9.3 Inversion and modeling techniques

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 resolution. 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
• Regularization 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
  • Conjugate gradient method 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
• Sensitivity analysis 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
• Cross-validation 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 filtering 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