Wave field extrapolation techniques in seismic migration; a tutorial
Wave field extrapolation techniques in seismic migration; a tutorial
复制标题
地震偏移中的波场外推技术；
DOI：
10.1190/1.1441172
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发表时间：
1981
期刊：
影响因子：
3.3
通讯作者：
A. Berkhout
中科院分区：
文献类型：
--
作者：
A. Berkhout
本文的目的是对地震模拟和地震偏移中所用的波场外推方法，即，Kirchhoff求和法、平面波法（k-f法）和有限差分法，并着重讨论了不同方法之间的关系。通过在空间-频率域（x，y，Ω域）中公式化问题，可以采用系统方法，其导致简单和简洁的表达式。这些表达式清楚地表明，前向外推由空间卷积过程描述，逆外推由空间反卷积过程描述。在横向速度变化的情况下，卷积（去）过程成为空间变化的。空间-频率域最适合于递归深度偏移。此外，该方法还可以很容易地处理吸收、色散和空间带宽等频率相关的性质，所有的外推方法都是基于两个方程：泰勒级数和波动方程。在基尔霍夫求和方法中，泰勒级数的所有项都被求和为一个精确的解析表达式--平面的基尔霍夫积分。它制定了外推过程中的空间卷积积分，必须在实际应用中离散。基尔霍夫积分的傅里叶变换形式用于平面波方法（k-f方法）。这实际上意味着x，y，Ω域中的空间卷积（去卷积）被转换成k，x，k，y，Ω域中的乘法.当然，如果外推算子是空间变化的，这是不允许的。在显式有限差分技术中，使用泰勒级数的截断版本，并对系数进行一些最佳调整。对于泰勒级数中的仅一个或两个项，必须应用空间低通滤波器来补偿高倾斜角处的幅度误差。显式方法是一种简单的方法，最适合于三维（3-D）的应用。在隐式有限差分格式中，波场外推器被写为一个显式的正向外推器和一个显式的逆外推器。适当设计的隐式格式不显示振幅误差，因此，不需要应用振幅校正滤波器。与显式格式相比，隐式格式对数据文件两端的不适当边界条件更敏感，它表明，正演地震模型可以用一个矩阵方程来描述，对向下和向上的行波使用单独的算子。使用该模型，逆外推包括一个矩阵求逆过程来补偿向下传播效应和一个矩阵求逆过程来补偿向上传播效应。
The objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration, i.e., the Kirchhoff-summation approach, the plane-wave method (k-f method), and the finite-difference technique.Particular emphasis is given to the relationship between the different methods. By formulating the problem in the space-frequency domain (x, y, omega -domain), a systems approach can be adopted which results in simple and concise expressions. These expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure. In the situation of lateral velocity variations, the (de)convolution procedure becomes space-variant. The space-frequency domain is most suitable for recursive depth migration. In addition, frequency dependent properties such as absorption, dispersion, and spatial bandwidth can be handled easily.It is shown that all extrapolation methods are based on two equations: Taylor series and wave equation. In the Kirchhoff-summation approach all terms of the Taylor series are summed to an exact analytical expression--the Kirchhoff-integral for plane surfaces. It formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications. The Fourier-transformed version of the Kirchhoff-integral is used in the plane wave method (k-f method). This actually means that spatial (de)convolution in the x, y, omega -domain is translated into multiplication in the k x , k y , omega -domain. Of course, this is not allowed if the extrapolation operators are space-variant.In explicit finite-difference techniques a truncated version of the Taylor series is used with some optimum adjustments of the coefficients. For only one or two terms in the Taylor series, a spatial low-pass filter must be applied to compensate for the amplitude errors at high tilt angles. Explicit methods are simple and most suitable for three-dimensional (3-D) applications.In implicit finite-difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator. Properly designed implicit schemes do not show amplitude errors and, therefore, amplitude correction filters need not be applied. In comparison with explicit schemes, implicit schemes are more sensitive to improper boundary conditions at both ends of the data file.It is shown that the forward seismic model can be elegantly described by a matrix equation, using separate operators for downward and upward traveling waves. Using this model, inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to compensate for the upward propagation effects.