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我们意在通过一个全新而详尽的单程叠前深度偏移方法来阐述狄氏方程在地震波场外推的适用性。这种方法理论上可精确到偏离垂向90°,而且容许水平速度的变化。关于这点是在空间频率域运用狄拉克(Dirac)方程进行偏移过程的波场外推中得到证实。狄拉克方程是一个准确的线性平方根波动方程,而且相当于保持泰勒级数或平方根算子的连分数展开式的无穷多项。这个新方法一个重要的方面就是区域速度和空间导数可以在外推算子中分解成独立的项。所以我们不必根据速度预先计算和存储大量的依赖于速度的褶积外推系数。这种方法最主要的缺点就是在每一个深度步进处必须消除损耗能量以保持数值的稳定性。 我们进行了两次偏移算法的数值试验。第一个利用泰勒级数近似值来取深度步进以及利用高阶有限差分计算空间导数;第二个是利用快速展开式和带有假频的数值差分法求取深度步进,成像的条件是Claerbout的U/D0原理。 在上述两次试验中,脉冲响应精确到偏离垂向180°。我们用取自一个简单断层模型的合成数据,在有水平速度变化的情况下试验了深度偏移。结果表明,本文建议的偏移方法将倾斜反射界面和断层面成像于准确的位置。
We intend to illustrate the applicability of the Dijk equation in extrapolation of seismic waves using a new and exhaustive one-way prestack depth migration method. This method can theoretically be offset by 90 ° from vertical and allows for changes in horizontal velocity. This is confirmed in the wavefield extrapolation using the Dirac equation for the offset process in the spatial frequency domain. The Dirac equation is an accurate linear root-mean-square wave equation and is equivalent to maintaining an infinite number of even-fraction expansion of Taylor series or square root operators. An important aspect of this new method is that the regional velocity and spatial derivatives can be decomposed into independent terms in the extrapolation operator. So we do not have to precompute and store a large number of velocity dependent convolution extrapolations based on velocity. The main disadvantage of this method is that the loss energy must be eliminated at each deep step to keep the numerical stability. We performed two numerical experiments of the migration algorithm. The first uses the Taylor series approximation to take deep stepping and the high-order finite difference to calculate the spatial derivatives. The second is to use the fast expansion and numerical difference method with aliasing to find the depth stepping. The condition of the imaging is Claerbout the U / D0 principle. In the two experiments above, the impulse response was accurate to 180 ° from vertical. Using synthetic data from a simple fault model, we tested the depth migration with horizontal velocity variations. The results show that the offset method proposed in this paper images the oblique reflection surface and the fault plane at the exact location.