Coplanarity condition

The previous results show that as few as four cross-line offsets might be sufficient to ``capture'' all the useful propagation paths, when the range of cross-line offset dips for the downward continuation is properly defined. However, boundary artifacts caused by either reflecting or circular boundary conditions are unavoidable when such a narrow range of offsets are used to propagate the data. The effects of boundary artifacts are more obvious in the full source-receiver migration results (Figure ) than in the narrow-azimuth migration results (Figure ), but, as I discuss in Biondi (2001), they might also become a problem when narrow-azimuth downward-continuation is used with constant sampling in the cross-line offset wavenumber (dk_yh). A potential solution to the problem caused by boundary effects can be the use of absorbing boundary conditions. However, effective absorbing boundary conditions require the addition of several grid points, and consequently can cause a substantial increase in the computational cost.

Fortunately, the boundary artifacts can be effectively attenuated by applying a post-processing filter on the prestack image that preserves only the events for which the source and receiver rays are coplanar at the imaging point. This condition must be fulfilled by all the events that are correctly focused at zero offset because two lines passing through the same point are coplanar. The coplanarity condition can be easily applied on the prestack image after transformation into the Fourier domain, possibly at the same time that ADCIGs are computed using a 3-D generalization of the method described by Sava and Fomel (2003), as presented in Tisserant and Biondi (2003).

The coplanarity condition can be derived by simple geometric considerations starting from the common-azimuth condition expressed as Biondi and Palacharla (1996):  

$$\widehat{k_{y_h}}= k_{y_m}\frac{\sqrt{ \frac{\omega^2}{v^2({... ...{{\bf s},z})} - \frac{1}{4} \left(k_{x_m}-k_{x_h}\right)^2 }}.$$ (1)

where $\omega$ is the temporal frequency, k_xm and k_ym are the midpoint wavenumbers, k_xh is the in-line offset wavenumbers.

As for the common-azimuth condition, the coplanarity condition can be expressed as a relationship that links the cross-line offset wavenumber k_yh to the other wavenumbers in the image. For events with azimuth aligned along the in-line direction (x_m in my notation), the expression of the coplanarity condition is:  

$$k_{y_h}=\frac{k_{y_m}k_{x_m}k_{x_h}}{k_z^2 + k_{y_m}^2},$$ (2)

where k_z is the vertical wavenumber.

The condition expressed in equation (2) can be easily generalized to be valid for an arbitrary azimuthal direction $\beta$Tisserant and Biondi (2003). The wavenumber axes are rotated by $\beta$ in both the midpoint wavenumber plane $\left(k_{x_m},k_{y_m}\right)$and the offset wavenumber plane $\left(k_{x_h},k_{y_h}\right)$.

As discussed in Tisserant and Biondi (2003), in a prestack image each event fulfills the coplanarity condition for one value of the azimuth. However, streamer data illuminate the reflectors only within a fairly narrow azimuthal range. Therefore, the coplanarity condition can be used to remove from the prestack image all the events that are imaged outside a given azimuthal range. For example, if we image only events within a range of $\pm 22.5$ degrees, we will remove the 75 % of the events in the image that are least likely to be real reflection events.