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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.

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.

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.

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.

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.

In this paper we adopt a reachability-based approach to deal with the pursuit-evasion differential game between one evader and multiple pursuers 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 pursuers and the evader. The time-optimal trajectories and the corresponding optimal strategies can subsequently be retrieved from the level sets. The pursuers are divided into active pursuers, guards and redundant pursuers according to their respective roles in the pursuit-evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flow fields, with and without obstacles.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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).

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.

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.

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.

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.

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.

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

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

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.

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.

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 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.

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 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.

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

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