Abstract

This chapter provides various perspectives on an important challenge in data assimilation: model error. While the overall goal is to understand the implication of model error of any type in data assimilation, we emphasize on the effect of model error from unresolved scales. In particular, connection to related subjects under different names in applied mathematics, such as the Mori-Zwanzig formalism and the averaging method, was discussed with the hope that the existing methods can be more accessible and eventually be used appropriately. We will classify existing methods into two groups: the statistical methods for those who directly estimate the low-order model error statistics; and the stochastic parameterizations for those who implicitly estimate all statistics by imposing stochastic models beyond the traditional unbiased white noise Gaussian processes. We will provide theory to justify why stochastic parameterization, as one of the main theme in this book, is an adequate tool for mitigating model error in data assimilation. Finally, we will also discuss challenges in lifting this approach in general applications and provide an alternative nonparametric approach.

Introduction

Data assimilation (or Bayesian filtering) is a statistical method to find the conditional distribution of the hidden variables of interest given noisy observations from nature. In application, the hidden variables of interest can be the state variables that are directly or indirectly observed or can even be some unobserved parameters in the models. In practice, data assimilation is typically realized by numerical schemes that produce conditional statistics of the state variables of interests, accounting for the information from the observations, rather than the corresponding conditional distribution; this gives a reasonable justification why we called it a “statistical method”. When observations are available at discrete times, Bayesian filtering is an iterative predictor-corrector scheme that adjusts the prior forecast (background) statistical estimates from a predictor (or dynamical model) to be more consistent with the current observations. This correction step is referred to as analysis in the atmospheric and ocean science (AOS) community. Subsequently, the posterior (corrected or analysis) statistical estimates are fed into the model as initial conditions for future time prior statistical estimates.