Best and Worst Case Prediction in Seismic Inversion
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Solution Overview
Problem
Current methods for predicting best and worst case scenarios in non-unique model inversion problems, such as sand and porosity distribution in reservoirs, are either time-consuming or biased and do not accurately produce 'best' and 'worst' case solutions.
Innovation Solution
A computer-implemented method that diagonalizes the G matrix using orthonormal basis vectors, identifies null vectors associated with insignificant elements, and applies Lp mathematical norms to determine upper and lower bounds for solutions, scaling linear combinations of null vectors to obtain 'best' and 'worst' case scenarios without affecting data fit.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If forward simulations of many different models are performed to determine alternative scenarios, then the accuracy of best and worst case predictions is improved, but the computational time and resources required increase significantly
Solution Approach 1:
The patent segments the solution space by decomposing the inverse problem into a most likely solution component and null space components. This segmentation allows the method to identify and exploit the structure of non-uniqueness without requiring exhaustive sampling of all possible models, thereby reducing computational time while maintaining prediction accuracy.
Solution Approach 2:
The patent extracts the null space from the complete solution space using singular value decomposition. By separating the null space vectors from the full model space, the method can directly characterize best and worst case scenarios without performing forward simulations of numerous models, thus resolving the contradiction between accuracy and computational efficiency.
2Device complexity
If the most likely model is simply scaled up and down to obtain best and worst case scenarios, then the computational complexity is reduced, but the accuracy of the predictions deteriorates as the scaled models do not match the observed data
Solution Approach 1:
The patent changes the parameter representation from simple scaling factors to null space vector combinations. By expressing alternative scenarios as linear combinations of null space vectors added to the most likely solution, the method maintains data consistency while capturing the true range of possible solutions, thereby improving accuracy without excessive computational complexity.
3Productivity
If forward models are selected based on user judgment to determine alternative scenarios, then the computational resources are conserved, but the results suffer from user bias and may not represent the true best and worst case solutions
Solution Approach 1:
The patent enables the system to self-identify the best and worst case solutions through mathematical optimization of null space vector combinations. The method objectively determines the extreme scenarios by maximizing or minimizing the objective function subject to data constraints, eliminating user bias while maintaining computational efficiency.
Data Source
Figure 1
Figure 2A~2C
AI summary
Method for determining best and worst cases for values of model parameters of porosity and shale volume fraction generated by matrix inversion of physical data of seismic reflection amplitudes (Figure 1). The matrix is diagonal zed and ortho-normal basis vectors associated with insignificant diagonal elements are used to generate upper and lower bounds on the solution (6). The best and worst case solutions are determined as linear combinations of the null basis vectors where the expansion coefficients are determined by making a best fit to the upper and lower bounds (J).