L 1 ‐regularisation for ill‐posed problems in variational data assimilation

Abstract

We consider four‐dimensional variational data assimilation (4DVar) and show that it can be interpreted as Tikhonov or L 2 ‐regularisation, a widely used method for solving ill‐posed inverse problems. It is known from image restoration and geophysical problems that an alternative regularisation, namely L 1 ‐norm regularisation, recovers sharp edges better than L 2 ‐norm regularisation. We apply this idea to 4DVar for problems where shocks and model error are present and give two examples which show that L 1 ‐norm regularisation performs much better than the standard L 2 ‐norm regularisation in 4DVar. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)