VTI介质P波非双曲时差分析

具有垂直对称轴的横向各向同性介质模型(VTI)是目前各向异性理论研究和多波多分量地震资料叠前成像处理中最常用的一种各向异性模型.VTI介质中反射 P波时距曲线一般不再是双曲线.基于不同的相速度近似公式会得到不同的时距关系式.文中对几种典型的非双曲时距曲线与射线追踪得到的准确时距曲线在不同各向异性强度下进行了对比，结果表明Muir等和Stovas等提出的非双曲时距公式由于过高地考虑了横波垂直速度的影响与精确的时距曲线有很大偏差；Tsvankin等提出的弱各向异性非双曲时距公式在ε-δ<0时误差增大；Alkhalifah等提出的非双曲时距公式在大炮检距任意各向异性强度下都具有较高的精度，适于在实际资料处理中应用.     

Traveltime approximations for q‐P waves in vertical transversely isotropy media*

As exploration targets have become deeper, cable lengths have increased accordingly, making the conventional two term hyperbolic traveltime approximation produce increasingly erroneous traveltimes. To overcome this problem, many traveltime formulas were proposed in the literature that provide approximations of different quality. In this paper, we concentrate on simple traveltime approximations that depend on a single anisotropy parameter. We give an overview of a collection of such traveltime approximations found in the literature and compare their quality. Moreover, we propose some new single‐parameter traveltime approximations based on the approximations found in the literature. The main advantage of our approximations is that some of them are rather simple analytic expressions that make them easy to use, while achieving the same quality as the better of the established formulas.     

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.     

Perturbation‐based moveout approximations in anisotropic media

The moveout approximations play an important role in seismic data processing. The standard hyperbolic moveout approximation is based on an elliptical background model with two velocities: vertical and normal moveout. We propose a new set of moveout approximations based on a perturbation series in terms of anellipticity parameters using the alternative elliptical background model defined by vertical and horizontal velocities. We start with a transversely isotropic medium with a vertical symmetry axis. Then, we extend this approach to a homogeneous orthorhombic medium. To define the perturbation coefficients for a new background, we solve the eikonal equation with horizontal velocities in transversely isotropic medium with a vertical symmetry axis and orthorhombic media. To stabilise the perturbation series and improve the accuracy, the Shanks transform is applied for all the cases. We select different parameterisations for both velocities and anellipticity parameters for an orthorhombic model. From the comparison in traveltime error, the new moveout approximations result in better accuracy comparing with the standard perturbation‐based methods and other approximations.     

Traveltime approximation in vertical transversely isotropic layered media

The well‐known asymptotic fractional four‐parameter traveltime approximation and the five‐parameter generalised traveltime approximation in stratified multi‐layer transversely isotropic elastic media with a vertical axis of symmetry have been widely used for pure‐mode and converted waves. The first three parameters of these traveltime expansions are zero‐offset traveltime, normal moveout velocity, and quartic coefficient, ensuring high accuracy of traveltimes at short offsets. The additional parameter within the four‐parameter approximation is an effective horizontal velocity accounting for large offsets, which is important to avoid traveltime divergence at large offsets. The two additional parameters in the above‐mentioned five‐parameter approximation ensure higher accuracy up to a given large finite offset with an exact match at this offset. In this paper, we propose two alternative five‐parameter traveltime approximations, which can be considered extensions of the four‐parameter approximation and an alternative to the five‐parameter approximation previously mentioned. The first three short‐offset parameters are the same as before, but the two additional long‐offset parameters are different and have specific physical meaning. One of them describes the propagation in the high‐velocity layer of the overburden (nearly horizontal propagation in the case of very large offsets), and the other characterises the intercept time corresponding to the critical slowness that includes contributions of the lower velocity layers only. Unlike the above‐mentioned approximations, both of the proposed traveltime approximations converge to the theoretical (asymptotic) linear traveltime at the limit case of very large (“infinite”) offsets. Their accuracy for moderate to very large offsets, for quasi‐compressional waves, converted waves, and shear waves polarised in the horizontal plane, is extremely high in cases where the overburden model contains at least one layer with a dominant higher velocity compared with the other layers. We consider the implementation of the proposed traveltime approximations in all classes of problems in which the above‐mentioned approximations are used, such as reflection and diffraction analysis and imaging.     

Traveltime parameters in a tilted elliptical anisotropic medium‡

In this paper, we derive analytical expressions for one‐way and two‐way kinematical parameters in elliptical tilted transverse isotropy media. We show that the homogeneous elliptical tilted transverse isotropy models result in hyperbolic moveout with a reflection point sideslip x₀, which can be considered as an additional traveltime parameter for one‐way wave propagation. For homogeneous elliptical tilted transverse isotropy models we show that the inversion of one‐way traveltime parameters suffers from the ambiguity for large tilts. It is shown that the accuracy of the inversion is sensitive to the error in x₀. We also derive and invert the traveltime parameters for a vertically heterogeneous elliptical tilted transverse isotropy model with a tilt gradually changing with depth. The a priori knowledge of parameter δ is very important for inversion. The wrong choise of this parameter results in significant errors in inverted model parameters.     

First‐arrival traveltime tomography with modified total‐variation regularization

First‐arrival traveltime tomography is a robust tool for near‐surface velocity estimation. A common approach to stabilizing the ill‐posed inverse problem is to apply Tikhonov regularization to the inversion. However, the Tikhonov regularization method recovers smooth local structures while blurring the sharp features in the model solution. We present a first‐arrival traveltime tomography method with modified total‐variation regularization to preserve sharp velocity contrasts and improve the accuracy of velocity inversion. To solve the minimization problem of the new traveltime tomography method, we decouple the original optimization problem into the two following subproblems: a standard traveltime tomography problem with the traditional Tikhonov regularization and a L₂ total‐variation problem. We apply the conjugate gradient method and split‐Bregman iterative method to solve these two subproblems, respectively. Our synthetic examples show that the new method produces higher resolution models than the conventional traveltime tomography with Tikhonov regularization, and creates less artefacts than the total variation regularization method for the models with sharp interfaces. For the field data, pre‐stack time migration sections show that the modified total‐variation traveltime tomography produces a near‐surface velocity model, which makes statics corrections more accurate.     

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.     

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.     

Multipathing in three‐parameter common‐image gathers from reverse‐time migration

Reverse‐time migration gives high‐quality, complete images by using full‐wave extrapolations. It is thus not subject to important limitations of other migrations that are based on high‐frequency or one‐way approximations. The cross‐correlation imaging condition in two‐dimensional pre‐stack reverse‐time migration of common‐source data explicitly sums the product of the (forward‐propagating) source and (backward‐propagating) receiver wavefields over all image times. The primary contribution at any image point travels a minimum‐time path that has only one (specular) reflection, and it usually corresponds to a local maximum amplitude. All other contributions at the same image point are various types of multipaths, including prismatic multi‐arrivals, free‐surface and internal multiples, converted waves, and all crosstalk noise, which are imaged at later times, and potentially create migration artefacts. A solution that facilitates inclusion of correctly imaged, non‐primary arrivals and removal of the related artefacts, is to save the depth versus incident angle slice at each image time (rather than automatically summing them). This results in a three‐parameter (incident angle, depth, and image time) common‐image volume that integrates, into a single unified representation, attributes that were previously computed by separate processes. The volume can be post‐processed by selecting any desired combination of primary and/or multipath data before stacking over image time. Separate images (with or without artifacts) and various projections can then be produced without having to remigrate the data, providing an efficient tool for optimization of migration images. A numerical example for a simple model shows how primary and prismatic multipath contributions merge into a single incident angle versus image time trajectory. A second example, using synthetic data from the Sigsbee2 model, shows that the contributions to subsalt images of primary and multipath (in this case, turning wave) reflections are different. The primary reflections contain most of the information in regions away from the salt, but both primary and multipath data contribute in the subsalt region.     

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.     

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.     

Generalized anisotropy parameters and approximations of attenuations and velocities in viscoelastic media of arbitrary anisotropy type – theoretical and experimental aspects*

Seismic anisotropy in geological media is now widely accepted. Parametrizations and explicit approximations for the velocities in such media, considered as purely elastic and moderately anisotropic, are now standards and have even been extended to arbitrary types of anisotropy. In the case of attenuating media, some authors have also recently published different parametrizations and velocity and attenuation approximations in viscoelastic anisotropic media of particular symmetry type (e.g., transversely isotropic or orthorhombic). This paper extends such work to media of arbitrary anisotropy type, that is to say to triclinic media. In the case of homogeneous waves and using the so‐called ‘correspondence principle’, it is shown that the viscoelastic equations (for the phase velocities, phase slownesses, moduli, wavenumbers, etc.) are formally identical to the corresponding purely elastic equations available in the literature provided that all the corresponding quantities are complex (except the unit vector in the propagation direction that remains real). In contrast to previous work, the new parametrization uses complex anisotropy parameters and constitutes a simple extension to viscoelastic media of previous work dealing with non‐attenuating elastic media of arbitrary anisotropy type. We make the link between these new complex anisotropy parameters and measurable parameters, as well as with previously published anisotropy parameters, demonstrating the usefulness of the new parametrization. We compute the explicit complete directional dependence of the exact and of the approximate (first and higher‐order perturbation) complex phase velocities of the three body waves (qP, qS1 and qS2). The exact equations are successfully compared with the ultrasonic phase velocities and phase attenuations of the three body waves measured in a strongly attenuating water‐saturated sample of Vosges sandstone exhibiting moderate velocity anisotropy but very strong attenuation anisotropy. The approximate formulas are checked on experimental data. Compared to the exact solutions, the errors observed on the first‐order approximate velocities are small (<1%) for qP‐waves and moderate (<10%) for qS‐waves. The corresponding errors on the quality factor Q are moderate (<6%) for qP‐waves but critically large (up to 160%) for the qS‐waves. The use of higher‐order approximations substantially improves the accuracy, for instance typical maximum relative errors do not exceed 0.06% on all the velocities and 0.6% on all the quality factors Q, for third‐order approximations. All the results obtained on other rock samples confirm the results obtained on this rock. The simplicity of the derivations and the generality of the results are striking and particularly convenient for practical applications.     

PARABOLIC AND HYPERBOLIC PARAXIAL TWO-POINT TRAVELTIMES IN 3D MEDIA1

The 4 × 4 T -propagator matrix of a 3D central ray determines, among other important seismic quantities, second-order (parabolic or hyperbolic) two-point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two-component vector. Here, we provide parabolic and hyperbolic paraxial two-point traveltime approximations using the T -propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three-component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial-ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.     

Range imaging: a new method for high‐resolution topographic measurements in small‐ and medium‐scale field sites

Topographic measurements are essential for the study of earth surface processes. Three‐dimensional data have been conventionally obtained through terrestrial laser scanning or photogrammetric methods. However, particularly in steep and rough terrain, high‐resolution field measurements remain challenging and often require new creative approaches. In this paper, range imaging is evaluated as an alternative method for obtaining surface data in such complex environments. Range imaging is an emerging time‐of‐flight technology, using phase shift measurements on a multi‐pixel sensor to generate a distance image of a surface. Its suitability for field measurements has yet not been tested. We found ambient light and surface reflectivity to be the main factors affecting error in distance measurements. Low‐reflectivity surfaces and strong illumination contrasts under direct exposure to sunlight lead to noisy distance measurements. However, regardless of lighting conditions, the accuracy of range imaging was markedly improved by averaging multiple images of the same scene. For medium ambient lighting (shade) and a light‐coloured surface the measurement uncertainty was approximately 9 mm. To further test the suitability of range imaging for field applications we measured a reach of a steep mountain stream with a horizontal resolution of approximately 1 cm (in the focal plane of the camera), allowing for the interpolation of a digital elevation model on a 2 cm grid. Comparison with an elevation model obtained from terrestrial laser scanning for the same site revealed that both models show similar degrees of topographic detail. Despite limitations in measurement range and accuracy, particularly at bright ambient lighting, range imaging offers three‐dimensional data in real time and video mode without the need of post‐processing. Therefore, range imaging is a useful complement or alternative to existing methods for high‐resolution measurements in small‐ to medium‐scale field sites. Copyright © 2012 John Wiley & Sons, Ltd.     

Anisotropy signature in reverse‐time migration extended images

Reverse‐time migration can accurately image complex geologic structures in anisotropic media. Extended images at selected locations in the Earth, i.e., at common‐image‐point gathers, carry rich information to characterize the angle‐dependent illumination and to provide measurements for migration velocity analysis. However, characterizing the anisotropy influence on such extended images is a challenge. Extended common‐image‐point gathers are cheap to evaluate since they sample the image at sparse locations indicated by the presence of strong reflectors. Such gathers are also sensitive to velocity error that manifests itself through moveout as a function of space and time lags. Furthermore, inaccurate anisotropy leaves a distinctive signature in common‐image‐point gathers, which can be used to evaluate anisotropy through techniques similar to the ones used in conventional wavefield tomography. It specifically admits a V‐shaped residual moveout with the slope of the “V” flanks depending on the anisotropic parameter η regardless of the complexity of the velocity model. It reflects the fourth‐order nature of the anisotropy influence on moveout as it manifests itself in this distinct signature in extended images after handling the velocity properly in the imaging process. Synthetic and real data observations support this assertion.     

3D post‐stack interval velocity analysis with effective use of datuming

Interval velocity analysis using post‐stack data has always been a desire, mainly for 3D data sets. In this study we present a method that uses the unique characteristics of migrated diffractions to enable interval velocity analysis from three‐dimensional zero‐offset time data. The idea is to perform a standard three‐dimensional prestack depth migration on stack cubes and generate three‐dimensional common image gathers that show great sensitivity to velocity errors. An efficient ‘top‐down’ scheme for updating the velocity is used to build the model. The effectiveness of the method is related to the incorporation of wave equation based post‐stack datuming in the model building process. The proposed method relies on the ability to identify diffractions along redatumed zero‐offset data and to analyse their flatness in the migrated local angle domain. The method can be considered as an additional tool for a complete, prestack depth migration based interval velocity analysis.     

Increasing resolution of reverse‐time migration using time‐shift gathers

Reverse‐time migration has become an industry standard for imaging in complex geological areas. We present an approach for increasing its imaging resolution by employing time‐shift gathers. The method consists of two steps: (i) migrating seismic data with the extended imaging condition to get time‐shift gathers and (ii) accumulating the information from time‐shift gathers after they are transformed to zero‐lag time‐shift by a post‐stack depth migration on a finer grid. The final image is generated on a grid, which is denser than that of the original image, thus improving the resolution of the migrated images. Our method is based on the observation that non‐zero‐lag time‐shift images recorded on the regular computing grid contain the information of zero‐lag time‐shift image on a denser grid, and such information can be continued to zero‐lag time‐shift and refocused at the correct locations on the denser grid. The extra computational cost of the proposed method amounts to the computational cost of zero‐offset migration and is almost negligible compared with the cost of pre‐stack shot‐record reverse‐time migration. Numerical tests on synthetic models demonstrate that the method can effectively improve reverse‐time migration resolution. It can also be regarded as an approach to improve the efficiency of reverse‐time migration by performing wavefield extrapolation on a coarse grid and by generating the final image on the desired fine grid.