A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme

Retrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean and numerical weather estimation, climate dynamics, and Earth system modeling. However, dealing with the high-dimensionality and nonlinearity of geophysical processes remains a major challenge in the development of Bayesian smoothers. Addressing this issue, a novel subspace smoothing methodology for high-dimensional stochastic fields governed by general nonlinear dynamics is obtained. Building on recent Bayesian filters and classic Kalman smoothers, the fundamental equations and forward-backward algorithms of new Gaussian Mixture Model (GMM) smoothers are derived, for both the full state space and dynamic subspace. For the latter, the stochastic Dynamically Orthogonal (DO) field equations and their time-evolving stochastic subspace are employed to predict the prior subspace probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the dominant stochastic subspace, after fitting semiparametric GMMs to joint subspace realizations. The theoretical properties, varied forms, and computational costs of the new GMM smoother equations are presented and discussed.United States. Office of Naval Research (N00014-09-1- 0676)United States. Office of Naval Research (N00014-14-1- 0476)United States. Office of Naval Research (N00014-13-1-0518)United States. Office of Naval Research ( N00014-14-1-0725

Quantifying, predicting, and exploiting uncertainties in marine environments

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 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 nonlinear 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 were published by Ocean Dynamics. The following section includes an overview of the conference themes and summary of the published manuscripts.United States. Office of Naval Research (grant N00014-08-1-0586 (QPE))United States. Office of Naval Research (grant N00014-08-1-1097 (ONR6.1))United States. Office of Naval Research (grant N00014-08-1-0680 (PLUS-SEAS)

A Coupled-Mode Shallow-Water Model for Tidal Analysis: Internal Tide Reflection and Refraction by the Gulf Stream

A hydrostatic, coupled-mode, shallow-water model (CSW) is described and used to diagnose and simulate tidal dynamics in the greater Mid-Atlantic Bight region. The reduced-physics model incorporates realistic stratification and topography, internal tide forcing from a priori estimates of the surface tide, and advection terms that describe first-order interactions of internal tides with slowly varying mean flow and mean buoyancy fields and their respective shear. The model is validated via comparisons with semianalytic models and nonlinear primitive equation models in several idealized and realistic simulations that include internal tide interactions with topography and mean flows. Then, 24 simulations of internal tide generation and propagation in the greater Mid-Atlantic Bight region are used to diagnose significant internal tide interactions with the Gulf Stream. The simulations indicate that locally generated mode-one internal tides refract and/or reflect at the Gulf Stream. The redirected internal tides often reappear at the shelf break, where their onshore energy fluxes are intermittent (i.e., noncoherent with surface tide) because meanders in the Gulf Stream alter their precise location, phase, and amplitude. These results provide an explanation for anomalous onshore energy fluxes that were previously observed at the New Jersey shelf break and linked to the irregular generation of nonlinear internal waves.National Science Foundation (U.S.) (Grant OCE-1061160 (ShelfIT))National Science Foundation (U.S.) (Grant OCE-1060430)United States. Office of Naval Research (Grants N000 14-11-1-0701 (MURI- IODA))United States. Office of Naval Research (N00014-12-1-0944 (ONR6.2)

Internal-tide interactions with the Gulf Stream and Middle Atlantic Bight shelfbreak front

Internal tides in the Middle Atlantic Bight region are found to be noticeably influenced by the presence of the shelfbreak front and the Gulf Stream, using a combination of observations, equations, and data-driven model simulations. To identify the dominant interactions of these waves with subtidal flows, vertical-mode momentum and energy partial differential equations are derived for small-amplitude waves in a horizontally and vertically sheared mean flow and in a horizontally and vertically variable density field. First, the energy balances are examined in idealized simulations with mode-1 internal tides propagating across and along the Gulf Stream. Next, the fully nonlinear dynamics of regional tide-mean-flow interactions are simulated with a primitive-equation model, which incorporates realistic summer-mesoscale features and atmospheric forcing. The shelfbreak front, which has horizontally variable stratification, decreases topographic internal-tide generation by about 10% and alters the wavelengths and arrival times of locally generated mode-1 internal tides on the shelf and in the abyss. The (sub)mesoscale variability at the front and on the shelf, as well as the summer stratification itself, also alter internal-tide propagation. The Gulf Stream produces anomalous regions of math formula(20 mW m−2) mode-1 internal-tide energy-flux divergence, which are explained by tide-mean-flow terms in the mode-1 energy balance. Advection explains most tide-mean-flow interaction, suggesting that geometric wave theory explains mode-1 reflection and refraction at the Gulf Stream. Geometric theory predicts that offshore-propagating mode-1 internal tides that strike the Gulf Stream at oblique angles (more than thirty degrees from normal) are reflected back to the coastal ocean, preventing their radiation into the central North Atlantic.National Science Foundation (U.S.) (grant OCE-1061160 (ShelfIT))United States. Office of Naval Research (grant N00014-11-1-0701 (MURI-IODA))National Science Foundation (U.S.) (N00014-12-1-0944 (ONR6.2))National Science Foundation (U.S.) (grant N00014-13-1-0518 (Multi-DA)

Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part I. Theory and Scheme

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.United States. Office of Naval Research (Grant N00014-08-1-1097)United States. Office of Naval Research (Grant 00014-09-1-0676)United States. Office of Naval Research (Grant N00014-08-1-0586

Optimizing velocities and transports for complex coastal regions and archipelagos

We derive and apply a methodology for the initialization of velocity and transport fields in complex multiply-connected regions with multiscale dynamics. The result is initial fields that are consistent with observations, complex geometry and dynamics, and that can simulate the evolution of ocean processes without large spurious initial transients. A class of constrained weighted least squares optimizations is defined to best fit first-guess velocities while satisfying the complex bathymetry, coastline and divergence strong constraints. A weak constraint towards the minimum inter-island transports that are in accord with the first-guess velocities provides important velocity corrections in complex archipelagos. In the optimization weights, the minimum distance and vertical area between pairs of coasts are computed using a Fast Marching Method. Additional information on velocity and transports are included as strong or weak constraints. We apply our methodology around the Hawaiian islands of Kauai/Niihau, in the Taiwan/Kuroshio region and in the Philippines Archipelago. Comparisons with other common initialization strategies, among hindcasts from these initial conditions (ICs), and with independent in situ observations show that our optimization corrects transports, satisfies boundary conditions and redirects currents. Differences between the hindcasts from these different ICs are found to grow for at least 2–3 weeks. When compared to independent in situ observations, simulations from our optimized ICs are shown to have the smallest errors

Bayesian Learning of Coupled Biogeochemical-Physical Models

Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, level of complexity needed, and thus in the state fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state fields and parameter values, as well as the joint PDFs of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by PDEs by using state augmentation and the computationally efficient GMM-DO filter. Our innovations include stochastic formulation and complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms. Our new methodology is generalizable, interpretable, and extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. The probabilities of known, uncertain, and unknown model formulations, and of state fields and parameters, are updated jointly using Bayes' law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and model formulations that best explain the data are identified. When observations are sufficiently informative, model complexity and functions are discovered.Comment: 45 pages; 18 figures; 2 table