Seismic data assimilation with an imperfect model

Data assimilation methods often assume perfect models and uncorrelated observation error. The assumption of a perfect model is probably always wrong for applications to real systems, and since model error is known to generally induce correlated effective observation errors, then the common assumption of uncorrelated observation errors is probably almost always wrong, too. The standard approach to dealing with correlated observation errors, which simply ignores the correlation, leads to suboptimal assimilation of observations. In this paper, we examine the consequences of model errors on assimilation of seismic data. We show how to recognize the existence of correlated error through model diagnostics modified for large numbers of data, how to estimate the correlation in the error, and how to use a model with correlated errors in a perturbed observation form of an iterative ensemble smoother to improve the quantification of a posteriori uncertainty. The methodologies for each of these aspects have been developed to allow application to problems with very large number of model parameters and large amounts of data with correlated observation error. We applied the methodologies to a small toy problem with linear relationship between data and model parameters, and to synthetic seismic data from the Norne Field model. To provide a controlled investigation in the seismic example, we investigate an application of data assimilation with two sources of model error-errors in seismic resolution and errors in the petro-elastic model. Both types of model errors result in effectively correlated observation errors, which must be accounted for in the data assimilation scheme. Although the data are synthetic, parameters of the seismic resolution and the observation noise are estimated from the actual inverted acoustic impedance data. Using a structured approach, we are able to assimilate approximately 115,000 observations with correlated total observation error efficiently without neglecting correlations. We show that the application of this methodology leads to less overfitting to the observations, and results in an ensemble estimate with smaller spread than the initial ensemble of predictions, but that the final estimate of uncertainty is consistent with the truth.