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arXiv 0810.2745physics.geo-ph
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反射地震地下波场偏移的数学研究

Migration of a subsurface wavefield in reflection seismics: A mathematical study

Arnab K. Ray

摘要:

在这一教学动机的工作,在反射地震迁移的过程已被认为是从严格的数学观点。具有恒定倾角的倾斜地下反射层已被证明会引起正常时差方程的移动,时差曲线的峰值跟踪椭圆轨迹。由于任何地下反射体实际上都具有不均匀的空间变化,因此通过采用波动方程式,使用更全面的偏移原理被认为是必要的。用这种方法导出了速度垂直变化的地下波场的振幅和相位的表达式。这种处理需要应用WKB近似,其自洽性已经由速度的对数变化在垂直方向上非常慢的事实建立,这是一个在越来越大的地下深度更强烈地支持的特征。最后,它已被证明,对于一个平面的地下波场,有一个等价的恒定速度Stolt迁移算法和固定相位近似方法(通过它的反射地下信号的起源被确定)。

英文摘要:

In this pedagogically motivated work, the process of migration in reflection seismics has been considered from a rigorously mathematical viewpoint. An inclined subsurface reflector with a constant dipping angle has been shown to cause a shift in the normal moveout equation, with the peak of the moveout curve tracing an elliptic locus. Since any subsurface reflector actually has a non-uniform spatial variation, the use of a more comprehensive principle of migration, by adopting the wave equation, has been argued to be necessary. By this approach an expression has been derived for both the amplitude and the phase of a subsurface wavefield with vertical velocity variation. This treatment has entailed the application of the WKB approximation, whose self-consistency has been established by the fact that the logarithmic variation of the velocity is very slow in the vertical direction, a feature that is much more strongly upheld at increasingly greater subsurface depths. Finally, it has been demonstrated that for a planar subsurface wavefield, there is an equivalence between the constant velocity Stolt Migration algorithm and the stationary phase approximation method (by which the origin of the reflected subsurface signals is determined).

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