Schleicher et al. (1997) reached similar conclusion using Fresnel zone. He claimed that it is crucial not to have migration apertures below a certain sizes. This aperture is also called the data space Fresnel zone corresponding to one image point. The difference between the traveltime of one trace inside the Fresnel zone and that of the trace at the stationary point is less than a period of the dominant seismic wavelet. But it seems to me that, to avoid the intrude of the migration artifacts and the amplification of artifacts by the true amplitude weight function, it is crucial not to have migration apertures over a certain size, which is the Fresnel zone.
We have to discuss the conditions under which the stationary phase method can be applied to weighted integrals. One condition stated by Schleicher et al. (1997) is that the integral kernel, i.e., the weight function, should vary slowly across the space of integration. This condition sometimes is not satisfied, especially for the image points in the shallow layers. In this case, the result of the integration can not be represented by the evaluation of the integration kernel at the stationary point. This was clearly shown in Figure 6.5, where the artifacts from the deeper part are amplified and over beat the amplitudes at the stationary point.
From Bleistein's derivation it is clear that the true amplitude
image ( equation 6.11) is the direct result of equation 6.7, which is the
evaluation of integration equation 6.1 at the stationary point.