Second-Order Wave Equation for Stable RTM in Tilted Orthorhombic Media
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Solution Overview
Problem
Current Reverse Time Migration (RTM) methods face numerical instability when applied to tilted orthorhombic media, which are common in geological environments with complex fracture systems, and existing solutions do not provide stable and high-quality images for such media.
Innovation Solution
A stable second-order wave equation is developed for reverse time migration in arbitrarily heterogeneous three-dimensional orthorhombic media with a tilted symmetry axis, using self-adjoint differential operators and constraints on Thomsen's parameters to ensure physical and numerical stability, allowing for high-order finite-difference algorithms to solve the system of equations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If first-order wave equations are used to handle wave propagation in anisotropic media, then the accuracy of modeling is improved, but computational intensity increases significantly
Solution Approach 1:
The patent replaces the first-order wave equations with a second-order wave equation formulation. This substitution transforms the mathematical model from a first-order system to a second-order system, which reduces computational intensity while maintaining the ability to handle anisotropic media through the use of Thomsen parameters and appropriate coordinate transformations.
2Device complexity
If P- and S-waves are decoupled in isotropic media using second-order wave equations, then computational complexity is reduced, but this approach fails in anisotropic media where P- and S-waves are coupled
Solution Approach 1:
The patent applies local quality by introducing direction-dependent parameters (Thomsen parameters) that vary with propagation direction in anisotropic media. The second-order wave equation is formulated with anisotropic velocity terms that locally adjust to the media properties in different directions, allowing P-wave and S-wave coupling to be properly represented while maintaining computational tractability through selective decoupling where applicable.
Solution Approach 2:
The patent changes the parameter representation from isotropic velocity to anisotropic velocity parameters (Thomsen parameters ε, δ, η). This parameter transformation allows the wave equation to adapt to anisotropic media by incorporating direction-dependent velocity variations, enabling the model to handle P- and S-wave coupling in tilted orthorhombic media while maintaining a second-order formulation.
3Productivity
If existing RTM methods are applied to tilted orthorhombic media, then processing capability is maintained, but numerical instability occurs
Solution Approach 1:
The patent applies preliminary anti-action by pre-establishing stability conditions through constraints on Thomsen parameters before performing the RTM computation. The formulation includes acaustic conditions and parameter constraints that prevent numerical instability from occurring in the first place, rather than attempting to correct instability after it arises. This proactive approach ensures numerical stability while maintaining processing capability for tilted orthorhombic media.
Data Source
AI summary
A computing device, computing medium and method for generating an image of a tilted orthorhombic medium. The method includes receiving seismic data related to the tilted orthorhombic medium; calculating a wave propagation with a processing device by applying a second-order equation for reverse time migration to the seismic data to generate a tilted orthorhombic wave propagation; and generating the image of the tilted orthorhombic medium based on the tilted orthorhombic wave propagation.