In this paper, geological prior information is incorporated in the classification of reservoir lithologies after the adoption of Markov random fields (MRFs). The prediction of hidden lithologies is based on measured observations, such as seismic inversion results, which are associated with the latent categorical variables, based on the assumption of Gaussian distributions. Compared with other statistical methods, such as the Gaussian mixture model or k-Means, which do not take spatial relationships into account, the hidden MRFs approach can connect the same or similar lithologies horizontally while ensuring a geologically reasonable vertical ordering. It is, therefore, able to exclude randomly appearing lithologies caused by errors in the inversion. The prior information consists of a Gibbs distribution function and transition probability matrices. The Gibbs distribution connects the same or similar lithologies internally, which does not need a geological definition from the outside. The transition matrices provide preferential transitions between different lithologies, and an estimation of them implicitly depends on the depositional environments and juxtaposition rules between different lithologies. Analog cross sections from the subsurface or outcrop studies can contribute to the construction of these matrices by a simple counting procedure.

Hidden Markov Models (HMMs) have been applied to predict reservoir lithologies using seismic inversion results as inputs. This approach takes into account the conditional probabilities between different lithologies, i.e. the vertical transitions in sedimentary sequences. These properties are used as prior geological information. In order to relate the seismic inversion results to the true well-log data, HMMs need to be trained based on the Expectation-Maximization theory. Application of the resulting model on a synthetic example from the Book Cliffs (Utah, USA) showed that most lithologies are classified correctly, even for some thin layers. A comparison with point-wise methods in which data samples are treated independently from each other, such as k-means and fuzzy logic classifiers, leads to the conclusion that the spatial correlation in HMMs allows better lithological predictions because the prior information accounts for the geological depositional processes. A real case study with data from the Vienna Basin (Austria) is performed, in which lithologies in a 3D cube are obtained based on properties from seismic inversions, via trained HMMs. While the vertical sequences are shown to be reasonably well predicted, the horizontal continuities are not. This indicates that the future research should focus on the lateral geological relationships.

The coupled Markov chain model can be used to simulate reservoir lithologies between wells, by conditioning them on the observed data in the cored wells. However, with this method, only the state at the same depth as the current cell is going to be used for conditioning, which may be a problem if the geological layers are dipping. This will cause the simulated lithological layers to be broken or to become discontinuous across the reservoir. In order to address this problem, an actively conditioned process is proposed here, in which a tolerance angle is predefined. The states contained in the region constrained by the tolerance angle will be employed for conditioning in the horizontal chain first, after which a coupling concept with the vertical chain is implemented. In order to use the same horizontal transition matrix for different future states, the tolerance angle has to be small. This allows the method to work in reservoirs without complex structures caused by depositional processes or tectonic deformations. Directional artefacts in the modeling process are avoided through a careful choice of the simulation path. The tolerance angle and dipping direction of the strata can be obtained from a correlation between wells, or from seismic data, which are available in most hydrocarbon reservoirs, either by interpretation or by inversion that can also assist the construction of a horizontal probability matrix.

We study a reconstruction algorithm for the general inverse scattering problem based on the estimate of not only medium properties, as in more conventional approaches, but also wavefields propagating inside the computational domain. This extended set of unknowns is justified as a way to prevent local minimum stagnation, which is a common issue for standard methods. At each iteration of the algorithm, (i) the model parameters are obtained by solution of a convex problem, formulated from a special bilinear relationship of the data with respect to properties and wavefields (where the wavefield is kept fixed), and (ii) a better estimate of the wavefield is calculated, based on the previously reconstructed properties. The resulting scheme is computationally convenient since step (i) can greatly benefit from parallelization and the wavefield update (ii) requires modeling only in the known background model, which can be sped up considerably by factorization-based direct methods. The inversion method is successfully tested on synthetic elastic datasets.