Implicit Finite‐Difference Plane Wave Migration in TTI Media

Abstract

Implicit finite-difference (IFD) extrapolation operator and plane wave migration in TTI media are studied in this paper. Compared to isotropic and VTI media, the dispersion relation of TTI media is much more complicated. It is difficult to get the explicit expression of the dispersion relations. It needs no longer only a symmetrical even function like isotropic and VTI media but also an odd function to express the dispersion relations. We design TTI medium IFD wave-field extrapolation operator and solve operator coefficients by nonlinear optimization method. We combine the theory of plane wave migration in isotropic media with the IFD operator in TTI media and extend plane wave migration to TTI media. Comparing the dispersion curves and migration impulse response of the IFD extrapolation operator with the theoretical solution verifies the high accuracy of the operator. Synthetic P-wave data from full elastic TTI equation modeling followed by polarization analysis verifies the accuracy and effectiveness of the method.

An approximation of stochastic hyperbolic equations: case with Wiener process

“…The method of operators as a tool for the investigation of the solution to stochastic parabolic equations in Hilbert and Banach spaces has been systematically developed by several authors (see and the references therein). Finite difference method for the solution of initial boundary value problem for stochastic partial differential equations has been studied extensively by many researchers (see and the references therein). However, initial boundary value problems for stochastic hyperbolic differential equations were not well investigated.…”

Numerical Solution of Stochastic Hyperbolic Equations

“…For example, the method of operators as a tool for investigation of the solution to stochastic equations in Hilbert and Banach spaces have been used systematically by several authors see, 1-7 and the references therein . Numerical methods and theory of solutions of initial boundary value problem for stochastic partial differential equations have been studied in [8][9][10][11][12][13][14][15][16] . Moreover, the authors of 17 presented a two-step difference scheme for the numerical solution of the following initial value problem: dv t −Av t dt f t dw t , 0 < t < T, v 0 0,v 0 0, 1.1 for stochastic hyperbolic differential equations.…”