Variational Data Assimilation and Regularization for Ill-posed Problems: A Common Framework
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Data assimilation and inverse methods for ill-posed problems find optimal estimates of states or parameters. Methods for both combine observations with a model, which here we assume is a partial differential equation (PDE). Finding a compromise between observations and model is challenging because the actual observations often have values significantly different than the corresponding PDE estimates. Neither the observations nor the PDE exactly characterize the state because each has error, and in the case of the PDE, this can be due to unknown forcings, initial or boundary conditions.
State estimates from data assimilation can vary significantly depending on specified errors in the PDE. In this work we estimate PDE errors by developing a common framework between variational data assimilation and regularization for ill-posed problems. This framework arises when weakly constrained variational data assimilation is viewed as regularizing the severely underdetermined data fitting problem in data assimilation. Within this framework we derive error estimates for data assimilation using regularization parameter selection methods including the L-curve, Generalized Cross Validation (GCV) and the Chi-squared method. Data assimilation results will be shown from a one dimensional transport model with simulated data, where the resulting state estimates can be viewed as air quality estimates.