Nonlinear, Nongaussian Ensemble Data Assimilation with Rank Regression and a Rank Histogram Filter

Serial implementations of ensemble Kalman filters can partially separate the challenges of nongaussian observation likelihoods and prior probability distributions from those associated with nonlinear relations between state variables and observed quantities. A number of serial algorithms have been developed that deal with nongaussianity in observation space. Here, the rank histogram ensemble filter which can represent arbitrary prior distributions is used.

There has been less work on dealing with the challenges of nonlinearity and most proposed algorithms have not provided significant improvements in large geophysical applications. The standard method for computing the ensemble increments for a model state variable given increments for an observation is standard linear regression which is implicitly used in ensemble filters like the EnKF and in the classical Kalman filter. A novel method in which regression is based on the Spearman Rank correlation as opposed to the standard Pearson correlation is described. This rank regression method is appropriate when the state variable being updated is a monotonic nonlinear function of the observed variable.

Comprehensive results in the Lorenz-96 model are presented to demonstrate the capabilities of standard and rank regression each combined with either standard gaussian or rank histogram updates for observed variables. A series of nonlinear forward operators with different concavity is examined to explore a variety of challenges with nonlinear ensemble assimilation. A discussion of the relative costs and capabilities of the algorithms is presented. Initial results in a simple atmosphere GCM with nonlinear forward operators demonstrate that the rank regression method can be effective in large applications. The challenges of recognizing and exploiting nonlinear relations in the presence of noise is highlighted.