Abstract
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.
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Acknowledgments
We are grateful to Geovariances for providing a license for the use of Isatis for factorial co-kriging. Additionally, the authors thank Equinor (operator of the Norne Field) and its license partners Eni Norge and Petoro for the release of the Norne data. The authors acknowledge the Center for Integrated Operations at NTNU for cooperation and coordination of the Norne Cases. The view expressed in this paper are the views of the authors and do not necessarily reflect the views of Equinor and the Norne license partners.
Funding
Primary support for the authors has been provided by the CIPR/IRIS cooperative research project “4D Seismic History Matching” which is funded by industry partners Eni Norge, Petrobras, and Total, as well as the Research Council of Norway through the Petromaks2 program.
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Appendix: Expectation for residuals with incorrect forward model
Appendix: Expectation for residuals with incorrect forward model
The differences between observations and predictions from models provide information about the magnitude of model error and the magnitude of observation error, but the relationship is not simple. In this appendix, we show that the expected covariance for data residuals based on the difference between predictions from a perturbed observation method of data assimilation and the actual observations takes a relatively simple form.
where \(\tilde {H}\) is an approximation to the ‘true’ forward model, H. 𝜃post is the ensemble of model parameters conditioned to the observations, dobs is the vector of observations, and rpost is the ensemble of data residuals.
For simplicity, we make the simplifying assumption that the “truth” can be obtained from a model that has the same parameters as our approximate model and that the model error is due to deficiency of the forward model. Define
where \(\epsilon _{d} \sim N(0, R)\) and \(\epsilon _{\theta ,2} \sim N(0,P)\).
Because we are assuming a gaussian prior and a linear observation operator with Gaussian errors, we use the method of randomized maximum likelihood (RML) to sample from the posterior conditional to observations [41, 45]. In this case, however, we allow for the possibility that our forward model is imperfect and that our estimate of the observation error is incorrect. A sample of the model parameters from the conditional pdf can be generated by minimizing an objective function
the solution of which is
where we defined the gain matrix based on the approximate forward model
1.1 A.1 The difference between predictions from calibrated models and observations
The difference between the calibration of perturbed observations (RML) and the actual observations is
The covariance of these residuals is approximately
where we assume that \((\tilde {H} - H) \theta ^{\text {pr}} = 0\). Note that if \(\tilde {H} = H\) and \(\tilde {R} = R\), then Eq. 14 simplifies to
which is what we expect.
Also, note that if \(\tilde {H} P \tilde {H}^{\text {T}} + \tilde {R} = H P H^{\text {T}} + R \), then \(E[ r r^{\text {T}} ] = \tilde {R}\). For this case, we have
1.2 A.2 Iteration
We use an iterative fixed-point scheme for estimation of the total observation covariance based on the residuals as shown in Eq. 14.
The iterative scheme has a stable fixed-point solution when
is positive definite. See the discussion of convergence on page 262 of Ménard [37].
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Alfonzo, M., Oliver, D.S. Seismic data assimilation with an imperfect model. Comput Geosci 24, 889–905 (2020). https://doi.org/10.1007/s10596-019-09849-0
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DOI: https://doi.org/10.1007/s10596-019-09849-0