Importance Weighting in Hybrid Iterative Ensemble Smoothers for Data Assimilation

Because it is generally impossible to completely characterize the uncertainty in complex model variables after assimilation of data, it is common to approximate the uncertainty by sampling from approximations of the posterior distribution for model variables. When minimization methods are used for the sampling, the weights on each of the samples depend on the magnitude of the data mismatch at the critical points and on the Jacobian of the transformation from the prior density to the sample proposal density. For standard iterative ensemble smoothers, the Jacobian is identical for all samples, and the weights depend only on the data mismatch. In this paper, a hybrid data assimilation method is proposed which makes it possible for each ensemble member to have a distinct Jacobian and for the approximation to the posterior density to be multimodal. For the proposed hybrid iterative ensemble smoother, it is necessary that a part of the mapping from the prior Gaussian random variable to the data be analytic. Examples might include analytic transformation from a latent Gaussian random variable to permeability followed by a black-box transformation from permeability to state variables in porous media flow, or a Gaussian hierarchical model for variables followed by a similar black-box transformation from permeability to state variables. In this paper, the application of weighting to both hybrid and standard iterative ensemble smoothers is investigated using a two-dimensional, two-phase flow problem in porous media with various degrees of nonlinearity. As expected, the weights in a standard iterative ensemble smoother become degenerate for problems with large amounts of data. In the examples, however, the weights for the hybrid iterative ensemble smoother were useful for improving forecast reliability.