Three-dimensional (3D) underwater sound field computations have been used for a few decades to understand sound propagation effects above sloped seabeds and in areas with strong 3D temperature and salinity variations. For an approximate simulation of effects in nature, the necessary 3D sound-speed field can be made from snapshots of temperature and salinity from an operational data-driven regional ocean model. However, these models invariably have resolution constraints and physics approximations that exclude features that can have strong effects on acoustics, example features being strong submesoscale fronts and nonhydrostatic nonlinear internal waves (NNIWs). Here, work to predict NNIW fields to improve 3D acoustic forecasts using an NNIW model nested in a tide-inclusive data-assimilating regional model is reported. The work was initiated under the Integrated Ocean Dynamics and Acoustics project. The project investigated ocean dynamical processes that affect important details of sound-propagation, with a focus on those with strong intermittency (high kurtosis) that are challenging to predict deterministically. Strong internal tides and NNIW are two such phenomena, with the former being precursors to NNIW, often feeding energy to them. Successful aspects of the modeling are reported along with weaknesses and unresolved issues identified in the course of the work.

MSEAS research on the development of methodologies to predict the most informative sampling sites in the ocean for a given mission and optimal paths to reach them was highlighted in Underwater Contractor International (UCi), a popular magazine for underwater professionals, in their Jan/Feb 2019 issue. This research was funded, in part, by the Office of Naval Research, the MIT Lincoln Laboratory, the MIT Tata Center, and the National Science Foundation. The full article in the magazine can be found here.

The entire Jan/Feb 2019 issue can be found here.

The full article in PDF format can be accessed here.

A generalization of the concepts of extrinsic curvature and Weingarten endomorphism is introduced to study a class of nonlinear maps over embedded matrix manifolds. These (nonlinear) oblique projections, generalize (nonlinear) orthogonal projections, i.e. applications mapping a point to its closest neighbor on a matrix manifold. Examples of such maps include the truncated SVD, the polar decomposition, and functions mapping symmetric and non-symmetric matrices to their linear eigenprojectors. This paper specifically investigates how oblique projections provide their image manifolds with a canonical extrinsic differential structure, over which a generalization of the Weingarten identity is available. By diagonalization of the corresponding Weingarten endomorphism, the manifold principal curvatures are explicitly characterized, which then enables us to (i) derive explicit formulas for the differential of oblique projections and (ii) study the global stability of a governing generic Ordinary Differential Equation (ODE) computing their values. This methodology, exploited for the truncated SVD in (Feppon 2018), is generalized to non-Euclidean settings, and applied to the four other maps mentioned above and their image manifolds: respectively, the Stiefel, the isospectral, the Grassmann manifolds, and the manifold of fixed rank (non-orthogonal) linear projectors. In all cases studied, the oblique projection of a target matrix is surprisingly the unique stable equilibrium point of the above gradient flow. Three numerical applications concerned with ODEs tracking dominant eigenspaces involving possibly multiple eigenvalues finally showcase the results.