Pre-stack Depth Migration By The Summation Of Delta Packets

We present a pre-stack depth migration method that uses the delta packet summation.By definition,a delta packet is the inverse Fourier transform of the Gaussian beam under consideration.Thus,the delta packet is the space-time domain counterpart of the Gaussian beam.If the Kirchhoff integral uses the delta packet summation as approximating Green's function,we obtain the space-time domain depth migration method by the summation of delta packets.In the space-time domain,our algorithm has no trouble with frequency dispersion or alias in frequency sampling process,which Gaussian beam migration need to confront,and also can overcome problems with caustics,handle all arrivals,yield good images of steep flanks.Comparison with Gaussian beam migration,the new depth migration has to pay more time to obtain the similar high-quality images.There is a deep research to speed up the migration algorithm from two aspects.One is the approximation of convolution with respect to time between the filtered seismogram and the delta packet.Fortunately,the delta packet has a form exactly the same as the Cauchy distribution function.Thus,it is delta-funtion-like and its convolution with the seismogram can be approximated by a product.If this approximation is used in migration,the CPU time can be at least saved by 20%.The other is the fast algorithm of depth migration that uses the steepest descent method to asymptotically approximate the integrals in image conditions.To image with the approximated image conditions,it can only use the delta packets emerging from the specific directions,instead of those from all directions,which has less computation time.There are totally four fast algorithms derived in common-shot and common-offset coordinates,and we can choose the appropriate algorithm to meet different requirements of image quality and calculation speed.In addition to the cross-correlation image condition,we also suggest a deconvolution image condition by the summation of delta packets,which is more closely linked to stratigraphic reflection coefficients.It has extensive application and broad prospects in migration velocity analysis,amplitude interpretation,seismic inversion,et.al.