Accuracy comparison

For the proposed hybrid pseudospectral/finite-difference method, the cost of derivative calculation is reduced at the expense of the precision, as well as the accuracy of the solution, because some of the wavenumber operators are substituted by finite-difference approximations. To demonstrate the resulting differences on accuracy, I conduct a 2D modeling test on a simple 5-layer TTI model. Figure 4.1 displays model parameters that used in this test. The spacial interval of the computational grid is , and the maximum frequency of the source wavelet is $30~Hz$ .

Figure 4.1: Anisotropic model parameters used in the accuracy test.

First, equation 4.10 using the standard pseudospectral method is implemented. Then equation 4.12 using the hybrid method is computed, where spatial derivatives in equations 4.11a, 4.11b, and 4.11c are approximated and calculated using second, fourth and eighth-order centred finite-difference schemes from Taylor series expansions, respectively. To check the amplitude differences, three wiggle traces at zero/middle/far offsets computed using different methods are plotted and comparied in Figure 4.2. As we can see from Figure 4.2, amplitudes computed from the pseudospectral method and the hybrid method with an eighth-order finite-difference scheme are perfectly matched. And the computational costs of these two methods are almost equivalent. When the fourth-order finite-difference scheme is used, all major amplitudes from shallow to deep are still well matched to the pseudospectral result, except that some tiny discrepancies start to appear due to numerical dispersion. However, the runtime is reduced by half. A more compact second-order finite-difference scheme may further improve the computational efficiency, however, both the amplitude discrepancy and phase error are maximized due to the strong dispersive behavior associated with smaller stencils.

According to the above analysis, in latter numerical examples, the fourth-order finite-difference scheme is chosen to compute spatial derivatives in the hybrid method in terms of accuracy and efficiency.

Figure 4.2: Three traces computed from the 2D 5-layer TTI model are displayed in wiggle mode. (a) compares the first (strongest) reflection event, while (b) compares lower (weak) reflections.