The stationary phase analysis of the Kirchhoff-type d emigrated field

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.

Implementation technology of Kirchhoff demigration

Demigration refers to directly applying a specific imaging technique to a migrated section. It is applied primarily to seismic data mapping. In a previous research study, a time-efficient implementation technology of demigration was expounded. In the present study, the Fast Marching Method （FMM） used for traveltime computation in the isochrone-staek demigration, is developed. Furthermore, other key techniques （ such as selection of aperture and antialiasing filtering factor） are analyzed in detail. Besides, the detail implementation method and program flow are given, which is shown their good computational efficiency and high-quality demi- gration effect. This implementation technique is illustrated with both the V（z） model and Marmousi model. It provides a basic method for implementing demigration in the application of seismic data mapping.

Internal Multiple Prediction Based on Imaging Profile Prediction and Kirchhoff Demigration

This paper introduces an internal multiple prediction method based on imaging profile prediction and Kirchhoff demigration.First,based on an inputted prestack time migration profile,the method predicts the prestack time migration profile that only includes internal multiples by inverse scattering series method.Second,the method uses velocity-weighted Kirchhoff demigration to create shot gathers that contains only internal multiples.Internal multiple prediction based on the prestack time migration profile effectively reduces the computational cost of multiple predictions,and the internal-multiple shot gathers created by Kirchhoff demigration remarkably reduces the complexity of the practical problem.Internal multiple elimination can be conducted through the combined adaptive multiple subtraction based on event tracing.Synthetic and field data tests show that the method effectively predicts internal multiples and possesses considerable potential in field data processing,particularly in areas where internal multiples develop seriously.

Data-Domain Wave Equation Reflection Traveltime Tomography

Estimation of an accurate macro velocity model plays an important role in seismic imag- ing and model parameter inversion. Full waveform inversion （FWI） is the classical data-domain inver- sion method. However, the misfit function of FWI is highly nonlinear, and the local optimization cannot prevent convergence of the misfit function toward local minima. To converge to the global minimum, FWI needs a good initial model or reliable low frequency component and long offset data. In this article, we present a wave-equation-based reflection traveltime tomography （WERTT） method, which can pro- vide a good background model （initial model） for FWI and （least-square） pre-stack depth migration （LS-PSDM）. First, the velocity model is decomposed into a low-wavenumber component （background velocity） and a high-wavenumber component （reflectivity）. Second, the primary reflection wave is pre- dicted by wave-equation demigration, and the reflection traveltime is calculated by an automatic picking method. Finally, the misfit function of the 12-norm of the reflection traveltime residuals is mini- mized by a gradient-based method. Numerical tests show that the proposed method can invert a good background model, which can be used as an initial model for LS-PSDM or FWI.