Information-Theoretic Sampling for Geological Image Recovery

The Question

Given a budget of N measurements, where should you place them to learn as much as possible about an unknown spatial field? This is the Optimal Sensor Placement problem — fundamental to mineral exploration (where to drill next), environmental monitoring (where to install sensors), and any domain where data collection is expensive.

The search space is C(H×W, K) — combinatorially explosive. For a modest 50×50 field with 20 measurements, that’s over 10²⁶ possible configurations. Exhaustive search is impossible.

The Information-Theoretic Approach

The thesis frames the problem through Shannon entropy and mutual information:

• H(X) = -Σ p(x) log p(x) — total uncertainty about the unknown field
• H(X^f | X_f) — residual uncertainty after observing at locations f
• I(X_f; X^f) = H(X^f) - H(X^f | X_f) — information gained by measuring at f

The goal: choose locations that maximize mutual information. The breakthrough: entropy maximization in this setting satisfies submodularity — adding a measurement to a small set yields more information gain than adding it to a large set. This mathematical property guarantees that greedy sequential selection achieves at least (1 - 1/e) ≈ 63.2% of the global optimum. Not an empirical observation — a provable bound.

The AdSEMES (Adaptive Sequential Empirical Maximum Entropy Sampling) algorithm implements this with spatial penalty functions that prevent clustering and three reconstruction methods (nearest neighbor, indicator kriging, entropy-weighted inverse distance) for recovering the full field from sparse observations.

Publications

1. “Sampling Strategies for Uncertainty Reduction in Categorical Random Fields” — Mathematical Geosciences, 2019
2. “Optimal Sampling Strategy for Spatial Estimation of Ore-Waste Contacts” — Natural Resources Research, 2020
3. “Geological Facies Recovery Based on Weighted L1-Regularization” — Mathematical Geosciences, 2019