Seismic Data Assimilation With An Imperfect Model

Abstract

Summary Data assimilation methods often assume perfect models and uncorrelated observation error. The assumption of a perfect model is probably always wrong for real problems, and since model error is known to generally induce correlated effective observation errors, then the assumption of uncorrelated observation errors is probably almost always wrong, too. Ignoring the correlation of observation errors, leads to suboptimal assimilation of observations. Common methods for dealing with correlated observation errors included thinning of data, creation of super-observations, and inflation of error variance. While those methods can reduce the tendency to underestimate uncertainty, they tend to exclude small-scale information in the data. In this paper, we examine the consequences of model errors on assimilation of seismic data. To provide a controlled investigation, we investigate two sources of model error -- errors in seismic resolution and errors in the petroelastic model. Both errors result in correlated total observation errors, which must be accounted for in the data assimilation scheme. We show how to recognize the existence of correlated error through model diagnostics, 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 estimates of uncertainty after assimilation of seismic data. The methodology is applied to synthetic seismic data from the Norne Field model. Parameters of the seismic resolution and the observation noise are estimated from the actual inverted impedance. Using this approach, we are able to assimilate approximately 115,000 observations with correlated total observation error efficiently without neglecting correlations. The examples show that the iterative estimation of total observation error compensates for the model error and improves forecasts. The method requires the observation error to be non-diagonal, but we show that this is easily handled even for large problems.