Efficient wavefield extrapolation in anisotropic media

Isotropic wavefield extrapolation is far more efficient than the anisotropic one, and this is especially true when the anisotropy exhibits tilted (from the vertical) symmetry axis. We use the kinematics of the wavefield, appropriately represented in the high-frequency asymptotic approximation by the eikonal equation, to develop effective isotropic models, which are used to efficiently extrapolate anisotropic wavefields using the isotropic, relatively cheaper, operators. These effective velocity models are source dependent and tend to embed the anisotropy influence in the inhomogeneity representation.

Specifically, the method is made up of calculating an effective source-dependent isotropic velocity model using the kinematic high-frequency geometrical representation of the anisotropic wavefield. This effective velocity model is equivalent to the instantaneous phase velocity estimated by matching the kinematics of an isotropic model to that of the anisotropic one along the wavefront. As soon as an effective velocity model is constructed for a particular source in the anisotropic media, it is used to solve the isotropic wave equation using this effective isotropic velocity model. Solving the wave equation includes wavefield extrapolating in inhomogeneous isotropic media.

The development is nearly completed. Developments in gearing the approach for imaging and inversion are under way. We tested the approach by comparing the approximate solution to the exact one obtained using the more expensive conventional methods. Part of the TI BP model is given by a complex salt body in the center as shown in the Figure below (a). Using the TI model, including the d, e, and q models, we solve for the traveltime map for a source located at depth of 4 km and lateral position of 32.75 km using a fast marching method applied to the finite difference approximation of the acoustic eikonal equation for TI media. The resulting traveltime is used to compute an effective velocity using equation 2, and we display that effective velocity in the Figure below (b). The difference between the vertical velocity for the BP model (a) and the computed effective velocity (b) is shown in the Figure below (c). This difference is related to the anisotropy and it is source location dependent.

For the same TI model and source location we use a finite difference method applied to an acoustic TI wave equation to obtain the snap shot of the wavefield shown below (a). The grid spacing in both directions is 25 m. The extrapolation time step is 0,8 msec. The peak frequency is 20 Hz. We also overlay the exact finite-difference eikonal solution on the snap shot of the wavefield at 1.28 s. Using this computed effective velocity we solve the acoustic isotropic wave equation 3 using a finite difference approach. Using the computed effective velocity above (b), we use the acoustic isotropic wave equation to solve for the wavefield for the same source and obtain the snap shot shown below (b). The difference between the two wavefields at 1.28s plotted at the same scale is shown below (c). The difference considering we are using the cheaper isotropic acoustic wave equation is relatively small. The cost of computing the wavefield shown in (b) is half of that in (a) and for 3D the difference is at least four fold.