The stationary phase analysis of the Kirchhoff-type demigrated field

Abstract

In the literature, stationary phase analysis of Kirchhoff-type demigrated fields is carried out mainly under the following two conditions: (1) The considered isochrone and the target reflector are tangential to each other; (2) The spatial duration of the wavelet of the depthmigrated image is short. For the isochrones that are not tangential to the target reflector and for the depth-migrated images that have a large spatial duration, the published stationary phase equation for the demigrated field will become invalid. For performing the stationary phase analysis of the Kirchhoff-type demigrated field under the conditions that the considered isochrone and the target reflector are not tangential to each other and that the spatial duration of the wavelet of the depth-migrated image is not short (the general conditions), I derive the formulas for the factors appearing in the stationary phase formula in two dimensions, from which I find that for different isochrones the horizontal coordinates of the stationary point of the depth difference function are different. Also, the equation for the Kirchhoff-type demigrated field consists of two parts. One is the true-amplitude demigrated signal and the other is the amplitude distortion factor. From these facts the following two conclusions can be drawn: (1) A demigrated signal is composed of many depth-migrated images and one depth-migrated image trace provides only one sample to the demigrated signal; and (2) The amplitude distortion effect is an effect inherent in the Kirchhoff-type demigration and cannot be eliminated during demigration. If this effect should be eliminated, one should do an amplitude correction after demigration.