Weak-anisotropy Approximation Research Articles

Parameterization of anisotropic media by A-parameters

AbstractMost common parameterization of anisotropic media is by twenty one independent elements aijkl of the density-normalized stiffness tensor or by twenty one independent elements Aαβ of the density-normalized matrix of elastic parameters in the Voigt notation. These parameters are commonly of significantly different sizes, are dimensional, in (km/s)2, often appear in combinations. We are offering an alternative parameterization by twenty one A-parameters (anisotropic parameters), which removes the mentioned disadvantages and possesses some additional useful properties. For example, axes or planes of coordinate systems, in which A-parameters are defined, need not be related to symmetry axes or planes of the considered anisotropy symmetry as required in other similar parameterizations. In combination with the first-order weak-anisotropy approximation, in which anisotropy is considered as the first-order perturbation of reference isotropy, parameterization by A-parameters yields insight into the role of individual A-parameters in the wave propagation problems. For example, it turns out that in the first-order weak-anisotropy approximation, P- and S-wave velocities are each controlled by fifteen A-parameters. A set of six of them appears only in the expression for P-wave velocity, a set of other six A-parameters appears only in S-waves velocity expressions. Remaining set of nine A-parameters is common for both waves. We present transformation of A-parameters, analogue to Bond transformation, and useful formulae for the weak-anisotropy approximation for anisotropy of any symmetry and arbitrary tilt.

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

Use of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%.

The inverse scattering imaging condition for anisotropic reverse time migration

The inverse scattering imaging condition (ISIC) is widely used in reverse time migration (RTM), which naturally eliminates the low-wavenumber noise by combining a time-derivative imaging condition with a spatial-derivative imaging condition. However, when applying ISIC to anisotropic RTM, we observe that the low-wavenumber noise is not entirely eliminated. This phenomenon is the leakage of low-wavenumber noise caused by anisotropy. In anisotropic media, due to the deviation of the vertical velocity from the phase velocity, the spatial-derivative imaging condition does not generate correct weights to cancel the low-wavenumber noise obtained by using the time-derivative imaging condition. By adding an anisotropic correction term to compensate for this deviation, we obtain the anisotropic ISIC which can fully eliminate the low-wavenumber noise. In addition, we construct the connection between the anisotropic ISIC and the scattering function. The scattering function describes the power of the media to generate the scattering field, and it encompasses the effect of model perturbations and their scattering patterns. Under the small perturbation assumption and weak anisotropy approximation, we theoretically show that the ISIC reverses the scattering process and recovers the scattering function from the scattering field. Through three numerical examples, we demonstrate the effectiveness of our method in suppressing the low-wavenumber noise for anisotropic RTM. The anisotropic ISIC is suitable for titled transverse isotropic or orthorhombic media, and it also has potential usages in inversion-based imaging algorithms.

Certain problems in constrained cubic quasicrystals: General solutions and infinite-space Green’s functions

The generalized elasticity is applied to the analysis of quasicrystals to account for the mechanical response of the phonon and phason fields. Using the operator method introduced by Lur’e, the general solutions are derived for three-dimensional constrained cubic quasicrystals, leading to the fundamental solution with the phonon-phason coupling effect. Based on the experimental observation that most three-dimensional quasicrystals are of weak anisotropy, we firstly introduce a weak anisotropy approximation on the phason field and the phono-phason coupling field to construct a new quasicrystal model with weak anisotropy. Using this new model, we derived the new general solutions like the generalized Boussinesq-Galerkin solution, the generalized Papkovich-Neuber solution and the generalized Kelvin solution. We identified the distinction between quasicrystals and classical elastic isotropic media when anisotropy is weak. These solutions present complicated mechanical behaviors that cannot be seen in traditional isotropic media, such as phonon-phason symmetry breaking. We find out that the phason field itself could lead to certain unique phenomena without involving complex anisotropy, and the loading ratio has a noticeable influence on the quasicrystals even with weak anisotropy. These findings could serve as the theoretical basis to guide the future experiments of quasicrystals.

Weak-anisotropy approximation of P-wave reflection coefficient in anisotropic media of arbitrary symmetry and tilt

Applicability of the approximate expression for the P-wave reflection coefficient at a weak-contrast reflector separating two weakly anisotropic half-spaces of arbitrary symmetry is extended to media of arbitrary symmetry and tilt. The reflection coefficient consists of the approximate P-wave reflection coefficient at a weak-contrast interface separating two reference isotropic half-spaces and a correction term due to anisotropy. Along an arbitrarily chosen profile, the “isotropic” term depends on the density and P- and S-wave contrasts, and the correction term depends linearly on the contrasts of four profile weak-anisotropy (WA) parameters specifying anisotropy along the profile. In addition, both terms depend on the polar angle of incidence. In each half-space, the four profile WA parameters can be expressed as a linear combination of 12 of 21 global WA parameters specifying anisotropy of the half-space. Coefficients of this linear combination are functions of the azimuth of the profile. WA parameters are a generalization of Thomsen parameters to arbitrary anisotropy and represent an alternative to 21 independent elements of the stiffness tensor. WA parameters can be used for the approximation of other related concepts such as the reflection moveout or the geometric spreading. Presented tests illustrate high accuracy and flexibility of the proposed formula for the P-wave reflection coefficient. They also show that very accurate results can be obtained for contrasts and anisotropy, which cannot be considered weak.

Estimation of parameters and thickness of a horizontal layer of arbitrary anisotropy from P-wave moveout

We present an inversion scheme to estimate anisotropy parameters and thickness of a homogeneous layer of arbitrary anisotropy from traveltimes of a reflected P wave. The scheme is based on the weak-anisotropy approximation of P-wave moveout formula. For the description of anisotropy, we use the so-called A-parameters, which represent an alternative to standard elastic moduli specifying anisotropic media, and allow specification of anisotropy of any symmetry, orientation and strength. The proposed scheme allows estimation of, at most, nine of fifteen A-parameters, which describe P-wave propagation in the weak-anisotropy approximation. Therefore, generally, the reconstruction of the P-wave phase-velocity surface from the estimated parameters is impossible. If we deal, however, with higher-symmetry anisotropy (transversely isotropic, orthorhombic) and if information about the orientation of its symmetry elements is available, reconstruction of the phase-velocity surface is possible. Synthetic tests indicate that the inversion scheme works well even in situations, in which the use of other commonly used moveout formulae leads to unsatisfactory results.

Weak-anisotropy approximation of P-wave geometric spreading in horizontally layered anisotropic media of arbitrary symmetry: Tilted transversely isotropic specification

Understanding the role of geometric spreading and estimating its effects on seismic wave propagation play an important role in several techniques used in seismic exploration. The spreading can be estimated through dynamic ray tracing or determined from reflection traveltime derivatives. In the latter case, derivatives of nonhyperbolic moveout approximations are often used. We offer an alternative approach based on the weak-anisotropy approximation. The resulting formula is applicable to P-waves reflected from the bottom of a stack of horizontal layers, in which each layer can be of arbitrary anisotropy. At an arbitrary surface point, the formula depends, in each layer, on the thickness of the layer, on the P-wave reference velocity used for the construction of reference rays, and on nine P-wave weak-anisotropy (WA) parameters specifying the layer anisotropy. Along an arbitrary surface profile, the number of WA parameters reduces to five parameters related to the profile. WA parameters represent an alternative to the elastic moduli and, as such, can be used for the description of any anisotropy. The relative error of the approximate formula for a multilayered structure consisting of layers of anisotropy between 8% and 20% is, at most, 10%. For models including layers of anisotropy stronger than 20%, the relative errors may reach, locally, even 30%. For any offset, relative errors remain under a finite limit, which varies with the anisotropy strength.

Practical concept of traveltime inversion of simulated P-wave vertical seismic profile data in weak to moderate arbitrary anisotropy

We have developed a practical concept of compressional wave (P-wave) traveltime inversion in weakly to moderately anisotropic media of arbitrary symmetry and orientation. The concept provides sufficient freedom to explain and reproduce observed anisotropic seismic signatures to a high degree of accuracy. The key to this concept is the proposed P-wave anisotropy parameterization (A-parameters) that, together with the use of the weak-anisotropy approximation, leads to a significantly simplified theory. Here, as an example, we use a simple and transparent formula relating P-wave traveltimes to 15 P-wave A-parameters describing anisotropy of arbitrary symmetry. The formula is used in the inversion scheme, which does not require any a priori information about anisotropy symmetry and its orientation, and it is applicable to weak and moderate anisotropy. As the first step, we test applicability of the proposed scheme on a blind inversion of synthetic P-wave traveltimes generated in vertical seismic profile experiments in homogeneous models. Three models of varying anisotropy are used: tilted orthorhombic and triclinic models of moderate anisotropy (approximately 10%) and an orthorhombic model of strong anisotropy (>25%) with a horizontal plane of symmetry. In all cases, the inversion yields the complete set of 15 P-wave A-parameters, which make reconstruction of corresponding phase-velocity surfaces possible with high accuracy. The inversion scheme is robust with respect to noise and the source distribution pattern. Its quality depends on the angular illumination of the medium; we determine how the absence of nearly horizontal propagation directions affects inversion accuracy. The results of the inversion are applicable, for example, in migration or as a starting model for inversion methods, such as full-waveform inversion, if a model refinement is desired. A similar procedure could be designed for the inversion of S-wave traveltimes in anisotropic media of arbitrary symmetry.

P-wave reflection-moveout approximation for horizontally layered media of tilted weak-to-moderate orthorhombic symmetry

An approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of weak-to-moderate orthorhombic or higher-symmetry anisotropy of arbitrary orientation is presented and tested. Weak-anisotropy approximation is used for the derivation of the formula, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters specifying the anisotropy of the layers. Due to the use of the weak-anisotropy approximation, the proposed formula is applicable to any offset. Its accuracy decreases with increasing strength of anisotropy. The relation between traveltimes and WA parameters in the formula is transparent and relatively simple. For any offset along an arbitrarily chosen single surface profile, the formula depends in each layer on its thickness, on the auxiliary reference P-wave velocity, on, at most, six WA parameters describing P-wave orthorhombic or higher-symmetry anisotropy, and on three Euler angles specifying the orientation of symmetry elements in the layer. Performed tests indicate that the maximum relative traveltime error does not exceed 2.5% for P-wave anisotropy about 25%. Often, the errors are considerably lower.

Efficient mixed-domain pure P-wave equations for 2D and 3D tilted transversely isotropic media

In transversely isotropic (TI) media, accurate and efficient pure P-wave extrapolation is the basis of seismic imaging and inversion algorithms. For pure P-wave equations in TI media, combining the finite-difference (FD) method with the pseudo-spectral (PS) method is an effective solution. However, if the mixed-domain pure P-wave equation contains multiple wavenumbers, using the hybrid approach will involve multiple Fourier transforms, which will result in high computational costs. Referring to the weak anisotropy approximation, we propose a new approximate P-wave phase velocity expression. We then use an optimization strategy to reduce the number of wavenumber terms in the corresponding dispersion relations and derive the mixed-domain pure P-wave equations in 2D and 3D tilted transversely isotropic (TTI) media. Through numerical experiments in 2D and 3D TTI models, we verify the feasibility and efficiency of the proposed mixed-domain P-wave equations.

Nonlinear amplitude versus angle inversion for transversely isotropic media with vertical symmetry axis using new weak anisotropy approximation equations

In VTI media, the conventional inversion methods based on the existing approximation formulas are difficult to accurately estimate the anisotropic parameters of reservoirs, even more so for unconventional reservoirs with strong seismic anisotropy. Theoretically, the above problems can be solved by utilizing the exact reflection coefficients equations. However, their complicated expression increases the difficulty in calculating the Jacobian matrix when applying them to the Bayesian deterministic inversion. Therefore, the new reduced approximation equations starting from the exact equations are derived here by linearizing the slowness expressions. The relatively simple form and satisfactory calculation accuracy make the reduced equations easy to apply for inversion while ensuring the accuracy of the inversion results. In addition, the blockiness constraint, which follows the differentiable Laplace distribution, is added to the prior model to improve contrasts between layers. Then, the concept of GLI and an iterative reweighted least-squares algorithm is combined to solve the objective function. Lastly, we obtain the iterative solution expression of the elastic parameters and anisotropy parameters and achieve nonlinear AVA inversion based on the reduced equations. The test results of synthetic data and field data show that the proposed method can accurately obtain the VTI parameters from prestack AVA seismic data.

P-wave reflection-moveout approximation for horizontally layered media of arbitrary moderate anisotropy

We present an approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of moderate or weak anisotropy of arbitrary symmetry and orientation. Anisotropy symmetry and its orientation may differ from layer to layer. Instead of commonly used Taylor-series expansion of the square of the reflection traveltime in terms of the square of the offset, we use the weak-anisotropy approximation, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters. The resulting formula is simple, and it provides a transparent relation between the traveltimes and WA parameters. Along an arbitrarily chosen single surface profile, it depends, in each layer, on the thickness of the layer, on the reference P-wave velocity used for the construction of reference rays, and on three WA parameters specified in the Cartesian coordinate system related to the profile. In each layer, these three “profile” WA parameters depend on “local” WA parameters specifying anisotropy of a given layer in a local coordinate system and on directional cosines specifying the orientation of the local coordinate system with respect to the profile one. The number of local P-wave WA parameters may vary from three for transverse isotropy or six for orthorhombic symmetry to nine for triclinic symmetry. Our tests of the accuracy indicate that the maximum relative traveltime errors do not exceed 0.5% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Moment tensors of double-couple microseismic sources in anisotropic formations

Moment tensors of kinematically double-couple microseismic events triggered in anisotropic formations are known to exhibit non-double-couple focal mechanisms. The weak anisotropy approximation of these mechanisms reveals the combinations of anisotropy coefficients of vertically transversely isotropic and orthorhombic focal regions responsible for the deviations of moment tensors from double couples. Numerical examples for models of typical unconventional shales indicate the non-double-couple components of moment tensors to be sufficiently large to cause misinterpretation of the nature of ruptures associated with hydraulic fracturing.

Anisotropic P-wave travel-time tomography implementing Thomsen's weak approximation in TOMO3D

Abstract. We present the implementation of Thomsen's weak anisotropy approximation for vertical transverse isotropy (VTI) media within TOMO3D, our code for 2-D and 3-D joint refraction and reflection travel-time tomographic inversion. In addition to the inversion of seismic P-wave velocity and reflector depth, the code can now retrieve models of Thomsen's parameters (δ and ε). Here, we test this new implementation following four different strategies on a canonical synthetic experiment in ideal conditions with the purpose of estimating the maximum capabilities and potential weak points of our modeling tool and strategies. First, we study the sensitivity of travel times to the presence of a 25 % anomaly in each of the parameters. Next, we invert for two combinations of parameters (v, δ, ε and v, δ, v⊥), following two inversion strategies, simultaneous and sequential, and compare the results to study their performance and discuss their advantages and disadvantages. Simultaneous inversion is the preferred strategy and the parameter combination (v, δ, ε) produces the best overall results. The only advantage of the parameter combination (v, δ, v⊥) is a better recovery of the magnitude of v. In each case, we derive the fourth parameter from the equation relating ε, v⊥ and v. Recovery of v, ε and v⊥ is satisfactory, whereas δ proves to be impossible to recover even in the most favorable scenario. However, this does not hinder the recovery of the other parameters, and we show that it is still possible to obtain a rough approximation of the δ distribution in the medium by sampling a reasonable range of homogeneous initial δ models and averaging the final δ models that are satisfactory in terms of data fit.

Estimation of in situ anisotropy parameters from two perpendicular walkaway VSP lines in South Pars field

Abstract Vertical phase slowness and polarization angle are the in situ parameters of P-wave propagation that can be derived from walkaway vertical seismic profiling (VSP) data. To use these data for estimating anisotropy parameters, we obtain an explicit equation of vertical slowness as a function of polarization angle for P-wave propagation in transversely isotropic with vertical symmetry axis (VTI) media. We use this equation to estimate anisotropy parameters of a target layer in the South Pars field, Iran. This field is one of the world's largest gas fields. We show that the orthorhombic symmetry is a reasonable assumption for this layer, providing some geological and petrophysical information. Two walkaway VSP lines along the symmetry axes of the presumed orthorhombic layer are used to estimate its parameters. Seven is the maximum number of parameters that can be estimated using P-wave data in this acquisition pattern. Of those estimated parameters are six of the Tsvankin style parameters for orthorhombic media, plus an approximate combination of two others that define vertical S-wave splitting. We show that a previous method, which is based on weak anisotropy approximation, leads to considerable errors, even in models where the magnitude of anisotropy parameters do not exceed 0.1. We design a numerical experiment to study the reliability of the estimated parameters by the exact approach and show the importance of acquisition pattern in this regard. To show applicability, these parameters are used to estimate the in situ fracture properties of the studied layer.

Simultaneous seismic inversion to identify prospective areas in carbonate rocks of the southeastern part of the West Siberian Platform

This paper focuses on the description of a technology and results of identifying and mapping areas in Devonian carbonate rocks prospective for hydrocarbons and associated with increased fracture-cavern porosity in the southeastern part of the West Siberian Platform. The lack of wide-azimuthal seismic data and recording of only the vertical component in the study area did not allow the use of the direct indicators of fracture systems based either on azimuthal anisotropy of the amplitudes of pressure waves or on splitting converted waves into fast and slow. Instead, abnormally low values of the ratio of pressure-wave and shear-wave velocities derived by deterministic simultaneous pre-stack seismic inversion were used as an indirect indicator of increased fracturing. The choice of this seismic signature of fractures is substantiated by a brief review of publications on its successful use in the identification and delineation of highly fractured and cavernous zones in carbonate reservoirs. The behaviour of the indicator showed good agreement with well productivity in the study area and, therefore, made it possible to predict a number of perspective pay zones, presumably associated with in-creased fracturing. At the same time, however, well log estimates of the velocity ratio related to fractured reservoirs exhibited an opposite trend, which mismatched the facts observed in the study area and reported in the literature. This apparent discrepancy is explained by the impact of near-vertical natural macro-fracturing, which, due to different measurement scales, might substantially reduce seismic estimates while producing no impact on well log estimates. In order to describe this phenomenon quantitatively, the weak-anisotropy approximation of P-wave reflection coefficients at a horizontal boundary derived by Rüger for transversely isotropic media with a horizontal axis of symmetry was used. This equation was rewritten in terms of impedances and density and then was analyzed analytically. To express the impact of fractures in terms of fracture density, the simplest model of thin, isolated, penny-shaped fractures was used.

Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer

A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

Moveout approximation for a P-SV wave in a moderately anisotropic homogeneous DTI layer

We derive and test the moveout formula for converted P-SV (or SV-P) waves in a homogeneous, weakly or moderately anisotropic, transversely isotropic layer with axis of symmetry perpendicular to a dipping reflector underlying it. Instead of commonly used Taylor series expansion of the squared traveltime with respect to the offset, we use weak-anisotropy approximation of the exact traveltime formula. We replace the exact traveltime along the actual ray of a converted wave by its first-order weak-anisotropy approximation along a ray in a nearby reference isotropic medium. The traveltime formula is derived in the plane defined by the source-receiver line and the normal to the reflector. The source and the receiver may be situated arbitrarily in the model, the flat reflector may have an arbitrary 3D orientation. The accuracy of the proposed moveout formula is tested on models with varying strength of anisotropy for a varying dip of the reflector. In isotropic media, the formula yields highly accurate results with relative traveltime errors not exceeding 0.5%. In anisotropic media of P- and SV-wave anisotropy of about 25% and 12%, respectively, relative traveltime errors do not exceed 2%.

Weak-anisotropy moveout approximations for P-waves in homogeneous TOR layers

We have developed approximate P-wave reflection moveout formulas for a homogeneous layer of orthorhombic symmetry overlying a horizontal or dipping reflector. The symmetry planes of an orthorhombic medium in the layer can be arbitrarily oriented. The formulas are based on the weak-anisotropy approximation. Thus, their accuracy depends not only on the strength of anisotropy, but also on the deviations of the phase and ray velocities. Nevertheless, the performed tests indicate that for P-wave anisotropy of approximately 25%, the maximum relative traveltime errors of the formulas do not exceed 3%. The formulas are expressed in terms of six P-wave weak-anisotropy parameters and three Euler angles specifying the orientation of the symmetry planes of the orthorhombic medium. In the case of a dipping reflector, the formulas also depend on the apparent dip angle of the reflector.

Fourth-order normal moveout velocity in elastic layered orthorhombic media — Part 1: Slowness-azimuth domain

Considering all types of pure-mode and converted waves, we derive the azimuthally dependent, fourth-order normal moveout (NMO) velocity functions, and hence the corresponding effective anellipticity functions, for horizontally layered orthorhombic media. We emphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. We do not assume weak anisotropy or acoustic approximation for P-waves. At each layer, the elastic parameters, thickness, and azimuth of the orthorhombic vertical symmetry planes are considered to be different. We distinguish between two different azimuths: slowness azimuth (part 1 of this paper) and offset azimuth (part 2 of this paper). In part 1, the slowness-azimuth domain NMO is approximated as a series of either infinitesimal horizontal slowness (slowness-azimuth/slowness domain) or infinitesimal offsets (slowness-azimuth/offset domain). Similarly, in part 2, we distinguish between two offset-azimuth domains: offset-azimuth/slowness and offset-azimuth/offset. Note that the azimuthally dependent NMO velocity functions of each of the four cases are different. The validity of the method is tested by introducing our derived azimuthally dependent, fourth-order effective anellipticity, into the well-known azimuthally dependent, asymptotic nonhyperbolic traveltime approximation, in which we compare the traveltime approximation versus exact numerical ray tracing for short to moderate offsets. It is clearly shown that for these types of azimuthally anisotropic layered models, the fourth-order terms are essential even for relatively small horizontal-slowness values or short offsets.