Model error in data assimilation

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.