TTI Seismic P-wave Modeling with Stabilized Mixed Derivatives
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Solution Overview
Problem
Current methods for seismic P-wave modeling in inhomogeneous transversely isotropic media with a tilted symmetry axis face stability issues and inaccuracies, particularly in complex media with rapidly varying anisotropic symmetry axes, leading to instabilities and artifacts in wave propagation.
Innovation Solution
A method involving the derivation of wave equations from first principles using the acoustic TI approximation, with a local rotation of stress and strain tensors to align with the tilted symmetry axis, and the use of centered finite-difference operators to discretize mixed and non-mixed second-order derivatives, ensuring stability by weighing down mixed derivatives.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If a non-zero shear velocity is introduced to relieve the physical constraint, then the range of negative ε−δ can be handled, but the resulting discretization schemes do not satisfy the principles of self-adjointness and may fail to yield stable results in synthetic experiments
Solution Approach 1:
The patent changes the parameter of shear velocity from non-zero to zero, thereby relieving the physical constraint while maintaining stability. By setting shear velocity to zero, the acoustic P-wave approximation is used, which satisfies the self-adjointness principle and ensures numerical stability in synthetic experiments.
2Measurement precision
If centered finite-difference operators are used to discretize second-order derivatives, then accuracy is improved, but mixed second-order derivatives require special handling to maintain stability
Solution Approach 1:
The patent segments the discretization process into separate treatments for mixed and non-mixed second-order derivatives. Mixed derivatives are discretized using centered first-order difference operators applied twice, while non-mixed derivatives use centered second-order difference operators. This segmentation allows maintaining accuracy while ensuring stability through appropriate operator selection.
Solution Approach 2:
The patent introduces an intermediary weighting factor for mixed second-order derivative terms. By weighing down the mixed derivatives with a factor less than one, the stability of the explicit time-stepping method is established while preserving the accuracy of the centered finite-difference discretization.
3Measurement precision
If the symmetry axis is tilted to represent complex subsurface structures, then the model accuracy is improved, but the complexity of the wave equation increases
Solution Approach 1:
The patent applies local quality by rotating the coordinate system to align with the tilted symmetry axis at each location. This local rotation simplifies the elastic tensor representation and wave equation formulation, allowing accurate modeling of complex subsurface structures without increasing overall equation complexity.
Solution Approach 2:
The patent handles the dynamic nature of tilted symmetry axes by using spatially varying rotation matrices that adapt to local medium properties. This dynamic approach allows the wave equation to accommodate complex subsurface geometries while maintaining a manageable mathematical form through local coordinate transformations.
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 provides stable and accurate seismic imaging of subsurface formations by minimizing modeling artifacts and maintaining physical meaningfulness, even in complex media with strong spatial variations in medium parameters.
Implementation Method 1
generating a pseudo-acoustic stress-strain relationship of a stress tensor and a strain tensor of modelled P-waves by means of a TI elastic tensor
Implementation Method 2
measuring seismic P-waves, excited by a seismic source, and propagated through the subsurface formation
Implementation Method 3
each of the tensors is expressed in a rotated local Cartesian coordinate frame which is aligned with the tilted symmetry axis of the TI medium
Implementation Method 4
discretizing first-order spatial derivatives and non-mixed second-order spatial derivatives by centered finite-differences
Data Source
AI summary
An improved method for P-wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media), suitable for anisotropic reverse-time migration, is based on an acoustic TI approximation. The resulting wave equations (2.20) & (2.21) are derived directly from first principles, Hooke's law and the equations of motion, and therefore make no assumptions on spatial variation of medium parameters. Like in the acoustic VTI case, the wave equations are written as a set of two second-order partial differential equations. However, unlike in the acoustic VTI case, the acoustic TTI wave equations contain mixed second-order derivatives. The discretization scheme uses centered finite-difference operators for first- and second-order derivative operators to approximate the mixed and non-mixed second-order derivatives in the wave equation. The discretization scheme is stabilized by slightly weighing down the mixed derivatives, with almost negligible effect on the wave field kinematics.