Magnetics Inversion

Documents the equivalent source gridding and full 3D magnetic inversion processes in docker/base-runner/mag_processes/ and the underlying inversion systems in the simplemag library.

Source Files

• docker/base-runner/mag_processes/equiv_source_process.py — Equivalent source process wrapper
• docker/base-runner/mag_processes/inversion_3d_process.py — 3D inversion process wrapper
• equivalent_source.py — MagEquivalentSourceSystem
• full_3d.py — MagInversion3DSystem
• sensitivity_cache.py — Hash-keyed sensitivity caching

Equivalent Source Inversion

Process: MagEquivSource → MagEquivalentSourceSystem

Physical Model

Inverts flight-line TMI (total magnetic intensity) data for an equivalent distribution of magnetic dipoles on a single flat horizontal layer. The layer is placed at layer__depth_below_flight metres below the minimum observed flight altitude.

The forward problem uses SimPEG's Simulation3DIntegral with full tensor Green's functions for magnetic dipoles in a homogeneous half-space. The integral solution computes the magnetic field at each receiver location as the sum of contributions from all source dipoles.

Mesh

A discretize.TensorMesh with: - Single vertical cell (layer__cell_thickness) - Horizontal cells of layer__cell_size (default 50 m) - layer__padding_cells padding cells around data extent (default 8 per side)

Model Types

1. "scalar" (susceptibility): Assumes induced magnetization aligned with Earth's field. One parameter per cell: M = χ·H₀. Models only induced magnetization.

2. "vector" (MVI): Three-component magnetization (Mx, My, Mz) per cell. Can represent remanent magnetization with arbitrary direction. n_params = 3 × n_cells.

Reference: Lelièvre & Oldenburg (2009) for MVI theory.

Inversion Formulation

• Data: TMI values from magcom column (nT)
• Uncertainty: σ = 0.02 × |TMI| + 1 nT (fractional + absolute floor)
• Data misfit: L2 norm with W_d = diag(1/σ)
• Regularization: Tikhonov:
• α_s (smallness): 10⁻⁴ — penalizes deviation from zero susceptibility
• α_x, α_y (horizontal smoothness): 1
• α_z: 0 (single cell, no vertical smoothness)
• Starting model: χ = 0 (neutral start)
• Optimizer: InexactGaussNewton (max 40 iterations, 20 CG per step)
• Beta schedule: BetaEstimate_ByEig + BetaSchedule(coolingFactor=2, coolingRate=1)
• Optional IRLS: Iterative reweighted least squares for compact/sparse solutions

Prediction on Output Grid

After inversion, the recovered dipole layer is used to forward-predict Bx, By, Bz, and TMI on a regular grid at output__altitude (default: mean flight altitude). This step uses Simulation3DIntegral with store_sensitivities="forward_only" (no caching — G is not needed again).

Output

An xarray Dataset with dimensions (y, x) containing variables for each requested component (tmi, bx, by, bz), plus CRS metadata (spatial_ref with WKT). Written as webxtile.

Full 3D Magnetic Inversion

Process: MagInversion3D → MagInversion3DSystem

Physical Model

Inverts gridded field components (output of the equivalent source step) for a 3D susceptibility or magnetization vector model using an OcTree (TreeMesh) discretization. The same Simulation3DIntegral forward operator drives this inversion.

Model Types

1. "scalar": Susceptibility χ (SI) per active cell, induced-only. Works with any single component (usually tmi).

2. "vector" (MVI): Three-component magnetization per active cell. Requires bx, by, bz input components.

3. "amplitude": Scalar susceptibility from the amplitude of the anomalous field: |B_a| = √(Bx² + By² + Bz²) The forward model predicts all three components and internally computes the amplitude. This approach is remanence-insensitive (the amplitude depends only on the total magnetization magnitude, not its direction) but does not determine magnetization direction.

Reference: Li et al. (2017) for amplitude magnetic inversion.

Mesh

Adaptive OcTree built with mesh_builder_xyz and refine_tree_xyz: - Core cell size: mesh__core_cell_size (default 50 m) - Surface refinement: Refined around topography using octree_levels_surf (default [2, 4]) - Observation refinement: Radial refinement around observation points using octree_levels_obs (default [2, 4]) - Depth: mesh__depth_core (default 500 m below observation level) - Horizontal padding: mesh__max_distance (default 5000 m beyond data bounds)

Active cells are determined by active_from_xyz(mesh, topography) — cells below the topography surface are active.

Inversion Formulation

• Data: Gridded components from equivalent source
• Uncertainty: Same fractional + floor model: σ = 0.02 × |data| + 1 nT
• Regularization: Tikhonov with full 3D smoothness:
• α_s = 10⁻⁴, α_x = α_y = α_z = 1
• m_ref = 0 (reference model: zero susceptibility)

• Sensitivity weighting: Compensates for the natural ~1/r³ decay of magnetic sensitivity with distance from receivers. UpdateSensitivityWeights directive scales the regularization so that all depths are penalized approximately equally. This is strongly recommended for 3D magnetic inversions to avoid bias toward near-surface structure.

• Optimizer: InexactGaussNewton (same schedule as equivalent source)

• Optional IRLS: For compact/sparse models

Output

OcTree cell centres are interpolated (nearest neighbour) onto a regular 3D grid. Output as xarray Dataset with dimensions (z, y, x): - Scalar/amplitude: susceptibility (SI units) - Vector: mx, my, mz (A/m units)

Written as webxtile.

Sensitivity Matrix Caching

Both inversion systems use sensitivity_hash() — a SHA-256 hash computed from: - Receiver locations (3D coordinates) - Earth field parameters (intensity, inclination, declination) - Mesh parameters (cell size, padding, depth) - Model type (scalar/vector)

The hash determines a subdirectory for on-disk .npy sensitivity matrices (store_sensitivities="disk"). The process wrappers sync this directory to/from blob storage (sensitivity_cache/equiv/ and sensitivity_cache/mag3d/), enabling reuse across pipeline runs when the survey geometry is unchanged.

References

• Lelièvre, P. G., & Oldenburg, D. W. (2009). "A 3D total magnetization inversion applicable when significant, complicated remanence is present." Geophysics, 74(3), L21-L30. DOI: 10.1190/1.3103249
• Li, Y., Oldenburg, D. W., Farquharson, C. G., & Shekhtman, R. (2017). "Magnetic amplitude inversion for determining magnetization." Geophysics, 82(2). DOI: 10.1190/geo2016-0302.1
• Farquharson, C. G., & Oldenburg, D. W. (1998). "Non-linear inversion using general measures of data misfit and model structure." Geophysical Journal International, 134(1), 213-227. DOI: 10.1046/j.1365-246x.1998.00555.x
• Cockett, R., Kang, S., Heagy, L. J., Pidlisecky, A., & Oldenburg, D. W. (2015). "SimPEG: An open source framework for simulation and gradient based parameter estimation in geophysical applications." Computers & Geosciences, 85, 142-154. DOI: 10.1016/j.cageo.2015.09.015