Waveform Inversion Gradient Normalization for Subsurface Imaging
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Solution Overview
Problem
Waveform inversion in subsurface exploration faces challenges due to the absence of low-frequency data, 2D acoustic approximations of 3D earth wave propagation, and noise in seismic data, leading to variations in gradient vectors for frequency and damping constants, which can distort gradient directions and require additional operations like line searches.
Innovation Solution
The method employs explicit weighting functions to normalize gradient vectors for each damping constant and all damping constants, allowing for equal distribution of frequency components and eliminating the need for line searches by determining step lengths without additional algorithms.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If waveform inversion is performed using conventional methods without weighting functions, then the gradient direction can be determined by adding gradient vectors for each frequency and damping constant, but the gradient direction becomes distorted due to unequal distribution of frequency components and damping constants
Solution Approach 1:
The patent applies weighting functions that modify the parameters of gradient vectors by introducing frequency-dependent and damping constant-dependent weights. This changes the contribution of each gradient vector component to the total gradient direction, ensuring equal distribution of frequency components and damping constants in the gradient calculation, thereby resolving the distortion issue while maintaining computational feasibility
2Manufacturing precision
If waveform inversion uses all frequency components including low-frequency data, then the subsurface velocity model becomes more detailed, but noise and computational complexity increase
Solution Approach 1:
The patent applies different weighting functions to different frequency components and damping constants. Low-frequency components and components with high damping constants receive appropriate weighting that balances their contribution, allowing them to be utilized effectively without allowing noise to dominate the inversion process. This local differentiation of weighting resolves the contradiction between utilizing all frequency components and managing noise
3Reliability
If waveform inversion performs additional operations like line search to determine step length, then the convergence can be improved, but the computational time and complexity increase
Solution Approach 1:
The patent designs the weighting functions and gradient vector normalization such that the step length can be determined directly from the normalized gradient vectors without requiring additional line search operations. The weighting scheme inherently provides the necessary scaling, allowing the inversion process to proceed efficiently with reduced computational overhead while maintaining convergence reliability
Data Source
AI summary
Provided are an apparatus and method for imaging the subsurface structure of a target area by using waveform inversion. In the apparatus and method, the subsurface structure of a target area is estimated using waveform inversion of a seismic signal in the frequency domain, the Laplace domain, or the Laplace-Fourier domain, and an objective function is defined by applying a weighting function such that the objective function makes a different contribution for each frequency, each Laplace damping constant, or each Laplace-Fourier damping constant. The objective function is not limited to a particular type of objective function and a weighting function can be automatically determined when a gradient vector for each frequency, each Laplace damping constant, or each Laplace-Fourier damping constant is normalized. In addition, a gradient direction for all frequencies can be defined by applying another weighting function to the sum of respective gradient vectors for all frequencies, all Laplace damping constants, or all Laplace-Fourier damping constants, wherein the weighting function can also be automatically determined by normalization.