A multiscale method for data assimilation

Abstract

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.

Multi-Level Inversion Based on Mesh Decoupling

“…The Multi-Level method allows one to reduce the degrees of freedom while searching for the solution and yet stay robust. The use of Multi-Level and upscaling methods are introduced in Moraes et al [14] and in Echeverria et al [5], this thesis illustrates the robustness of upscaling in the design variable space. Further, this thesis demonstrates the use of inversion to find the reservoir characteristics.…”

“…Numerical Optimization and the Newton with Trust Region method are described at [4], [15]. Multi-Level appraoches help reduce computation and yet stay robust and similar ideas are introduced in [5], [14], [17]. The use of Julia and applications is described at [1], [6], [13] In chapter 1, an introduction to the topic is mentioned.…”

Iterative multilevel assimilation of inverted seismic data

“…We choose to construct them by spatial coarsening of the conventional simulation grid to several levels of coarseness, and correspondingly upscale the associated grid-based parameter functions. Multilevel Data Assimilation (MLDA) [8,14,15,22,23,32,35] utilizes multilevel models in the forecast step. Since inverted seismic data are given on the conventional grid (denoted the fine grid from now on), MLDA with such data must be able to handle differences in grid levels between data and model forecasts.…”

Simulation of the inelastic deformation of porous reservoirs under cyclic loading relevant for underground hydrogen storage

“…Development of the multiscale strategy is not only important for advancing the computational efficiency, as formerly addressed for linear mechanics but also crucial to allow for generating a consistent map between fine and coarse scale systems without any need of upscaled parameters. This can also allow for the connection of the coarse scale observation data directly to the fine-scale computational system, to reduce the uncertainty, as shown for flow in porous media 73 . For convenient integration within a given simulation framework, the developed multiscale strategy is also formulated and implemented algebraically.…”