Bayesian model evaluation for multiple scenarios

Traditional uncertainty analysis for subsurface models is typically based on a single dynamic model with a number of uncertain parameters. Improved and more robust forecasting can be obtained by combining several models in a Bayesian setting using model averaging. The traditional Bayesian Model Averaging (BMA), however, suffers from several drawbacks, such as too large sensitivity to prior model assumptions and instability with respect to measurement perturbations, especially when the number of measurements is large. We suggest a modified version of BMA (MBMA) where the calculations are stabilized using an ensemble of measurements. Bayesian stacking (BS) is a method that is directly focused on the performance of the combined predictive distribution of several models. The original version of BS (BSLOO) is based on leave-one-out cross-validation and requires a Bayesian inversion for each data point which may be very time consuming. We suggest a modified version of stacking (MBS) that requires only a single history match and uses an ensemble of measurements. MBS may be used with either prior (MBS-pri) or posterior (MBS-post) predictive distributions. The behavior of the methods is illustrated using three synthetic, linear examples. One is a simple mixture model. The other two are inspired by 4D seismic data. The results with MBS-pri are very similar to the results with MBMA. The results with MBS-post are similar to those of BSLOO when the data are uncorrelated. MBS can take into account correlated data or measurement errors, while correlations are neglected in the BSLOO weight calculations.publishedVersio

A binary level set model for elliptic inverse problems with discontinuous coefficients

In this paper we propose a variant of a binary level set approach for solving elliptic problems with piecewise constant coefficients. The inverse problem is solved by a variational augmented Lagrangian approach with a to-tal variation regularisation. In the binary formulation, the seeked interfaces between the domains with different values of the coefficient are represented by discontinuities of the level set functions. The level set functions shall only take two discrete values, i.e. 1 and-1, but the minimisation functional is smooth. Our formulation can, under moderate amount of noise in the observations, re-cover rather complicated geometries without requiring any initial curves of the geometries, only a reasonable guess of the constant levels is needed. Numerical results show that our implementation of this formulation has a faster conver-gence than the traditional level set formulation used on the same problems