Tomographic Inversion via Matrix Transformation for Grid Stability
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Current seismic tomographic inversion methods face instability and high computational costs due to the use of uniform grids, which fail to effectively remove long-wavelength artifacts and are inefficient in ray tracing, while non-uniform grids stabilize but are slow and difficult to add smoothing constraints to.
Innovation Solution
A method that transforms a uniform grid-based linear system of tomographic equations into a non-uniform parameterization using a linear mapping matrix, reducing the number of model parameters and stabilizing the inversion process while maintaining efficient ray tracing in the uniform grid.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If a uniform grid is used to parameterize the model, then ray tracing is computationally efficient, but the inversion becomes unstable and fails to remove long-wavelength artifacts
Solution Approach 1:
The patent segments the model parameterization into two distinct representations: a uniform grid system for efficient ray tracing computations and a non-uniform grid system for stable inversion. The uniform grid handles the forward modeling while the non-uniform grid with fewer parameters handles the inversion stability, effectively dividing the conflicting requirements into separate domains.
Solution Approach 2:
The patent introduces a linear mapping matrix as an intermediary that connects the uniform grid representation (used for ray tracing) with the non-uniform grid representation (used for inversion). This mapping matrix enables data to be transformed between the two representations, allowing the system to leverage the computational efficiency of the uniform grid while achieving the stability of the non-uniform grid during inversion.
2Reliability
If a non-uniform grid is used to parameterize the model, then the inversion is stabilized by reducing the number of parameters, but ray tracing becomes much slower
Solution Approach 1:
The patent segments the computational workflow into two distinct phases using two different grid representations: ray tracing is performed on the uniform grid where it is computationally efficient, while inversion is performed on the non-uniform grid where it is stable. This segmentation allows each operation to be performed in the most suitable representation without compromising the other.
Solution Approach 2:
The linear mapping matrix serves as an intermediary that enables transformation between the uniform and non-uniform representations. The workflow uses the mapping to convert data from the uniform grid to the non-uniform grid for inversion, and can convert results back if needed, allowing the system to exploit the speed of uniform grid ray tracing while achieving non-uniform grid inversion stability.
3Manufacturing precision
If smoothing constraints are added to a uniform grid model, then short-wavelength artifacts are removed, but long-wavelength artifacts remain and the method fails when artifacts have the same wavelength as true velocity structure
Solution Approach 1:
The patent changes the fundamental parameterization of the model from a uniform grid with many parameters to a non-uniform grid with fewer parameters. This parameter change fundamentally alters the wavelength characteristics of the model representation, allowing the inversion to naturally handle long-wavelength artifacts without relying on smoothing constraints that fail at certain wavelength scales.
Data Source
AI summary
Method for reducing instability and increasing computational efficiency in tomographic inversion for velocity model building. A system of tomographic equations is developed for a uniform grid. A non-uniform parameterization is found for which a linear mapping exists between the space of the uniform grid and the space of the non-uniform grid. The matrix that relates velocity to the tomographic data in the non-uniform representation is then given by the matrix product of the corresponding matrix in the uniform grid representation and the mapping matrix. Inversion can then be performed for the non-uniform parameterization on a smaller, more stable matrix.