Resolution of sharp fronts in the presence of model error in variational data assimilation

Abstract

We show that the four-dimensional variational data assimilation method (4DVar) can be interpreted as a form of Tikhonov regularization, a very familiar method for solving ill-posed inverse problems. It is known from image restoration problems that L₁-norm penalty regularization recovers sharp edges in the image more accurately than Tikhonov, or L₂-norm, penalty regularization. We apply this idea from stationary inverse problems to 4DVar, a dynamical inverse problem, and give examples for an L₁-norm penalty approach and a mixed total variation (TV) L₁–L₂-norm penalty approach. For problems with model error where sharp fronts are present and the background and observation error covariances are known, the mixed TV L₁–L₂-norm penalty performs better than either the L₁-norm method or the strong-constraint 4DVar (L₂-norm) method. A strength of the mixed TV L₁–L₂-norm regularization is that in the case where a simplified form of the background error covariance matrix is used it produces a much more accurate analysis than 4DVar. The method thus has the potential in numerical weather prediction to overcome operational problems with poorly tuned background error covariance matrices. Copyright © 2012 Royal Meteorological Society