Numerical solutions to the parabolic wave equation are plagued by the curse of dimensionality coupled with the Nyquist criterion. As a remedy, a new range-dynamical low-rank split-step Fourier methodology is developed. Our integration scheme scales sub-linearly with the number of classical degrees of freedom in the transverse directions. It is orders of magnitude faster than the classic full-rank split-step Fourier algorithm and also saves copious amounts of storage space. This enables numerical solutions of the parabolic wave equation at higher frequencies and on larger domains, and simulations may be performed on laptops rather than high-performance computing clusters. By using a rank-adaptive scheme to further optimize the low-rank equations, we ensure our approximate solution is highly accurate and efficient. The methodology and algorithms are demonstrated on realistic high-resolution data-assimilative ocean fields in Massachusetts Bay for three-dimensional acoustic configurations with different source locations and frequencies. The acoustic pressure, transmission loss, and phase solutions are analyzed in geometries with seamounts and canyons across and along Stellwagen Bank. The convergence with the rank of the subspace and the properties of the rank-adaptive scheme are demonstrated, and all results are successfully compared with those of the full-rank method when feasible.

The stochastic dynamically orthogonal (DO) narrow-angle parabolic equations (NAPEs) are exemplified and their properties and capabilities are described using three new 2D stochastic range-independent and range-dependent test cases with uncertain sound speed field, bathymetry, and source location. We validate results against ground-truth deterministic analytical solutions and direct Monte Carlo predictions of acoustic pressure and transmission loss fields. We verify the stochastic convergence and computational advantages of the DO-NAPEs and discuss the differences with normal mode approaches. Results show that a single DO-NAPE simulation can accurately predict stochastic range-dependent acoustic fields and their non-Gaussian probability distributions, with computational savings of several orders of magnitude when compared to direct Monte Carlo methods. With their coupling properties and their adaptation in range to the dominant uncertainties, the DO-NAPEs are shown to predict accurate statistics, from mean and variance to multiple modes and full probability distributions, and to provide excellent reconstructed realizations, from amplitudes and phases to other specific properties of complex realization fields.

Robust informative acoustic predictions require precise knowledge of ocean physics, bathymetry, seabed, and acoustic parameters. However, in realistic applications, this information is uncertain due to sparse and heterogeneous measurements and complex ocean physics. Efficient techniques are thus needed to quantify these uncertainties and predict the stochastic acoustic wave fields. In this work, we derive and implement new stochastic differential equations that predict the acoustic pressure fields and their probability distributions. We start from the stochastic acoustic parabolic equation (PE) and employ the instantaneously-optimal Dynamically Orthogonal (DO) equations theory. We derive stochastic DO-PEs that dynamically reduce and march the dominant multi-dimensional uncertainties respecting the nonlinear governing equations and non-Gaussian statistics. We develop the dynamical reduced-order DO-PEs theory for the Narrow-Angle PE (NAPE) and implement numerical schemes for discretizing and integrating the stochastic acoustic fields.

We develop two new sets of stable, rank-adaptive Dynamically Orthogonal Runge-Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while preserving mode continuity using newly derived retractions. We show that arbitrarily high-order optimal perturbative retractions can be obtained, and we prove that these new retractions are stable. In addition, we demonstrate that repeatedly applying retractions yields a gradient-descent algorithm on the low-rank manifold that converges geometrically when approximating a low-rank matrix. When approximating a higher-rank matrix, iterations converge linearly to the best low-rank approximation. We then develop a rank-adaptive retraction that is robust to overapproximation. Building off of these retractions, we derive two novel, rank-adaptive integration schemes that dynamically update the subspace upon which the system dynamics is projected within each time-step: the stable, optimal Dynamically Orthogonal Runge-Kutta (so-DORK) and gradient-descent Dynamically Orthogonal Runge-Kutta (gd-DORK) schemes. These integration schemes are numerically evaluated and compared on an ill-conditioned matrix differential equation, an advection-diffusion partial differential equation, and a nonlinear, stochastic reaction-diffusion partial differential equation. Results show a reduced error accumulation rate with the new stable, optimal and gradient-descent integrators. In addition, we find that rank adaptation allows for highly accurate solutions while preserving computational efficiency.

In recent years, the Gulf of Mexico Loop Current System has received increased attention. Its dynamics and the warm water it transports from the Caribbean influence the local weather and ecosystems. The high velocities of the Loop Current and the eddies it sheds can disrupt important industries. Accurate forecasting of the Loop Current system is challenging, in part because of the lack of data over long enough periods of time, which leads to considerable uncertainty. In this work, we describe and apply our MIT Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS) and Error Subspace Statistical Estimation (ESSE) ensemble forecasting methodology and software to estimate such uncertainty and to inform data collection in a quantitative manner. The ensemble forecasts allow for mitigating risks and optimizing data collection. We demonstrate that our probabilistic system has qualitative skill for over a month. We show that uncertainty grows along and around the Loop Current and its eddies, and transfers to depth from the shelf and slope. Using information theory, we find that our probabilistic hindcasts can have predictive capabilities for one to three months, with a slower loss of predictability in the quieter Loop Current states. Through the use of correlation and mutual information fields, we optimize future sampling by predicting the impacts and information content of observations. We find that the most informative data are those that either directly sample dynamically relevant areas or sample coastal modes that are correlated with these areas. Subsurface data are shown to have more impact on forecasts of one month or longer.

Improving the predictive capability and computational cost of dynamical models is often at the heart of augmenting computational physics with machine learning (ML). However, most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters. In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. Our flexible modeling framework provides full autonomy for the design of the unknown closure terms such as using any linear-, shallow-, or deep-NN architectures, selecting the span of the input function libraries, and using either or both Markovian and non-Markovian closure terms, all in accord with prior knowledge. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. Our learned gnCMs discover missing physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models. Finally, we analyze the computational advantages of our new framework.

Whether due to the sheer size of a computational domain, the fine resolution required, or the multiples scales and stochasticity of the dynamics, the dimensionality of a system must often be reduced so that problems of interest become computationally tractable. In this paper, we develop retractions for time-integration schemes that efficiently and accurately evolve the dynamics of a system’s low-rank approximation. Through differential geometry, we analyze the error incurred at each time-step due to the high-order curvature of the manifold of fixed-rank matrices. We first obtain a novel, explicit, computationally inexpensive set of algorithms that we refer to as perturbative retractions and show that the set converges to an ideal retraction that projects optimally and exactly to the manifold of fixed-rank matrices by reducing what we define as the projection-retraction error. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. Using perturbative retractions, we then develop a new class of integration techniques that we refer to as dynamically orthogonal Runge–Kutta (DORK) schemes. DORK schemes integrate along the nonlinear manifold, updating the subspace upon which we project the system’s dynamics as it is integrated. Through numerical test cases, we demonstrate our schemes for matrix addition, real-time data compression, and deterministic and stochastic partial differential equations. We find that DORK schemes are highly accurate by incorporating knowledge of the dynamic, nonlinear manifold’s high-order curvature, and they are computationally efficient by limiting the growing rank needed to represent the evolving dynamics.

Onboard forecasting is challenging but essential for unmanned autonomous ocean platforms. Due to the numerous operational constraints of these platforms, efficient adaptive Reduced-Order Models (ROMs) are needed. In this work, we employ the incremental Low-Rank Dynamic Mode Decomposition (iLRDMD), which is an adaptive, data-driven, DMD-based ROM that enables efficient forecast compression, transmission, and onboard forecasting. We demonstrate the algorithm on 3D multivariate Hybrid Coordinate Ocean Model (HYCOM) ocean fields in the Middle Atlantic Ridge (MAR) region. We further demonstrate that these iLRDMD ocean forecasts can be used for interdisciplinary applications such as underwater acoustics predictions. Here, acoustics fields computed from the ocean iLRDMD forecasts are compared to those computed from HYCOM fields. We also illustrate the application of a joint ocean-acoustics iLRDMD model for predetermined acoustics configurations. In the MAR region, we find that iLRDMD models are sufficiently accurate and efficient for onboard ocean and acoustic forecasting of temperature, salinity, velocity, and transmission loss fields.

Onboard probabilistic forecasting and data assimilation is challenging for unmanned autonomous platforms. Due to the operational constraints, efficient adaptive reduced order models (ROMs) are needed. To extend the duration for which Dynamic Mode Decomposition (DMD) predictions are accurate, we utilize and augment incremental methods that update the reduced order state but also adapt the DMD. Our adaptive ROM methods are dynamic and stochastic. They update the state, parameters, and basis functions, in response to the changing forecasts, possibly computed in remote centers, and to observations made by the autonomous platforms and by other assets. For the latter, to allow learning even when observations are sparse and multivariate, we employ Bayesian data assimilation. Specifically, we extend the Gaussian Mixture Model – Dynamically Orthogonal (GMM-DO) filter to stochastic DMD forecasts and Bayesian GMM updates of the DMD coefficients, state, and parameters, learning from the limited gappy observation data sets.

Accurate and computationally efficient acoustic models are needed for varied marine applications. In this paper, we focus our attention on forward models, which are essential to inverse problems such as imaging and mapping. First, we introduce new dynamically orthogonal (DO) equations for the acoustic wave equation in full generality, allowing for stochastic and spatially heterogeneous parameters. These equations may be spatially discretized and integrated in time numerically. Alternatively, the DO equations may be discretized themselves, admitting a non-intrusive reduced-order approach to solve the stochastic wave equation. We demonstrate the latter with a test case of an acoustic pulse traveling through the ocean with an uncertain sound speed. Second, we adapt the spatially discrete DO approach, typically used to reduce the stochastic dimension, to efficient reduced-order modeling of deterministic 3D acoustic propagation. We solve the 3D parabolic wave equation and show that low-rank solutions rapidly converge to the full-rank solution. Together, these approaches offer novel ways to solve stochastic and deterministic problems with strong or weak scattering at a reduced computational cost.

We address the problem of finding the closest matrix Ũ to a given U under the constraint that a prescribed second-moment matrix P̃ must be matched, i.e. Ũ^TŨ=P̃. We obtain a closed-form formula for the unique global optimizer Ũ for the full-rank case, that is related to U by an SPD (symmetric positive definite) linear transform. This result is generalized to rank-deficient cases as well as to infinite dimensions. We highlight the geometric intuition behind the theory and study the problem’s rich connections to minimum congruence transform, generalized polar decomposition, optimal transport, and rank-deficient data assimilation. In the special case of P̃=I, minimum-correction second-moment matching reduces to the well-studied optimal orthonormalization problem. We investigate the general strategies for numerically computing the optimizer and analyze existing polar decomposition and matrix square root algorithms. We modify and stabilize two Newton iterations previously deemed unstable for computing the matrix square root, such that they can now be used to efficiently compute both the orthogonal polar factor and the SPD square root. We then verify the higher performance of the various new algorithms using benchmark cases with randomly generated matrices. Lastly, we complete two applications for the stochastic Lorenz-96 dynamical system in a chaotic regime. In reduced subspace tracking using dynamically orthogonal equations, we maintain the numerical orthonormality and continuity of time-varying base vectors. In ensemble square root filtering for data assimilation, the prior samples are transformed into posterior ones by matching the covariance given by the Kalman update while also minimizing the corrections to the prior samples.

The widely-used Global Positioning System (GPS) does not work underwater. This presents a severe limitation on the communication capabilities and deployment options for undersea assets such as AUVs and UUVs. To address this challenge, the Positioning System for Deep Ocean Navigation (POSYDON) program aims to develop an undersea system that provides omnipresent, robust positioning across ocean basins. To do so, it is critically important to accurately model sound waves and signals under diverse, and often uncertain, undersea environmental conditions. Probabilistic estimates of the four-dimensional variability of the fields of sound speed, salinity, temperature, and currents are thus needed. In this paper, we employ our MSEAS primitive-equation and error subspace data-assimilative ensemble ocean forecasting system during two real-time POSYDON sea exercises, one in winter 2017 and another in August 2018. We provide real-time high-resolution estimates of sound speed fields and their uncertainty, and describe the ocean conditions from submesoscales eddies and internal tides to warm core rings and larger-scale circulations. We verify our results against independent data of opportunity; in all cases, we show that our probabilistic forecasts demonstrate skill.

The reliability of sonar systems in the littoral environment is greatly affected by the variability of the surrounding nonlinear ocean dynamics. This variability occurs on multiple scales in space and time, and involves multiple interacting processes, from internal tides and waves to meandering fronts, eddies, boundary layers, and strong air-sea interactions. We utilize our high-resolution MSEAS-PE ocean modeling system to hindcast the ocean physical environment off the New Jersey continental shelf for the end of June 2009, and then utilize our new MSEAS probabilistic acoustic NAPE and WAPE solvers in a coupled ocean physics-acoustic modeling fashion to predict the transmission and integrated transmission losses, respectively. The coupled models are described, and their predictions verified against independent ocean physics observations and sound propagation measurements from acoustic sources and receivers in the region. Our high-resolution ocean simulations are shown to substantial reduce the RMSE and bias of the coarser simulations. Our acoustic simulations of deterministic and stochastic TL fields also show significant skill.

We describe and investigate several Dynamic Mode Decomposition (DMD) and reduced order projection methods for regional stochastic ocean predictions. We then showcase some of their results as applied to a 300-member set of ensemble forecasts from the POSYDON-POINT sea experiment in the Middle Atlantic–New York Bight region for the period 23–27 August 2018 as well as to a 42-day data-driven reanalysis from the AWACS–SW06 sea experiment in the Middle Atlantic Bight region for the period 14 August to 24 September 2006. We discuss these results for use by autonomous platforms in uncertain scenarios as well the combination of DMD with ideas from large-ensemble forecasting and Dynamically-Orthogonal (DO) differential equations.

Covering the vast majority of our planet, the ocean is still largely unmapped and unexplored. Various imaging techniques researched and developed over the past decades, ranging from echo-sounders on ships to LIDAR systems in the air, have only systematically mapped a small fraction of the seafloor at medium resolution. This, in turn, has spurred recent ambitious efforts to map the remaining ocean at high resolution. New approaches are needed since existing systems are neither cost nor time effective. One such approach consists of a sparse aperture mapping technique using autonomous surface vehicles to allow for efficient imaging of wide areas of the ocean floor. Central to the operation of this approach is the need for robust, accurate, and efficient inference methods that effectively provide reliable estimates of the seafloor profile from the measured data. In this work, we utilize such a stochastic prediction and Bayesian inversion and demonstrate results on benchmark problems. We first outline efficient schemes for deterministic and stochastic acoustic modeling using the parabolic wave equation and the optimally-reduced Dynamically Orthogonal equations and showcase results on stochastic test cases. We then present our Bayesian inversion schemes and its results for rigorous nonlinear assimilation and joint bathymetry-ocean physics-acoustics inversion.

A generalization of the concepts of extrinsic curvature and Weingarten endomorphism is introduced to study a class of nonlinear maps over embedded matrix manifolds. These (nonlinear) oblique projections, generalize (nonlinear) orthogonal projections, i.e. applications mapping a point to its closest neighbor on a matrix manifold. Examples of such maps include the truncated SVD, the polar decomposition, and functions mapping symmetric and non-symmetric matrices to their linear eigenprojectors. This paper specifically investigates how oblique projections provide their image manifolds with a canonical extrinsic differential structure, over which a generalization of the Weingarten identity is available. By diagonalization of the corresponding Weingarten endomorphism, the manifold principal curvatures are explicitly characterized, which then enables us to (i) derive explicit formulas for the differential of oblique projections and (ii) study the global stability of a governing generic Ordinary Differential Equation (ODE) computing their values. This methodology, exploited for the truncated SVD in (Feppon 2018), is generalized to non-Euclidean settings, and applied to the four other maps mentioned above and their image manifolds: respectively, the Stiefel, the isospectral, the Grassmann manifolds, and the manifold of fixed rank (non-orthogonal) linear projectors. In all cases studied, the oblique projection of a target matrix is surprisingly the unique stable equilibrium point of the above gradient flow. Three numerical applications concerned with ODEs tracking dominant eigenspaces involving possibly multiple eigenvalues finally showcase the results.

We combine decision theory with fundamental stochastic time-optimal path planning to develop partial-differential-equations-based schemes for risk-optimal path planning in uncertain, strong and dynamic flows. The path planning proceeds in three steps: (i) predict the probability distribution of environmental flows, (ii) compute the distribution of exact time-optimal paths for the above flow distribution by solving stochastic dynamically orthogonal level set equations, and (iii) compute the risk of being suboptimal given the uncertain time-optimal path predictions and determine the plan that minimizes the risk. We showcase our theory and schemes by planning risk-optimal paths of unmanned and/or autonomous vehicles in illustrative idealized canonical flow scenarios commonly encountered in the coastal oceans and urban environments. The step-by-step procedure for computing the risk-optimal paths is presented and the key properties of the risk-optimal paths are analyzed.

Ocean data assimilation is increasingly recognized as crucial for the accuracy of the real-time ocean prediction systems. Here, the current status of ocean data assimilation in support of the operational demands of analysis and forecasting is reviewed, focusing on the methods currently adopted in operational prediction systems. Significant challenges associated with the most commonly employed approaches are identified and discussed. Overarching issues faced by ocean data assimilation in general are also addressed, and important future directions in response to scientific advances, evolving and forthcoming ocean observing systems and the needs of stakeholders and downstream applications are presented.

Recent advances in probabilistic forecasting of regional ocean dynamics, and stochastic optimal path planning with massive ensembles motivate principled analysis of their large datasets. Specifically, stochastic time-optimal path planning in strong, dynamic and uncertain ocean flows produces a massive dataset of the stochastic distribution of exact timeoptimal trajectories. To synthesize such big data and draw insights, we apply machine learning and data mining algorithms. Particularly, clustering of the time-optimal trajectories is important to describe their PDFs, identify representative paths, and compute and optimize risk of following these paths. In the present paper, we explore the use of hierarchical clustering algorithms along with a dissimilarity matrix computed from the pairwise discrete Frechet distance between all the optimal trajectories. We apply the algorithms to two datasets of massive ensembles of vehicle trajectories in a stochastic flow past a circular island and stochastic wind driven double gyre flow. These paths are computed by solving our dynamically orthogonal level set equations. Hierarchical clustering is applied to the two datasets, and results are qualitatively and quantitatively analyzed.

Quantifying the uncertainty of Lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities, or equivalently a stochastic transport partial differential equation (PDE) for the ensemble of flow-maps. The Dynamically Orthogonal (DO) decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and Lagrangian transport. Its interpretation as the method that applies instantaneously the truncated SVD on the matrix discretization of the original stochastic PDE is used to obtain new numerical schemes. Fully linear, explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency, accuracy, stability, and direct consistency between the original deterministic and stochastic DO advections and flow-maps. Various strategies are presented for selecting a time-stepping that accounts for the curvature of the fixed rank manifold and the error related to closely singular coefficient matrices. Efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time. Finally, the new schemes are applied to quantify the uncertain Lagrangian motions of a 2D double gyre flow with random frequency and of a stochastic flow past a cylinder.

Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The Dynamically Orthogonal (DO) approximation is the canonical reduced order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed.  The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map.  Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated Singular Value Decomposition (SVD)  of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values.  The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution.  Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix.  The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial data.

Accounting for uncertainty in optimal path planning is essential for many applications. We present and apply stochastic level-set partial differential equations that govern the stochastic time-optimal reachability fronts and time-optimal paths for vehicles navigating in uncertain, strong, and dynamic flow fields. To solve these equations efficiently, we obtain and employ their dynamically orthogonal reduced-order projections, maintaining accuracy while achieving several orders of magnitude in computational speed-up when compared to classic Monte Carlo methods. We utilize the new equations to complete stochastic reachability and time-optimal path planning in three test cases: (i) a canonical stochastic steady-front with uncertain flow strength, (ii) a stochastic barotropic quasi-geostrophic double-gyre circulation, and (iii) a stochastic flow past a circular island. For all the three test cases, we analyze the results with a focus on studying the effect of flow uncertainty on the reachability fronts and time-optimal paths, and their probabilistic properties. With the first test case, we demonstrate the approach and verify the accuracy of our solutions by comparing them with the Monte Carlo solutions.With the second, we show that different flow field realizations can result in paths with high spatial dissimilarity but with similar arrival times. With the third, we provide an example where time-optimal path variability can be very high and sensitive to uncertainty in eddy shedding direction downstream of the island. Keywords: Stochastic Path Planning, Level Set Equations, Dynamically Orthogonal, Ocean Modeling, AUV, Uncertainty Quantification

We provide a new framework for the study of fl‡uid ‡flows presenting complex uncertain behavior. Our approach is based on the stochastic reduction and analysis of the governing equations using the dynamically orthogonal field equations. By numerically solving these equations we evolve in a fully coupled way the mean fl‡ow and the statistical and spatial characteristics of the stochastic fl‡uctuations. This set of equations is formulated for the general case of stochastic boundary conditions and allows for the application of projection methods that reduce considerably the computational cost. We analyze the transformation of energy from stochastic modes to mean dynamics, and vice-versa, by deriving exact expressions that quantify the interaction among different components of the fl‡ow. The developed framework is illustrated through specifi…c fl‡ows in unstable regimes. In particular, we consider the ‡flow behind a disk and the Rayleigh–-Bénard convection, for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for different parameters associated with the considered ‡flows. We reveal the low-dimensionality of the underlying stochastic attractor.

The properties and capabilities of the GMM-DO filter are assessed and exemplified by applications to two dynamical systems: (1) the Double Well Diffusion and (2) Sudden Expansion flows; both of which admit far-from-Gaussian statistics. The former test case, or twin experiment, validates the use of the EM algorithm and Bayesian Information Criterion with Gaussian Mixture Models in a filtering context; the latter further exemplifies its ability to efficiently handle state vectors of non-trivial dimensionality and dynamics with jets and eddies. For each test case, qualitative and quantitative comparisons are made with contemporary filters. The sensitivity to input parameters is illustrated and discussed. Properties of the filter are examined and its estimates are described, including: the equation-based and adaptive prediction of the probability densities; the evolution of the mean field, stochastic subspace modes and stochastic coefficients; the fitting of Gaussian Mixture Models; and, the efficient and analytical Bayesian updates at assimilation times and the corresponding data impacts. The advantages of respecting nonlinear dynamics and preserving non-Gaussian statistics are brought to light. For realistic test cases admitting complex distributions and with sparse or noisy measurements, the GMM-DO filter is shown to fundamentally improve the filtering skill, outperforming simpler schemes invoking the Gaussian parametric distribution.

This work introduces and derives an efficient, data-driven assimilation scheme, focused on a time-dependent stochastic subspace, that respects nonlinear dynamics and captures non-Gaussian statistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical applications but that also rigorously utilizes the governing dynamical equations with information theory and learning theory for efficient Bayesian data assimilation. Building on the foundations of classical filters, the underlying theory and algorithmic implementation of the new filter are developed and derived. The stochastic Dynamically Orthogonal (DO) field equations and their adaptive stochastic subspace are employed to predict prior probabilities for the full dynamical state, effectively approximating the Fokker-Planck equation. At assimilation times, the DO realizations are fit to semiparametric Gaussian mixture models (GMMs) using the Expectation-Maximization algorithm and the Bayesian Information Criterion. Bayes’ Law is then efficiently carried out analytically within the evolving stochastic subspace. The resulting GMM-DO filter is illustrated in a very simple example. Variations of the GMM-DO filter are also provided along with comparisons with related schemes.

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non- homogeneous fluid and ocean flows. The Dynamically Orthogonal (DO) field equations provide an efficient time-dependent adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the orthonormal modes that define a basis for the evolving DO subspace, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with Total Variation Diminishing schemes for the advection terms. Other results specific to the DO equations include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size in- stead of quadratic; (ii) symmetric Total Variation Diminishing-based advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify the correctness of our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.

Following the scientific, technical and field trial initiatives ongoing since the Maritime Rapid Environmental Assessment (MREA) conferences in 2003, 2004 and 2007, the MREA10 conference provided a timely opportunity to review the progress on various aspects of MREA, with a particular emphasis on marine environmental uncertainty management. A key objective of the conference was to review the present state-of-the art in Quantifying, Predicting and Exploiting (QPE) marine environmental uncertainties. The integration of emerging environmental monitoring and modeling techniques into data assimilation streams and their subsequent exploitation at an operational level involves a complex chain of non-linear uncertainty transfers, including human factors. Accordingly the themes for the MREA10 conference were selected to develop a better understanding of uncertainty, from its inception in the properties being measured and instrumentation employed, to its eventual impact in the applications that rely upon environmental information.

Contributions from the scientific community were encouraged on all aspects of environmental uncertainties: their quantification, prediction, understanding and exploitation. Contributions from operational communities, the consumers of environmental information who have to cope with uncertainty, were also encouraged. All temporal and spatial scales were relevant: tactical, operational, and strategic, including uncertainty studies for topics with long-term implications. Manuscripts reporting new technical and theoretical developments in MREA, but acknowledging effects of uncertainties to be accounted for in future research, were also included.

The response was excellent with 87 oral presentations (11 of which were invited keynote speakers) and 24 poster presentations during the conference. A subset of these presentations was submitted to this topical issue and 22 manuscripts have been published by Ocean Dynamics.

We estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations, within the context of dynamically orthogonal (DO) field equations. Transient nonlinear dynamics, irregular data and non-stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation. To efficiently quantify uncertainties in such systems, it is essential to vary the dimensionality of the stochastic subspace with time. An objective here is to provide criteria to do so, working directly with the original equations of the dynamical system under study and its DO representation. We first analyze the scaling of the computational cost of these DO equations with the stochastic dimensionality and show that unlike many other stochastic methods the DO equations do not suffer from the curse of dimensionality. Subsequently, we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i) stability arguments and ii) Bayesian data updates. We then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2D stochastic fluid flows, specifically a double-gyre wind-driven circulation and a lid-driven cavity flow in a basin. In these two applications, we focus on the growth of uncertainty due to internal instabilities in deterministic flows. We consider a range of flow conditions described by varied Reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions.

An important element of present oceanographic research is the assessment and quantification of uncertainty. These studies are challenging in the coastal ocean due to the wide variety of physical processes occurring on a broad range of spatial and temporal scales. In order to assess new methods for quantifying and predicting uncertainty, a joint Taiwan-US field program was undertaken in August/ September 2009 to compare model forecasts of uncertainties in ocean circulation and acoustic propagation, with high-resolution in situ observations. The geographical setting was the continental shelf and slope northeast of Taiwan, where a feature called the “cold dome” frequently forms. Even though it is hypothesized that Kuroshio subsurface intrusions are the water sources for the cold dome, the dome’s dynamics are highly uncertain, involving multiple scales and many interacting ocean features. During the experiment, a combination of near-surface and profiling drifters, broadscale and high-resolution hydrography, mooring arrays, remote sensing, and regional ocean model forecasts of fields and uncertainties were used to assess mean fields and uncertainties in the region. River runoff from Typhoon Morakot, which hit Taiwan August 7-8, 2009, strongly affected shelf stratification. In addition to the river runoff, a cold cyclonic eddy advected into the region north of the Kuroshio, resulting in a cold dome formation event. Uncertainty forecasts were successfully employed to guide the hydrographic sampling plans. Measurements and forecasts also shed light on the evolution of cold dome waters, including the frequency of eddy shedding to the north-northeast, and interactions with the Kuroshio and tides. For the first time in such a complex region, comparisons between uncertainty forecasts and the model skill at measurement locations validated uncertainty forecasts. To complement the real-time model simulations, historical simulations with another model show that large Kuroshio intrusions were associated with low sea surface height anomalies east of Taiwan, suggesting that there may be some degree of predictability for Kuroshio intrusions.

Uncertainty prediction for ocean and climate predictions is essential for multiple applications today. Many-Task Computing can play a significant role in making such predictions feasible. In this manuscript, we focus on ocean uncertainty prediction using the Error Subspace Statistical Estimation (ESSE) approach. In ESSE, uncertainties are represented by an error subspace of variable size. To predict these uncertainties, we perturb an initial state based on the initial error subspace and integrate the corresponding ensemble of initial conditions forward in time, including stochastic forcing during each simulation. The dominant error covariance (generated via SVD of the ensemble) is used for data assimilation. The resulting ocean fields are used as inputs for predictions of underwater sound propagation. ESSE is a classic case of Many Task Computing: It uses dynamic heterogeneous workflows and ESSE ensembles are data intensive applications. We first study the execution characteristics of a distributed ESSE workflow on a medium size dedicated cluster, examine in more detail the I/O patterns exhibited and throughputs achieved by its components as well as the overall ensemble performance seen in practice. We then study the performance/usability challenges of employing Amazon EC2 and the Teragrid to augment our ESSE ensembles and provide better solutions faster.

In this paper, we quantify the dynamical causes and uncertainties of striking differences in acoustic transmission data collected on the shelf and shelfbreak in the northeastern Taiwan region within the context of the 2008 Quantifying, Predicting, and Exploiting Uncertainty (QPE 2008) pilot experiment. To do so, we employ our coupled oceanographic (4-D) and acoustic (Nx2-D) modeling systems with ocean data assimilation and a best-fit depth-dependent geoacoustic model. Predictions are compared to the measured acoustic data, showing skill. Using an ensemble approach, we study the sensitivity of our results to uncertainties in several factors, including geoacoustic parameters, bottom layer thickness, bathymetry, and ocean conditions. We find that the lack of signal received on the shelfbreak is due to a 20-dB increase in transmission loss (TL) caused by bottom trapping of sound energy during up-slope transmissions over the complex and deeper bathymetry. Sensitivity studies on sediment properties show larger but isotropic TL variations on the shelf and smaller but more anisotropic TL variations over the shelfbreak. Sediment sound-speed uncertainties affect the shape of the probability density functions of the TLs more than uncertainties in sediment densities and attenuations. Diverse thicknesses of sediments lead to only limited effects on the TL. The small bathymetric data uncertainty is modeled and also leads to small TL variations. We discover that the initial transport conditions in the Taiwan Strait can affect acoustic transmissions downstream more than 100 km away, especially above the shelfbreak. Simulations also reveal internal tides and we quantify their spatial and temporal effects on the ocean and acoustic fields. One type of predicted waves are semidiurnal shelfbreak internal tides propagating up-slope with wavelengths around 40-80 km, horizontal phase speeds of 0.5-1 m/s, and vertical peak-to-peak displacements of isotherms of 20-60 m. These waves lead to variations of broadband TL estimates over 5-6-km range that are more isotropic and on bearing average larger (up to 5-8-dB amplitudes) on the shelf than on the complex shelfbreak where the TL varies rapidly with bearing angles.

An ensemble data assimilation scheme, Error Subspace Statistical Estimation (ESSE), is utilized to investigate the seasonal ecosystem dynamics of the Lagoon of Venice and provide guidance on the monitoring and management of the Lagoon, combining a rich data set with a physical-biogeochemical numerical estuary-coastal model. Novel stochastic ecosystem modeling components are developed to represent prior uncertainties in the Lagoon dynamics model, measurement model, and boundary forcing by rivers, open-sea inlets, and industrial discharges. The formulation and parameters of these additive and multiplicative stochastic error models are optimized based on data-model forecast misfits. The sensitivity to initial and boundary conditions is quantified and analyzed. Half-decay characteristic times are estimated for key ecosystem variables, and their spatial and temporal variability are studied. General results of our uncertainty analyses are that boundary forcing and internal mixing have a significant control on the Lagoon dynamics and that data assimilation is needed to reduce prior uncertainties. The error models are used in the ESSE scheme for ensemble uncertainty predictions and data assimilation, and an optimal ensemble dimension is estimated. Overall, higher prior uncertainties are predicted in the central and northern regions of the Lagoon. On the basis of the dominant singular vectors of the ESSE ensemble, the two major northern rivers are the biggest sources of dissolved inorganic nitrogen (DIN) uncertainty in the Lagoon. Other boundary sources such as the southern rivers and industrial discharges can dominate uncertainty modes on certain months. For dissolved inorganic phosphorus (DIP) and phytoplankton, dominant modes are also linked to external boundaries, but internal dynamics effects are more significant than those for DIN. Our posterior estimates of the seasonal biogeochemical fields and of their uncertainties in 2001 cover the whole Lagoon. They provide the means to describe the ecosystem and guide local environmental policies. Specifically, our findings and results based on these fields include the temporal and spatial variability of nutrient and plankton gradients in the Lagoon; dynamical connections among ecosystem fields and their variability; strengths, gradients and mechanisms of the plankton blooms in late spring, summer, and fall; reductions of uncertainties by data assimilation and thus a quantification of data impacts and data needs; and, finally, an assessment of the water quality in the Lagoon in light of the local environmental legislation.

During the August-September 2003 Autonomous Ocean Sampling Network-II experiment, the Harvard Ocean Prediction System (HOPS) and Error Subspace Statistical Estimation (ESSE) system were utilized in real-time to forecast physical fields and uncertainties, assimilate various ocean measurements (CTD, AUVs, gliders and SST data), provide suggestions for adaptive sampling, and guide dynamical investigations. The qualitative evaluations of the forecasts showed that many of the surface ocean features were predicted, but that their detailed positions and shapes were less accurate. The root-mean-square errors of the real-time forecasts showed that the forecasts had skill out to two days. Mean one-day forecast temperature RMS error was 0.26^oC less than persistence RMS error. Mean two-day forecast temperature RMS error was 0.13^oC less than persistence RMS error. Mean one- or two-day salinity RMS error was 0.036 PSU less than persistence RMS error. The real-time skill in the surface was found to be greater than the skill at depth. Pattern correlation coefficient comparisons showed, on average, greater skill than the RMS errors. For simulations lasting 10 or more days, uncertainties in the boundaries could lead to errors in the Monterey Bay region.

Following the real-time experiment, a reanalysis was performed in which improvements were made in the selection of model parameters and in the open-boundary conditions. The result of the reanalysis was improved long-term stability of the simulations and improved quantitative skill, especially the skill in the main thermocline (RMS simulation error 1^oC less than persistence RMS error out to five days). This allowed for an improved description of the ocean features. During the experiment there were two-week to 10-day long upwelling events. Two types of upwelling events were observed: one with plumes extending westward at point Ano Nuevo (AN) and Point Sur (PS); the other with a thinner band of upwelled water parallel to the coast and across Monterey Bay. During strong upwelling events the flows in the upper 10-20 m had scales similar to atmospheric scales. During relaxation, kinetic energy becomes available and leads to the development of mesoscale features. At 100-300 m depths, broad northward flows were observed, sometimes with a coastal branch following topographic features. An anticyclone was often observed in the subsurface fields in the mouth of Monterey Bay.

Variabilities in the coastal ocean environment span a wide range of spatial and temporal scales. From an acoustic viewpoint, the limited oceanographic measurements and today’s ocean computational capabilities are not always able to provide oceanic-acoustic predictions in high-resolution and with enough accuracy. Adaptive Rapid Environmental Assessment (AREA) is an adaptive sampling concept being developed in connection with the emergence of Autonomous Ocean Sampling Networks and interdisciplinary ensemble predictions and adaptive sampling via Error Subspace Statistical Estimation (ESSE). By adaptively and optimally deploying in situ sampling resources and assimilating these data into coupled nested ocean and acoustic models, AREA can dramatically improve the estimation of ocean fields that matter for acoustic predictions. These concepts are outlined and a methodology is developed and illustrated based on the Focused Acoustic Forecasting-05 (FAF05) exercise in the northern Tyrrhenian sea. The methodology first couples the data-assimilative environmental and acoustic propagation ensemble modeling. An adaptive sampling plan is then predicted, using the uncertainty of the acoustic predictions as input to an optimization scheme which finds the parameter values of autonomous sampling behaviors that optimally reduce this forecast of the acoustic uncertainty. To compute this reduction, the expected statistics of unknown data to be sampled by different candidate sampling behaviors are assimilated. The predicted-optimal parameter values are then fed to the sampling vehicles. A second adaptation of these parameters is ultimately carried out in the water by the sampling vehicles using onboard routing, in response to the real ocean data that they acquire. The autonomy architecture and algorithms used to implement this methodology are also described. Results from a number of real-time AREA simulations using data collected during the Focused Acoustic Forecasting (FAF05) exercise are presented and discussed for the case of a single Autonomous Underwater Vehicle (AUV). For FAF05, the main AREA-ESSE application was the optimal tracking of the ocean thermocline based on ocean-acoustic ensemble prediction, adaptive sampling plans for vertical Yo-Yo behaviors and subsequent onboard Yo-Yo routing.

In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where the stochasticity `lives’ as well as a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new evolution equations are derived directly from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of the solution. If additional restrictions are assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two cases of two-dimensional viscous fluid flows described by the NavierStokes equations and we compare our results with Monte Carlo simulations.

For efficient progress, model properties and measurement needs can adapt to oceanic events and interactions as they occur. The combination of models and data via data assimilation can also be adaptive. These adaptive concepts are discussed and exemplified within the context of comprehensive real-time ocean observing and prediction systems. Novel adaptive modeling approaches based on simplified maximum likelihood principles are developed and applied to physical and physical-biogeochemical dynamics. In the regional examples shown, they allow the joint calibration of parameter values and model structures. Adaptable components of the Error Subspace Statistical Estimation (ESSE) system are reviewed and illustrated. Results indicate that error estimates, ensemble sizes, error subspace ranks, covariance tapering parameters and stochastic error models can be calibrated by such quantitative adaptation. New adaptive sampling approaches and schemes are outlined. Illustrations suggest that these adaptive schemes can be used in real time with the potential for most efficient sampling.

The problem of how to optimally deploy a suite of sensors to estimate the oceanographic environment is addressed. An optimal way to estimate (nowcast) and predict (forecast) the ocean environment is to assimilate measurements from dynamic and uncertain regions into a dynamical ocean model. In order to determine the sensor deployment strategy that optimally samples the regions of uncertainty, a Genetic Algorithm (GA) approach is presented. The scalar cost function is defined as a weighted combination of a sensor suite’s sampling of the ocean variability, ocean dynamics, transmission loss sensitivity, modeled temperature uncertainty (and others). The benefit of the GA approach is that the user can determine “optimal” via a weighting of constituent cost functions, which can include ocean dynamics, acoustics, cost, time, etc. A numerical example with three gliders, two powered AUVs, and three moorings is presented to illustrate the optimization approach in the complex shelfbreak region south of New England.

Variabilities in the coastal ocean environment span a wide range of spatial and temporal scales. From an acoustic viewpoint, the limited oceanographic measurements and today’s ocean modeling capabilities can’t always provide oceanic-acoustic predictions in sufficient detail and with enough accuracy. Adaptive Rapid Environmental Assessment (AREA) is a new adaptive sampling concept being developed in connection with the emergence of the Autonomous Ocean Sampling Network (AOSN) technology. By adaptively and optimally deploying in-situ measurement resources and assimilating these data in coupled nested ocean and acoustic models, AREA can dramatically improve the ocean estimation that matters for acoustic predictions and so be essential for such predictions. These concepts are outlined and preliminary methods are developed and illustrated based on the Focused Acoustic Forecasting-05 (FAF05) exercise. During FAF05, AREA simulations were run in real-time and engineering tests carried out, within the context of an at-sea experiment with Autonomous Underwater Vehicles (AUV) in the northern Tyrrhenian sea, on the eastern side of the Corsican channel.

Scientific computations for the quantification, estimation and prediction of uncertainties for ocean dynamics are developed and exemplified. Primary characteristics of ocean data, models and uncertainties are reviewed and quantitative data assimilation concepts defined. Challenges involved in realistic data-driven simulations of uncertainties for four-dimensional interdisciplinary ocean processes are emphasized. Equations governing uncertainties in the Bayesian probabilistic sense are summarized. Stochastic forcing formulations are introduced and a new stochastic-deterministic ocean model is presented. The computational methodology and numerical system, Error Subspace Statistical Estimation, that is used for the efficient estimation and prediction of oceanic uncertainties based on these equations is then outlined. Capabilities of the ESSE system are illustrated in three data-assimilative applications: estimation of uncertainties for physical-biogeochemical fields, transfers of ocean physics uncertainties to acoustics, and real-time stochastic ensemble predictions with assimilation of a wide range of data types. Relationships with other modern uncertainty quantification schemes and promising research directions are discussed.

A multitude of physical and biological processes occur in the ocean over a wide range of temporal and spatial scales. Many of these processes are nonlinear and highly variable, and involve interactions across several scales and oceanic disciplines. For example, sound propagation is influenced by physical and biological properties of the water column and by the seabed. From observations and conservation laws, ocean scientists formulate models that aim to explain and predict dynamics of the sea. This formulation is intricate because it is challenging to observe the ocean on a sustained basis and to transform basic laws into generic but usable models. There are imperfections in both data and model estimates. It is important to quantify such uncertainties to understand limitations and identify the research needed to increase accuracies, which will lead to fundamental progress. There are several sources of uncertainties in ocean modeling. First, to simplify models (thereby reducing computational expenses), explicit calculations are only performed on a restricted range of spatial and temporal scales (referred to as the “scale window”) (Nihoul and Djenidi, 1998). Influences of scales outside this window are neglected, parameterized, or provided at boundaries. Such simplifications and scale reductions are a source of error. Second, uncertainties also arise from the limited knowledge of processes within the scale window, which leads to approximate representations or parameterizations. Third, ocean data are required for model initialization and parameter values; however, raw measurements are limited in coverage and accuracy, and they are often processed with the aim of extracting information within a predetermined scale window. Initial conditions and model parameters are thus inexact. Fourth, models of interactions between the ocean and Earth system are approximate and ocean boundary conditions are inexact. For example, effects of uncertain atmospheric fluxes can dominate oceanic uncertainty. Fifth, miscalculations occur due to numerical implementations. All of the above leads to differences between the actual values (unknown) and the measured or modeled values of physical, biological, and geo-acoustical fields and properties.

The observation, computation and study of “Lagrangian Coherent Structures” (LCS) in turbulent geophysical flows have been active areas of research in fluid mechanics for the last 30 years. Growing evidence for the existence of LCSs in geophysical flows (e.g., eddies, oscillating jets, chaotic mixing) and other fluid flows (e.g., separation prole at the surface of an airfoil, entrainment and detrainment by a vortex) generates an increasing interest for the extraction and understanding of these structures as well as their properties. In parallel, realistic ocean modeling with dense data assimilation has developed in the past decades and is now able to provide accurate nowcasts and predictions of ocean flow fields to study coherent structures. Robust numerical methods and sufficiently fast hardware are now available to compute real-time forecasts of oceanographic states and render associated coherent structures. It is therefore natural to expect the direct predictions of LCSs based on these advanced models. The impact of uncertainties on the coherent structures is becoming an increasingly important question for practical applications. The transfer of these uncertainties from the ocean state to the LCSs is an unexplored but intriguing scientific problem. These two questions are the motivation and focus of this presentation. Using the classic formalism of continuous-discrete estimation [1], the spatially discretized dynamics of the ocean state vector x and observations are described by (1a) dx =M(x; t) + d yok (1b) = H(xk; tk) + k where M and H are the model and measurement model operator, respectively. The stochastic forcings d and k are Wiener/Brownian motion processes,   N(0;Q(t)), and white Gaussian sequences, k  N(0;Rk), respectively. In other words, Efd(t)d T (t)g := Q(t) dt. The initial conditions are also uncertain and x(t0) is random with a prior PDF, p(x(t0)), i.e. x(t0) = bx0 + n(0) with n(0) random. Of course, vectors and operators in Eqs. (1a-b) are multivariate which impacts the PDFs: e.g. their moments are also multivariate. The estimation problem at time t consists of combining all available information on x(t), the dynamics and data (Eqs. 1a-b), their prior distributions and the initial conditions p(x(t0)). Defining the set of all observations prior to time t by yt

The uncertainties in the predicted acoustic wavefield associated with the transmission of low- frequency sound from the continental slope, through the shelfbreak front, onto the continental shelf are examined. The locale and sensor geometry being investigated is that of the New England continental shelfbreak with a moored low-frequency sound source on the slope. Our method of investigation employs computational fluid mechanics coupled with computational acoustics. The coupled methodology for uncertainty estimation is that of Error Subspace Statistical Estimation. Specifically, based on observed oceanographic data during the 1996 Shelfbreak Primer Experiment, the Harvard University primitive-equation ocean model is initialized with many realizations of physical fields and then integrated to produce many realizations of a five-day regional forecast of the sound speed field. In doing so, the initial physical realizations are obtained by perturbing the physical initial conditions in statistical accord with a realistic error subspace. The different forecast realizations of the sound speed field are then fed into a Naval Postgraduate School coupled-mode sound propagation model to produce realizations of the predicted acoustic wavefield in a vertical plane across the shelfbreak frontal zone. The combined ocean and acoustic results from this Monte Carlo simulation study provide insights into the relations between the uncertainties in the ocean and acoustic estimates. The modeled uncertainties in the transmission loss estimate and their relations to the error statistics in the ocean estimate are discussed.

The estimation of oceanic environmental and acoustical fields is considered as a single coupled data assimilation problem. The four-dimensional data assimilation methodology employed is Error Subspace Statistical Estimation. Environmental fields and their dominant uncertainties are predicted by an ocean dynamical model and transferred to acoustical fields and uncertainties by an acoustic propagation model. The resulting coupled dominant uncertainties define the error subspace. The available physical and acoustical data are then assimilated into the predicted fields in accord with the error subspace and all data uncertainties. The criterion for data assimilation is presently to correct the predicted fields such that the total error variance in the error subspace is minimized. The approach is exemplified for the New England continental shelfbreak region, using data collected during the 1996 Shelfbreak Primer Experiment. The methodology is discussed, computational issues are outlined and the assimilation of model-simulated acoustical data is carried out. Results are encouraging and provide some insights into the dominant variability and uncertainty properties of acoustical fields.

The efficient interdisciplinary 4D data assimilation with nonlinear models via Error Subspace Statistical Estimation (ESSE) is reviewed and exemplified. ESSE is based on evolving an error subspace, of variable size, that spans and tracks the scales and processes where the dominant errors occur. A specific focus here is the use of ESSE in interdisciplinary smoothing which allows the correction of past estimates based on future data, dynamics and model errors. ESSE is useful for a wide range of purposes which are illustrated by three investigations: (i) smoothing estimation of physical ocean fields in the Eastern Mediterranean, (ii) coupled physical-acoustical data assimilation in the Middle Atlantic Bight shelfbreak, and (iii) coupled physical-biological smoothing and dynamics in Massachusetts Bay.

An interdisciplinary team of scientists is collaborating to enhance the understanding of the uncertainty in the ocean environment, including the sea bottom, and characterize its impact on tactical system performance. To accomplish these goals quantitatively an end-to-end system approach is necessary. The conceptual basis of this approach and the framework of the end-to-end system, including its components, is the subject of this presentation. Specifically, we present a generic approach to characterize variabilities and uncertainties arising from regional scales and processes, construct uncertainty models for a generic sonar system, and transfer uncertainties from the acoustic environment to the sonar and its signal processing. Illustrative examples are presented to highlight recent progress toward the development of the methodology and components of the system.

A data and dynamics driven approach to estimate, decompose, organize and analyze the evolving three-dimensional variability of ocean fields is outlined. Variability refers here to the statistics of the differences between ocean states and a reference state. In general, these statistics evolve in time and space. For a first endeavor, the variability subspace defined by the dominant eigendecomposition of a normalized form of the variability covariance is evolved. A multiscale methodology for its initialization and forecast is outlined. It combines data and primitive equation dynamics within a Monte-Carlo approach. The methodology is applied to part of a multidisciplinary experiment that occurred in Massachusetts Bay in late summer and early fall of 1998. For a 4-day time period, the three-dimensional and multivariate properties of the variability standard deviations and dominant eigenvectors are studied. Two variability patterns are discussed in detail. One relates to a displacement of the Gulf of Maine coastal current offshore from Cape Ann, with the creation of adjacent mesoscale recirculation cells. The other relates to a Bay-wide coastal upwelling mode from Barnstable Harbor to Gloucester in response to strong southerly winds. Snapshots and tendencies of physical fields and trajectories of simulated Lagrangian drifters are employed to diagnose and illustrate the use of the dominant variability covariance. The variability subspace is shown to guide the dynamical analysis of the physical fields. For the stratified conditions, it is found that strong wind events can alter the structures of the buoyancy flow and that circulation features are more variable than previously described, on multiple scales. In several locations, the factors estimated to be important include some or all of the atmospheric and surface pressure forcings, and associated Ekman transports and downwelling/upwelling processes, the Coriolis force, the pressure force, inertia and mixing.

A basis is outlined for the first-guess spatial mapping of three-dimensional multivariate and multiscale geophysical fields and their dominant errors. The a priori error statistics are characterized by covariance matrices and the mapping obtained by solving a minimum-error-variance estimation problem. The size of the problem is reduced efficiently by focusing on the error subspace, here the dominant eigendecomposition of the a priori error covariance. The first estimate of this a priori error subspace is constructed in two parts. For the “observed” portions of the subspace, the covariance of the a priori missing variability is directly specified and eigendecomposed. For the “non-observed” portions, an ensemble of adjustment dynamical integrations is utilized, building the nonobserved covariances in statistical accord with the observed ones. This error subspace construction is exemplified and studied in a Middle Atlantic Bight simulation and in the eastern Mediterranean. Its use allows an accurate, global, multiscale and multivariate, three-dimensional analysis of primitive-equation fields and their errors, in real time. The a posteriori error covariance is computed and indicates complex data-variability influences. The error and variability subspaces obtained can also confirm or reveal the features of dominant variability, such as the Ierapetra Eddy in the Levantine basin.

Considering mesoscale variability in the Strait of Sicily during September 1996, the four-dimensional physical fields and their dominant variability and error covariances are estimated and studied. The methodology applied in real-time combines an intensive data survey and primitive equation dynamics based on the error subspace statistical estimation approach. A sequence of filtering and prediction problems are solved for a period of 10 days, with adaptive learning of the dominant errors. Intercomparisons with optimal interpolation fields, clear sea surface temperature images and available in situ data are utilized for qualitative and quantitative evaluations. The present estimation system is shown to be a comprehensive nonlinear and adaptive assimilation scheme, capable of providing real-time forecasts of ocean fields and associated dominant variability and error covariances. The initialization and evolution of the error subspace is explained. The dominant error eigenvectors, variance and covariance fields are illustrated and their multivariate, multiscale properties described. Five coupled features associated with the dominant variability in the Strait during August-September 1996 emerge from the dominant decomposition of the initial PE variability covariance matrix: the Adventure Bank Vortex, Maltese Channel Crest, Ionian Shelf Break Vortex, Strait of Messina Vortex, and subbasin-scale temperature and salinity fronts of the Ionian slope. From the evolution of the estimated fields and dominant predictability error covariance decompositions, several of the primitive equation processes associated with the variations of these features are revealed, decomposed and studied. In general, the estimation of the evolving dominant decompositions of the multivariate predictability error and variability covariances appears promising for ocean sciences and technology. The practical feedbacks of the present approach which include the determination of data optimals and the refinements of dynamical and measurement models are considered.

Identical twin experiments are utilized to assess and exemplify the capabilities of error subspace statistical estimation (ESSE). The experiments consists of nonlinear, primitive equation-based, idealized Middle Atlantic Bight shelfbreak front simulations. Qualitative and quantitative comparisons with an optimal interpolation (OI) scheme are made. Essential components of ESSE are illustrated. The evolution of the error subspace, in agreement with the initial conditions, dynamics, and data properties, is analyzed. The three-dimensional multivariate minimum variance melding in the error subspace is compared to the OI melding. Several advantages and properties of ESSE are discussed and evaluated. The continuous singular value decomposition of the nonlinearly evolving variations of variability and the possibilities of ESSE for dominant process analysis are illustrated and emphasized.

A rational approach is used to identify efficient schemes for data assimilation in nonlinear ocean-atmosphere models. The conditional mean, a minimum of several cost functionals, is chosen for an optimal estimate. After stating the present goals and describing some of the existing schemes, the constraints and issues particular to ocean-atmosphere data assimilation are emphasized. An approximation to the optimal criterion satisfying the goals and addressing the issues is obtained using heuristic characteristics of geophysical measurements and models. This leads to the notion of an evolving error subspace, of variable size, that spans and tracks the scales and processes where the dominant errors occur. The concept of error subspace statistical estimation (ESSE) is defined. In the present minimum error variance approach, the suboptimal criterion is based on a continued and energetically optimal reduction of the dimension of error covariance matrices. The evolving error subspace is characterized by error singular vectors and values, or in other words, the error principal components and coefficients. Schemes for filtering and smoothing via ESSE are derived. The data-forecast melding minimizes variance in the error subspace. Nonlinear Monte Carlo forecasts integrate the error subspace in time. The smoothing is based on a statistical approximation approach. Comparisons with existing filtering and smoothing procedures are made. The theoretical and practical advantages of ESSE are discussed. The concepts introduced by the subspace approach are as useful as the practical benefits. The formalism forms a theoretical basis for the intercomparison of reduced dimension assimilation methods and for the validation of specific assumptions for tailored applications. The subspace approach is useful for a wide range of purposes, including nonlinear field and error forecasting, predictability and stability studies, objective analyses, data-driven simulations, model improvements, adaptive sampling, and parameter estimation.