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本文旨在以下两个方面建立有效算法:(1)基于时间偏移速度建立地震速度模型;(2)将时间偏移成像结果转换到深度域。本文借助于旁轴射线追踪理论,分别在2D和3D情况下建立了时间偏移速度和地震速度之间的关系。2D情况下的理论分析表明,常规Dix速度是地震层速度与成像射线的几何扩散值的比值。文章通过公式推导建立了由Dix速度得到地震速度的反问题,并且找到了解决的数值计算方法。方法包括两个步骤:①计算成像射线的几何扩散值,并由Dix速度计算时间域坐标系下的真地震速度;②计算时间域坐标系到深度域的转换矩阵,实现真地震速度时间域到深度域的转换。在步骤1中,我们推导出联系DIX速度和成像射线的几何扩散值的偏微分方程(PDE)。这是一个非线性椭圆偏微分方程。结合此方程的物理意义我们提出一个柯西问题。这个问题是不适定的,但是我们可以在要求的时段从两个方面得到其数值解,不过这个时段要足够小。一种是受Lax-Friedrichs法的启发得到的有限差分法;另一种是频谱—契比雪夫法。而在步骤2中,受Sethian的快速前进法启发建立了一种有效的似Dijkstra解决方案。最后结合一个合成数据模型和野外实例对上述数值计算方法进行了测试,结果表明该地震速度分析法较常规Dix反演,可明显提高计算精确性。该算法可以用于深度偏移中速度模型的建立。
This paper aims to establish an effective algorithm in the following two aspects: (1) to establish the seismic velocity model based on the time migration velocity; (2) to convert the time migration imaging result to the depth region. In this paper, by means of paraxial ray tracing theory, the relationship between time migration velocity and seismic velocity is established respectively in 2D and 3D cases. Theoretical analysis in the 2D case shows that the conventional Dix velocity is the ratio of the velocity of the seismic layer to the geometric spread of the imaging ray. The article deduces the inverse problem of obtaining the seismic velocity by the Dix velocity through the derivation of the formula, and finds a solution to the numerical calculation method. The method consists of two steps: ① Calculate the geometrical diffusion value of the imaging ray and calculate the true seismic velocity under the time-domain coordinate system by the Dix velocity; ② Calculate the transformation matrix from the time domain coordinate system to the depth domain and achieve the true seismic velocity time domain Deep domain conversion. In step 1, we derive a partial differential equation (PDE) that relates the geometric diffusion values of DIX velocity and imaging rays. This is a nonlinear elliptic partial differential equation. Combining the physical meaning of this equation we propose a Cauchy problem. This problem is ill-posed, but we can obtain its numerical solution in two ways at the required time, but this time should be small enough. One is the finite difference method inspired by the Lax-Friedrichs method; the other is the spectrum-Chebyshev method. In Step 2, inspired by Sethian’s fast-forward method, an effective Dijkstra-like solution was built. Finally, a synthetic data model and a field example are used to test the above numerical calculation method. The results show that the seismic velocity analysis method is more accurate than Dix inversion, which can obviously improve the calculation accuracy. The algorithm can be used to establish the velocity model in depth migration.