Search

Inverse problems where relationships are linear arise in many fields of science and engineering and, consequently, algorithms for solving them are widespread. However, when the size of the problem increases, the computational challenge becomes huge, hence, unless some simplifying assumptions are yielded, it becomes impossible to solve the problem. Our study addresses these issues for large linear(izable) inverse problems when the uncertainties can be modeled as Gaussian and the forward relationship and covariance matrices can be expressed in terms of Kronecker products. Under these conditions, we illustrate an algorithm capable of addressing very large problems with very limited storage requirements and much faster than the traditional approach. The result is a complete characterization of the posterior distribution in a probabilistic sense in terms of mean and covariance. We extend this method also to nonlinear problems where a Gauss-Newton algorithm is employed. Applications to reflection seismology, magnetic anomaly inversion and image restoration are presented.

Calibration of the rock-frame elastic properties is an important step in establishing the rock-physics models for quantitative seismic reservoir characterization. While several empirical and theoretical rock-frame models have been proposed in the literature, their applicability to carbonate rocks is challenged due to their specific textural and varied pore type characteristics. Here, the suitability of an empirical model (Nur’s critical porosity), and a semi-empirical inclusion-based model (Keys-Xu) for a hard rock carbonate reservoir in south-west Iran is examined. Wireline log data in conjunction with deterministic and probabilistic optimization algorithms are employed to calibrate the free parameters in these models, as well as the minerals’ effective elastic moduli and density. The optimum values derived from the very fast simulated annealing algorithm were comparable to the estimated values from a probabilistic calibration workflow (Markov chain Monte Carlo sampling-based method). The uncertainty quantification via the probabilistic approach indicates higher uncertainty is associated with the calibrated clay elastic parameters (10%–13%) compared to the uncertainty associated with the calcite elastic moduli (1%–3%) in both rock-frame models. This could be attributed to the dominance of the calcite mineral and lack of variability in clay volume fraction in the interval of interest. The uncertainty associated with critical porosity and pore aspect ratio are 3% and 7%, respectively. The results of this case study show that simple rock frame models with only one free parameter (critical porosity or pore aspect ratio) could successfully model the elastic logs without resorting to more sophisticated rock-frame models.