Square Root and Perturbed Observation Ensemble Generation Techniques in Kalman and Quadratic Ensemble Filtering Algorithms

Abstract

he main goal of this work is to present a new square root ensemble generation technique that is consistent with a recently developed extension of Kalman-based linear regression algorithms such that they may perform nonlinear polynomial regression (i.e., includes a quadratically nonlinear term in the mean update equation) and that is applicable to ensemble data assimilation in the geosciences. Along the way the authors present a unification of the theories of square root and perturbed observation (sometimes referred to as stochastic) ensemble generation in data assimilation algorithms configured to perform both linear (Kalman) regression as well as quadratic nonlinear regression. The performance of linear and nonlinear regression algorithms with both ensemble generation techniques is explored in the three-variable Lorenz model as well as in a nonlinear model configured to simulate shear layer instabilities.