Transversely Isotropic Research Articles

A new three-dimensional strength criterion considering the transverse isotropy based on the α-Spatial Mobilized Plane

Natural geotechnical materials are affected by sedimentation and exhibit significant anisotropy. To study the transverse isotropy characteristics of soil, the influence of intermediate principal stress and loading direction must be considered. Currently, research on transverse isotropy primarily focuses on the modified stress space, which is cumbersome to apply in multi-yield surface constitutive models. To describe the three-dimensional mechanical properties of geomaterials in real stress space, the α-Spatial Mobilized Plane strength criterion is introduced. Then, combined with the structure tensor, the transverse isotropic three-dimensional strength criterion can account for the effect of the loading angle. Finally, the three-dimensional strengths of Fukakusa clay, unsaturated SP-SC soils, uncemented Monterey sands, Yamaguchi marble, San Francisco Bay mud, Toyoura sand, and Santa Monica Beach sand are predicted on the π-plane. The results show that the αmn-SMP criterion, in the context of transverse isotropy, can describe the three-dimensional mechanical properties reasonably, and it can provide an accurate strength criterion for geotechnical engineering practice.

Fractional Viscoelastic Analysis of Transversely Isotropic Surrounding Rock Along Shallow Buried Elliptical Tunnel

ABSTRACTThis study introduces the fractional order Merchant model to analytically solve the stress and displacement fields of the transversely isotropic viscoelastic surrounding rock along shallow elliptical tunnels. First, the stress and displacement solutions of fractional order viscoelastic and transversely isotropic half planes under arbitrary loads are obtained using the Laplace transform and the elastic‐viscoelastic correspondence principle. Second, based on the above half plane solution and the solution of the deep buried elliptical tunnel problem, the Schwarz alternating method is introduced to obtain the analytical solution of the shallow buried elliptical tunnel. A MATLAB program is developed, and the accuracy of the theory and program in this study is verified by comparing it with the results of ABAQUS. Finally, the effects of transversely isotropic parameters, tunnel burial depth, and viscoelastic parameters on the stress and displacement of tunnel surrounding rock are analyzed.

Symmetries and symmetry-generated averages of elastic constants up to the sixth order of nonlinearity for all crystal classes, isotropy and transverse isotropy.

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non-crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low-symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank-deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database.

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

Experimental Study Of Anisotropic And Dispersion Signature Of Shale Reservoirs: Implications For Synthetic Seismic Well Tie

To investigate velocity anisotropy and dispersion in lacustrine shales, we conducted laboratory measurements of acoustic properties at seismic frequencies (2–100 Hz) and ultrasonic frequencies (1 MHz) under varying effective pressures. Our study reveals significant transverse isotropy, accompanied by notable velocity dispersion and attenuation, with anisotropy coefficients ranging from 0.45 to 0.74 for P-waves and 0.44 to 0.95 for S-waves. Attenuation exhibited weak dependence on frequency and pressure but showed a strong positive correlation with clay content. We interpreted the data by integrating geological and physical analyses. Using rock physics modeling and the propagation matrix method, we found that velocity dispersion primarily caused the mismatch between seismic data and synthetic seismograms, while anisotropy had minimal impact on near-offset seismic data. Our findings suggest that anisotropy and dispersion observed at the core scale persist at the seismic scale. Incorporating these effects into seismic workflows is crucial for improving data processing, inversion, and reservoir characterization. This study provides valuable insights for exploration and production strategies in the study area and similar settings.

Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation

This paper investigates and analyzes the dynamic response of asphalt pavement under pavement-unevenness excitation based on an orthogonal vector function system and efficient DVP (dual variable and position) method. Firstly, starting from the pavement unevenness of the vehicle excitation source, the pavement-unevenness excitation is established by using the filtered white-noise method, and the random load of the vehicle model is obtained by simulation. Then, based on the basic governing equation of the road-surface problem under the random load, the analytical solution of the road-surface mechanical response is obtained by using the orthogonal vector function system and DVP method. The effects of pavement-unevenness grade, vehicle speed, vehicle load, interlayer contact condition, and transverse isotropy on the mechanical response of the road surface are analyzed via the analytical results. The results show that DVP can effectively solve the dynamic response of pavements under the excitation of pavement unevenness; in addition, it can also be applied to certain situations, such as transverse isotropy of materials and interface conditions. The results show that the pavement unevenness does not affect the average stress and strain of each layer but has a significant effect on the peak value and dispersion degree. An increase in vehicle speed causes a peak in strain and a larger coefficient of variation. Poor bonding between interfaces can lead to increased stress and strain at the bottom of the surface layer.

3-D parallel anisotropic inversion of controlled-source electromagnetic data using nested tetrahedral grids

SUMMARY Geophysicists today face the challenge of quickly and reliably interpreting extensive controlled-source electromagnetic (CSEM) data sets to map subsurface conductivity structures within realistic geological environments. An ideal 3-D CSEM inversion algorithm using tetrahedral grids should be capable of distinguishing different resolution requirements between forward modelling and inversion grids, have an optimal parallel strategy that fully exploits the inherent independence of CSEM data sets while also possessing the capability to handle large-scale geo-electrical models, and incorporate conductivity anisotropy which should be a common characteristic in realistic subsurface environments. However, existing tools in the geo-electromagnetic community often fall short of these three demands. Addressing this gap, our study introduces a scalable and parallel anisotropic inversion technique for CSEM data, capitalizing on the potential of unstructured tetrahedral grids. We first apply the tetrahedral longest-edge bisection method to create a refined dense, heterogeneous forward modelling grid from a coarse inversion grid. This refinement, focused on areas around transmitters and receivers, is seamlessly integrated within the coarser inversion grid’s topology, enabling precise conductivity mapping and preserving electromagnetic response accuracy during model updates. We further innovate with a source-mesh double-level parallel strategy, utilizing the message passing interface technique for parallel handling of independent CSEM data sets and large-scale geo-electrical models. Externally, we dedicate a processor for inversion model updates employing the Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and divide other processors into groups, each associated with specific transmitting sources and frequencies. Internally, in each group, we employ a domain-decomposition-based scalable and robust iterative solvers using the Auxiliary-Space Maxwell pre-conditioner to parallel quickly calculate the electromagnetic responses from its assigned source-frequency set. Additionally, recognizing the potential for electrical conductivity anisotropy in field data, we incorporate the case of vertical transverse isotropy. We validate the effectiveness of our method through examples, including an isotropic land model with undulating topography, an anisotropic marine model and a real-field data case. Results from both synthetic and field data inversions underscore our method’s significant advancements in efficiency and practicality, particularly in addressing large-scale 3-D CSEM data sets inversion challenges in realistic geological environments.

Dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading

This paper conducts the dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading. Based on the Biot theory and transversely isotropic (TI) parameter expression of the geogrid reinforced subgrade, the governing equations of the poroelastic reinforced subgrade are established in the wavenumber domain by the double Fourier transform. Considering the viscosity of the soil skeleton and the flow-dependent viscosity between the soil skeleton and pore water, the governing equations are extended to the fractional poroviscoelastic medium by introducing the Zener viscoelastic model, fractional calculus theory and the dynamic elastic-viscoelastic correspondence principle. Combining boundary conditions and interlayer continuity conditions, the extended precise integration method (PIM) and double Fourier integral transform are employed to obtain the solution of fractional poroviscoelastic reinforced subgrade in the spatial domain. After the numerical validation, a sensitivity analysis of the relaxation time, permeability, reinforcement ratio and the load velocity are conducted.

Effects of background elastic and permeability anisotropy on dynamic seismic signatures of a fluid-saturated porous rock with aligned fractures

Seismic dispersion, attenuation, and frequency-dependent anisotropy due to wave-induced fluid flow (WIFF) in fluid-saturated porous and fractured rocks have been widely studied. However, most of these studies do not consider the effects of background elastic and permeability anisotropy. In this work, we study such effects by considering a fluid-saturated porous rock with penny-shaped fractures embedded in the isotropic plane of a transversely isotropic (TI) background. Starting with P-waves propagating perpendicularly to the fracture plane, we obtain the wavefields in the fluid-saturated porous TI background scattered from a single fracture. This allows for the calculation of effective P-wavenumber through the Foldy approximation and the derivation of a frequency-dependent fracture compliance matrix. The angle-dependent seismic dispersion and attenuation, as well as frequency-dependent anisotropy, can thus be calculated. Using this model, we study a fractured tight sandstone with a TI background. The results show that the background elastic anisotropy primarily affects the seismic frequency-dependent anisotropy, while the background permeability anisotropy not only affects seismic frequency-dependent anisotropy but also influences seismic dispersion and attenuation. In particular, the background permeability anisotropy alters the rate of change in seismic velocities and velocity anisotropy parameters with frequencies and the peaks of seismic attenuation and attenuation anisotropy parameters. By comparing the theoretical predictions of our model to other theories and also to two sets of experimental data, our model is validated. Our model can be applied to characterize the fractured, layered earth.

Three-phase-lag analysis of transversely isotropic double porous thermoelastic waves with liquid medium

This manuscript presents a mathematical framework for a double porous, thermoelastic solid half-space with transverse isotropy in contact with an inviscid liquid half-space. The analysis takes into account the three-phase-lag model within thermoelasticity theory. To gain deeper insights, we formulate the governing equations in a two-dimensional representation and subsequently employ normal mode analysis for their rigorous solution. The governing equations are influenced by both types of voids, anisotropy, thermal effect, and inviscid liquid. It has been discovered that thermoelastic half-space exhibits five coupled longitudinal waves, and in the liquid half-space, there is one mechanical wave. Secular equations are obtained by using thermal and mechanical conditions, resulting in a compact representation of phase velocity, specific loss, penetrating depth, and attenuation coefficient. Analytical presentations have been made to showcase the effects of anisotropy, voids, and liquid on various wave parameters. These effects are compared by illustrating the results graphically using the MATLAB software, allowing for a visual understanding of the comparative outcomes. This study has practical applications in engineering, materials science, geophysics, and environmental studies, contributing to advancements in various industries and scientific fields.

Torsional dynamic response of a large-diameter pipe pile in transversely isotropic unsaturated soils considering construction disturbance

Large-diameter pipe piles are widely applied in various civil engineering fields due to their outstanding load-bearing capability. The unsaturated characteristics, anisotropy, and heterogeneity of the soil jointly affect the dynamic response of the pipe pile. However, most previous studies were limited to single-phase or two-phase soil. This paper develops an analytical model for the torsional vibration of a pipe pile in transversely isotropic unsaturated soils considering construction disturbance. Based on transversely isotropic unsaturated soil theory, a pipe pile-soil interaction model has been developed, while the effect of construction disturbance is simulated by the radial heterogeneity of the soil. The general solution for the unsaturated soil is obtained using the separation of variables method with the boundary conditions. Then, the solution for the whole pipe pile-soil system is derived by considering the pile-soil interface conditions. The accuracy of the proposed solution is verified through comparisons with previous research results. The results show that unsaturated characteristics, construction disturbance, and transverse isotropy of the soil have significant effects on the impedance of large-diameter pipe piles. Specifically, with a low degree of saturation, there will be significant prediction errors when using previous works based on single-phase or two-phase soil theory to predict the dynamic response of large-diameter pipe piles.

Seismic anisotropy of granitic rocks from a fracture stimulation well at Utah FORGE using ultrasonic measurements

For effective fracture stimulation in Enhanced Geothermal Systems (EGS), characterization of subsurface rock elastic properties is essential. Because seismic anisotropy significantly influences hydraulic fracturing, locating and characterizing microseismic events, and forecasting stress trajectories, our research undertakes laboratory experiments to study seismic anisotropy parameters of core samples from the Utah FORGE EGS site. We evaluated two rock samples from well 58–32, at two depths of 2074.16 m (6,805 feet) and 2268.01 m (7,441 feet), respectively. We analyze the mineralogy and classify the shallower rock as syenogranite, and the deeper rock as orthogneiss and monzonite. We cut cylindrical plugs from these two core samples, imaged them by computed tomography (CT) X-ray imaging, and measured their seismic anisotropy using ultrasonic waves at 1 MHz and under varying confining pressures (5–50 MPa) in ambient temperature conditions. With the assumption of the transversely isotropic (TI) medium, we measured three P-wave velocities and two S-wave velocities along different propagation directions to compute the Thomsen anisotropy parameters. Our results showed that the measured seismic velocities of each rock sample increase with increasing effective pressures, a behavior likely caused by closure of microcracks identified in our CT images. We also find that the measured anisotropies are higher at lower effective pressures. The maximum measured anisotropies for P-waves and S-waves are ∼23% and ∼20%, respectively. Therefore, we expect in the field operation that anisotropy should increase with increasing pore pressure caused by injection, which could be valuable in future EGS reservoir management and monitoring.

Pure quasi-P-wave modeling and imaging using an approximated space-domain pseudo-differential operator in vertical transverse isotropy media

Incorporating anisotropy in seismic modeling and imaging is important to produce the correct locations of subsurface reflectors. Traditional wave equations for quasi-P-wave in transverse isotropic media either suffer from S-wave artifacts or require complicated and expensive computation strategies. To mitigate this issue, we develop a novel pure quasi-P-wave equation with an approximated space-domain pseudo-differential operator in the vertical transverse isotropic (VTI) medium. For the pure quasi-P-wave equation, we first simplify it to an elliptical anisotropy equation with an additional pseudo-differential correction term. Then, we directly approximate the pseudo-differential term with a space-domain convolution operator that is calculated by solving a nonlinear inverse problem. Phase-velocity analysis and numerical modeling show that the new space-domain pseudo-differential operator has good accuracy in describing wave propagation in the VTI medium. In addition, it is more suitable for parallel computation with domain-decomposition than the Fourier transform, which is necessary for solving traditional pseudo-differential operators. Finally, we apply our quasi-P-wave propagator to reverse time migration to correct the anisotropic effects in seismic imaging. Numerical experiments for the benchmark models and a land survey demonstrate the feasibility and adaptability of our method.

Lateral Behavior Analysis of a Rectangular Barrette in Layered Soil with Transverse Isotropy

Lateral Behavior Analysis of a Rectangular Barrette in Layered Soil with Transverse Isotropy

Numerical and experimental study on anisotropic heat transfer behaviors of quartz fabric composite preforms: Multiple micro‐scale models method

AbstractThis paper presents a comprehensive study on anisotropic heat transmission behaviors in quartz fiber fabrics. Considering the random distribution characteristic of fibers within the yarn, an innovative two‐scale finite element method (tFEM) is introduced, incorporating multiple micro‐scale fiber models (MMFM) based on scanning electron microscope (SEM) observations. MMFM are divided into 8 distinct models to capture intricate fiber arrangements. The macro‐scale fabric model is constructed based on the geometric structure analysis by Micro‐CT technology. Both micro‐ and macro‐scale models are assembled with the air matrix to form the two‐phase composite model. The Hot‐Disk thermal analysis instrument is applied to measure the anisotropic thermal conductivity (ATC). Numerical results from MMFM shows more excellent agreements with the experimental ones at 3D orthogonal directions of fabrics, i.e., the error rates of the thermal conductivity in the warp, weft and thickness directions between numerical and experimental methods are all less than 3%, which indicates the tFEM including MMFM in this paper is more accurate than the tFEM including OSMFM. In addition, the temperature distribution and heat transmission differences due to fiber arrangements are simulated and illustrated through the MMFM. Afterwards, temperature drop and isothermal characteristics are demonstrated in this study.Highlights Effects of fiber arrangements on steady heat transfer behaviors at micro‐scale are illustrated. Isothermal and temperature‐drop characteristics at micro‐ and macro‐scale are simulated and studied. The transverse isotropy of thermal conductivity at micro‐scale and the anisotropy of that at macro‐scale are revealed.

Horizontal Transverse Isotropy Studied Using Tools for Full-Azimuth Seismic Data Processing and an Advanced Well Logging Complex on the Example of Famennian Deposits of the Yugomashevskoye Field

Abstract —This paper presents a solution to the problem of searching for cavernous fractured zones and the zones that determine reservoir heterogeneity in Famennian deposits of one of the licensed sections of the Republic of Bashkortostan. The use of an integrated approach to the study of azimuthal horizontal-transverse isotropy of rocks using wellbore data and modern methods for processing areal seismic data for carbonate deposits associated with the development of organogenic structures is described. The results of using the full-azimuth angular migration technology are obtained. The introduction of high-tech approaches at the research site using full-azimuth seismic data processing, spectral decomposition, AVA/AVAZ inversion, and complex analysis of dynamic properties of the wave field in conjunction with well logging and core data makes it possible to identify the distorted zones of the carbonate reservoir associated with changes in the characteristics of the rock skeleton, the presence of a porous-cavernous-fracture void space and lithological replacement zones, as well as secondary processes, thereby reducing the risks associated with subsequent production drilling.

Lateral and vertical variations of seismic anisotropy in the lithosphere–asthenosphere system underneath Central Europe from long-term splitting measurements

SUMMARY Backazimuthal variations in the shear wave splitting of core-refracted shear waves (SKS, SKKS and PKS phases, jointly referred to as XKS) at the Black Forest Observatory (BFO, Southwest Germany) indicate small-scale lateral and partly vertical variations of the seismic anisotropy. However, existing anisotropy studies and models for the nearby Upper Rhine Graben (URG) area in the northern Alpine foreland are mostly based on short-term recordings and by this suffer from a limited backazimuthal coverage and averaging over a wide or the whole backazimuth range. To identify and delimit laterally confined anisotropy regimes in this region, we carry out XKS splitting measurements at six neighbouring (semi-)permanent broad-band seismological recording stations (interstation distance 10–80 km). We manually analyse long-term (partly > 20 yr) recordings to achieve a sufficient backazimuthal coverage to resolve complex anisotropy. The splitting parameters (fast polarization direction $\phi $, delay time $\delta t$) are determined in a single- and multi-event analysis. We test structural anisotropy models with one layer with horizontal or tilted symmetry axis and with two layers with horizontal symmetry axes (transverse isotropy). To account for lateral variations around a single recording site, modelling is compared for the whole and for limited backazimuth ranges. Based on this, we provide a 3-D block model with spatial variation of anisotropic properties. Based on delay times > 0.3 s and missing discrepancies between SKS and SKKS phases, which do not support lower mantle anisotropy, the found anisotropy is placed in the lithosphere and asthenosphere. The spatial distribution as well as the lateral and backazimuthal variations of the splitting parameters confirm lateral and partly vertical variations in anisotropy. On the east side of the URG, we suggest two anisotropic layers in the Moldanubian Zone (south) and one anisotropic layer in the Saxothuringian Zone (north). In the Moldanubian Zone, a change of the fast polarization directions is observed between the east and the west side of the URG, indicating different textures. At the boundary between the two terranes, an inclined anisotropy is modelled which may be related with deformation during Variscan subduction. Regarding the observation of numerous null measurements and inconsistent splitting parameters, especially (southwest of BFO) in the southern URG, different hypothesis are tested: scattering of the seismic wavefield due to small-scale lateral heterogeneities, a vertical a-axis due to a vertical mantle flow related to the Kaiserstuhl Volcanic Complex, as well as a different preferred orientation of the olivine crystals (not A-type, but C-type) due to specific ambient conditions (high temperature, water content).

Anisotropy in shale and its impact on AVO modelling and prospect de-risk: a case study of a Central North Sea field

The inherent anisotropy in shale is an attribute associated with its clay minerals platelets alignment and layering and very common with transversely isotropic (TI) medium. This implies that velocity is fastest along the horizontal axis and slowest along the vertical axis, which could affect velocity signature from sonic tools measurements in deviated wells. This may lead to wrong amplitude variation with offset (AVO) signature, inaccurate depth conversions, seismic-to-well tie difficulty etc. Nineteen (19) wells in Forties field andthirteen (13) in Huntington field were investigated and interpreted for the presence of anisotropy effect within the caprock over the reservoir Forties sands of the Sele (shales) Formation. In order to analyse the Thomsen weak anisotropy parameters (ϵ, γ and δ) per field, wells with closer Sele depth range were considered. The well inclinations ( θ ), average compressional (α) and shear (β) velocities per well were utilised for forward and inverse modelling of Thomsen anisotropy equations. The estimated anisotropy (VTI) parameters and the P-wave ( α o ) and S-wave velocities (βo ) at zero inclinations in Sele Formation for the studied wells are: α o = 2448 ± 15.9 m/s, β o = 1003 ± 34.8 m/s, ϵ = 0.15 ± 0.02, γ = 0.33 ± 0.10, δ = 0.030 ± 0.015 in the Forties field Sele, while α o = 2775 ± 23.6 m/s, β o = 1306 ± 47.9 m/s, ϵ = 0.22 ± 0.02, γ = 0.43 ± 0.14, δ = 0.022 ± 0.018 in the Huntington field. AVO responses were compared, assuming both isotropic and anisotropic P–P reflectivity models. Result shows an increase in AVO gradient for all anisotropic case in both fields. AVO signatures were either completely or averagely masked in both fields using the isotropic assumptions. There is also an increase in AVO gradient for all anisotropic cases in both fields. Velocity decreases of about 1.2% and 3.5% were observed in the Sele Formation for the Forties and Huntington fields respectively for anisotropy. The corrected two-way traveltime (TWT) should in principle allow for a better seismic-to-well tie estimation. Similarly, using the faster deviated velocity (i.e. uncorrected velocity) in any depth conversion would result in under estimation of the actual depth across the field.

3D reverse-time migration for pure P-wave in orthorhombic media

Compared with the transverse isotropic (TI) medium, the orthorhombic anisotropic medium has both horizontal and vertical symmetry axes and it can be approximated as a set of vertical fissures developed in a group of horizontal strata. Although the full-elastic orthorhombic anisotropic wave equation can accurately simulate seismic wave propagation in the underground media, a huge computational cost is required in seismic modeling, migration, and inversion. The conventional coupled pseudo-acoustic wave equations based on acoustic approximation can be used to significantly reduce the cost of calculation. However, these equations usually suffer from unwanted shear wave artifacts during wave propagation, and the presence of these artifacts can significantly degrade the imaging quality. To solve these problems, we derived a new pure P-wave equation for orthorhombic media that eliminates shear wave artifacts while compromising computational efficiency and accuracy. In addition, the derived equation involves pseudo-differential operators and it must be solved by 3D FFT algorithms. In order to reduce the number of 3D FFT, we utilized the finite difference and pseudo-spectral methods to conduct 3D forward modeling. Furthermore, we simplified the equation by using elliptic approximation and implemented 3D reverse-time migration (RTM). Forward modeling tests on several homogeneous and heterogeneous models confirm that the accuracy of the new equation is better than that of conventional methods. 3D RTM imaging tests on three-layer and SEG/EAGE 3D salt models confirm that the ORT media have better imaging quality.

Weakly anelliptical traveltime analysis: Ambiguity between subsurface and elasticity

The building of a subsurface and (anisotropic) velocity model from a single gather of reflection traveltime (kinematic) data is inherently ambiguous because the processing of such data can only determine a horizontal slowness component, not a vertical one. Based thereon, I derive a simple algorithm that generates an infinite series of combinations of the subsurface-velocity models, all of which will show nearly the same seismic kinematic response, as further demonstrated by simulating wave propagation through a model with different interface dips. This algorithm assumes, first, that all interface dips remain constant over the distance considered and, second, that an approximation of the elasticity model — that is, the linearization of a phase velocity — valid for weak anisotropy can be used. Furthermore, when applied to the classic and analytically solvable case of traveltime analysis for a stack of flat layers with weak transverse isotropy, the algorithm theoretically explains the combination of anisotropy parameters that govern the nonhyperbolic term of a traveltime series: the established [Formula: see text] and its new counterpart [Formula: see text] for a [Formula: see text] and [Formula: see text] wave, respectively.