REVIEW OF PROFILE IMAGING

One of the most frequently used prestack depth migration techniques is the profile imaging scheme (Claerbout, 1985). The profile imaging concept says that reflectors exist in the earth at places where the downgoing wave is time-coincident with an upcoming wave. In profile migration methods, both the downgoing wave and the upcoming wave are extrapolated to a depth and followed by imaging.

The most straightforward imaging technique is to extract the zero lag of the cross-correlation of the two waves. The image is then created by displaying the zero-lagged cross-correlation everywhere in (x,z)-space. The image obtained by the cross-correlation imaging condition doesn't tell us much about the reflector itself because the amplitude of the zero-lagged cross-correlation depends not only on the reflection coefficient but also on the amplitude of the downgoing and upcoming wave at the reflector.

Let $U(\omega,x,z)$ and $D(\omega,x,z)$ be the upcoming and downgoing waves in the frequency domain, respectively. The zero lag of the cross-correlation can be represented by

$$Image(x,z) = \sum_{\omega} U(\omega,x,z)D^\ast(\omega,x,z)$$ (1)

where $\ast$ represents the complex conjugate. Since the upcoming wave can be approximated by the convolution of the downgoing wave with the reflectivity function, R(x,z), Equation (1) can be rewritten as

$$Image(x,z) = R(x,z) \sum_{\omega} D(\omega,x,z)D^\ast(\omega,x,z)$$ (2)

The above equation tells us that the the image obtained by the zero lag cross-correlation represents the reflection coefficient scaled by the energy of the downgoing wave. The better reflection coefficient can be obtained from Equation (1) and (2) by

$$R(x,z) = { {\sum_{\omega} U(\omega,x,z)D^\ast(\omega,x,z) } ... ...\sum_{\omega} D(\omega,x,z)D^\ast(\omega,x,z) } + \epsilon^2} }$$ (3)

In the above equation, $\epsilon^2$ represents a small positive value that is introduces to stabilize because $D(\omega,x,z)$ may contain zeros.

The final image of the full prestack depth migration is usually obtained by summing all the shot profile images. Let's assume a spherical wavefront for each shot profile and a relatively gentle dip of the reflectors. Then the final image can be interpreted as the integral of the reflection coefficient over the incident angle because each shot has a different incident angle for the same reflection point. The real earth, however, does not involve such assumptions. The spherical wavefront is only valid in a constant velocity medium. Even in the constant medium, a reflector with complex shape does not allow regular samples in the incident angle for regularly spaced shots. Thus, the image of the conventional prestack migration only provide the reflector locations by mapping a value which is summation of many reflection coefficients shown from every shots.

This difficulty can be solved if we can control the incident angle at the reflector.