loader graphic

Loading content ...

Pursuit-Evasion Games in Dynamic Flow Fields via Reachability Set Analysis

Sun, W., P. Tsiotras, T. Lolla, D. N. Subramani, and P. F. J. Lermusiaux, 2017. Pursuit-Evasion Games in Dynamic Flow Fields via Reachability Set Analysis. 2017 American Control Conference. In press.

In this paper, we adopt a reachability-based approach to deal with the pursuit-evasion differential game between two players in the presence of dynamic environmental disturbances (e.g., winds, sea currents). We give conditions for the game to be terminated in terms of reachable set inclusions. Level set equations are defined and solved to generate the reachable sets of the pursuer and the evader. The corresponding time-optimal trajectories and optimal strategies can be retrieved immediately afterwards. We validate our method by applying it to a pursuit-evasion game in a simple flow field, for which an analytical solution is available.We then implement the proposed scheme to a problem with a more realistic flow field.

Dynamically Orthogonal numerical schemes for efficient stochastic advection and Lagrangian transport.

Feppon, F. and P.F.J. Lermusiaux, 2017. Dynamically Orthogonal numerical schemes for efficient stochastic advection and Lagrangian transport. SIAM Review, sub-judice.

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.

Autonomy for Surface Ship Interception

Mirabito, C., D.N. Subramani, T. Lolla, P.J. Haley, Jr., A. Jain, P.F.J. Lermusiaux, C. Li, D.K.P. Yue, Y. Liu, F.S. Hover, N. Pulsone, J. Edwards, K.E. Railey, G. Shaw, 2017. Autonomy for Surface Ship Interception. In: Oceans '17 MTS/IEEE Aberdeen, 19-22 June 2017, Sub-judice.

The optimal interception of ships sailing on the ocean surface has numerous applications, including search and rescue operations, inspections of ship’s hulls, ship repair and refueling, naval operations and planning, and recovery of underwater platforms. Interest in utilizing autonomous undersea vehicles (AUVs) for these operations has been increasing in recent years. In that case, the optimal recovery of these underwater vehicles by surface ships is also crucial. The time-sensitive nature of these operations render the search for an optimal route from a given point of deployment to a (possibly moving) target of paramount importance. However, numerous factors, including complex coastal geometry, time-varying and complicated currents, and a moving ship wake (further disrupting the local near-ship currents) make this a very challenging problem. Our present research motivation is thus to apply and extend our theory and schemes for optimal path planning of autonomous vehicles operating for long durations in strong and dynamic currents to the optimal interception of surface vessels. The long-term goal is to develop autonomy for AUVs to enable intercept and proximity operations with underway surface vessels, predicting and optimally using dynamic wakes, surface waves, and underwater currents. After extending our time-optimal path planning to the ship interception problem, we study a set of simulated experiments for the Buzzards Bay, Vineyard Sound, and Elizabeth Islands region in Massachusetts. We combine realistic data-assimilative ocean modeling with rigorous time- optimal control and simple ship and wake modeling. To show the versatility of the autonomy approach and also illustrate how it is needed even for the simplest of the cases, we consider several different scenarios: environments with no flow at all but with several straits, cases with time-varying currents, and finally proximity operations considering the effects of ship wakes. We extended our time-optimal path planning to ship interception and illustrated results for varied scenarios in the southern littoral of Massachusetts for varied ship and AUV speeds, start locations, and behaviors, with and without currents, and with and without ship wake effects.

Data-driven Learning and Modeling of AUV Operational Characteristics for Optimal Path Planning

Edwards, J., J. Smith, A. Girard, D. Wickman, D. N. Subramani, C. S. Kulkarni, P.J. Haley, Jr., C. Mirabito, S. Jana, P. F. J. Lermusiaux, 2017. Data-driven Learning and Modeling of AUV Operational Characteristics for Optimal Path Planning. In: Oceans '17 MTS/IEEE Aberdeen, 19-22 June 2017, Sub-judice.

The utilization of Autonomous Underwater Vehicles (AUVs) such as propelled vehicles, gliders, and floats is rapidly growing for a wide range of missions and ocean regions. For optimized utilization, the operational characteristics of the AUVs need to be modeled as accurately as needed by the optimization and specific needs of the ocean missions considered. The advent of machine learning and data sciences provides an opportunity to augment the classic engineering modeling and laboratory analyses by learning the AUV operational characteristics in situ, during and after each sea operations. Such data-driven learning is critical because, from mission to mission, the AUV usage frequently differs, the dynamic ocean environment changes, and the configuration of the AUV itself changes. For the latter, considering propelled vehicles, it is for example very common for fins and buoyancy to be modified, for payloads to be changed, and for the internal content and overall body of the AUVs to be altered. We illustrated the use of in-situ-data-driven learning and modeling of operational characteristics of AUVs for path planning. The operations and learning experiments were conducted in the Buzzards Bay, Vineyard Sound, and Martha Vineyard’s region for several AUV configurations, missions, and ocean conditions. Specifically, we identified and applied simple methods to estimate the relationships between thruster RPM with forward vehicle speed and to confirm that the specific fin configuration affects the net forward speed of the REMUS 600. Such data-based learning should be completed in real-time so as to ensure accurate F(t) models and thus time-optimal performances. These results can be employed for other types of optimal path planning and AUV missions, including energy, sensing, and surveillance optimality.

Time-Optimal Path Planning: Real-Time Sea Exercises

Subramani, D. N., P. F. J. Lermusiaux, P.J. Haley, Jr., C. Mirabito, S. Jana, C. S. Kulkarni, A. Girard, D. Wickman, J. Edwards, J. Smith, 2017. Time-Optimal Path Planning: Real-Time Sea Exercises. In: Oceans '17 MTS/IEEE Aberdeen, 19-22 June 2017, Sub-judice.

Autonomous underwater vehicles (AUVs) are employed in many applications such as ocean sensing, search and rescue operations, acoustic surveillance, and oil and gas exploitation. With advances in AUV capability and increasing mission complexity, there is a demand for predicting all reachable locations, prolonging endurance, and reducing operational costs by optimally utilizing ocean flow forecasts for navigation. For such optimal navigation, we recently developed new theory, schemes, and computational systems for exact partial differential equation-based path planning. This new level-set path planning was applied in realistic re-analysis simulations for the sustained coordinated operation of multiple collaborative AUVs for time-, coordination- and energy- optimal missions. In the present paper, our goal is to demonstrate our level-set path planning in real-time sea exercises with real AUVs in shallow coastal ocean regions with strong and dynamic currents. Our specific objectives are to report the (i) improvements to our 4-D primitive equation ocean modeling system for accurately forecasting the currents in the Buzzard’s Bay and Vineyard Sound region, (ii) results of the time-optimal path planning of REMUS 600 AUVs using our fundamental theory and real-time forecasts, (iii) portability of our software systems for real-time optimal path prediction in multiple regions and its ability to work with the AUV navigation software. These exercises were the first sea tests of our new theory and software. Our ocean forecasts had skill and time-optimal path forecasts worked with REMUS 600’s. We also identified relationships between the REMUS 600’s rpm and nominal in-water speed. The results open a new era of optimal AUV missions.

A Geometric Approach to Dynamical Model–Order Reduction

Feppon, F. and P.F.J. Lermusiaux, 2017. A Geometric Approach to Dynamical Model-Order Reduction. SIAM Journal on Matrix Analysis and Applications, sub-judice.

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. Geodesic equations are derived and extrinsic curvatures are characterized through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit formula for the differential of the truncated Singular Value Decomposition (SVD). A similar analysis applied to the group of orthogonal matrices yields the differential of the polar decomposition. 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. Numerically, the DO approximation is also the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. The geometric analysis is used to provide improved numerical time-integration schemes. Riemannian matrix optimization including gradient and Newton methods allows to adaptively track the best low rank approximation of dynamical matrices.

Energy-Optimal Path Planning in the Coastal Ocean

Subramani, D.N., P.J. Haley Jr. and P.F.J. Lermusiaux, 2017. Energy-optimal Path Planning in the Coastal Ocean. Journal of Geophysical Research: Oceans. Sub-judice.

We integrate data-driven ocean modeling with the stochastic Dynamically Orthogonal (DO) level-set optimization methodology to compute and study energy-optimal paths, speeds, and headings for ocean vehicles in the Middle-Atlantic Bight (MAB) region. We hindcast the energy-optimal paths from among exact time-optimal paths for the period 28 August 2006 to 9 September 2006. To do so, we first obtain a data-assimilative multiscale re-analysis, combining ocean observations with implicit two-way nested multiresolution primitive-equation simulations of the tidal-to-mesoscale dynamics in the region. Second, we solve the reduced-order stochastic DO level-set partial differential equations (PDEs) to compute the joint probability of minimum arrival-time, vehicle-speed time-series, and total energy utilized. Third, for each arrival time, we select the vehiclespeed time-series that minimize the total energy utilization from the marginal probability of vehicle-speed and total energy. The corresponding energy-optimal path and headings are obtained through a particle backtracking equation. Theoretically, the present methodology is PDE-based and provides fundamental energy-optimal predictions without heuristics. Computationally, it is three- to four-orders of magnitude faster than direct Monte Carlo methods. For the missions considered, we analyze the effects of the regional tidal currents, strong wind events, coastal jets, shelfbreak front, and other local circulations on the energy-optimal paths. Results showcase the opportunities for vehicles that intelligently utilize the ocean environment to minimize energy usage, rigorously integrating ocean forecasting with optimal control of autonomous vehicles.

Multiple-Pursuer-One-Evader Pursuit Evasion Game in Dynamic Flow Fields

Sun, W., P. Tsiotras, T. Lolla, D. N. Subramani, and P. F. J. Lermusiaux, 2017. Multiple-Pursuer-One-Evader Pursuit Evasion Game in Dynamic Flow Fields. Journal of Guidance, Control and Dynamics. In press.

In this paper a reachability-based approach is adopted to deal with the pursuit-evasion di erential game between one evader and multiple pursuers in the presence of dynamic environmental disturbances (e.g., winds, sea currents). Conditions for the game to be terminated are given in terms of reachable set inclusions. Level set equations are defi ned and solved to generate the forward reachable sets of the pursuers and the evader. The time-optimal trajectories and the corresponding optimal strategies are sub- sequently retrieved from these level sets. The pursuers are divided into active pursuers, guards, and redundant pursuers according to their respec- tive roles in the pursuit-evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flow fi elds, with and without obstacles.

A Forward Reachability Equation for Minimum-Time Path Planning in Strong Dynamic Flows

Lolla, T. and P.F.J. Lermusiaux, 2017. A Forward Reachability Equation for Minimum-Time Path Planning in Strong Dynamic Flows. SIAM Journal on Control and Optimization, sub-judice.

A theoretical synthesis of forward reachability for minimum–time control of anisotropic vehicles operating in strong and dynamic flows is provided. The synthesis relies on the computation of the forward reachable set of states. Using ideas rooted in the theory of non–smooth calculus, we prove that this set is governed by the viscosity solution of an unsteady Hamilton–Jacobi (HJ) equation. We show that the minimum arrival time satisfies a static HJ equation, when a special local controllability condition holds. Results are exemplified by applications to a sailboat moving in a uniform wind–field and autonomous underwater gliders operating in the Sulu Archipelago.

A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Applications

Lolla, T. and P.F.J. Lermusiaux, 2017b. A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Applications. Monthly Weather Review. doi:10.1175/MWR-D-16-0065.1.

The Gaussian–Mixture–Model Dynamically–Orthogonal (GMM–DO) smoother is exemplified and contrasted with other smoothers by applications to three dynamical systems, all of which admit far–from–Gaussian statistics. A double–well–diffusion experiment is first used to examine the capabilities of the smoother and compare its performance to that of the Ensemble Kalman Smoother. A passive tracer advected by a reversible shear flow is then employed. The exact smoothed solution is obtained and utilized to validate the GMM–DO smoother and its results. Finally, the third example illustrates the applicability of the smoother in more complex ocean flows consisting of variable jets and eddies. To illustrate the non-Gaussian effects, comparisons are then made with the update of the Error Subspace Statistical Estimation smoother. In each application, the properties of the GMM–DO smoother and of its posterior probabilities are studied and quantified. Rigorous evaluation of Bayesian smoothers for nonlinear high-dimensional dynamical systems is challenging in itself. The present three dynamical system examples provide complementary and effective benchmarks for such evaluation.

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

Lolla, T. and P.F.J. Lermusiaux, 2017a. A Gaussian--Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme. Monthly Weather Review. doi:10.1175/MWR-D-16-0064.1.

Retrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean estimation, numerical weather prediction, 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, we obtain a novel smoothing methodology for high– dimensional stochastic fields governed by general nonlinear dynamics. Building on recent Bayesian filters and classic Kalman smoothers, the equations and forward–backward algorithm of the new smoother are derived. The smoother uses the stochastic Dynamically–Orthogonal (DO) field equations and their time–evolving stochastic subspace to predict the prior probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the dominant DO subspace, after fitting semi–parametric Gaussian Mixture Models (GMMs) to joint DO realizations. The theoretical properties and computational cost of the new GMM-DO smoother are presented and discussed.

Validation of Genetic Algorithm Based Optimal Sampling for Ocean Data Assimilation

Heaney, K. D., P. F. J. Lermusiaux, T. F. Duda and P. J. Haley Jr., 2016.Validation of Genetic Algorithm Based Optimal Sampling for Ocean Data Assimilation. Ocean Dynamics. 66: 1209-1229. doi:10.1007/s10236-016-0976-5.

Regional ocean models are capable of forecasting conditions for usefully long intervals of time (days) provided that initial and ongoing conditions can be measured. In resource-limited circumstances, the placement of sensors in optimal locations is essential. Here, a nonlinear optimization approach to determine optimal adaptive sampling that uses the Genetic Algorithm (GA) method is presented. The method determines sampling strategies that minimize a user-defined physics-based cost function. The method is evaluated using identical twin experiments, comparing hindcasts from an ensemble of simulations that assimilate data selected using the GA adaptive sampling and other methods. For skill metrics, we employ the reduction of the ensemble root-mean-square-error (RMSE) between the “true” data-assimilative ocean simulation and the different ensembles of data-assimilative hindcasts. A 5-glider optimal sampling study is set up for a 400 km x 400 km domain in the Middle Atlantic Bight region, along the New Jersey shelf-break. Results are compared for several ocean and atmospheric forcing conditions.

Energy-optimal Path Planning by Stochastic Dynamically Orthogonal Level-Set Optimization

Subramani, D.N. and P.F.J. Lermusiaux, 2016. Energy-optimal Path Planning by Stochastic Dynamically Orthogonal Level-Set Optimization. Ocean Modeling, 100, 57–77. DOI: 10.1016/j.ocemod.2016.01.006

A stochastic optimization methodology is formulated for computing energy–optimal paths from among time–optimal paths of autonomous vehicles navigating in a dynamic flow field. Based on partial differential equations, the methodology rigorously leverages the level–set equation that governs time–optimal reachability fronts for a given relative vehicle speed function. To set up the energy optimization, the relative vehicle speed is considered to be stochastic and new stochastic Dynamically Orthogonal (DO) level–set equations are derived. Their solution provides the distribution of time–optimal reachability fronts and corresponding distribution of time–optimal paths. An optimization is then performed on the vehicle’s energy–time joint distribution to select the energy–optimal paths for each arrival time, among all stochastic time–optimal paths for that arrival time. Numerical schemes to solve the reduced stochastic DO level–set equations are obtained and accuracy and efficiency considerations are discussed. These reduced equations are first shown to be efficient at solving the governing stochastic level-sets, in part by comparisons with direct Monte Carlo simulations.To validate the methodology and illustrate its overall accuracy, comparisons with `semi–analytical’ energy–optimal path solutions are then completed. In particular, we consider the energy–optimal crossing of a canonical steady front and set up its `semi–analytical’ solution using a dual energy–time nested nonlinear optimization scheme. We then showcase the inner workings and nuances of the energy–optimal path planning, considering different mission scenarios. Finally, we study and discuss results of energy-optimal missions in a strong dynamic double–gyre flow field.

Science of Autonomy: Time-Optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles

Lermusiaux P.F.J, T. Lolla, P.J. Haley. Jr., K. Yigit, M.P. Ueckermann, T. Sondergaard and W.G. Leslie, 2016. Science of Autonomy: Time-Optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles. Chapter 21, Springer Handbook of Ocean Engineering: Autonomous Ocean Vehicles, Subsystems and Control, Tom Curtin (Ed.), pp. 481-498. doi:10.1007/978-3-319-16649-0_21.

The science of autonomy is the systematic development of fundamental knowledge about autonomous decision making and task completing in the form of testable autonomous methods, models and systems. In ocean applications, it involves varied disciplines that are not often connected. However, marine autonomy applications are rapidly growing, both in numbers and in complexity. This new paradigm in ocean science and operations motivates the need to carry out interdisciplinary research in the science of autonomy. This chapter reviews some recent results and research directions in time-optimal path planning and optimal adaptive sampling. The aim is to set a basis for a large number of vehicles forming heterogeneous and collaborative underwater swarms that are smart, i.e. knowledgeable about the predicted environment and their uncertainties, and about the predicted effects of autonomous sensing on future operations. The methodologies are generic and applicable to any swarm that moves and senses dynamic environmental fields. However, our focus is underwater path planning and adaptive sampling with a range of vehicles such as AUVs, gliders, ships or remote sensing platforms.

A Stochastic Optimization Method for Energy-based Path Planning

Subramani, D. N., Lolla, T., Haley Jr., P. J., Lermusiaux, P. F. J., 2015. A stochastic optimization method for energy-based path planning. In: Ravela, S., Sandu, A. (Eds.), DyDESS 2014. Vol. 8964 of LNCS. Springer, pp. 347-358.

We present a novel stochastic optimization method to compute energy-optimal paths, among all time-optimal paths, for vehicles traveling in dynamic unsteady currents. The method defines a stochastic class of instantaneous nominal vehicle speeds and then obtains the energy-optimal paths within the class by minimizing the total time-integrated energy usage while still satisfying the strong-constraint time-optimal level set equation. This resulting stochastic level set equation is solved using a dynamically orthogonal decomposition and the energy-optimal paths are then selected for each arrival time, among all stochastic time-optimal paths. The first application computes energy-optimal paths for crossing a steady front. Results are validated using a semi-analytical solution obtained by solving a dual nonlinear energy-time optimization problem. The second application computes energy-optimal paths for a realistic mission in the Middle Atlantic Bight and New Jersey Shelf/Hudson Canyon region, using dynamic data-driven ocean field estimates.

Path Planning in Multi-scale Ocean Flows: Coordination and Dynamic Obstacles

Lolla, T., P.J. Haley. Jr. and P.F.J. Lermusiaux, 2015. Path Planning in Multi-scale Ocean Flows: Coordination and Dynamic Obstacles. Ocean Modelling, 94, 46-66. DOI: 10.1016/j.ocemod.2015.07.013.

As the concurrent use of multiple autonomous vehicles in ocean missions grows, systematic control for their coordinated operation is becoming a necessity. Many ocean vehicles, especially those used in longer–range missions, possess limited operating speeds and are thus sensitive to ocean currents. Yet, the effect of currents on their trajectories is ignored by many coordination techniques. To address this issue, we first derive a rigorous level-set methodology for distance–based coordination of vehicles operating in minimum time within strong and dynamic ocean currents. The new methodology integrates ocean modeling, time-optimal level-sets and optimization schemes to predict the ocean currents, the short-term reachability sets, and the optimal headings for the desired coordination. Schemes are developed for dynamic formation control, where multiple vehicles achieve and maintain a given geometric pattern as they carry out their missions. Secondly, we obtain an efficient, non–intrusive technique for level-set-based time–optimal path planning in the presence of moving obstacles. The results are time-optimal path forecasts that rigorously avoid moving obstacles and sustain the desired coordination. They are exemplified and investigated for a variety of simulated ocean flows. A wind–driven double–gyre flow is used to study time-optimal dynamic formation control. Currents exiting an idealized strait or estuary are employed to explore dynamic obstacle avoidance. Finally, results are analyzed for the complex geometry and multi–scale ocean flows of the Philippine Archipelago.

Autonomous & Adaptive Oceanographic Front Tracking On Board Autonomous Underwater Vehicles

Petillo, S., H. Schmidt, P.F.J. Lermusiaux, D. Yoerger and A. Balasuriya, 2015. Autonomous & Adaptive Oceanographic Front Tracking On Board Autonomous Underwater Vehicles. Proceedings of IEEE OCEANS'15 Conference, Genoa, Italy, 18-21 May, 2015.

Oceanic fronts, similar to atmospheric fronts, occur at the interface of two fluid (water) masses of varying characteristics. In regions such as these where there are quantifiable physical, chemical, or biological changes in the ocean environment, it is possible—with the proper instrumentation—to track, or map, the front boundary.

In this paper, the front is approximated as an isotherm that is tracked autonomously and adaptively in 2D (horizontal) and 3D space by an autonomous underwater vehicle (AUV) running MOOS-IvP autonomy. The basic, 2D (constant depth) front tracking method developed in this work has three phases: detection, classification, and tracking, and results in the AUV tracing a zigzag path along and across the front. The 3D AUV front tracking method presented here results in a helical motion around a central axis that is aligned along the front in the horizontal plane, tracing a 3D path that resembles a slinky stretched out along the front.

To test and evaluate these front tracking methods (implemented as autonomy behaviors), virtual experiments were conducted with simulated AUVs in a spatiotemporally dynamic MIT MSEAS ocean model environment of the Mid-Atlantic Bight region, where a distinct temperature front is present along the shelfbreak. A number of performance metrics were developed to evaluate the performance of the AUVs running these front tracking behaviors, and the results are presented herein.

Adaptive Sampling Using Fleets of Underwater Gliders in the Presence of Fixed Buoys using a Constrained Clustering Algorithm

Cococcioni M., B. Lazzerini and P.F.J. Lermusiaux, 2015. Adaptive Sampling Using Fleets of Underwater Gliders in the Presence of Fixed Buoys using a Constrained Clustering Algorithm. Proceedings of IEEE OCEANS'15 Conference, Genoa, Italy, 18-21 May, 2015.

This paper presents a novel way to approach the problem of how to adaptively sample the ocean using fleets of underwater gliders. The technique is particularly suited for those situations where the covariance of the field to sample is unknown or unreliable but some information on the variance is known. The proposed algorithm, which is a variant of the well-known fuzzy C-means clustering algorithm, is able to exploit the presence of non-maneuverable assets, such as fixed buoys. We modified the fuzzy C-means optimization problem statement by including additional constraints. Then we provided an algorithmic solution to the new, constrained problem.

Time-Optimal Path Planning in Dynamic Flows using Level Set Equations: Realistic Applications

Lolla, T., P.J. Haley, Jr. and P.F.J. Lermusiaux, 2014. Time-Optimal Path Planning in Dynamic Flows using Level Set Equations: Realistic Applications. Ocean Dynamics, 64, 10:1399–1417. DOI: 10.1007/s10236-014-0760-3.

The level set methodology for time-optimal path planning is employed to predict collision-free and fastest time trajectories for swarms of underwater vehicles deployed in the Philippine Archipelago region. To simulate the multiscale ocean flows in this complex region, a data-assimilative primitive-equation ocean modeling system is employed with telescoping domains that are interconnected by implicit two-way nesting. These data-driven multiresolution simulations provide a realistic flow environment, including variable large-scale currents, strong jets, eddies, wind-driven currents and tides. The properties and capabilities of the rigorous level set methodology are illustrated and assessed quantitatively for several vehicle types and mission scenarios. Feasibility studies of all-to-all broadcast missions, leading to minimal time transmission between source and receiver locations, are performed using a large number of vehicles. The results with gliders and faster propelled vehicles are compared. Reachability studies, i.e.~determining the boundaries of regions that can be reached by vehicles for exploratory missions, are then exemplified and analyzed. Finally, the methodology is used to determine the optimal strategies for fastest time pick-up of deployed gliders by means of underway surface vessels or stationary platforms. The results highlight the complex effects of multiscale flows on the optimal paths, the need to utilize the ocean environment for more efficient autonomous missions and the benefits of including ocean forecasts in the planning of time-optimal paths.

Time-Optimal Path Planning in Dynamic Flows using Level Set Equations: Theory and Schemes

Lolla, T., P.F.J. Lermusiaux, M.P. Ueckermann and P.J. Haley, Jr., 2014. Time-Optimal Path Planning in Dynamic Flows using Level Set Equations: Theory and Schemes. Ocean Dynamics, 64, 10:1373–1397. DOI: 10.1007/s10236-014-0757-y.

We develop an accurate partial differential equation based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow-fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned, even in the presence of complex flows in domains with obstacles. Finally, we present, and support through illustrations, several remarks that describe specific features of our methodology.

Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part II: Applications

Sondergaard, T. and P.F.J. Lermusiaux, 2013b. Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part II: Applications. Monthly Weather Review, 141, 6, 1761-1785, doi:10.1175/MWR-D-11-00296.1.

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.

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

Sondergaard, T. and P.F.J. Lermusiaux, 2013a. Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part I. Theory and Scheme. Monthly Weather Review, 141, 6, 1737-1760, doi:10.1175/MWR-D-11-00295.1.

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.

Numerical Schemes for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows

Ueckermann, M.P., P.F.J. Lermusiaux and T.P. Sapsis, 2013. Numerical Schemes for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows. J. Comp. Phys., 233, 272-294, doi: 10.1016/j.jcp.2012.08.041.

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.

Path Planning in Time Dependent Flow Fields using Level Set Methods

Lolla, T.; Ueckermann, M.P.; Yigit, K.; Haley, P.J.; Lermusiaux, P.F.J., 2012, Path planning in time dependent flow fields using level set methods, 2012 IEEE International Conference on Robotics and Automation (ICRA), 166-173, 14-18 May 2012, doi: 10.1109/ICRA.2012.6225364.

We develop and illustrate an efficient but rigorous methodology that predicts the time-optimal paths of ocean vehicles in dynamic continuous flows. The goal is to best utilize or avoid currents, without limitation on these currents nor on the number of vehicles. The methodology employs a new modified level set equation to evolve a wavefront from the starting point of vehicles until they reach their desired goal locations, combining flow advection with nominal vehicle motions. The optimal paths of vehicles are then computed by solving particle tracking equations backwards in time. The computational cost is linear with the number of vehicles and geometric with spatial dimensions. The methodology is applicable to any continuous flows and many vehicles scenarios. Present illustrations consist of the crossing of a canonical uniform jet and its validation with an optimization problem, as well as more complex time varying 2D flow fields, including jets, eddies and forbidden regions.

Quantifying, predicting, and exploiting uncertainties in marine environments

Rixen, M., P.F.J. Lermusiaux and J. Osler, 2012. Quantifying, Predicting and Exploiting Uncertainties in Marine Environments, Ocean Dynamics, 62(3):495–499, doi: 10.1007/s10236-012-0526-8.

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.

Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty

Sapsis, T.P. and P.F.J. Lermusiaux, 2012. Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty. Physica D, 241(1), 60-76, doi:10.1016/j.physd.2011.10.001.

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.

Acoustically Focused Adaptive Sampling and On-board Routing for Marine Rapid Environmental Assessment

Wang, D., P.F.J. Lermusiaux, P.J. Haley, D. Eickstedt, W.G. Leslie and H. Schmidt, 2009. Acoustically Focused Adaptive Sampling and On-board Routing for Marine Rapid Environmental Assessment. Special issue of Journal of Marine Systems on "Coastal processes: challenges for monitoring and prediction", Drs. J.W. Book, Prof. M. Orlic and Michel Rixen (Guest Eds), 78, S393-S407, doi: 10.1016/j.jmarsys.2009.01.037.

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.

Dynamically orthogonal field equations for continuous stochastic dynamical systems

Sapsis, T.P. and P.F.J. Lermusiaux, 2009. Dynamically orthogonal field equations for continuous stochastic dynamical systems. Physica D, 238, 2347-2360, doi:10.1016/j.physd.2009.09.017.

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.

Path Planning of Autonomous Underwater Vehicles for Adaptive Sampling Using Mixed Integer Linear Programming

Yilmaz, N.K., C. Evangelinos, P.F.J. Lermusiaux and N. Patrikalakis, 2008. Path Planning of Autonomous Underwater Vehicles for Adaptive Sampling Using Mixed Integer Linear Programming. IEEE Transactions, Journal of Oceanic Engineering, 33 (4), 522-537. doi: 10.1109/JOE.2008.2002105.

The goal of adaptive sampling in the ocean is to predict the types and locations of additional ocean measurements that would be most useful to collect. Quantitatively, what is most useful is defined by an objective function and the goal is then to optimize this objective under the constraints of the available observing network. Examples of objectives are better oceanic understanding, to improve forecast quality, or to sample regions of high interest. This work provides a new path-planning scheme for the adaptive sampling problem. We define the path-planning problem in terms of an optimization framework and propose a method based on mixed integer linear programming (MILP). The mathematical goal is to find the vehicle path that maximizes the line integral of the uncertainty of field estimates along this path. Sampling this path can improve the accuracy of the field estimates the most. While achieving this objective, several constraints must be satisfied and are implemented. They relate to vehicle motion, intervehicle coordination, communication, collision avoidance, etc. The MILP formulation is quite powerful to handle different problem constraints and flexible enough to allow easy extensions of the problem. The formulation covers single- and multiple-vehicle cases as well as singleand multiple-day formulations. The need for a multiple-day formulation arises when the ocean sampling mission is optimized for several days ahead. We first introduce the details of the formulation, then elaborate on the objective function and constraints, and finally, present a varied set of examples to illustrate the applicability of the proposed method.

Adaptive Modeling, Adaptive Data Assimilation and Adaptive Sampling.

Lermusiaux, P.F.J, 2007. Adaptive Modeling, Adaptive Data Assimilation and Adaptive Sampling. Refereed invited manuscript. Special issue on "Mathematical Issues and Challenges in Data Assimilation for Geophysical Systems: Interdisciplinary Perspectives". C.K.R.T. Jones and K. Ide, Eds. Physica D, Vol 230, 172-196, doi: 10.1016/j.physd.2007.02.014.

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.

Environmental Prediction, Path Planning and Adaptive Sampling: Sensing and Modeling for Efficient Ocean Monitoring, Management and Pollution Control

Lermusiaux, P.F.J., P.J. Haley Jr. and N.K. Yilmaz, 2007. Environmental Prediction, Path Planning and Adaptive Sampling: Sensing and Modeling for Efficient Ocean Monitoring, Management and Pollution Control. Sea Technology, 48(9), 35-38.

Non-linear Optimization of Autonomous Undersea Vehicle Sampling Strategies for Oceanographic Data-Assimilation

Heaney, K.D., G. Gawarkiewicz, T.F. Duda and P.F.J. Lermusiaux, 2007. Non-linear Optimization of Autonomous Undersea Vehicle Sampling Strategies for Oceanographic Data-Assimilation. Special issue on "Underwater Robotics", Journal of Field Robotics, 24(6), 437-448, doi:10.1002/rob.20183.

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.

Adaptive Acoustical-Environmental Assessment for the Focused Acoustic Field-05 At-sea Exercise

Wang, D., P.F.J. Lermusiaux, P.J. Haley, W.G. Leslie and H. Schmidt, 2006. Adaptive Acoustical-Environmental Assessment for the Focused Acoustic Field-05 At-sea Exercise, Oceans 2006, 6pp, Boston, MA, 18-21 Sept. 2006, doi: 10.1109/OCEANS.2006.306904.

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.

Path Planning Methods for Adaptive Sampling of Environmental and Acoustical Ocean Fields

Yilmaz, N.K., C. Evangelinos, N.M. Patrikalakis, P.F.J. Lermusiaux, P.J. Haley, W.G. Leslie, A.R. Robinson, D. Wang and H. Schmidt, 2006a. Path Planning Methods for Adaptive Sampling of Environmental and Acoustical Ocean Fields, Oceans 2006, 6pp, Boston, MA, 18-21 Sept. 2006, doi: 10.1109/OCEANS.2006.306841.

Adaptive sampling aims to predict the types and locations of additional observations that are most useful for specific objectives, under the constraints of the available observing network. Path planning refers to the computation of the routes of the assets that are part of the adaptive component of the observing network. In this paper, we present two path planning methods based on Mixed Integer Linear Programming (MILP). The methods are illustrated with some examples based on environmental ocean fields and compared to highlight their strengths and weaknesses. The stronger method is further demonstrated on a number of examples covering multi-vehicle and multi-day path planning, based on simulations for the Monterey Bay region. The framework presented is powerful and flexible enough to accommodate changes in scenarios. To demonstrate this feature, acoustical path planning is also discussed.

Progress and Prospects of U.S. Data Assimilation in Ocean Research

Lermusiaux, P.F.J., P. Malanotte-Rizzoli, D. Stammer, J. Carton, J. Cummings and A.M. Moore, 2006. "Progress and Prospects of U.S. Data Assimilation in Ocean Research". Oceanography, Special issue on "Advances in Computational Oceanography", T. Paluszkiewicz and S. Harper, Eds., 19, 1, 172-183.

THIS REPORT summarizes goals, activities, and recommendations of a workshop on data assimilation held in Williamsburg, Virginia on September 9-11, 2003, and sponsored by the U.S. Office of Naval Research (ONR) and National Science Foundation (NSF). The overall goal of the workshop was to synthesize research directions for ocean data assimilation (DA) and outline efforts required during the next 10 years and beyond to evolve DA into an integral and sustained component of global, regional, and coastal ocean science and observing and prediction systems. The workshop built on the success of recent and existing DA activities such as those sponsored by the National Oceanographic Partnership Program (NOPP) and NSF-Information Technology Research (NSF-ITR). DA is a quantitative approach to optimally combine models and observations. The combination is usually consistent with model and data uncertainties, which need to be represented. Ocean DA can extract maximum knowledge from the sparse and expensive measurements of the highly variable ocean dynamics. The ultimate goal is to better understand and predict these dynamics on multiple spatial and temporal scales, including interactions with other components of the climate system. There are many applications that involve DA or build on its results, including: coastal, regional, seasonal, and inter-annual ocean and climate dynamics; carbon and biogeochemical cycles; ecosystem dynamics; ocean engineering; observing-system design; coastal management; fisheries; pollution control; naval operations; and defense and security. These applications have different requirements that lead to variations in the DA schemes utilized. For literature on DA, we refer to Ghil and Malanotte-Rizzoli (1991), the National Research Council (1991), Bennett (1992), Malanotte- Rizzoli (1996), Wunsch (1996), Robinson et al. (1998), Robinson and Lermusiaux (2002), and Kalnay (2003). We also refer to the U.S. Global Ocean Data Assimilation Experiment (GODAE) workshop on Global Ocean Data Assimilation: Prospects and Strategies (Rienecker et al., 2001); U.S. National Oceanic and Atmospheric Administration-Office of Global Programs (NOAA-OGP) workshop on Coupled Data Assimilation (Rienecker, 2003); and, NOAA-NASA-NSF workshop on Ongoing Analysis of the Climate System (Arkin et al., 2003).

Uncertainty Estimation and Prediction for Interdisciplinary Ocean Dynamics

Lermusiaux, P.F.J., 2006. Uncertainty Estimation and Prediction for Interdisciplinary Ocean Dynamics. Refereed manuscript, Special issue on "Uncertainty Quantification". J. Glimm and G. Karniadakis, Eds. Journal of Computational Physics, 217, 176-199. doi: 10.1016/j.jcp.2006.02.010.

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.

Quantifying Uncertainties in Ocean Predictions

Lermusiaux, P.F.J., C.-S. Chiu, G.G. Gawarkiewicz, P. Abbot, A.R. Robinson, R.N. Miller, P.J. Haley, W.G. Leslie, S.J. Majumdar, A. Pang and F. Lekien, 2006. Quantifying Uncertainties in Ocean Predictions. Refereed invited manuscript. Oceanography, Special issue on "Advances in Computational Oceanography", T. Paluszkiewicz and S. Harper (Office of Naval Research), Eds., 19, 1, 92-105, doi: 10.5670/oceanog.2006.93.

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.

Dynamics and Lagrangian Coherent Structures in the Ocean and their Uncertainty

Lermusiaux, P.F.J. and F. Lekien, 2005. Dynamics and Lagrangian Coherent Structures in the Ocean and their Uncertainty. Extended Abstract in report of the "Dynamical System Methods in Fluid Dynamics" Oberwolfach Workshop. Jerrold E. Marsden and Jurgen Scheurle (Eds.), Mathematisches Forschungsinstitut Oberwolfach, July 31st - August 6th, 2005, Germany. 2pp.

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

Adaptive Coupled Physical and Biogeochemical Ocean Predictions: A Conceptual Basis

Lermusiaux, P.F.J, C. Evangelinos, R. Tian, P.J. Haley, J.J. McCarthy, N.M. Patrikalakis, A.R. Robinson and H. Schmidt, 2004. Adaptive Coupled Physical and Biogeochemical Ocean Predictions: A Conceptual Basis. Refereed invited manuscript, F. Darema (Ed.), Lecture Notes in Computer Science, 3038, 685-692.

Physical and biogeochemical ocean dynamics can be intermittent and highly variable, and involve interactions on multiple scales. In general, the oceanic fields, processes and interactions that matter thus vary in time and space. For efficient forecasting, the structures and parameters of models must evolve and respond dynamically to new data injected into the executing prediction system. The conceptual basis of this adaptive modeling and corresponding computational scheme is the subject of this presentation. Specifically, we discuss the process of adaptive modeling for coupled physical and biogeochemical ocean models. The adaptivity is introduced within an interdisciplinary prediction system. Model-data misfits and data assimilation schemes are used to provide feedback from measurements to applications and modify the runtime behavior of the prediction system. Illustrative examples in Massachusetts Bay and Monterey Bay are presented to highlight ongoing progress.

The use of data assimilation in coupled hydrodynamic, ecological and bio-geo-chemical models of the ocean

Gregoire, M., P. Brasseur and P.F.J. Lermusiaux (Guest Eds.), 2003. The use of data assimilation in coupled hydrodynamic, ecological and bio-geo-chemical models of the ocean. Journal of Marine Systems, 40, 1-3.

The International Lie`ge Colloquium on Ocean Dynamics is organized annually. The topic differs from year to year in an attempt to address, as much as possible, recent problems and incentive new subjects in oceanography. Assembling a group of active and eminent scientists from various countries and often different disciplines, the Colloquia provide a forum for discussion and foster a mutually beneficial exchange of information opening on to a survey of recent discoveries, essential mechanisms, impelling question marks and valuable recommendations for future research. The objective of the 2001 Colloquium was to evaluate the progress of data assimilation methods in marine science and, in particular, in coupled hydrodynamic, ecological and bio-geo-chemical models of the ocean. The past decades have seen important advances in the understanding and modelling of key processes of the ocean circulation and bio-geo-chemical cycles. The increasing capabilities of data and models, and their combination, are allowing the study of multidisciplinary interactions that occur dynamically, in multiple ways, on multiscales and with feedbacks. The capacity of dynamical models to simulate interdisciplinary ocean processes over specific space- time windows and thus forecast their evolution over predictable time scales is also conditioned upon the availability of relevant observations to: initialise and continually update the physical and bio-geo-chemical sectors of the ocean state; provide relevant atmospheric and boundary forcing; calibrate the parameterizations of sub-grid scale processes, growth rates and reaction rates; construct interdisciplinary and multiscale correlation and feature models; identify and estimate the main sources of errors in the models; control or correct for mis-represented or neglected processes. The access to multivariate data sets requires the implementation, exploitation and management of dedicated ocean observing and prediction systems. However, the available data are often limited and, for instance, seldom in a form to be directly compatible or directly inserted into the numerical models. To relate the data to the ocean state on all scales and regions that matter, evolving three-dimensional and multivariate (measurement) models are becoming important. Equally significant is the reduction of observational requirements by design of sampling strategies via Observation System Simulation Experiments and adaptive sampling. Data assimilation is a quantitative approach to extract adequate information content from the data and to improve the consistency between data sets and model estimates. It is also a methodology to dynamically interpolate between data scattered in space and time, allowing comprehensive interpretation of multivariate observations. In general, the goals of data assimilation are to: control the growth of predictability errors; correct dynamical deficiencies; estimate model parameters, including the forcings, initial and boundary conditions; characterise key processes by analysis of four- 0924-7963/03/$ – see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-7963(03)00027-7 www.elsevier.com/locate/jmarsys The use of data assimilation in coupled hydrodynamic, ecological and bio-geo-chemical models of the ocean Journal of Marine Systems 40-41 (2003) 1-3 dimensional fields and their statistics (balances of terms, etc.); carry out advanced sensitivity studies and Observation System Simulation Experiments, and conduct efficient operations, management and monitoring. The theoretical framework of data assimilation for marine sciences is now relatively well established, routed in control theory, estimation theory or inverse techniques, from variational to sequential approaches. Ongoing research efforts of special importance for interdisciplinary applications include the: stochastic representation of processes and determination of model and data errors; treatment of (open) boundary conditions and strong nonlinearities; space-time, multivariate extrapolation of limited and noisy data and determination of measurement models; demonstration that bio-geo-chemical models are valid enough and of adequate structures for their deficiencies to be controlled by data assimilation; and finally, ability to provide accurate estimates of fields, parameters, variabilities and errors, with large and complex dynamical models and data sets. Operationally, major engineering and computational challenges for the coming years include the: development of theoretically sound methods into useful, practical and reliable techniques at affordable costs; implementation of scalable, seamless and automated systems linking observing systems, numerical models and assimilation schemes; adequate mix of integrated and distributed (Web-based) networks; construction of user-friendly architectures and establishment of standards for the description of data and software (metadata) for efficient communication, dissemination and management. In addition to addressing the above items, the 33rd Lie`ge Colloquium has offered the opportunity to: – review the status and current progress of data assimilation methodologies utilised in the physical, acoustical, optical and bio-geo-chemical scientific communities; – demonstrate the potentials of data assimilation systems developed for coupled physical/ecosystem models, from scientific to management inquiries; – examine the impact of data assimilation and inverse modelling in improving model parameterisations; – discuss the observability and controllability properties of, and identify the missing gaps in current observing and prediction systems; and exchange the results of and the learnings from preoperational marine exercises. The presentations given during the Colloquium lead to discussions on a series of topics organized within the following sections: (1) Interdisciplinary research progress and issues: data, models, data assimilation criteria. (2) Observations for interdisciplinary data assimilation. (3) Advanced fields estimation for interdisciplinary systems. (4) Estimation of interdisciplinary parameters and model structures. (5) Assimilation methodologies for physical and interdisciplinary systems. (6) Toward operational interdisciplinary oceanography and data assimilation. A subset of these presentations is reported in the present Special Issue. As was pointed out during the Colloquium, coupled biological-physical data assimilation is in its infancy and much can be accomplished now by the immediate application of existing methods. Data assimilation intimately links dynamical models and observations, and it can play a critical role in the important area of fundamental biological oceanographic dynamical model development and validation over a hierarchy of complexities. Since coupled assimilation for coupled processes is challenging and can be complicated, care must be exercised in understanding, modeling and controlling errors and in performing sensitivity analyses to establish the robustness of results. Compatible interdisciplinary data sets are essential and data assimilation should iteratively define data impact and data requirements. Based on the results presented during the Colloquium, data assimilation is expected to enable future marine technologies and naval operations otherwise impossible or not feasible. Interdisciplinary predictability research, multiscale in both space and time, is required. State and parameter estimation via data assimilation is central to the successful establishment of advanced interdisciplinary ocean observing and prediction systems which, functioning in real time, will contribute to novel and efficient capabilities to manage, and to operate in our oceans. The Scientific Committee and the participants to the 33rd Lie`ge Colloquium wish to express their 2 Preface gratitude to the Ministe`re de l’Enseignement Supe’rieur et de la Recherche Scientifique de la Communaute – Francaise de Belgique, the Fonds National de la Recherche Scientifique de Belgique (F.N.R.S., Belgium), the Ministe`re de l’Emploi et de la Formation du Gouvernement Wallon, the University of Lie`ge, the Commission of European Union, the Scientific Committee on Oceanographic Research (SCOR), the International Oceanographic Commission of the UNESCO, the US Office of Naval Research, the National Science Foundation (NSF, USA) and the International Association for the Physical Sciences of the Ocean (IAPSO) for their most valuable support.

Four-dimensional data assimilation for coupled physical-acoustical fields

Lermusiaux, P.F.J. and C.-S. Chiu, 2002. Four-dimensional data assimilation for coupled physical-acoustical fields. In "Acoustic Variability, 2002". N.G. Pace and F.B. Jensen (Eds.), Saclantcen. Kluwer Academic Press, 417-424.

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.

Advanced interdisciplinary data assimilation: Filtering and smoothing via error subspace statistical estimation.

Lermusiaux, P.F.J., A.R. Robinson, P.J. Haley and W.G. Leslie, 2002. Advanced interdisciplinary data assimilation: Filtering and smoothing via error subspace statistical estimation. Proceedings of "The OCEANS 2002 MTS/IEEE" conference, Holland Publications, 795-802.

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.

Data assimilation for modeling and predicting coupled physical-biological interactions in the sea

Robinson, A.R. and P.F.J. Lermusiaux, 2002. Data assimilation for modeling and predicting coupled physical-biological interactions in the sea. In "The Sea, Vol. 12: Biological-Physical Interactions in the Ocean", Robinson A.R., J.R. McCarthy and B.J. Rothschild (Eds.). 475-536.

Data assimilation is a modern methodology of relating natural data and dynamical models. The general dynamics of a model is combined or melded with a set of observations. All dynamical models are to some extent approximate, and all data sets are finite and to some extent limited by error bounds. The purpose of data assimilation is to provide estimates of nature which are better estimates than can be obtained by using only the observational data or the dynamical model. There are a number of specific approaches to data assimilation which are suitable for estimation of the state of nature, including natural parameters, and for evaluation of the dynamical approximations. Progress is accelerating in understanding the dynamics of real ocean biological- physical interactive processes. Although most biophysical processes in the sea await discovery, new techniques and novel interdisciplinary studies are evolving ocean science to a new level of realism. Generally, understanding proceeds from a quantitative description of four-dimensional structures and events, through the identification of specific dynamics, to the formulation of simple generalizations. The emergence of realistic interdisciplinary four-dimensional data assimilative ocean models and systems is contributing significantly and increasingly to this progress.

On the mapping of multivariate geophysical fields: sensitivity to size, scales and dynamics

Lermusiaux, P.F.J., 2002. On the mapping of multivariate geophysical fields: sensitivity to size, scales and dynamics. Journal of Atmospheric and Oceanic Technology, 19, 1602-1637.

The effects of a priori parameters on the error subspace estimation and mapping methodology introduced by P. F. J. Lermusiaux et al. is investigated. The approach is three-dimensional, multivariate, and multiscale. The sensitivities of the subspace and a posteriori fields to the size of the subspace, scales considered, and nonlinearities in the dynamical adjustments are studied. Applications focus on the mesoscale to subbasin-scale physics in the northwestern Levantine Sea during 10 February-15 March and 19 March-16 April 1995. Forecasts generated from various analyzed fields are compared to in situ and satellite data. The sensitivities to size show that the truncation to a subspace is efficient. The use of criteria to determine adequate sizes is emphasized and a backof- the-envelope rule is outlined. The sensitivities to scales confirm that, for a given region, smaller scales usually require larger subspaces because of spectral redness. However, synoptic conditions are also shown to strongly influence the ordering of scales. The sensitivities to the dynamical adjustment reveal that nonlinearities can modify the variability decomposition, especially the dominant eigenvectors, and that changes are largest for the features and regions with high shears. Based on the estimated variability variance fields, eigenvalue spectra, multivariate eigenvectors and (cross)-covariance functions, dominant dynamical balances and the spatial distribution of hydrographic and velocity characteristic scales are obtained for primary regional features. In particular, the Ierapetra Eddy is found to be close to gradient-wind balance and coastal-trapped waves are anticipated to occur along the northern escarpment of the basin.

Transfer of uncertainties through physical-acoustical-sonar end-to-end systems: A conceptual basis

Robinson, A.R., P. Abbot, P.F.J. Lermusiaux and L. Dillman, 2002. Transfer of uncertainties through physical-acoustical-sonar end-to-end systems: A conceptual basis. In "Acoustic Variability, 2002:. N.G. Pace and F.B. Jensen (Eds.), SACLANTCEN. Kluwer Academic Press, 603-610.

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.

Data Assimilation in Models

Robinson, A.R. and P.F.J. Lermusiaux, 2001. Data Assimilation in Models. Encyclopedia of Ocean Sciences, Academic Press Ltd., London, 623-634.

Data assimilation is a novel, versatile methodology for estimating oceanic variables. The estimation of a quantity of interest via data assimilation involves the combination of observational data with the underlying dynamical principles governing the system under observation. The melding of data and dynamics is a powerful methodology which makes possible efRcient, accurate, and realistic estimations otherwise not feasible. It is providing rapid advances in important aspects of both basic ocean science and applied marine technology and operations. The following sections introduce concepts, describe purposes, present applications to regional dynamics and forecasting, overview formalism and methods, and provide a selected range of examples.

Evolving the subspace of the three-dimensional multiscale ocean variability: Massachusetts Bay

Lermusiaux, P.F.J., 2001. Evolving the subspace of the three-dimensional multiscale ocean variability: Massachusetts Bay. Journal of Marine Systems, Special issue on "Three-dimensional ocean circulation: Lagrangian measurements and diagnostic analyses", 29/1-4, 385-422, doi: 10.1016/S0924-7963(01)00025-2.

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.

On the mapping of multivariate geophysical fields: error and variability subspace estimates

Lermusiaux, P.F.J., D.G.M. Anderson and C.J. Lozano, 2000. On the mapping of multivariate geophysical fields: error and variability subspace estimates. The Quarterly Journal of the Royal Meteorological Society, April B, 1387-1430.

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.

Data assimilation via Error Subspace Statistical Estimation. Part II: Middle Atlantic Bight shelfbreak front simulations and ESSE validation

Lermusiaux, P.F.J., 1999a. Data assimilation via Error Subspace Statistical Estimation. Part II: Middle Atlantic Bight shelfbreak front simulations and ESSE validation. Monthly Weather Review, 127(7), 1408-1432, doi: 10.1175/1520-0493(1999)127<1408:DAVESS> 2.0.CO;2.

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.

Data assimilation via Error Subspace Statistical Estimation. Part I: Theory and schemes

Lermusiaux, P.F.J. and A.R. Robinson, 1999. Data assimilation via Error Subspace Statistical Estimation. Part I: Theory and schemes. Monthly Weather Review, 127(7), 1385-1407, doi: 10.1175/1520-0493(1999) 127<1385:DAVESS>2.0.CO;2.

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.

Data Assimilation

Robinson, A.R., P.F.J. Lermusiaux and N.Q. Sloan, III, 1998. Data Assimilation. In "The Sea: The Global Coastal Ocean I", Processes and Methods (K.H. Brink and A.R. Robinson, Editors), Volume 10, John Wiley and Sons, New York, NY, 541-594