基于delta波包叠加的时间域深度偏移

The time-domain depth migration by the summation of delta packets

delta波包可由高斯波束经傅里叶逆变换得到,是高斯波束在时空域的表达.它最早出现在合成理论地震图的研究中,本文将其应用于偏移领域.通过delta波包叠加表达时间域格林函数,可将高斯波束偏移由频率域转换到时间域,再结合Rayleigh积分和激励时间成像条件,本文给出了基于delta波包叠加的深度偏移算法.该偏移算法可在时间域直接计算,但因包含褶积运算,成像时将耗费大量的计算时间.针对这一问题,本文提出了将褶积简化为乘积的近似公式.近似后的偏移算法,不仅保留了高斯波束偏移的优点,而且计算效率得到显著提升.文中通过两个数值算例验证了上述结论.

Delta packets may serve as building blocks of a wavefield in space-time domain,just as Gaussian beams do in space-frequency domain.This paper will present a depth migration using the summation of delta packets and develop an approximate method that can speed up the convolution between the data and the delta packet.By definition,a delta packet is the inverse Fourier transformation of Gaussian beam under consideration.Thus,the Green′s function in the space-time domain can be asymptotically expressed as an integral superposition of delta packets.In the Rayleigh integral,the choice of Green′s function determines the migration method.Choosing a Green′s function of the delta packet summation form will lead to a new depth migration in space-time domain.The space-time formulation contains a convolution with respect to time between the filtered seismogram and the delta packet,which makes the new depth migration very time-consuming.Here the convolution integration is rewritten in terms of Taylor expansion,and is further reduced to product formula.In the approximate expression of convolution,the time-shifted seismogram only uses multiplication by the termAattenuation,so that the computational cost is much lower than before approximation.The term Aattenuationindicates attenuation degree of amplitudes,and depends on ratio of parameterΔt（the half width of the integration interval）and imaginary part of the complex travel time.When the value ofΔt takes nomore than several time sampling intervals,such as Δt=0.005 s,there will be a satisfactory approximation.Numerical tests using the Marmousi and the Sigsbee2 A models show that the migration quality for the approximated method is comparable to the one with the exact method.However,the computation time can reduce by about 75%.The depth migration using the summation of delta packets can be recognized as a flexible and effective migration technique.And it is also easily extended to 3-D depth migration and suitable for different kinds of acquisition geometries.