论波前构建法中的几个计算问题
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摘要
鉴于波前构建法是一种产生于1993年的经典方法,本论文首先对该算法的研究现状、基本思想、数值解法、算法流程等进行了讨论,然后详细的讨论了波前构建法目前存在和探讨较少的以下4个问题:
(1)波前构建过程中非网格节点处速度以及速度导数的插值计算:通过对比分析数字地形、数字导航、计算机图形学中常用的二维插值算法,引入二维三次卷积插值实现非网格节点处速度及导数的计算,提高射线追踪的效率和精度;
(2)模型的光滑处理对地震波走时、射线路径及振幅信息带来的影响:应用不同的光滑算子、同一光滑算子不同光滑次数及不同光滑因子讨论了波前构建法中模型的光滑处理对地震波走时、射线路径及振幅信息的影响,并在浅层模型和深层模型了进行了走时和射线路径的对比分析;
(3)波前非规则的四边形网格与规则的矩形网格节点相对定位及转换:通过对比计算机图形学、计算机图像学中常用的几种定位算法并在前人工作的基础上从4个方面对其进行改进和完善,提高了网格转换的精度和效率;
(4)波前构建法在起伏地表条件下的应用:通过对强起伏地表条件(山顶、山谷、山坡)下震源初始化、射线追踪范围以及边界条件的具体改进,使波前构建法可以应用到起伏地表条件下来计算射线传播的路径。
The traveltimes, ray path and amplitudes of the seismic wave are several more important physical quantities in exploration seismology. They have many useful purposes such as Kirchhoff’s migration and demigration, tomography, and modeling. Wavefront construction method is a kind of fast algorithm which calculated these physical quantities, this method consider the calculation of the ray family which come from the entire wave field, make the classical geometric ray theory application in exploration seismology to be broken through, make it become possible to calculate hundreds and thousands of rays as well as traveltimes and amplitudes attach to they at the same time. Compared with the other algorithms of ray tracing, this method has incomparable superiority:①It can count traveltimes, ray path and amplitude of seismic wave at the same time;②It can improve rays coverage in study range, with the initial wavefront, we can through the wavefront propagation and ray interpolation to achieve the coverage of the entire study range;③It can calculate Multi-valued traveltimes, according to geometry diffraction theory, ray in caustic region will be split, a ray can produce many rays of diffractions (scattering), tracing these rays at the same time, we will be achieve the Multi-valued traveltimes at the same point;④It can adapt to the comparatively complicated velocity model and no limitation in observation system;⑤It has very high computational efficiency and accuracy.
Seeing that wavefront construction method is a kind of classical method from 1993, in this paper, we first introduce the research situation, basic theory, numerical solution and algorithm implementation scheme of wavefront construction method, then sum up and summarize at present and still exist problems in research current situation, to these questions, this paper will carry on thorough researching and analysing from 4 following respects:
(1) The interpolation problem of velocity and velocity derivative in non-grid nodes; By the way of calculating in numerical solution can be used to achieve the spread of wavefront, in the course of calculating, wavefront expansion outwards according to the time step by step, calculate each time step needs to know the velocity and derivative value in x direction and z direction in this step , but this step is not might just leave the regular grid node , so we need to know velocity and derivative of any point in the velocity model, and the model that often provide the velocity value with nodal regular grid, so we must use the interpolation algorithm to get velocity and derivative of any point value in the model. In the process of wavefront construction method ray tracing, the accuracy of the velocity and velocity derivative of interpolation points in the model influences the accuracy of ray path and traveltimes, so the choice of the interpolation method is influencing the efficiency and accuracy. In this paper, we analysis and comparison two-dimensional interpolation algorithm of digital topography, digital navigation and computer graphics, use bicubic convolution interpolation to calculate the value of the velocity and velocity derivative of non-grids node in the model, because this algorithm adopts 16 grid nodes around the interpolation point to calculate the value, and obtain the value of velocity and velocity derivative at once time ,so, this method have been improved computational efficiency and accuracy relative to other interpolation algorithm. This kind of interpolation algorithm using the wavefront construction make the ray tracing become more faster and accuracy.
(2)The problem of the smooth processing of the velocity model influence of traveltimes ray path and amplitude of seismic wave;
The smooth processing of the velocity model is an essential problem to wavefront construction method, firstly, it can ensure the ray tracing successful, Secondly, for using the smooth processing of the velocity model, thus avoid using snell law in the velocity interface of the model, improved the computational efficiency of ray tracing. But the problem of the smooth processing of the velocity model, we must take into account the relationship between the spatial distribution of velocity model at wavenumber region, computational speed and efficiency, and the number of the smooth times. As to this, we use different smooth operators, the different smooth number of times and different smooth factors of the same smooth operators, we are detailed from 3 respects to discuss the question of smooth processing of the velocity model:
①The problem of the smooth processing of the velocity model influence of ray path;
②The problem of the smooth processing of the velocity model influence of traveltimes;
③The problem of the smooth processing of the velocity model influence of amplitudes.
Through the above 3 aspects of analysis and discussion, we can get:
①In the data processing of the seismic, we should choose a suitable smooth operator, so it can reduce the error caused by smooth processing of the model of ray path traveltimes and amplitude information;
②For the choice number of smooth times, we must take into account comprehensive influence of ray path traveltimes and amplitude information, choose the proper smooth times, it can improve efficiency and accuracy of ray tracing;
③For convolution smooth operator, because the value of factor is different, it can effect on ray path traveltimes and amplitude information at the same.
④Through the comparative analysis in the shallow and deep velocity model, the smooth processing of the velocity model influence of traveltimes on the shallow model at less than deep model, but the influence of ray path on the shallow model more than deep model. This conclusion has a great role in practical application.
(3)The relative position between the non- regular quadrangle and regular rectangle grid node and question of changing in wavefront construction method;
For this question, the article in the previous studies mainly by use of vector product approach to determine the regular of the rectangular grid node is located in a nonregular quadrilateral grid of wavefront, but does not take the rules of the other relative position of rectangle grid nodes and nonregular quadrilateral grid of wavefront, so the accuracy of the conversion property will be influenced by the fact. in order to improve the accuracy of the conversion, this paper through the comparative analysis of several computer graphics grid point positioning algorithm as well as application and error analysis in wavefront construction, we studying the relative position of rectangular grid nodes and non-rules of wavefront quadrilateral grid following 4 respects and by using the different interpolation algorithm corresponding to calculate traveltimes:
①Rectangle grid point is located in nonregular quadrilateral grids of wavefront, we propose a distance weighted averaging of extrapolated traveltimes;
②Rectangle grid point is located on two rays in nonregular quadrilateral grids of wavefront, we using linear interpolation algorithm for calculating the traveltimes;
③Rectangle grid point is located on the intersection of ray and wavefront, this situation needn't interpolation, the value of Rectangle grid point is equal to the value of intersection of ray and wavefront;
④Rectangle grid point is located on the boundary of non-ray in nonregular quadrilateral grids of wavefront, we can use two algorithm to achieve: The first kind of method is expanding the region, to incorporate other nonregular quadrilateral grids of wavefront, and using the distance weighted interpolation algorithm for calculating the traveltimes; The second kind of method is insert new ray between two adjacent rays, and using linear interpolation algorithm for calculating the traveltimes.
Through the above four methods and calculations to determine the nonregular quadrilateral grids of wavefront and rectangle grid point, this kind of method using the wavefront construction make the conversion of property become more faster and accuracy.
(4)The application of wavefront construction method under ragged surface condition According to the application of wavefront construction method, because the algorithm is based on the level of surface made by a fast speed computing traveltimes ray path of seismic wave, the algorithm should be applied to ragged surface condition, we must change source initialization, the range and border of ray tracing, make the wavefront construction method to calculate traveltimes and ray path under ragged surface condition.
(1)波前构建过程中非网格节点处速度以及速度导数的插值计算:通过对比分析数字地形、数字导航、计算机图形学中常用的二维插值算法,引入二维三次卷积插值实现非网格节点处速度及导数的计算,提高射线追踪的效率和精度;
(2)模型的光滑处理对地震波走时、射线路径及振幅信息带来的影响:应用不同的光滑算子、同一光滑算子不同光滑次数及不同光滑因子讨论了波前构建法中模型的光滑处理对地震波走时、射线路径及振幅信息的影响,并在浅层模型和深层模型了进行了走时和射线路径的对比分析;
(3)波前非规则的四边形网格与规则的矩形网格节点相对定位及转换:通过对比计算机图形学、计算机图像学中常用的几种定位算法并在前人工作的基础上从4个方面对其进行改进和完善,提高了网格转换的精度和效率;
(4)波前构建法在起伏地表条件下的应用:通过对强起伏地表条件(山顶、山谷、山坡)下震源初始化、射线追踪范围以及边界条件的具体改进,使波前构建法可以应用到起伏地表条件下来计算射线传播的路径。
The traveltimes, ray path and amplitudes of the seismic wave are several more important physical quantities in exploration seismology. They have many useful purposes such as Kirchhoff’s migration and demigration, tomography, and modeling. Wavefront construction method is a kind of fast algorithm which calculated these physical quantities, this method consider the calculation of the ray family which come from the entire wave field, make the classical geometric ray theory application in exploration seismology to be broken through, make it become possible to calculate hundreds and thousands of rays as well as traveltimes and amplitudes attach to they at the same time. Compared with the other algorithms of ray tracing, this method has incomparable superiority:①It can count traveltimes, ray path and amplitude of seismic wave at the same time;②It can improve rays coverage in study range, with the initial wavefront, we can through the wavefront propagation and ray interpolation to achieve the coverage of the entire study range;③It can calculate Multi-valued traveltimes, according to geometry diffraction theory, ray in caustic region will be split, a ray can produce many rays of diffractions (scattering), tracing these rays at the same time, we will be achieve the Multi-valued traveltimes at the same point;④It can adapt to the comparatively complicated velocity model and no limitation in observation system;⑤It has very high computational efficiency and accuracy.
Seeing that wavefront construction method is a kind of classical method from 1993, in this paper, we first introduce the research situation, basic theory, numerical solution and algorithm implementation scheme of wavefront construction method, then sum up and summarize at present and still exist problems in research current situation, to these questions, this paper will carry on thorough researching and analysing from 4 following respects:
(1) The interpolation problem of velocity and velocity derivative in non-grid nodes; By the way of calculating in numerical solution can be used to achieve the spread of wavefront, in the course of calculating, wavefront expansion outwards according to the time step by step, calculate each time step needs to know the velocity and derivative value in x direction and z direction in this step , but this step is not might just leave the regular grid node , so we need to know velocity and derivative of any point in the velocity model, and the model that often provide the velocity value with nodal regular grid, so we must use the interpolation algorithm to get velocity and derivative of any point value in the model. In the process of wavefront construction method ray tracing, the accuracy of the velocity and velocity derivative of interpolation points in the model influences the accuracy of ray path and traveltimes, so the choice of the interpolation method is influencing the efficiency and accuracy. In this paper, we analysis and comparison two-dimensional interpolation algorithm of digital topography, digital navigation and computer graphics, use bicubic convolution interpolation to calculate the value of the velocity and velocity derivative of non-grids node in the model, because this algorithm adopts 16 grid nodes around the interpolation point to calculate the value, and obtain the value of velocity and velocity derivative at once time ,so, this method have been improved computational efficiency and accuracy relative to other interpolation algorithm. This kind of interpolation algorithm using the wavefront construction make the ray tracing become more faster and accuracy.
(2)The problem of the smooth processing of the velocity model influence of traveltimes ray path and amplitude of seismic wave;
The smooth processing of the velocity model is an essential problem to wavefront construction method, firstly, it can ensure the ray tracing successful, Secondly, for using the smooth processing of the velocity model, thus avoid using snell law in the velocity interface of the model, improved the computational efficiency of ray tracing. But the problem of the smooth processing of the velocity model, we must take into account the relationship between the spatial distribution of velocity model at wavenumber region, computational speed and efficiency, and the number of the smooth times. As to this, we use different smooth operators, the different smooth number of times and different smooth factors of the same smooth operators, we are detailed from 3 respects to discuss the question of smooth processing of the velocity model:
①The problem of the smooth processing of the velocity model influence of ray path;
②The problem of the smooth processing of the velocity model influence of traveltimes;
③The problem of the smooth processing of the velocity model influence of amplitudes.
Through the above 3 aspects of analysis and discussion, we can get:
①In the data processing of the seismic, we should choose a suitable smooth operator, so it can reduce the error caused by smooth processing of the model of ray path traveltimes and amplitude information;
②For the choice number of smooth times, we must take into account comprehensive influence of ray path traveltimes and amplitude information, choose the proper smooth times, it can improve efficiency and accuracy of ray tracing;
③For convolution smooth operator, because the value of factor is different, it can effect on ray path traveltimes and amplitude information at the same.
④Through the comparative analysis in the shallow and deep velocity model, the smooth processing of the velocity model influence of traveltimes on the shallow model at less than deep model, but the influence of ray path on the shallow model more than deep model. This conclusion has a great role in practical application.
(3)The relative position between the non- regular quadrangle and regular rectangle grid node and question of changing in wavefront construction method;
For this question, the article in the previous studies mainly by use of vector product approach to determine the regular of the rectangular grid node is located in a nonregular quadrilateral grid of wavefront, but does not take the rules of the other relative position of rectangle grid nodes and nonregular quadrilateral grid of wavefront, so the accuracy of the conversion property will be influenced by the fact. in order to improve the accuracy of the conversion, this paper through the comparative analysis of several computer graphics grid point positioning algorithm as well as application and error analysis in wavefront construction, we studying the relative position of rectangular grid nodes and non-rules of wavefront quadrilateral grid following 4 respects and by using the different interpolation algorithm corresponding to calculate traveltimes:
①Rectangle grid point is located in nonregular quadrilateral grids of wavefront, we propose a distance weighted averaging of extrapolated traveltimes;
②Rectangle grid point is located on two rays in nonregular quadrilateral grids of wavefront, we using linear interpolation algorithm for calculating the traveltimes;
③Rectangle grid point is located on the intersection of ray and wavefront, this situation needn't interpolation, the value of Rectangle grid point is equal to the value of intersection of ray and wavefront;
④Rectangle grid point is located on the boundary of non-ray in nonregular quadrilateral grids of wavefront, we can use two algorithm to achieve: The first kind of method is expanding the region, to incorporate other nonregular quadrilateral grids of wavefront, and using the distance weighted interpolation algorithm for calculating the traveltimes; The second kind of method is insert new ray between two adjacent rays, and using linear interpolation algorithm for calculating the traveltimes.
Through the above four methods and calculations to determine the nonregular quadrilateral grids of wavefront and rectangle grid point, this kind of method using the wavefront construction make the conversion of property become more faster and accuracy.
(4)The application of wavefront construction method under ragged surface condition According to the application of wavefront construction method, because the algorithm is based on the level of surface made by a fast speed computing traveltimes ray path of seismic wave, the algorithm should be applied to ragged surface condition, we must change source initialization, the range and border of ray tracing, make the wavefront construction method to calculate traveltimes and ray path under ragged surface condition.
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