TTI RTM using Variable Grid in Depth

Abstract

Abstract Summary In this abstract, we discuss some practical issues related to the implementation of reverse time migration (RTM) in a tilted transversely isotropic (TTI) medium. We have proposed a methodology to effectively reduce the cost. Through optimization of the coefficients of high-order difference scheme on the variable grid, larger depth steps at greater depth can be used instead of fixed step size. Synthetic data examples are used to demonstrate the effectiveness of the methodology. Introduction Reverse time migration (RTM) based on solving a two-way wave equation has been one of the most accurate imaging tools available in dealing with large lateral velocity variations and complicated geology such as large dips. RTM for isotropic and vertical transversely isotropic (VTI) medium has been routinely in use in the industry, but the application of RTM in TTI medium is still in the early stage. This lag results from the difficulties in numerical formulations for non-vertical symmetric axes and the subsequent instabilities. The TTI implementation also leads to higher computational costs compared with its isotropic or VTI counterparts. Instead of solving the original anisotropic wave equation, several modified quasi-P wave equations were derived by different authors, that can be used to perform anisotropic RTM for P-waves imaging without cause significant S wave scattering. Alkhalifah (2000) derived a 4th-order partial differential equation (PDE) in time to describe TTI propagation. To avoid solving a high order PDE, Zhou et al. (2006) split Alkhaifah's equation into two coupled 2nd-order PDEs. However it has been observed that their equations cause instability in practice when large dip is at presence. To circumvent this stability issue, Zhang et al. (2009) pursue an equivalent TTI formulation that preserves physical energy with the kinematics retained. Fletcher et al. (2009) proposed adding non-zero shear wave velocity terms to mitigate the instability problem. Unfortunately, their solution adds numerical complexity. In our implementation, to deal with both the stability and accuracy issues in TTI environment, we have to cut the cost dramatically to run it in production environment. To reduce the computational cost of TTI RTM, we use the least-square-root method in the Fourier domain to construct optimized finite difference (FD) schemes of variable grid in depth to approximate the first and second order derivatives. With larger steps in space in deep layers of higher velocity while keeping the same criteria for controlling the numerical dispersion, the total number of grid point is reduced. We analyzed the stability condition for this proposed FD scheme to determine the size of time step, which is related to minimum grid space, maximum velocity as well as the coefficients of FD scheme. With this implementation our RTM has been enhanced greatly in efficiency, and gained success in numerous 3D land imaging projects. Here we will demonstrate the effectiveness of this method using the 2D BP TTI model as a test case.