Simultaneous Inversion for Velocity and Reflector Geometry Using Multi-phase Fresnel Volume Rays

Traditional ray tomography methods based on the high frequency assumption are sometimes unable to obtain a high resolution tomographic picture due to a deficient coverage of ray paths in real applications, especially for low velocity anomalous regions. In contrast, finite-frequency ray theory is more suitable for handling real seismic propagation problems because the travel time depends not only on the velocity distribution along a central ray (or traditional geometric ray), but also on the velocity values within a region (referred to as the first Fresnel Volume) which incorporates the central ray. In this study, we develop an algorithm to calculate multi-phase Fresnel Volume finite-frequency rays, and then present an inversion method to simultaneous invert for both velocity and reflector geometry by using these multi-phase Fresnel Volume finite-frequency rays. Using synthetic data examples, we compare the reconstructions of the velocity field and the reflector orientation using the Fresnel Volume ray tomographic methods and the traditional ray tomography approach. Results show that the former is advantageous over the latter, especially when the ray density is relatively low. An additional benefit of the Fresnel Volume finite-frequency ray tomographic method is that it can start with a low frequency to capture the coarse velocity structure, thereby mitigating the local minimum trapping problem, and then be tuned to a high frequency for delineating the fine velocity structure.     

变阻尼约束层析成像及其在VSP资料中的应用(英文)

初至波走时层析成像已经取得了广泛的应用,然而,由于观测系统的限制,射线在模型中分布不均匀,导致层析结果的分辨能力不足。变阻尼约束方法应用不均匀的先验信息来匹配不均匀的数据分布,可以减小速度模型校正量与射线覆盖程度的相关性。本文将变阻尼约束方法应用于初至波旅行时层析成像中,并将平滑约束方法加入正则化方程组中来避免单独使用变阻尼约束带来的不稳定性,利用阿尔法滤波器对反演中间迭代结果进行平滑和去噪,采用LSQR算法求解线性方程组来提高收敛速度和压制误差传递。本文应用上述层析成像算法对VSP观测系统进行速度反演,分别应用于检测板速度模型数据和实际VSP资料速度反演中,结果表咀变阻尼约束层析成像可以改善射线不均匀覆盖带来的影响,从而提高速度反演结果的质量;VSP资料检波点附近的速度反演结果可靠性高。     

煤矿巷间地震层析成像技术探测煤体构造

本文对煤矿井下巷间地震波透射观测资料在平滑弯曲射线的基础上，利用联合迭代法（ＳＩＲＴ）进行煤体波速场的反演成像，并结合数值计算结果给以验证。其结果表明：采用平滑弯曲射线追中学地震波射线可以大大提高成像质量；地震层析成像技术用于探测煤体中的构造是一种有效且可行的方法。     

基于交叉梯度交替结构约束的二维地震走时与全通道直流电阻率联合反演

在利用不同的地球物理勘探方法对地下复杂介质成像时,因观测系统的非完备性及数据本身对某些岩石物性的不敏感性,单独成像的结果存在较大的不确定性和不一致性.对于地震体波走时成像与直流电阻率成像,均面临着成像阴影区问题.对于地震走时成像,地震射线对低速区域覆盖较差形成阴影区,造成低速区域分辨率降低.对于电阻率成像,电场线在高阻区域分布较少,造成高阻区域分辨率较低.为了提高地下介质成像的精度,Gallado和Meju(2003)提出了基于交叉梯度结构约束的联合地球物理成像方法.在要求不同的物性模型拟合各自对应的数据同时,模型之间的结构要求一致,即交叉梯度趋于零.为了更有效地实现基于交叉梯度的结构约束,我们提出了一种新的交替结构约束的联合反演流程,即交替反演不同的数据而且在反演一种数据时要求对应的模型与另一个模型结构一致.新的算法能够更容易地把单独的反演系统耦合在一起,而且也更容易建立结构约束和数据拟合之间的平衡.基于新的联合反演流程,我们测试了基于交叉梯度结构约束的二维跨孔地震走时和直流电阻率联合成像.合成数据测试表明,我们提出的交替结构约束流程能够很好地实现基于交叉梯度结构约束的联合成像.与单独成像结果相比,地震走时和全通道电阻率联合成像更可靠地确定了速度和电阻率异常.     

Estimation of geological dip and curvature from time‐migrated zero‐offset reflections in heterogeneous anisotropic media

Starting from a given time‐migrated zero‐offset data volume and time‐migration velocity, recent literature has shown that it is possible to simultaneously trace image rays in depth and reconstruct the depth‐velocity model along them. This, in turn, allows image‐ray migration, namely to map time‐migrated reflections into depth by tracing the image ray until half of the reflection time is consumed. As known since the 1980s, image‐ray migration can be made more complete if, besides reflection time, also estimates of its first and second derivatives with respect to the time‐migration datum coordinates are available. Such information provides, in addition to the location and dip of the reflectors in depth, also an estimation of their curvature. The expressions explicitly relate geological dip and curvature to first and second derivatives of reflection time with respect to time‐migration datum coordinates. Such quantitative relationships can provide useful constraints for improved construction of reflectors at depth in the presence of uncertainty. Furthermore, the results of image‐ray migration can be used to verify and improve time‐migration algorithms and can therefore be considered complementary to those of normal‐ray migration. So far, image‐ray migration algorithms have been restricted to layered models with isotropic smooth velocities within the layers. Using the methodology of surface‐to‐surface paraxial matrices, we obtain a natural extension to smooth or layered anisotropic media.     

Offset continuation (OCO) ray tracing using OCO trajectories

Offset continuation (OCO) is a seismic configuration transform designed to simulate a seismic section as if obtained with a certain source-receiver offset using the data measured with another offset. Since OCO is dependent on the velocity model used in the process, comparison of the simulated section to an acquired section allows for the extraction of velocity information. An algorithm for such a horizon-oriented velocity analysis is based on so-called OCO rays. These OCO rays describe the output point of an OCO as a function of the Root Mean Square (RMS) velocity. The intersection point of an OCO ray with the picked traveltime curve in the acquired data corresponding to the output half-offset defines the RMS velocity at that position. We theoretically relate the OCO rays to the kinematic properties of OCO image waves that describe the continuous transformation of the common-offset reflection event from one offset to another. By applying the method of characteristics to the OCO image-wave equation, we obtain a raytracing-like procedure that allows to construct OCO trajectories describing the position of the OCO output point under varying offset. The endpoints of these OCO trajectories for a single input point and different values of the RMS velocity form then the OCO rays. A numerical example demonstrates that the developed ray-tracing procedure leads to reliable OCO rays, which in turn provide high-quality RMS velocities. The proposed procedure can be carried out fully automatically, while conventional velocity analysis needs human intervention. Moreover, since velocities are extracted using offset sections, more redundancy is available or, alternatively, OCO velocities can be studied as a function of offset.     

True amplitude elastic Gaussian beam imaging of multicomponent walkaway vertical seismic profiling data

We develop the true‐amplitude prestack migration of multicomponent data based on the use of elastic Gaussian beams for walkaway vertical seismic profile (VSP) acquisition systems. It consists in a weighted summation of multishot data with specific weights, computed by tracing elastic Gaussian beams from each imaging point of the target area towards the sources and receivers. Each pair of beams may be connected with either a pair of P‐rays (PP‐image) or the P‐ray towards sources and the S‐ray to receivers (PS‐image) and is uniquely determined by dip (the angle of the bisector between the rays and the vertical direction) and opening (the angle between the rays) angles. Shooting from the bottom towards the acquisition system helps to avoid well‐known troubles, in particular multipathing for the imaging conditions in complex velocity models. The ability to fix the dip angle and implement summation over opening angles leads to the so‐called selective images that contain mostly interfaces with desired slopes. On the other hand, a set of images computed for a range of opening angles by summation over all available dip angles is used as input of an AVO‐like inversion procedure for the recovery of elastic parameters. The feasibility of this imaging procedure is verified by synthetic data for 2D realistic elastic models.     

First-Order Perturbation Theory for Seismic Isochrons

A first-order perturbation theory for seismic isochrons is presented in a model independent form. Two ray concepts are fundamental in this theory, the isochron ray and the velocity ray, for which I obtain first-order approximations to position vectors and slowness vectors. Furthermore, isochron points are connected to a shot and receiver by conventional ray fields. Based on independent perturbation of the shot and receiver ray I obtain first-order approximations to velocity rays. The theory is applicable for 3D inhomogeneous anisotropic media, given that the shot and receiver rays, as well as their perturbations, can be generated with such model generality. The theory has applications in sensitivity analysis of prestack depth migration and in velocity model updating. Numerical examples of isochron and velocity rays are shown for a 2D homogeneous VTI model. The general impression is that the first-order approximation is, with some exceptions, sufficiently accurate for practical applications using an anisotropic velocity model.     

Tomography and the Herglotz-Wiechert inverse formulation

In this paper, linearized tomography and the Herglotz-Wiechert inverse formulation are compared. Tomographic inversions for 2-D or 3-D velocity structure use line integrals along rays and can be written in terms of Radon transforms. For radially concentric structures, Radon transforms are shown to reduce to Abel transforms. Therefore, for straight ray paths, the Abel transform of travel-time is a tomographic algorithm specialized to a one-dimensional radially concentric medium. The Herglotz-Wiechert formulation uses seismic travel-time data to invert for one-dimensional earth structure and is derived using exact ray trajectories by applying an Abel transform. This is of historical interest since it would imply that a specialized tomographic-like algorithm has been used in seismology since the early part of the century (seeHerglotz, 1907;Wiechert, 1910). Numerical examples are performed comparing the Herglotz-Wiechert algorithm and linearized tomography along straight rays. Since the Herglotz-Wiechert algorithm is applicable under specific conditions, (the absence of low velocity zones) to non-straight ray paths, the association with tomography may prove to be useful in assessing the uniqueness of tomographic results generalized to curved ray geometries.     

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.     

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.     

Finite-frequency tomography in a crustal environment: Application to the western part of the Gulf of Corinth

In this paper we investigate finite-frequency effects in crustal tomography. We developed an inversion procedure based on an exact numerical computation of the sensitivity kernels. In this approach we compute the 3D travel-time sensitivity kernels by using (1) graph theory and an additional bending to estimate accurately both rays and travel-times between source/receiver and diffraction points and (2) paraxial ray theory to estimate the amplitude along theses rays. We invert both the velocity and the hypocentre parameters, using these so-called banana-doughnut kernels and the LSQR iterative solver. We compare the ray-theoretical and the finite-frequency tomography to image the intermediate structures beneath the Gulf of Corinth (Greece), which has long been recognized as the most active continental rifting zone in the Mediterranean region. Our dataset consists of 451 local events with 9233 P- first-arrival times recorded in the western part of the Gulf (Aigion area) in the framework of the 3F-Corinth European project. Previous tomographic images showed a complex velocity crustal model and a low-dip surface that may accommodate the deformation. Accurate velocity models will help to better constrain the rifting process, which is still a subject of debate. The main results of this study show that finite-frequency tomography improves crustal tomographic images by providing better resolved images of the 3D complicated velocity structure. Because the kernels spread the information over a volume, finite-frequency tomography results in a sharpening of layer boundaries as we observed for the shallower part of the crust (down to 5 km depth) beneath the Gulf of Corinth.     

Comparison of the anisotropic-ray-theory rays and anisotropic common S-wave rays with the SH and SV reference rays in a velocity model with a split intersection singularity

We describe the behaviour of the anisotropic–ray–theory S–wave rays in a velocity model with a split intersection singularity. The anisotropic–ray–theory S–wave rays crossing the split intersection singularity are smoothly but very sharply bent. While the initial–value rays can be safely traced by solving Hamilton’s equations of rays, it is often impossible to determine the coefficients of the equations of geodesic deviation (paraxial ray equations, dynamic ray tracing equations) and to solve them numerically. As a result, we often know neither the matrix of geometrical spreading, nor the phase shift due to caustics. We demonstrate the abrupt changes of the geometrical spreading and wavefront curvature of the fast anisotropic–ray–theory S wave. We also demonstrate the formation of caustics and wavefront triplication of the slow anisotropic–ray–theory S wave.Since the actual S waves propagate approximately along the SH and SV reference rays in this velocity model, we compare the anisotropic–ray–theory S–wave rays with the SH and SV reference rays. Since the coupling ray theory is usually calculated along the anisotropic common S–wave rays, we also compare the anisotropic common S–wave rays with the SH and SV reference rays.     

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.     

Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous,Anisotropic and Anelastic Media

Diffraction and anelasticity problems involving decaying, “evanescent” or “inhomogeneous” waves can be studied and modelled using the notion of “complex rays”. The wavefront or “eikonal” equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates.In perfectly elastic media complex rays are associated with, for example, evanescent waves in the shadow of a caustic. More generally, caustics in anelastic media may lie just outside the real coordinate subspace and one must trace complex rays around the complex caustic in order to obtain accurate waveforms nearby or the turning waves at greater distances into the lit region. The complex extension of the Maslov method for computing such waveforms is described. It uses the complex extension of the Legendre transformation and the extra freedom of complex rays makes pseudocaustics avoidable. There is no need to introduce a Maslov/KMAH index to account for caustics in the geometrical ray approximation, the complex amplitude being generally continuous. Other singular ray problems, such as the strong coupling around acoustic axes in anisotropic media, may also be addressed using complex rays.Complex rays are insightful and practical for simple models (e.g. homogeneous layers). For more complicated numerical work, though, it would be desirable to confine attention to real position coordinates. Furthermore, anelasticity implies dispersion so that complex rays are generally frequency dependent. The concept of group velocity as the velocity of a spatial or temporal maximum of a narrow-band wave packet does lead to real ray/Hamilton equations. However, envelope-maximum tracking does not itself yield enough information to compute synthetic seismogramsFor anelasticity which is weak in certain precise senses, one can set up a theory of real, dispersive wave-packet tracking suitable for synthetic seismogram calculations in linearly visco-elastic media. The seismologically-accepiable constant-Q rheology of Liu et al. (1976), for example, satisfies the requirements of this wave-packet theory, which is adapted from electromagnetics and presented as a reasonable physical and mathematical basis for ray modelling in inhomogeneous, anisotropic, anelastic media. Dispersion means that one may need to do more work than for elastic media. However, one can envisage perturbation analyses based on the ray theory presented here, as well as extensions like Maslov's which are based on the Hamiltonian properties.     

Ray tracing and geodesic deviation of the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately uniaxial

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. If we know that a medium is close to uniaxial (transversely isotropic), it may be advantageous to trace reference rays which resemble the SH–wave and SV–wave rays. This paper is devoted to defining and tracing these SH and SV reference rays of elastic S waves in a heterogeneous generally anisotropic medium which is approximately uniaxial (approximately transversely isotropic), and to the corresponding equations of geodesic deviation (dynamic ray tracing). All presented equations are simultaneously applicable to ordinary and extraordinary reference rays of electromagnetic waves in a generally bianisotropic medium which is approximately uniaxially anisotropic. The improvement of the coupling–ray–theory seismograms calculated along the proposed SH and SV reference rays, compared to the coupling–ray–theory seismograms calculated along the anisotropic common reference rays, has already been numerically demonstrated by the authors in four approximately uniaxial velocity models.     

多震相菲涅耳体射线走时同时反演成像

高频假设下的地震射线理论以及相应的地震成像理论表明,在射线稀疏条件下,不可能得到较高分辨率的构造成像;而有限频射线理论更符合实际地震的传播规律,即地震波的走时不仅与中心射线(传统的几何射线)上的速度分布有关,而且与中心射线附近一定范围(称其为第一菲涅耳体)内的速度异常分布有关.鉴于此,本文提出了计算多震相地震波菲涅耳体有限频射线的方法,并定义了走时敏感核函数,同时给出了利用多震相菲涅耳体有限频射线进行速度模型和反射界面同时反演成像的公式.利用多震相走时资料,使用传统射线层析成像方法与有限频射线层析成像方法进行了速度和界面的同时反演成像.结果表明,当射线密度较小时,无论是对速度模型的重建还是对反射界面几何形状的更新,有限频射线层析成像方法均优于传统射线层析成像方法, 而变频有限频射线层析成像则是实际地震层析成像的首选反演算法.      

3D multivalued travel time and amplitude maps

An algorithm for computing multivalued maps for travel time, amplitude and any other ray related variable in 3D smooth velocity models is presented. It is based on the construction of successive isochrons by tracing a uniformly dense discrete set of rays by fixed travel-time steps. Ray tracing is based on Hamiltonian formulation and includes computation of paraxial matrices. A ray density criterion ensures uniform ray density along isochrons over the entire ray field including caustics. Applications to complex models are shown.     

Seismic ray path variations in a 3D global velocity model

A three-dimensional (3D) ray tracing technique is used to investigate ray path variations of P, PcP, pP and PP phases in a global tomographic model with P wave velocity changing in three dimensions and with lateral depth variations of the Moho, 410 and 660 km discontinuities. The results show that ray paths in the 3D velocity model deviate considerably from those in the average 1D model. For a PcP wave in Western Pacific to East Asia where the high-velocity (1-2%) Pacific slab is subducting beneath the Eurasian continent, the ray path change amounts to 27 km. For a PcP ray in South Pacific where very slow (−2%) velocity anomalies (the Pacific superplume) exist in the whole mantle, the maximum ray path deviation amounts to 77 km. Ray paths of other phases (P, pP, PP) are also displaced by tens of kilometers. Changes in travel time are as large as 3.9 s. These results suggest that although the maximal velocity anomalies of the global tomographic model are only 1-2%, rays passing through regions with strong lateral heterogeneity (in velocity and/or discontinuity topography) can have significant deviations from those in a 1D model because rays have very long trajectories in the global case. If the blocks or grid nodes adopted for inversion are relatively large (3-5°) and only a low-resolution 3D model is estimated, 1D ray tracing may be feasible. But if fine blocks or grid nodes are used to determine a high-resolution model, 3D ray tracing becomes necessary and important for the global tomography.     

ANISOTROPIC TRAVELTIME TOMOGRAPHY

Velocity estimation technique using seismic data is often based on time/distance equations which are independent of direction, and even though we now know that many rocks are quite anisotropic, useful results have been obtained over the years from these isotropic estimates. Nevertheless, if velocities are significantly direction-dependent, then the isotropic assumption may lead to serious structural interpretation errors. Additionally, information on angle-dependence may lead to a better understanding of the lithology of the rocks under measurement. VSP and cross-well data may each lack the necessary aperture to estimate more than two velocity parameters for each wave type, and if the data straddle a symmetry axis, then these may be usefully chosen to be the direct velocities (from time-and-distance measurements along the axis) and NMO velocities (from differential time-offset measurements). These sets of two parameters define ellipses, and provide intermediate models for the variation of velocity with angle which can later be assembled and translated into estimates of the elastic moduli of the rocks under scrutiny. If the aperture of the measurements is large enough e.g. we have access to both VSP and cross-well data, we divide the procedure into two independent steps, first fitting best ellipses around one symmetry axis and then fitting another set around the orthogonal axis. These sets of four elliptical parameters are then combined into a new, double elliptical approximation. This approximation keeps the useful properties of elliptical anisotropy, in particular the simple relation between group and phase velocities which simplifies the route from the traveltimes measurements to the elastic constants of the medium. The inversion proposed in this paper is a simple extension of well-known isotropic schemes and it is conceptually identical for all wave types. Examples are shown to illustrate the application of the technique to cross-well synthetic and field P-wave data. The examples demonstrate three important points: (a) When velocity anisotropy is estimated by iterative techniques such as conjugate gradients, early termination of the iterations may produce artificial anisotropy. (b) Different components of the velocity are subject to different type of artifacts because of differences in ray coverage, (c) Even though most rocks do not have elliptical dispersion relations, our elliptical schemes represent a useful intermediate step in the full characterization of the elastic properties.