摘要：针对传统射线方法在奇异区成像精度不高,而2D频率域高斯波束叠前深度偏移需要计算成像点处每个频率的格林函数,影响计算效率的问题,本文通过使用复走时代替实走时,改变频率域下成像公式的积分顺序,给出了在时间域下进行高斯波束偏移的方法和计算公式.本文使用复杂数值模型验证了2D时间域高斯波束叠前偏移方法的正确性,并同传统射线偏移成像结果做了对比.对比结果表明时间域高斯波束偏移在成像精度上优于传统射线偏移.

Abstract: The traditional ray method don't perform well in the caustic and shadow region because its image accuracy is low. While the 2D Gaussian beam prestack depth migration in the frequency domain is less efficient because it needs calculation of Green function of all frequencies at imaging points. To solve these problems, we propose a novel method of 2D Gaussian beam migration of common-shot records in the time domain.In Gaussian beam migration, the source and recorded wavefield are both based on the Green functions staked by Gaussian beam from the source point and the receive point. Thus we can use the source wavefield and the recorded wavefield to generate the migrated image. Amplitude and exponential terms are both complex numbers, as Green function is complex and the traveltime expressed by the exponential term is called complex traveltime. This complex traveltime at the image point is determined by the second-order Taylor expansion of the nearest point's traveltime in the center ray. With the image condition, the migration formula is a triple integral of the horizontal slowness, vertical slowness and frequency, and the integrand is also a complex number. Focused on the real part of the integrand, the triple integral is reduced to a double integral by transforming the integral variable time to frequency using the Fourier transform. In the triple integral with the real part of the integrand, we change the sequence, and transform the frequency to the time by the Fourier transform to get the result that is two-fold integral of the horizontal slowness and the vertical slowness at the image time. The image time in the formula is the summation of the complex traveltime from the source point to the image point and that from the receiver point to the image point. Because the amplitude is a complex number, so the result with the imaginary of the amplitude is needed to transform the product of the Green function and the seismic data with the Hilbert transform. Then the triple integral becomes a double integral of the horizon slowness and the vertical slowness, and the image formula becomes the summation of two double integrals.The validity of the new method is proved on a complex numerical model. Using the Marmousi data set, the imaging result and computational efficiency of Gaussian beam migration in the time domain is compared with the method in the frequency domain. We can find that the image results of these two methods are similar, while the migration algorithm in the time domain takes less computing time. Then we compare the imaging result of Gaussian beam migration with that of Kirchhoff migration on the Sigsbee model. The migration result of Gaussian beam migration is clearer, especially for the region under the high-velocity geologic body. Gaussian beam migration section can show clear interfaces and even faults under this body which appears as migration illusion in the Kirchhoff migration. The result compared with the migrated imaging result based on the ray method indicates that Gaussian beam migration in the time domain has a higher accuracy.In this article, we present a novel method of 2D common-shot Gaussian beam migration in the time domain by employing the Fourier transform and the Hilbert transform. The ability of this method to deal with complicated models has been validated using the Marmousi data set and Sigsbee data set. Furthermore, because of holding the similar integral kernels, this method can be applied to other types of Gaussian beam migration straightforwardly, such as common-offset, common-receiver, common-midpoint and even 3D conditions.