Anisotropic velocity analysis1

Results from walkaway VSP and shale laboratory experiments show that shale anisotropy can be significantly anelliptic. Heterogeneity and anellipticity both lead to non-hyperbolic moveout curves and the resulting ambiguity in velocity analysis is investigated for the case of a factorizable anisotropic medium with a linear dependence of velocity on depth. More information can be obtained if there are several reflectors. The method of Dellinger et al. for anisotropic velocity analysis in layered transversely isotropic media is examined and is shown to be restricted to media having relatively small anellipticity. A new scheme, based on an expansion of the inverse-squared group velocity in spherical harmonics, is presented. This scheme can be used for larger anellipticity, and is applicable for horizontal layers having monoclinic symmetry with the symmetry plane parallel to the layers. The method is applied to invert the results of anisotropic ray tracing on a model Sand/shale sequence. For transversely isotropic media with small anisotropy, the scheme reduces to the method of Byun et al. and Byun and Corrigan. The expansion in spherical harmonics allows the P-phase slowness surface of each layer to be determined in analytic form from the layer parameters obtained by inversion without the need to assume that the anisotropy is weak.     

Fractal travel time estimates for dispersive contaminants

Alternative fractional models of contaminant transport lead to a new travel time formula for arbitrary concentration levels. For an evolving contaminant plume in a highly heterogeneous aquifer, the new formula predicts much earlier arrival at low concentrations. Travel times of contaminant fronts and plumes are often obtained from Darcy's law calculations using estimates of average pore velocities. These estimates only provide information about the travel time of the average concentration (or peak, for contaminant pulses). Recently, it has been shown that finding the travel times of arbitrary concentration levels is a straightforward process, and equations were developed for other portions of the breakthrough curve for a nonreactive contaminant. In this paper, we generalize those equations to include alternative fractional models of contaminant transport.     

Anisotropic attenuation of stratified viscoelastic media

An important cause of seismic anisotropic attenuation is the interbedding of thin viscoelastic layers. However, much less attention has been devoted to layer‐induced anisotropic attenuation. Here, we derive a group of unified weighted average forms for effective attenuation from a binary isotropic, transversely isotropic‐ and orthorhombic‐layered medium in the zero‐frequency limit by using the Backus averaging/upscaling method and analyse the influence of interval parameters on effective attenuation. Besides the corresponding interval attenuation and the real part of stiffness, the contrast in the real part of the complex stiffness is also a key factor influencing effective attenuation. A simple linear approximation can be obtained to calculate effective attenuation if the contrast in the real part of stiffness is very small. In a viscoelastic medium, attenuation anisotropy and velocity anisotropy may have different orientations of symmetry planes, and the symmetry class of the former is not lower than that of the latter. We define a group of more general attenuation‐anisotropy parameters to characterize not only the anisotropic attenuation with different symmetry classes from the anisotropic velocity but also the elastic case. Numerical tests reveal the influence of interval attenuation anisotropy, interval velocity anisotropy and the contrast in the real part of stiffness on effective attenuation anisotropy. Types of effective attenuation anisotropy for interval orthorhombic attenuation and interval transversely isotropic attenuation with a vertical symmetry (vertical transversely isotropic attenuation) are controlled only by the interval attenuation anisotropy. A type of effective attenuation anisotropy for interval TI attenuation with a horizontal symmetry (horizontal transversely isotropic attenuation) is controlled by the interval attenuation anisotropy and the contrast in the real part of stiffness. The type of effective attenuation anisotropy for interval isotropic attenuation is controlled by all three factors. The magnitude of effective attenuation anisotropy is positively correlated with the contrast in the real part of the stiffness. Effective attenuation even in isotropic layers with identical isotropic attenuation is anisotropic if the contrast in the real part of stiffness is non‐zero. In addition, if the contrast in the real part of stiffness is very small, a simple linear approximation also can be performed to calculate effective attenuation‐anisotropy parameters for interval anisotropic attenuation.     

Anisotropic viscoelastic models with singular memory

The physical background of singular memory models and in particular the Cole–Cole model is discussed. Three models of anisotropic linear viscoelasticity with frequency-dependent stiffness coefficients are considered. The models are constructed in such a way that anisotropic properties are separated from anelastic effects. Two of these models represent finite-speed wave propagation with singularities at the wavefronts (the exponential relaxation model) and without singularities at the wavefronts (the Cole–Cole model), while a third model called the fractional model is related to the constant Q with unbounded propagation speed. The Cole–Cole and fractional models belong to the class of singular memory models studied earlier because of their applications in polymer rheology, poroelasticity, poroacoustics, seismic wave propagation and other applications. Well-posedness of initial boundary value problems with mixed Dirichlet–Neumann boundary conditions is established for the three models. Regularity properties of the three models are examined.     

Anisotropic models of the upper mantle

Long period Rayleigh wave and Love wave dispersion data, particularly for oceanic areas, have not been simultaneously satisfied by an isotropic structure. In this paper available phase and group velocity data are inverted by a procedure which includes the effects of transverse anisotropy, anelastic dispersion, sphericity, and gravity. We assume that the surface wave data represents an azimuthal average of actual velocities. Thus, we can treat the mantle as transversely isotropic. The resulting models for average Earth, average ocean, and oceanic regions divided according to the age of the ocean floor, are quite different from previous results which ignore the above effects. The models show a low-velocity zone with age dependent anisotropy and velocities higher than derived in previous surface wave studies. The correspondence between the anisotropy variation with age and a physical model based on flow aligned olivine is suggestive. For most of the Earth SH > SV in the vicinity of the low-velocity zone. Neat the East Pacific Rise, however, SV > SH at depth, consistent with ascending flow. Anisotropy is as important as temperature in causing radial and lateral variations in velocity. The models have a high velocity nearly isotropic layer at the top of the mantle that thickens with age. This layer defines the LID, or seismic lithosphere. In the Pacific, the LID thickens with age to a maximum thickness of ～50 km. This thickness is comparable to the thickness of the elastic lithosphere. The LID thickness is thinner than derived using isotropic or pseudo-isotropic procedures. A new model for average Earth is obtained which includes a thin LID. This model extends the fit of a PREM, type model to shorter period surface waves.     

Anisotropic tomography of the Atlantic Ocean

We present the first regional three-dimensional model of the Atlantic Ocean with anisotropy. The model, derived from Rayleigh and Love wave phase velocity measurements, is defined from the Moho down to 300 km depth with a lateral resolution of about 500 km and is presented in terms of average isotropic S-wave velocity, azimuthal anisotropy and transverse isotropy.The cratons beneath North America, Brazil and Africa are clearly associated with fast S-wave velocity anomalies. The mid-Atlantic ridge (MAR) is a shallow structure in the north Atlantic corresponding to a negative velocity anomaly down to about 150 km depth. In contrast, the ridge negative signature is visible in the south Atlantic down to the deepest depth inverted, that is 300 km depth. This difference is probably related to the presence of hot-spots along or close to the ridge axis in the south Atlantic and may indicate a different mechanism for the ridge between the north and south Atlantic. Negative velocity anomalies are clearly associated with hot-spots from the surface down to at least 300 km depth, they are much broader than the supposed size of the hot-spots and seem to be connected along a north-south direction.Down to 100 km depth, a fast S-wave velocity anomaly is extenting from Africa into the Atlantic Ocean within the zone defined as the Africa superswell area. This result indicates that the hot material rising from below does not reach the surface in this area but may be pushing the lithosphere upward.In most parts of the Atlantic, the azimuthal anisotropy directions remain stable with increasing depth. Close to the ridge, the fast S-wave velocity direction is roughly parallel to the sea floor spreading direction. The hot-spot anisotropy signature is striking beneath Bermuda, Cape Verde and Fernando Noronha islands where the fast S-wave velocity direction seems to diverge radially from the hot-spots.The Atlantic average radial anisotropy is similar to that of the PREM model, that is positive down to about 220 km, but with slightly smaller amplitude and null deeper. Cratons have a lower than average radial anisotropy. As for the velocities, there is a difference between north and south Atlantic. Most hot-spots and the south-Atlantic ridge are associated with positive radial anisotropy perturbation whereas the north-Atlantic ridge corresponds to negative radial anisotropy perturbation.     

Inverse modeling for seawater intrusion in coastal aquifers: Insights about parameter sensitivities,variances, correlations and estimation procedures derived from the Henry problem

Inverse modeling studies employing data collected from the classic Henry seawater intrusion problem give insight into several important aspects of inverse modeling of seawater intrusion problems and effective measurement strategies for estimation of parameters for seawater intrusion. Despite the simplicity of the Henry problem, it embodies the behavior of a typical seawater intrusion situation in a single aquifer. Data collected from the numerical problem solution are employed without added noise in order to focus on the aspects of inverse modeling strategies dictated by the physics of variable-density flow and solute transport during seawater intrusion. Covariances of model parameters that can be estimated are strongly dependent on the physics. The insights gained from this type of analysis may be directly applied to field problems in the presence of data errors, using standard inverse modeling approaches to deal with uncertainty in data.     

Seminumerical simulation of singly dispersive convection in groundwater

Geothermal activity may lead to convection currents in groundwater. Such currents considerably affect transport phenomena in the aquifer. In this study, the applicability of a seminumerical approach for the simulation of flow conditions in such an aquifer is presented. Flow field variables are expanded through truncated sets of eigenfunctions, which leads to a system of general first-order differential equations that can be solved by applying available subroutines. Criteria of flow field stability, parameters affecting flow conditions, and quantitative analysis of transport processes in the aquifer are studied.     

Classes of Anisotropic Media: A Tutorial

The purpose of the present article is to give a precise definition and analysis from first principals of anisotropy, as the term applies to elastic media, taking care to avoid unnecessary assumptions. Two fundamental concepts, material invariance and symmetry group of a material, are defined purely in terms of the stress-strain relation. The implications of material symmetry, or in other words, of anisotropy, for the structure of the stiffness tensor are then investigated. Using the reduced notation of Voigt, these results are presented as the well-known simplifications in the form taken by the six-by-six stiffness matrix that represents the material's stiffness tensor. A new, simple proof is given for the remarkable fact that an elastic medium cannot have rotational symmetry by an angle of less than 90° without being transversely isotropic. In addition, the mutual relation that the notions of elastic symmetry and crystal symmetry have with respect to the so-called orthogonal group is sketched. Despite the historical association between anisotropic elastic materials and the study of crystals, the given presentation shows that conceptually the notion of anisotropy in elastic media is entirely independent of that of crystal symmetry.     

Anisotropic reverse-time migration for tilted TI media

Seismic anisotropy in dipping shales results in imaging and positioning problems for underlying structures. We develop an anisotropic reverse‐time depth migration approach for P‐wave and SV‐wave seismic data in transversely isotropic (TI) media with a tilted axis of symmetry normal to bedding. Based on an accurate phase velocity formula and dispersion relationships for weak anisotropy, we derive the wave equation for P‐wave and SV‐wave propagation in tilted transversely isotropic (TTI) media. The accuracy of the P‐wave equation and the SV‐wave equation is analyzed and compared with other acoustic wave equations for TTI media. Using this analysis and the pseudo‐spectral method, we apply reverse‐time migration to numerical and physical‐model data. According to the comparison between the isotropic and anisotropic migration results, the anisotropic reverse‐time depth migration offers significant improvements in positioning and reflector continuity over those obtained using isotropic algorithms.     

Synthetic Experiments Dealing with Anisotropic Model Building

It has been shown in the past that the interval-NMO velocity and the non-ellipticity parameter largely control the P-wave reflection time moveout of VTI media. To invert for these two parameters, one needs either reasonably large offsets, or some structure in the subsurface in combination with relatively mild lateral velocity variation.This paper deals with a simulation of an inversion approach, building on the assumption that accurately measured V _NMO, as defined by small offset asymptotics for a particular reflector, were available. Instead of such measurements we take synthetically computed data. First, an isotropic model is constructed which explains these V _NMO. Subsequently, residual moveout in common image gathers is modelled by ray tracing (replacing real data), along with its sensitivity for changes in the interval-NMO velocity and the non-ellipticity parameter under the constraint that V _NMO is preserved. This enables iterative updating of the non-ellipticity parameter and the interval-NMO velocity in a layer that can be laterally inhomogeneous.This approach is successfully applied for a mildly dipping reflector at the bottom of a layer with laterally varying medium parameters. With the exact V _NMO assumed to be given, lateral inhomogeneity and anisotropy can be distinguished for such a situation. However, for another example with a homogeneous VTI layer overlying a curved reflector with dip up to 30°, there appears to be an ambiguity which can be understood by theoretical analysis. Consistently with existing theory using the NMO-ellipse, the presented approach is successfully applied to the latter example if V _NMO in the strike direction is combined with residual moveout in dip direction.     

应力诱导各向异性介质中地震波的传播

主要讨论了应力变化如何影响各向异性介质中波速度的问题。推导了一般各向异性介质在初始应力下的Christoffel方程,得到介质中3种波的相速度和初始应力的关系表达式;通过实验数据验证了单轴应力能够诱导各向异性,当施加单轴应力时,速度在沿应力的方向增加最大,在垂直应力的方向增加最小,实验结果与理论推导一致;用Christoffel方程的数值解模拟在3种对称情况下的弹性各向异性介质中初始应力对波速度的影响。数值结果表明:初始应力对各向异性介质中波传播速度的影响,随着各向异性强度的增加而增大,而且速度越慢,影响越大。     

Realistic Models of Anisotropic Laminated Lower Crust

—The genesis of the laminated lower crust has been attributed to extensional processes leading to structural and textural ordering. This implies that the lower crust might be anisotropic. Laboratory measurements of lower crustal rock samples and xenolithes show evidence of anisotropy in these rocks due to oriented structure.¶In this paper we investigate the seismic shear-wave response of realistic anisotropic lower crustal models using the anisotropic reflectivity method. Our models are based on representative petrophysical data obtained from exposed lower crustal sections in Calabria (South Italy), Val Strona and Val Sesia (Ivrea Zone, Northern Italy). The models consist of stacks of anisotropic layers characterized by quantified elastic tensors derived from representative rock samples which provide alternating high and low velocity layers.¶The seismic signature of the data is comparable to seismic observations of in situ lower crust. For the models based on the Calabria and Val Strona sequences shear-wave splitting occurs for the Moho reflection at offsets beyond 70 km with travel-time delays up to 300 and 500 ms, respectively. The leading shear wave is predominantly horizontally polarized and followed by a predominantly vertically polarized shear wave. Contrastingly, the Val Sesia model shows no clear evidence of birefringence. Isotropic versus anisotropic modelling demonstrates that the shear-wave splitting is clearly related to the intrinsic anisotropy of the lower crustal rocks for the Val Strona sequence. No evidence of birefringence caused by thin layering is found.     

利用介质的不均匀分布构建各向异性模型

在复杂区域应力场的作用下, 大范围扩容各向异性(extensive-dilatancy anisotropy, EDA)造成的裂隙定向排列形成了地壳内部介质的不均匀性。 因此, 地壳介质各向异性的描述不仅限于横向各向同性(transversely isotropic, TI)的理论。 我们首先利用自相关函数随机扰动得到水平向小尺度速度不均匀, 然后利用这种不均匀性描述地壳介质中EDA裂隙定向排列所造成的介质各向异性, 利用数值模拟弹性波在该介质中的传播过程验证了利用介质的不均匀性构建介质的各向异性是一个有效、 可靠的方法。 可以将此方法应用到复杂地壳介质结构中, 了解地震波在复杂地壳各向异性介质中的传播特征。     

Fermat's Variational Principle for Anisotropic Inhomogeneous Media

Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional (l) stationary. Fermat's functional (l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral (l) = (x ^k,x ^k ')du, taken along curve l, where (x ^k,x ^k ') is the relevant Lagrangian, x ^k are coordinates, u is a parameter used to specify the position of points along l, and x ^k ' = dx ^k÷du. If Lagrangian (x ^k,x ^k ') is a homogeneous function of the first degree in x ^k ', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k '). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k ') to derive the relevant Hamiltonian ⁽¹⁾(x ^k,p _k), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians ⁽¹⁾(x ^k,x ^k '). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians ⁽²⁾. Parameter u along the curves is taken to correspond to travel time , and the second-degree Lagrangian ⁽²⁾(x ^k, ^k ) is then introduced by the relation ⁽²⁾(x ^k, ^k ) = [⁽¹⁾(x ^k, ^k )]², with ^k = dx ^k÷d. The second-degree Lagrangian ⁽²⁾(x ^k, ^k ) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian ⁽²⁾(x ^k,p _k) from Lagrangian ⁽²⁾(x ^k, ^k ), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both ⁽²⁾(x ^k, ^k ) and ⁽²⁾(x ^k,p _k) can be expressed in terms of the wave propagation metric tensor g _ij(x ^k, ^k ), which depends not only on position x ^k, but also on the direction of vector ^k . It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian ⁽²⁾(x ^k, ^k ), not to the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). It is also shown that relations ⁽²⁾(x ^k, ^k ) = ; and ⁽²⁾(x ^k,p _k) = are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates x ⁱ. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.     

利用地脉动台阵观测推算Love波频散曲线

土层剪切波速结构是工程场地条件的重要的基本参数，许多研究者关注利用从地脉动台阵中推得的Rayleigh波或Love波的相速度频散曲线反演实际工程场地剪切波速度结构。探讨从地脉动台阵观测的记录中提取Love波相速度的频散曲线的方法，从厦门某工程场地的地脉动台阵观测三分量记录中提取了Love波相速度的频散曲线，将提取的结果与根据实际场地资料构成的水平成层介质模型计算的Love波相速度的频散曲线进行了对比，为下一步的反演奠定了基础。