Ocean acoustic tomography as a data assimilation problem

The estimation of sound speed fields using ocean acoustic tomography (OAT) in operational systems is limited because user intensive post processing and manual association of arrivals with ray paths are required. Furthermore, inversion techniques typically rely on CTD casts or historical data to create reference sound speed profiles and Gaussian statistical characterizations for the inversion, which may be either unavailable, out of date, or too approximate due to the intermittent ocean dynamics. A method is presented for real-time implementation of OAT, including the tracking of acoustic arrivals, model-data association of ray paths to arrivals (ray identification), and tomographic inversions. One- and two-dimensional tomographic inversions for sound speed fields are performed using stochastic ensemble forecasts of the sound speed field generated via the Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS) Primitive-Equation probabilistic ocean modeling system. The method is demonstrated on simulated and real data obtained from moored tomographic sources and receivers, deployed in deep water 100 km South of Nantucket, MA. [This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA). ]

Classification of sound speed fields in the Western Pacific Ocean based on tensor decomposition

Observing and predicting the ocean’s meso‐scale circulation presents a formidable task for oceanographers. Ocean acoustic tomography uses travel times between acoustic sources and receivers to measure the ocean’s low wave‐number sound speed (temperature) and current fields. Experiments are being performed with simulated ocean data to determine the relative effectiveness of tomographic and point measurements in constraining a numerical circulation model. The primary tool is data assimilation, i.e., an optimized combination of model output and observations. Relative weights of observation and model output are based on a priori estimates of the model and observation error covariance. Results from identical twin simulations using tomographic and point measurements are presented. In the twin experiment, two assimilation schemes are examined, one in which the estimated model error covariance is fixed (optimal interpolation) and another in which a Monte Carlo [G. Evensen, J. Geophys. Res. 99, 10 143–10 162 (1994)] method is used to estimate the model error covariance dynamically. Error statistics of the energy and flux fields are used to quantify the performance for each observation and updating scheme. The number of measurements required to achieve a target error level is estimated for each method. [Work supported by ONR and by the Strategic Environmental Research and Development Program through ARPA.]

Wavefront distortion can affect the performance of underwater acoustic communication (a-comms) and imaging systems. A measure of wavefront distortion, the background structure functions (BSFs) are functions of sound speed profiles (SSPs) that are functions of depth. Long term, BSFs can use established commercial software to reduce wavefront distortion and forecast phase statistics for coherent systems. Historically, a suite of path-integral, ray tracing, and modal methods replaced the initial methods for statistical acoustics developed in the latter half of the last century. Underwater (UW) acoustics methods provide the amalgamation of deterministic and statistical expressions to help estimate Doppler and delay for UW acoustics by integrating the classical structure functions from the source to the receiver. In the interim years, non-acoustic communications industries used the older methods with algorithms and hardware that alleviated some of the old shortcomings in order to forecast communication performance. For acoustics, the use of databases of BSFs and SSPs recently helped simulate proof-of-concept forecasting of a-comms operating envelopes. These purely statistical BSFs have less than a quarter of the degrees of freedom (DOFs) in classical UW structure functions. Symmetries also drastically reduce the required background DOFs. For decades industry used these purely statistical structure functions to estimate wavefront wander, lensing (ducting), and fade. Commercial methods to estimate comms fade assume the medium has BSFs that are sufficiently symmetric in order to simplify the calculations. While these metrics can require re-calculation at different stations along the propagation path, extra phase screens are compatible with the evolution of computer technology during the past half century. Conditions of acoustical symmetry and invariance allow simplification of the acoustical calculations in most of the deep-water ocean and in littorals under iso-speed conditions. Results from shallow-sea measurements show that background BSFs in the littorals validate the simplified calculations for sufficiently symmetric sound speed fields. However, the complex calculations that arise from removing the simplifications are tractable with recent algorithms. Adaptive wavefront methods can use BSFs with arrays to guide the wavefront through less lossy paths. Nevertheless, even the value of the ability to forecast phase and throughput statistics in locations where measurements or estimates of sound speed fields exist is of adequate value to provide this foundation for the future use of these purely statistical methods.

Modern Approaches to Data Assimilation in Ocean Modeling

To reconstruct the ocean sound speed field (SSF), a matrix in horizontal could be formed by gridding the sea area, where some sparse entries were filled by the observation sound speed profiles (SSPs). The reconstruction of the SSF can be modeled as a matrix completion problem, using limited and noisy observation values to reconstruct the entire matrix. Furthermore, the spatial correlation of the SSF could be modeled by graph space model, utilizing with the low-rank property of the SSF matrix, the problem of graph regularization low-rank matrix completion was developed. The observed average SSPs in six cross section were estimated from the four underwater acoustic tomography moorings, and nine local SSPs were obtained from pressure inverted echo sounders (PIESs) inversion. By combining the method of deep matrix factorization and the graph regularization, the Graph Regularized Deep Matrix Factorization (GRDMF) method was proposed. The simulation results showed that the GRDMF performs better than the spatial interpolation algorithm in SSF reconstruction. Considering the application scenarios of deep-sea acoustic tomography, an average constraint (AC) item was designed additionally, and then a customized algorithm named GRDMF-AC is proposed. Finally, data of the experiment at the South China Sea in 2021 is processed to verify GRDMF-AC, the results showed that the reconstructed SSF captures mesoscale eddy in the experiment.

The recent prevalence of low cost robotic platforms such as oceanographic gliders has increased the availability of long--term measurements of the ocean environment. Gliders can take direct measurements of the ocean sound speed environment, which is of interest in many ocean acoustic problems, including source localization and tomography. These measurements, however, have a low spatial--temporal resolution that makes them difficult to use directly. These measurements have the potential to provide an accurate environmental parameterization for acoustic inversions, which could in turn be used to measure the sound speed field at a much higher spatial--temporal resolution. This study uses glider measurements to provide the environmental parameterization used in the adjoint inversion method. The adjoint method calculates the gradient of a cost function describing the mismatch between observed data and acoustic model predictions with respect to the ocean sound speed. This gradient is a measure of how changing the sound speed at any point in the acoustic environment would affect this misfit. This cost function and its gradient information is then used as inputs to a numerical optimization routine, which efficiently finds a local minimum. There are two challenges of this method addressed in this study; the first is restricting the search space of this inversion. Proper parameterization of the inversion will ensure that the local minimum found in the numerical optimization routine is the correct result of the inversion. This parameterization allows for the combination of the relative strengths of both methods of measuring the sound speed field, the robust direct measurement of the glider and the near instantaneous result of an acoustic inversion. A covariance matrix is created from glider measurements of the range dependent sound speed field, which is then decomposed into an empirical orthogonal function (EOF) base. The mean profile and the significant EOF bases then form the search space of the adjoint method. The second issue is the proper treatment of the acoustic interaction between the ocean and its sea floor. The adjoint method uses the implicit finite difference form of the Parabolic Equation, which has a few possible bottom treatments. Two simple bottom interface treatments are the local boundary conditions of McDaniel and Lee [J. Acoustic. Soc. Am. 71 , 855-858 (1992)] and the non--local boundary conditions of Papadakis \emph{et al.} [J. Acoustic. Soc. Am. 92 , 2030-2038 (1992)]. Local boundary conditions treat the interface by altering sound speed values at the interface to account for

Underwater sound speed exerts a significant influence on underwater acoustic communication by altering the propagation path of sound waves. Existing sound speed inversions rely on remote sensing data. These methods predominantly reflect variations of the ocean surface, resulting in low accuracy in the main thermocline layer. To address this issue, a spatio-temporal imputation and forecast (STIF) method is proposed, which conceptualizes the inversion of sound speed fields (SSFs) as a sequence forecast problem independent of remote sensing data. First, to overcome the obstacle in three-dimensional data forecast posed by the sparsity of SSFs, the spatio-temporal imputation is applied to reconstruct complete SSFs. It employs the alternating direction method of multipliers (ADMM) to break the reconstruction problem down into a series of denoising sub-problems. Secondly, the convolutional long short-term network (ConvLSTM) is utilized for future SSFs forecast to capture the dynamic features of temporal and spatial variations. Numerical experimental results demonstrate that the proposed STIF method effectively addresses the constraint of inversion imposed by remote sensing data. Meanwhile, STIF exhibits lower inversion errors under conditions of sparse spatial.

Ocean dynamics can result in significant sound speed variability that are not well captured by deterministic numerical models and can impact differently high to low frequency acoustic propagation and bottom interactions by constraining horizontal and vertical paths. Uncertainty in the sound speed can, therefore, cause inaccuracies in acoustic based tactical decision tools and severely impact the accuracy of algorithms using sound propagation to estimate ocean volume states (e.g., ocean tomography or data assimilation). In this work, we propose to use a non-intrusive Reduced Order Modelling solution that consists of building a dictionary of static modes from historical ocean simulations with different settings and resolutions and climatology to derive an expedite uncertainty model along a central deterministic forecast. The system will then, through Orthogonal Matching Pursuit along the dictionary, use the available ocean model-data mismatches and/or acoustic innovations (e.g., arrival numbers and times, acoustic energy distribution, etc.) to define an ensemble of possible sound speed volume distributions and associated acoustic propagation outcomes. We will show results using a simplified simulation experiment that extracts “observations” from a nature reference field to document the procedure and benchmark results to reconstruct 3D sound speed fields from ocean-acoustic measurements.

Ocean acoustic tomography (OAT) technique can obtain oceanographic information and has been received a lot of interest. High frequency OAT (in few kHz range) can be used for small and confined areas such as estuaries and bays with complicated hydrological conditions. In this study, we investigate the application of the high-frequency reciprocal transmission OAT to assess the temperature and current in Xiamen sea area using computer simulations and sea experiments. Based on the temperature data obtained from remote sensing, high frequency OAT is employed to reconstruct two-dimensional temperature, sound speed, and current field of the 1.2 km × 1.2 km region. The results show that increasing the number of acoustic stations decreases the travel-time errors of high frequency OAT; however, excessively increasing stations can not significantly improve the inversion accuracy. Furthermore, this method has also been examined by a sea experiment on monitoring the current and water temperature of Wuyuan Bay. High frequency OAT might provide an effective method on temperature and current observation at coastal estuary areas. (This work was financially supported by the National Science Foundation of China (Grant No. 11174240) and the Fundamental Research Funds for the Central Universities (2011121010))

This work is devoted to the statistical approach to ocean acoustic tomography. Maximum-likelihood (ML) estimate of sound speed field is considered. Deterministic, quasi-deterministic and gaussian signals in the presence of gaussian background noise are examined. A rigorous expression for the functional derivative of the Green function with respect to the sound speed field is applied to find the ML estimate. Fisher information matrices (FIM) giving the Cramer-Rao bounds are obtained for various signal models. The modal, ray and interference tomography methods for global, large scale and small scale inhomogeneities are derived using the statistical approach. FIM of estimate of mode or ray delay are obtained. Some numerical evaluations of potential accuracy of sound-speed field reconstruction in typical propagation conditions are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

Environment-aware underwater acoustic detection and communications demand precise forecasting of the sound speed field (SSF) both temporally and spatially. Toward this goal, recent machine learning models, such as recurrent neural networks and Gaussian process regressions, have outperformed classical autoregressive models. However, from the unified theoretical perspective of conditional distribution learning, there is still significant room for improvement, as existing works have not fully learned the conditional distribution of future SSFs given past SSFs. To address these limitations, in this paper, we leverage the use of diffusion models, the foundation of recent successful deep generative models, such as DALL-E 2 and SORA, to learn the conditional distribution even under limited training data, through careful neural architecture and training strategy design. Our experiments, conducted on real-life South China Sea datasets, confirm that our proposed model outperforms the state-of-the-art baselines in forecasting range-dependent SSFs and the associated underwater transmission losses. Additionally, our model provides reliable confidence intervals that quantify the uncertainties of predictions.

ABSTRACTAn original data assimilation (DA) scheme with a general dynamics model is considered. It is shown that this scheme can be approximated by the stochastic diffusion process. The sufficient conditions to provide this approximation are formulated. Based on this algorithm a new DA method is developed. The method combines variational and statistical approaches commonly used in DA theory and minimizes the variance of the trajectory of a diffusion process in conjunction with a dynamics numerical model. In this sense the method is optimal in contrast to other DA approaches. The proposed scheme takes the model dynamics into account and in this way it differs from the well-known Kalman filter. Furthermore, the derived DA method can be applied to a very wide field of dynamical systems, for example, gas dynamics, fluid dynamics and other disciplines. However, the current study deals with oceanography and DA in oceanography specifically. Then the method is applied to the HYbrid Coordinate Ocean Model and assimilates satellite sea level anomaly data from the Archiving, Validating and Interpolating Satellite Oceanography Data over the Atlantic Ocean to correct the model state. Several numerical experiments have been performed. The experiments show that the method substantially changes the synoptic and mesoscale structure of ocean dynamics. Also, the distribution of the obtained result is estimated through the solution of the Fokker–Planck–Kolmogorov equation.

A latent space method with maximum entropy deep reinforcement learning for data assimilation