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1.
Paraxial ray methods for anisotropic inhomogeneous media 总被引：1，自引：0，他引：1
A new formalism of surface-to-surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user-selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface-to-surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non-eikonal and ray-tangent solutions of the dynamic ray-tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface-to-surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface-to-surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary-value problems of a four-parametric system of paraxial rays, connecting the two surfaces, similarly as the well-known surface-to-surface matrices in isotropic media in ray-centred coordinates. Applications of such boundary-value problems include the two-point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more. 相似文献
2.
Luděk Klime? 《Studia Geophysica et Geodaetica》2012,56(2):373-382
Although synthetic seismograms can, in many cases, be generated by direct numerical methods, approximate analytical ray-theory solutions are often very useful in forward and especially inverse problems of seismic wave propagation in complex 3-D media. In this paper, the frequency-domain zero-order ray-theory Green tensor in a heterogeneous anisotropic elastic medium is derived from the zero-order ray-theory approximation using the representation theorem applied in ray-centred coordinates. 相似文献
3.
Peidong Shi Doug Angus Andy Nowacki Sanyi Yuan Yanyan Wang 《Surveys in Geophysics》2018,39(4):567-611
Seismic anisotropy which is common in shale and fractured rocks will cause travel-time and amplitude discrepancy in different propagation directions. For microseismic monitoring which is often implemented in shale or fractured rocks, seismic anisotropy needs to be carefully accounted for in source location and mechanism determination. We have developed an efficient finite-difference full waveform modeling tool with an arbitrary moment tensor source. The modeling tool is suitable for simulating wave propagation in anisotropic media for microseismic monitoring. As both dislocation and non-double-couple source are often observed in microseismic monitoring, an arbitrary moment tensor source is implemented in our forward modeling tool. The increments of shear stress are equally distributed on the staggered grid to implement an accurate and symmetric moment tensor source. Our modeling tool provides an efficient way to obtain the Green’s function in anisotropic media, which is the key of anisotropic moment tensor inversion and source mechanism characterization in microseismic monitoring. In our research, wavefields in anisotropic media have been carefully simulated and analyzed in both surface array and downhole array. The variation characteristics of travel-time and amplitude of direct P- and S-wave in vertical transverse isotropic media and horizontal transverse isotropic media are distinct, thus providing a feasible way to distinguish and identify the anisotropic type of the subsurface. Analyzing the travel-times and amplitudes of the microseismic data is a feasible way to estimate the orientation and density of the induced cracks in hydraulic fracturing. Our anisotropic modeling tool can be used to generate and analyze microseismic full wavefield with full moment tensor source in anisotropic media, which can help promote the anisotropic interpretation and inversion of field data. 相似文献
4.
L. Klimeš 《Studia Geophysica et Geodaetica》2006,50(3):431-447
Whereas the ray-centred coordinates for isotropic media by Popov and Pšenčík are uniquely defined by the selection of the
basis vectors at one point along the ray, there is considerable freedom in selecting the ray-centred coordinates for anisotropic
media. We describe the properties common to all ray-centred coordinate systems for anisotropic media and general conditions,
which may be imposed on the basis vectors. We then discuss six different particular choices of ray-centred coordinates in
an anisotropic medium. This overview may be useful in choosing the ray-centred coordinates best suited for a particular application.
The equations are derived for a general homogeneous Hamiltonian of an arbitrary degree and are thus applicable both to the
anisotropic-ray-theory rays and anisotropic common S-wave rays. 相似文献
5.
V. M. Pavlov 《Izvestiya Physics of the Solid Earth》2017,53(4):556-564
The problem of calculating complete synthetic seismograms from a point dipole with an arbitrary seismic moment tensor in a plane parallel medium composed of homogeneous elastic isotropic layers is considered. It is established that the solutions of the system of ordinary differential equations for the motion–stress vector have a reciprocity property, which allows obtaining a compact formula for the derivative of the motion vector with respect to the source depth. The reciprocity theorem for Green’s functions with respect to the interchange of the source and receiver is obtained for a medium with cylindrical boundary. The differentiation of Green’s functions with respect to the coordinates of the source leads to the same calculation formulas as the algorithm developed in the previous work (Pavlov, 2013). A new algorithm appears when the derivatives with respect to the horizontal coordinates of the source is replaced by the derivatives with respect to the horizontal coordinates of the receiver (with the minus sign). This algorithm is more transparent, compact, and economic than the previous one. It requires calculating the wavenumbers associated with Bessel function’s roots of order 0 and order 1, whereas the previous algorithm additionally requires the second order roots. 相似文献
6.
Vlastislav Červený 《Studia Geophysica et Geodaetica》2013,57(2):267-275
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. 相似文献
7.
We present a new depth‐imaging method for seismic data in heterogeneous anisotropic media. This recursive explicit method uses a non‐stationary extrapolation operator to allow lateral velocity variation, and it uses the relationship between phase angle and the spectral coordinates of seismic data to allow velocity variation with phase angle. A qualitative comparison of migration impulse responses suggests that, for an equivalent cost, the symmetric non‐stationary phase‐shift (SNPS) operator is superior to the phase‐shift plus interpolation (PSPI) operator, for very large depth intervals. To demonstrate the potential of the new method, seismic data from a physical model acquired over a transversely isotropic medium are imaged using a shot‐record migration based on the SNPS operator. 相似文献
8.
由所建立的三维qP波相速度表示式出发，导出并解析求解各向异性介质中的频散方程，得到三维各向异性介质中的相移算子，进而将以相移算子为基础的对称非平稳相移方法推广到各向异性介质，发展了一个三维各向异性介质的深度偏移方法. 文中使用的各向异性介质的速度模型与现行的各向异性构造的速度估计方法一致，将各向同性、弱各向异性及强各向异性统一在一个模型中. 所建立的各向异性介质对称非平稳相移波场延拓算子可以同时适应速度及各向异性参数横向变化；文中给出的算例虽然是针对二维VTI介质的，但所提出的算法同样适用于三维TI介质. 相似文献
9.
目前大地电磁(MT)测深资料反演主要基于各向同性介质,但随着MT实际应用的需要,各向异性研究已逐渐引起关注.我们采用广泛应用的广义逆法对一维MT水平层状各向异性介质模型反演进行了探索性研究,并实现了MT全张量响应(即所有的阻抗张量的视电阻率和相位)的一维各向异性反演.理论模型试验表明,无论理论观测值中是否含有噪声,这种方法都能够较好地恢复真实模型,验证了其正确性和有效性.将此方法用于MT实测资料时,能够同时拟合4对视电阻率和阻抗相位曲线,说明本方法可以用于实测资料的处理解释,具有一定的实用价值. 相似文献
10.
介质的弹性常数为三维四阶张量的分量,共有８１个,由于应力张量和应变张量的对称性及能量密度是应变的二次函数,一般各向异常性介质的独立弹性常数可减为２１个,如果介质具有较高的对称性,独立弹性常数的数目会更少。 对于地壳和上地幔,具有５个独立弹性常数的横向各向同性介质是一个非常好的近似,本研究中横向各向同性介质的对称轴方向可以是任意的（即对称轴可以不平等于铅直方向）,在此情况下,需要进行坐标变换,如果已知介质在某一坐标系（其坐标轴平行或垂直于介质的对称轴）中的弹性常数,我们能够容易地利用变换公式得到变换后新坐标系中的弹性常数。 本文提出了一种方案,利用伪谱法既能模拟横向各向同性介质中的平面波,也能模拟点源激发的波场。在勘探地球物理和地震学中,模拟横向各向同性介拮中传播的平面波及区域源产生的波是最重要的研究课题之一。然而在一般各向异性介质中,很难或不可能确定弹性波的相速度和偏振方向,但在横向各向同性介质中,则可以通过坐标变换来实现,这里我们所提出的方法可以用于横向各向同性介质中弹性波的模拟。 相似文献
11.
Numerous experimental studies indicate that as a result of shear stress, the elastic behavior of granular media becomes both non-linear and anisotropic. This paper presents a simple constitutive model for sands with respect to anisotropic elasticity. To this aim, using the concept of second order fabric tensor, a simplified elasticity theory is presented which is capable of considering the effect of induced anisotropy on the elastic response. SANISAND is the name used for a family of simple anisotropic sand models developed in the framework of critical state soil mechanics and bounding surface plasticity. An existing SANISAND model is modified in order to include the proposed anisotropic elasticity. The modified model simulations are compared with those obtained from the other members of this family. It is shown that considering anisotropic elasticity can take part in explanation of drastic loss of mean principal stress when sand is subjected to reverse loading in dilative branch of behavior and as a result, improve the liquefaction simulations. 相似文献
12.
A note on dynamic ray tracing in ray-centered coordinates in anisotropic inhomogeneous media 总被引：1，自引：0，他引：1
V. Červený 《Studia Geophysica et Geodaetica》2007,51(3):411-422
Dynamic ray tracing plays an important role in paraxial ray methods. In this paper, dynamic ray tracing systems for inhomogeneous
anisotropic media, consisting of four linear ordinary differential equations of the first order along the reference ray, are
studied. The main attention is devoted to systems expressed in a particularly simple choice of ray-centered coordinates, here
referred to as the standard ray-centered coordinates, and in wavefront orthonormal coordinates. These two systems, known from
the literature, were derived independently and were given in different forms. In this paper it is proved that both systems
are fully equivalent. Consequently, the dynamic ray tracing system, consisting of four equations in wavefront orthonormal
coordinates, can also be used if we work in ray-centered coordinates, and vice versa.
vcerveny@seis.karlov.mff.cuni.cz 相似文献
13.
The phase and group velocity surfaces are essential for wave propagation in anisotropic media. These surfaces have certain features that, especially, for shear waves result in complications for modelling and inversion of recorded wavefields. To analyse wave propagation in an anisotropic model, it is important to identify these features in both the phase and group domains. We propose few characteristics for this analysis: the energy flux angle, decomposed in the polar and azimuth angle correction angles and enhancement factor, which is able to characterize both singularity points and triplication zones. The very simple equation that controls the triplications is derived in the phase domain. The proposed characteristics are illustrated for elastic and acoustic anisotropic models of different symmetry classes. 相似文献
14.
New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity 总被引：2，自引：0，他引：2
Many heterogeneous media and environmental processes are statistically anisotropic. In this paper we focus on range anisotropy,
that is, stochastic processes with variograms that have direction dependent correlation lengths and direction independent
sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after
rotation and rescaling operations. Class (B) models can be separated into a product of one-dimensional functions oriented
along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface
hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal
functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy,
i.e., the ratio(s) of the correlation lengths. This information reduces the degrees of freedom of anisotropic variograms and
thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic and Class (B) models to
one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which
can be estimated from the data, and the covariance tensor. The procedure is conceptually simple and numerically efficient.
It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions.
In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its
implementation with numerical simulations. 相似文献
15.
Luděk Klimeš 《Studia Geophysica et Geodaetica》2016,60(3):451-470
We consider the partial derivatives of travel time with respect to both spatial coordinates and perturbation parameters. These derivatives are very important in studying wave propagation and have already found various applications in smooth media without interfaces. In order to extend the applications to media composed of layers and blocks, we derive the explicit equations for transforming these travel–time derivatives of arbitrary orders at a general smooth curved interface between two arbitrary media. The equations are applicable to both real–valued and complex–valued travel time. The equations are expressed in terms of a general Hamiltonian function and are applicable to the transformation of travel–time derivatives in both isotropic and anisotropic media. The interface is specified by an implicit equation. No local coordinates are needed for the transformation. 相似文献
16.
地壳介质各向异性电性本构关系讨论 总被引：11，自引：3，他引：8
讨论了地壳介质中常见的两类电性各向异性。推导了长波长条件下薄中向异性电导率张量;提出了一种新的电导率物理模型并由此推导了与ＥＤＡ裂隙相适应的各向异性电导率张量。从直流电法和大地电磁测深法的角度分别讨论了这两种电性各向异性所显现的地表视电阻率、阻抗和电位特征。 相似文献
17.
Characterizing the expressions of seismic waves in elastic anisotropic media depends on multiparameters. To reduce the complexity, decomposing the P-mode wave from elastic seismic data is an effective way to describe the considerably accurate kinematics with fewer parameters. The acoustic approximation for transversely isotropic media is widely used to obtain P-mode wave by setting the axial S-wave phase velocity to zero. However, the separated pure P-wave of this approach is coupled with undesired S-wave in anisotropic media called S-wave artefacts. To eliminate the S-wave artefacts in acoustic waves for anisotropic media, we set the vertical S-wave phase velocity as a function related to propagation directions. Then, we derive a pure P-wave equation in transversely isotropic media with a horizontal symmetry axis by introducing the expression of vertical S-wave phase velocity. The differential form of new expression for pure P-wave is reduced to second-order by inserting the expression of S-wave phase velocity as an auxiliary operator. The results of numerical simulation examples by finite difference illustrate the stability and accuracy of the derived pure P-wave equation. 相似文献
18.
The effect of the electrical anisotropy on the response of helicopter-borne frequency-domain electromagnetic systems 总被引：4，自引：0，他引：4
Helicopter electromagnetic (HEM) systems are commonly used for conductivity mapping and the data are often interpreted using an isotropic horizontally layered earth model. However, in regions with distinct dipping stratification, it is useful to extend the model to a layered earth with general anisotropy by assigning each layer a symmetrical 3 × 3 resistivity tensor. The electromagnetic (EM) field is represented by two scalar potentials, which describe the poloidal and toroidal parts of the magnetic field. Via a 2D Fourier transform, we obtain two coupled ordinary differential equations in the vertical coordinate. To stabilize the numerical calculation, the wavenumber domain is divided into two parts associated with small and large wavenumbers. The EM field for small wavenumbers is continued from layer to layer with the continuity conditions. For large wavenumbers, the EM field behaves like a DC field and therefore cannot be sensed by airborne EM systems. Thus, the contribution from the large wavenumbers is simply ignored. The magnetic fields are calculated for the vertical coaxial (VCX), horizontal coplanar (HCP) and vertical coplanar (VCP) coil configurations for a helicopter EM system. The apparent resistivities defined from the VCX, VCP and HCP coil responses, when plotted in polar coordinates, clearly identify the principal anisotropic axes of an anisotropic earth. The field example from the Edwards Aquifer recharge area in Texas confirms that the polar plots of the apparent resistivities identify the principal anisotropic axes that coincide well with the direction of the underground structures. 相似文献
19.
A. C. LIAKOPOULOS Dr. Eng. 《水文科学杂志》2013,58(3):41-48
Synopsis The dynamic equation of motion that governs the laminar flow of water through soils is the empirical equation of Darcy. According to Darcy's equation the velocity of the flowing water is proportional to the hydraulic gradient under which the water is flowing, with the constant of proportionality being the coefficient of permeability. The interesting question arising is whether or not the coefficient of permeability is a scalar quantity (having only a magnitude) or a vector (having both magnitude and direction). It is proved, in the present paper, that the permeability coefficient is neither a scalar nor a vector but a symmetric tensor of second rank. The fact that the permeability tensor is symmetric gives rise to great simplifications and permits a simple graphical construction of the tensor ellipsoid. Having the tensor ellipsoid, the determination of the direction at which the water will flow under a known imposed hydraulic gradient can be found graphically. In case of isotropic soils (the permeability coefficient has the same value along any direction) the ellipsoid reduces to a sphere and the tensor becomes a scalar. In the general case of anisotropic soils the permeability tensor is an entity with nine elements, six of which are independent representing pure extension or contraction along the three principal coordinate axes, thus transforming the permeability sphere into an ellipsoid and vice versa. It should be noted that in anisotropic soils the only directions along which the flow takes place in the direction of the hydraulic gradient are those of the principal axes of the tensor ellipsoid. Permeability tests were conducted on anisotropic sandstone samples taken at different directions with respect to rectangular coordinates. The permeability coefficient values plotted on a two-dimensional polar coordinate graph paper give rise to an ellipse substantiating therefore the tensor concept of the permeability coefficient. The graphical construction of the tensor ellipse and the use of it in order to obtain the direction of flow by knowing the direction of the hydraulic gradient is also shown. 相似文献
20.
The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented. 相似文献