Coherence is a ubiquitous feature of natural phenomena in general and geophysical fluids in particular. Reflecting many elements of the underlying flow, appearance and geometry of coherent features are often used to characterize the apparent dynamics of fluids and describe them phenomenologically. However, the quantitative use of coherence in inference problems associated with geophysical fluids, such as data assimilation (estimation), uncertainty quantification, model reduction or induction, and downscaling, among others, remains poorly developed.

In 2003, we proposed a theory that  Symbiotically couples Transport with Inference for Coherent Signals (STICS), wherein fluids are organized dynamical systems of distributions of position-amplitude-scale "generalized coherence features" (GCF). One way to incoprorate GCFs (coherent structures) is to analyze them as sparse features at preferred locations, amplitudes, and scales from physical fields (observed or modeled), carry out estimation or inference to update the distributions in response to observations, for example, using non-Gaussian data assimilation and synthesize physical fields as smooth, dynamic reconstructions from the updated features.  In many problems, however, explicitly detecting GCFs, tracking them across time which may require establishing correspondence, and resynthesizing the fluid field is quite difficult. For example, Koopman operators allow us to go into the latent space but returning to the physical space is often impossible.

Here, we show that the featuredness in coherent signals can be taken into account "implicitly" and "non-parametrically", by defining 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 and couple what we know as Optimal Transport with Optimal Estimation (and Inference) in ways that neither along suffice for. Here you will find methodology for synthesis, analysis and inference and their applications, along with codes.