BP TTI Model

To demonstrate the instability problem, I do a second test on a small region of the 2D BP TTI model. The P-wave velocity and three extra anisotropy distributions are shown in Figure 2.3. Rapid variation of the tilt angle around the salt presents challenges to TTI RTM. The benchmark dataset is originally created by an elastic FD modeling code, which approximates the field data.

Figure 2.4c shows a snapshot of the wavefield from the forward modeling simulation. The variation of the tilt angle around the right salt flank blows up the amplitudes of the wavefield. By checking the gradient of theta (Figure 2.4a), I found that regions of large gradients excite these instabilities. The gradient shown here is weighted by the difference of epsilon and delta. Following Yoon's (2010) method, I first pick up the high gradient points by filtering Figure 2.4a with a given threshold. Here, the word "filtering" means that points with a value greater than the threshold are marked as one, otherwise they are marked as zero. Values of the threshold are empirically chosen and need to be tested before migration starts. The plot of the filtered gradient is shown in Figure 2.4b. Then I do a equating of $\varepsilon = \delta$ around the selected high gradient points. Instead of changing $\varepsilon$ or changing $\delta$ , I operate on the model by setting $(\varepsilon-\delta)=0$ along high gradient points. The pre-processed anisotropic model is then spatially smoothed by a 2D filter that is about 5 wavelength wide along each coordinate direction. The smoothed version of is used as an input model in equation 2.1. Figure 2.4d shows a stable snapshot at the same time step as Figure 2.4c after parameter equating is employed.

Figure 2.3: Partial region of the 2D BP TTI model. Top left shows the P-wave velocity distribution, top right and the bottom left show the Thomsen anisotropy parameter model. The bottom right is the input dip angle relative to the vertical symmetry.

Figure 2.5 shows the comparison of the RTM images in this test region. Due to the presence of anisotropy, imaging of dipping events such as the salt flank is severely affected and mispositioned in the conventional RTM image. TTI RTM produces a superior image at both the salt flank and sedimentary layers around the salt. Significant imaging differences observed in Figure 2.5 tell us that TTI RTM is more accurate than conventional isotropic RTM in the presence of anisotropy.

Figure 2.4: (a) is the gradient of the dip angle theta weighted by . (b) shows the filtered gradient by a given threshold. White corresponds to 0 and black corresponds to 1. Parameter equating of epsilon and delta is done along the black curve. (c) and (d) are wavefield snapshots without and with anisotropic parameter equating. Wavefield "blow up" is accommodated by equating epsilon and delta around the high gradient points.

Figure 2.5: The TTI RTM image (right) for the 2D BP TTI benchmark dataset. Left panel shows the isotropic RTM image. The salt flank is well delineated by TTI RTM but is mispositioned in the isotropic RTM image.