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Dynamically Orthogonal Equations for Stochastic Underwater Sound Propagation

Ali, W.H., 2019. Dynamically Orthogonal Equations for Stochastic Underwater Sound Propagation. SM Thesis, Massachusetts Institute of Technology, Mechanical Engineering, September 2019.

Grand challenges in ocean acoustic propagation are to accurately capture the dynamic environmental uncertainties and to predict the evolving probability density function of stochastic acoustic waves. This is due to the complex ocean physics and acoustics dynamics, nonlinearities, multiple scales, and large dimensions. There are several sources of uncertainty including: the initial and boundary conditions of the ocean physics and acoustic dynamics, the bathymetry and seabed fields; the models parameters; and, the models themselves. In the present work, we start addressing these challenges by deriving, implementing and verifying new optimally-reduced Dynamically Orthogonal (DO) differential equations that govern the propagation of stochastic acoustic waves for underwater sound propagation in an uncertain ocean environment. The developed methodology allows modeling environmental uncertainties in a rigorous probabilistic framework and predicting the uncertainties of acoustic fields, fully respecting the nonlinear governing equations and non-Gaussian statistics of the sound speed and acoustic state variables. The methodology is applied to the standard narrow-angle parabolic equation and is utilized to predict acoustic field uncertainties for three new stochastic idealized test cases: (1) an uncertain Pekeris waveguide with penetrable bottom, (2) an uncertain horizontal interface problem, and (3) an uncertain range-dependent sloping interface problem. For the first case, the solutions of the DO acoustic equations are validated against those obtained using standard Monte Carlo sampling. The second test case showcases results for predicting acoustic field probabilities due to uncertainties in the location of a sound speed channel. For the third test case, the advantages of the DO acoustic equations in predicting uncertainties in complex range-dependent environments are highlighted.


Efficient Matrix-Free Implementation and Automated Verification of Hybridizable Discontinuous Galerkin Finite Element Methods

Foucart, C., 2019. Efficient Matrix-Free Implementation and Automated Verification of Hybridizable Discontinuous Galerkin Finite Element Methods. SM Thesis, Massachusetts Institute of Technology, Mechanical Engineering, June 2019.

This work focuses on developing efficient and robust implementation methods for hybridizable discontinuous Galerkin (HDG) schemes for fluid and ocean dynamics. In the first part, we compare choices in weak formulations and their numerical consequences. We address details in making the leap from the mathematical formulation to the implementation, including the different spaces and mappings, discretization of the integral operators, boundary conditions, and assembly of the linear systems. We provide a flexible mapping procedure amenable to both quadrature-free and quadrature-based discretizations, and compare the accuracy of the two on different problem geometries. We verify the quadrature-free approach, demonstrating that optimal orders of convergence can be obtained, even on non-affine and curvilinear geometries. The second part of the work investigates the scalability of HDG schemes, identifying memory and time-to-solution bottlenecks. The form of the quadrature-free integral operators is exploited to develop a novel and efficient matrix-free approach to solving the global linear system that arises from HDG discretizations. Additional manipulations to improve numerical robustness are discussed. To mitigate the complexity of the implementation, we provide an automated and computationally efficient verification procedure for the HDG methodologies discussed, using a hierarchical approach to provide diagnostic information and isolate problems. Finally, challenges related to the effective visualization of high-order, discontinuous HDG-FEM data for fluid and ocean applications are illustrated and strategies are provided to address them.


Path Planning and Adaptive Sampling in the Coastal Ocean

Lolla, T., 2016. Path Planning and Adaptive Sampling in the Coastal Ocean. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2016.

When humans or robots operate in complex dynamic environments, the planning of paths and the collection of observations are basic, indispensable problems. In the oceanic and atmospheric environments, the concurrent use of multiple mobile sensing platforms in unmanned missions is growing very rapidly. Opportunities for a paradigm shift in the science of autonomy involve the development of fundamental theories to optimally collect information, learn, collaborate and make decisions under uncertainty while persistently adapting to and utilizing the dynamic environment. To address such pressing needs, this thesis derives governing equations and develops rigorous methodologies for optimal path planning and optimal sampling using collaborative swarms of autonomous mobile platforms. The application focus is the coastal ocean where currents can be much larger than platform speeds, but the fundamental results also apply to other dynamic environments.

We first undertake a theoretical synthesis of minimum-time control of vehicles operating in general dynamic flows. Using various ideas rooted in non-smooth calculus, we prove that an unsteady Hamilton-Jacobi equation governs the forward reachable sets in any type of Lipschitz-continuous flow. Next, we show that with a suitable modification to the Hamiltonian, the results can be rigorously generalized to perform time-optimal path planning with anisotropic motion constraints and with moving obstacles and unsafe ‘forbidden’ regions. We then derive a level-set methodology for distance-based coordination of swarms of vehicles operating in minimum time within strong and dynamic ocean currents. The results are illustrated for varied fluid and ocean flow simulations. Finally, the new path planning system is applied to swarms of vehicles operating in the complex geometry of the Philippine Archipelago, utilizing realistic multi-scale current predictions from a data-assimilative ocean modeling system.

In the second part of the thesis, we derive a theory for adaptive sampling that exploits the governing nonlinear dynamics of the system and captures the non-Gaussian structure of the random state fields. Optimal observation locations are determined by maximizing the mutual information between the candidate observations and the variables of interest. We develop a novel Bayesian smoother for high-dimensional continuous stochastic fields governed by general nonlinear dynamics. This smoother combines the adaptive reduced-order Dynamically-Orthogonal equations with Gaussian Mixture Models, extending linearized Gaussian backward pass updates to a nonlinear, non-Gaussian setting. The Bayesian information transfer, both forward and backward in time, is efficiently carried out in the evolving dominant stochastic subspace. Building on the foundations of the smoother, we then derive an efficient technique to quantify the spatially and temporally varying mutual information field in general nonlinear dynamical systems. The globally optimal sequence of future sampling locations is rigorously determined by a novel dynamic programming approach that combines this computation of mutual information fields with the predictions of the forward reachable set. All the results are exemplified and their performance is quantitatively assessed using a variety of simulated fluid and ocean flows.

The above novel theories and schemes are integrated so as to provide real-time computational intelligence for collaborative swarms of autonomous sensing vehicles. The integrated system guides groups of vehicles along predicted optimal trajectories and continuously improves field estimates as the observations predicted to be most informative are collected and assimilated. The optimal sampling locations and optimal trajectories are continuously forecast, all in an autonomous and coordinated fashion.


Internal Tides Near Steep Topographies

Sroka, S.G., 2016. Internal Tides Near Steep Topographies. SM Thesis, Massachusetts Institute of Technology, Mechanical Engineering, September 2016.

The primary contributions of this thesis include the first stages of development of a 2D, finitevolume, non-hydrostatic, sigma-coordinate code and beginning to apply the Dynamically Orthogonal field equations to study the sensitivity of internal tides to perturbations in the density field. First, we ensure that the 2D Finite Volume (2DFV) code that we use can accurately capture non-hydrostatic internal tides since these dynamics have not yet been carefully evaluated for accuracy in this framework. We find that, for low-aspect ratio topographies, the -coordinate mesh in the 2DFV code produces numerical artifacts near the bathymetry. To ameliorate these staircasing effects, and to develop the framework towards a moving mesh with free-surface dynamics, we have begun to implement a non-hydrostatic sigma-coordinate framework which significantly improves the representation of the internal tides for low-aspect ratio topographies. Finally we investigate the applicability of stochastic density perturbations in an internal tide field. We utilize the Dynamically Orthogonal field equations for this investigation because they achieve substantial model order reduction over ensemble Monte-Carlo methods.

Ocean Acoustic Uncertainty for Submarine Applications

Swezey, M., 2016. Ocean Acoustic Uncertainty for Submarine Applications. SM Thesis, Massachusetts Institute of Technology, MechE-USN Joint Program, June 2016.

The focus of this research is to study the uncertainties forecast by multi-resolution ocean models and quantify how those uncertainties affect the pressure fields estimated by coupled ocean models. The quantified uncertainty can then be used to provide enhanced sonar performance predictions for tactical decision aides. High fidelity robust modeling of the oceans can resolve various scale processes from tidal shifts to mesoscale phenomena. These ocean models can be coupled with acoustic models that account for variations in the ocean environment and complex bathymetry to yield accurate acoustic field representations that are both range and time independent. Utilizing the MIT Multidisciplinary Environmental Assimilation System (MSEAS) implicit two-way nested primitive-equation ocean model and Error Subspace Statistical Estimation scheme (ESSE), coupled with three-dimensional-inspace (3D) parabolic equation acoustic models, we conduct a study to understand and determine the effects of ocean state uncertainty on the acoustic transmission loss. The region of study is focused on the ocean waters surrounding Taiwan in the East China Sea. This region contains complex ocean dynamics and topography along the critical shelf-break region where the ocean acoustic interaction is driven by several uncertainties. The resulting ocean acoustic uncertainty is modeled and analyzed to quantify sonar performance and uncertainty characteristics with respect to submarine counter detection. Utilizing cluster based data analysis techniques, the relationship between the resulting acoustic field and the uncertainty in the ocean model can be characterized. Furthermore, the dynamic transitioning between the clustered acoustic states can be modeled as Markov processes. This analysis can be used to enhance not only submarine counter detection aides, but it may also be used for several applications to enhance understanding of the capabilities and behavior of uncertainties of acoustic systems operating in the complex ocean environment.

An Iterative Pressure-Correction Method for the Unsteady Incompressible Navier-Stokes Equation

Aoussou, J.P., 2016. An Iterative Pressure-Correction Method for the Unsteady Incompressible Navier-Stokes Equation. SM Thesis, Massachusetts Institute of Technology, Computation for Design and Optimization Graduate Program, June 2016.

The pressure-correction projection method for the incompressible Navier-Stokes equation is approached as a preconditioned Richardson iterative method for the pressure- Schur complement equation. Typical pressure correction methods perform only one iteration and suffer from a splitting error that results in a spurious numerical boundary layer, and a limited order of convergence in time. We investigate the benefit of performing more than one iteration. We show that that not only performing more iterations attenuates the effects of the splitting error, but also that it can be more computationally efficient than reducing the time step, for the same level of accuracy. We also devise a stopping criterion that helps achieve a desired order of temporal convergence, and implement our method with multi-stage and multi-step time integration schemes. In order to further reduce the computational cost of our iterative method, we combine it with an Aitken acceleration scheme. Our theoretical results are validated and illustrated by numerical test cases for the Stokes and Navier-Stokes equations, using Implicit-Explicit Backwards Difference Formula and Runge-Kutta time integration solvers. The test cases comprises a now classical manufactured solution in the projection method literature and a modified version of a more recently proposed manufactured solution.

Time-Optimal Path Planning in Uncertain Flow Fields Using Stochastic Dynamically Orthogonal Level Set Equations

Wei, Q.J., 2015. Time-Optimal Path Planning in Uncertain Flow Fields Using Stochastic Dynamically Orthogonal Level Set Equations, B.S. Thesis, Massachusetts Institute of Technology, Dept. of Mechanical Engineering, June 2015.

Path-planning has many applications, ranging from self-driving cars to flying drones, and to our daily commute to work. Path-planning for autonomous underwater vehicles presents an interesting problem: the ocean flow is dynamic and unsteady. Additionally, we may not have perfect knowledge of the ocean flow. Our goal is to develop a rigorous and computationally efficient methodology to perform path-planning in uncertain flow fields. We obtain new stochastic Dynamically Orthogonal (DO) Level Set equations to account for uncertainty in the flow field. We first review existing path-planning work: time-optimal path planning using the level set method, and energy-optimal path planning using stochastic DO level set equations. We build on these methods by treating the velocity field as a stochastic variable and deriving new stochastic DO level set equations. We use the new DO equations to simulate a simple canonical flow, the stochastic highway. We verify that our results are correct by comparing to corresponding Monte Carlo results. We explore novel methods of visualizing the results of the equations. Finally we apply our methodology to an idealized ocean simulation using Double-Gyre flows.


Energy Optimal Path Planning Using Stochastic Dynamically Orthogonal Level Set Equations

Subramani, D.N., 2014. Energy Optimal Path Planning Using Stochastic Dynamically Orthogonal Level Set Equations. SM Thesis, Massachusetts Institute of Technology, Computation for Design and Optimization Graduate Program, September 2014.

The growing use of autonomous underwater vehicles and underwater gliders for a variety of applications gives rise to new requirements in the operation of these vehicles. One such important requirement is optimization of energy required for undertaking missions that will enable longer endurance and lower operational costs. Our goal in this thesis is to develop a computationally efficient, and rigorous methodology that can predict energy-optimal paths from among all time-optimal paths to complete an underwater mission. For this, we develop rigorous a new stochastic Dynamically Orthogonal Level Set optimization methodology. In our thesis, after a review of existing path planning methodologies with a focus on energy optimality, we present the background of time-optimal path planning using the level set method. We then lay out the questions that inspired the present thesis, provide the goal of the current work and explain an extension of the time-optimal path planning methodology to the time-optimal path planning in the case of variable nominal engine thrust. We then proceed to state the problem statement formally. Thereafter, we develop the new methodology for solving the optimization problem through stochastic optimization and derive new Dynamically Orthogonal Level Set Field equations. We then carefully present different approaches to handle the non-polynomial non-linearity in the stochastic Level Set Hamilton-Jacobi equations and also discuss the computational efficiency of the algorithm. We then illustrate the inner-workings and nuances of our new stochastic DO level set energy optimal path planning algorithm through two simple, yet important, canonical steady flows that simulate a steady front and a steady eddy. We formulate a double energy-time minimization to obtain a semi-analytical energy optimal path for the steady front crossing test case and compare the results to these of our stochastic DO level set scheme. We then apply our methodology to an idealized ocean simulation using Double Gyre flows, and finally show an application with real ocean data for completing a mission in the Middle Atlantic Bight and New Jersey Shelf/Hudson Canyon region.

Time-optimal Path Planning for Sea-surface Vehicles Under the Effects of Strong Currents and Winds

Hessels, B., 2014. Time-optimal Path Planning for Sea-surface Vehicles Under the Effects of Strong Currents and Winds.. BS Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, June 2014.

A path-planning methodology that takes into account sea state fields, specifically wind forcing, is discussed and exemplified in this thesis. This general methodology has been explored by the Multidisciplinary Simulation, Estimation, and Assimilation Systems group (MSEAS) at MIT, however this is the first instance of wind effects being taken into account. Previous research explored vessels and isotropy, where the nominal speed of the vessel is uniform in all directions. This thesis explores the non-isotropic case, where the maximum speed of the vessel varies with direction, such as a sailboat. Our goal in this work is to predict the time-optimal path between a set of coordinates, taking into account flow currents and wind speeds. This thesis reviews the literature on a modified level set method that governs the path in any continuous flow to minimize travel time. This new level set method, pioneered by MSEAS, evolves a front from the starting coordinate until any point on that front reaches the destination. The vehicles optimal path is then gained by solving a particle back tracking equation. This methodology is general and applicable to any vehicle, ranging from underwater vessels to aircraft, as it rigorously takes into account the advection effects due to any type of environmental flow fields such as time-dependent currents and dynamic wind fields.

High Order Hybrid Discontinuous Galerkin Regional Ocean Modeling

Ueckermann, M.P., 2014. High Order Hybrid Discontinuous Galerkin Regional Ocean Modeling. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2014.

Accurate modeling of physical and biogeochemical dynamics in coastal ocean regions is required for multiple scientific and societal applications, covering a wide range of time and space scales. However, in light of the strong nonlinearities observed in coastal regions and in biological processes, such modeling is challenging. An important subject that has been largely overlooked is the numerical requirements for regional ocean simulation studies. Major objectives of this thesis are to address such computational questions for non-hydrostatic multiscale flows and for biogeochemical interactions, and to derive and develop numerical schemes that meet these requirements, utilizing the latest advances in computational fluid dynamics.

We are interested in studying nonlinear, transient, and multiscale ocean dynamics over complex geometries with steep bathymetry and intricate coastlines, from sub-mesoscales to basin-scales. These dynamical interests, when combined with our requirements for accurate, efficient and flexible ocean modeling, led us to develop new variable resolution, higher-order and non-hydrostatic ocean modeling schemes. Specifically, we derived, developed and applied new numerical schemes based on the novel hybrid discontinuous Galerkin (HDG) method in combination with projection methods.

The new numerical schemes are first derived for the Navier-Stokes equations. To ensure mass conservation, we define numerical fluxes that are consistent with the discrete divergence equation. To improve stability and accuracy, we derive a consistent HDG stability parameter for the pressure-correction equation. We also apply a new boundary condition for the pressure-corrector, and show the form and origin of the projection method’s time-splitting error for a case with implicit diffusion and explicit advection. Our scheme is implemented for arbitrary, mixed-element unstructured grids using a novel quadrature-free integration method for a nodal basis, which is consistent with the HDG method. To prevent numerical oscillations, we design a selective high-order nodal limiter. We demonstrate the correctness of our new schemes using a tracer advection benchmark, a manufactured solution for the steady diffusion and stokes equations, and the 2D lock-exchange problem.

These numerical schemes are then extended for non-hydrostatic, free-surface, variable-density regional ocean dynamics. The time-splitting procedure using projection methods is derived for non-hydrostatic or hydrostatic, and nonlinear free-surface or rigid-lid, versions of the model. We also derive consistent HDG stability parameters for the free-surface and non-hydrostatic pressure-corrector equations to ensure stability and accuracy. New boundary conditions for the free-surface-corrector and pressure-corrector are also introduced. We prove that these conditions lead to consistent boundary conditions for the free-surface and pressure proper. To ensure discrete mass conservation with a moving free-surface, we use an arbitrary LagrangianEulerian (ALE) moving mesh algorithm. These schemes are again verified, this time using a tidal flow problem with analytical solutions and a 3D lock-exchange benchmark.

We apply our new numerical schemes to evaluate the numerical requirements of the coupled biological-physical dynamics. We find that higher-order schemes are more accurate at the same efficiency compared to lower-order (e.g. second-order) accurate schemes when modeling a biological patch. Due to decreased numerical dissipation, the higher-order schemes are capable of modeling biological patchiness over a sustained duration, while the lower-order schemes can lose significant biomass after a few non-dimensional times and can thus solve erroneous nonlinear dynamics.

Finally, inspired by Stellwagen Bank in Massachusetts Bay, we study the effect of non-hydrostatic physics on biological productivity and phytoplankton fields for tidally-driven flows over an idealized bank. We find that the non-hydrostatic pressure and flows are important for biological dynamics, especially when flows are supercritical. That is, when the slope of the topography is larger than the slope of internal wave rays at the tidal frequency. The non-hydrostatic effects increase with increasing nonlinearity, both when the internal Froude number and criticality parameter increase. Even in cases where the instantaneous biological productivity is not largely modified, we find that the total biomass, spatial variability and patchiness of phytoplankton can be significantly altered by non-hydrostatic processes.

Our ultimate dynamics motivation is to allow quantitative simulation studies of fundamental nonlinear biological-physical dynamics in coastal regions with complex bathymetric features such as straits, sills, ridges and shelfbreaks. This thesis develops the necessary numerical schemes that meet the stringent accuracy requirements for these types of flows and dynamics.


Coastal Ocean Variability off the Coast of Taiwan in Response to Typhoon Morakot: River Forcing, Atmospheric Forcing and Cold Dome Dynamics

Landry, J.J., 2014. Coastal Ocean Variability off the Coast of Taiwan in Response to Typhoon Morakot: River Forcing, Atmospheric Forcing and Cold Dome Dynamics. SM Thesis, MIT-WHOI Joint Program, September 2014.

The ocean is a complex, constantly changing, highly dynamical system. Prediction capabilities are constantly being improved in order to better understand and forecast ocean properties for applications in science, industry, and maritime interests. Our overarching goal is to better predict the ocean environment in regions of complex topography with a continental shelf, shelfbreak, canyons and steep slopes using the MIT Multidisciplinary Simulation, Estimation and Assimilation Systems (MSEAS) primitive-equation ocean model. We did this by focusing on the complex region surrounding Taiwan, and the period of time immediately following the passage of Typhoon Morakot. This area and period were studied extensively as part of the intense observation period during August – September 2009 of the joint U.S. – Taiwan program Quantifying, Predicting, and Exploiting Uncertainty Department Research Initiative (QPE DRI). Typhoon Morakot brought an unprecedented amount of rainfall within a very short time period and in this research, we model and study the effects of this rainfall on Taiwan’s coastal oceans as a result of river discharge. We do this through the use of a river discharge model and a bulk river-ocean mixing model. We complete a sensitivity study of the primitive-equation ocean model simulations to the different parameters of these models. By varying the shape, size, and depth of the bulk mixing model footprint, and examining the resulting impacts on ocean salinity forecasts, we are able to determine an optimal combination of salinity relaxation factors for highest accuracy.

Generation of High Quality 2D Meshes for Given Bathymetry

Colmenero J., 2014. Generation of High Quality Meshes for Given Bathymetry. BS Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, June 2014.

This thesis develops and applies a procedure to generate high quality 2D meshes for any given ocean region with complex coastlines. The different criteria used in determining mesh element sizes for a given domain are discussed, especially sizing criteria that depend on local properties of the bathymetry and relevant dynamical scales. Two different smoothing techniques, Laplacian conditioning and targeted averaging, were applied to the fields involved in calculating the sizing matrix. The L^2 norm was used to quantify which technique had the greatest preservation of the original field. In both the reduced gradient and gradient cases, targeted averaging had a lower L^2 norm. The sizing matrices were used as inputs for two mesh generators, Distmesh and GMSH, and their meshing results were presented over a set of ocean domains in the Gulf of Maine and Massachusetts Bay region. Further research into the capabilities of each mesh generator are needed to provide a detailed evaluation. Mesh quality issues near coastlines revealed the need for small scale feature size recognition algorithms that could be implemented and studied in the future.

Missiles & Misconceptions: Why We Know More About the Dark Side of the Moon than the Depths of the Ocean

Young, G.C., 2014. Missiles & Misconceptions: Why We Know More About the Dark Side of the Moon than the Depths of the Ocean. BS Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, June 2014.

We know more about the dark side of the moon than the depths of the ocean. This is startling, considering how much more tangible the ocean is than space, and more importantly, how much more critical it is to the health and survival of humanity. Tens of billions of dollars are spent on manned and unmanned missions probing deeper into space, while 95% of Earth’s oceans remain unexplored. The result is a perilous dearth in knowledge about our planet at a time when rapid changes in our marine ecosystems profoundly affect its habitability.
The more intensive focus on space exploration is a historically recent phenomenon. For millennia until the mid-20th century, space and ocean exploration proceeded roughly at the same pace, driven by curiosity, military, and commerce. Both date back to early civilization when star-gazers scanned the skies, and sailors and free-divers scoured the seas. Since the 1960s when Don Walsh and Jacques Piccard descended to the deepest point on the ocean floor, and Neil Armstrong ascended to the moon, however, the trajectories of exploration diverged dramatically. Cold War-inspired geopolitical-military imperatives propelled space research to en extraordinary level, while ocean exploration stagnated in comparison. Moreover, although the Cold War ended more than 20 years ago, the disparity in effort remains vast despite evidence that accelerating changes in our marine ecosystems directly threatens our well being. Misconception about the relative importance of space and ocean exploration caused, and continues to sustain, this knowledge disparity to our peril.
In this thesis, we first review in section 2 the history of space and ocean exploration before the Cold War, when the pace of exploration in each sector was more or less comparable for thousands of years. We show in section 3, however, how the relative paces and trajectories of exploration diverged dramatically during the Cold War and continue to the present. In section 4 we seek to dispel the persistent misconceptions that have led to the disparity in resources allocated between space and ocean exploration, and argue for prioritizing ocean research. Finally, in section 5 we highlight the urgent imperative for expanding our understanding of the ocean.

Uncertainty Quantification and Prediction for Non-autonomous Linear and Nonlinear Systems

Phadnis, A., 2013. Uncertainty Quantification and Prediction for Non-autonomous Linear and Nonlinear Systems. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2013.

p> The science of uncertainty quantification has gained a lot of attention over recent years. This is because models of real processes always contain some elements of uncertainty, and also because real systems can be better described using stochastic components. Stochastic models can therefore be utilized to provide a most informative prediction of possible future states of the system. In light of the multiple scales, nonlinearities and uncertainties in ocean dynamics, stochastic models can be most useful to describe ocean systems.

Uncertainty quantification schemes developed in recent years include order reduction methods (e.g. proper orthogonal decomposition (POD)), error subspace statistical estimation (ESSE), polynomial chaos (PC) schemes and dynamically orthogonal (DO) field equations. In this thesis, we focus our attention on DO and various PC schemes for quantifying and predicting uncertainty in systems with external stochastic forcing. We develop and implement these schemes in a generic stochastic solver for a class of non-autonomous linear and nonlinear dynamical systems. This class of systems encapsulates most systems encountered in classic nonlinear dynamics and ocean modeling, including flows modeled by Navier-Stokes equations. We first study systems with uncertainty in input parameters (e.g. stochastic decay models and Kraichnan-Orszag system) and then with external stochastic forcing (autonomous and non-autonomous self-engineered nonlinear systems). For time-integration of system dynamics, stochastic numerical schemes of varied order are employed and compared. Using our generic stochastic solver, the Monte Carlo, DO and polynomial chaos schemes are intercompared in terms of accuracy of solution and computational cost.

To allow accurate time-integration of uncertainty due to external stochastic forcing, we also derive two novel PC schemes, namely, the reduced space KLgPC scheme and the modified TDgPC (MTDgPC) scheme. We utilize a set of numerical examples to show that the two new PC schemes and the DO scheme can integrate both additive and multiplicative stochastic forcing over significant time intervals. For the final example, we consider shallow water ocean surface waves and the modeling of these waves by deterministic dynamics and stochastic forcing components. Specifically, we time-integrate the Korteweg-de Vries (KdV) equation with external stochastic forcing, comparing the performance of the DO and Monte Carlo schemes. We find that the DO scheme is computationally efficient to integrate uncertainty in such systems with external stochastic forcing.


Bayesian inference of stochastic dynamical models

Lu, P., 2013. Bayesian inference of stochastic dynamical models. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2013.

A new methodology for Bayesian inference of stochastic dynamical models is developed. The methodology leverages the dynamically orthogonal (DO) evolution equations for reduced-dimension uncertainty evolution and the Gaussian mixture model DO filtering algorithm for nonlinear reduced-dimension state variable inference to perform parallelized computation of marginal likelihoods for multiple candidate models, enabling efficient Bayesian update of model distributions. The methodology also employs reduced-dimension state augmentation to accommodate models featuring uncertain parameters. The methodology is applied successfully to two high-dimensional, nonlinear simulated fluid and ocean systems. Successful joint inference of an uncertain spatial geometry, one uncertain model parameter, and 0(105) uncertain state variables is achieved for the first. Successful joint inference of an uncertain stochastic dynamical equation and 0(105) uncertain state variables is achieved for the second. Extensions to adaptive modeling and adaptive sampling are discussed.


Stochastic Modeling of Flows behind a Square Cylinder with uncertain Reynolds numbers

Wamala, J., 2012. Stochastic Modeling of Flows behind a Square Cylinder with uncertain Reynolds numbers. BS Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, June 2012.

In this thesis, we explore the use of stochastic Navier-Stokes equations through the Dynamically Orthogonal (DO) methodology developed at MIT in the Multidisciplinary Simulation, Estimation, and Assimilation Systems Group. Specifically, we examine the effects of the Reynolds number on stochastic fluid flows behind a square cylinder and evaluate computational schemes to do so. We review existing literature, examine our simulation results and validate the numerical solution. The thesis uses a novel open boundary condition formulation for DO stochastic Navier-Stokes equations, which allows the modeling of a wide range of random inlet boundary conditions with a single DO simulation of low stochastic dimensions, reducing computational costs by orders of magnitude. We first test the numerical convergence and validating the numerics. We then study the sensitivity of the results to several parameters, focusing for the dynamics on the sensitivity to the Reynolds number. For the method, we focus on the sensitivity to the: resolution of in the stochastic subspace, resolution in the physical space and number of open boundary conditions DO modes. Finally, we evaluate and study how key dynamical characteristics of the flow such as the recirculation length and the vortex shedding period vary with the Reynolds number.

Technological Review of Deep Ocean Manned Submersibles

Vaskov, A.K., 2012. Technological Review of Deep Ocean Manned Submersibles. BS Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, June 2012.

James Cameron’s dive to the Challenger Deep in the Deepsea Challenger in March of 2012 marked the first time man had returned to the Mariana Trench since the Bathyscaphe Trieste’s 1960 dive. Currently little is known about the geological processes and ecosystems of the deep ocean. The Deepsea Challenger is equipped with a plethora of instrumentation to collect scientific data and samples. The development of the Deepsea Challenger has sparked a renewed interest in manned exploration of the deep ocean.
Due to the immense pressure at full ocean depth, a variety of advanced systems and materials are used on Cameron’s dive craft. This paper provides an overview of the many novel features of the Deepsea Challenger as well as related features of past vehicles that have reached the Challenger Deep. Four key areas of innovation are identified: buoyancy materials, pilot sphere construction/instrument housings, lighting, and battery power. An in depth review of technological development in these areas is provided, as well as a glimpse into future manned submersibles and their technologies of choice.

Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations

Sondergaard, T., 2011. Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2011.

Data assimilation, as presented in this thesis, is the statistical merging of sparse observational data with computational models so as to optimally improve the probabilistic description of the field of interest, thereby reducing uncertainties. The centerpiece of this thesis is the introduction of a novel such scheme that overcomes prior shortcomings observed within the community. Adopting techniques prevalent in Machine Learning and Pattern Recognition, and building on the foundations of classical assimilation schemes, we introduce the GMM-DO filter: Data Assimilation with Gaussian mixture models using the Dynamically Orthogonal field equations.

We combine the use of Gaussian mixture models, the EM algorithm and the Bayesian Information Criterion to accurately approximate distributions based on Monte Carlo data in a framework that allows for efficient Bayesian inference. We give detailed descriptions of each of these techniques, supporting their application by recent literature. One novelty of the GMM-DO filter lies in coupling these concepts with an efficient representation of the evolving probabilistic description of the uncertain dynamical field: the Dynamically Orthogonal field equations. By limiting our attention to a dominant evolving stochastic subspace of the total state space, we bridge an important gap previously identified in the literature caused by the dimensionality of the state space.

We successfully apply the GMM-DO filter to two test cases: (1) the Double Well Diffusion Experiment and (2) the Sudden Expansion fluid flow. With the former, we prove the validity of utilizing Gaussian mixture models, the EM algorithm and the Bayesian Information Criterion in a dynamical systems setting. With the application of the GMM-DO filter to the two-dimensional Sudden Expansion fluid flow, we further show its applicability to realistic test cases of non-trivial dimensionality. The GMM-DO filter is shown to consistently capture and retain the far-from-Gaussian statistics that arise, both prior and posterior to the assimilation of data, resulting in its superior performance over contemporary filters. We present the GMM-DO filter as an efficient, data-driven assimilation scheme, focused on a dominant evolving stochastic subspace of the total state space, that respects nonlinear dynamics and captures non-Gaussian statistics, obviating the use of heuristic arguments.


Dynamically Orthogonal Field Equations for Stochastic Fluid Flows and Particle Dynamics

Sapsis, Themis, 2011. Dynamically Orthogonal Field Equations for Stochastic Fluid Flows and Particle Dynamics. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2011.

In the past decades an increasing number of problems in continuum theory have been treated using stochastic dynamical theories. This is because dynamical systems governing real processes always contain some elements characterized by uncertainty or stochasticity. Uncertainties may arise in the system parameters, the boundary and initial conditions, and also in the external forcing processes. Also, many problems are treated through the stochastic framework due to the incomplete or partial understanding of the governing physical laws. In all of the above cases the existence of random perturbations, combined with the com- plex dynamical mechanisms of the system often leads to their rapid growth which causes distribution of energy to a broadband spectrum of scales both in space and time, making the system state particularly complex. Such problems are mainly described by Stochastic Partial Differential Equations and they arise in a number of areas including fluid mechanics, elasticity, and wave theory, describing phenomena such as turbulence, random vibrations, flow through porous media, and wave propagation through random media. This is but a partial listing of applications and it is clear that almost any phenomenon described by a field equation has an important subclass of problems that may profitably be treated from a stochastic point of view.

In this work, we develop a new methodology for the representation and evolution of the complete probabilistic response of infinite-dimensional, random, dynamical systems. More specifically, we derive an exact, closed set of evolution equations for general nonlinear continuous stochastic fields described by a Stochastic Partial Differential Equation. The derivation is based on a novel condition, the Dynamical Orthogonality (DO), on the representation of the solution. This condition is the “key” to overcome the redundancy issues of the full representation used while it does not restrict its generic features. Based on the DO condition 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 uncertainty “lives” as well as a system of Stochastic Differential Equations that defines how the uncertainty evolves in the time varying stochastic subspace. If additional restrictions are assumed on the form of the representation, we recover both the Proper-Orthogonal-Decomposition (POD) equations and the generalized Polynomial- Chaos (PC) equations; thus the new methodology generalizes these two approaches. For the efficient treatment of the strongly transient character on the systems described above we derive adaptive criteria for the variation of the stochastic dimensionality that characterizes the system response. Those criteria follow directly from the dynamical equations describing the system.

We illustrate and validate this novel technique by solving the 2D stochastic Navier-Stokes equations in various geometries and compare with direct Monte Carlo simulations. We also apply the derived framework for the study of the statistical responses of an idealized “double gyre” model, which has elements of ocean, atmospheric and climate instability behaviors.

Finally, we use our new stochastic description for flow fields to study the motion of inertial particles in flows with uncertainties. Inertial or finite-size particles in fluid flows are commonly encountered in nature (e.g., contaminant dispersion in the ocean and atmosphere) as well as in technological applications (e.g., chemical systems involving particulate reactant mixing). As it has been observed both numerically and experimentally, their dynamics can differ markedly from infinitesimal particle dynamics. Here we use recent results from stochastic singular perturbation theory in combination with the DO representation of the random flow, in order to derive a reduced order inertial equation that will describe efficiently the stochastic dynamics of inertial particles in arbitrary random flows.


Upwelling Dynamics off Monterey Bay: Heat Flux and Temperature Variability, and their Sensitivities

Kaufman, M.R.S., 2010. Upwelling Dynamics off Monterey Bay: Heat Flux and Temperature Variability, and their Sensitivities. BS Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, May 2010.

Understanding the complex dynamics of coastal upwelling is essential for coastal ocean dynamics, phytoplankton blooms, and pollution transport. Atmospheric- driven coastal upwelling often occurs when strong alongshore winds and the Coriolis force combine to displace warmer surface waters offshore, leading to upward motions of deeper cooler, nutrient-dense waters to replace these surface waters. Using the models of the MIT Multidisciplinary Simulation, Estimation, and Assimilation System (MSEAS) group, we conduct a large set of simulation sensitivity studies to determine which variables are dominant controls for upwelling events in the Monterey Bay region. Our motivations include determining the dominant atmospheric fluxes and the causes of high-frequency fluctuations found in ocean thermal balances. We focus on the first upwelling event from August 1- 5, 2006 in Monterey Bay that occurred during the Monterey Bay 06 (MB06) at-sea experiment, for which MSEAS data-assimilative baseline simulations already existed.

Using the thermal energy (temperature), salinity and momentum (velocity) conservation equations, full ocean fields in the region as well as both control volume (flux) balances and local differential term-by-term balances for the upwelling event events were computed. The studies of ocean fields concentrate on specific depths: surface-0m, thermocline-30m and undercurrent-150m. Effects of differing atmospheric forcing contributions (wind stress, surface heating/cooling, and evaporation-precipitation) on these full fields and on the volume and term-by-term balances are analyzed. Tidal effects are quantified utilizing pairs of simulations in which tides are either included or not. Effects of data assimilation are also examined.

We find that the wind stress forcing is the most important dynamical parameter in explaining the extent and shape of the upwelling event. This is verified using our large set of sensitivity studies and examining the heat flux balances. The assimilation of data has also an impact because this first upwelling event occurs during the initialization. Tidal forcing and, to a lesser extent, the daily atmospheric and data assimilation cycles explain the higher frequency fluctuations found in the volume averaged time rate of change of thermal energy.


Towards Next Generation Ocean Models: Novel Discontinuous Galerkin Schemes for 2D unsteady biogeochemical models

Ueckermann, M.P., 2009. Towards Next Generation Ocean Models: Novel Discontinuous Galerkin Schemes for 2D unsteady biogeochemical models. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2009.

A new generation of efficient parallel, multi-scale, and interdisciplinary ocean models is required for better understanding and accurate predictions. The purpose of this thesis is to quantitatively identify promising numerical methods that are suitable to such predictions. In order to fulfill this purpose, current efforts towards creating new ocean models are reviewed, an understanding of the most promising methods used by other researchers is developed, the most promising existing methods are studied and applied to idealized cases, new methods are incubated and evaluated by solving test problems, and important numerical issues related to efficiency are examined. The results of other research groups towards developing the second generation of ocean models are first reviewed. Next, the Discontinuous Galerkin (DG) method for solving advection-diffusion problems is described, including a discussion on schemes for solving higher order derivatives. The discrete formulation for advection-diffusion problems is detailed and implementation issues are discussed. The Hybrid Discon- tinuous Galerkin (HDG) Finite Element Method (FEM) is identified as a promising new numerical scheme for ocean simulations. For the first time, a DG FEM scheme is used to solve ocean biogeochemical advection-diffusion-reaction equations on a two- dimensional idealized domain, and p-adaptivity across constituents is examined. Each aspect of the numerical solution is examined separately, and p-adaptive strategies are explored. Finally, numerous solver-preconditioner combinations are benchmarked to identify an efficient solution method for inverting matrices, which is necessary for implicit time integration schemes. From our quantitative incubation of numerical schemes, a number of recommendations on the tools necessary to solve dynamical equations for multiscale ocean predictions are provided.

Statistical Field Estimation and Scale Estimation for Complex Coastal Regions and Archipelagos

Agarwal, A., 2009. Statistical Field Estimation and Scale Estimation for Complex Coastal Regions and Archipelagos. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, May 2009.

A fundamental requirement in realistic computational geophysical fluid dynamics is the optimal estimation of gridded fields and of spatial-temporal scales directly from the spatially irregular and multivariate data sets that are collected by varied instruments and sampling schemes. In this work, we derive and utilize new schemes for the mapping and dynamical inference of ocean fields in complex multiply-connected domains, study the computational properties of our new mapping schemes, and derive and investigate new schemes for adaptive estimation of spatial and temporal scales.

Objective Analysis (OA) is the statistical estimation of fields using the Bayesian- based Gauss-Markov theorem, i.e. the update step of the Kalman Filter. The existing multi-scale OA approach of the Multidisciplinary Simulation, Estimation and Assimilation System consists of the successive utilization of Kalman update steps, one for each scale and for each correlation across scales. In the present work, the approach is extended to field mapping in complex, multiply-connected, coastal regions and archipelagos. A reasonably accurate correlation function often requires an estimate of the distance between data and model points, without going across complex land- forms. New methods for OA based on estimating the length of optimal shortest sea paths using the Level Set Method (LSM) and Fast Marching Method (FMM) are derived, implemented and utilized in general idealized and realistic ocean cases. Our new methodologies could improve widely-used gridded databases such as the climatological gridded fields of the World Ocean Atlas (WOA) since these oceanic maps were computed without accounting for coastline constraints. A new FMM-based methodology for the estimation of absolute velocity under geostrophic balance in complicated domains is also outlined. Our new schemes are compared with other approaches, including the use of stochastically forced differential equations (SDE). We find that our FMM-based scheme for complex, multiply-connected, coastal regions is more efficient and accurate than the SDE approach. We also show that the field maps obtained using our FMM-based scheme do not require postprocessing (smoothing) of fields. The computational properties of the new mapping schemes are studied in detail. We find that higher-order schemes improve the accuracy of distance estimates. We also show that the covariance matrices we estimate are not necessarily positive definite because the Weiner Khinchin and Bochner relationships for positive definiteness are only valid for convex simply-connected domains. Several approaches to overcome this issue are discussed and qualitatively evaluated. The solutions we propose include introducing a small process noise or reducing the covariance matrix based on the dominant singular value decomposition. We have also developed and utilized novel methodologies for the adaptive estimation of spatial-temporal scales from irregularly spaced ocean data. The three novel methodologies are based on the use of structure functions, short term Fourier transform and second generation wavelets. To our knowledge, this is the first time that adaptive methodologies for the spatial-temporal scale estimation are proposed. The ultimate goal of all these methods would be to create maps of spatial and temporal scales that evolve as new ocean data are fed to the scheme. This would potentially be a significant advance to the ocean community for better understanding and sampling of ocean processes.


Modeling Coupled Physics and Biology in Ocean Straits with Application to the San Bernardino Strait in the Philippine Archipelago

Burton, L.J., 2009. Modeling Coupled Physics and Biology in Ocean Straits with Application to the San Bernardino Strait in the Philippine Archipelago. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, May 2009.

In this thesis, we conduct research toward understanding coupled physics-biology processes in ocean straits. Our focus is on new analytical studies and higher-order simulations of idealized dynamics that are relevant to generic biological processes. The details of coupled physics-biology models are reviewed and an in-depth global equilibrium and local stability analysis of a Nutrient-Phytoplankton-Zooplankton (NPZ) model is performed. This analysis includes parameter studies and methods to evaluate parameter sensitivity, especially in the case where some system parameters are unknown. As an initial step toward investigating the interaction between physics and biology in ocean straits, we develop and verify a new coupled physics-biology model for two-dimensional idealized physical processes including tides and apply it to the San Bernardino Strait in the Philippine Archipelago. This two-dimensional numerical model is created on a structured grid using operator splitting and masking. This model is able to accurately represent biology for various physical flows, including advection-dominated flows over discontinuities, by using the Weighted Essentially Non-Oscillatory (WENO) scheme. The numerical model is verified against a Discontinuous-Galerkin (DG) numerical scheme on an unstructured grid. Several simulations of tidal flow are completed using bathymetry and flow magnitudes com- parable to those found in the San Bernardino Strait with different sets of parameters, tidal periods, and levels of diffusion. Results are discussed and compared to those of a three-dimensional modeling system. New results include: new methods for analyzing stability, the robust two-dimensional model designed to best represent advection-dominant flows with minimal numerical diffusion and computational time, and a novel technique to initialize three-dimensional biology fields using satellite data. Additionally, application of the two-dimensional model with tidal forcing to the San Bernardino Strait reveals that flow frequencies have strong influence on biology, as very fast oscillations act to stabilize biology in the water column, while slower frequencies provide sufficient transport for increased biological activity.

Parameter Estimation and Adaptive Modeling Studies in Ocean Mixing

Heubel, E., 2008. Parameter Estimation and Adaptive Modeling Studies in Ocean Mixing. SM Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2008.

In this thesis, we explore the different methods for parameter estimation in straightforward diffusion problems and develop ideas and distributed computational schemes for the automated evaluation of physical and numerical parameters of ocean models. This is one step of “adaptive modeling”. Adaptive modeling consists of the automated adjustment of self-evaluating models in order to best represent an observed system. In the case of dynamic parameterizations, self-modifying schemes are used to learn the correct model for a particular regime as the physics change and evolve in time.

The parameter estimation methods are tested and evaluated on one-dimensional tracer diffusion problems. Existing state estimation methods and new filters, such as the unscented transform Kalman filter, are utilized in carrying out parameter estimation. These include the popular Extended Kalman Filter (EKF), the Ensemble Kalman Filter (EnKF) and other ensemble methods such as Error Subspace Statistical Estimation (ESSE) and Ensemble Adjustment Kalman Filter (EAKF), and the Unscented Kalman Filter (UKF). Among the aforementioned recursive state estimation methods, the so-called “adjoint method” is also applied to this simple study.

Finally, real data is examined for the applicability of such schemes in real-time fore- casting using the MIT Multidisciplinary Simulation, Estimation, and Assimilation System (MSEAS). The MSEAS model currently contains the free surface hydrostatic primitive equation model from the Harvard Ocean Prediction System (HOPS), a barotropic tidal prediction scheme, and an objective analysis scheme, among other models and developing routines. The experiment chosen for this study is one which involved the Monterey Bay region off the coast of California in 2006 (MB06). Accurate vertical mixing parameterizations are essential in this well known upwelling region of the Pacific. In this realistic case, parallel computing will be utilized by scripting code runs in C-shell. The performance of the simulations with different parameters is evaluated quantitatively using Pattern Correlation Coefficient, Root Mean Squared error, and bias error. Comparisons quantitatively determined the most adequate model setup.


Adaptive Rapid Environmental Assessment

Ding Wang, 2007. Adaptive Rapid Environmental Assessment. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2007 (Co-supervised with Prof. Henrik Schmidt).

In shallow water, a large part of underwater acoustic prediction uncertainties are in- duced by sub-meso-to-small scale oceanographic variabilities. Conventional oceano- graphic measurements for capturing such ocean-acoustic environmental variabilities face the classical conflict between resolution and coverage. The Adaptive Rapid En- vironmental Assessment (AREA) project was proposed to resolve this conflict by optimizing the location of in-situ measurements in an adaptive manner. In this thesis, ideas, concepts and performance limits in AREA are clarified. Both an engineering and a mathematical model for AREA are developed. A modularized AREA simulator was developed and implemented in C++. Philosophies in AREA are discussed. Presumptions about the ocean are made to bridge the gap between the viewpoint in the oceanography community, where the ocean environment is consid- ered to be a deterministic but very complicated system, and that of the underwater acoustic community, where the ocean environment is treated as a random system. At present, how to optimally locate the in-situ measurements made by a single AUV carrying a CTD (conductivity, temperature and depth) sensor is considered in AREA. In this thesis, the AUV path planning is modeled as a Shortest Path problem. However, due to the sound velocity correlation effect, the size of this problem can be very large. A method is developed to simplify the graph for a fast solution. As a significant step, a linear approximation for acoustic Transmission Loss (TL) is investigated numerically and analytically. In addition to following a predetermined path, an AUV can also adaptively gener- ate its path on-board. This adaptive on-board AUV routing problem is modeled using Dynamic Programming (DP) in this thesis. A method based on an optimized prede- termined path is developed to reduce the size of the DP problem and approximately yet efficiently solve it using Pattern Recognition. As a special case, a thermocline- oriented AUV yoyo control and control parameter optimization methods for AREA are also developed. 2 Finally, some AUV control algorithms for capturing fronts are developed. A frame- work for real-time TL forecasts is developed. This is the first time that TL forecasts have been linked with ocean forecasts in real-time. All of the above ideas and methods developed were tested in two experiments, FAF05 in the northern Tyrrhenian Sea in 2005 and MB06 in Monterey Bay, CA in 2006. The latter MB06 sea exercise was a major field experiment sponsored by the Office of Naval Research and the thesis compiles significant findings from this effort.