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

Abstract

AbstractWe 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