A Lagrangian methodology based on a Discrete Kernel Filter (DKF) and the Ensemble Bred Vector (EBV) that uses local estimators that preserve significant dynamical features, detected by observations and in numerical simulations, is developed for the Forward Lagrangian Trajectory Prediction (LTP) and inverse source estimation problems that are fundamental in many different scientific disciplines. These nonlinear filtering problems are non-Gaussian and can have large-dimensional state spaces. DKF is based upon a particle filter and it does not suffer ensemble collapse because it has a built-in regeneration process in the parameterization of the diffusion process that defines the primary branches for prediction. The method linearizes about branches of prediction, yet makes no Gaussian assumption in the analysis stage. The EBV algorithm is used to find the best choices for branches of prediction, thus increasing the efficiency of the method significantly for application to important real-world problems such as oil spill modeling and pollution source identification, transport and dispersion of radioactive gases in the atmosphere, fish larvae transport and fishery connectivity, predicting sea ice motion, human colonization, mapping invasive species, and monitoring asteroid movements, to name a few.
Observations of fluid flows and the trajectories of the flows are called Eulerian if the observations are taken from points fixed independent of the flows, and Lagrangian if they are taken from points that move with the flows themselves. In the same way, a computational simulation of a flow is called Eulerian if the flow moves through a fixed computational grid, and Lagrangian if the grid moves with the flow. The investigators develop a method to predict the trajectories of fluid flows, which are subject to random perturbations. They apply the method to two problems where capturing features is critical: hurricane/typhoon tracking, and US Coast Guard search and rescue. Both of these problems are of great societal importance because more optimal solutions with tighter error bars can save both lives and money. For reasons of practicality they seek an efficient method that is also capable of (1) fusing multi-platform observations and numerical model simulations of ocean circulation for improving both the Eulerian, diagnostic variables of the model and prediction of Lagrangian trajectories, as well as the associated estimation uncertainties, and (2) handling already existing ensembles, fusing hurricane track forecast ensembles from the leading operational models (such as NCEP GFS, GFDL, UKMET, ECMWF, NOGAPS, others) to improve predictions of the path of tropical cyclones (hurricane, typhoon). The method can help produce state-of-the-art uncertainty maps that are critical in search and rescue flight planning and for reducing the "cone of uncertainty" for operational hurricane predictions.