Occam Inversion via Jacobian Decomposition
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Solution Overview
Problem
Current geophysical inversion methods, such as Occam's inversion, are computationally expensive and inefficient for large-scale problems, particularly in 3D inversion, due to the need for multiple large matrix inversions during the Lagrange parameter line search, and struggle with joint inversion of disparate data types requiring proper relative weighting.
Innovation Solution
The method involves linear remapping of the subsurface property model to represent the regularization operator as an identity matrix, followed by singular value decomposition (SVD) to reduce computational complexity, allowing for a single matrix factorization per iteration and adaptive weighting of data types through dual Lagrange parameters.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional Occam inversion is used with multiple large matrix inversions during Lagrange parameter line search, then model accuracy is improved, but computational time and cost increase significantly
Solution Approach 1:
The patent applies preliminary action by performing a single matrix factorization (decomposition) at the beginning of each iteration, before the Lagrange parameter line search. This pre-computed decomposition is then reused throughout the line search process, eliminating the need to perform expensive matrix inversions multiple times during the parameter search, thus reducing computational time while maintaining model accuracy
Solution Approach 2:
The patent changes the mathematical approach by transforming the inversion problem into a decomposition problem. Instead of directly inverting large matrices multiple times, the method uses a single matrix decomposition per iteration and leverages this decomposition for all subsequent calculations during the line search, fundamentally changing how the computational burden is distributed
2Measurement precision
If conventional Occam inversion performs multiple large matrix inversions for each Lagrange parameter value, then data misfit reduction is improved, but computational complexity increases
Solution Approach 1:
The patent performs the computationally intensive matrix factorization once per iteration before the line search begins. This preliminary decomposition structure is then reused for evaluating different Lagrange parameter values, reducing the computational complexity from multiple full inversions to much cheaper operations that leverage the pre-computed decomposition
Solution Approach 2:
The patent creates a computational template through the single matrix decomposition that can be copied and reused for multiple Lagrange parameter evaluations. Rather than重新 computing inversions for each parameter value, the method uses the decomposed structure to efficiently evaluate different scenarios, significantly reducing overall computational complexity
3Measurement precision
If 3D inversion with large number of model parameters is performed using conventional methods, then model resolution is improved, but inversion speed deteriorates
Solution Approach 1:
The patent applies preliminary matrix decomposition at the start of each iteration, creating a computational framework that can handle large numbers of model parameters efficiently. This single decomposition serves all subsequent calculations during the line search, maintaining high model resolution while dramatically improving inversion speed for 3D problems
Solution Approach 2:
The patent transforms the computational approach from multiple full matrix inversions to a single decomposition followed by efficient parameter updates. This parameter change in the mathematical methodology allows the system to handle large-scale 3D inversion problems with many parameters at speeds previously unattainable while preserving model resolution
Data Source
AI summary
A method including: generating an updated subsurface property model of a subsurface region, with a computer, from an initial estimate of the subsurface property model by performing an iterative inversion, which includes inverting geophysical data to infer the updated subsurface property model, wherein the generating the updated subsurface property model includes linearly remapping the initial estimate of the subsurface property model based on an inverse of a regularization operator, included with the initial estimate of the subsurface property model, into one in which the regularization operator is represented by an identity matrix, performing a unitary matrix decomposition in order to group the identity matrix with sparse matrices output from the unitary matrix decomposition, and performing a search over at least one trade-off parameter to reduce a misfit between simulated data generated from a most recent estimate of the subsurface property model and the geophysical data until a predetermined stopping criteria is satisfied; and generating, with a computer, an image of the subsurface region using the updated subsurface property model.