# Spatial Gaussian Mixture Model for Geological Predictive Mapping

Herbert Rakotonirina, Théophile Lohier, Julien Baptiste, Hélène Tissoux

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Herbert Rakotonirina, Théophile Lohier, Julien Baptiste, Hélène Tissoux. Spatial Gaussian Mixture Model for Geological Predictive Mapping. VGC 2025 - Geosciences Reimagined: Digital Innovations for a Changing World, Sep 2025, Lausanne, Switzerland. hal-05147489

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https://brgm.hal.science/hal-05147489v1

Submitted on 7 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Spatial Gaussian Mixture Model for Geological Predictive Mapping

Herbert Rakotonirina ¹*, Théophile Lohier ¹, Julien Baptiste¹, Hélène Tissoux ¹

¹ BRGM French Geological Survey, 3 Av Claude Guillemin, Orléans, h.rakotonirina@brgm.fr

**Keywords:** Predictive Mapping, Machine Learning, Gaussian Mixture Model, Unsupervised Learning

Predictive mapping is a cornerstone task in geoscience, aiming to identify the spatial distribution of geological features or resources based on indirect or partial observations. It plays a critical role in exploration, environmental assessment, and risk management. Traditionally, supervised machine learning methods have shown promise in this domain, but their performance strongly depends on the availability of labeled training data, a significant limitation in geoscientific contexts where ground truth is sparse or expensive to obtain.

To address this, unsupervised learning approaches such as Gaussian Mixture Models (GMMs) offer a compelling alternative by discovering hidden structures in the data without the need for labeled samples. GMMs model the data as a mixture of Gaussian components, each representing a latent cluster with its own mean and variance. However, standard GMMs only consider the attribute space and ignore spatial relationships, which are crucial in geological systems that often exhibit strong spatial coherence

In this study, we propose a novel spatial extension to GMMs that enhances predictive mapping by incorporating spatial context. Our method adjusts the posterior probabilities of a classical GMM by averaging them over a sliding window across the spatial domain, thereby integrating local neighborhood information while preserving the probabilistic nature of the model. We evaluate the approach on synthetic datasets characterized by clear spatial patterns. Comparative results against classical GMM, classical Kmeans and two state-of-the-art spatial GMM variants (ZHAO et al., 2016; ZHOU et al., 2020) demonstrate the superior ability of our method to produce geologically plausible and spatially consistent predictions.

<figure><img src="image_2.png" /><figcaption>Figure 1: Clustering results on three synthetic spatial datasets (sp1, sp2, sp3). Each row shows the input data, ground-truth clusters, and results from six methods: classical GMM, GMM on (value, i, j), KMeans on values, KMeans on (value, i, j), Zhou et al.'s spatial GMM, and the proposed spatial GMM. Adjusted Rand Index (ARI) scores indicate clustering accuracy. The proposed method better preserves spatial structure and achieves higher ARI scores overall.</figcaption></figure>

**Acknowledgements:** This research project is funded by PEPR PC4 Digital Earth ANR-22-EXSS-0005. We are grateful for their support and commitment to advancing research and innovation.
References

B. ZHAO, Y. ZHONG, A. MA AND L. ZHANG, "A SPATIAL GAUSSIAN MIXTURE MODEL FOR OPTICAL REMOTE SENSING IMAGE CLUSTERING," IN IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 9, NO. 12, PP. 5748-5759, DEC. 2016, DOI: 10.1109/JSTARS.2016.2546918

ZHOU, S., BONDELL, H., TORDESILLAS, A., RUBINSTEIN, B. I. P., & BAILEY, J. (2020). EARLY IDENTIFICATION OF AN IMPENDING ROCKSLIDE LOCATION VIA A SPATIALLY-AIDED GAUSSIAN MIXTURE MODEL. THE ANNALS OF APPLIED STATISTICS, 14(2), 977-992. HTTPS://WWW.JSTOR.ORG/STABLE/27119672