Coherence is a ubiquitous feature of natural phenomena in general and of geophysical fluids in particular. Reflecting many elements of the underlying flow, the appearance, geometry and apparent dynamics of coherent features are often used to describe or simulate evolution. However, we don't quite understand how to use them in inference problems pertaining to geophysical fluids such as data assimilation and uncertainty quantification, among others.
A Statistical Theory of Inference for Coherent Structures (STICS) is proposed, wherein fluids are organized as a spatial distribution of position-amplitude-scale features with embedded dynamical constraints. Features at preferred locations, amplitudes and scales are analyzed from physical fields, which we call Generalized Coherent Features(GCF). The GCFs are sparse. They are updated in response to observations, for example, using non-Gaussian data assimilation, and the fluid is synthesized as smooth, dynamically-sound reconstructions from the sparse features.
In many problems, however, the explicit detection of GCFs, establishing correspondence, and tracking them is difficult. We show that features can be taken into account "non-parametrically", as an auxiliary deformation field, so that the joint amplitude-deformation vector becomes the state variable upon which inference operates. This leads to a new formulation for Data Assimilation, Ensemble Statistics, Quantifying Uncertainty, EOF/POD analysis and Downscaling, among others documented on this site. The applications are truly broad. Here you will find methodology for synthesis, analysis and inference and their applications, along with codes.