Graph-Based Seismic Velocity Estimation with Second Eigenvalue Analysis
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Solution Overview
Problem
Existing seismic velocity estimation methods for subsurface layers are computationally burdensome and prone to errors, particularly when using eigenvalues to calculate velocities, which can lead to inaccurate results.
Innovation Solution
A system and method utilizing graph theory to estimate seismic velocity by forming a weighted adjacency matrix, calculating eigenvalues of a normalized weighted Laplacian matrix, and employing a quartic regression polynomial to determine the second largest eigenvalue, thereby improving accuracy and reducing computational complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional methods (Hermitian matrix with Laplacian operator) are used to determine wave velocity, then velocity estimation can be performed, but computational cost increases significantly due to complex matrix operations
Solution Approach 1:
The patent extracts only the essential information needed for velocity estimation from the complex seismic data, specifically using the second largest eigenvalue of the normalized Laplacian matrix rather than performing complete Hermitian matrix computations. This extraction approach maintains measurement precision while significantly reducing computational complexity by focusing on the critical eigenvalue component.
Solution Approach 2:
The patent changes the mathematical parameter used for velocity estimation from the complete Hermitian matrix spectrum to specifically the second largest eigenvalue of the normalized Laplacian matrix. This parameter change simplifies the computational procedure while preserving the ability to accurately estimate wave velocity through the relationship between eigenvalues and seismic wave propagation characteristics.
2Measurement precision
If iterative methods with multiple domain computations are used to determine velocity, then velocity can be calculated, but computation time increases due to multiple domain transformations
Solution Approach 1:
The patent segments the velocity estimation process into distinct steps: computing the normalized Laplacian matrix, extracting the second largest eigenvalue, and applying the velocity formula. This segmentation eliminates the need for iterative computations across multiple domains (CSG, CRG, CMP, COG) while maintaining velocity calculation accuracy through the graph theory-based approach.
Solution Approach 2:
The patent substitutes the mechanical iterative computation process with a mathematical graph theory approach. Instead of performing multiple domain transformations and iterative optimizations, the method uses spectral graph theory to directly compute velocity from the second largest eigenvalue, significantly reducing computation time while preserving measurement precision.
3Device complexity
If highest eigenvalue is used to calculate velocity, then computation is simplified, but accuracy decreases due to higher error rates
Solution Approach 1:
The patent inverts the conventional approach by using the second largest eigenvalue rather than the highest eigenvalue. This inversion of the selection criterion maintains computational simplicity while significantly improving accuracy, as the second largest eigenvalue provides a more reliable indicator of wave velocity in the subsurface layers.
Solution Approach 2:
The patent incorporates feedback through the use of the second largest eigenvalue, which has been shown to provide more accurate velocity estimates. This feedback mechanism allows the system to automatically select the most appropriate eigenvalue for velocity calculation, improving measurement precision while maintaining computational efficiency.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
The method achieves accurate seismic velocity estimation with a relative error of less than 0.1%, significantly improving upon prior methods that had errors of 4% or more, while reducing computational burden.
Implementation Method 1
Following the principles of wave reflection and refraction, seismic waves, upon encountering different types of rock or soil, may undergo partial refraction and reflection
Implementation Method 2
Following the principles of wave reflection and refraction, seismic waves, upon encountering different types of rock or soil, may undergo partial refraction and reflection
Implementation Method 3
forming a normalized weighted Laplacian matrix Lsymw by calculating... calculating eigenvalues of the normalized weighted Laplacian matrix
Data Source
AI summary
A method and a system for estimating a seismic velocity of a subsurface layer of a geologic formation is described. A seismic source is configured to direct seismic shots into the geologic formation. Multiple seismic receivers placed at certain distance from the seismic source are configured to receive seismic waves refracted from the subsurface layer, convert the received seismic waves into seismic traces and transmit seismic traces to a signal processing circuitry which identifies a common midpoint of the seismic traces and applies a graph-based computation on at least three seismic traces to identify a second largest eigenvalue corresponding to multiple transmission points within the subsurface layer. The second largest eigenvalue is used in a quartic regression polynomial, which uses the selected seismic traces to determine each coefficient of the quartic regression polynomial in order to identify a velocity of seismic waves and the corresponding material of the subsurface layer.