergence in high-frequency regimes. Future research directions are identified, emphasizing a multi-faceted strategy that systematically addresses limitations at both the physical formulation level and the neural network architecture level. By integrating advanced hybrid physics-data modeling and scalable training algorithms, this review highlights the pathway toward bridging the gap between theoretical frameworks and realistic ocean applications.

This study introduces a convolutional neural network based method for rapid prediction of underwater acoustic propagation fields, addressing the high computational cost of traditional methods. By analyzing regional terrain features and constructing a training dataset, the model learns acoustic transmission loss patterns across various terrain conditions. Tests in the Western Pacific demonstrate a root mean square error of 3.48 dB for non-smoothed fields, with an average prediction time of 1.95 ms per batch (10 samples). This method highlights the potential for fast acoustic propagation predictions using simplified inputs, offering a promising direction for real-time applications.

This study investigates underwater acoustic propagation patterns under mesoscale eddy conditions through numerical modeling and parametric analysis. A mathematical model of mesoscale eddies was developed, and acoustic transmission loss was computed using the BELLHOP ray-tracing model. Systematic simulations were conducted to examine the effects of source depth, eddy polarity (cold/warm), eddy intensity, and seabed topography. The results reveal distinct acoustic behaviors: cold-core eddies shift convergence zones forward, reduce their width, elevate their depth, and enhance convergence gain within certain ranges. In contrast, warm-core eddies displace convergence zones backward, broaden their width, and can induce surface duct formation. Furthermore, seabed topography exerts minimal influence on acoustic propagation under cold-core eddies but significantly modulates propagation under warm-core eddies, with different topographies producing markedly distinct effects. These findings provide valuable insights for marine scientific research and engineering applications leveraging mesoscale eddy phenomena.

Semi-analytical solution for three-dimensional underwater acoustic propagation from a directional source

Shipping expansion, offshore energy generation, fish farming, and construction work radiate high levels of underwater noise, which may critically stress marine ecosystems. Tools for simulating, analyzing, and forecasting underwater noise can be of great help in understanding the impact of underwater radiated noise both on the environment and on man-made equipment, such as underwater communication and telemetry systems. To address this challenge, we developed a web-based Marine Noise Analysis Tool (MNAT) that models, simulates, and predicts underwater radiated noise levels. To reproduce realistic shipping conditions, MNAT combines real-time Automatic Identification System data with environmental data using broadly accepted underwater acoustic propagation models, including Bellhop and RAM. Moreover, MNAT can simulate other kinds of noise sources, such as seismic airguns. It features an intuitive interface enabling real-time tracking, noise impact assessment, and interactive visualizations. MNAT’s noise modeling capabilities allow the user to design resilient communication systems in different noise conditions, analyze maritime noise data, and forecast future noise levels, with potential contributions to the design of noise-resilient systems, to the optimization of environmental monitoring device deployments, and to noise mitigation policymaking. MNAT has been made available for the community at a public GIT repository.

Underwater acoustic propagation is influenced by water column properties, seabed topography, and source frequency, with existing numerical models exhibiting varied performance across different conditions. This study evaluates the frequency adaptability of three acoustic models—BELLHOP (geometric ray-based), RAM (parabolic equation), and KRAKEN (coupled mode)—under diverse seabed topographies, including deep-sea-flat (25–2000 Hz), shallow-sea-flat (25–10000 Hz), and gentle/steep-slope seabed (25–800 Hz). Flat seabed scenarios use the Scooter model as a benchmark, while sloping seabed scenarios are compared against analytical solutions. Results indicate that in a 200 m deep flat ocean environment, BELLHOP achieves high accuracy for frequencies above 200 Hz, KRAKEN performs comparably to RAM below 50 Hz, and RAM excels below 200 Hz. In a 4000 m deep flat ocean, RAM outperforms at frequencies below 100 Hz, while BELLHOP performs well above 100 Hz. For sloping seabed environments with slopes less than 6.5°, RAM demonstrates stability below 100 Hz, while BELLHOP performs better above 100 Hz; for slopes greater than 6.5°, RAM remains stable below 50 Hz, with BELLHOP outperforming above 50 Hz. KRAKEN is found unsuitable for sloping seabed simulations. These findings provide quantitative guidance for selecting acoustic models based on frequency and seabed topography.

The deep-sea convergence zone (CZ) is a critical phenomenon for long-range underwater acoustic propagation. Accurate prediction of its distance, width, and gain is essential for enhancing sonar detection performance. However, conventional ray-tracing models, which assume vertically stratified sound speed profiles (SSPs), fail to account for horizontal sound speed gradients in the mixed layer, leading to significant prediction errors. To address this, we propose a novel ray-tracing model that incorporates horizontally inhomogeneous SSPs in the mixed layer. Our approach combines empirical orthogonal function (EOF) decomposition with the Del Grosso sound speed formula to construct a continuous 3D sound speed field. We further derive a modified ray equation including horizontal gradient terms and solve it using a fourth-order Runge–Kutta method. Simulation and experimental validation in the South China Sea demonstrate that our model reduces the prediction error for the first CZ distance by 2.26%, width by 2.66%, and gain deviation by 5.85% compared to the Bellhop model. These results confirm the effectiveness of our method in improving CZ parameter prediction accuracy.