Effective ellipsoidal models for wavefield extrapolation in tilted orthorhombic media

Wavefield computations using the ellipsoidally anisotropic extrapolation operator offer significant cost reduction compared to that for the orthorhombic case, especially when the symmetry planes are tilted and/or rotated. However, ellipsoidal anisotropy does not provide accurate wavefield representation or imaging for media of orthorhombic symmetry. Therefore, we propose the use of ‘effective ellipsoidally anisotropic’ models that correctly capture the kinematic behaviour of wavefields for tilted orthorhombic (TOR) media. We compute effective velocities for the ellipsoidally anisotropic medium using kinematic high-frequency representation of the TOR wavefield, obtained by solving the TOR eikonal equation. The effective model allows us to use the cheaper ellipsoidally anisotropic wave extrapolation operators. Although the effective models are obtained by kinematic matching using high-frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The proposed methodology offers a much better cost versus accuracy trade-off for wavefield computations in TOR media, particularly for media of low to moderate anisotropic strength. Furthermore, the computed wavefield solution is free from shear-wave artefacts as opposed to the conventional finite-difference based TOR wave extrapolation scheme. We demonstrate applicability and usefulness of our formulation through numerical tests on synthetic TOR models.     

Modeling multiple elementary waves in layered media using ray field maps

Seismic ray tracing in layered media becomes complicated and demanding when modeling for multiple ray codes (reflection/transmission sequences) and/or dense acquisition geometries. However, we observe some redundancies in current algorithms: (a) the same layers are crossed repeatedly by similar ray segments, and (b) the effort of tracing through a layer is determined by variations in the incoming wavefront rather than the medium. We deal with these redundancies by separating the modeling process in two stages: (Stage 1) compute ray field maps representing all ray segments between each pair of adjacent interfaces, then (Stage 2) for each desired ray code assemble the complete ray field from ray segments by iterative lookup in the ray field maps.     

Propagation of elastic waves in layered media resulting from an impulsive point source

Summary The problems of Cagniard and Abramovici-Alterman, regarding propagation of seismic pulses in horizontally layered media, are solved by a direct method without involving integral transforms.     

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%.     

Apparent resistivity of azimuthal anisotropy layered media

Introduction The method of apparent resistivity with direct current (DC) measurement, adopted from geophysical exploration, is one of the most important precursory approaches in earthquake prediction, in which the geo-electric constitutive model is illustrated by the scalar theory Archies law and correspondingly a set of calculation methods for apparent resistivity of layered media has been established (QIAN, et al, 1985; YAO, 1989). But this is not consistent with the fact that in earthqu…     

Moveout analysis for tranversely isotropic media with a tilted symmetry axis

The transversely isotropic (TI) model with a tilted axis of symmetry may be typical, for instance, for sediments near the flanks of salt domes. This work is devoted to an analysis of reflection moveout from horizontal and dipping reflectors in the symmetry plane of TI media that contains the symmetry axis. While for vertical and horizontal transverse isotropy zero-offset reflections exist for the full range of dips up to 90°, this is no longer the case for intermediate axis orientations. For typical homogeneous models with a symmetry axis tilted towards the reflector, wavefront distortions make it impossible to generate specular zero-offset reflected rays from steep interfaces. The ‘missing’ dipping planes can be imaged only in vertically inhomogeneous media by using turning waves. These unusual phenomena may have serious implications in salt imaging. In non-elliptical TI media, the tilt of the symmetry axis may have a drastic influence on normal-moveout (NMO) velocity from horizontal reflectors, as well as on the dependence of NMO velocity on the ray parameter p (the ‘dip-moveout (DMO) signature’). The DMO signature retains the same character as for vertical transverse isotropy only for near-vertical and near-horizontal orientation of the symmetry axis. The behaviour of NMO velocity rapidly changes if the symmetry axis is tilted away from the vertical, with a tilt of ±20° being almost sufficient to eliminate the influence of the anisotropy on the DMO signature. For larger tilt angles and typical positive values of the difference between the anisotropic parameters ε and δ, the NMO velocity increases with p more slowly than in homogeneous isotropic media; a dependence usually caused by a vertical velocity gradient. Dip-moveout processing for a wide range of tilt angles requires application of anisotropic DMO algorithms. The strong influence of the tilt angle on P-wave moveout can be used to constrain the tilt using P-wave NMO velocity in the plane that includes the symmetry axis. However, if the azimuth of the axis is unknown, the inversion for the axis orientation cannot be performed without a 3D analysis of reflection traveltimes on lines with different azimuthal directions.     

Dispersion function of Rayleigh waves in porous layered half-space system

Rayleigh wave exploration is based on an elastic layered half-space model. If practical formations contain porous layers, these layers need to be simplified as an elastic medium. We studied the effects of this simplification on the results of Rayleigh wave exploration. Using a half-space model with coexisting porous and elastic layers, we derived the dispersion functions of Rayleigh waves in a porous layered half-space system with porous layers at different depths, and the problem of transferring variables to matrices of different orders is solved. To solve the significant digit overflow in the multiplication of transfer matrices, we propose a simple, effective method. Results suggest that dispersion curves differ in a lowfrequency region when a porous layer is at the surface; otherwise, the difference is small.     

基于地震正交各向异性的裂缝介质岩石力学建模研究

岩石力学模型是描述地层原地应力状态的基础,而常规各向同性模型难以刻画地层本征横观各向同性(VTI)和天然高角度裂缝的耦合作用,建立更为准确的正交各向异性模型变得尤为重要.本文利用VTI介质中发育单组垂直缝的Schoenberg裂缝等效模型,简化了一般正交各向异性介质的表征参数,推导了由背景VTI介质弹性参数和裂缝参数的各向异性杨氏模量、泊松比和水平地应力的精确方程.借鉴Thomsen弱各向异性近似思路,舍去裂缝弱度参数高阶扰动项,推导了正交各向异性介质岩石力学近似方程,由裂缝弱度表示的近似方程更为直观地反映了裂缝对背景介质力学性质的影响,杨氏模量和泊松比值随裂缝法向弱度的增大而显著减小.选取页岩实验数据进行数值实验,结果表明本文方程与实际物理规律相吻合,随裂缝弱度的增加,形成相同应变所需的水平地应力值降低,且近似方程与模拟结果相对误差小于5%,有助于提高岩石力学参数估算和地应力预测精度.     

层状半空间中洞室对平面SH波的放大作用

利用间接边界元法,求解了弹性层状半空间中洞室对入射平面SH波的放大作用问题,并以基岩上单一土层中洞室对入射平面SH波的放大作用为例进行了数值计算分析.本文模型的特点之一是考虑了层状场地的动力特性,因而更接近于实际工程;特点之二是计算精度非常高.研究表明,层状半空间中洞室对波的放大作用与均匀半空间中情况有着本质的差别;层状半空间中洞室附近地表动力响应由土层动力特性和洞室对波的散射二者共同决定.土层动力特性不仅影响洞室附近地表位移的幅值,还会影响地表位移的频谱.在土层的前几阶共振频率附近,随着基岩与土层剪切波速比的增大,土层的影响随之增大,而随着土层厚度的增加,土层的影响随之减小,并逐渐趋于均匀半空间情况.     

Surface waves in magneto-elastic initially stressed conducting media

Summary The object of the present paper is to investigate the magneto-elastic surface waves in an initially stressed conducting medium. The theory of magneto-elastic surface waves in an initially stressed conducting medium has firstly been deduced and then it has been employed in investigating the particular cases of surface waves such as (i) Rayleigh waves (ii) Love waves and (iii) Stoneley waves. The wave-velocity equations obtained in different cases are in agreement with the corresponding classical results when the solid is initially unstressed and the magnetic field is absent or the material is non-magnetic.     

Dynamic soil-tunnel interaction in layered half-space for incident plane SH waves

The dynamic soil-tunnel interaction is studied by indirect boundary element method (IBEM), using the model of a rigid tunnel in layered half-space, which is simplified to a single soil layer on elastic bedrock, subjected to incident plane SH waves. The accuracy of the results is verified through comparison with the analytical solution. It is shown that soil-tunnel interaction in layered half-space is larger than that in homogeneous half-space and this interaction mechanism is essentially different from that of soil-foundation-superstructure interaction.     

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.     

Stratigraphic filtering of SH waves in a spherical,layered earth

We present a derivation of the formula for filtering a transmitted SH wavelet by shortperiod multiples in a spherically layered Earth. We use a continuous, rather than a discrete formulation and regard the impedance and the velocity as random variables. The mean shear displacement represents the propagating wavelet as modified by short-period multiples. Standard procedures and approximations Lead to the dispersion relation of the mean displacement. To describe the stratigraphic filtering we introduce a complex quantityF such that a wavelet which has travelled a time T is modified by the filter exp {iFT}. The impact of the higher angular harmonic modes is shown to produce a relative enhancement of those modes over the low angular harmonic modes due to fluctuations in the shear-wave propagation velocity. Numerical estimates indicate that the sizes of the apparent attenuation of the mean field and the time delay introduced by the short-period multiples sit squarely in the regime where they produce a nonnegligible distortion of the SH modes of propagation in both phase and amplitude.     

Surface and plate waves in biaxially-stressed elastic media

Summary The propagation of waves of small amplitude in an incompressible elastic medium subjected to a large homogeneous equibiaxial stress is investigated. The dispersion equation for surface waves on a semi-infinite medium is obtained explicitly, and the dispersion equation for waves in a slab of finite depth. In the latter case, approximate values, valid for waves whose lengths are much greater than the slab thickness, are determined for the phase-velocities of the different modes.     

Rocking response of flexible circular foundations on layered media

The analysis of the response of a flexible circular foundation on layered media due to an arbitrarily distributed vertical loading is presented. The analysis is based on the ‘ring method’ approach, i.e. discretization of the foundation in a set of concentric rings. The arbitrarily distributed loading is expanded in the circumferential direction in a Fourier series. The influence coefficient matrix of soil for each element of the series is evaluated utilizing the stiffness matrix approach. The stiffness matrix of the foundation is obtained from the finite difference energy method approach. Numerical examples illustrate the influence of several soil-foundation parameters on the rocking response of a foundation. Results are presented in terms of displacement and soil reaction distributions and impedance functions point to significantly different responses of flexible and rigid foundations.     

Vertical vibration of three-dimensional rigid foundations on layered media

A numerical method of analysis is presented for the determination of the steady-state vertical vibration of rigid foundations with arbitrary three-dimensional geometries resting on the surface of a layered soil medium. The method utilizes the flexibility concept applied to steady-state periodic problems and it is solved in the frequency domain. The accuracy of the method is verified by comparison with several published solutions for massless, smooth rigid rectangular foundations on a homogeneous, isotropic elastic half-space. Parametric solutions are presented to study the dynamic behaviour of massless, smooth rigid rectangular foundations on a homogeneous, elastic stratum.     

Dynamic response of single piles embedded in transversely isotropic layered media

Dynamic response of single piles embedded in transversely isotropic layered media is investigated using the finite element method combined with dynamic stiffness matrices of the soil derived from Green's functions for ring loads. The influence of soil anisotropy on the dynamic behaviour of piles is examined through a series of parametric studies.