MATLAB Code Implementation for Seismic Inversion

Seismic inversion is a technique that reconstructs subsurface structural parameters from observed data, with significant applications in petroleum exploration and geological research. MATLAB's robust matrix computation capabilities make it particularly suitable for implementing the core computational components of seismic inversion.

In practical implementation, seismic inversion typically involves handling two types of equations: Over-determined systems (more equations than unknowns) These problems are commonly solved using least squares methods. In MATLAB, optimal solutions can be directly computed using the left division operator () or functions like `lsqr`. The core concept is to identify model parameters that minimize the sum of squared errors between predicted data and actual observed measurements. Implementation often involves constructing the Jacobian matrix and applying QR decomposition for numerical stability.

Under-determined systems (fewer equations than unknowns) These require the introduction of regularization terms to formulate optimization problems, such as Tikhonov regularization to control solution smoothness. MATLAB's Optimization Toolbox (e.g., `fmincon`) or third-party packages like `l1_ls` can solve these problems, with the key challenge being to balance data fitting accuracy against model complexity. The regularization parameter can be optimized using L-curve analysis or cross-validation techniques.

Practical programming considerations: Utilize sparse storage formats (e.g., `sparse()`) for large-scale sparse matrices to enhance computational efficiency Analyze solution stability through singular value decomposition (SVD) using `svd()` function Validate inversion rationality by visualizing intermediate results with MATLAB's plotting functions (`plot3`, `imagesc`)

Advanced considerations: Integrate deep learning frameworks (e.g., MATLAB's Deep Learning Toolbox) to develop hybrid inversion models using neural networks for nonlinear pattern recognition Implement parallel computing (Parallel Computing Toolbox) with `parfor` loops to accelerate large-scale problem solving Consider incorporating Bayesian inversion methods for uncertainty quantification using Markov Chain Monte Carlo (MCMC) sampling