Conventional prestack imaging

In conventional prestack imaging as in shot profile imaging Claerbout (1985), the reflectivity is obtained by deconvolving the downgoing wave field from the upcoming wave field in the $(x,\omega)$ domain as follows:  

$$X(x,z_n,\omega) = {{ {\bf g}(x,z_n,\omega) {\bf s}^\ast(x,z_... ...{\bf s}(x,z_n,\omega){\bf s}^\ast(x,z_n,\omega) + \epsilon^2}},$$ (50)

where $\epsilon^2$ represents a small positive value introduced for stability because ${\bf s}(x,z_n,\omega)$may contain zeros. Then imaging is carried out by summation over all frequencies to extract the zero-time component of the reflectivity as follows:  

$${\bf r}(x,z_n) = \sum_{\omega} X(x,z_n,\omega).$$ (51)

Therefore we obtain one reflection coefficient value for each point of the subsurface.

Since we performed the deconvolution in the $(x,\omega)$ domain by dividing the source wave from received wave in equation (), we implicitly assumed the reflectivity matrix R is a diagonal matrix. The diagonality of the reflectivity matrix R implies locally reacting reflection coefficient. If we assume that source wave field acts like a planewave locally at reflector, the imaging condition used in the profile imaging will produce the reflection coefficient of the angle that corresponds to the local incidence angle of the source wave field at each point of the subsurface.