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

The nonlinear Gaussian Mixture Model Dynamically Orthogonal (GMM–DO) smoother for high-dimensional stochastic fields is exemplified and contrasted with other smoothers by applications to three dynamical systems, all of which admit far-from-Gaussian distributions. The capabilities of the smoother are first illustrated using a double-well stochastic diffusion experiment. Comparisons with the original and improved versions of the ensemble Kalman smoother explain the detailed mechanics of GMM–DO smoothing and show that its accuracy arises from the joint GMM distributions across successive observation times. Next, the smoother is validated using the advection of a passive stochastic tracer by a reversible shear flow. This example admits an exact smoothed solution, whose derivation is also provided. Results show that the GMM–DO smoother accurately captures the full smoothed distributions and not just the mean states. The final example showcases the smoother in more complex nonlinear fluid dynamics caused by a barotropic jet flowing through a sudden expansion and leading to variable jets and eddies. The accuracy of the GMM–DO smoother is compared to that of the Error Subspace Statistical Estimation smoother. It is shown that even when the dynamics result in only slightly multimodal joint distributions, Gaussian smoothing can lead to a severe loss of information. The three examples show that the backward inferences of the GMM–DO smoother are skillful and efficient. Accurate evaluation of Bayesian smoothers for nonlinear high-dimensional dynamical systems is challenging in itself. The present three examples—stochastic low dimension, reversible high dimension, and irreversible high dimension—provide complementary and effective benchmarks for such evaluation.

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 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.

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 path planningadaptivesamplingocean 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 autonomous underwater vehicles (AUVautonomousunderwater vehicle (AUV)s), gliders, ships or remote sensing platforms.

Autonomous underwater vehicles such as gliders have emerged as valuable scientific platforms due to their increasing uses in several oceanic applications, ranging from security, acoustic surveillance and military reconnaissance to collection of ocean data at specific locations for ocean prediction, monitoring and dynamics investigation. Gliders exhibit high levels of autonomy and are ideal for long range missions. As these gliders become more reliable and affordable, multi-vehicle coordination and sampling missions are expected to become very common in the near future. This endurance of gliders however, comes at an expense of being susceptible to typical coastal ocean currents. Due to the physical limitations of underwater vehicles and the highly dynamic nature of the coastal ocean, path planning to generate safe and fast vehicle trajectories becomes crucial for their successful operation. As a result, our motivation in this thesis is to develop a computationally efficient and rigorous methodology that can predict the time-optimal paths of underwater vehicles navigating in continuous, strong and dynamic ow-fields. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid ow currents to minimize their travel time.In this thesis, we fist review existing path planning methods and discuss their advantages and drawbacks. Then, we discuss the theory of level set methods and their utility in solving front tracking problems. Then, we present a rigorous (partial differential equation based) methodology based on the level set method, which can compute time-optimal paths of swarms of underwater vehicles, obviating the need for any heuristic control based approaches. We state and prove a theorem, along with several corollaries, that forms the foundation of our approach for path planning. We show that our algorithm is computationally efficient - the computational cost grows linearly with the number of vehicles and geometrically with spatial directions. We illustrate the working and capabilities of our path planning algorithm by means of a number of applications. First, we validate our approach through simple benchmark applications, and later apply our methodology to more complex, realistic and numerically simulated ow-fields, which include eddies, jets, obstacles and forbidden regions. Finally, we extend our methodology to solve problems of coordinated motion of multiple vehicles in strong dynamic ow-fields. Here, coordination refers to maintenance of specific geometric patterns by the vehicles. The level-set based control scheme that we derive is shown to provide substantial advantages to a local control approach. Specifically, the illustrations show that the resulting coordinated vehicle motions can maintain specific patterns in dynamic flow fields with strong and complex spatial gradients.