Traveltime computation by wavefront-orientated ray tracing

For multivalued traveltime computation on dense grids, we propose a wavefront‐orientated ray‐tracing (WRT) technique. At the source, we start with a few rays which are propagated stepwise through a smooth two‐dimensional (2D) velocity model. The ray field is examined at wavefronts and a new ray might be inserted between two adjacent rays if one of the following criteria is satisfied: (1) the distance between the two rays is larger than a predefined threshold; (2) the difference in wavefront curvature between the rays is larger than a predefined threshold; (3) the adjacent rays intersect. The last two criteria may lead to oversampling by rays in caustic regions. To avoid this oversampling, we do not insert a ray if the distance between adjacent rays is smaller than a predefined threshold. We insert the new ray by tracing it from the source. This approach leads to an improved accuracy compared with the insertion of a new ray by interpolation, which is the method usually applied in wavefront construction. The traveltimes computed along the rays are used for the estimation of traveltimes on a rectangular grid. This estimation is carried out within a region bounded by adjacent wavefronts and rays. As for the insertion criterion, we consider the wavefront curvature and extrapolate the traveltimes, up to the second order, from the intersection points between rays and wavefronts to a gridpoint. The extrapolated values are weighted with respect to the distances to wavefronts and rays. Because dynamic ray tracing is not applied, we approximate the wavefront curvature at a given point using the slowness vector at this point and an adjacent point on the same wavefront. The efficiency of the WRT technique is strongly dependent on the input parameters which control the wavefront and ray densities. On the basis of traveltimes computed in a smoothed Marmousi model, we analyse these dependences and suggest some rules for a correct choice of input parameters. With suitable input parameters, the WRT technique allows an accurate traveltime computation using a small number of rays and wavefronts.     

Joint 3D traveltime calculation based on fast marching method and wavefront construction

3D traveltime calculation is widely used in seismic exploration technologies such as seismic migration and tomography. The fast marching method (FMM) is useful for calculating 3D traveltime and has proven to be efficient and stable. However, it has low calculation accuracy near the source, which thus gives it low overall accuracy. This paper proposes a joint traveltime calculation method to solve this problem. The method firstly employs the wavefront construction method (WFC), which has a higher calculation accuracy than FMM in calculating traveltime in the small area near the source, and secondly adopts FMM to calculate traveltime for the remaining grid nodes. Due to the increase in calculation precision of grid nodes near the source, this new algorithm is shown to have good calculation precision while maintaining the high calculation efficiency of FMM, which is employed in most of the computational area. Results are verified using various numerical models.     

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.     

A new 3-D ray tracing method based on LTI using successive partitioning of cell interfaces and traveltime gradients

We present a new method of three-dimensional (3-D) seismic ray tracing, based on an improvement to the linear traveltime interpolation (LTI) ray tracing algorithm. This new technique involves two separate steps. The first involves a forward calculation based on the LTI method and the dynamic successive partitioning scheme, which is applied to calculate traveltimes on cell boundaries and assumes a wavefront that expands from the source to all grid nodes in the computational domain. We locate several dynamic successive partition points on a cell's surface, the traveltimes of which can be calculated by linear interpolation between the vertices of the cell's boundary. The second is a backward step that uses Fermat's principle and the fact that the ray path is always perpendicular to the wavefront and follows the negative traveltime gradient. In this process, the first-arriving ray path can be traced from the receiver to the source along the negative traveltime gradient, which can be calculated by reconstructing the continuous traveltime field with cubic B-spline interpolation. This new 3-D ray tracing method is compared with the LTI method and the shortest path method (SPM) through a number of numerical experiments. These comparisons show obvious improvements to computed traveltimes and ray paths, both in precision and computational efficiency.     

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.     

THE INTERPRETATION OF TRAVELTIME FIELDS IN REFRACTION SEISMOLOGY*

We consider multiply covered traveltimes of first or later arrivals which are gathered along a refraction seismic profile. The two-dimensional distribution of these traveltimes above a coordinate frame generated by the shotpoint axis and the geophone axis or by the common midpoint axis and the offset axis is named a traveltime field. The application of the principle of reciprocity to the traveltime field implies that for each traveltime value with a negative offset there is a corresponding equal value with positive offset. In appendix A procedures are demonstrated which minimize the observational errors of traveltimes inherent in particular traveltime branches or complete common shotpoint sections. The application of the principle of parallelism to an area of the traveltime field associated with a particular refractor can be formulated as a partial differential equation corresponding to the type of the vibrating string. The solution of this equation signifies that the two-dimensional distribution of these traveltimes may be generated by the sum of two one-dimensional functions which depend on the shotpoint coordinate and the geophone coordinate. Physically, these two functions may be interpreted as the mean traveltime branches of the reverse and the normal shot. In appendix B procedures are described which compute these two functions from real traveltime observations by a least-squares fit. The application of these regressed traveltime field data to known time-to-depth conversion methods is straightforward and more accurate and flexible than the use of individual traveltime branches. The wavefront method, the plus-minus method, the generalized reciprocal method and a ray tracing method are considered in detail. A field example demonstrates the adjustment of regressed traveltime fields to observed traveltime data. A time-to-depth conversion is also demonstrated applying a ray tracing method.     

复杂地表条件下基于线性插值和窄带技术的地震波走时计算

为了在复杂地表条件下实现地震波走时计算,提出了一种基于线性插值和窄带技术的走时计算新方法.其中,线性插值用于局部走时计算,窄带技术用于局部波前捕获和追踪.为了逼近起伏地表,采用三角网和矩形网相结合的方法对速度模型进行剖分.为了得到局部走时计算公式,利用费马（Fermat）原理和关于入射点位置的限定条件.有关编程实践和数值试验表明：新方法不仅可以有效、灵活地处理地表高程的剧烈变化,而且还具有很好的适应性和稳定性,得到的计算结果满足波前传播规律.     

Efficient traveltime compression for 3D prestack Kirchhoff migration

Kirchhoff 3D prestack migration, as part of its execution, usually requires repeated access to a large traveltime table data base. Access to this data base implies either a memory intensive or I/O bounded solution to the storage problem. Proper compression of the traveltime table allows efficient 3D prestack migration without relying on the usually slow access to the computer hard drive. Such compression also allows for faster access to desirable parts of the traveltime table. Compression is applied to the traveltime field for each source location on the surface on a regular grid using 3D Chebyshev polynomial or cosine transforms of the traveltime field represented in the spherical coordinates or the Celerity domain. We obtain practical compression levels up to and exceeding 20 to 1. In fact, because of the smaller size traveltime table, we obtain exceptional traveltime extraction speed during migration that exceeds conventional methods. Additional features of the compression include better interpolation of traveltime tables and more stable estimates of amplitudes from traveltime curvatures. Further compression is achieved using bit encoding, by representing compression parameters values with fewer bits.     

基于MSFM的复杂近地表模型走时计算

地震走时层析成像方法是解决复杂近地表模型速度建模问题的重要技术.该方法是一种迭代反演方法,在反演过程中需要反复计算地震射线走时.故而,高效高精度且能适应复杂模型的走时计算方法是地震走时层析成像实用化的关键技术之一.本文引入医学成像领域研究的MSFM(Multi-stencils Fast Marching Methods)用于地震层析反演中的走时计算.该方法在标准FMM(Fast Marching Methods)基础上利用坐标旋转生成新的FMM计算模板,使计算网格点对角方向邻点参与计算,改善了标准FMM存在对角方向误差大的缺陷.本文分析对比了MSFM和标准FMM的计算精度和计算效率;针对地震层析成像技术解决的起伏地表模型建模问题,研究了起伏地表模型地震走时计算的MSFM实现方法;采用炮点邻近区域局部细分网格技术只需增加很少的计算量即可大幅提高计算精度.理论分析和模型试算表明MSFM算法明显改善了FMM的计算精度,同时保持了FMM算法的高效性.文章通过对崎岖地表模型的正演和层析反演试算,验证了基于MSFM的地震走时计算方法对复杂模型有很强的适应能力.研究表明该方法作为地震走时层析反演中高效高精度的正演算法,有很好的应用价值.     

Passive seismic source localization via common-reflection-surface attributes

The common-reflection-surface (CRS) stack can be viewed as a physically justified extension of the classical common-midpoint (CMP) stack, utilizing redundant information not only in a single, but in several neighboring CMP gathers. The zero-offset CRS moveout is parameterized in terms of kinematic attributes, which utilize reciprocity and raypath symmetries to describe the two-way process of the actual wave propagation in active seismic experiments by the propagation of auxiliary one-way wavefronts. For the diffraction case, only the attributes of a single one-way wavefront, originating from the diffractor are sufficient to explain the traveltime differences observed at the surface. While paraxial ray theory gives rise to a second-order approximation of the CRS traveltime, many higher-order approximations were subsequently introduced either by squaring the second-order expression or by employing principles of optics and geometry. It was recently discovered that all of these higher-order operators can be formulated either for the optical projection or in an auxiliary medium of a constant effective velocity. Utilizing this duality and the one-way nature of the CRS parameters, we present a simple data-driven stacking scheme that allows for the estimation of the a priori unknown excitation time of a passive seismic source. In addition, we demonstrate with a simple data example that the output of the suggested workflow can directly be used for subsequent focusing-based normal-incidence-point (NIP) tomography, leading to a reliable localization in depth.     

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.     

有序波前重建法的射线追踪

建立了一种新的计算最小走时和射线路径的方法——有序波前重建法. 文中算法按照波前面的实际扩展顺序外推计算走时，采用以计算点为中心的走时计算策略，直接记录计算点获取最小走时的前一节点坐标，同步计算最小走时和射线路径，得到一种全局算法. 该方法具有原理简单、易于实现、不受介质速度差异大小限制、计算速度快等优点. 数值实验表明有序波前重建法具有较高的计算精度和运行效率.     

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.     

RAYS AND WAVEFRONT CHARTS IN GRADIENT MEDIA1

Wavefront charts in anisotropic gradient media are a useful tool in ray geometric constructions, particular in shear-wave exploration. They can be constructed by: (i) a family of wavefronts that contains a vertical plane as member - it is convenient to choose constant time increments; (ii) tracing one ray that makes everywhere the angle with the normal to the wavefront that is required by the anisotropy of the medium; (iii) scaling this ray to obtain a set of rays with different ray parameters; (iv) shifting these rays (with wavefront elements attached) so that they pass through a common source point; (v) interpolating the wavefronts between the elements. The construction is particularly simple in linear-gradient media, since here all members of the family of wavefronts are planes. Since the ray makes everywhere the angle prescribed by the anisotropy with the normal of the (plane) wavefronts, the ray has the shape of the slowness curve rotated by ?π/2. For isotropic media the slowness curve is a circle, and thus rays are circular arcs. The circles themselves intersect in the source point and in a second point above the surface of the earth. This provides a simple proof that wavefronts emanating from a point source in an isotropic linear-gradient medium are spheres: inversion of the set of circular rays with the source as centre maps the pencil of circular rays into a pencil of straight lines passing through a point. A pencil of concentric spheres around this point is perpendicular to the pencil of straight lines. On inverting back the pencil of spheres is mapped into another pencil of spheres that is perpendicular to the circular rays.     

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.     

3-D simultaneous inversion for velocity and reflector geometry using multiple classes of arrivals in a spherical coordinate frame

The two key requirements in conducting 3-D simultaneous traveltime tomography on real data at the regional and global scale with multiple classes of arrival time information are (1) it needs an efficient and accurate arrival tracking algorithm for multiply transmitted, reflected (or refracted) and converted waves in a 3-D variable velocity model with embedded velocity discontinuities (or subsurface interfaces), and (2) a subdimensional inversion solver is required which can easily search for different types of model parameters to balance the trade-off between the different types of model parameter updated in the simultaneous inversion process. For these purposes, we first extend a popular grid/cell-based wavefront expanding ray tracing algorithm (the multistage irregular shortest-path ray tracing method), which previously worked only in Cartesian coordinate at the local scale, to spherical coordinates appropriate to the regional or global scale. We then incorporated a fashionable inversion solver (the subspace method) to formulate a simultaneous inversion algorithm, in which the multiple classes of arrivals (including direct and reflected arrivals from different velocity discontinuities) can be used to simultaneously update both the velocity fields and the reflector geometries. Numerical tests indicate that the new inversion method is both applicable and flexible in terms of computational efficiency and solution accuracy, and is not sensitive to a modest level of noise in the traveltime data. It offers several potential benefits over existing schemes for real data seismic imaging.     

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.     

Second-order interpolation of later-arrival traveltimes

The performance of a 3D prestack migration of the Kirchhoff type can be significantly enhanced if the computation of the required stacking surface is replaced by an efficient and accurate method for the interpolation of diffraction traveltimes. Thus, input traveltimes need only be computed and stored on coarse grids, leading to considerable savings in CPU time and computer storage. However, interpolation methods based on a local approximation of the traveltime functions fail in the presence of triplications of the wavefront or later arrivals. This paper suggests a strategy to overcome this problem by employing the coefficients of a hyperbolic traveltime expansion to locate triplications and correct for the resulting errors in the interpolated traveltime tables of first and later arrivals.     

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.     

椭球坐标系下多震相地震射线追踪

地震射线追踪方法技术在地震学领域有着较为广泛的应用,然而大多数算法建立在直角坐标系或球坐标系下,实际地球并非完美的球体,而是两极略扁的椭球体,因此,球坐标系下计算结果与真实情况存在一定误差.传统的做法一般是在球坐标系下进行计算,而后进行椭球校正.本文提出了一种直接在椭球体模型中采用分区多步最短路径算法进行多震相地震射线追踪的方法技术,实现了椭球坐标系下多震相地震波射线路径追踪和走时计算.与解析解的对比表明:该算法具有较高的计算精度,适用于任意形状的椭球体,且不需要进行额外的走时校正.数值模拟结果表明,计算所得P波和PcP反射波的走时与AK135走时表的误差小于0.1 s.当震中距较大时,使用球对称模型和椭球体模型计算所得的走时差异显著,说明采用椭球坐标系的必要性.