Implicit Finite-Difference Wavefield Modeling Without Saturation
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Solution Overview
Problem
Existing finite-difference methods for solving the wave equation in seismic data processing suffer from the 'saturation effect' and numerical dispersion, leading to inefficient and inaccurate modeling of seismic waves, which hinders the identification of hydrocarbon reservoirs and wellbore planning.
Innovation Solution
A method using deconvolution filters to evaluate an inverse matrix for each spatial partial differential equation within the wave equation, mitigating the saturation effect and numerical dispersion while maintaining computational efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If explicit finite-difference methods are used to solve the wave equation, then computational performance increases with operator length up to a certain point, but performance saturates and ceases to increase significantly beyond that operator length
Solution Approach 1:
The patent transitions from explicit finite-difference methods to implicit finite-difference methods, fundamentally changing the computational approach. This parameter change allows the use of shorter operators while maintaining or improving accuracy, thereby resolving the saturation effect where longer operators no longer improve performance
Solution Approach 2:
The patent replaces the explicit computational mechanism with an implicit computational mechanism. By using implicit finite-difference methods with deconvolution filters, the system achieves better performance without requiring increased operator length, thus overcoming the saturation limitation
2Productivity
If other finite-difference methods are used to solve the wave equation, then computational performance may be improved, but the methods become computationally expensive or produce numerical dispersion
Solution Approach 1:
The patent changes the computational method from explicit to implicit finite-differences and applies deconvolution filtering. This parameter change achieves high computational performance while maintaining numerical accuracy and avoiding dispersion errors that plague other methods
Solution Approach 2:
The patent introduces deconvolution filters as an intermediary component in the implicit finite-difference scheme. These filters act as mediators that correct numerical errors and prevent dispersion, thereby maintaining reliability while achieving improved computational performance
Data Source
AI summary
Systems and methods are disclosed. The method includes, for each of a plurality of spatial partial differential equations (sPDEs) within a wave equation, determining a linear system of equations using an approximate solution at a plurality of grid nodes. The linear system of equations includes an inverse matrix, a first vector, and a second vector. The method further includes, for each of the plurality of sPDEs, evaluating the inverse matrix by evaluating a first portion of the inverse matrix using a first deconvolution filter and evaluating a second portion of the inverse matrix using a second deconvolution filter. The method further still includes, for each of the plurality of sPDEs, evaluating the first vector using the evaluated inverse matrix and the second vector as well as determining the wavefield using the evaluated first vector for each of the plurality of sPDEs, a velocity model, and the wave equation.