Stereographic imaging condition for wave-equation migration

Conventional imaging condition

Under the single scattering (Born) approximation, seismic migration consists of two components: wavefield reconstruction and imaging.

Wavefield reconstruction forms solutions to the considered (acoustic) wave-equation with recorded data as boundary condition. We can consider many different numeric solutions to the acoustic wave-equation, which are distinguished, for example, by implementation domain (space-time, frequency-wavenumber, etc.) or type of numeric solution (differential, integral, etc.). Irrespective of numeric implementation, we reconstruct two wavefields, one forward-propagated from the source and one backward-propagated from the receiver locations. Those wavefields can be represented as four-dimensional objects function of position in space ${ \bf x}=\left (x,y,z\right)$ and time $t$

$$u_s = u_s\left ({ \bf x},t \right)$$ (1)

$$u_r = u_r\left ({ \bf x},t \right)\;,$$ (2)

where $u_s$ and $u_r$ denote source and receiver wavefields. For the remainder of this paper, we can assume that the two wavefields have been reconstructed with one of the numerical methods mentioned earlier.

The second migration component is the imaging condition which is designed to extract from the extrapolated wavefields ($u_s$ and $u_r$) the locations where reflectors occur in the subsurface. The image $r$ can be extracted from the extrapolated wavefields by evaluating the match between the source and receiver wavefields at every location in the subsurface. The wavefield match can be evaluated using an extended imaging condition (Sava and Fomel, 2006,2005), where image $r$ represents an estimate of the similarity between the source and receiver wavefields in all $4$ dimensions, space (${ \bf x}$) and time ($t$):

$$r\left ({ \bf x}, \mathbf{\lambda}, {\tau}\right)= \!\!\!\i... ... u_r\left ({ \bf x}+ \mathbf{\lambda}, t+{\tau}\right) dt \;.$$ (3)

The quantities $\mathbf{\lambda}$ and ${\tau}$ represent the spatial and temporal cross-correlation lags between the source and receiver wavefields. The source and receiver wavefields are coincident (i.e. form an image) if the local cross-correlation between the source and receiver wavefields maximizes at zero-lag on all four dimensions. Other extended imaging conditions (Rickett and Sava, 2002; Biondi and Symes, 2004) represent special cases of the extended imaging condition corresponding to horizontal $\mathbf{\lambda}=\left ( \lambda_x,\lambda_y,0\right)$, or vertical $\mathbf{\lambda}=\left (0,0,\lambda_z\right)$ space lags, respectively. The conventional imaging condition Claerbout (1985) is also a special case of the extended imaging condition 3, corresponding to zero cross-correlation lag in space ( $\mathbf{\lambda}=0$) and time (${\tau}=0$):

$$r\left ({ \bf x}\right)= \!\!\!\int\!\! u_s\left ({ \bf x},t \right) u_r\left ({ \bf x},t \right) dt \;.$$ (4)

velo,refl,dd
Figure 1. Constant velocity model (a), reflectivity model (b), data (c) and shot locations at $x=600,1000,1200$ m).

velo,refl,dd
Figure 2. Velocity model with a negative Gaussian anomaly (a), reflectivity model (b), data (c) and shot location at $x=1000$ m).

The four-dimensional cross-correlation 3 maximizes at zero lag if the source and receiver wavefields are correctly reconstructed. If this is not true, either because we are using an approximate extrapolation operator (e.g. one-way extrapolator with limited angular accuracy), or because the velocity used for extrapolation is inaccurate, the four-dimensional cross-correlation does not maximize at zero lag and part of the cross-correlation energy is smeared over space and time lags ( $\mathbf{\lambda}$ and ${\tau}$). Therefore, extended imaging conditions can be used to evaluate imaging accuracy, for example by decomposition of reflectivity function of scattering angle at every image location (Sava and Fomel, 2006,2003; Biondi and Symes, 2004). Angle-domain images carry information useful for migration velocity analysis (Sava and Biondi, 2004a; Shen et al., 2005; Biondi and Sava, 1999; Sava and Biondi, 2004b), or for amplitude analysis (Sava et al., 2001), or for attenuation of multiples (Sava and Guitton, 2005; Artman et al., 2007)

The conventional imaging condition 4 is the focus of this paper. As discussed above, assuming accurate extrapolation, this imaging condition should produce accurate images at zero cross-correlation lags. However, this conclusion does not always hold true, as illustrated next.

Figures 1(a) and 1(b) represent a simple model of constant velocity with a horizontal reflector. Data in this model are simulated from $3$ sources triggered simultaneously at coordinates $x=600,1000,1200$ m. Using the standard imaging procedure outlined in the preceding paragraphs, we can reconstruct the source and receiver wavefields, $u_s$ and $u_r$, and apply the conventional imaging condition equation 4 to obtain the image in Figure 3(a). The image shows the horizontal reflector superposed with linear artifacts of comparable strength.

Figures 2(a) and 2(b) represent another simple model of spatially variable velocity with a horizontal reflector. Data in this model are simulated from a source located at coordinate $x=1000$ m. The negative Gaussian velocity anomaly present in the velocity model creates triplications of the source and receiver wavefields. Using the same standard imaging procedure outlined in the preceding paragraphs, we obtain the image in Figure 5(a). The image also shows the horizontal reflector superposed with complex artifacts of comparable strength.

In both cases discussed above, the velocity model is perfectly known and the acoustic wave equation is solved with the same finite-difference operator implemented in the space-time domain. Therefore, the artifacts are caused only by properties of the conventional imaging condition used to produce the migrated image and not by inaccuracies of wavefield extrapolation or of the velocity model.

The cause of artifacts is cross-talk between events present in the source and receiver wavefields, which are not supposed to match. For example, cross-talk can occur between wavefields corresponding to multiple sources, as illustrated in the example shown in Figures 1(a)-1(b), multiple branches of a wavefield corresponding to one source, as illustrate in the example shown in Figures 2(a)-2(b), events that correspond to multiple reflections in the subsurface, or multiple wave modes of an elastic wavefield, for example between PP and PS reflections, etc.

ii,kk
Figure 3. Images obtained for the model in Figures 1(a)-1(c) using the conventional imaging condition (a) and the stereographic imaging condition (b).

Stereographic imaging condition for wave-equation migration