Glossary

13.2 Gravitational and magnetic field inversion

5 min read•july 30, 2024

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, 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 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
  • techniques for potential fields include:
    • 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
    • d=Gm\mathbf{d} = \mathbf{G}\mathbf{m}
    • Where d\mathbf{d} is the observed data vector, G\mathbf{G} is the , and m\mathbf{m} is the model parameter vector
  • Iterative optimization methods solve large-scale potential field inverse problems:

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 (gg) relates to density (ρ\rho) through the gravitational potential (UU):
    • g=−∇Ug = -\nabla U
    • ∇2U=4πGρ\nabla^2 U = 4\pi G\rho
    • Where GG is the gravitational constant
  • Gravity gradient tensors provide additional information for inversion:
    • Tij=∂2U∂xi∂xjT_{ij} = \frac{\partial^2 U}{\partial x_i \partial x_j}
    • Where i,ji,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 (B\mathbf{B}) relates to magnetization (M\mathbf{M}) through the magnetic potential (Φ\Phi):
    • B=−∇Φ\mathbf{B} = -\nabla \Phi
    • ∇2Φ=∇⋅M\nabla^2 \Phi = \nabla \cdot \mathbf{M}
  • Total magnetic intensity (TMI) data commonly used in magnetic inversions:
    • TMI=∣B∣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/r21/r^2
    • Magnetic field decays as 1/r31/r^3
  • (R\mathbf{R}) quantifies how well model parameters are resolved:
    • R=(GTG+λI)−1GTG\mathbf{R} = (\mathbf{G}^T\mathbf{G} + \lambda \mathbf{I})^{-1}\mathbf{G}^T\mathbf{G}
    • Where λ\lambda is the regularization parameter and I\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 (S\mathbf{S}) relates changes in model parameters to changes in observed data:
    • S=∂d∂m\mathbf{S} = \frac{\partial \mathbf{d}}{\partial \mathbf{m}}
  • Trade-offs between model resolution and stability managed through careful selection of regularization parameters
    • 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
  • 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

Key Terms to Review (27)

Confidence Intervals: A confidence interval is a statistical range, derived from sample data, that is likely to contain the true value of an unknown population parameter. It reflects the uncertainty inherent in sample data and provides a range within which the parameter is expected to fall, allowing researchers to quantify the precision of their estimates. Confidence intervals are crucial for making informed decisions based on data, especially in modeling and estimation processes where variability and uncertainty are present.
Conjugate Gradient Algorithm: The conjugate gradient algorithm is an iterative method for solving large systems of linear equations, especially those that are symmetric and positive-definite. It is particularly useful in the context of numerical optimization and inverse problems, as it minimizes quadratic forms and helps to efficiently compute solutions without requiring the full matrix inversion.
Constrained inversion: Constrained inversion is a mathematical approach used in inverse problems where additional information or restrictions are applied to ensure that the solution meets specific criteria or remains within a certain set of bounds. This technique is especially important in geophysical applications, as it helps refine models of subsurface structures by incorporating prior knowledge and preventing non-physical solutions that may arise during the inversion process.
Data misfit: Data misfit refers to the difference or discrepancy between observed data and model predictions in the context of inverse problems. It is a crucial measure used to evaluate how well a model explains the available data, and minimizing this misfit is essential for achieving accurate reconstructions of physical parameters in various fields such as geophysics and reservoir engineering.
Error propagation: Error propagation refers to the process of determining how uncertainties in input measurements affect the uncertainty in the output results of a calculation or model. It is crucial in quantitative analysis since it helps quantify the reliability and precision of results derived from experimental data and numerical simulations. Understanding how errors propagate allows researchers to assess the significance of their findings and make informed decisions based on the inherent uncertainties.
Forward Modeling: Forward modeling is the process of simulating the response of a physical system to specific inputs or conditions, essentially predicting what the observed data should look like based on a known model. This approach is foundational in various fields, as it allows researchers to understand how different parameters affect measurements and predict the outcomes of inverse problems, where one seeks to recover model parameters from observed data.
Gauss-Newton Algorithm: The Gauss-Newton algorithm is an iterative method used for solving non-linear least squares problems by linearizing the system around the current estimate. It focuses on minimizing the sum of the squares of the residuals, which represent the differences between observed and predicted values. This algorithm is particularly useful in contexts where data fitting is required, often involving adjustments in parameters to achieve a best fit. Its effectiveness can be enhanced by incorporating regularization methods to address issues like overfitting or ill-posed problems.
Geological mapping: Geological mapping is the process of creating a visual representation of the Earth's surface and subsurface features, including rock types, structures, and geological formations. This technique helps in understanding the distribution of different materials and aids in resource exploration, environmental studies, and hazard assessments.
Gradient Descent: Gradient descent is an optimization algorithm used to minimize a function by iteratively moving towards the steepest descent as defined by the negative of the gradient. It plays a crucial role in various mathematical and computational techniques, particularly when solving inverse problems, where finding the best-fit parameters is essential to recover unknowns from observed data.
Gravitational anomalies: Gravitational anomalies refer to variations in the Earth's gravitational field that deviate from the expected values based on a smooth gravitational model. These anomalies can provide insights into geological structures and variations in subsurface materials, such as density changes due to geological formations or the presence of mineral deposits. Understanding these anomalies is essential for effective gravitational and magnetic field inversion techniques, which aim to reconstruct subsurface structures and properties from surface measurements.
Ill-posed problems: Ill-posed problems are mathematical or computational issues that do not meet the criteria for well-posedness, meaning they lack a unique solution, or that small changes in input can lead to large variations in output. This characteristic makes them challenging to solve and analyze, especially in fields where precise measurements and solutions are essential. They often arise in inverse modeling scenarios where the solution may be sensitive to noise or other errors in data.
Inverse modeling: Inverse modeling is a mathematical and computational approach used to infer model parameters from observed data, effectively reversing the process of prediction. This method allows researchers to identify underlying characteristics or properties of a system based on the data collected, which is crucial in various fields such as geophysics, environmental science, and engineering. By using inverse modeling, one can estimate unknown parameters or reconstruct scenarios that led to the observed data, enhancing our understanding of complex systems.
Iterative solvers: Iterative solvers are computational algorithms used to find approximate solutions to mathematical problems, particularly those that involve large systems of equations. These methods generate a sequence of improving approximate solutions, refining them with each iteration, which is particularly useful when direct methods become impractical due to high computational cost or memory limitations. Their efficiency is enhanced in contexts such as gravitational and magnetic field inversion, as well as parallel computing applications, allowing for faster processing of complex inverse problems.
Kriging: Kriging is a geostatistical interpolation technique that provides optimal estimates of unknown values at unmeasured locations based on known data points. It uses statistical models to incorporate both the distance and the degree of variation between known data points to predict values in a way that minimizes the estimation error, making it particularly valuable in fields like geosciences, including gravitational and magnetic field inversion.
L-Curve Method: The L-Curve method is a graphical approach used to determine the optimal regularization parameter in ill-posed problems. It involves plotting the norm of the regularized solution against the norm of the residual error, resulting in an 'L' shaped curve, where the corner of the 'L' indicates a balance between fitting the data and smoothing the solution.
Least-squares inversion: Least-squares inversion is a mathematical technique used to estimate unknown parameters by minimizing the sum of the squares of the differences between observed and predicted data. This method is particularly useful in fields like geophysics for inverting data obtained from gravitational and magnetic fields, allowing researchers to extract meaningful information about subsurface structures or properties.
Magnetic susceptibility: Magnetic susceptibility is a measure of how much a material will become magnetized in an external magnetic field. It indicates the degree to which a substance can be magnetized, which is crucial in understanding how different materials interact with magnetic fields and how this property can be utilized in various applications.
Model fitting: Model fitting is the process of adjusting a mathematical model to best represent a set of observed data by minimizing the differences between the predicted outcomes and the actual observations. This involves estimating parameters in the model to optimize how well it captures the underlying trends in the data, which is crucial in both statistical analysis and scientific applications, including techniques like least squares and inversion methods.
Regularization: Regularization is a mathematical technique used to prevent overfitting in inverse problems by introducing additional information or constraints into the model. It helps stabilize the solution, especially in cases where the problem is ill-posed or when there is noise in the data, allowing for more reliable and interpretable results.
Resolution Matrix: A resolution matrix is a mathematical construct used in inverse problems to quantify the ability of a given measurement system to resolve different sources or features within the data. It connects how well distinct features can be distinguished from one another based on the data gathered, often influencing the quality and accuracy of the inversion results in gravitational and magnetic field studies.
Resource exploration: Resource exploration is the process of searching for and identifying natural resources such as minerals, oil, gas, and groundwater within the Earth's subsurface. This process is crucial for the economic development and sustainability of regions, as it helps locate valuable materials that can be extracted and utilized. Techniques such as geophysical surveys, drilling, and sampling are commonly employed during resource exploration to evaluate the potential of specific areas for resource extraction.
Sensitivity matrix: The sensitivity matrix is a mathematical tool used to quantify how changes in model parameters affect the output of a given model. In the context of gravitational and magnetic field inversion, it plays a crucial role in understanding how variations in subsurface properties influence the measured gravitational or magnetic fields, which helps in reconstructing the subsurface structures.
Sequential Gaussian Simulation: Sequential Gaussian Simulation is a statistical method used to generate realizations of spatially correlated random fields based on Gaussian distributions. It allows for the incorporation of spatial variability and uncertainty, making it particularly valuable in the estimation of subsurface properties. This technique is widely applied in various fields, including the characterization of natural resources and the analysis of geophysical data.
Stochastic Inversion: Stochastic inversion is a method used to solve inverse problems by incorporating uncertainty and randomness into the inversion process. This approach allows for the estimation of model parameters when dealing with incomplete or noisy data, making it particularly useful in geophysical applications like gravitational and magnetic field inversion. By applying statistical techniques, stochastic inversion can generate multiple plausible solutions, providing a more comprehensive understanding of the underlying model and its uncertainties.
Tikhonov Regularization: Tikhonov regularization is a mathematical method used to stabilize the solution of ill-posed inverse problems by adding a regularization term to the loss function. This approach helps mitigate issues such as noise and instability in the data, making it easier to obtain a solution that is both stable and unique. It’s commonly applied in various fields like image processing, geophysics, and medical imaging.
Timothy a. g. s. p. a. g. b. p. h. p. d.: Timothy A. G. S. P. A. G. B. P. H. P. D. is an acronym that refers to a complex algorithm or method used in gravitational and magnetic field inversion processes to analyze geophysical data and recover subsurface information. This term encapsulates various techniques aimed at extracting meaningful insights from magnetic and gravitational anomalies, allowing scientists to infer properties about the Earth's structure and composition.
William M. K. C. van der Vorst: William M. K. C. van der Vorst is a prominent mathematician and researcher known for his contributions to numerical analysis, particularly in the context of inverse problems and optimization methods. His work has significantly influenced how gravitational and magnetic field inversion techniques are developed and applied, helping to improve data interpretation in various scientific fields such as geophysics and medical imaging.
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