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.     

Nonlinear ray perturbation theory with its applications to ray tracing and inversion in anisotropic media

A comprehensive approach, based on the general nonlinear ray perturbation theory (Druzhinin, 1991), is proposed for both a fast and accurate uniform asymptotic solution of forward and inverse kinematic problems in anisotropic media. It has been developed to modify the standard ray linearization procedures when they become inconsistent, by providing a predictable truncation error of ray perturbation series. The theoretical background consists in a set of recurrent expressions for the perturbations of all orders for calculating approximately the body wave phase and group velocities, polarization, travel times, ray trajectories, paraxial rays and also the slowness vectors or reflected/transmitted waves in terms of elastic tensor perturbations. We assume that any elastic medium can be used as an unperturbed medium. A total 2-D numerical testing of these expressions has been established within the transverse isotropy to verify the accuracy and convergence of perturbation series when the elastic constants are perturbed. Seismological applications to determine crack-induced anisotropy parameters on VSP travel times for the different wave types in homogeneous and horizontally layered, transversally isotropic and orthorhombic structures are also presented. A number of numerical tests shows that this method is in general stable with respect to the choice of the reference model and the errors in the input data. A proof of uniqueness is provided by an interactive analysis of the sensitivity functions, which are also used for choosing optimum source/receiver locations. Finally, software has been developed for a desktop computer and applied to interpreting specific real VSP observations as well as explaining the results of physical modelling for a 3-D crack model with the estimation of crack parameters.     

Computation of additional components of the first-order ray approximation in isotropic media

This study shows that the use of the first-order additional components of the ray method in the seismic wave field modeling is easy and that it can bring a substantial improvement of the standard ray results obtained with the zero-order ray approximation only. For the calculation of a first-order additional component, spatial derivatives of the parameters of the medium and spatial derivatives of the zero-order ray amplitude term are necessary. The evaluation of the former derivatives is straightforward; the latter derivatives can be calculated approximately from neighboring rays by substituting the derivatives by finite differences. This allows an effective calculation of the first-order additional terms in arbitrary laterally varying layered media.The importance of the first-order additional terms is demonstrated by the study of individual higher-order terms of the ray series representing elementaryP andS elastodynamic Green functions for a homogeneous isotropic medium. The study shows clearly that the consideration of the first-order additional terms leads to a more substantial decrease of the difference between approximate and exact elementary Green functions than any other higher-order term. With this in mind, effects of the first-order additional terms on the ray synthetic seismograms for aVSP configuration are studied. It is shown that the use of the additional terms leads to such phenomena, unknown in the zero-order approximation of the ray method, like quasi-elliptical and transverse polarization of a singleP wave or longitudinal polarization of a singleS wave.     

Approximate Relation Between the Ray Vector and the Wave Normal in Weakly Anisotropic Media

Determination of the ray vector (the unit vector specifying the direction of the group velocity vector) corresponding to a given wave normal (the unit vector parallel to the phase velocity vector or slowness vector) in an arbitrary anisotropic medium can be performed using the exact formula following from the ray tracing equations. The determination of the wave normal from the ray vector is, generally, a more complicated task, which is usually solved iteratively. We present a first-order perturbation formula for the approximate determination of the ray vector from a given wave normal and vice versa. The formula is applicable to qP as well as qS waves in directions, in which the waves can be dealt with separately (i.e. outside singular directions of qS waves). Performance of the approximate formulae is illustrated on models of transversely isotropic and orthorhombic symmetry. We show that the formula for the determination of the ray vector from the wave normal yields rather accurate results even for strong anisotropy. The formula for the determination of the wave normal from the ray vector works reasonably well in directions, in which the considered waves have convex slowness surfaces. Otherwise, it can yield, especially for stronger anisotropy, rather distorted results.     

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 seismograms. For 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.     

Focusing in Prestack Isochrone Migration Using Instantaneous Slowness Information

—Prestack migration finds increasing application in processing crustal seismic data. However, less effort has been made to incorporate slowness information in the imaging process. The combination of slowness information with migration leads to an improved image in the depth domain, especially by reducing migration artefacts and noise. A slowness-driven isochrone migration scheme is introduced for migration of 2-D seismic data. Instantaneous slowness information p(x, t) is extracted from the data using correlation analysis in moving time and space windows. Slowness values resulting from spatial coherent energy (signal) and incoherent background noise are distinguished by the simultaneous evaluation of an instantaneous coherence criterion g(x, t). In slowness-driven isochrone migration this information is used for locally weighting the amplitude A(x, t) smearing on the isochrone surface. In particular, slowness p and coherence criterion g determine position and sharpness of a Gaussian weighting function. The method is demonstrated using two synthetic data examples and is subsequently applied to two deep crustal data sets, one wide-angle (along DEKORP4) and one steep-angle reflection seismic observation (KTB8506). Both data sets were collected in the surroundings of the KTB drill site, Oberpfalz, as part of the German DEKORP project.     

多震相层析成像与在介质中的多波型射线跟踪

本文介绍了多震相的层析成像的思路和算法，通过穿透和反射走时可以同时作出２维和３维慢度（速度的倒数）重建。我们分析了在穿透和反射数据中确定速度和深度的不确定性，并认识到深度扰动对反射走时异常比慢度扰动更敏感。由不同波类型所提供的对速度和深度的约束，这个算法实际上减少了在一般反射层析成像在速度和深度之间的不确定性，并且也避免了在穿透层析成像中的不确定问题。线性化反演是通过从反射界面深度由分离速度参数迭代进行的。使用一个快速的２－Ｄ和３－Ｄ射线跟踪算法来计算穿透和反射走时和对幔度及反射界面深度的偏导数。深度和速度都用立方Ｂ样条函数来进行参数化。合成例子表明，当同时考虑穿透和反射时间，层析成像的结果得到改进。这个方法也应用到英国煤炭测量局（ＢｒｉｔｉｓｈＣｏａｌＭｅａｓｕｒｅｓ）沿跨线排列所记录的逆ＶＳＰ数据组。通过使用波形配合技术，用同时确定时间延迟和叠加权，可以自动拾取旅行时间。所观察到的逆ＶＳＰ层析成像可比周围介质具有较低速度的两个断层区域成像。断层的位置由附近的反射测线所确定。本文还讨论了在复杂２－Ｄ和３－Ｄ非均匀各向同性介质中地震射线跟踪方法。界面的几何形状和水平速度场都通过使用非均匀步长立方Ｂ－样条节点     

Estimating the elastic parameters of anisotropic media using a joint inversion of P-wave and SV-wave traveltime error

A method is presented to estimate the elastic parameters and thickness of media that are locally laterally homogeneous using P‐wave and vertically polarized shear‐wave (SV‐wave) data. This method is a ‘layer‐stripping’ technique, and it uses many aspects of common focal point (CFP) technology. For each layer, a focusing operator is computed using a model of the elastic parameters with which a CFP gather can be constructed using the seismic data. Assuming local homogeneity, the resulting differential time shifts (DTSs) represent error in the model due to anisotropy and error in thickness. In the (τ?p) domain, DTSs are traveltimes Δτ that connect error in layer thickness z, vertical slowness q, and ray parameter p. Series expansion is used to linearize Δτ with respect to error in the elastic parameters and thickness, and least‐squares inversion is used to update the model. For stability, joint inversion of P and SV data is employed and, as pure SV data are relatively rare, the use of mode‐converted (PSV) data to represent SV in the joint inversion is proposed. Analytic and synthetic examples are used to demonstrate the utility and practicality of this inversion.     

二步法射线追踪

本文提出的二步法射线追踪,能够在复杂地质模型下有效、准确地确定射线在炮点的出射角,使得射线恰好到达检波点.射线到达地面的位置是射线在炮点的出射角的函数,称为出射函数.二步法射线追踪通过搜索出射函数的极值点来确定其单调变化的区间;然后通过比较检波点位置与出射函数极值的大小,确定与检波点相对应的出射角究竟落在出射函数的哪一个单调变化的区间里;最后,利用出射函数在局部区域里单调变化的性质,迅速、准确地确定与检波点相对应的出射角.出射函数的极值点,对于共炮集中每一个检波点和检波点中每一个不同方向的入射射线都是适用的,因此,二步法射线追踪只需要进行一次一维全局搜索（极值点搜索）.另外,本文采用B-样条的线性组合来表示速度函数,通过提高B-样条的阶数,可以改善速度函数的光滑程度,从而既满足射线追踪对于速度函数光滑性的要求,又加强射线在炮点的出射角对于射线的控制作用.     

Exact Elastodynamic Green Functions for Simple Types of Anisotropy Derived from Higher-Order Ray Theory

Using higher-order ray theory, we derived exact elastodynamic Green functions for three simple types of homogeneous anisotropy. The first type displays an orthorhombic symmetry, the other two types display transverse isotropy. In all cases, the slowness surfaces of waves are either ellipsoids, spheroids or spheres. All three Green functions are expressed by a ray series with a finite number of terms. The Green functions can be written in explicit and elementary form similar to the Stokes solution for isotropy. In two Green functions, the higher-order ray approximations form a near-singularity term, which is significant near a kiss singularity. In the third Green function, the higher-order ray approximations also form a near-field term, which is significant near the point source. No effect connected with the line singularity was observed.     

Marine seismic wavefield measurement to remove sea-surface multiples

We propose a new method for removing sea-surface multiples from marine seismic reflection data in which, in essence, the reflection response of the earth, referred to a plane just above the sea-floor, is computed as the ratio of the plane-wave components of the upgoing wave and the downgoing wave. Using source measurements of the wavefield made during data acquisition, three problems associated with earlier work are solved: (i) the method accommodates source arrays, rather than point sources; (ii) the incident field is removed without simultaneously removing part of the scattered field; and (iii) the minimum-energy criterion to find a wavelet is eliminated. Pressure measurements are made in a horizontal plane in the water. The source can be a conventional array of airguns, but must have both in-line and cross-line symmetry, and its wavefield must be measured and be repeatable from shot to shot. The problem is formulated for multiple shots in a two-dimensional configuration for each receiver, and for multiple receivers in a two-dimensional configuration for each shot. The scattered field is obtained from the measurements by subtracting the incident field, known from measurements at the source. The scattered field response to a single incident plane wave at a single receiver is obtained by transforming the common-receiver gather to the frequency–wavenumber domain, and a single component of this response is obtained by Fourier transforming over all receiver coordinates. Each scattered field component is separated into an upgoing wave and a downgoing wave using the zero-pressure condition at the water-surface. The upgoing wave may then be expressed as a reflection coefficient multiplied by the incident downgoing wave plus a sum of scattered downgoing plane waves, each multiplied by the corresponding reflection coefficient. Keeping the upgoing scattered wave fixed, and using all possible incident plane waves for a given frequency, yields a set of linear simultaneous equations for the reflection coefficients which are solved for each plane wave and for each frequency. To create the shot records that would have been measured if the sea-surface had been absent, each reflection coefficient is multiplied by complex amplitude and phase factors, for source and receiver terms, before the five-dimensional Fourier transformation back to the space–time domain.     

叠前地震数据特征波场分解、偏移成像与层析反演

本文提出了一套叠前地震数据稀疏表达(特征波场合成)、深度偏移成像和层析成像的处理流程.不同于传统的变换域中的数据稀疏表达理论,本文利用局部平面波的传播方向(慢度矢量),在中心炮检点处同时进行波束合成,从而将地震数据投影到局部平面波域(高维空间)中.由于波束合成后的地震数据描述了局部平面波的方向特征,因此称之为特征波场.然而波束合成算法需要估计局部平面波的慢度矢量.当地震数据受噪声干扰时,难以在常规τ-p谱中自动估计局部平面波的射线参数(慢度矢量).本文提出了基于反演理论的特征波场合成方法,可以同时反演局部平面波及其传播方向,从而提高特征波合成的自动化程度并保持方法的稳健性.通过特征波场合成,可以将地震数据分解为单独的震相(波形).这样的数据可以直接用来成像及反演.在局部平面波域中,由于局部平面波的入射与出射射线参数已知,传统的Kirchhoff叠前深度偏移(PSDM)和高斯束/控制束PSDM可以实现从"沿等时面的画弧"到"向反射点(段)的直接投影"的转变,叠前偏移的效率以及成像质量可以同时提高.此外,特征波场与地下反射点(段)的一对一映射关系使得叠前深度偏移与层析成像融为一体,可以极大地提高速度反演的效率.数值试验证明了特征波场合成、叠前深度成像以及层析反演的有效性.     

Radiation patterns of point sources situated close to structural interfaces and to the earth's surface

The seismic wave field is considerably influenced by local structures close to the source and to the receiver. This applies to sources and receivers situated close to localized inhomogeneities, to structural interfaces, to the earth's surface, etc. In this paper we concentrate our attention mainly to the ray-theoretical radiation patterns of point sources situated close to the structural interfaces and to the earth's surface. In numerical modeling of high-frequency seismic wave fields by the ray method, the interaction of the source with the earth's surface has not usually been taken into account.The proposed procedure of the computation of the radiation patterns of point sources situated directly on structural interfaces and on the earth's surface is based on the zero-order approximation of the ray method, assuming that the length of the ray between the source and the receiver is long. The derived equations are extended to point sources located close to structural interface, to the earth's surface and to thin transition layers using the hybrid ray-reflectivity method, seeervený (1989). The thin layer need not be homogeneous; it may include an arbitrary inner layering (transition layers, laminas, etc.) The only requirement is for the layer to be thin. Roughly speaking, we require its thickness to be less than one quarter of the prevailing wavelength. The hybrid ray-reflectivity method describes well even certain non-ray effects (tunneling.S ^* waves, etc.). Explicit analytical expressions for radiation patterns for all above listed point sources are found. These expression have a local character and may be easily implemented into computer codes designed for the routine computation of ray amplitudes and synthetic ray seismograms in 2-D and 3-D, laterally varying isotropic layered and block structures by the ray method.Numerical examples of radiation patterns ofP andS waves of point sources situated close to the earth's surface and to a thin low-velocity surface layer are presented and discussed. The explosive point source (center of dilatation) and the vertical and horizontal single force point sources are considered. It has been ascertained that the radiation patterns of point sources depend drastically on the depth of the source below the surface even if the depths vary within one quarter of the prevailing wavelength.     

利用深反射地震数据构建的多阶面波频散曲线反演近地表横波速度结构——以跨班公湖—怒江缝合带深反射地震资料为例

深反射地震剖面法为了获取深部结构特征常常采取大的偏移距采集数据.目前公开发表的相关资料中,鲜有利用深反射地震炮集数据获取近地表的结构特征.为此,本文通过正演测试了相关数据处理流程,即利用有限差分正演了起伏地表模型的大偏移距地震单炮弹性波场特征,通过共检波点域面波信号F-K频谱叠加构建新方法,从深反射地震数据集中提取了高品质的多阶面波频散曲线,再利用多阶面波联合反演获得了近地表的结构特征.在前述正演流程基础上,利用跨越班公湖—怒江缝合带的SinoProbe深反射地震剖面中的实际炮集数据,求取了基阶和一阶瑞利波频散曲线,联合反演后得到近地表横波速度结构.该结果与初至波走时反演获取的纵波速度结构具有较好的一致性,且在近地表的浅层分辨率较纵波速度结构特征更高,而更与已有地质认识相吻合.本文提供的相关数据处理流程表明利用深反射地震炮集数据,也能够获取近地表浅层的横波速度结构.     

Local Determination of Weak Anisotropy Parameters from Walkaway VSP qP-Wave Data in the Java Sea Region

We apply the inversion scheme of Zheng and Peník (2002) to the walkaway VSP data of Horne and Leaney (2000) collected in the Java Sea region. The goal is a local determination of parameters of the medium surrounding the borehole receiver array. The inversion scheme is based on linearized equations expressing qP-wave slowness and polarization vectors in terms of weak anisotropy (WA) parameters. It thus represents an alternative approach to Horne and Leaney (2000), who based their procedure on inversion of the Christoffel equation using a global optimization method. The presented inversion scheme is independent of structural complexities in the overburden and of the orientation of the borehole. The inverson formula is local, and has therefore potential to separate effects of anisotropy from effects of inhomogeneity. The data used are components of the slowness vector along the receiver array and polarization vectors. The inversion is performed without any assumptions concerning the remaining components of the slowness vector. The inversion is made (a) assuming arbitrary anisotropy, i.e., without any assumptions about symmetry of the medium, (b) assuming transverse isotropy with a vertical axis of symmetry and (c) assuming isotropy of the medium. Inverted are the raw data as well as data, in which weighting is used to reduce the effect of outliers. It is found that the WA parameters _z, ₁₅ and ₃₅ are considerably more stable than the parameters _x and _x. The latter two parameters are also found to be strongly correlated. Weaker correlation is also found between the mentioned two parameters and _z. The results of inversion show clearly that the studied medium is not isotropic. They also seem to indicate that the studied medium does not possess the VTI symmetry.     

An analytical migration method for crustal wide angle reflections

Summary An analytical migration method is developed for inverting wide angle seismic reflection travel time data with proper migration. In this method the concept of effective velocity (V _eff ) in the vertically inhomogeneous earth has been used, which takes into account the refraction effect. It is shown that the migration process in this case becomes extremely rapid as curved ray paths are replaced by equivalent straight paths. A post migration sliding correction is found to be necessary, which arises due to the non-collinearity of the recording point, reflection point and the image point in the equivalent straight path geometry for dipping reflectors. Calculations for some representative dips show that, for the depths and recording ranges of interest in DSS, the vertical component of the sliding correction is negligible compared to the horizontal component. However, even the horizontal component is not significant for reflectors with gentle dips. The effect of certain simplifying assumptions in the estimation of V _eff is shown to be within the error limits of velocity determination. A computer programme package has been written for carrying out the migration. The method yields more accurate results in various theoretical and field cases than the conventional graphical wave front methods which are at present widely used for crustal reflection work.     

Modeling of Wide-Angle Seismic Attributes Using the Gaussian Beam Method

In this study, observed seismic attributes from shot gather 11 of the SAREX experiment are used to derive a preliminary velocity and attenuation model for the northern end of the profile in southern Alberta. Shot gather 11 was selected because of its prominent Pn arrivals and good signal to noise ratio. The 2-D Gaussian beam method was used to perform the modeling of the seismic attributes including travel times, peak envelope amplitudes and pulse instantaneous frequencies for selected phases. The preliminary model was obtained from the seismic attributes from shot gather 11 starting from prior tomographic results. The amplitudes and instantaneous frequencies were used to constrain the velocity and attenuation structure, with the amplitudes being more sensitive to the velocity gradients and the instantaneous frequencies more sensitive to the attenuation structure. The resulting velocity model has a velocity discontinuity between the upper and lower crust, and lower velocity gradients in the upper and lower crust compared to earlier studies. The attenuation model has Q _p ^-1 values between 0.011 and 0.004 in the upper crust, 0.0019 in the lower crust and a laterally variable Q _p ^-1 in the upper mantle. The Q _p ^-1 values are similar to those found in Archean terranes from other studies. Although the results from a single gather are non-unique, the initial model derived here provides a self-consistent starting point for a more complete seismic attribute inversion for the velocity and attenuation structure.     

一阶多次波聚焦变换成像

将多次波转换成反射波并按传统反射波偏移算法成像,是多次波成像的一种方法.聚焦变换能准确的将多次波转换为纵向分辨率更高的新波场记录,其中一阶多次波转换为反射波.本文对聚焦变换提出了两点改进:1)提出局部聚焦变换,以减小存储量和计算量,增强该方法对检波点随炮点移动的采集数据的适应性;2)引入加权矩阵,理论上证明原始记录的炮点比检波点稀疏时,共检波点道集域的局部聚焦变换可以将多次波准确转换成炮点与检波点有相同采样频率的新波场记录.本文在第一个数值实验中对比了对包含反射波与多次波的原始记录做局部聚焦变换和直接对预测的多次波做局部聚焦变换两种方案,验证了第二种方案转换得到的波场记录信噪比更高且避免了第一个方案中切聚焦点这项比较繁杂的工作.第二个数值实验表明:在炮点采样较为稀疏时,该方法能有效的将一阶多次波转换成反射波;转换的反射波能提供更丰富的波场信息,成像结果更均衡、在局部有更高的信噪比,以及较高的纵向分辨率.     

Slowness-domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre-critical slowness match

Part II of this paper is a direct continuation of Part I, where we consider the same types of orthorhombic layered media and the same types of pure-mode and converted waves. Like in Part I, the approximations for the slowness-domain kinematical characteristics are obtained by combining power series coefficients in the vicinity of both the normal-incidence ray and an additional wide-angle ray. In Part I, the wide-angle ray was set to be the critical ray (‘critical slowness match’), whereas in Part II we consider a finite long offset associated with a given pre-critical ray (‘pre-critical slowness match’). Unlike the critical slowness match, the approximations in the pre-critical slowness match are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. Moreover, for the pre-critical slowness match, there is no need to distinguish between the high-velocity layer and the other, low-velocity layers. The form of the approximations in both critical and pre-critical slowness matches is the same, where only the wide-angle power series coefficients are different. Comparing the approximated kinematical characteristics with those obtained by exact numerical ray tracing, we demonstrate high accuracy. Furthermore, we show that for all wave types, the accuracy of the pre-critical slowness match is essentially higher than that of the critical slowness match, even for matching slowness values close to the critical slowness. Both approaches can be valuable for implementation, depending on the target offset range and the nature of the subsurface model. The pre-critical slowness match is more accurate for simulating reflection data with conventional offsets. The critical slowness match can be attractive for models with a dominant high-velocity layer, for simulating, for example, refraction events with very long offsets.