贴体网格各向异性对坐标变换法求解起伏地表下地震初至波走时的影响

笛卡尔坐标系中的经典程函方程在静校正、叠前偏移、走时反演、地震定位、层析成像等很多地球物理工作中都有应用,然而用其计算起伏地表的地震波走时却比较困难.本文通过把曲线坐标系中的矩形网格映射到笛卡尔坐标系的贴体网格,推导出曲线坐标中的程函方程,而后,用Lax-Friedrichs快速扫描算法求解曲线坐标系的程函方程.研究表明本文方法能有效处理地表起伏的情况,得到准确稳定的计算结果.由于地表起伏,导致与之拟合的贴体网格在空间上的展布呈各向异性,且这种各向异性的强弱对坐标变换法求解地震初至波的走时具有重要影响.本文研究表明,随着贴体网格的各向异性增强,用坐标变换法求解地表起伏区域的走时计算误差增大,且计算效率降低,这在实际应用具有指导意义.     

An upwind fast sweeping scheme for calculating seismic wave first‐arrival travel times for models with an irregular free surface

The topography‐dependent eikonal equation formulated in a curvilinear coordinate system has recently been established and revealed as being effective in calculating first‐arrival travel times of seismic waves in an Earth model with an irregular free surface. The Lax–Friedrichs sweeping scheme, widely used in previous studies as for approximating the topography‐dependent eikonal equation viscosity solutions, is more dissipative and needs a much higher number of iterations to converge. Furthermore, the required number of iterations grows with the grid refinement and results in heavy computation in dense grids, which hampers the application of the Lax–Friedrichs sweeping scheme to seismic wave travel‐time calculation and high‐resolution imaging. In this paper, we introduce a new upwind fast sweeping solver by discretising the Legendre transform of the numerical Hamiltonian of the topography‐dependent eikonal equation using an explicit formula. The minimisation related to the Legendre transform in the sweeping scheme is solved analytically, which proved to be much more efficient than the Lax–Friedrichs algorithm in solving the topography‐dependent eikonal equation. Several numerical experiments demonstrate that the new upwind fast sweeping method converges and achieves much better accuracy after a finite number of iterations, independently of the mesh size, which makes it an efficient and robust tool for calculating travel times in the presence of a non‐flat free surface.     

Secondorder interpolation of traveltimes

To carry out a 3D prestack migration of the Kirchhoff type is still a task of enormous computational effort. Its efficiency can be significantly enhanced by employing a fast traveltime interpolation algorithm. High accuracy can be achieved if secondorder spatial derivatives of traveltimes are included in order to account for the curvature of the wavefront. We suggest a hyperbolic traveltime interpolation scheme that permits the determination of the hyperbolic coefficients directly from traveltimes sampled on a coarse grid, thus reducing the requirements in data storage. This approach is closely related to the paraxial ray approximation and corresponds to an extension of the wellknown     method to arbitrary heterogeneous and complex media in 3D. Application to various velocity models, including a 3D version of the Marmousi model, confirms the superiority of our method over the popular trilinear interpolation. This is especially true for regions with strong curvature of the local wavefront. In contrast to trilinear interpolation, our method also provides the possibility of interpolating source positions, and it is 56 times faster than the calculation of traveltime tables using a fast finitedifference eikonal solver.     

Traveltime computation with the linearized eikonal equation for anisotropic media

A linearized eikonal equation is developed for transversely isotropic (TI) media with a vertical symmetry axis (VTI). It is linear with respect to perturbations in the horizontal velocity or the anisotropy parameter η. An iterative linearization of the eikonal equation is used as the basis for an algorithm of finite-difference traveltime computations. A practical implementation of this iterative technique is to start with a background model that consists of an elliptically anisotropic, inhomogeneous medium, since traveltimes for this type of medium can be calculated efficiently using eikonal solvers, such as the fast marching method. This constrains the perturbation to changes in the anisotropy parameter η (the parameter most responsible for imaging improvements in anisotropic media). The iterative implementation includes repetitive calculation of η from traveltimes, which is then used to evaluate the perturbation needed for the next round of traveltime calculations using the linearized eikonal equation. Unlike isotropic media, interpolation is needed to estimate η in areas where the traveltime field is independent of η, such as areas where the wave propagates vertically.
Typically, two to three iterations can give sufficient accuracy in traveltimes for imaging applications. The cost of each iteration is slightly less than the cost of a typical eikonal solver. However, this method will ultimately provide traveltime solutions for VTI media. The main limitation of the method is that some smoothness of the medium is required for the iterative implementation to work, especially since we evaluate derivatives of the traveltime field as part of the iterative approach. If a single perturbation is sufficient for the traveltime calculation, which may be the case for weak anisotropy, no smoothness of the medium is necessary. Numerical tests demonstrate the robustness and efficiency of this approach.     

3D P‐wave traveltime computation in transversely isotropic media using layer‐by‐layer wavefront marching

Subsurface rocks (e.g. shale) may induce seismic anisotropy, such as transverse isotropy. Traveltime computation is an essential component of depth imaging and tomography in transversely isotropic media. It is natural to compute the traveltime using the wavefront marching method. However, tracking the 3D wavefront is expensive, especially in anisotropic media. Besides, the wavefront marching method usually computes the traveltime using the eikonal equation. However, the anisotropic eikonal equation is highly non‐linear and it is challenging to solve. To address these issues, we present a layer‐by‐layer wavefront marching method to compute the P‐wave traveltime in 3D transversely isotropic media. To simplify the wavefront tracking, it uses the traveltime of the previous depth as the boundary condition to compute that of the next depth based on the wavefront marching. A strategy of traveltime computation is designed to guarantee the causality of wave propagation. To avoid solving the non‐linear eikonal equation, it updates traveltime along the expanding wavefront by Fermat's principle. To compute the traveltime using Fermat's principle, an approximate group velocity with high accuracy in transversely isotropic media is adopted to describe the ray propagation. Numerical examples on 3D vertical transverse isotropy and tilted transverse isotropy models show that the proposed method computes the traveltime with high accuracy. It can find applications in modelling and depth migration.     

基于HAFMM的无射线追踪跨孔雷达走时层析成像

本文使用最小二乘线性迭代反演方法对跨孔雷达直达波初至时数据进行反演,每次迭代过程中,用有限差分法求解走时程函方程,并用高精度快速推进方法(HAFMM)进行波前扩展,通过追踪波前避免了进行射线追踪.为了验证该方案,我们对三组合成数据进行了测试,分析了单位矩阵算子、一阶差分算子和拉普拉斯算子等三种不同模型参数加权算子对模型的约束和平滑效果;讨论了FMM和HAFMM对反演精度的影响;测试了LSQR,GMRES和BICGSTAB等三种矩阵反演算法的反演效果.此外,我们还对一组野外实测数据进行了反演,对比了基于本方案以及基于平直射线追踪和弯曲射线追踪的走时层析成像反演效果.对比分析结果表明,使用拉普拉斯算子和HAFMM进行反演能较好地进行目标体重建,而三种矩阵反演方法对反演效果的影响差别不大;并且通过对波前等时线图的分析可以定性地判断异常体的性质和位置;而在对实测数据目标体的重建上,本方案能达到甚至优于弯曲射线算法的重建效果.     

三维起伏地表条件下地震波走时计算的不等距迎风差分法

三维起伏地表条件下的地震波走时计算技术是研究三维起伏地表地区很多地震数据处理技术的基础性工具.为了获得适应于任意三维起伏地表且计算精度高的走时算法,提出三维不等距迎风差分法.该方法采用不等距网格剖分三维起伏地表模型,通过在迎风差分格式中引入不等距差分格式、Huygens原理及Fermat原理来建立地表附近的局部走时计算公式,并通过在窄带技术中设定新的网格节点类型来获得三维起伏地表条件下算法的整体实现步骤.精度及算例分析表明:三维不等距迎风差分法具有很高的计算精度且能够适应于任意三维起伏地表模型.     

曲线坐标系因式分解程函方程及其走时计算

地震波走时广泛应用于静校正、层析成像、Kirchhoff偏移成像、地震定位等研究.复杂地表条件是影响走时计算精度的重要因素.近年来,发展的曲线坐标系程函方程为精细刻画起伏地表条件下的地震波走时场特征提供了新的思路.然而,基于有限差分程函方程的求解方法不可避免地受到震源奇异性的影响,即震源附近波前的曲率较大,此时使用平面波近似假设的差分格式会导致较大误差.而震源误差会随着波前的传播到达整个计算区域,从而影响整个区域的求解精度.针对该问题,本文借鉴因式分解的思想,推导建立了曲线坐标系因式分解程函方程,并针对性地发展了其数值求解方法,从根源上解决了复杂模型走时计算中的震源奇异性问题.数值实例表明因式分解法能够有效降低震源误差,显著提高起伏地表走时计算的精度和效率,为起伏地表地震波走时计算提供更佳的选择,在复杂模型的地震资料处理中展现出广泛的应用前景.     

基于快速推进迎风双线性插值法的三维地震波走时计算

三维地震波走时计算技术是三维地震反演、层析成像、偏移成像等诸多地震数据处理技术中非常重要的正演计算工具.为了获得精度高且兼顾效率的三维走时计算方法:首先,在常规双线性插值公式推导过程中,充分利用平面波双线性假设的结论,获得了二元极小值超越方程的解析解,进而推导出了准确的局部走时计算公式,同时构造性地证明了该计算公式满足地震波的传播规律和Eikonal方程;其次,引入迎风差分的基本思想,提出迎风双线性插值的局部走时计算策略,该计算策略能简化算法、提高效率且保证无条件稳定性;然后,将上述计算公式和迎风双线性插值策略与常规快速推进法中的窄带技术结合,获得了一种新的基于快速推进迎风双线性插值法的三维地震波走时计算方法;最后,通过精度和效率分析检验了新算法的精度、效率和正确性,并通过计算实例验证了算法在面对复杂介质时的稳定性和有效性.     

Calculating complex‐valued P‐wave first‐arrival traveltimes in attenuative vertical transversely isotropic media with an irregular surface

The complex‐valued first‐arrival traveltime can be used to describe the properties of both velocity and attenuation as seismic waves propagate in attenuative elastic media. The real part of the complex‐valued traveltime corresponds to phase arrival and the imaginary part is associated with the amplitude decay due to energy absorption. The eikonal equation for attenuative vertical transversely isotropic media discretized with rectangular grids has been proven effective and precise to calculate the complex‐valued traveltime, but less accurate and efficient for irregular models. By using the perturbation method, the complex‐valued eikonal equation can be decomposed into two real‐valued equations, namely the zeroth‐ and first‐order traveltime governing equations. Here, we first present the topography‐dependent zeroth‐ and first‐order governing equations for attenuative VTI media, which are obtained by using the coordinate transformation from the Cartesian coordinates to the curvilinear coordinates. Then, we apply the Lax–Friedrichs sweeping method for solving the topography‐dependent traveltime governing equations in order to approximate the viscosity solutions, namely the real and imaginary parts of the complex‐valued traveltime. Several numerical tests demonstrate that the proposed scheme is efficient and accurate in calculating the complex‐valued P‐wave first‐arrival traveltime in attenuative VTI media with an irregular surface.     

A Fast Sweeping Scheme for Calculating P Wave First-Arrival Travel Times in Transversely Isotropic Media with an Irregular Surface

Seismic wave propagation shows anisotropic characteristics in many sedimentary rocks. Modern seismic exploration in mountainous areas makes it important to calculate P wave travel times in anisotropic media with irregular surfaces. The challenges in this context are mainly from two aspects. First is how to tackle the irregular surface in a Cartesian coordinate system, and the other lies in solving the anisotropic eikonal equation. Since for anisotropic media the ray (group) velocity direction is not the same as the direction of the travel-time gradient, the travel-time gradient no longer serves as an indicator of the group velocity direction in extrapolating the travel-time field. Recently, a topography-dependent eikonal equation formulated in a curvilinear coordinate system has been established, which is effective for calculating first-arrival travel times in an isotropic model with an irregular surface. Here, we extend the above equation from isotropy to transverse isotropy (TI) by formulating a topography-dependent eikonal equation in TI media in the curvilinear coordinate system, and then use a fast sweeping scheme to solve the topography-dependent anisotropic eikonal equation in the curvilinear coordinate system. Numerical experiments demonstrate the feasibility and accuracy of the scheme in calculating P wave travel times in TI models with an irregular surface.     

三角网波行面扩展最小走时射线追踪全局算法

To address the problem of subdividing inflexible rectangular grid models and their poor definition of velocity interfaces, we propose a complex structure triangular net for a minimum traveltime ray tracing global algorithm. Our procedure is: (1) Subdivide a triangle grid based on the Delaunay triangular subdivision criterion and the relationships of the points, lines, and the surfaces in the subdividing area. (2) Define the topology relationships and related concepts of triangular unit ray tracing. (3) The source point and wave arrival points at any time compose the propagating plane wave and the minimum traveltime and secondary source positions are calculated during the plane wave propagation. We adopt the hyperbolic approximation global algorithm for secondary source retrieving. (4) By minimum traveltime ray tracing, collect the path from receiver to source points with the neighborhood point’s traveltime and the direction of the secondary source. Numerical simulation examples are given to test the algorithm. The results show that the triangular net ray tracing method demonstrates model subdivision flexibility, precise velocity discontinuity interfaces, and accurate computations.     

基于走时的保幅偏移方法

振幅随偏移距变化是描述储层特征的重要方法之一,保幅偏移方法就是使偏移剖面能够反映出振幅随偏移距的变化.本论文中的保幅偏移是以走时为基础,主要的方法是采用走时的双曲线展开法,通过走时的二阶空间导数来确定波前曲率.该方法通过建立在大网格上的走时表来确定插值系数,将大网格插值成为较为精细的网格,这样就节省了数据的存储空间.对于相同的网格密度,通过插值来计算走时表比采用程函方程有限差分法直接计算走时要节省5至6倍的时间.走时的插值系数还可以用来计算几何扩散因子、权函数,不仅提高了成像质量,还大大节省了计算时间.     

叠前逆时深度偏移中的激发时间成像条件

与其他偏移方法相比,逆时偏移基于精确的波动方程而不是对其近似,用时间外推来代替深度外推.因此,它具有良好的精度,不受地下构造倾角和介质横向速度变化的限制.激发时间成像条件的求取是叠前逆时偏移的难点之一,本文采用求解程函方程的方法得到地下各点的初至波走时,以此作为叠前逆时偏移的成像条件.基于任意矩形网格和局部平面波前近似的有限差分初至波走时计算方法精度较高并适用于强纵横向变速的复杂介质.试算结果表明,在复杂介质模型中利用叠前逆时深度偏移收到了很好的成像效果.     

初至波相位走时层析

近地表速度结构通常是利用射线走时层析或菲涅尔体走时层析等反演方法得到的,但它们的目标函数仍利用射线走时残差构建,导致反演精度不高.为此,本文提出了基于散射积分算法的初至波相位走时层析成像方法.该方法的核心是:(1)提出了依赖于频率的相位走时概念;(2)利用依赖于频率的相位走时信息,而非单一的无限频率射线走时;(3)发展了一种改进的相位展开方法,即通过监测相位不连续性和2π周期判定来消除相位折叠现象;(4)考虑了地震波传播的有限频特征,即基于波动理论而非传统的射线路径或有限空间的菲涅尔体构建核函数.通过利用Overthrust模型的数值实验及与传统射线走时层析和菲涅尔体走时层析的对比表明:本文提出的方法是一种有效的初至波走时反演方法.同时,基于Overthrust模型的数值试验还证明了下列结论,即通过挖掘更多的走时信息的确可以获得更高的反演精度和分辨率.     

地震初至波走时的有限差分计算

本文提出了用有限差分解程函方程求取地震初至波走时的快速、精确方法。算法考虑了首波，散射波开采新的延拓方法。在任意复杂的速度结构中能得到精确的结果。本方法对叠前偏移、层析成像是非常适宜的。     

SEISMIC RAY TRACING USING LINEAR TRAVELTIME INTERPOLATION1

A new ray-tracing method called linear traveltime interpolation (LTI) is proposed. This method computes traveltimes and raypaths in a 2D velocity structure more rapidly and accurately than other conventional methods. The LTI method is formulated for a 2D cell model, and calculations of traveltimes and raypaths are carried out only on cell boundaries. Therefore a raypath is considered to be always straight in a cell with uniform velocity. This approach is suitable to tomography analysis. The algorithm of LTI consists of two separate steps: step 1 calculates traveltimes on all cell boundaries; step 2 traces raypaths for all pairs of receivers and the shot. A traveltime at an arbitrary point on a cell boundary is assumed to be linearly interpolated between traveltimes at the adjacent discrete points at which we calculate traveltimes. Fermat's principle is used as the criterion for choosing the correct traveltimes and raypaths from several candidates routinely. The LTI method has been compared numerically with the shooting method and the finite-difference method (FDM) of the eikonal equation. The results show that the LTI method has great advantages of high speed and high accuracy in the calculation of both traveltimes and raypaths. The LTI method can be regarded as an advanced version of the conventional FDM of the eikonal equation because the formulae of FDM are independently derived from LTI. In the process of derivation, it is shown theoretically that LTI is more accurate than FDM. Moreover in the LTI method, we can avoid the numerical instability that occurs in Vidale's method where the velocity changes abruptly.     

Traveltime computation for 3D anisotropic media by a finite-difference perturbation method

The first-order perturbation theory is used for fast 3D computation of quasi-compressional (qP)-wave traveltimes in arbitrarily anisotropic media. For efficiency we implement the perturbation approach using a finite-difference (FD) eikonal solver. Traveltimes in the unperturbed reference medium are computed with an FD eikonal solver, while perturbed traveltimes are obtained by adding a traveltime correction to the traveltimes of the reference medium. The traveltime correction must be computed along the raypath in the reference medium. Since the raypath is not determined in FD eikonal solvers, we approximate rays by linear segments corresponding to the direction of the phase normal of plane wavefronts in each cell. An isotropic medium as a reference medium works well for weak anisotropy. Using a medium with ellipsoidal anisotropy as a background medium in the perturbation approach allows us to consider stronger anisotropy without losing computational speed. The traveltime computation in media with ellipsoidal anisotropy using an FD eikonal solver is fast and accurate. The relative error is below 0.5% for the models investigated in this study. Numerical examples show that the reference model with ellipsoidal anisotropy allows us to compute the traveltime for models with strong anisotropy with an improved accuracy compared with the isotropic reference medium.