Hessian Vector Approximation in Full Wavefield Inversion
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Solution Overview
Problem
Full wavefield inversion in seismic data processing requires computationally intensive evaluation of the Hessian times vector, which is often approximated using iterative methods that demand repeated evaluations of the Hessian matrix, leading to high memory requirements and slow convergence rates.
Innovation Solution
Approximate the Hessian times vector using a single forward-wave propagation and a single gradient computation in a modified subsurface model, leveraging the Born approximation to reduce computational burden and improve convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If iterative methods are used to evaluate the Hessian times vector, then the convergence rate of full wavefield inversion is improved, but the computational time and memory requirements increase significantly
Solution Approach 1:
The patent changes the parameter representation by introducing a modified Hessian operator that acts on the gradient vector through a different mathematical formulation. Instead of directly computing Hg, the method uses an alternative expression involving the wave equation solver that achieves the same result with fewer computational operations, thereby reducing time loss while maintaining convergence improvement.
Solution Approach 2:
The patent extracts and eliminates the computationally expensive repeated Hessian matrix evaluations from the iterative process. By formulating the Hessian times vector operation in a way that requires only a single wave equation solve rather than multiple iterations, the method removes the time-consuming element while preserving the beneficial convergence acceleration.
2Quantity of substance
If the Hessian matrix is computed iteratively using conjugate gradient method, then memory space requirements are reduced, but the computational complexity and time increase
Solution Approach 1:
The patent transforms the computational parameter from iterative Hessian-vector multiplications to a single wave equation solve with modified source terms. This parameter change reduces the operational complexity from O(N×M) iterations to O(N) single pass, decreasing computational complexity while maintaining the memory efficiency of not storing the full Hessian matrix.
Solution Approach 2:
The method creates an equivalent computational representation of the Hessian times vector operation that can be obtained through a different physical process (single wave propagation with modified sources rather than iterative linear algebra). This copying approach achieves the same mathematical result with reduced computational complexity.
3Measurement precision
If repeated evaluations of the Hessian matrix are performed, then accurate model updates are achieved, but the convergence speed decreases due to computational burden
Solution Approach 1:
The patent modifies the computational parameters by changing from repeated Hessian evaluations to a single evaluation with a modified computation path. The alternative formulation uses different intermediate parameters (wave fields from modified sources) that lead to the same final result (Hessian times gradient) but with improved productivity through reduced computational steps.
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
This approach significantly reduces computational time and enhances convergence rates by eliminating the need for repeated Hessian evaluations, achieving a 40% speedup in convergence compared to traditional methods.
Implementation Method 1
Approximate the Hessian times vector using a single forward-wave propagation and a single gradient computation in a modified subsurface model, leveraging the Born approximation to reduce computational burden and improve convergence.
Data Source
AI summary
Method for estimating the Hessian of the objective function, times a vector, in order to compute an update in an iterative optimization solution to a partial differential equation such as the wave equation, used for example in full wave field inversion of seismic data. The Hessian times vector operation is approximated as one forward wave propagation (24) and one gradient computation (25) in a modified subsurface model (23). The modified subsurface model may be a linear combination of the current subsurface model (20) and the vector (21) to be multiplied by the Hessian matrix. The forward-modeled data from the modified model are treated as a field measurement in the data residual of the objective function for the gradient computation in the modified model. In model parameter estimation by iterative inversion of geophysical data, the vector in the first iteration may be the gradient of the objective function.