Tilted Transversely Isotropic Media Research Articles

Modeling of pure qP- and qSV-waves in tilted transversely isotropic media with the optimal quadratic approximation

Seismic forward modeling in tilted transverse isotropic (TTI) media is crucial for the application of reverse time migration and full-waveform inversion. Modeling based on conventional coupled pseudoacoustic wave equations not only generates SV-wave artifacts, but it also suffers from instabilities in which the anisotropy parameter [Formula: see text]. To address these issues, we have started with the exact vertical transversely isotropic phase velocity formula and developed novel pure qP- and qSV-wave governing equations in TTI media by using the optimal quadratic approximation. For the convenience of using finite-difference (FD) method to solve the new pure qP- and qSV-wave wave equations, we decompose the equations into a combination of a time-space-domain wave equation that can be solved by the FD method and a Poisson equation that can be solved by the pseudospectral method. We find that the high-frequency errors caused by the pseudospectral method and the usual truncation errors in FD schemes may be responsible for the instability of the numerical simulations. To stabilize the computation, we design a 2D low-pass filtering operator to eliminate severe high-frequency numerical noise. Several numerical examples demonstrate that modeling using the new pure qP-wave equations does not have qSV-wave artifacts interference and is stable for [Formula: see text]. Our results indicate that our method can achieve highly accurate and stable modeling results even in extremely complex TTI media.

Dispersion and Stability Condition of Seismic Wave Simulation in TTI Media

For seismic waveform simulation in tilted transversely isotropic (TTI) media, we derive explicitly the numerical dispersion relation and the stability condition for the computation of a 2D pseudo-acoustic wave equation. The numerical dispersion relation indicates that the number of sampling points per wavelength has the greatest influence on the dispersion, while the anisotropic parameters of the TTI media and the mesh rotation angle have little influence on the dispersion. Given an appropriate spatial sampling, the stability condition is for the selection of the time step for the implementation of the TTI wave equation. We partition a numerical model using quadrangle grids in Cartesian coordinates, and map it to a computing model in which any non-rectangular meshes in Cartesian coordinates become rectangular meshes. Then we reformulate the pseudo-acoustic wave equation for the TTI media accordingly in the computational space. We implement seismic waveform simulation using the second-order finite-difference method straightforwardly, and show examples with a desirable accuracy using a model with non-rectangular meshes in Cartesian coordinates along a curved surface and fluctuating interfaces in the TTI media.

Calculation of slowness vectors from ray directions for qP-, qSV-, and qSH-waves in tilted transversely isotropic media

Kinematic ray tracing is an effective way to simulate the seismic wave propagation in isotropic and anisotropic media. It is essential to know the ray velocity when tracing seismic rays. But in anisotropic media, the ray velocity is a function of the direction of the slowness vector instead of the ray direction and it often deviates from the phase velocity. In this case, it causes a critical problem for ray tracing, which is how to calculate the ray velocity from a known ray direction. If we could calculate the phase slowness vector from ray directions, the ray velocity could be computed. We have evaluated a previous method in the first place. Then, we developed two new methods to solve two existing problems of the previous method: (1) It leads to complex and multiple solutions of the slowness vector and (2) it mixes up the qP- and qSV-wave modes. Our first method solves the two problems by applying eigenvalues to separate the wave modes and decrease the two unknowns ([Formula: see text] and [Formula: see text]) to only one unknown in two equations. Our second method is based on the general relationship between the slowness and ray-velocity vectors and shows that only one unknown is involved in one equation for tilted transversely isotropic (TTI) media. After obtaining the slowness vector, the ray velocity can be computed easily. A 2D model is designed to test the feasibility of the new methods. Using the results for the model, we found that the presented approaches were applicable for ray tracing in TTI media.

Pseudoacoustic tilted transversely isotropic modeling with optimal k-space operator-based implicit finite-difference schemes

Current temporal high-order finite-difference (FD) stencils are mainly designed for isotropic wave equations, which cannot be directly extended to pseudoacoustic wave equations (PWEs) in tilted transversely isotropic (TTI) media. Moreover, it is difficult to obtain the time-space domain FD coefficients for anisotropic PWEs based on nonlinear dispersion relations in which anisotropy parameters are coupled with FD coefficients. Therefore, a second-order FD for temporal derivatives and a high-order FD for spatial derivatives are commonly used to discretize PWEs in TTI media. To improve the temporal and spatial modeling accuracy further, we have developed several effective FD schemes for modeling PWEs in TTI media. Through combining the [Formula: see text] (wavenumber)-space operators with the conventional implicit FD stencils (i.e., regular-grid [RG], staggered-grid [SG], and rotated SG [RSG]), we establish novel dispersion relations and determine FD coefficients using least-squares (LS). Based on [Formula: see text]-space operator compensation, we adopt the modified LS-based implicit RG-FD, implicit SG-FD, and implicit RSG-FD methods to respectively solve the second- and first-order PWEs in TTI media. Dispersion analyses indicate that the modified implicit FD schemes based on [Formula: see text]-space operator compensation can greatly increase the numerical accuracy at large wavenumbers. Modeling examples in TTI media demonstrate that the proposed FD schemes can adopt a short FD operator to simultaneously achieve high temporal and spatial modeling accuracy, thus significantly improve the computational efficiency compared with the conventional methods.

Acoustic wave propagation in tilted transversely isotropic media: Incorporating topography

Simulating two-way acoustic wavefield propagation directly from a free-surface boundary in the presence of topography remains a computational challenge for applications of reverse time migration (RTM) or full-waveform inversion (FWI). For land-seismic settings involving heavily reworked geology (e.g., fold and thrust belts), two-way wavefield propagation operators should also handle commonly observed complex anisotropy including tilted transversely isotropic (TTI) media. To address these issues, I have extended a system of coupled partial differential equations used to model 3D acoustic TTI wave propagation in Cartesian coordinates to more generalized 3D geometries, including a deformed computational mesh with a domain boundary conformal to free-surface topography. A generalized curvilinear transformation is used to specify a system of equations governing 3D acoustic TTI wave propagation in the “topographic” coordinate system. The developed finite-difference time-domain numerical solution adapts existing Cartesian TTI operators to this more generalized geometry with little additional computational overhead. Numerical evaluations illustrate that 2D and 3D impulse responses are well-matched to those simulated on Cartesian meshes and analytic traveltimes for homogeneous elliptical TTI media. Accordingly, these generalized acoustic TTI propagators and their numerical adjoints are useful for undertaking most RTM or FWI applications using computational domains conforming to free-surface topography.

Modeling of pure acoustic wave in tilted transversely isotropic media using optimized pseudo-differential operators

Conventional modeling and imaging for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear artifacts due to the coupling of the P- and SV-wave modes in the coupled equations. On the contrary, the decoupled equations for TTI media provide a more stable solution due to the separated P- and S-wave modes. Because the decoupled equations involve complicated pseudo-differential operators in space, it is more convenient to apply the pseudo-spectral method to these equations. However, the second-order time-stepping scheme of the pseudo-spectral method may suffer from time-stepping errors and instabilities for a large time step. We have developed an optimized pseudo-differential operator (OPO) that incorporates not only the spectral evaluation of the pseudo-differential operator but also a temporal correction term that would effectively compensate the time-stepping errors associated with the time-wavenumber domain extrapolation of the wave equation. The OPO was constructed through multiplying the symbol of the OPO by the normalized pseudo-Laplacian operator, which contains a variable compensation velocity. It was efficiently solved through the low-rank decomposition. We have applied OPOs to solve the TTI decoupled equation to simulate the pure acoustic wave. Our 2D and 3D synthetic results demonstrate that the proposed method has high accuracy in time and space with relaxed stability conditions compared with the conventional pseudo-spectral method. The low rank of symbols of OPOs makes the proposed method more efficient than the dispersion relation-based low-rank wave extrapolation and pseudo-analytical methods.

Imaging tilted transversely isotropic media with a generalised screen propagator

One-way wave equation migration is computationally efficient compared with reverse time migration, and it provides a better subsurface image than ray-based migration algorithms when imaging complex structures. Among many one-way wave-based migration algorithms, we adopted the generalised screen propagator (GSP) to build the migration algorithm. When the wavefield propagates through the large velocity variation in lateral or steeply dipping structures, GSP increases the accuracy of the wavefield in wide angle by adopting higher-order terms induced from expansion of the vertical slowness in Taylor series with each perturbation term. To apply the migration algorithm to a more realistic geological structure, we considered tilted transversely isotropic (TTI) media. The new GSP, which contains the tilting angle as a symmetric axis of the anisotropic media, was derived by modifying the GSP designed for vertical transversely isotropic (VTI) media. To verify the developed TTI-GSP, we analysed the accuracy of wave propagation, especially for the new perturbation parameters and the tilting angle; the results clearly showed that the perturbation term of the tilting angle in TTI media has considerable effects on proper propagation. In addition, through numerical tests, we demonstrated that the developed TTI-GS migration algorithm could successfully image a steeply dipping salt flank with high velocity variation around anisotropic layers.

Optimal rotated staggered-grid finite-difference schemes for elastic wave modeling in TTI media

The rotated staggered-grid finite-difference (RSFD) is an effective approach for numerical modeling to study the wavefield characteristics in tilted transversely isotropic (TTI) media. But it surfaces from serious numerical dispersion, which directly affects the modeling accuracy. In this paper, we propose two different optimal RSFD schemes based on the sampling approximation (SA) method and the least-squares (LS) method respectively to overcome this problem. We first briefly introduce the RSFD theory, based on which we respectively derive the SA-based RSFD scheme and the LS-based RSFD scheme. Then different forms of analysis are used to compare the SA-based RSFD scheme and the LS-based RSFD scheme with the conventional RSFD scheme, which is based on the Taylor-series expansion (TE) method. The contrast in numerical accuracy analysis verifies the greater accuracy of the two proposed optimal schemes, and indicates that these schemes can effectively widen the wavenumber range with great accuracy compared with the TE-based RSFD scheme. Further comparisons between these two optimal schemes show that at small wavenumbers, the SA-based RSFD scheme performs better, while at large wavenumbers, the LS-based RSFD scheme leads to a smaller error. Finally, the modeling results demonstrate that for the same operator length, the SA-based RSFD scheme and the LS-based RSFD scheme can achieve greater accuracy than the TE-based RSFD scheme, while for the same accuracy, the optimal schemes can adopt shorter difference operators to save computing time.

Alternative stable qP wave equations in TTI media with their applications for reverse time migration

Numerical instabilities may arise if the spatial variation of symmetry axis is handled improperly when implementing P-wave modeling and reverse time migration in heterogeneous tilted transversely isotropic (TTI) media, especially in the cases where fast changes exist in TTI symmetry axis' directions. Based on the pseudo-acoustic approximation to anisotropic elastic wave equations in Cartesian coordinates, alternative second order qP (quasi-P) wave equations in TTI media are derived in this paper. Compared with conventional stable qP wave equations, the proposed equations written in stress components contain only spatial derivatives of wavefield variables (stress components) and are free from spatial derivatives involving media parameters. These lead to an easy and efficient implementation for stable P-wave modeling and imaging. Numerical experiments demonstrate the stability and computational efficiency of the presented equations in complex TTI media.

Common-reflection-point migration velocity analysis of 2D P-wave data from TTI media

We have developed a common-reflection-point (CRP)-based kinematic migration velocity analysis for 2D P-wave reflection data to estimate the four transversely isotropic (TI) parameters [Formula: see text], [Formula: see text], and [Formula: see text], and the tilt angle [Formula: see text] of the symmetry axis in a TI medium. In each iteration, the tomographic parameter was updated alternately with prestack anisotropic ray-based migration. Iterations initially used layer stripping to reduce the number of degrees of freedom; after convergence was reached, a couple of more iterations over all parameters and all CRPs ensured global interlayer coupling and parameter interaction. The TI symmetry axis orientation was constrained to be locally perpendicular to the reflectors. The [Formula: see text] dominated the inversion, and so it was weighted less than [Formula: see text] and [Formula: see text] in the parameter updates. Estimates of [Formula: see text] and [Formula: see text] were influenced if the error in [Formula: see text] was [Formula: see text]; estimates of [Formula: see text] were also influenced if the error in [Formula: see text] was [Formula: see text]. Examples included data for a simple model with a homogeneous TI layer whose dips allowed recovery of all anisotropy parameters from noise-free data, and a more realistic model (the BP tilted transversely isotropic (TTI) model) for which only [Formula: see text], [Formula: see text], and [Formula: see text] were recoverable. The adequacy of the traveltimes predicted by the inverted anisotropic models was tested by comparing migrated images and common image gathers, with those produced using the known velocity models.

Ray tracing of multiple transmitted/reflected/converted waves in 2-D/3-D layered anisotropic TTI media and application to crosswell traveltime tomography

To overcome the deficiency of some current grid-/cell-based ray tracing algorithms, which are only able to handle first arrivals or primary reflections (or conversions) in anisotropic media, we have extended the functionality of the multistage irregular shortest-path method to 2-D/3-D tilted transversely isotropic (TTI) media. The new approach is able to track multiple transmitted/reflected/converted arrivals composed of any kind of combinations of transmissions, reflections and mode conversions. The basic principle is that the seven parameters (five elastic parameters plus two polar angles defining the tilt of the symmetry axis) of the TTI media are sampled at primary nodes, and the group velocity values at secondary nodes are obtained by tri-linear interpolation of the primary nodes across each cell, from which the group velocities of the three wave modes (qP, qSV and qSH) are calculated. Finally, we conduct grid-/cell-based wave front expansion to trace multiple transmitted/reflected/converted arrivals from one region to the next. The results of calculations in uniform anisotropic media indicate that the numerical results agree with the analytical solutions except in directions of SV-wave triplications, at which only the lowest velocity value is selected at the singularity points by the multistage irregular shortest-path anisotropic ray tracing method. This verifies the accuracy of the methodology. Several simulation results show that the new method is able to efficiently and accurately approximate situations involving continuous velocity variations and undulating discontinuities, and that it is suitable for any combination of multiple transmitted/reflected/converted arrival tracking in TTI media of arbitrary strength and tilt. Crosshole synthetic traveltime tomographic tests have been performed, which highlight the importance of using such code when the medium is distinctly anisotropic.

Seismic wavefield simulation in 2D elastic and viscoelastic tilted transversely isotropic media: comparisons between four different kinds of finite-difference grid schemes

In this paper, we use the staggered grid, the auxiliary grid, the rotated staggered grid and the non-staggered grid finite-difference methods to simulate the wavefield propagation in 2D elastic tilted transversely isotropic (TTI) and viscoelastic TTI media, respectively. Under the stability conditions, we choose different spatial and temporal intervals to get wavefront snapshots and synthetic seismograms to compare the four algorithms in terms of computational accuracy, CPU time, phase shift, frequency dispersion and amplitude preservation. The numerical results show that: (1) the rotated staggered grid scheme has the least memory cost and the fastest running speed; (2) the non-staggered grid scheme has the highest computational accuracy and least phase shift; (3) the staggered grid has less frequency dispersion even when the spatial interval becomes larger.

Anisotropic parsimonious prestack depth migration

An efficient Kirchhoff-style prestack depth migration, called “parsimonious” migration, was developed a decade ago for isotropic 2D and 3D media by using measured slownesses to reduce the amount of ray tracing by orders of magnitude. It is conceptually similar to “map” migration, but its implementation has some differences. We have extended this approach to 2D tilted transversely isotropic (TTI) media and illustrated it with synthetic P-wave data. Although the framework of isotropic parsimonious may be retained, the extension to TTI media requires redevelopment of each of the numerical components, calculation of the phase and group velocity for TTI media, development of a new two-point anisotropic ray tracer, and substitution of an initial-angle isotropic shooting ray-trace algorithm for an anisotropic one. The model parameterization consists of Thomsen’s parameters ([Formula: see text], [Formula: see text], [Formula: see text]) and the tilt angle of the symmetry axis of the TI medium. The parsimonious anisotropic migration algorithm is successfully applied to synthetic data from a TTI version of the Marmousi2 model. The quality of the image improves by weighting the impulse response by the calculation of the anisotropic Fresnel radius. The accuracy and speed of this migration makes it useful for anisotropic velocity model building. The elapsed computing time for 101 shots for the Marmousi2 TTI model is 35 s per shot (each with 501 traces) in 32 Opteron cores.

Migration velocity analysis for TI media in the presence of quadratic lateral velocity variation

One of the most serious problems in anisotropic velocity analysis is the trade-off between anisotropy and lateral heterogeneity, especially if velocity varies on a scale smaller than the maximum offset. We have developed a P-wave MVA (migration velocity analysis) algorithm for transversely isotropic (TI) models that include layers with small-scale lateral heterogeneity. Each layer is described by constant Thomsen parameters [Formula: see text] and [Formula: see text] and the symmetry-direction velocity [Formula: see text] that varies as a quadratic function of the distance along the layer boundaries. For tilted TI media (TTI), the symmetry axis is taken orthogonal to the reflectors. We analyzed the influence of lateral heterogeneity on image gathers obtained after prestack depth migration and found that quadratic lateral velocity variation in the overburden can significantly distort the moveout of the target reflection. Consequently, medium parameters beneath the heterogeneous layer(s) are estimated with substantial error, even when borehole information (e.g., check shots or sonic logs) is available. Because residual moveout in the image gathers is highly sensitive to lateral heterogeneity in the overburden, our algorithm simultaneously inverts for the interval parameters of all layers. Synthetic tests for models with a gently dipping overburden demonstrate that if the vertical profile of the symmetry-direction velocity [Formula: see text] is known at one location, the algorithm can reconstruct the other relevant parameters of TI models. The proposed approach helps increase the robustness of anisotropic velocity model-building and enhance image quality in the presence of small-scale lateral heterogeneity in the overburden.

Acceleration of stable TTI P-wave reverse-time migration with GPUs

When a pseudo-acoustic TTI (tilted transversely isotropic) coupled wave equation is used to implement reverse-time migration (RTM), shear wave energy is significantly included in the migration image. Because anisotropy has intrinsic elastic characteristics, coupling P-wave and S-wave modes in the pseudo-acoustic wave equation is inevitable. In RTM with only primary energy or the P-wave mode in seismic data, the S-wave energy is regarded as noise for the migration image. To solve this problem, we derive a pure P-wave equation for TTI media that excludes the S-wave energy. Additionally, we apply the rapid expansion method (REM) based on a Chebyshev expansion and a pseudo-spectral method (PSM) to calculate spatial derivatives in the wave equation. When REM is incorporated with the PSM for the spatial derivatives, wavefields with high numerical accuracy can be obtained without grid dispersion when performing numerical wave modeling. Another problem in the implementation of TTI RTM is that wavefields in an area with high gradients of dip or azimuth angles can be blown up in the progression of the forward and backward algorithms of the RTM. We stabilize the wavefields by applying a spatial-frequency domain high-cut filter when calculating the spatial derivatives using the PSM. In addition, to increase performance speed, the graphic processing unit (GPU) architecture is used instead of traditional CPU architecture. To confirm the degree of acceleration compared to the CPU version on our RTM, we then analyze the performance measurements according to the number of GPUs employed.

On the instability in second-order systems for acoustic VTI and TTI media

We studied second-order wave propagation systems for vertical transversely isotropic (VTI) and tilted transversely isotropic (TTI) acoustic media with variable axes of symmetry that have their shear-wave speeds set to zero. Acoustic TTI systems are commonly used in reverse-time migration, but these second-order systems are susceptible to instablities appearing as nonphysical stationary noise growing linearly in time, particularly in variable-tilt TTI media. We found an explanation of the cause of this phenomenon. The instabilities are not caused only by the numerical schemes; they are inherent to the differential equations. These instabilities are present even in homogeneous VTI media. These instabilities are caused by zero wave speeds at a wide variety of wavenumbers — a direct consequence of setting the shear-wave speeds to zero — coupled with the second time derivative in these systems. Although the second-order isotropic wave equation allows smooth time-growing solutions, a larger class of time-growing solutions exists for the second-order acoustic TI systems, including nonsmooth solutions. Boundary conditions appear to be less effective in controlling these time-growing solutions than they are for the isotropic wave equation. These systems conserve an incomplete energy that does not prevent the instabilities. The corresponding steady-state systems are no longer elliptic differential equations and can have nonsmooth solutions that are related to the instabilities. We started initially with homogeneous VTI media, and then extended these results to heterogeneous variable-tilt TTI media. We also developed a second-order acoustic system for heterogeneous variable-tilt TTI media derived directly from the full-elastic system for heterogeneous variable-tilt TTI media. All second-order systems with a dispersion relation obtained by setting the shear-wave speeds to zero in the elastic dispersion relation allowed these nonphysical time-growing solutions; however, knowing the cause of these instabilities, it may be possible to prevent or control the activation of these solutions.

Decoupled equations for reverse time migration in tilted transversely isotropic media

Conventional modeling and migration for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear wave artifacts due to the coupling of the P-wave and SV-wave modes in the TTI coupled equations. Starting with the separated P- and SV-phase velocity expressions for vertical transversely isotropic (VTI) media, we extend these decoupled equations for modeling and reverse time migration (RTM) in acoustic TTI media. Compared with the TTI coupled equations published in the geophysical literature, the new TTI decoupled equations provide a more stable solution due to the complete separation of the P-wave and SV-wave modes. The pseudospectral method is the most convenient method to implement these equations due to the form of wavenumber expressions and has the added benefit of being highly accurate and thus avoiding numerical dispersion. The rapid expansion method (REM) in time is employed to produce a broad band numerically stable time evolution of the wavefields. Synthetic results validate the proposed TTI decoupled equations and show that modeling and RTM in TTI media with the decoupled equations remain numerically stable even for models with strong anisotropy and sharp contrasts.

Approximation of pure acoustic seismic wave propagation in TTI media

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

TTI reverse time migration using the pseudo-analytic method

Reverse time migration (RTM) was first introduced in the early 1980s (Whitmore, 1983), but was seemingly dormant until recent advances in computer hardware helped propel it onto the stage as a powerful depth-imaging method. RTM is now standard for areas where large velocity contrasts and/or steep dips pose a challenge, for instance below salt in the Gulf of Mexico. In recent years, where called for by the data, the migration tool of choice has gone from isotropic RTM to anisotropic RTM. The most common representations of anisotropy in sedimentary rocks are VTI (transverse isotropy with a vertical axis of symmetry) and TTI (tilted transverse isotropy). While isotropic and VTI RTM have become somewhat routine, TTI RTM remains challenging due to the complexity, stability, computational cost, and the difficulty in estimating the anisotropic parameters for TTI media.

Analysis of Phase Velocity and Polarization Features for Elastic Waves in TTI Media

Phase velocity and polarization direction are the important parameters for studying seismic wave propagation and describing medium properties, which play essential roles in theory and application. We assume that the symmetry axis of tilted transverse isotropy (TTI) media lies in XOZ plane in the observation coordinates system and directly derive an analytic expression for phase velocity and polarization direction in TTI media in this system. Based on the Thomson dimensionless anisotropic parameters and weak anisotropy approximation theory, we also derive the approximate expressions of phase velocity for elastic waves, polarization direction for qP wave and qSV wave. Theoretical analysis and numerical examples indicate that the change of velocity for qP wave and qSH wave is gentle, and the change of velocity for qSV wave is dramatic. The errors of approximate values of phase velocity are small, so the approximate expression is a good approximation for precise phase velocity. The polarization direction of SH wave is the function of propagation angle and tilted angle of symmetry axis instead of anisotropic parameters. The polarization directions of SH wave is not only perpendicular to propagation direction, but also perpendicular to the direction of symmetry axis. The polarization directions of qP wave and qSV wave are neither parallel nor perpendicular to propagation direction except some special directions, and changes with the angle of symmetry axis in TTI media. For both precise and approximate expressions, the polarization directions of qP wave and qSV wave are always perpendicular. In a given range of errors, the weak anisotropy approximation and analytic expressions match well.