O the Topological Complexity of the Cost Function in Variational Data Assimilation

We investigate here the causes and implications of topological complexity in the cost function in the context of the variational data assimilation. This complexity, which can take many forms from multiple minima to very flat regions with almost no curvature, causes severe problems for minimization algorithms and can lead to a retrieved state that is strongly related to the initial guess, i.e., non-uniqueness. To determine the origin of multiple minima, we first utilize guess, simple dynamical systems: the chaotic logistic equation, Duffing's equation, and Burgers equation. We demonstrate that multiple minima are associated with nonlinear dynamics, and that the length of the assimilation window and data availability also exert a strong influence over the topology of the cost function. This finding is validated for the somewhat more complicated resonant Rossby wave model. Having established the principal causes of topological complexity in the cost function, we extend our work to the convective regime by performing retrievals using simulated data from a 3-D Boussinesq model and its adjoint. We find that the complexity of the cost function is not as serious a problem as shown in the simple nonlinear systems. Only in an extreme case, where the first guess is chosen to differ substantially from the true solution, does the retrieved state depart from the control, and even then it does so only superficially. This result could be due to the high dimensionality (i.e., greater degrees of freedom) of the Boussinesq flow. By including a penalty term, consisting of second-order temporal derivatives of the model state variables, the cost function is regularized and an improved retrieval is obtained. Such a penalty term can also improve the conditioning of the Hessian and thus the efficiency of the minimization process. The role of diffusion in data retrieval is also examined, and a linear analysis based on the one-dimensional diffusion equation shows that the retrieved initial state will be amplified (smoothed) if the diffusion in the prediction model is larger (smaller) than that of the observations. This amplification (smoothing) increases dramatically as the length scale of the features under consideration decreases. From a topological point of view, errors in the diffusion processes essentially shift the locations of minima in the cost function. Further, because of the irreversible loss of information associated with diffusion, our analysis suggests that diffusive processes place a limit on the length of the assimilation window. The 3-D convection model and its adjoint are again used to extend and verify our suppositions in a more physically relevant context.