On Localization and Multiscale in Data Assimilation

The main goal of inverse problems is the reconstruction of unknown quantities, often they are located in an inaccessible part of space. Data assimilation, which is the integration of inverse problems techniques with dynamical systems with the task of forecasting the evolution of some quantity, has evolved strongly, with important applications in meteorology and atmospherical sciences. Localization is an essential part of ensemble based assimilation schemes. The size of an ensemble is always much smaller than the dimension of the state space for real numerical predictions. It is necessary to ensure a sufficent number of degree's of freedom when generating the analysis ensemble and, thus, increase the rank of the set of analysis equations. Also, the small ensemble size gives an insufficent estimate of the background error correlations. Localization effectivly eliminates spurious correlations in the background ensemble between distant state variables. The choice of the localization radius needs to depend on the number of ensemble members for the short-range forcasts used to calculate the background error for the analysis step as well as on the number of observations and the observation error. However, a challenge arises when the observation operator under consideration is non-local (e.g. satellite radiance data), the localization which is applicable to the problem can be severly limited, with strong effects on the quality of the assimilation step. We study a transformation approach to change non-local operators to local operators in transformed space, such that localization becomes applicable. We interpret this approach as a generalized localization and study its general algebraic formulation. Examples are provided for a compact integral operator and a non-local matrix observation operator to demonstrate the feasibility of the approach and study the quality of the assimilation by transformation. In particular, we apply the approach to temperature profile reconstruction from infrared measurements given by the IASI Infrared Sounder and show that the approach is feasible for this important data type in atmospheric analysis and forecasting. We also believe that our derivations will work in a similar way for particle filters. If we do localization we have to adjust the localization radius to the scale of the problem. In this respect choosing large radius mean the data we have is important for large scale and vice versa. Thus localization is linked to the scale of the meteorological processes under consideration. Inherently when we choose localization, we chose scale. This leads to the topic of scales, which usually comes under the umbrella of multiscale methods. The broad idea of a multiscale approach is to decompose your problem into different scales or levels and to use these decompositions either for constructing appropriate approximations or to solve smaller problems on each of these levels, leading to increased stability or increased efficiency. Our goal is to analyse the sequential multiscale approach applied to an inversion or state estimation problem. We work in a generic setup given by a Hilbert space environment. We work out the analysis both for an unregularized and a regularized sequential multiscale inversion. In general the sequential multiscale approach is not equivalent to a full solution, but we show that under appropriate assumptions we obtain convergence of an iterative sequential multiscale version of the method. For the regularized case we develop a strategy to appropriately adapt the regularization when an iterative approach is taken. We demonstrate the validity of the iterative sequential multiscale approach by testing the method on an integral equation as it appears for atmospheric temperature retrieval from infrared satellite radiances.