Pixel-MPS: Stochastic Embedding and Density-Based Clustering of Image Patterns for Pixel-Based Multiple-Point Geostatistical Simulation

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Multiple-point geostatistics (MPS) is an established tool for the uncertainty quantification of Earth systems modeling, particularly when dealing with the complexity and heterogeneity of geological data. This study presents a novel pixel-based MPS method for modeling spatial data using advanced machine-learning algorithms. Pixel-based multiple-point simulation implies the sequential modeling of individual points on the simulation grid, one at a time, by borrowing spatial information from the training image and honoring the conditioning data points. The developed methodology is based on the mapping of the training image patterns database using the t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm for dimensionality reduction, and the clustering of patterns by applying the Density-based Spatial Clustering of Applications with Noise (DBSCAN) algorithm, as an efficient unsupervised classification technique. For the automation, optimization, and input parameter tuning, multiple stages are implemented, including entropy-based determination of the template size and a k-nearest neighbors search for clustering parameter selection, to ensure the proposed method does not require the user’s interference. The proposed model is validated using synthetic two- and three-dimensional datasets, both for conditional and unconditional simulations, and runtime information is provided. Finally, the method is applied to a case study gold mine for stochastic orebody modeling. To demonstrate the computational efficiency and accuracy of the proposed method, a two-dimensional training image with 101 by 101 pixels is simulated for 100 conditional realizations in 453 s (~4.5 s per realization) using only 361 hard data points (~3.5% of the simulation grid), and the resulting average simulation has a good visual match and only an 11.8% pixel-wise mismatch with the training image.