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Joint 3D Gravity and Magnetic Inversion via Rectified Flow and Ginzburg-Landau Guidance

Dhruman Gupta, Yashas Shende, Aritra Das, Chanda Grover Kamra, Debayan Gupta

Abstract

Subsurface ore detection is of paramount importance given the gradual depletion of shallow mineral resources in recent years. It is crucial to explore approaches that go beyond the limitations of traditional geological exploration methods. One such promising new method is joint magnetic and gravitational inversion. Given magnetic and gravitational data on a surface, jointly reconstructing the underlying densities that generate them remains an ill-posed inverse problem. Although joint inversion of multiple properties mitigates the non-uniqueness problem in magnetic and gravitational data, deterministic algorithms converge to a single regularized solution and thus do not capture the distribution of possible solutions. Similarly, most machine learning based techniques predict a single solution without modelling the entire distribution. In this paper, we introduce a novel framework that reframes 3D gravity and magnetic joint inversion as a rectified flow on the Noddyverse dataset, the largest physics-based dataset for inversion. We introduce a Ginzburg-Landau (GL) regularizer, a generalized version of the Ising model that aids in ore identification, enabling physics-aware training. We also propose a guidance methodology based on GL theory that can be used as a plug-and-play module with existing unconditional denoisers. Lastly, we also train and release a VAE for the 3D densities, which facilitates downstream work in the field.

Joint 3D Gravity and Magnetic Inversion via Rectified Flow and Ginzburg-Landau Guidance

Abstract

Subsurface ore detection is of paramount importance given the gradual depletion of shallow mineral resources in recent years. It is crucial to explore approaches that go beyond the limitations of traditional geological exploration methods. One such promising new method is joint magnetic and gravitational inversion. Given magnetic and gravitational data on a surface, jointly reconstructing the underlying densities that generate them remains an ill-posed inverse problem. Although joint inversion of multiple properties mitigates the non-uniqueness problem in magnetic and gravitational data, deterministic algorithms converge to a single regularized solution and thus do not capture the distribution of possible solutions. Similarly, most machine learning based techniques predict a single solution without modelling the entire distribution. In this paper, we introduce a novel framework that reframes 3D gravity and magnetic joint inversion as a rectified flow on the Noddyverse dataset, the largest physics-based dataset for inversion. We introduce a Ginzburg-Landau (GL) regularizer, a generalized version of the Ising model that aids in ore identification, enabling physics-aware training. We also propose a guidance methodology based on GL theory that can be used as a plug-and-play module with existing unconditional denoisers. Lastly, we also train and release a VAE for the 3D densities, which facilitates downstream work in the field.
Paper Structure (17 sections, 6 theorems, 54 equations, 6 figures)

This paper contains 17 sections, 6 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Theoretical Background
  4. Gravitational Forward Problem
  5. Magnetic Forward Problem
  6. Joint Forward Model
  7. Ginzburg-Landau Energy for Continuous Ising-Style Dynamics
  8. Dataset and Pre-processing
  9. Model Architectures and Training
  10. VAE
  11. Diffusion and Flow Model
  12. DPS for Inversion
  13. Baseline Flow
  14. Flow with GL Regularizer
  15. Flow with GL Guidance
  16. ...and 2 more sections

Key Result

Proposition 1

For a rectangular prism with corners at $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, the gravity kernel element evaluated at observation point $(x_0, y_0, z_0)$ is:

Figures (6)

  • Figure 1: Overview of our VAE architecture (generated with AI).
  • Figure 2: Example generated 3D volume visualized across multiple $z$-slices.
  • Figure 3: Per-observation qualitative comparison of observed and predicted fields.
  • Figure 4: Quantitative comparison relative to the baseline flow model. Positive $\Delta$RMSE indicates lower error than the baseline.
  • Figure 5: Figure \ref{['fig:quant_plots']} (continued).
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1: Gravity Anomaly
  • Proposition 1: Analytical Kernel for Rectangular Prisms
  • proof
  • Definition 2: Total Magnetic Intensity Anomaly
  • Definition 3: Ginzburg-Landau Free Energy
  • Definition 4: Double-Well Potential
  • Theorem 1: Ising-GL Correspondence (Modica--Mortola)
  • proof
  • Proposition 2: GL Energy Gradient
  • proof
  • ...and 11 more