An Adjoint‐Free Alternating Direction Method for Four‐Dimensional Variational Data Assimilation With Multiple Parameter Tikhonov Regularization

Abstract

AbstractTikhonov regularization is critical for accurately specifying both the background (B) and observational (R) error covariances in four‐dimensional variational data assimilation (4DVar). The ratio of the background and observation error variances (referred to as the B‐R ratio) is the key to ensuring that 4DVar maximizes the information extracted from the observations. However, it is difficult to specify the regularization parameters in a high‐dimensional variational data assimilation (VDA) system for both the single‐ and multiple‐parameter regularization schemes. In this study, we used a maximum likelihood estimation (MLE)‐based inflation scheme that originated from the ensemble Kalman filter (EnKF) community and proposed an alternating direction method (ADM) to minimize the 4DVar cost function with the iterative application of multiple regularization parameters to simultaneously optimize the regularization parameters and model states under the framework of the nonlinear least‐squares 4‐D ensemble variational data assimilation method (NLS‐4DVar). The big‐data‐driven version of NLS‐4DVar (BD‐NLS4DVar) with multiple‐parameter Tikhonov regularization was able to adjust the B‐R ratios more accurately. Several groups of observing system simulation experiments (OSSEs) based on 2‐D shallow‐water equations demonstrated that BD‐NLS4DVar with multiple‐parameter Tikhonov regularization produced a substantial performance improvement over the standard BD‐NLS4DVar method with no regularization.