Choice of v

Without loss of generality, we consider experiments with a receiver array spreading on the right side of the shot. (A split-spreading experiment can be decomposed into two one-side spreading experiments). Let us first look at a simple model that consists of a flat reflector with an overburden of a constant velocity medium. We want to investigate how the image of the reflector will be distorted as the parameter v in ${\bf P}(v)$ deviates from the true velocity v_t of the medium.

Goldin (1982) proved that when v is equal to $\sqrt{2}v_t$ the crossing points of the caustics of rays and the image of the reflector are located right below the shot location. In Figure 1, a sequence of distorted images are drawn with $v=0.9v_t \sim \sqrt{2}v_t$. We see that when $v=\sqrt{2}v_t$, the image of the reflector is completely migrated to the left side of the shot. Let us demonstrate this fact by graphics.

 

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Figure 1 The distorted images of the interface between two flat layers. The star indicates the shot position and the dashed line represents the receiver array. When v=0.9v_t, the distorted image bends upwards. As v increases, the distorted image gradually moves to the left. Finally when $v= \protect\sqrt{2}v_t$, the distorted image is completely migrated to the left side of the shot.

As I mentioned earlier, ${\bf P}(v)$ is a Kirchhoff migration operator. Its output is the superposition of a set of migration ellipses. Therefore the image I(t₀,x) is constructed by the envelopes of these ellipses. Because the principle axes of migration ellipses depend on the parameter v, so do their envelopes. In Figure 2, I show a set of migration ellipses and their envelopes. When $v=\sqrt{2}v_t$, the envelope of the migration ellipses appears entirely on the left side of the shot. In contrast, when v=0.9v_t, the envelope of the migration ellipses appears entirely on the right side of the shot. In fact, it can be mathematically proved that, after the transformation with operator ${\bf P}(v)$, the images of the events whose velocities are equal to $v/\sqrt{2}$ will be completely separable from the images of the events whose velocities are greater than v.

 

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Figure 2 The distorted image of a reflector is defined by the envelopes of the migration ellipses that are v-dependent. The star indicates the position of the shot and the dashed line represents the receiver array. (a) When $v= \protect\sqrt{2}v_t$, the envelope of the migration ellipses is entirely on the left side of the shot; (b) when v= 0.9v_t, the envelope of the migration ellipses is entirely on the right side of the shot.