A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

An analytical model of seismic travel times for a spherically asymmetric Earth

Summary A general analytical model for travel times of seismic waves propagating in a radially asymmetric Earth, is suggested. It is represented by a series of irreducible spherical tensor products with bipolar spherical coefficients. The main term of the series describes the travel times in a radially symmetric Earth, the others represent corrections due to the spherical assymmetry. The method of least squares is suggested for determining the bipolar spherical coefficients from observed seismic travel times. Since the proposed theory assumes that the analytical representation is related to the reference earth, the corrections to the non-zero focal depth and non-zero sea-level height of the seismographic station must be introduced.

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A note on the measurement of travel times using bender and extender elements

By using the bender and extender elements tests, the travel times of the shear (S) and the primary (P) waves were measured for dry sand samples at different relative densities and effective confining pressures. Three methods of interpretations, namely, (i) the first time of arrival, (ii) the first peak to peak, and (iii) the cross-correlation method, were employed. All the methods provide almost a unique answer associated with the P-wave measurements. On contrary, a difference was noted in the arrival times obtained from the different methods for the S-wave due to the near field effect. The resonant column tests in the torsional mode were also performed to check indirectly the travel time of the shear wave. The study reveals that as compared to the S-wave, it is more reliable to depend on the arrival times’ measurement for the P-wave.     

A test of new Earth models against PcP and PmKP travel times

PcP and PmKP travel times are computed for three simple or parametric Earth models, based on free-oscillation and travel-time data B₁, PEM-A and HB₁ and compared with PcP and PmKP travel times from different sources. This comparison is made only for the region above and below the core-mantle boundary and is of interest because of the current search for a standard Earth model. The comparison shows that only model B₁ does not need a correction for its PcP travel times. For the PmKP travel times for the three models, corrections of the form Δt = a + bm were obtained. The models need the following corrections for b: ?1.3 for B₁, 2.8 for HB₁ and 0.6 for PEM-A. The corrections a are shown to be equal to the observed corrections for PcP at large epicentral distances. The inversions of free-oscillation data to obtain Earth models are most successful when body-wave phases that interact with the core are included.     

Peculiarities of Rayleigh wave displacement in a transversely isotropic half space

We consider a transversely isotropic medium with vertical axis of symmetry (VTI). Rayleigh wave displacement components in a homogeneous VTI medium contain exp(±kr_jz), where z is the vertical coordinate, k is the wave number, and j?=?1, 2; r_j may be considered as depth-decay factor. In a VTI medium, r_j can be a real or imaginary as in an isotropic medium, or it can be a complex depending on the elastic parameters of the VTI medium; if complex, r₁ and r₂ are complex conjugates. In a homogeneous VTI half space, Rayleigh wave displacement is significantly modified with a phase shift when r_j changes from real to complex with variation of VTI parameters; at the transition, the displacement becomes zero when r₁?=?r₂. In a liquid layer over a VTI half space, Rayleigh waves are dispersive. Here, also Rayleigh wave displacement significantly modified with a phase shift when r_j changes from real to complex with a decrease of period. At very low period, phase velocity of Rayleigh waves becomes less than P-wave velocity in the liquid layer giving rise to Scholte waves (interface waves). The amplitudes of Scholte waves with a VTI half space are found to be significantly larger than those with an isotropic half space.     

Torsional vibration of a pipe pile in transversely isotropic saturated soil

This study considers the torsional vibration of a pipe pile in a transversely isotropic saturated soil layer. Based on Biot’s poroelastic theory and the constitutive relations of the transversely isotropic medium, the dynamic governing equations of the outer and inner transversely isotropic saturated soil layers are derived. The Laplace transform is used to solve the governing equations of the outer and inner soil layers. The dynamic torsional response of the pipe pile in the frequency domain is derived utilizing 1D elastic theory and the continuous conditions at the interfaces between the pipe pile and the soils. The time domain solution is obtained by Fourier inverse transform. A parametric study is conducted to demonstrate the influence of the anisotropies of the outer and inner soil on the torsional dynamic response of the pipe pile.     

Reflection and transmission coefficients of a fracture in transversely isotropic media

We obtain the reflection and transmission coefficients at a fracture in transversely isotropic media, whose symmetry axis is perpendicular to the fracture surface. We consider dissimilar upper and lower media. The fracture is modeled as an interface with boundary discontinuities in the displacement and the particle velocity. The stress components are proportional to the displacement and velocity discontinuities through the specific stiffnesses and specific viscosities, respectively. The stiffness introduces frequency-dependence and phase changes in the fracture response and the viscosity is related to the energy loss. We also calculate the energy balance at the fracture and the dissipated energy. The theory is illustrated by computing the reflection coefficient of a fracture present in the Antarctic ice cap. In this case, the reflection coefficient decreases with increasing incidence angle and then approaches 1 at grazing angle.     

Separation of diffracted waves in transversely isotropic media

Imaging diffracted waves can provide useful information about complex subsurface geology and fracture networks. Separation of diffractions from typically more intensive reflected events can be done based on specularity, which measures deviation from Snell’s law. Here, we analyze two formulations of specularity and their applicability to diffraction processing in the presence of anisotropy. We show that the most common definition of specularity, originally introduced for pure modes in isotropic media, remains valid for both pure and converted waves in arbitrarily anisotropic models. The other formulation operates directly with the difference between the slowness projections onto the reflector for the incident and reflected waves. Testing on a VTI (transversely isotropic with a vertical symmetry axis) diffraction ramp model demonstrates that both formulations produce satisfactory results for anisotropic media with appropriate tapering of the specularity gathers. Then separation and imaging of diffractions is performed for the structurally complex VTI Marmousi model. We also analyze the sensitivity of diffractions in the specularity gathers to errors in the symmetry-direction velocity and anellipticity parameter η.     

横向各向同性地层随钻多极子测井声场的分波分析与特性研究

随钻声场在各向异性地层中的传播一直是随钻声波测井的重要课题.该问题中钻铤与井孔和地层的相互作用极大地影响着声场,特别是系统中的模式波的传播特征.因此,本文采用分波计算法来详细地研究在横向各向同性（VTI）地层中随钻多极子声波的传播规律,将声场全波分解成来自波数域中的奇异点（包括支点和极点）的分波的贡献,以此模拟分析了声场中各个分波的传播特征,包括模式波在频域中的频散、激发响应及时域中的分波波形.结果表明：在快速VTI地层中,钻铤的存在使得单极伪瑞利波和二阶四极地层螺旋波的频散大为降低,可以此来测量地层垂直向横波速度;慢速VTI地层中仅存在单一四极地层螺旋波,其低频速度为地层垂直向横波速度.值得注意的是,在各向异性参数满足δ>ε+c₄₄/2c₃₃的地层中,上述模式波的低频截止速度均小于地层的垂向横波速度,这一结果与已知的裸眼井情况一致.

     

3-D isotropic and anisotropic tomography of P-wave travel times from the Anhui airgun experiment in the Yangtze River

The Middle-Lower Yangtze River is a typical transition region between the nearly NW-oriented Tethys and NE-trending Pacific tectonic regimes.Structures of different periods and directions overlap strongly during these processes.The NE-trending Yangtze River compound structural belt and NW-trending Tongling-Hangzhou struc-tural belt both control the magmatic activities and distribu-tions of the metallogenic belts in the area.Here,we obtain 3-D high-resolution isotropic and azimuthally anisotropic velocity structures at depths of 1-10 km using the first arrivals from airgun sources.The velocity maps correspond well with the tectonic structures,with high-velocity anomalies distributed in ore-concentrated districts and low-velocity anomalies distributed along the Yangtze River.The fast directions are generally consistent with the fault strike,indicating that the azimuthal anisotropy is mainly dominated by the fault and fracture trends in the upper crust.The complicated fast directions near the Luzong and Tongling ore deposits reveal complex deformations in the upper crust,which are mainly caused by the intersection of the Yangtze River compound and Tongling-Hangzhou structural belts.The magma intrusion beneath the two ore deposits(Luzong and Tongling)are connected at depths of 5-10 km.     

Earthquake location in transversely isotropic media with a tilted symmetry axis

The conventional intersection method for earthquake location in isotropic media is developed in the case of transversely isotropic media with a tilted symmetry axis (TTI media). The hypocenter is determined using its loci, which are calculated through a minimum travel time tree algorithm for ray tracing in TTI media. There are no restrictions on the structural complexity of the model or on the anisotropy strength of the medium. The location method is validated by its application to determine the hypocenter and origin time of an event in a complex TTI structure, in accordance with four hypotheses or study cases: (a) accurate model and arrival times, (b) perturbed model with randomly variable elastic parameter, (c) noisy arrival time data, and (d) incomplete set of observations from the seismic stations. Furthermore, several numerical tests demonstrate that the orientation of the symmetry axis has a significant effect on the hypocenter location when the seismic anisotropy is not very weak. Moreover, if the hypocentral determination is based on an isotropic reference model while the real medium is anisotropic, the resultant location errors can be considerable even though the anisotropy strength does not exceed 6.10%.     

Axisymmetric dynamic response of the multi-layered transversely isotropic medium

By virtue of the precise integration method (PIM) and the technique of mixed variable formulations, solutions for the dynamic response of the multi-layered transversely isotropic medium subjected to the axisymmetric time-harmonic forces are presented. The planes of cross anisotropy are assumed to be parallel to the horizontal surface of the stratified media. Four kinds of vertically acting axisymmetric loads are prescribed either at the external surface or in the interior of the soil system. Thicknesses and number of the medium strata are not limited. Employing the Hankel integral transform in cylindrical coordinate, the axisymmetric governing equations in terms of displacements of the multi-layered media are uncoupled. Applying mixed variable formulations, more concise first-order ordinary differential matrix equations from the uncoupled motion equations can be obtained. Solutions of the ordinary differential matrix equations in the transformed domain are acquired by utilizing the approach of PIM. Since PIM is highly accurate to solve the sets of first-order ordinary differential equations, any desired accuracy of the solutions can be achieved. All calculations are based on the corresponding algebraic operations and computational efforts can be reduced to a great extent. Comparisons with the existing numerical solutions are made to confirm the accuracy of the present solutions proposed by this procedure. Several examples are illustrated to explore the influences of the type and degree of material anisotropy, the frequency of excitation and loading positions on the dynamic response of the stratified medium.     

横向各向同性地层中倾斜线圈系响应特征的数值模拟

详细推导了模拟倾斜线圈系在横向各向同性地层中响应的垂向数值模式匹配方法,在轴对称模型下,该方法采用垂直本征模式描述三维激励源,使得三维问题可以在二维条件下快速解决,另外,本文引入的高阶基函数进一步提高了算法的计算速度.为了验证算法的可靠性与准确性,分别与均匀无限厚各向同性地层的解析解和横向各向同性地层的有限差分解进行对比,结果表明,模式匹配方法与它们吻合的较好.在此基础上,利用模式匹配方法模拟了倾斜线圈系和水平线圈系在横向各向同性地层的响应特征,计算结果表明:在横向各向同性地层中,倾斜线圈系的响应既受水平电导率的影响,也受垂直电导率的影响,而水平线圈系则只与水平电导率有关,由此,可以通过倾斜线圈系评价地层的各向异性特征.此外,本文还分析了钻铤对倾斜线圈系响应的影响,模拟结果表明,虽然钻铤没有改变相位值和幅度值的变化趋势,但对它们的绝对值有一定的影响,而且这种影响与不同法向角度也有关系,因此当组合不同倾斜线圈系进行地层评价时,需要对钻铤的影响进行校正.最后研究了不同法向角度线圈系在横向各向同性地层中的响应特征,得出不同法向角度线圈系的探测特性和对各向异性的敏感性都有较大的差异.本文的研究表明倾斜线圈系的响应与水平线圈系的响应不同,通过利用不同倾斜线圈系组合可以进行油气储层评价.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

Torsional vibration of a finite cylindrical cavity in a two-layer transversely isotropic half-space

The aim of this paper is to present a rigorous investigation for a two-layered transversely isotropic linear elastic half-space containing a circular cylindrical cavity of length equal to the top layer undergoing mono-harmonic ring shape shear stress applied either on the vertical cylindrical surface or on the base of the cavity. To this end, a combination of Fourier cosine integral transform for depth and Hankel integral transform for radial distance are used, which translate the boundary value problem to a singular integral equation for the shear stress comes out from the continuity of two layers. The integral equation is solved for some collocation points with a smoothed variable of distance, which is adapted with the use of a free parameter. It is shown that, although the shear stress is highly singular, it does not highly depend on this free parameter. Both the analytical and numerical results are verified with both the static isotropic and dynamic transversely isotropic homogeneous cases. In addition, some new graphical results are presented for more understanding in engineering point of view.     

Time-harmonic response of transversely isotropic multilayered half-space in a cylindrical coordinate system

With the aid of the analytical layer-element method, a comprehensive analytical derivation of the response of transversely isotropic multilayered half-space subjected to time-harmonic excitations is presented in a cylindrical coordinate system. Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the Fourier expansion, Hankel and Laplace integral transform, analytical layer-elements for a finite layer and a half-space are derived. Considering the continuity conditions on adjacent layers׳ interfaces and the boundary conditions, the global stiffness matrix equations for multilayered half-space are assembled and solved. Finally, some numerical examples are given to make a comparison with the existing solution and to demonstrate the influence of parameters on the dynamic response of the medium.     

Diffraction of SH-waves by topographic features in a layered transversely isotropic half-space

The scattering of plane SH-waves by topographic features in a layered transversely isotropic (TI) half-space is investigated by using an indirect boundary element method (IBEM). Firstly, the anti-plane dynamic stiffness matrix of the layered TI half-space is established and the free fields are solved by using the direct stiffness method. Then, Green’s functions are derived for uniformly distributed loads acting on an inclined line in a layered TI half-space and the scattered fields are constructed with the deduced Green’s functions. Finally, the free fields are added to the scattered ones to obtain the global dynamic responses. The method is verified by comparing results with the published isotropic ones. Both the steady-state and transient dynamic responses are evaluated and discussed. Numerical results in the frequency domain show that surface motions for the TI media can be significantly different from those for the isotropic case, which are strongly dependent on the anisotropy property, incident angle and incident frequency. Results in the time domain show that the material anisotropy has important effects on the maximum duration and maximum amplitudes of the time histories.     

Caustics in a periodically layered transversely isotropic medium with vertical symmetry axis

Wave propagation in a finely layered medium is a very important topic in seismic modelling and inversion. Here we analyse non‐vertical wave propagation in a periodically layered transversely isotropic (VTI) medium and show that the evanescent (attenuation) zones in the frequency‐horizontal slowness domain result in caustics in the group velocity domain. These caustics, which may appear for both the quasi‐compressional (qP) and quasi‐shear (qSV) wave surfaces are frequency dependent but display weak dependence at low frequencies. The caustics computed for a specific frequency differ from those observed at the low‐ and high‐frequency limits. We illustrate these caustics with a few numerical examples and snapshots computed for both qP‐ and qSV‐wave types.     

Upper-mantle travel times in Australia — A preliminary report

This paper gives a preliminary account of an experiment to determine upper-mantle travel times in Australia. A set of fifteen 14-track tape-recording units are being used along a profile from Darwin to Alice Springs in central Australia. In addition to the 14-track recorders, eleven 6-track units are being used both on-line and off-line as monitors. The sources used are natural earthquakes occurring to the north of Australia, especially Banda Sea events.Preliminary analysis of the records from the first sub-array has confirmed the high apparent velocities for P and S previously found for the distance range 800–1,800 km in central Australia. It is inferred that the high apparent velocities are due to a sharp increase in the proportion of garnet at a depth of about 90 km.Neither the P nor S arrivals from 800 to 1,800 km show any evidence of the shadow zones or offsets in the travel-time curve which would be associated with an upper-mantle low-velocity zone. It is suggested, however, that a low-velocity zone is not precluded by the observations, but that the associated offset may not occur in the first-arrival part of the travel-time curve.