起伏地表条件下的地震波走时与射线路径计算
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摘要
在当今世界各地,尤其是在中国的西部地区,大量与石油、天然气及矿产等资源相关的地震勘探工作在起伏地表地区进行。与平原地区的地震勘探相比,在山地开展地震勘探要面临一些特殊问题,例如：数据采集困难且采集到的数据质量差、散射干扰严重、静校正不准、成像效果不好甚至不成像等。造成这些问题的主要原因有：①起伏地表地区复杂的地质条件造成地震波场的结构也很复杂,而长期以来人们对这一特定区域地震波的传播规律认识不够深入,很多理论和实际问题还没有得到很好的解决；②由于地质条件的复杂和缺乏大量的正演模拟数据的支持,起伏地表地区地震数据采集的有效性和针对性相对较差；③传统的地震勘探理论和处理技术都是基于地表为水平的假设,对于起伏地表问题常采用静校正来处理,然而在地表剧烈起伏地区即使采用了细致的静校正技术也很难取得好的成像效果。
解决上述问题的一个基本途径是增加对地震波场的认识,其主要方法有两种,即波动方程数值模拟和射线追踪。本文选择后者来研究该问题,主要研究对象是起伏地表条件下的地震波走时与射线路径的计算问题,主要研究内容包括：
①为了给起伏地表条件下的地震波走时与射线路径计算寻找合适的算法,首先对常规走时与射线路径的计算方法进行了分析及对比：分别阐述了两点射线追踪法、最短路径射线追踪法、有限差分法、线性插值法、波前构建法以及其它方法的基本原理、发展历程等问题,并基于相关文献对这些算法进行了综合对比,最后基于对比的结果选择快速推进法和线性插值法作为本文研究问题的算法基础。
②针对二维起伏地表条件下地震波走时的计算问题,对常规快速推进法进行改进,提出了阶梯网格迎风差分法和不等距差分法：阶梯网格迎风差分法运用阶梯网格建立起伏地表模型,采用改进后的迎风差分格式进行局部走时计算。不等距差分法运用不等距网格建立起伏地表模型,通过在迎风差分格式中引入不等距差分格式并综合应用Huygens原理和Fermat原理进行局部走时计算。两种方法在整体实现时均采用起伏地表条件下的窄带技术作为波前扩展方式。计算精度分析表明两种算法均能达到很好的计算精度。
③为了计算二维起伏地表条件下的地震波走时,对常规线性插值法进行改进,提出了一种混合网格线性插值法：对常规线性插值法的局部实现策略和波前扩展方式均进行一定程度的改进,然后采用混合网格建立起伏地表模型,采用三角网格和正方形网格中的插值公式进行局部走时计算,并利用起伏地表条件下的窄带技术作为波前扩展方式。计算精度分析表明混合网格线性插值法能够达到很好的计算精度。
④对二维起伏地表条件下的曲网格法进行了初步的研究：推导了坐标变换法和正交贴体网格法在计算空间中的程函方程,给出了求解这两个方程的数值实现策略,并对坐标变换法进行了初步的计算精度分析,最后讨论了曲网格法存在的一些技术难点。
⑤对②、③、④中提出的几种二维起伏地表条件下的地震波走时算法进行了对比分析,并给出了相应的计算实例：首先对常规快速推进法和线性插值法进行精度和效率的对比,然后对笛卡尔坐标系下的三种算法进行对比,最后对曲网格法和笛卡尔坐标系下的算法进行了对比。
⑥提出二维起伏地表条件下的地震波射线路径的计算方法,并对算法进行了相应的计算精度、稳定性及模型的适应性分析：起伏地表条件下的地震波射线路径计算是从已计算出的起伏地表条件下的地震波走时分布信息出发利用线性插值法来完成的。通过算法分析和具体的计算实例表明算法能达到很好的计算精度,同时算法也能稳定地适应任意起伏地形和复杂介质模型。
⑦提出三维起伏地表条件下的地震波走时计算方法,并给出了算法的稳定性及计算精度分析和计算实例：采用不等距网格建立三维起伏地表模型,推导了三维不等距网格中的不等距差分公式。算法的稳定性、计算精度及计算实例分析表明基于不等距差分法的三维起伏地表条件下的地震波走时计算方法简单易行且能达到很好的计算精度,同时也能稳定地适应任意的三维起伏地形和复杂介质模型。
⑧利用上述③、⑥中提出的起伏地表条件下的地震波走时与射线路径的计算方法,有针对性地计算了起伏地表条件下各种波型的走时与射线路径,并研究了起伏地表条件下的地震波的一些传播规律：利用③中提出的混合网格线性插值法和⑥中提出的射线路径计算方法分别计算了起伏地表条件下初至波、首波、透射波、绕射波、反射波以及多次反射波的走时与射线路径,通过对这些波的走时与射线路径分布情况的分析得出了一些起伏地表条件下地震波的传播规律。
基于上述研究内容,本论文主要取得了如下研究成果：①提出了一套以阶梯网格迎风差分法、不等距差分法以及混合网格线性插值法为主体的二维起伏地表条件下的地震波走时计算方法,这些方法均能适应二维任意起伏地形和任意复杂介质且兼顾计算精度及效率；②引入曲网格法解决起伏地表问题,建立了曲网格中用于走时计算的基本方程,并提出了相应的数值实现策略,同时还分析了算法在实现时还有待解决的一些技术难点；③提出了一种基于线性插值的起伏地表条件下的射线路径计算方法,该方法能在保证一定计算精度的条件下适应二维任意起伏地形和任意复杂介质；④提出了一种基于不等距差分法的三维起伏地表条件下的地震波走时计算方法,该方法能在保证一定计算精度的条件下适应三维任意起伏地形和任意复杂介质；⑤提出了起伏地表条件下的各种波型的走时与射线路径的计算方案,并得出了一些起伏地表条件下地震波的传播规律。
综上所述,本论文所做的研究工作和所得出的一些结论及研究成果对于起伏地表地区地震勘探效果的提高有着一定的理论意义和实际价值,同时也有着很广阔的应用前景。
A large number of seismic explorations for petroleum, natural gas, mineral, etc resources are carried out in mountainous regions in current world, especially in western China. Comparing with the seismic explorations in plain, there are some special problems in mountainous regions should be solved, such as:①We have not done some deep enough research on the propagation law of seismic wave in mountainous regions, and many theoretical and practical problems have not been solved for a long time;③There are many difficulties for collecting seismic data, and the quality of the seismic data which is collected in mountainous regions is relatively poor;③The traditional theory of seismic wave and data processing techniques are both based on the assumption that the earth's surface is plane. The static correction is the main data processing technique for treating the undulating earth's surface problems, but it has been shown that most of the effects caused by surface topography cannot be removed by static correction. Specifically, the imaging quality is sometimes still poor, although a careful static correction has been performed.
A basic way for solving the above problems is to increase the study on the seismic wave field's feature in mountainous regions by the following two main ways: numerical simulation of wave equation and ray tracing. In this paper, we study the methods for calculating the seismic traveltime and raypath under undulating earth's surface condition. The main contents of our thesis as following:
①To select the methods which are suitable for solving the irregular surface problem, we review the conventional methods including two-point ray tracing method, shortest path ray tracing method, finite difference method, linear interpolation method, wavefront construction method and the other methods that are used for calculating seismic traveltime and raypath. We also make a comprehensive comparison among these methods. At last, we choose the fast marching method and linear interpolation method as the basis algorithm for our following research works.
②For solving the eikonal equation in 2D undulating earth's surface condition, we present two schemes:one is an upwind finite difference scheme with ladder-like grids, and another is an finite difference scheme with non-uniform grid spacing. Specifically, the first scheme use the upwind finite difference scheme with ladder-like grids as the local traveltime formulas, and the second scheme deduce the local traveltime formulas by introducing the finite difference scheme with non-uniform grid spacing into upwind finite difference scheme and by integrated applying the Huygens principle and the Fermat principle. Specifically, the two schemes both use the narrow band technique under undulating earth's surface condition as the wavefront expansion scheme. We analyze the stability and accuracy of the two new methods, and make a conclusion that the two methods both have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.
③To calculate the seismic traveltime in 2D undulating earth's surface condition, we present a new linear interpolation scheme by improving the conventional linear interpolation method with the hybrid grids that are combined by the triangular and the rectangular grids. This new scheme also uses the narrow band technique under undulating earth's surface condition as the wavefront expansion scheme. We also analyze the stability and accuracy of the new scheme, and make a conclusion that the the new scheme has an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.
④We do a tentative research on the curvilinear grid methods for calculating the seismic traveltime under 2D undulating earth's surface condition. We first deduce some theoretical formulas by coordinate transformation method and orthogonal curvilinear grid method and secondly propose the corresponding numerical implementation schemes for these two curvilinear grid methods. Finally, we discuss the technical difficulties of the curvilinear grid methods in their numerical implementation process.
⑤We make a comprehensive comparison among the methods (These methods include an upwind finite difference scheme with ladder-like grids, an finite difference scheme with non-uniform grid spacing, a linear interpolation scheme with hybrid grids, and two curvilinear grid methods) are proposed in our thesis for calculating the seismic traveltime under 2D undulating earth's surface condition, and give some numerical examples:Firstly, we make the accuracy and efficiency comparison between the conventional fast marching method and linear interpolation method. Secondly, we make a comparison among the three methods which are all in the Cartesian coordinates. At last, we make a comparison among the methods which are all in the Cartesian coordinates and the curvilinear grid methods.
⑥To calculate seismic raypath in 2D undulating earth's surface condition, we study the methods in various grids base on the linear interpolation scheme. The raypath is calculated by linear interpolation scheme basing on the distribution information of the traveltime. Algorithm analysis and some numerical examples show that the method is proposed here have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.
⑦For calculating the seismic traveltime in 3D undulating earth's surface condition, we present a 3D finite difference scheme with non-uniform grid spacing. This scheme uses non-uniform grids for building up the 3D topography model, and deducing the local traveltime formulas by introducing the finite difference scheme with non-uniform grid spacing into upwind finite difference scheme. We also analyze the stability and accuracy of the new scheme, and make a conclusion that the new scheme has an unconditional stability and good accuracy, and good adaptability to any 3D rugged terrain and any 3D complex media.
⑧To obtain some propagation laws of seismic wave under the undulating earth's surface condition, we study the methods for calculating traveltime and raypath of some types of seismic wave in complex media under undulating earth's surface condition. Here, the seismic wave types include first-arrival, refraction, transmission, diffraction, reflection, multiple reflection, etc.
Based on the above research, we obtain the following results:①We propose three methods including an upwind finite difference scheme with ladder-like grid, an finite difference scheme with non-uniform grid spacing, and a linear interpolation scheme with hybrid grid for calculating the seismic traveltime under 2D undulating earth's surface condition in the Cartesian coordinates. These methods all have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media;②We propose two curvilinear grid methods including coordinate transformation and orthogonal curvilinear grids for calculating the seismic traveltime under 2D undulating earth's surface condition and discuss the technical difficulties of the curvilinear grid method in their numerical implementation process;③We present a method for calculating the raypath in various grid types in undulating earth's surface condition basing on the linear interpolation method which has an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media;④We propose a 3D finite difference scheme with non-uniform grid spacing for calculating the seismic traveltime under 3D undulating earth's surface condition, and this method has an unconditional stability, good accuracy, and good adaptability to any 3D rugged terrain and any 3D complex media;⑤ccording to the calculation result from the methods that are used for calculating the traveltime and raypath of some types of seismic wave in complex media under undulating earth's surface condition, we obtain some laws of seismic wave propagation laws under undulating earth's surface condition. The studying works and some conclusions in this thesis are of benefit to deeply research some propagation laws of seismic wave under undulating earth's surface condition, and to prove some reference information for the design of seismic data acquisition system under undulating earth's surface condition. The methods are proposed in our thesis can provide a set of tools for improving the effect of seismic data processing and for presenting some new seismic data processing techniques under undulating earth's surface condition.
解决上述问题的一个基本途径是增加对地震波场的认识,其主要方法有两种,即波动方程数值模拟和射线追踪。本文选择后者来研究该问题,主要研究对象是起伏地表条件下的地震波走时与射线路径的计算问题,主要研究内容包括：
①为了给起伏地表条件下的地震波走时与射线路径计算寻找合适的算法,首先对常规走时与射线路径的计算方法进行了分析及对比：分别阐述了两点射线追踪法、最短路径射线追踪法、有限差分法、线性插值法、波前构建法以及其它方法的基本原理、发展历程等问题,并基于相关文献对这些算法进行了综合对比,最后基于对比的结果选择快速推进法和线性插值法作为本文研究问题的算法基础。
②针对二维起伏地表条件下地震波走时的计算问题,对常规快速推进法进行改进,提出了阶梯网格迎风差分法和不等距差分法：阶梯网格迎风差分法运用阶梯网格建立起伏地表模型,采用改进后的迎风差分格式进行局部走时计算。不等距差分法运用不等距网格建立起伏地表模型,通过在迎风差分格式中引入不等距差分格式并综合应用Huygens原理和Fermat原理进行局部走时计算。两种方法在整体实现时均采用起伏地表条件下的窄带技术作为波前扩展方式。计算精度分析表明两种算法均能达到很好的计算精度。
③为了计算二维起伏地表条件下的地震波走时,对常规线性插值法进行改进,提出了一种混合网格线性插值法：对常规线性插值法的局部实现策略和波前扩展方式均进行一定程度的改进,然后采用混合网格建立起伏地表模型,采用三角网格和正方形网格中的插值公式进行局部走时计算,并利用起伏地表条件下的窄带技术作为波前扩展方式。计算精度分析表明混合网格线性插值法能够达到很好的计算精度。
④对二维起伏地表条件下的曲网格法进行了初步的研究：推导了坐标变换法和正交贴体网格法在计算空间中的程函方程,给出了求解这两个方程的数值实现策略,并对坐标变换法进行了初步的计算精度分析,最后讨论了曲网格法存在的一些技术难点。
⑤对②、③、④中提出的几种二维起伏地表条件下的地震波走时算法进行了对比分析,并给出了相应的计算实例：首先对常规快速推进法和线性插值法进行精度和效率的对比,然后对笛卡尔坐标系下的三种算法进行对比,最后对曲网格法和笛卡尔坐标系下的算法进行了对比。
⑥提出二维起伏地表条件下的地震波射线路径的计算方法,并对算法进行了相应的计算精度、稳定性及模型的适应性分析：起伏地表条件下的地震波射线路径计算是从已计算出的起伏地表条件下的地震波走时分布信息出发利用线性插值法来完成的。通过算法分析和具体的计算实例表明算法能达到很好的计算精度,同时算法也能稳定地适应任意起伏地形和复杂介质模型。
⑦提出三维起伏地表条件下的地震波走时计算方法,并给出了算法的稳定性及计算精度分析和计算实例：采用不等距网格建立三维起伏地表模型,推导了三维不等距网格中的不等距差分公式。算法的稳定性、计算精度及计算实例分析表明基于不等距差分法的三维起伏地表条件下的地震波走时计算方法简单易行且能达到很好的计算精度,同时也能稳定地适应任意的三维起伏地形和复杂介质模型。
⑧利用上述③、⑥中提出的起伏地表条件下的地震波走时与射线路径的计算方法,有针对性地计算了起伏地表条件下各种波型的走时与射线路径,并研究了起伏地表条件下的地震波的一些传播规律：利用③中提出的混合网格线性插值法和⑥中提出的射线路径计算方法分别计算了起伏地表条件下初至波、首波、透射波、绕射波、反射波以及多次反射波的走时与射线路径,通过对这些波的走时与射线路径分布情况的分析得出了一些起伏地表条件下地震波的传播规律。
基于上述研究内容,本论文主要取得了如下研究成果：①提出了一套以阶梯网格迎风差分法、不等距差分法以及混合网格线性插值法为主体的二维起伏地表条件下的地震波走时计算方法,这些方法均能适应二维任意起伏地形和任意复杂介质且兼顾计算精度及效率；②引入曲网格法解决起伏地表问题,建立了曲网格中用于走时计算的基本方程,并提出了相应的数值实现策略,同时还分析了算法在实现时还有待解决的一些技术难点；③提出了一种基于线性插值的起伏地表条件下的射线路径计算方法,该方法能在保证一定计算精度的条件下适应二维任意起伏地形和任意复杂介质；④提出了一种基于不等距差分法的三维起伏地表条件下的地震波走时计算方法,该方法能在保证一定计算精度的条件下适应三维任意起伏地形和任意复杂介质；⑤提出了起伏地表条件下的各种波型的走时与射线路径的计算方案,并得出了一些起伏地表条件下地震波的传播规律。
综上所述,本论文所做的研究工作和所得出的一些结论及研究成果对于起伏地表地区地震勘探效果的提高有着一定的理论意义和实际价值,同时也有着很广阔的应用前景。
A large number of seismic explorations for petroleum, natural gas, mineral, etc resources are carried out in mountainous regions in current world, especially in western China. Comparing with the seismic explorations in plain, there are some special problems in mountainous regions should be solved, such as:①We have not done some deep enough research on the propagation law of seismic wave in mountainous regions, and many theoretical and practical problems have not been solved for a long time;③There are many difficulties for collecting seismic data, and the quality of the seismic data which is collected in mountainous regions is relatively poor;③The traditional theory of seismic wave and data processing techniques are both based on the assumption that the earth's surface is plane. The static correction is the main data processing technique for treating the undulating earth's surface problems, but it has been shown that most of the effects caused by surface topography cannot be removed by static correction. Specifically, the imaging quality is sometimes still poor, although a careful static correction has been performed.
A basic way for solving the above problems is to increase the study on the seismic wave field's feature in mountainous regions by the following two main ways: numerical simulation of wave equation and ray tracing. In this paper, we study the methods for calculating the seismic traveltime and raypath under undulating earth's surface condition. The main contents of our thesis as following:
①To select the methods which are suitable for solving the irregular surface problem, we review the conventional methods including two-point ray tracing method, shortest path ray tracing method, finite difference method, linear interpolation method, wavefront construction method and the other methods that are used for calculating seismic traveltime and raypath. We also make a comprehensive comparison among these methods. At last, we choose the fast marching method and linear interpolation method as the basis algorithm for our following research works.
②For solving the eikonal equation in 2D undulating earth's surface condition, we present two schemes:one is an upwind finite difference scheme with ladder-like grids, and another is an finite difference scheme with non-uniform grid spacing. Specifically, the first scheme use the upwind finite difference scheme with ladder-like grids as the local traveltime formulas, and the second scheme deduce the local traveltime formulas by introducing the finite difference scheme with non-uniform grid spacing into upwind finite difference scheme and by integrated applying the Huygens principle and the Fermat principle. Specifically, the two schemes both use the narrow band technique under undulating earth's surface condition as the wavefront expansion scheme. We analyze the stability and accuracy of the two new methods, and make a conclusion that the two methods both have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.
③To calculate the seismic traveltime in 2D undulating earth's surface condition, we present a new linear interpolation scheme by improving the conventional linear interpolation method with the hybrid grids that are combined by the triangular and the rectangular grids. This new scheme also uses the narrow band technique under undulating earth's surface condition as the wavefront expansion scheme. We also analyze the stability and accuracy of the new scheme, and make a conclusion that the the new scheme has an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.
④We do a tentative research on the curvilinear grid methods for calculating the seismic traveltime under 2D undulating earth's surface condition. We first deduce some theoretical formulas by coordinate transformation method and orthogonal curvilinear grid method and secondly propose the corresponding numerical implementation schemes for these two curvilinear grid methods. Finally, we discuss the technical difficulties of the curvilinear grid methods in their numerical implementation process.
⑤We make a comprehensive comparison among the methods (These methods include an upwind finite difference scheme with ladder-like grids, an finite difference scheme with non-uniform grid spacing, a linear interpolation scheme with hybrid grids, and two curvilinear grid methods) are proposed in our thesis for calculating the seismic traveltime under 2D undulating earth's surface condition, and give some numerical examples:Firstly, we make the accuracy and efficiency comparison between the conventional fast marching method and linear interpolation method. Secondly, we make a comparison among the three methods which are all in the Cartesian coordinates. At last, we make a comparison among the methods which are all in the Cartesian coordinates and the curvilinear grid methods.
⑥To calculate seismic raypath in 2D undulating earth's surface condition, we study the methods in various grids base on the linear interpolation scheme. The raypath is calculated by linear interpolation scheme basing on the distribution information of the traveltime. Algorithm analysis and some numerical examples show that the method is proposed here have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.
⑦For calculating the seismic traveltime in 3D undulating earth's surface condition, we present a 3D finite difference scheme with non-uniform grid spacing. This scheme uses non-uniform grids for building up the 3D topography model, and deducing the local traveltime formulas by introducing the finite difference scheme with non-uniform grid spacing into upwind finite difference scheme. We also analyze the stability and accuracy of the new scheme, and make a conclusion that the new scheme has an unconditional stability and good accuracy, and good adaptability to any 3D rugged terrain and any 3D complex media.
⑧To obtain some propagation laws of seismic wave under the undulating earth's surface condition, we study the methods for calculating traveltime and raypath of some types of seismic wave in complex media under undulating earth's surface condition. Here, the seismic wave types include first-arrival, refraction, transmission, diffraction, reflection, multiple reflection, etc.
Based on the above research, we obtain the following results:①We propose three methods including an upwind finite difference scheme with ladder-like grid, an finite difference scheme with non-uniform grid spacing, and a linear interpolation scheme with hybrid grid for calculating the seismic traveltime under 2D undulating earth's surface condition in the Cartesian coordinates. These methods all have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media;②We propose two curvilinear grid methods including coordinate transformation and orthogonal curvilinear grids for calculating the seismic traveltime under 2D undulating earth's surface condition and discuss the technical difficulties of the curvilinear grid method in their numerical implementation process;③We present a method for calculating the raypath in various grid types in undulating earth's surface condition basing on the linear interpolation method which has an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media;④We propose a 3D finite difference scheme with non-uniform grid spacing for calculating the seismic traveltime under 3D undulating earth's surface condition, and this method has an unconditional stability, good accuracy, and good adaptability to any 3D rugged terrain and any 3D complex media;⑤ccording to the calculation result from the methods that are used for calculating the traveltime and raypath of some types of seismic wave in complex media under undulating earth's surface condition, we obtain some laws of seismic wave propagation laws under undulating earth's surface condition. The studying works and some conclusions in this thesis are of benefit to deeply research some propagation laws of seismic wave under undulating earth's surface condition, and to prove some reference information for the design of seismic data acquisition system under undulating earth's surface condition. The methods are proposed in our thesis can provide a set of tools for improving the effect of seismic data processing and for presenting some new seismic data processing techniques under undulating earth's surface condition.
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