Glossary

9.3 Inversion and modeling techniques

5 min read•august 14, 2024

Geophysical inversion techniques are crucial for understanding Earth's subsurface. They help scientists estimate properties like density and velocity from measured data. This process involves solving complex mathematical problems to find the best-fitting model.

Inversion methods face challenges like non-uniqueness and limited . Scientists use various approaches, from deterministic to probabilistic, to tackle these issues and quantify uncertainties in their results.

Forward vs Inverse Modeling in Geophysics

Defining Forward and Inverse Modeling

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  • Forward modeling predicts geophysical data based on a given subsurface model
  • Inverse modeling estimates the subsurface model based on observed geophysical data
  • Forward modeling requires a mathematical description of the physical processes that relate the subsurface properties to the geophysical measurements
  • Examples of forward modeling include:
    • Calculating gravity anomalies from a given density distribution
    • Computing seismic travel times through a specified velocity model

Challenges in Inverse Modeling

  • Inverse modeling aims to find a subsurface model that best explains the observed geophysical data by minimizing the difference between the predicted and observed data
  • Inverse problems are typically ill-posed
    • Multiple subsurface models can explain the observed data equally well
    • The solution may not be unique or stable
  • The relationship between the subsurface model parameters and the geophysical data is often non-linear
    • Non-linearity can make the inverse problem challenging to solve
    • Iterative optimization techniques are often required to estimate the model parameters

Inversion Techniques for Subsurface Properties

Mathematical Formulation of Inversion

  • Inversion techniques are mathematical methods used to estimate subsurface properties from geophysical data by solving the inverse problem
  • The objective function in an inversion quantifies the misfit between the predicted and observed data
    • The goal is to minimize this misfit by adjusting the subsurface model parameters
    • Common objective functions include the least-squares misfit and the L1-norm misfit
  • techniques are often used to stabilize the inversion process and to incorporate prior information about the subsurface
    • Tikhonov regularization adds a penalty term to the objective function to favor smooth or simple models
    • Total variation regularization promotes models with sharp boundaries or discontinuities

Optimization Algorithms and Probabilistic Inversion

  • Gradient-based optimization algorithms are often employed to iteratively update the subsurface model parameters and minimize the objective function
    • Steepest descent method updates the model parameters in the direction of the negative gradient of the objective function
    • improves the convergence rate by using a set of conjugate search directions
  • Markov chain Monte Carlo (MCMC) methods can be used for probabilistic inversion
    • The goal is to estimate the posterior probability distribution of the subsurface model parameters given the observed data and prior information
    • MCMC methods generate an ensemble of possible subsurface models that are consistent with the data and prior knowledge
    • Examples of MCMC algorithms include the Metropolis-Hastings algorithm and the Gibbs sampler

Limitations of Inversion Results

Non-Uniqueness and Resolution

  • Inversion results are inherently non-unique
    • Multiple subsurface models may explain the observed data equally well
    • The true subsurface structure may not be uniquely determined by the available data
  • The resolution of the inverted subsurface model is limited by the spatial and temporal sampling of the geophysical data, as well as the physics of the imaging process
    • The resolution matrix can be used to quantify the spatial resolution of the inverted model
    • High-resolution models require dense spatial sampling and high-frequency data

Uncertainty Quantification and Model Validation

  • Uncertainty in the inverted model arises from various sources
    • Measurement errors in the geophysical data
    • Modeling errors due to simplifying assumptions or incomplete physics
    • The ill-posed nature of the inverse problem
  • The model covariance matrix can be used to quantify the uncertainty in the estimated subsurface properties
    • Diagonal elements represent the variances of the model parameters
    • Off-diagonal elements capture the correlations between different parameters
  • can be performed to assess how changes in the input data or model parameters affect the inversion results
    • Perturbing the input data or model parameters and observing the corresponding changes in the inverted model
  • techniques can be used to evaluate the robustness and predictive performance of the inverted model
    • Leave-one-out cross-validation involves removing one data point at a time and inverting the remaining data
    • K-fold cross-validation divides the data into K subsets and uses each subset as a validation set while inverting the remaining data

Deterministic vs Probabilistic Inversion

Deterministic Inversion

  • Deterministic inversion aims to find a single "best" subsurface model that minimizes the misfit between the predicted and observed data
  • Deterministic inversion typically relies on gradient-based optimization algorithms to update the model parameters iteratively
    • The model parameters are adjusted in the direction that reduces the objective function
    • The inversion proceeds until a convergence criterion is met or a maximum number of iterations is reached
  • The result of a deterministic inversion is a single subsurface model that represents the most likely or optimal solution given the data and the chosen objective function
    • The inverted model provides a point estimate of the subsurface properties
    • Uncertainty quantification is often limited in deterministic inversion

Probabilistic Inversion

  • Probabilistic inversion seeks to estimate the posterior probability distribution of the subsurface model parameters given the observed data and prior information
  • Probabilistic inversion often uses sampling-based methods, such as Markov chain Monte Carlo (MCMC) algorithms, to explore the model parameter space
    • MCMC methods generate an ensemble of possible subsurface models that are consistent with the data and prior knowledge
    • The ensemble of models represents the uncertainty in the estimated subsurface properties
  • The result of a probabilistic inversion is a probability distribution that quantifies the uncertainty in the estimated subsurface properties
    • The probability distribution provides a more complete characterization of the model uncertainty
    • Marginal distributions and confidence intervals can be derived from the ensemble of models

Comparison and Hybrid Approaches

  • Deterministic inversion is computationally more efficient but provides only a single solution
    • Suitable for large-scale problems or real-time applications where computational resources are limited
  • Probabilistic inversion is more computationally intensive but provides a more complete characterization of the model uncertainty
    • Suitable for problems where quantifying uncertainty is crucial for decision-making or risk assessment
  • Hybrid approaches combine elements of both deterministic and probabilistic inversion to balance computational efficiency and uncertainty quantification
    • Ensemble Kalman uses an ensemble of models to approximate the posterior distribution while updating the models sequentially with new data
    • Particle swarm optimization uses a swarm of particles to explore the model parameter space and converge towards the optimal solution

Key Terms to Review (18)

Boundary Element Method: The boundary element method (BEM) is a numerical computational technique used to solve linear partial differential equations, especially those related to boundary value problems. This method simplifies complex geometries by focusing on the boundaries of a domain rather than the entire volume, making it particularly efficient for problems in geophysics like modeling subsurface structures and inversion techniques.
COMSOL Multiphysics: COMSOL Multiphysics is a simulation software platform that allows users to model and simulate various physical phenomena through multiphysics applications. It integrates different physical models, enabling users to solve complex problems in fields such as engineering, physics, and geophysics by coupling multiple physics phenomena in a single environment, making it especially useful for inversion and modeling techniques.
Conjugate Gradient Method: The conjugate gradient method is an efficient algorithm used for solving large systems of linear equations, especially those that arise from the discretization of continuous problems. It is particularly valuable in geophysics for inversion and modeling techniques, as it minimizes the quadratic form associated with a positive-definite matrix, allowing for rapid convergence towards the solution.
Cross-validation: Cross-validation is a statistical method used to assess how the results of a statistical analysis will generalize to an independent data set. It is often used in the context of model validation, where the goal is to ensure that a predictive model performs well not just on training data but also on unseen data, making it crucial for inversion and modeling techniques, integration of data sets, and ensuring quality control in geophysical surveys.
Extrapolation: Extrapolation is a statistical technique used to estimate unknown values by extending a known sequence of data points. This method assumes that the trends observed within the existing data will continue beyond the range of those data points. In modeling and inversion techniques, extrapolation is crucial for predicting geophysical properties and behaviors in unmeasured areas based on available measurements.
Filtering: Filtering is a process used to remove unwanted components or noise from a signal, allowing for a clearer representation of the desired data. This technique is crucial in various fields, particularly in processing acoustic and seismic data, where distinguishing between relevant signals and background noise can significantly enhance data interpretation. Filtering can also aid in improving the quality of modeled data, facilitating more accurate inversion processes.
Finite element modeling: Finite element modeling (FEM) is a numerical technique used to obtain approximate solutions to boundary value problems for partial differential equations. It involves breaking down complex structures into smaller, simpler parts called finite elements, which can be analyzed individually. This method allows for detailed modeling of physical phenomena, making it essential in fields like engineering, physics, and geophysics, particularly when dealing with complex inversion and modeling techniques.
Gauss-Newton Algorithm: The Gauss-Newton algorithm is an iterative method used for solving nonlinear least squares problems. It is particularly useful in optimization tasks where the objective is to minimize the sum of the squares of residuals between observed and predicted values. This algorithm is a key technique in data fitting, enabling researchers to refine models based on empirical data and improve the accuracy of their predictions.
Gravity data: Gravity data refers to the measurements of gravitational acceleration at various locations on the Earth's surface, which are used to infer subsurface geological structures and density variations. These measurements help geophysicists understand the distribution of mass beneath the Earth's surface and are essential for modeling geological features like mineral deposits, oil reserves, and fault lines.
Interpretation: Interpretation refers to the process of analyzing and explaining data obtained from geophysical measurements to derive meaningful insights about subsurface structures and processes. This involves integrating various modeling techniques and inversion methods to translate raw data into understandable geological information, enabling scientists and engineers to make informed decisions about natural resources or hazards.
Linear inversion: Linear inversion is a mathematical technique used to deduce unknown parameters or properties from observed data by establishing a linear relationship between the input and output variables. It is a vital tool in geophysics for modeling and interpreting geophysical data, allowing scientists to construct models of subsurface features based on surface measurements.
Matlab: MATLAB is a high-level programming language and environment designed for numerical computing, data analysis, and algorithm development. It provides powerful tools for processing and visualizing data, making it especially useful for tasks like digital signal processing, inversion and modeling techniques, and integrating complex geophysical datasets. With its extensive libraries and user-friendly interface, MATLAB is a go-to choice for researchers and engineers working in various scientific fields.
Model comparison: Model comparison is the process of evaluating different models to determine which one best explains a given set of data. This is particularly important in the context of inversion and modeling techniques, as it allows researchers to assess the validity and predictive power of various models against observed measurements. Effective model comparison helps in identifying the most accurate representation of geological or geophysical phenomena, guiding decisions in research and application.
Non-linear inversion: Non-linear inversion is a mathematical technique used to recover model parameters from observed data where the relationship between the data and the model is not linear. This approach is essential in geophysics for accurately modeling complex geological structures and processes, as it accommodates the intricacies that linear methods cannot capture. By utilizing iterative algorithms and advanced optimization techniques, non-linear inversion can produce more realistic models that fit observed data better than simpler linear approaches.
Regularization: Regularization is a mathematical technique used to prevent overfitting in inversion problems by introducing additional information or constraints into the model. It aims to find a solution that balances fidelity to the data with smoothness or simplicity of the model, making it crucial in scenarios where data is noisy or incomplete. By adding a penalty term to the loss function, regularization helps ensure that the inversion process produces more stable and reliable results.
Resolution: Resolution refers to the ability of a geophysical survey or imaging technique to distinguish between two closely spaced features in the subsurface. In this context, it directly impacts how accurately we can interpret geophysical data, affecting inversion and modeling processes as well as quality control in data management. The higher the resolution, the clearer and more detailed the image of subsurface structures will be, leading to better decision-making and analysis.
Seismic data: Seismic data refers to the information gathered from seismic waves generated by natural or artificial sources, used to investigate the properties of the Earth's subsurface. This data is crucial for understanding geological structures and processes, and it plays a key role in inversion and modeling techniques, which aim to reconstruct subsurface features based on the recorded seismic signals. Analyzing seismic data allows scientists to make informed predictions about the Earth's behavior, which can be essential for resource exploration and hazard assessment.
Sensitivity analysis: Sensitivity analysis is a technique used to determine how the different values of an independent variable affect a particular dependent variable under a given set of assumptions. This method helps in understanding the robustness of a model by examining how changes in input parameters impact the output, providing insights into the relationships and dependencies within the model.
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