11.2 Inverse theory and parameter estimation

Inverse theory and parameter estimation are crucial tools in geophysics for extracting meaningful information from data. These techniques allow scientists to infer Earth's properties and structures from various measurements, bridging the gap between observations and underlying physical processes.

This section delves into the mathematical foundations and practical applications of inverse problems in geophysics. It covers key concepts like least-squares estimation, regularization techniques, and uncertainty quantification, essential for interpreting geophysical data and building accurate Earth models.

Solving Inverse Problems in Geophysics

Formulation and Characteristics of Inverse Problems

Top images from around the web for Formulation and Characteristics of Inverse Problems

• 

  ACP - Inverse modeling of SO2 and NOx emissions over China using multisensor satellite data ... View original

  Is this image relevant?

• 

  SE - Integration of geoscientific uncertainty into geophysical inversion by means of local ... View original

  Is this image relevant?

• 

  Frontiers | Joint Seismic and Gravity Data Inversion to Image Intra-Crustal Structures: The ... View original

  Is this image relevant?

• 

  ACP - Inverse modeling of SO2 and NOx emissions over China using multisensor satellite data ... View original

  Is this image relevant?

• 

  SE - Integration of geoscientific uncertainty into geophysical inversion by means of local ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Formulation and Characteristics of Inverse Problems

• 

  ACP - Inverse modeling of SO2 and NOx emissions over China using multisensor satellite data ... View original

  Is this image relevant?

• 

  SE - Integration of geoscientific uncertainty into geophysical inversion by means of local ... View original

  Is this image relevant?

• 

  Frontiers | Joint Seismic and Gravity Data Inversion to Image Intra-Crustal Structures: The ... View original

  Is this image relevant?

• 

  ACP - Inverse modeling of SO2 and NOx emissions over China using multisensor satellite data ... View original

  Is this image relevant?

• 

  SE - Integration of geoscientific uncertainty into geophysical inversion by means of local ... View original

  Is this image relevant?

1 of 3

• Inverse problems involve estimating model parameters from observed data, in contrast to forward problems that predict data from known model parameters
• The general formulation of a linear inverse problem is $d = Gm$, where $d$ is the data vector, $m$ is the model parameter vector, and $G$ is the forward modeling operator or sensitivity matrix
• Inverse problems are typically ill-posed, meaning they may have non-unique solutions (multiple models fitting the data equally well), be sensitive to small changes in data (small perturbations leading to large changes in estimated parameters), or have solutions that do not depend continuously on the data (instability)
• Examples of inverse problems in geophysics include estimating subsurface velocity structure from seismic travel times, inferring Earth's interior density distribution from gravity measurements, and determining fault slip distribution from surface deformation observations

Least-Squares Estimation and Singular Value Decomposition

• Least-squares estimation is a common approach to solve inverse problems, minimizing the misfit between observed and predicted data
• The least-squares solution minimizes the objective function $\Phi(m) = ||Gm - d||^2$, where $||\cdot||$ denotes the Euclidean norm
• The normal equations $(G^TG)m = G^Td$ provide the least-squares solution, where $G^T$ is the transpose of the sensitivity matrix
• Singular Value Decomposition (SVD) is a powerful tool for analyzing and solving linear inverse problems, providing insight into model resolution and uncertainty
• SVD decomposes the sensitivity matrix as $G = U\Sigma V^T$, where $U$ and $V$ are orthogonal matrices containing the data and model space eigenvectors, and $\Sigma$ is a diagonal matrix containing the singular values
• The least-squares solution can be expressed using the SVD as $m = V\Sigma^{-1}U^Td$, where $\Sigma^{-1}$ is the inverse of the singular value matrix

Regularization Techniques for Ill-Posed Problems

Tikhonov Regularization and Regularization Parameter Selection

• Tikhonov regularization is a widely used technique that adds a smoothness constraint to the least-squares objective function, balancing data misfit and model complexity
• The regularized objective function is $\Phi(m) = ||Gm - d||^2 + \lambda||Lm||^2$, where $L$ is a regularization operator (e.g., identity matrix or finite-difference approximation of derivatives) and $\lambda$ is the regularization parameter
• The regularization parameter $\lambda$ controls the trade-off between fitting the data and satisfying the smoothness constraint, with larger values favoring smoother models
• The L-curve method plots the data misfit against the model norm for different values of $\lambda$, and the optimal $\lambda$ is chosen at the corner of the L-shaped curve, balancing the two terms
• Cross-validation techniques, such as leave-one-out or k-fold cross-validation, estimate the predictive performance of the model for different $\lambda$ values, selecting the one that minimizes the validation error

Alternative Regularization Approaches and Bayesian Inference

• Total variation regularization preserves sharp discontinuities in the model while promoting piecewise smoothness, making it suitable for models with abrupt changes (e.g., layered Earth structure)
• Sparsity-promoting regularization, such as L1-norm or Lasso regularization, favors models with a small number of non-zero coefficients, useful for identifying a few key model parameters (e.g., sparse source localization)
• Bayesian inference provides a probabilistic framework for incorporating prior information and quantifying uncertainty in inverse problems
• In the Bayesian approach, model parameters are treated as random variables with a prior probability distribution, and the solution is given by the posterior distribution, which combines the prior information with the data likelihood
• Markov chain Monte Carlo (MCMC) methods, such as the Metropolis-Hastings algorithm or Gibbs sampling, can sample from the posterior distribution to estimate model parameters and their uncertainties

Model Parameter Resolution and Uncertainty

Assessing Model Resolution and Trade-offs

• Model resolution describes the ability to distinguish between different model parameters and is influenced by the data coverage, noise level, and forward modeling operator
• The model resolution matrix $R = VV^T$ quantifies the spatial resolution and trade-offs between model parameters
• Diagonal elements of $R$ close to 1 indicate well-resolved parameters, while off-diagonal elements show the trade-offs or "blurring" between parameters
• The spread of the resolution matrix around its diagonal provides a measure of the spatial resolution, with narrow spreads indicating high resolution and wide spreads suggesting low resolution
• Trade-offs between model parameters can be visualized using the off-diagonal elements of $R$ or by plotting the rows of $V$, which represent the "averaging kernels" or "resolution functions"

Quantifying Uncertainty and Sensitivity Analysis

• Uncertainty in estimated model parameters arises from measurement errors, modeling errors, and the inherent non-uniqueness of inverse problems
• The covariance matrix $C_m = \sigma^2V\Sigma^{-2}V^T$ describes the uncertainty and correlation between model parameters, where $\sigma^2$ is the data variance
• Diagonal elements of $C_m$ give the variances of individual model parameters, while off-diagonal elements show their covariances or correlations
• Error propagation techniques, such as linear error propagation or Monte Carlo methods, can be used to estimate the uncertainty in derived quantities (e.g., spatial averages or gradients) from the model parameter uncertainties
• Sensitivity analysis investigates how changes in data or model assumptions affect the estimated model parameters and can guide data acquisition and experimental design
• Sensitivity matrices, such as the Jacobian or Fréchet derivatives, quantify the change in model parameters with respect to perturbations in the data or forward modeling operator
• Singular vectors and singular values from the SVD provide insight into the most influential data combinations and the relative importance of different model parameters

Optimization Algorithms for Parameter Estimation

Gradient-Based and Newton-Type Methods

• Gradient-based methods use the gradient of the objective function to iteratively update the model parameters towards the minimum
• Steepest descent updates the model in the direction of the negative gradient, with a step size determined by a line search or fixed value
• Conjugate gradient methods improve convergence by searching along conjugate directions that avoid the zigzagging behavior of steepest descent
• Newton's method uses the Hessian matrix (second-order derivatives) to scale the gradient steps, providing quadratic convergence near the minimum
• Gauss-Newton methods approximate the Hessian using the Jacobian matrix, which is more efficient to compute and update than the full Hessian
• Quasi-Newton methods, like the BFGS algorithm, approximate the Hessian using gradient information from previous iterations, providing a balance between convergence speed and computational cost

Global Optimization and Computational Considerations

• Global optimization methods can escape local minima and explore a wider range of the model space, which is useful for non-linear and non-convex problems
• Simulated annealing mimics the annealing process in metallurgy, accepting worse solutions with a decreasing probability to avoid getting trapped in local minima
• Genetic algorithms simulate biological evolution, using operators like mutation and crossover to generate new solutions and select the fittest individuals
• Particle swarm optimization moves a population of candidate solutions (particles) through the model space, updating their positions and velocities based on individual and group best solutions
• The choice of optimization algorithm depends on the problem size, complexity, and available computational resources
• Large-scale inverse problems may require parallel computing, matrix-free methods, or iterative solvers to handle high-dimensional model spaces and large datasets
• Regularization and preconditioning techniques can improve the convergence and stability of optimization algorithms by modifying the objective function or search directions
• Hybrid methods that combine global and local optimization strategies, or use different algorithms in different stages of the inversion, can balance exploration and exploitation to efficiently find the best-fitting model parameters