Regularized inversion returns images that are linear smoothings of an underlying field. The smoothing operator is known but nonstationary. It is the model resolution matrix of the inverse problem. When several such images are available, each with a different resolution matrix and each responding nonlinearly to a different combination of the same variables, classical change-of-support methods no longer apply. We formulate the problem in latent space. Point-support latent fields carry geostatistical priors and are observed through a petrophysical relationship composed with an image-specific resolution matrix. Sparse direct measurements enter as point observations, and iterated linear-Gaussian updating gives the posterior. Two properties are central, and both hold under linear relationships, Gaussian errors and a correctly specified model. The estimator is sufficient in the following sense. For every regularization weight, an image together with its resolution matrix and propagated error covariance yields the same posterior as the measurements used to produce the image. The estimator also provides a diagnostic. Pointwise likelihood identifiability follows from a rank condition on the stacked resolution-weighted Jacobian, a condition that can be evaluated before acquisition. Two further properties link the estimator to existing methods, and synthetic studies confirm each. Over an ensemble of realizations, latent joint estimation recovers porosity at point support with a coefficient of determination of 0.77 ± 0.08 , against 0.61 ± 0.16 and 0.38 ± 0.20 for two established practices. The formulation applies wherever latent fields are observed through known, nonstationary smoothing operators. Its central consequence is that an inverted image reported with its resolution matrix is a sufficient statistic for the measurements behind it, whereas an image reported without one is strictly less informative.
We present a reaction–diffusion (RD) formulation for geostatistical simulation of anisotropic stochastic fields on structured simulation grids. The proposed framework treats stochastic field generation as a variational formulation in which spatial correlation emerges through gradient-flow evolution, without requiring explicit covariance construction. Variogram and histogram measures are incorporated into a composite energy functional, and the governing RD dynamics are formulated as a gradient flow driven by its variational derivative, which yields an energy-dissipating evolution. Anisotropic spatial correlation is introduced through a variogram-based metric defined by principal ranges and orientations, while an edge-preserving gradient-dependent diffusivity regulates smoothing across sharp contrasts. Geological features and hard data may be imposed as fixed structural constraints with controlled boundary interaction. Starting from uncorrelated Gaussian initial conditions, the RD evolution produces stochastic realizations that satisfy prescribed geostatistical targets while preserving imposed geological structure. The formulation provides a dynamical alternative to classical covariance-based simulation by incorporating variogram and distribution measures directly within the evolution equation. Numerical examples demonstrate the generation of anisotropic stochastic fields on corner-point grids and illustrate how statistical targets and structural conditioning are integrated within a unified variational framework.
Due to the inherent complexity of structural geological modeling, it is common for surface-based models to exhibit overlapping surfaces, unrealistic intersections, or simply the need for some kind of editing. These challenges often arise during the interpretation of seismic data or in situations where complex geological phenomena, such as salt diapirism, generate structures with geometric incompatibilities. This study presents a methodology for geological surface deformation based on a combination of radial basis functions (RBF) and particle swarm optimization (PSO) methods. This approach enables the deformation of one or more surfaces guided by special geometric conditions without requiring strong boundary conditions. RBF is used to ensure deformed surfaces with minimum curvature variation, consistent with physical principles such as generalized elasticity. PSO is used to optimize the positioning of the control points that define the deformation pattern, satisfying two complex constraints: displacement along a target polyline and displacement through restriction surfaces. The effectiveness of the methodology was tested by means of two case studies. The first case addresses the adjustment of sedimentary horizons and salt structures in a model with diapirism. The second case focuses on structural restoration, eliminating fault slips with compatibility constraints between horizons and faults. In both scenarios, the methodology produced consistent results, showing its potentiality as an optimization solver for various applications, such as structural modeling, structural restoration, machine learning model generation, and geological simulations.
Mineral resource classification, based on confidence levels associated with tonnages and grades, is essential for public disclosure and for assessing deposit maturity and related risks. Although geometric classification methods are transparent and straightforward, they do not explicitly account for local variations in uncertainty. In contrast, geostatistical approaches focus primarily on the quality of grade estimation. The scorecard methodology attempts to integrate multiple parameters relevant to mineral resource classification; however, elements such as the definition of uncertainty classes and the weighting of individual criteria often rely on subjective judgment. This study proposes a hybrid methodology for mineral resource classification that integrates traditional geometric approaches with a quantitative risk map. The risk map incorporates three main sources of uncertainty: (1) geological uncertainty, (2) database quality, and (3) grade estimation quality. To minimize subjectivity in the integration of these factors, the risk map is constructed using a k-means clustering approach, resulting in three risk categories: low, medium, and high. The final classification is obtained through a hierarchical combination of geometric classification and the risk map. A machine learning post-processing step is applied to mitigate local classification artifacts, commonly referred to as "salt-and-pepper" or "spotted dog" effects. The proposed methodology was applied to a stratigraphic deposit case study and yielded robust results. It effectively integrates conventional geometric classification with key sources of uncertainty, reduces subjectivity compared with traditional scorecard methodology, and ensures reproducibility in mineral resource classification.
The synchrony estimator with parameter β for the correlation between two time series observed at different time points is defined as the standard Pearson correlation coefficient computed using a fraction β of the pairs of the two series corresponding to the smallest time differences. Two problems limit the use of the estimator in applications. First, no rule is available for selecting β . Second, it has been argued in the literature that the estimator only partially exploits the information in the data, since it does not use all available observations. In this paper, we address these issues under the assumption that the data-generating process is a bivariate first-order autoregressive [AR(1)] model with positive autocorrelation coefficients. This assumption follows from embedding and identifiability considerations and the widespread use of AR(1) models in geoscience applications. The main contribution is an algorithm for selecting the β parameter of the synchrony estimator. We illustrate the performance of the proposed procedure using simulated data. We found evidence that, contrary to what has been claimed in the literature, not using all available observations individually is a strength of the synchrony estimator rather than a weakness. As such, the synchrony estimator is a valid option for estimating the correlation. The proposed procedure is also extended to the selection of the β parameter of the synchrony estimator of the cross-correlation function. Relevant aspects of its implementation are presented and illustrated with examples using simulated and real climatological data.
