Eigenrays in 3D heterogeneous anisotropic media,Part II: Dynamics

This paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.     

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.     

Geometrical spreading of seismic body waves in laterally inhomogeneous media with curved interfaces of the second order

Summary Seismic rays and geometrical spreading in laterally inhomogeneous media are described by a system of ordinary differential equations of the first order. When the ray crosses an interface of the second order, certain quantities in the system change discontinuously. The initial conditions for the system at the point of intersection of the ray with the interface are presented.     

Eigenrays in 3D heterogeneous anisotropic media,Part I: Kinematics

We present a new ray bending approach, referred to as the Eigenray method, for solving two‐point boundary‐value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial‐value ray shooting methods, followed by numerical convergence techniques, become challenging. The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second‐order Euler–Lagrange equation. The solution is based on a finite‐element approach, applying the weak formulation that reduces the Euler–Lagrange second‐order ordinary differential equation to the first‐order weighted‐residual nonlinear algebraic equation set. For the kinematic finite‐element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation. The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle‐point solution (due to caustics). The minimization process involves the computation of the global (all‐node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analysing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter, however, is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics. Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples.     

Complex rays applied to wave propagation in a viscoelastic medium

In order to trace a ray between known source and receiver locations in a perfectly elastic medium, the take-off angle must be determined, or equialently, the ray parameter. In a viscoelastic medium, the initial value of a second angle, the attenuation angle (the angle between the normal to the plane wavefront and the direction of maximum attenuation), must also be determined. There seems to be no agreement in the literature as to how this should be done. In computing anelastic synthetic seismograms, some authors have simply chosen arbitrary numerical values for the initial attenuation angle, resulting in different raypaths for different choices. There exists, however, a procedure in which the arbitrariness is not present, i.e., in which the raypath is uniquely determined. It consists of computing the value of the anelastic ray parameter for which the phase function is stationary (Fermat's principle). This unique value of the ray parameter gives unique values for the take-off and attenuation angles. The coordinates of points on these stationary raypaths are complex numbers. Such rays are known as complex rays. They have been used to study electromagnetic wave propagation in lossy media. However, ray-synthetic seismograms can be computed by this procedure without concern for the details of complex raypath coordinates. To clarify the nature of complex rays, we study two examples involving a ray passing through a vertically inhomogeneous medium. In the first example, the medium consists of a sequence of discrete homogeneous layers. We find that the coordinates of points on the ray are generally complex (other than the source and receiver points which are usually assumed to lie in real space), except for a ray which is symmetric about an axis down its center, in which case the center point of the ray lies in real space. In the second example, the velocity varies continuously and linearly with depth. We show that, in geneneral, the turning point of the ray lies in complex space (unlike the symmetric ray in the discrete layer case), except if the ratio of the velocity gradient to the complex frequency-dependent velocity at the surface is a real number. We also present a numerical example which demonstrates that the differences between parameters, such as arrival time and raypath angles, for the stationary ray and for rays computed by the above-mentioned arbitrary approaches can be substantial.     

Prevailing-frequency approximation of the coupling ray theory along the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately uniaxial

The behaviour of the actual polarization of an electromagnetic wave or elastic S–wave is described by the coupling ray theory, which represents the generalization of both the zero–order isotropic and anisotropic ray theories and provides continuous transition between them. The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. In a generally anisotropic or bianisotropic medium, the actual wave paths may be approximated by the anisotropic–ray–theory rays if these rays behave reasonably. In an approximately uniaxial (approximately transversely isotropic) anisotropic medium, we can define and trace the SH (ordinary) and SV (extraordinary) reference rays, and use them as reference rays for the prevailing–frequency approximation of the coupling ray theory. In both cases, i.e. for the anisotropic–ray–theory rays or the SH and SV reference rays, we have two sets of reference rays. We thus obtain two arrivals along each reference ray of the first set and have to select the correct one. Analogously, we obtain two arrivals along each reference ray of the second set and have to select the correct one. In this paper, we suggest the way of selecting the correct arrivals. We then demonstrate the accuracy of the resulting prevailing–frequency approximation of the coupling ray theory using elastic S waves along the SH and SV reference rays in four different approximately uniaxial (approximately transversely isotropic) velocity models.     

TI介质局部角度域射线追踪与叠前深度偏移成像

研究与实践表明,对于长偏移距、宽方位地震数据,忽略各向异性会明显降低成像质量,影响储层预测与描述的精度.针对典型的横向各向同性(TI)介质,本文面向深度域构造成像与偏移速度分析的需要,研究基于射线理论的局部角度域叠前深度偏移成像方法.它除了像传统Kirchhoff叠前深度偏移那样输出成像剖面和炮检距域的共成像点道集,还遵循地震波在成像点处的局部方向特征、基于扩展的脉冲响应叠加原理获得入射角度域和照明角度域的成像结果.为了方便快捷地实现TI介质射线走时与局部角度信息的计算,文中讨论和对比了两种改进的射线追踪方法:一种采用从经典各向异性介质射线方程演变而来的由相速度表征的简便形式;另一种采用由对称轴垂直的TI(即VTI)介质声学近似qP波波动方程推导出来的射线方程.文中通过坐标旋转将其扩展到了对称轴倾斜的TI(即TTI)介质.国际上通用的理论模型合成数据偏移试验表明,本文方法既适用于复杂构造成像,又可为TI介质深度域偏移速度分析与模型建立提供高效的偏移引擎.     

Efficient and accurate computation of seismic traveltimes and amplitudes

We describe two practicable approaches for an efficient computation of seismic traveltimes and amplitudes. The first approach is based on a combined finite‐difference solution of the eikonal equation and the transport equation (the ‘FD approach’). These equations are formulated as hyperbolic conservation laws; the eikonal equation is solved numerically by a third‐order ENO–Godunov scheme for the traveltimes whereas the transport equation is solved by a first‐order upwind scheme for the amplitudes. The schemes are implemented in 2D using polar coordinates. The results are first‐arrival traveltimes and the corresponding amplitudes. The second approach uses ray tracing (the ‘ray approach’) and employs a wavefront construction (WFC) method to calculate the traveltimes. Geometrical spreading factors are then computed from these traveltimes via the ray propagator without the need for dynamic ray tracing or numerical differentiation. With this procedure it is also possible to obtain multivalued traveltimes and the corresponding geometrical spreading factors. Both methods are compared using the Marmousi model. The results show that the FD eikonal traveltimes are highly accurate and perfectly match the WFC traveltimes. The resulting FD amplitudes are smooth and consistent with the geometrical spreading factors obtained from the ray approach. Hence, both approaches can be used for fast and reliable computation of seismic first‐arrival traveltimes and amplitudes in complex models. In addition, the capabilities of the ray approach for computing traveltimes and spreading factors of later arrivals are demonstrated with the help of the Shell benchmark model.     

对共反射面叠加的进一步探讨

共反射面元（CRS）叠加考虑了反射层的局部特征和第一菲涅耳带内的全部反射,更充分挖掘了多次覆盖数据的潜力.但在地下介质复杂并存在倾斜层时存在反射点分散的情况,从而影响了CRS叠加效果.本文从射线理论出发,在考虑反射层局部特征的情况下,推导了水平地表和起伏地表情况下计算真实反射点分散程度的公式,最终将反射点分散程度定量表达出来.通过对反射点分散程度的控制,从CRS道集中抽取出共反射点（CRP）道集,在CRP道集中而不是在CRS道集中实现叠加,其效果应比传统的CRS叠加效果要好.利用水平地表和起伏地表的模型验证了本文所推导的公式的正确性和有效性.该公式在实际资料处理中的运用尚待进一步研究.     

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.     

COMPUTATION OF ZERO-OFFSET VERTICAL SEISMIC PROFILES INCLUDING GEOMETRICAL SPREADING AND ABSORPTION*

Synthetic vertical seismic profiles (VSP) provide a useful tool in the interpretation of VSP data, allowing the interpreter to analyze the propagation of seismic waves in the different layers. A zero-offset VSP modeling program can also be used as part of an inversion program for estimating the parameters in a layered model of the subsurface. Proposed methods for computing synthetic VSP are mostly based on plane waves in a horizontally layered elastic or anelastic medium. In order to compare these synthetic VSP with real data a common method is to scale the data with the spherical spreading factor of the primary reflections. This will in most cases lead to artificial enhancement of multiple reflections. We apply the ray series method to the equations of motion for a linear viscoelastic medium after having done a Fourier transformation with respect to the time variable. This results in a complex eikonal equation which, in general, appears to be difficult to solve. For vertically traveling waves in a horizontally layered viscoelastic medium the solution is easily found to be the integral along the ray of the inverse of the complex propagation velocity. The spherical spreading due to a point source is also complex, and it is equal to the integral along the ray of the complex propagation velocity. Synthetic data examples illustrate the differences between spherical, cylindrical, and plane waves in elastic and viscoelastic layered media.     

A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

AN ANALYTICAL RAYPATH APPROACH TO THE REFRACTION WAVEFRONT METHOD1

Seismic refraction surveying is still an important tool for determining the geometries and elastic wave propagation velocities of near-surface layers. Many analytical and graphical methods have been developed over the years for refraction interpretation, and these can be classified into two basic groups. The first group visualizes critically refracted rays converging on a common surface position, while the second group, which includes the wavefront methods, makes use of the critical rays emerging from a common point on the refractor. The method described in this paper is an analytical approach to the wavefront methods. The reverse refracted ray received by a geophone is intersected by the forward refracted rays received by subsequent geophones and a common critical refraction point on the refractor is estimated after a series of comparisons. This process is repeated for each geophone to yield the geometry and the velocity of the refractor. Several interpolations are performed to achieve a better accuracy. Palmer's models are used to test the efficiency of the algorithm. The results are presented together with those of other methods applied to the same models.     

Derivatives of reflection point coordinates with respect to model parameters

The motivation for this paper is to provide expressions for first-order partial derivatives of reflection point coordinates, taken with respect to model parameters. Such derivatives are expected to be useful for processes dealing with the problem of estimating velocities for depth migration of seismic data.The subject of the paper is a particular aspect of ray perturbation theory, where observed parameters—two-way reflection time and horizontal components of slowness, are constraining the ray path when parameters of the reference velocity model are perturbed. The methodology described here is applicable to general rays in a 3D isotropic, heterogeneous medium. Each ray is divided into a shot ray and a receiver ray, i.e., the ray portions between the shot/receiver and the reflection point, respectively. Furthermore, by freezing the initial horizontal slowness of these subrays as the model is perturbed,elementary perturbation quantities may be obtained, comprising derivatives of ray hit positions within theisochrone tangent plane, as well as corresponding time derivatives. The elementary quantities may be estimated numerically, by use of ray perturbation theory, or in some cases, analytically. In particular, when the layer above the reflection point is homogeneous, explicit formulas can be derived. When the elementary quantities are known,reflection point derivatives can be obtained efficiently from a set of linear expressions.The method is applicable for a common shot, receiver or offset data sorting. For these gather types, reflection point perturbationlaterally with respect to the isochrone is essentially different. However, in theperpendicular direction, a first-order perturbation is shown to beindependent of gather type.To evaluate the theory, reflection point derivatives were estimated analytically and numerically. I also compared first-order approximations to true reflection point curves, obtained by retracing rays for a number of model perturbations. The results are promising, especially with respect to applications in sensitivity analysis for prestack depth migration and in velocity model updating.     

Migration to multiple offset and velocity analysis1

The stacking velocity best characterizes the normal moveout curves in a common-mid-point gather, while the migration velocity characterizes the diffraction curves in a zero-offset section as well as in a common-midpoint gather. For horizontally layered media, the two velocity types coincide due to the conformance of the normal and the image ray. In the case of dipping subsurface structures, stacking velocities depend on the dip of the reflector and relate to normal rays, but with a dip-dependent lateral smear of the reflection point. After dip-moveout correction, the stacking velocities are reduced while the reflection-point smear vanishes, focusing the rays on the common reflection points. For homogeneous media the dip-moveout correction is independent of the actual velocity and can be applied as a dip-moveout correction to multiple offset before velocity analysis. Migration to multiple offset is a prestack, time-migration technique, which presents data sets which mimic high-fold, bin-centre adjusted, common-midpoint gathers. This method is independent of velocity and can migrate any 2D or 3D data set with arbitrary acquisition geometry. The gathers generated can be analysed for normal-moveout velocities using traditional methods such as the interpretation of multivelocity-function stacks. These stacks, however, are equivalent to multi-velocity-function time migrations and the derived velocities are migration velocities.     

Three-dimensional Travel-time Calculation Based on Fermat's Principle

— We present a new travel-time calculation method based on Fermat's principle. In the method, travel times are recursively calculated on horizontal planes of increasing depth. For typical configurations of exploration geophysics, the distance between the planes is on the order of 100 m. The travel times on the first plane are calculated by connecting straight ray segments from the source point to the grid points on the plane and integrating the slowness along each segment. The times on the other planes are calculated by finding the minimum of the combination of the times on the plane above plus the additional time along segments connecting grid points on the two planes. The travel-time calculation method is designed for calculating either the first arrival times, or the time of the shortest travel path arrival. The method is extended to handle vertically transverse isotropic (VTI) media by an approach which increases the computing time only slightly. The algorithm is tested against synthetic examples for isotropic and VTI wave propagation.     

CGP叠加柱面波偏移成像方法

针对松辽盆地薄砂层油藏的地震勘探问题,提出了一种共检波器接收点(CGP)叠加柱面波偏移成像方法。该方法对小炮点距、小检波器点距、中间放炮观测系统采集地震资料,经CGP道集叠加组合成柱面波剖面;采用下行波射线法向下延拓和上行波波动方程向下延拓的方法,使柱面波剖面偏移成像。通过模型分析和对松辽盆地TK8157测线资料进行处理证明该方法的地震分辨率和保真度较高,在发现小砂体、小断层、地层尖灭等方面有较好的效果。     

A NON-GEOMETRICAL SH-ARRIVAL1

The existence of‘*-waves’has, in recent years, prompted a renewed interest in these non-geometrical arrivals, which are generated by point sources located adajcent to plane interfaces. It has led to the re-evaluation of seismic data aquisition techniques and to the question of how to use this real phenomena in enhancing existing seismic interpretation methods. This paper considers a non-geometrical SH-arrival which is generated by a point torque source unrealistically buried within a half-space. The method of solution is essentially the same as presented in an earlier paper, with the modification that the limitation placed on the distance of the source from the interface has been removed in the saddle point method used to obtain a high-frequency approximate solution. In the earlier paper, a preliminary assumption forced the saddle point, which corresponded to the *-wave arrival, to be real when it is generally complex. However, for offsets removed from the distinct ray, the imaginary part of this complex quantity is negligible. A problem which arose when comparing exact synthetic traces with those obtained using zero-order saddle point methods, was the inability to match either the amplitude or phase of the geometrical arrival in the range of offsets when the *-wave and this corresponding geometrical ray were well separated. For this range of offsets the geometrical arrival was approaching grazing incidence and another term in the saddle point expansion of the integral was necessary to rectify this error. This method is also being used to validate the results for higher order terms obtained using asymptotic ray theory. Analytical formulae are given for both the *-wave and the higher order expansion of the geometrical event, together with a comparison of synthetic seismograms using the method developed here and a numerical integration algorithm.     

Quadratic normal moveouts of symmetric reflections in elastic media: A quick tutorial

The design of reflection traveltime approximations for optimal stacking and inversion has always been a subject of much interest in seismic processing. A most prominent role is played by quadratic normal moveouts, namely reflection traveltimes around zero-offset computed as second-order Taylor expansions in midpoint and offset coordinates. Quadratic normal moveouts are best employed to model symmetric reflections, for which the ray code in the downgoing direction coincides with the ray code in the upgoing direction in reverse order. Besides pure (non-converted) primaries, many multiply reflected and converted waves give rise to symmetric reflections. We show that the quadratic normal moveout of a symmetric reflection admits a natural decomposition into a midpoint term and an offset term. These, in turn, can be be formulated as the traveltimes of the one-way normal (N) and normal-incidence-point (NIP) waves, respectively. With the help of this decomposition, which is valid for propagation in isotropic and anisotropic elastic media, we are able to derive, in a simple and didactic way, a unified expression for the quadratic normal moveout of a symmetric reflection in its most general form in 3D. The obtained expression allows for a direct interpretation of its various terms and fully encompasses the effects of velocity gradients and Earth surface topography.