L-BFGS Optimization with Adaptive Gradient for Seismic Inversion
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Solution Overview
Problem
Current seismic inversion methods face challenges with scaling and cross-talk issues, leading to slow convergence rates and inaccurate parameter estimation due to differences in parameter scales and illumination, especially in multi-parameter inversions.
Innovation Solution
The method employs second-order optimization techniques, specifically modifying the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization by substituting a scaled identity matrix with an alternative scaled matrix derived from previous iteration data and prior knowledge, and using a convolutional operator to estimate the inverse Hessian matrix, which improves parameter scaling and reduces cross-talk.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional L-BFGS optimization with scaled identity matrix is used, then the inversion process can be performed, but convergence is slow and parameter estimation is inaccurate due to scaling and cross-talk issues
Solution Approach 1:
The patent modifies the optimization matrix from a scaled identity matrix to a scaled permutation matrix that reorders parameters to group together those with similar scaling characteristics and illumination conditions. This parameter reordering transforms the optimization landscape to reduce cross-talk between parameters and improve convergence behavior, directly addressing the slow convergence and inaccuracy issues.
Solution Approach 2:
The patent segments the parameter vector into groups based on their scaling and illumination characteristics. By permuting parameters into ordered groups, the method creates a structured optimization problem where parameters with similar properties are processed together, reducing interactions (cross-talk) between different parameter types and improving overall estimation accuracy.
2Adaptability or versatility
If multi-parameter inversion is performed to estimate multiple subsurface properties, then more comprehensive geological information is obtained, but scaling differences and illumination variations cause cross-talk and slow convergence
Solution Approach 1:
The patent applies parameter permutation to reorder the multi-parameter vector according to scaling and illumination characteristics. This transformation groups parameters with similar properties together, reducing cross-talk effects and improving the convergence rate of the inversion process while maintaining the ability to estimate multiple subsurface properties simultaneously.
Solution Approach 2:
The patent introduces a dynamic permutation matrix that adapts the parameter ordering based on the specific inversion problem characteristics. This dynamic reordering strategy allows the optimization process to automatically adjust to different scaling and illumination conditions, maintaining high convergence rates across various multi-parameter inversion scenarios.
3Device complexity
If standard optimization algorithms are used without preconditioning, then the inversion process is simpler, but illumination compensation is insufficient leading to inaccurate parameter estimation
Solution Approach 1:
The patent uses parameter permutation to implicitly perform illumination compensation without requiring explicit preconditioning operators. By reordering parameters based on their illumination characteristics, the method automatically adjusts the optimization landscape to account for varying illumination conditions, improving estimation accuracy while keeping the process relatively simple.
Data Source
AI summary
A method for determining spatial distribution of properties of formations in a subsurface volume using geophysical sensor signals recorded proximate the volume includes inversion processing an initial model of the spatial distribution. The inversion processing comprises at least second order optimizing. The second order optimizing comprises calculating a scalar for the identity matrix in limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization, modifying the scalar using an adaptive gradient type scheme to estimate an inverse Hessian matrix, and using the modified plurality of scalars to optimize the inversion processing. The method of estimating the inverse Hessian matrix in L-BFGS can be further extended to include convolutional operators.