Vertical Transversely Isotropic Media Research Articles

High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media

Pseudoacoustic algorithms are very fast in comparison with full elastic ones for vertical transversely isotropic (VTI) modeling, so they are suitable for many applications, especially reverse time migration. Finite differences using simple grids are commonly used to solve pseudoacoustic equations. We have developed and implemented general high-order 3D pseudoacoustic transversely isotropic formulations. The focus is the development of staggered-grid finite-difference algorithms, known for their superior numerical properties. The staggered-grid schemes based on first-order velocity-stress wave equations are developed in detail as well as schemes based on direct application to second-order stress equations. This last case uses the recently presented equivalent staggered-grid theory, resulting in a staggered-grid scheme that overcomes the problem of large memory requirement. Two examples are presented: a 3D simulation and a prestack reverse time migration application, and we perform a numerical analysis regarding computational cost and precision. The errors of the new schemes are smaller than the existing nonstaggered-grid schemes. In comparison with existing staggered-grid schemes, they require 25% less memory and only have slightly greater computational cost.

Efficient wave-mode separation in vertical transversely isotropic media

Wave-mode separation can be achieved by projecting elastic wavefields onto mutually orthogonal polarization directions. In isotropic media, because the P-wave’s polarization vectors are consistent with wave vectors, the isotropic separation operators are represented by divergence and curl operators, which are easy to realize. In anisotropic media, polarization vectors deviate from wave vectors based on local anisotropic strength and separation operators lose their simplicity. Conventionally, anisotropic wave-mode separation is implemented either by direct filtering in the wavenumber domain or nonstationary filtering in the space domain, which are computationally expensive. Moreover, in conventional anisotropic separation, correcting for amplitude and phase changes of waveforms by applying separation operators is also more difficult than in an isotropic case. We have developed new operators for efficient wave-mode separation in vertical transversely isotropic (VTI) media. Our separation operators are constructed by local rotation of wave vectors to directions where the quasi-P (qP) wave is polarized. The deviation angles between the wave vectors and the qP-wave’s polarization vectors are explicitly estimated using the Poynting vectors. Obtaining polarization directions by rotating wave vectors yields separation operators in VTI media with the same forms as divergence and curl operators, except that the spatial derivatives are now rotated to implement wavefield projections in accurate polarization directions. The main increase in computational cost relative to isotropic separation operators is the estimation of the Poynting vectors, which is relatively small within elastic-wave extrapolation. As a result, applying the proposed operators is efficient. In the meantime, the waveforms corrections for divergence and curl operators can be directly extended for our new operators due to the similarities between these operators. By numerical exercises, we have determined that wave modes can be well-separated with small numerical cost using the present separation operators. The conservation of energy in wave-mode separation by applying waveform corrections was also verified.

The stability problem of reverse time migration for viscoacoustic VTI media

In real strata anisotropy and viscosity extensively exists. They degraded waveforms in amplitude, resulting in which reducing of image resolution. To obtain high-precision imaging of deep reservoirs, we extended the separated viscous and anisotropic reverse time migration (RTM) to a stable viscoacoustic anisotropic RTM for vertical transverse isotropic (VTI) media, based on single generalized standard and linear solid (GSLS) media theory.. We used a pseudo-spectral method to develop the numerical simulation. By introducing a regularization operator to eliminate the high-frequency instability problem, we built a stable inverse propagator and achieved viscoacoustic VTI media RTM. High-resolution imaging results were obtained after correcting for the effects of anisotropy and viscosity. Synthetic tests verify the validity and accuracy of algorithm.

A workflow to model anisotropy in a vertical transverse isotropic medium

Velocity anisotropy, which is known as the directional dependency of velocities, is becoming increasingly important in subsurface imaging and characterization. Most elasticity theories consider an isotropic medium to describe the phenomena in the field of reservoir geophysics. This assumption is challenged by the reality of the subsurface which is subject to a complex geological history such as tectonic movements and changes in the differential stress that can typically introduce fractures. In some cases, these factors can make the subsurface highly anisotropic. In general, four classes of anisotropy can be defined, ranging between the two extremes of a completely isotropic medium (with two elastic constants) and a completely anisotropic medium (with 21 elastic constants). The four classes refer to specific conditions where we can reduce the number of elements of the elastic stiffness tensor. These are known as Cubic (with three independent elastic constants), Transverse Isotropic or TI (five independent elastic constants), Orthorhombic (nine independent elastic constants) and Monoclinic (13 independent elastic constants). TI is the most often used to describe sedimentary rock. Anisotropy as an extension to isotropic approaches is usually dealt with using Thomsen (1986) parameters as approximations. Thomsen (1986) suggested three parameters to correct for anisotropy effects in weakly anisotropic media. These parameters, ε, δ and γ, are regularly used in all reservoir geophysics disciplines to address anisotropy effects. However, determining these three parameters is not straightforward and requires information such as laboratory data or well logs acquired in boreholes in different directions with respect to the symmetry axis of the anisotropy. The purpose of this paper is to review anisotropy in vertically isotropic media and use existing theory to model changes in the elastic stiffness tensor based on conventional well logs. Furthermore, this elastic stiffness tensor can be used to calculate the Thomsen parameters or even to model anisotropic velocities directly.

Velocity-independent estimation of kinematic attributes in vertical transverse isotropy media using local slopes and predictive painting

Agood seismic velocity model is required for many routine seismic imaging techniques. Velocity model building from seismic data is often labor intensive and time consuming. The process becomes more complicated by taking nonhyperbolic traveltime estimations into account. An alternative to the conventional time-domain imaging algorithms is to use techniques based on the local event slopes, which contain sufficient information about the traveltime moveout for velocity estimation and characterization of the subsurface geologic structures. Given the local slopes, there is no need for a prior knowledge of a velocity model. That is why the term “velocity independent” is commonly used for such techniques. We improved upon and simplified the previous versions of velocity-independent nonhyperbolic approximations for horizontally layered vertical transverse isotropy (VTI) media by removing one order of differentiation with respect to offset from the imaging kinematic attributes. These kinematic attributes are derived in terms of the local event slopes and zero-offset two-way traveltime (TWTT). We proposed the use of predictive painting, which keeps all the attributes curvature independent, to estimate the zero-offset TWTT. The theoretical contents and performance of the proposed approach were evaluated on synthetic and field data examples. We also studied the accuracy of moveout attributes for shifted hyperbola, rational, three-parameter, and acceleration approximations on a synthetic example. Our results show that regardless of the approximation types, NMO velocity estimate has higher accuracy than the nonhyperbolicity attribute. Computational time and accuracy of the inversion of kinematic attributes in VTI media using our approach were compared with routine/conventional multiparameter semblance inversion and with the previous velocity-independent inversion techniques.

An effective anisotropic poroelastic model for elastic wave propagation in finely layered media

Mesoscopic-scale heterogeneities in porous media cause attenuation and dispersion at seismic frequencies. Effective models are often used to account for this. We have developed a new effective poroelastic model for finely layered media, and we evaluated its impact focusing on the angle-dependent attenuation behavior. To enable this, an exact solution was obtained for the response of a periodically layered medium to a surface point load using Floquet’s theory. We compared this solution with that of the new model and the equivalent viscoelastic vertical transverse isotropic medium available from existing literature. We have observed that the quasi-P (qP) wave dispersion and attenuation was predicted with high accuracy by the new effective poroelastic model. For the quasi-S (qS) wave, the effective poroelastic model provides a perceptibly better prediction of the attenuation, resulting in closer to the exact waveforms. The qS-wave attenuation is underestimated by the effective viscoelastic model, whereas for the qP-wave, the model gives accurate predictions in all cases except for highly permeable weak-frame media.

Near-surface anisotropic structure characterization by Love wave inversion for assessing ground conditions in Urban Areas

In many geophysical applications, neglecting of anisotropy is somehow an oversimplification. The mismatch between prediction based on isotropic theory and near-surface seismic observations indicates the need for the inclusion of medium anisotropy. In this paper, surface wave (Love wave) dispersion properties are used to estimate the anisotropic structure of the near-surface layered earth, which is modeled as media possess vertical transverse isotropy (VTI), a reasonable assumption for near-surface sedimentary layers. Our approach utilizes multi-mode surface waves to estimate both the velocity structure and the anisotropy structure. This approach consists of three parts. First, the dispersion analysis is used to extract dispersion curves from real data. Second, the forward modeling is carried out based on the dispersion equation of Love wave in a multi-layered VTI medium. Dispersion curves of multi-modes, which are the numerical solutions of the dispersion equation, are obtained by a graphic-based method. Finally, the very fast simulated annealing (VFSA) algorithm is used to invert velocity structure and anisotropy structure simultaneously. Our approach is verified by the synthetic dispersion curve generated by a VTI medium model. The estimation of shear wave velocity and anisotropy structure of surface wave data acquired at Rentschler Field, an urban center site on sediments in the Connecticut River valley reveals a simple structure of the sediment layer over a bedrock half space. The results are verified by other inversion results based on different data set obtained on the same site. The consistency of inversion results shows the feasibility and efficiency of the approach.

A pure viscoacoustic equation for VTI media applied in anisotropic RTM

Neglecting anisotropy and attenuation in seismic wave propagation will result in inaccurate images, with problems such as diffracted wave convergence and seismic wave attenuation. It is therefore important to take anisotropy and viscosity into account. In this paper, based on the dissipation mechanism of the standard linear solid model, we derive a pure viscoacoustic wave equation for vertical transversely isotropic (VTI) media in the time–space domain, which describes the attenuation characteristics of seismic waves through a pseudo-differential operator in the equations. The numerical results show that the VTI viscoacoustic wave equation not only describes the propagation of waves in anisotropic media accurately, but it also reflects the effects of absorption and attenuation. The reverse time migration results for the VTI_HESS model show that we can obtain accurate and reliable amplitude-preserved images based on the anisotropic viscoacoustic wave equation.

Forward Modeling of VTI Media in the Frequency‐Space Domain Based on an Average‐Derivative Optimal Method

AbstractThis paper proposed a new kind of qP wave equation second‐order 9‐point scheme of two‐dimensional vertical transversely isotropic (VTI) media in the frequency‐space domain based on an average‐derivative optimal method (ADM for short). This scheme represents the finite‐difference approximations of the second‐order centered spatial‐derivative terms of two‐dimensional VTI media qP wave equation as the weighted average of 3 grid points in orthogonal directions and the acceleration term as the weighted average of all 9 grid points. Determining the optimal weighting coefficients of the second‐order centered spatial‐derivative terms and the acceleration term by using the least‐square optimal method to minimize numerical dispersion, the number of grid points per wavelength bounded by a phase velocity error range of ±1% is only 3.57. But the VTI conventional 9‐point scheme needs approximately 12 grid points per wavelength bounded by the same error range of ±1%, so the ADM method significantly increases the computational accuracy. The numerical example of the complex BP2007 2D VTI ocean standard model also proves the validity and precision of the VTI media ADM 9‐point scheme.

The electromagnetic response in a layered vertical transverse isotropic medium: A new look at an old problem

ABSTRACTWe determined that the electromagnetic vertical transverse isotropic response in a layered earth can be obtained by solving two equivalent scalar equations, which were for the vertical electric field and for the vertical magnetic field, involving only a scalar global reflection coefficient. Besides the complete derivation of the full electromagnetic response, we also developed the corresponding computer code called EMmod, which models the full electromagnetic fields including internal multiples in the frequency-wavenumber domain and obtains the frequency-space domain solutions through a Hankel transformation by computing the Hankel integral using a 61-point Gauss-Kronrod integration routine. The code is able to model the 3D electromagnetic field in a 1D earth for diffusive methods such as controlled source electromagnetics as well as for wave methods such as ground penetrating radar. The user has complete freedom to place the source and the receivers in any layer. The modeling is illustrated with ...

Theory of equivalent staggered‐grid schemes: application to rotated and standard grids in anisotropic media

ABSTRACTThe previous finite‐difference numerical schemes designed for direct application to second‐order elastic wave equations in terms of displacement components are strongly dependent on Poisson's ratio. This fact makes theses schemes useless for modelling in offshore regions or even in onshore regions where there is a high Poisson's ratio material. As is well known, the use of staggered‐grid formulations solves this drawback. The most common staggered‐grid algorithms apply central‐difference operators to the first‐order velocity–stress wave equations. They have been one of the most successfully applied numerical algorithms for seismic modelling, although these schemes require more computational memory than those mentioned based on second‐order wave equations. The goal of the present paper is to develop a general theory that enables one to formulate equivalent staggered‐grid schemes for direct application to hyperbolic second‐order wave equations. All the theory necessary to formulate these schemes is presented in detail, including issues regarding source application, providing a general method to construct staggered‐grid formulations to a wide range of cases. Afterwards, the equivalent staggered‐grid theory is applied to anisotropic elastic wave equations in terms of only velocity components (or similar displacements) for two important cases: general anisotropic media and vertical transverse isotropy media using, respectively, the rotated and the standard staggered‐grid configurations. For sake of simplicity, we present the schemes in terms of velocities in the second‐ and fourth‐order spatial approximations, with second‐order approximation in time for 2D media. However, the theory developed is general and can be applied to any set of second‐order equations (in terms of only displacement, velocity, or even stress components), using any staggered‐grid configuration with any spatial approximation order in 2D or 3D cases. Some of these equivalent staggered‐grid schemes require less computer memory than the corresponding standard staggered‐grid formulation, although the programming is more evolved. As will be shown in theory and practice, with numerical examples, the equivalent staggered‐grid schemes produce results equivalent to corresponding standard staggered‐grid schemes with computational advantages. Finally, it is important to emphasize that the equivalent staggered‐grid theory is general and can be applied to other modelling contexts, e.g., in electrodynamical and poroelastic wave propagation problems in a systematic and simple way.

The electromagnetic response in a layered vertical transverse isotropic medium: A new look at an old problem

We determined that the electromagnetic vertical transverse isotropic response in a layered earth can be obtained by solving two equivalent scalar equations, which were for the vertical electric field and for the vertical magnetic field, involving only a scalar global reflection coefficient. Besides the complete derivation of the full electromagnetic response, we also developed the corresponding computer code called EMmod, which models the full electromagnetic fields including internal multiples in the frequency-wavenumber domain and obtains the frequency-space domain solutions through a Hankel transformation by computing the Hankel integral using a 61-point Gauss-Kronrod integration routine. The code is able to model the 3D electromagnetic field in a 1D earth for diffusive methods such as controlled source electromagnetics as well as for wave methods such as ground penetrating radar. The user has complete freedom to place the source and the receivers in any layer. The modeling is illustrated with three examples, which aim to present the different capabilities of EMmod, while assessing its correctness.

Seismic wave modeling in viscoelastic VTI media using spectral element method

Spectral element method (SEM) for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries. It is an advanced choice for wave simulations. Due to anelasticity of earth media, SEM for elastic media is no longer appropriate. On fundamental of the second-order elastic SEM, this work takes the viscoelastic wave equations and the vertical transversely isotropic (VTI) media into consideration, and establishes the second-order SEM for wave modeling in viscoelastic VTI media. The second-order perfectly matched layer for viscoelastic VTI media is also introduced. The problem of handling the overlapped absorbed corners is solved. A comparison with the analytical solution in a two-dimensional viscoelastic homogeneous medium shows that the method is accurate in the wave-field modeling. Furtherly, numerical validation also presents its great flexibility in solving wave propagation problems in complex heterogeneous media. This second-order SEM with perfectly matched layer for viscoelastic VTI media can be easily applied in wave modeling in a limited region.

A new parameterization for frequency-domain elastic full waveform inversion for VTI media

Abstract To improve full waveform inversion (FWI) results for 2-D vertical transverse isotropic (VTI) media, we propose a new parameterization scheme that is based on Lame constants. In the new parameterization approach, elastic constants of C11, C13, C33, and C44 are replaced with Lame constants and newly introduced parameters of ξ1 and ξ2 for anisotropic properties. Multi-parametric inversion for elastic VTI media has been conducted using a two-step sequential inversion strategy, in which primary parameters (that are well inverted) are inverted first, and secondary parameters are inverted using the inverted primary parameters. In the conventional VTI parameterization approach, C33 and C44 are regarded as the primary parameters, whereas in the new parameterization approach, Lame constants are primary parameters. The new parameterization scheme is distinguished from the conventional VTI parameterization approach in scattering patterns of partial derivative wavefields with respect to primary parameters. Scattering patterns of Lame constants are similar to those of the isotropic case. Because Lame constants are well inverted in the isotropic case, they may also be well inverted in the new parameterization. On the other hand, we do not expect any direct improvements for anisotropic properties because ξ1 and ξ2 produce the same scattering patterns as those of C11 and C13. Gradient directions for a two-layered model show that we can improve the resolution of gradient direction for μ (one of the primary parameters). Although ξ1 and ξ2 of the new parameterization retain characteristics of C11 and C13 in the conventional parameterization, improved primary parameters (Lame constants) affect anisotropic parameters positively. Consequently, inversion results for all of the parameters can be improved. Numerical examples indicate that the new parameterization scheme provides more reliable inversion results than the conventional and sequential inversion approaches. In addition, the new parameterization approach exhibits higher degrees of computational efficiency than the sequential inversion approach because it involves simultaneous inversion of all parameters and does not require an additional inversion stage. Our new parameterization scheme can therefore overcome the intrinsic disadvantages of elastic inversion for VTI media. In applying the new parameterization to real seismic data, it is necessary to increase its applicability to noisy and low-frequency missing data and it also needs to be extended to 3-D case.

Error estimates of elastic components in stress-dependent VTI media

Abstract This work examines the ranges of physically acceptable elastic components for a vertical transversely isotropic (VTI) laboratory shale data set. A stochastic rock-physics approach combined with physically based acceptance and rejection criteria determined the ranges. The importance of this work is to demonstrate that multiple constrained models explain independently calculated measurement error bars. The data set consisted of pressure- and directional-dependent velocity measurements conducted on a low porosity, brine-saturated hard shale. Error bars were calculated for all five elastic stiffnesses and compliances as a function of pressure. The rock physics model is pressure dependent and represents simultaneously five elastic compliances for a VTI medium. A non-linear least squares fitting routine established a best-fit model to the five compliances at all pressures. Perturbations of the best-fit model provided the statistical parameter space. Twelve physical constraints or data-set-specific conditions comprised the acceptance/rejection criteria. These constraints and conditions included strain-energy requirements, inequalities among stiffnesses and anisotropy parameters, and rates of change of moduli with pressure. The largest number of rejected models resulted from violating a criterion relating a compressional and shear stiffness. Minimum misfits between the accepted models and the data illustrate that a fraction of the accepted models best explain the data. The misfits between these accepted models and data explain the error in the data and/or inhomogeneities at the measurement scale. The ranges of acceptable elastic component values and the corresponding uncertainty estimates could be incorporated into seismic-inversion, imaging, and velocity-modeling schemes.

The analysis and application of simplified two-parameter moveout equation for C-waves in VTI anisotropy media

Several parameters are needed to describe the converted-wave (C-wave) moveout in processing multi-component seismic data, because of asymmetric raypaths and anisotropy. As the number of parameters increases, the converted wave data processing and analysis becomes more complex. This paper develops a new moveout equation with two parameters for C-waves in vertical transverse isotropy (VTI) media. The two parameters are the C-wave stacking velocity (VC2) and the squared velocity ratio (γvti) between the horizontal P-wave velocity and C-wave stacking velocity. The new equation has fewer parameters, but retains the same applicability as previous ones. The applicability of the new equation and the accuracy of the parameter estimation are checked using model and real data. The form of the new equation is the same as that for layered isotropic media. The new equation can simplify the procedure for C-wave processing and parameter estimation in VTI media, and can be applied to real C-wave processing and interpretation. Accurate VC2 and γvti can be deduced from C-wave data alone using the double-scanning method, and the velocity ratio model is suitable for event matching between P- and C-wave data.

A new strategy for 2D VTI seismic full waveform inversion

We develop an inversion strategy for seismic full waveform inversion (FWI) for 2D vertical transversely isotropic media (VTI). In our strategy, we use LamA© constants (I» and I¼) instead of C33 (= I» + 2I¼) and C44 (= I¼). For C11 and C13, we redefine them by LamA© constants and residuals between anisotropic parameters (C11 and C13) and LamA© constants. Accordingly, these residuals represent anisotropic characteristics. In FWI, we invert LamA© constants and residuals, and then we extract C11, C13, C33 and C44 from the inverted parameters. We validate our new strategy by comparing inversion results obtained by the new inversion strategy to those obtained by the isotropic inversion and the conventional VTI inversion for a modified version of the overthrust geological model. Results obtained by the isotropic inversion are reliable only for isotropic layers. On the other hand, anisotropic layers cannot be recovered well by only I» and I¼. In case of the conventional VTI inversion, C33 and C44 are well inverted. These results agree well with the isotropic inversion results. The C11 and C13 models inverted by the conventional VTI inversion are reasonable, but their resolutions are poorer than those of the C33 and C44. On the other hand, the C11 and C13 models inverted by the new inversion strategy are much improved compared to those of the conventional VTI inversion. From these results, we note that anisotropic properties in subsurface media cannot be recovered by the isotropic inversion, whereas the developed inversion strategy can provide reasonable anisotropic characteristics. In addition, elastic parameters by the new inversion strategy are more precise and stable than those of the conventional isotropic and VTI FWIs.

Amplitude-preserved pre-stack time migration in 3D VTI media

We present a new method of amplitude-preserved pre-stack time migration in 3D vertical transversely isotropic (VTI) media. The travel time and amplitude of a wave propagating in VTI media are calculated by using one-way wave operators and the stationary-phase theory. An imaging weight from the deconvolution imaging condition is introduced to produce amplitude-preserved image gathers for hydrocarbon and fluid detection. Velocities and stacking anisotropic parameters are estimated by a semblance scanning method. A time-dependent migration aperture corresponding to the dip of subsurface structure is then obtained to balance imaging quality and imaging noises. Numerical examples demonstrate that our method is accurate, robust and easy to implement.

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.

First-order systems for elastic and acoustic variable-tilt TI media

We derive and compare first-order wave propagation systems for variable-tilt elastic and acoustic tilted transversely isotropic (TTI) media. Acoustic TTI systems are commonly used in reverse-time migration. Starting initially with homogeneous vertical transversely isotropic (VTI) media, and then extending to heterogeneous variable-tilt TI media, we derive a pseudoacoustic [Formula: see text] first-order system of differential equations by setting the shear-wave speeds to zero and simplifying the full-elastic system accordingly. This [Formula: see text] system conserves a complete energy, but only when the anelliptic anisotropy parameter [Formula: see text]. For [Formula: see text] (including isotropic media), the system allows linearly time-growing and spatially nonpropagating nonphysical solutions frequently taken for numerical noise. We modified this [Formula: see text] acoustic first-order system by changing the stress variables to obtain a system that stays stable for [Formula: see text]. This system for homogeneous VTI media is generalized to heterogeneous variable-tilt TI media by rotating the stress and strain variables in the full elastic system before setting the shear-wave speeds to zero; the system obtained can be greatly simplified by combining the rotational terms, resulting in only one rotation and extra lower-order terms compared to the [Formula: see text] first-order acoustic system for VTI media. This new system can be simplified further by neglecting the lower-order terms. Both systems (with and without lower-order terms) conserve the same complete energy. Finally, the corresponding [Formula: see text] full elastic system for variable-tilt acoustic TI media can be used for the purposes of benchmarking.