Traveltimes In Anisotropic Media Research Articles

Homotopy solutions of the acoustic eikonal equation for strongly attenuating transversely isotropic media with a vertical symmetry axis

Many applications in seismology involve the modeling of seismic-wave traveltimes in anisotropic media. We have developed homotopy solutions of the acoustic eikonal equation for P-wave traveltimes in attenuating transversely isotropic media with a vertical symmetry axis. Instead of the commonly used perturbation theory, we use the homotopy analysis method to express the traveltimes by a Taylor series expansion over powers of an embedding parameter. For the derivation, we first perform homotopy analysis of the eikonal equation and derive the linearized ordinary differential equations for the coefficients of the Taylor series expansion. Then, we obtain the homotopy solutions for the traveltimes by solving the linearized ordinary differential equations. Results of our investigation with approximate formulas demonstrate that the analytical expressions are efficient methods for the computation of traveltimes from the eikonal equation. In addition, these formulas are also effective methods for benchmarking approximate numerical solutions in strongly attenuating anisotropic media.

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

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

Tracing Seismic Shortest Path Rays in Anisotropic Medium with Rolling Surface

AbstractIt has very important theoretical and practical values to take rolling surface and seismic anisotropy into consideration in seismic forward modeling and inversion. Based on previous researches, the shortest path calculation algorithm is introduced in this paper to calculate seismic travel times in anisotropic media with rolling surface. In the process of model gridding, the whole model is divided into square cells firstly and rolling boundary is regarded as the approximation of units with irregular shapes. Then shortest path calculation is achieved on irregular grids by setting irregular nodes. Sena's group velocity approximation is used to calculate seismic travel times in anisotropic media during ray tracing, and seismic shortest path ray tracing in a complex anisotropic medium with rolling surface is achieved in the end. Theoretical model calculations show that the method in this paper can be reliably applied to complex anisotropic models with higher calculation accuracy.

A Fast Sweeping Scheme for Calculating P Wave First-Arrival Travel Times in Transversely Isotropic Media with an Irregular Surface

Seismic wave propagation shows anisotropic characteristics in many sedimentary rocks. Modern seismic exploration in mountainous areas makes it important to calculate P wave travel times in anisotropic media with irregular surfaces. The challenges in this context are mainly from two aspects. First is how to tackle the irregular surface in a Cartesian coordinate system, and the other lies in solving the anisotropic eikonal equation. Since for anisotropic media the ray (group) velocity direction is not the same as the direction of the travel-time gradient, the travel-time gradient no longer serves as an indicator of the group velocity direction in extrapolating the travel-time field. Recently, a topography-dependent eikonal equation formulated in a curvilinear coordinate system has been established, which is effective for calculating first-arrival travel times in an isotropic model with an irregular surface. Here, we extend the above equation from isotropy to transverse isotropy (TI) by formulating a topography-dependent eikonal equation in TI media in the curvilinear coordinate system, and then use a fast sweeping scheme to solve the topography-dependent anisotropic eikonal equation in the curvilinear coordinate system. Numerical experiments demonstrate the feasibility and accuracy of the scheme in calculating P wave travel times in TI models with an irregular surface.

Feasibility of nonhyperbolic moveout inversion in transversely isotropic media

Inversion of reflection traveltimes in anisotropic media can provide estimates of anisotropic coefficients required for seismic processing and lithology discrimination. Nonhyperbolic P-wave moveout for transverse isotropy with a vertical symmetry axis (VTI media) is controlled by the parameter η (or, alternatively, by the horizontal velocity Vhor), which is also responsible for the influence of anisotropy on all time‐processing steps, including dip‐moveout (DMO) correction and time migration. Here, we recast the nonhyperbolic moveout equation, originally developed by Tsvankin and Thomsen, as a function of Vhorand normal‐moveout (NMO) velocity Vnmoand introduce a correction factor in the denominator that increases the accuracy at intermediate offsets. Then we apply this equation to obtain Vhorand η from nonhyperbolic semblance analysis on long common midpoint (CMP) spreads and study the accuracy and stability of the inversion procedure. Our error analysis shows that the horizontal velocity becomes relatively well‐constrained by reflection traveltimes if the spreadlength exceeds twice the reflector depth. There is, however, a certain degree of tradeoff between Vhorand Vnmocaused by the interplay between the quadratic and quartic term of the moveout series. Since the errors in Vhorand Vnmohave opposite signs, the absolute error in the parameter η (which depends on the ratio Vhor/Vnmo) turns out to be at least two times bigger than the percentage error in Vhor. Therefore, the inverted value of η is highly sensitive to small correlated errors in reflection traveltimes, with moveout distortions on the order of 3–4 ms leading to errors in η up to ±0.1—even in the simplest model of a single VTI layer. Similar conclusions apply to vertically inhomogeneous media, in which the interval horizontal velocity can be obtained with an accuracy often comparable to that of the NMO velocity, while the interval values of η are distorted by the tradeoff between Vhorand Vnmothat gets amplified by the Dix‐type differentiation procedure. We applied nonhyperbolic semblance analysis to a walkaway VSP data set acquired at Vacuum field, New Mexico, and obtained a significant value of η = 0.19 indicative of nonnegligible anisotropy in this area. Then we combined moveout inversion results with the known vertical velocity to resolve the anisotropic coefficients ε and δ. However, in agreement with our modeling results, η estimation was significantly compounded by the scatter in the measured traveltimes. Certain instability in η inversion has no influence on the results of anisotropic poststack time migration because all kinematically equivalent models obtained from nonhyperbolic moveout give an adequate description of long‐spread reflection traveltimes. Also, inversion of nonhyperbolic moveout provides a relatively accurate horizontal‐velocity function that can be combined with the vertical velocity (if it is available) to estimate the anisotropic coefficient ε. However, η represents a valuable lithology indicator that can be obtained from surface P-wave data. Therefore, for purposes of lithology discrimination, it is preferable to find η by means of the more stable DMO method of Alkhalifah and Tsvankin.

First‐arrival traveltime calculation for anisotropic media

Huygens’s principle is used to derive a method for calculating first arrival traveltimes in anisotropic media. Numerical results for several transverse isotropic (TI) models show that calculated Huygens traveltimes are in good agreement with traveltimes computed by a finite‐difference solution to the anisotropic wave equation. This Huygens method is stable and accurate for the test models and so it may be useful in anisotropic data inversion and wave propagation visualization.

Finite difference calculation of first traveltimes in anisotropic media

SUMMARY I propose a finite difference approach, in a general sense, to calculate the traveltimes of the first arrivals in anisotropic media at any point of a regular grid. This method is based on the application of Huygens principle and is similar to an earlier developed method for isotropic cases. The procedure takes explicitly into account the existence of different waves, in the high-frequency approximation, and can deal with any peculiar waves such as diffracted waves or headwaves which may sometimes be the fastest ones. 2-D calculations for anisotropic materials with cubic, hexagonal, tetragonal or orthorhombic symmetry are presented. However, since this approach is based on the same algorithm as used in isotropic media for both 2-D and 3-D cases, it may be extended to 3-D anisotropic media. Moreover, the general algorithm does not depend on the symmetry system and is therefore applicable in any other case of anisotropy. I suggest, in a first attempt, to apply the finite difference calculation of traveltimes for qP waves, because qS waves may have very peculiar propagation in anisotropic media.