13.2 Gravitational and magnetic field inversion

Gravitational and magnetic field inversion is a crucial technique in geophysics for uncovering hidden structures beneath Earth's surface. By analyzing gravity and magnetic data, scientists can reconstruct subsurface density and magnetic properties, offering valuable insights into mineral deposits, oil reservoirs, and crustal structure.

This topic delves into the principles, challenges, and methods of potential field inversion. It explores non-uniqueness issues, regularization techniques, and applications in various geological settings. Understanding these concepts is essential for interpreting geophysical data and creating accurate subsurface models.

Gravitational and Magnetic Field Inversion

Principles and Challenges

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• Gravitational and magnetic field inversion reconstructs subsurface density or magnetic susceptibility distributions from observed potential field data
• Potential field theory fundamental principle states gravitational and magnetic fields obey Laplace's equation in source-free regions
• Non-uniqueness presents an inherent challenge in potential field inversion as multiple subsurface models can produce the same observed field
• Regularization techniques (smoothness constraints, depth weighting) stabilize the inversion process and produce geologically reasonable solutions
• Forward problem in potential field methods calculates the gravitational or magnetic response of a given subsurface model
  • Example: Calculating the gravitational acceleration at the surface due to a buried dense sphere
• Inverse modeling techniques for potential fields include:
  • Least-squares inversion
  • Stochastic inversion
  • Constrained inversion approaches

Applications and Methods

• Common applications of gravitational and magnetic field inversion:
  • Mineral exploration (locating ore deposits)
  • Hydrocarbon reservoir characterization (identifying potential oil and gas traps)
  • Crustal structure mapping (determining the thickness of the Earth's crust)
• Discretization of the subsurface into cells or voxels parameterizes the inverse problem
  • Example: Dividing the subsurface into a 3D grid of cubic cells, each with its own density or magnetic susceptibility value
• Linear relationship between potential field data and subsurface properties allows formulation of the inverse problem as a system of linear equations
  • $\mathbf{d} = \mathbf{G}\mathbf{m}$
  • Where $\mathbf{d}$ is the observed data vector, $\mathbf{G}$ is the sensitivity matrix, and $\mathbf{m}$ is the model parameter vector
• Iterative optimization methods solve large-scale potential field inverse problems:
  • Conjugate gradient algorithm
  • Gauss-Newton algorithm

Inverse Problems for Gravity and Magnetic Data

Gravity Inversion

• Gravity inverse problem determines subsurface density distribution from observed gravitational acceleration or gradient measurements
• Gravitational acceleration ($g$) relates to density ($\rho$) through the gravitational potential ($U$):
  • $g = -\nabla U$
  • $\nabla^2 U = 4\pi G\rho$
  • Where $G$ is the gravitational constant
• Gravity gradient tensors provide additional information for inversion:
  • $T_{ij} = \frac{\partial^2 U}{\partial x_i \partial x_j}$
  • Where $i,j$ represent the x, y, and z directions
• Example: Using gravity data to map salt domes in sedimentary basins for hydrocarbon exploration

Magnetic Inversion

• Magnetic field inversion reconstructs the distribution of magnetic susceptibility or remanent magnetization from total field or vector component measurements
• Magnetic field ($\mathbf{B}$) relates to magnetization ($\mathbf{M}$) through the magnetic potential ($\Phi$):
  • $\mathbf{B} = -\nabla \Phi$
  • $\nabla^2 \Phi = \nabla \cdot \mathbf{M}$
• Total magnetic intensity (TMI) data commonly used in magnetic inversions:
  • $TMI = |\mathbf{B}|$
• Example: Mapping the Curie depth (depth at which rocks lose their magnetic properties) to estimate crustal temperatures

Joint Inversion and Constraints

• Joint inversion of gravity and magnetic data provides complementary information and reduces ambiguity in resulting subsurface models
• Incorporation of a priori information improves reliability and geological relevance of inversion results:
  • Geological constraints (fault locations, layer boundaries)
  • Petrophysical relationships (density-susceptibility correlations)
• Example: Combining gravity and magnetic data to differentiate between dense, magnetic basement rocks and sedimentary cover

Resolution and Sensitivity of Inversion Methods

Resolution Analysis

• Resolution in potential field inversion distinguishes between closely spaced subsurface features or anomalies
• Depth of investigation for potential field methods limited by decay of gravitational and magnetic fields with distance from source
  • Gravity field decays as $1/r^2$
  • Magnetic field decays as $1/r^3$
• Resolution matrix ($\mathbf{R}$) quantifies how well model parameters are resolved:
  • $\mathbf{R} = (\mathbf{G}^T\mathbf{G} + \lambda \mathbf{I})^{-1}\mathbf{G}^T\mathbf{G}$
  • Where $\lambda$ is the regularization parameter and $\mathbf{I}$ is the identity matrix
• Concept of equivalent sources in potential field theory impacts non-uniqueness of inversion solutions and limits achievable resolution
  • Example: A deep, large mass can produce the same gravitational effect as a shallow, small mass

Sensitivity Analysis

• Sensitivity analysis quantifies how changes in subsurface model parameters affect observed potential field data
• Sensitivity matrix ($\mathbf{S}$) relates changes in model parameters to changes in observed data:
  • $\mathbf{S} = \frac{\partial \mathbf{d}}{\partial \mathbf{m}}$
• Trade-offs between model resolution and stability managed through careful selection of regularization parameters
  • L-curve method for optimal regularization parameter selection
• Synthetic model studies and resolution tests assess capabilities and limitations of specific inversion algorithms or survey designs
  • Example: Creating a synthetic subsurface model, generating synthetic data, and inverting to compare with the original model

Integrating Inversion Results with Other Data

Complementary Geophysical Methods

• Integration of potential field inversion results with seismic data provides complementary information on subsurface structure and composition
  • Example: Using gravity inversion to constrain densities in seismic velocity models
• Electromagnetic methods (magnetotellurics) combined with potential field inversions improve constraints on crustal properties
  • Example: Joint inversion of MT and gravity data to map conductivity and density variations in the crust
• Joint inversion of multiple geophysical datasets reduces ambiguity and improves overall reliability of subsurface models
  • Structural coupling: Enforcing similar geometries across different physical property models
  • Petrophysical coupling: Using relationships between different physical properties to constrain the inversion

Geological Integration and Visualization

• Petrophysical relationships between density, magnetic susceptibility, and other rock properties link potential field inversion results with other geophysical parameters
  • Example: Using the Gardner equation to relate seismic velocity to density
• Geological information from well logs, core samples, or surface mapping incorporated as constraints or validation for potential field inversion models
• Geostatistical methods integrate potential field inversion results with other spatially distributed geophysical or geological data
  • Kriging
  • Sequential Gaussian simulation
• Visualization and interpretation techniques essential for effectively combining potential field inversion results with other geophysical datasets:
  • 3D modeling software (GOCAD, Petrel)
  • Data fusion techniques (RGB color blending, transparency overlays)
  • Example: Creating a 3D geological model integrating inverted density and susceptibility distributions with seismic horizons and well data