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.
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.
Multivariate geostatistical cosimulation is commonly used to reproduce spatial uncertainty and cross-variable relationships. However, conventional approaches based on simple or ordinary cokriging can become problematic under partially heterotopic sampling, particularly when an important variable is strongly undersampled or absent in part of the domain. This study proposes a two-step turning bands cosimulation framework that incorporates ordinary cokriging with related means. Non-conditional multivariate Gaussian realizations are first generated and subsequently conditioned to the available data by cokriging the residuals, while linear relationships between the unknown Gaussian means are incorporated into the conditioning system. The method is applied to an iron deposit where Fe, P, and S are widely sampled, whereas FeO is available only in the eastern part of the deposit. FeO is particularly important in this deposit because the Fe/FeO ratio helps distinguish magnetite-dominated preferred ore from hematite-dominated material and therefore contributes to mineralogical dilution assessment. To do so, the conditional realizations are generated at fine-scale discretization blocks, back-transformed to the original data units, and subsequently aggregated to selective mining-unit support for mineralogical dilution assessment. The proposed approach is compared with equivalent cosimulation frameworks based on simple and ordinary cokriging using spatial patterns, global statistics, distribution and correlation reproduction, trend analysis, cross-validation, and dilution maps. The related-mean approach maintained complete spatial coverage and provided competitive predictive performance, while ordinary cokriging became unstable in poorly sampled areas. This improved reproduction is transferred to the mineral dilution assessment, providing a more consistent characterization of mineralogical variability across the deposit. The proposed framework offers a practical approach for cosimulating undersampled mineralogical or geometallurgical variables when informative relationships with more densely sampled variables are available.
