Explicit extrapolation operators have proved useful in isotropic wavefield extrapolation Blacquiere et al. (1989); Holberg (1988); Thorbecke (1997). The dispersion relation in a tilted TI medium is very complicated, and it is very difficult to design an implicit extrapolation operator for it. However, explicit operators can still handle in the same way as isotropic media. In 3D, the circular symmetry of the isotropic or VTI media allows us to design a 1D algorithm to replace the 2D convolution operator by McClellen transformations Hale (1991a,b); Zhang et al. (2001b). For tilted TI media, the deviation of the symmetry axis from the vertical direction breaks that circular symmetry. As a result, a 2D convolution operator has to be designed for the wavefield extrapolation in 3D tilted TI media.
Tilted coordinates Shan and Biondi (2004a) are used to extrapolate wavefields in a direction close to the wave propagation direction. We can use tilted coordinates to get good accuracy for high-angle energy using a less accurate operator. A VTI medium in Cartesian coordinates becomes a tilted TI medium in tilted coordinates. Thus to extrapolate wavefields in tilted coordinates in a VTI medium, we need an extrapolation operator for tilted TI media.
In this paper, we extrapolate the wavefield in 3D tilted TI media using an implicit isotropic operator with an explicit anisotropic correction Shan and Biondi (2004b). We begin by first deriving the 3D dispersion relation in tilted TI media. Then we discuss how to design 2D antisymmetric convolution operators in the Fourier domain for tilted TI media. We discuss how the length of the filter affects the accuracy of the operator and propose a way to design short 2D filters. Finally, we present 3D impulse response for a tilted TI medium of our algorithm.