Cycle skipping in image perturbations

 

WEP1.imag
Figure 4 Comparison of image perturbations obtained as a difference between two migrated images (b) and as the result of the forward WEMVA operator applied to the known slowness perturbation (c). Panel (a) depicts the background image corresponding to the background slowness. Since the slowness perturbation is small ($0.1\%$), the image perturbations in panels (b) and (c) are practically identical.

I illustrate the WEMVA method with a simple model depicted in WEP1.imag. The velocity is constant and the data are represented by an impulse in space and time. I consider two slowness models: one regarded as the correct slowness s, and the other as the background slowness $\tilde{s}$. The two slownesses are related by a scale factor $\frac{s}{\tilde{s}}=\rho$.For this example, I consider $\rho=1.001$ to ensure that I do not violate the requirements imposed by the Born approximation.

Next, I migrate the data with the background slowness $\tilde{s}$and store the extrapolated wavefield at all depth levels. WEP1.imaga shows the image corresponding to the background slowness $\widetilde{\RR}$.I also migrate the data with the correct slowness and obtain a second image $\RR$. A simple subtraction of the two images gives the image perturbation in WEP1.imagb.

Finally, I compute an image perturbation by a simple application of the forward WEMVA operator defined in WEMVAobj to the slowness perturbation $\Delta s=s-\tilde{s}$ (WEP1.imagc). Since the slowness perturbation is very small, the requirements imposed by the Born approximation are fulfilled, and the two images in WEP1.imagb and WEP1.imagc are identical. The image perturbations are phase-shifted by $90^\circ$ relative to the background image.

A simple illustration of the adjoint operator $\Lop$ defined in WEMVAobj is depicted in WEP1.rays. Panel (a) shows the background image, panels (b) and (c) show image perturbations, and panels (d) and (e) show slowness perturbations. I extract a small subset of each image perturbation to create the impulsive image perturbations in WEP1.raysb and WEP1.raysc. The left panels (b and d) correspond to the image perturbation computed as an image difference, while the panels on the right (c and e) correspond to the image perturbation computed with the forward WEMVA operator. In this way, the data corresponds to a single point on the surface, and the image perturbation corresponds to a single point in the subsurface. By backprojecting the image perturbations in WEP1.raysb and WEP1.raysc with the adjoint WEMVA operator, I obtain identical wavepaths or ``fat rays'' shown in WEP1.raysd and WEP1.rayse, respectively.

 

WEP1.rays
Figure 5 Comparison of slowness backprojections using the WEMVA operator applied to image perturbations computed as a difference between two migrated images (b,d) and as the result of the forward WEMVA operator applied to a known slowness perturbation (c,e). Panel (a) depicts the background image corresponding to the background slowness. Since the slowness perturbation is small ($0.1\%$), the image perturbations in panels (b) and (c), and the fat rays in panels (d) and (e) are practically identical.

Prestack Stolt Residual Migration (storm) can be used to create image perturbations. Given an image migrated with the background velocity, I can construct another image by using an operator $\Kop$ function of a parameter $\rho$ which represents the ratio of the original and modified velocities. The improved velocity map is unknown explicitly, although it is described indirectly by the ratio map of the two velocities:  

$$\RR = \Kop \lp \rho \rp \lb \widetilde{\RR}\rb \;.$$ (67)

The simplest form of an image perturbation can be constructed as a difference between an improved image ($\RR$) and the background image ($\widetilde{\RR}$):

$$\Delta \RR= \RR - \widetilde{\RR}\;.$$ (68)

The main challenge with this method of constructing image perturbations for WEMVA is that the two images can be phase-shifted too much with respect to one-another. Thus, we can violate the requirements of the Born approximation and risk subtracting images that are out of phase. This problem is common for all wavefield-based velocity analysis or tomographic methods using the Born approximation (114; 28; 72).

A simple illustration of this problem is depicted in WEP2.imag and WEP2.rays. This example is similar with the one in WEP1.imag and WEP1.rays, except that the velocity ratio linking the two slownesses is much larger: $\rho=1.20$.In this case, the background and correct images are not at all in phase, and when I subtract them I obtain two distinct events, as shown in WEP2.imagb. In contrast, the image perturbation obtained by the forward WEMVA operator, WEP2.imagc, shows only one event as in the previous example. The only difference between the image perturbations in WEP1.imagc and WEP2.imagc is a scale factor related to the magnitude of the slowness anomaly.

WEP2.rays depicts fat rays for each kind of image perturbation: on the left, the image perturbations obtained by subtraction of the two images, and on the right, the image perturbation obtained with the forward WEMVA operator. The fat rays corresponding to the ideal image perturbation (panels c and e) do not change from the previous example, except for a scale factor. However, in case we use image differences (panels b and d), we can violate the requirements of the Born approximation. In this case, we see slowness backprojections of opposite sign relative to the true anomaly, and also the two characteristic migration ellipsoidal side-events indicating cycle-skipping (114).