A Multiscale Method for Data Assimilation

In data assimilation problems, various types of data are naturally linked to different spatial resolutions (e.g., seismic and electromagnetic data), and these scales are usually not coincident to the subsurface simulation model scale. Alternatives like upscaling/downscaling of the data and/or the simulation model can be used, but with potential loss of important information. Such alternatives introduce additional uncertainties which are not in the nature of the problem description, but the result of the post processing of the data or the geo-model. To address this issue, a novel multiscale (MS) data assimilation method is introduced. The overall idea of the method is to keep uncertain parameters and observed data at their original representation scale, avoiding upscaling/downscaling of any quantity. The method relies on a recently developed mathematical framework to compute adjoint gradients via a MS strategy in an algebraic framework. The fine-scale uncertain parameters are directly updated and the MS grid is constructed in a resolution that meets the observed data resolution. This formulation therefore enables a consistent assimilation of data represented at a coarser scale than the simulation model. The misfit objective function is constructed to keep the MS nature of the problem. The regularization term is represented at the simulation model (fine) scale, whereas the data misfit term is represented at the observed data (coarse) scale. The computational aspects of the method are investigated in a simple synthetic model, including an elaborate uncertainty quantification step, and compared to upscaling/downscaling strategies. The experiment shows that the MS strategy provides several potential advantages compared to more traditional scale conciliation strategies: (1) expensive operations are only performed at the coarse scale; (2) the matched uncertain parameter distribution is closer to the "truth"; (3) faster convergence behavior occurs due to faster gradient computation; and (4) better uncertainty quantification results are obtained. The proof-of-concept example considered in this paper sheds new lights on how one can reduce uncertainty within fine-scale geo-model parameters with coarse-scale data, without the necessity of upscaling/downscaling the data nor the geo-model. The developments demonstrate how to consistently formulate such a gradient-based MS data assimilation strategy in an algebraic framework which allows for implementation in available computational platforms.