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1.
We describe two practicable approaches for an efficient computation of seismic traveltimes and amplitudes. The first approach is based on a combined finite‐difference solution of the eikonal equation and the transport equation (the ‘FD approach’). These equations are formulated as hyperbolic conservation laws; the eikonal equation is solved numerically by a third‐order ENO–Godunov scheme for the traveltimes whereas the transport equation is solved by a first‐order upwind scheme for the amplitudes. The schemes are implemented in 2D using polar coordinates. The results are first‐arrival traveltimes and the corresponding amplitudes. The second approach uses ray tracing (the ‘ray approach’) and employs a wavefront construction (WFC) method to calculate the traveltimes. Geometrical spreading factors are then computed from these traveltimes via the ray propagator without the need for dynamic ray tracing or numerical differentiation. With this procedure it is also possible to obtain multivalued traveltimes and the corresponding geometrical spreading factors. Both methods are compared using the Marmousi model. The results show that the FD eikonal traveltimes are highly accurate and perfectly match the WFC traveltimes. The resulting FD amplitudes are smooth and consistent with the geometrical spreading factors obtained from the ray approach. Hence, both approaches can be used for fast and reliable computation of seismic first‐arrival traveltimes and amplitudes in complex models. In addition, the capabilities of the ray approach for computing traveltimes and spreading factors of later arrivals are demonstrated with the help of the Shell benchmark model. 相似文献
2.
Multiple vertical fracture sets, possibly combined with horizontal fine layering, produce an equivalent medium of monoclinic symmetry with a horizontal symmetry plane. Although monoclinic models may be rather common for fractured formations, they have hardly been used in seismic methods of fracture detection due to the large number of independent elements in the stiffness tensor. Here, we show that multicomponent wide-azimuth reflection data (combined with known vertical velocity or reflector depth) or multi-azimuth walkaway VSP surveys provide enough information to invert for all but one anisotropic parameters of monoclinic media. In order to facilitate the inversion procedure, we introduce a Thomsen-style parametrization for monoclinic media that includes the vertical velocities of the P-wave and one of the split S-waves and a set of dimensionless anisotropic coefficients. Our notation, defined for the coordinate frame associated with the polarization directions of the vertically propagating shear waves, captures the combinations of the stiffnesses responsible for the normal-moveout (NMO) ellipses of all three pure modes. The first group of the anisotropic parameters contains seven coefficients (ε(1,2), δ(1,2,3) and γ(1,2)) analogous to those defined by Tsvankin for the higher-symmetry orthorhombic model. The parameters ε(1,2), δ(1,2) and γ(1,2) are primarily responsible for the pure-mode NMO velocities along the coordinate axes x1 and x2 (i.e. in the shear-wave polarization directions). The remaining coefficient δ(3) is not constrained by conventional-spread reflection traveltimes in a horizontal monoclinic layer. The second parameter group consists of the newly introduced coefficients ζ(1,2,3) which control the rotation of the P-, S1- and S2-wave NMO ellipses with respect to the horizontal coordinate axes. Misalignment of the P-wave NMO ellipse and shear-wave polarization directions was recently observed on field data by Pérez et al. Our parameter-estimation algorithm, based on NMO equations valid for any strength of the anisotropy, is designed to obtain anisotropic parameters of monoclinic media by inverting the vertical velocities and NMO ellipses of the P-, S1- and S2-waves. A Dix-type representation of the NMO velocity of mode-converted waves makes it possible to replace the pure shear modes in reflection surveys with the PS1- and PS2-waves. Numerical tests show that our method yields stable estimates of all relevant parameters for both a single layer and a horizontally stratified monoclinic medium. 相似文献
3.
Analytic expressions for the velocity sensitivity to the elastic moduli for the most general anisotropic media 总被引：1，自引：0，他引：1
For non‐linear kinematic inversion of elastic anisotropy parameters and related investigations of the sensitivity of seismic data, the derivatives of the wavespeed (phase velocity and group velocity) with respect to the individual elastic moduli are required. This paper presents two analytic methods, called the eigenvalue and eigenvector methods, to compute the derivatives of the wavespeeds for wave propagation in a general anisotropic medium, which may be defined by up to 21 density‐normalized elastic moduli. The first method employs a simple and compact form of the eigenvalue (phase velocity) and a general form of the group velocity, and directly yields general expressions of the derivatives for the three wave modes (qP, qS1, qS2). The second method applies simple eigenvector solutions of the three wave modes and leads to other general forms of the derivatives. These analytic formulae show that the derivatives are, in general, functions of the 21 elastic moduli as well as the wave propagation direction, and they reflect the sensitivity of the wavespeeds to the individual elastic moduli. Meanwhile, we give results of numerical investigations with some examples for particular simplified forms of anisotropy. They show that the eigenvalue method is suitable for the qP‐, qS1‐ and qS2‐wave computations and mitigates the singularity problem for the two quasi‐shear waves. The eigenvector method is preferable to the eigenvalue method for the group velocity and the derivative of the phase velocity because it involves simpler expressions and independent computations, but for the derivative of the group velocity the derivative of the eigenvector is required. Both methods tackle the singularity problem and are applicable to any degree of seismic anisotropy for all three wave modes. 相似文献
4.
5.
Traveltime computation by wavefront-orientated ray tracing 总被引：1，自引：0，他引：1
For multivalued traveltime computation on dense grids, we propose a wavefront‐orientated ray‐tracing (WRT) technique. At the source, we start with a few rays which are propagated stepwise through a smooth two‐dimensional (2D) velocity model. The ray field is examined at wavefronts and a new ray might be inserted between two adjacent rays if one of the following criteria is satisfied: (1) the distance between the two rays is larger than a predefined threshold; (2) the difference in wavefront curvature between the rays is larger than a predefined threshold; (3) the adjacent rays intersect. The last two criteria may lead to oversampling by rays in caustic regions. To avoid this oversampling, we do not insert a ray if the distance between adjacent rays is smaller than a predefined threshold. We insert the new ray by tracing it from the source. This approach leads to an improved accuracy compared with the insertion of a new ray by interpolation, which is the method usually applied in wavefront construction. The traveltimes computed along the rays are used for the estimation of traveltimes on a rectangular grid. This estimation is carried out within a region bounded by adjacent wavefronts and rays. As for the insertion criterion, we consider the wavefront curvature and extrapolate the traveltimes, up to the second order, from the intersection points between rays and wavefronts to a gridpoint. The extrapolated values are weighted with respect to the distances to wavefronts and rays. Because dynamic ray tracing is not applied, we approximate the wavefront curvature at a given point using the slowness vector at this point and an adjacent point on the same wavefront. The efficiency of the WRT technique is strongly dependent on the input parameters which control the wavefront and ray densities. On the basis of traveltimes computed in a smoothed Marmousi model, we analyse these dependences and suggest some rules for a correct choice of input parameters. With suitable input parameters, the WRT technique allows an accurate traveltime computation using a small number of rays and wavefronts. 相似文献
6.
A. Hanyga 《Journal of Applied Geophysics》2003,54(3-4):411
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. 相似文献
7.
Time horizons can be depth-migrated when interval velocities are known; on the other hand, the velocity distribution can be found when traveltimes and NMO velocities at zero offset are known (wavefront curvatures; Shah 1973). Using these concepts, exact recursive inversion formulae for the calculation of interval velocities are given. The assumption of rectilinear raypath propagation within each layer is made; interval velocities and curvatures of the interfaces between layers can be found if traveltimes together with their gradients and curvatures and very precise VNMO velocities at zero offset are known. However, the available stacking velocity is a numerical quantity which has no direct physical significance; its deviation from zero offset NMO velocity is examined in terms of horizon curvatures, cable length and lateral velocity inhomogeneities. A method has been derived to estimate the geological depth model by searching, iteratively, for the best solution that minimizes the difference between stacking velocities from the real data and from the structural model. Results show the limits and capabilities of the approach; perhaps, owing to the low resolution of conventional velocity analyses, a simplified version of the given formulae would be more robust. 相似文献
8.
Byron McKavanagh Bruce Boreham Kevin McCue Gary Gibson Jennifer Hafner George Klenowski 《Pure and Applied Geophysics》1995,144(1):39-57
Digital seismograms continuously recorded from 1988 to 1992 by two stations of the RESNOM seismic network in northern Baja
California, Mexico, were used to search for probable shear-wave anisotropic characteristics in the region of the Cerro Prieto
fault. Shear-wave splitting was identified in many of the three-component records analyzed. We measured the polarization direction
of the leadingS wave inside theS-wave window as well as the delay times between fast and slow phases on those records displaying shear-wave splitting. For
station CPX, which is nearest the Imperial Valley region to the north, the preferred polarization direction found in this
study (azimuth 180°±10°) coincides with the direction of the regional maximum compressive stress determined for the region.
This polarization direction can be interpreted in terms of the “Extensive Dilatancy Anisotropy” model as the effect of vertical
parallel aligned cracks. The preferred polarization direction measured at LMX, however, gives an azimuth of 45°±5°. Thus,
it appears that faults and fractures aligned oblique to the main tectonic trend have a greater influence on the anisotropic
characteristics of the crust south of Cerro Prieto volcano than that of the regional stress field. Time delays between slow
and fastS waves observed at CPX appear constant from 1988 to 1992 while delays measured at LMX for the same interval indicate a small
increase with time which cannot be attributed to azimuthal variations of paths. 相似文献
9.
Inversion of Travel Times in Weakly Anisotropic Rock Samples 总被引：1，自引：0，他引：1
Based on the perturbation theory, inversion formulae for travel time of qP and qS waves in arbitrary weak anisotropy media are presented. The inversion formulae are linear expressions of elastic parameters expressed in terms of weak anisotropy (WA) parameters. The formulae of qS1 and qS2 waves have the same form and they can be used without identifying which wave is considered. A synthetic experiment similar to the measurement of rock sample in the laboratory is carried out to illustrate the efficiency of the presented inversion formulae. Two data sets for qP wave travel time from rock samples in the laboratory are inverted and 15 WA parameters are obtained. 相似文献
10.
Summary In this short contribution,SH-wave propagation in an anisotropic inhomogeneous crustal layer lying on an yielding and rigid isotropic half space is considered. The inhomogeneity is assumed to be present in the directional rigidities and the density. The type of variation in elastic parameters considered herein, is such that the velocities ofSH-waves in horizontal and vertical directions are constant. The frequency equations, governing propagation ofSH-waves are derived for both the cases, i.e. (I) when the lower medium is yielding half space and (II) when it is rigid. Dispersion curves for these cases are also presented. 相似文献
11.
A new mesh refinement strategy for generating high quality unstructured meshes of the Northwestern European continental shelf, the continental slope and the neighbouring ocean is presented. Our objective is to demonstrate the ability of anisotropic unstructured meshes to adequately address the challenge of simulating the hydrodynamics occurring in these three regions within a unique mesh. The refinement criteria blend several hydrodynamic considerations as the tidal wave propagation on the continental shelf and the hydrostatic consistency condition in steep areas. Several meshes illustrate both the validity and the efficiency of the refinement strategy. The selection of the refinement parameters is discussed. Finally, an attempt is made to take into account tidal ellipses, providing another cause for anisotropy in the mesh. 相似文献
12.
Determination of the ray vector (the unit vector specifying the direction of the group velocity vector) corresponding to a given wave normal (the unit vector parallel to the phase velocity vector or slowness vector) in an arbitrary anisotropic medium can be performed using the exact formula following from the ray tracing equations. The determination of the wave normal from the ray vector is, generally, a more complicated task, which is usually solved iteratively. We present a first-order perturbation formula for the approximate determination of the ray vector from a given wave normal and vice versa. The formula is applicable to qP as well as qS waves in directions, in which the waves can be dealt with separately (i.e. outside singular directions of qS waves). Performance of the approximate formulae is illustrated on models of transversely isotropic and orthorhombic symmetry. We show that the formula for the determination of the ray vector from the wave normal yields rather accurate results even for strong anisotropy. The formula for the determination of the wave normal from the ray vector works reasonably well in directions, in which the considered waves have convex slowness surfaces. Otherwise, it can yield, especially for stronger anisotropy, rather distorted results. 相似文献
13.
The standard ray theory (RT) for inhomogeneous anisotropic media does not work properly or even fails when applied to S-wave
propagation in inhomogeneous weakly anisotropic media or in the vicinity of shear-wave singularities. In both cases, the two
shear waves propagate with similar phase velocities. The coupling ray theory was proposed to avoid this problem. In it, amplitudes
of the two S waves are computed by solving two coupled, frequency-dependent differential equations along a common S-wave ray.
In this paper, we test the recently developed approximation of coupling ray theory (CRT) based on the common S-wave rays obtained
by first-order ray tracing (FORT). As a reference, we use the Fourier pseudospectral method (FM), which does not suffer from
the limitations of the ray method and yields very accurate results. We study the behaviour of shear waves in weakly anisotropic
media as well as in the vicinity of intersection, kiss or conical singularities. By comparing CRT and RT results with results
of the FM, we demonstrate the clear superiority of CRT over RT in the mentioned regions as well as the dangers of using RT
there. 相似文献
14.
The most difficult part of multicomponent processing is the estimation of the shear-wave velocity map for migration. We used refracted shear waves and a simple iterative method called wavefield continuation (WFC) to evaluate the shallow shear-wave velocity profile on a real data example. The WFC was developed in 1981 by Clayton and McMechan to determine compressional-wave velocity profiles from refracted compressional waves. The application to refracted shear waves is straightforward. The real data example shows that shear structure can be easily determined independently of the compressional structure. 相似文献
15.
Kirchhoff–Helmholtz (KH) theory is extended to synthesize two-way elastic wave propagation in 3D laterally heterogeneous, anisotropic media. I have developed and tested numerically a specialized algorithm for the generation of three-component synthetic seismograms in multi-layered isotropic and transversely isotropic (TI) media with dipping interfaces and tilted axes of symmetry. This algorithm can be applied to vertical seismic profile (VSP) geometries and works well when the receiver is located near the reflector interface. It is superior to ray methods in predicting elliptical polarization effects observed on radial and transverse components. The algorithm is used to study converted-wave propagation for determining fracture-related shear-wave anisotropy in realistic reservoir models. Results show that all wavefront attributes are strongly affected by the anisotropy. However, it is necessary to resolve a trade-off between the effects of fractures and formation dip prior to converted-wave interpretation. These results provide some assurance that the present scheme is sufficiently versatile to handle shear wave behaviour due to various generalized rays propagating in complex geological models. 相似文献
16.
Seismic Anisotropy and Velocity Variations in the Mantle beneath the Saxothuringicum-Moldanubicum Contact in Central Europe 总被引：1，自引：0，他引：1
—We report on results of a passive seismic experiment undertaken to study the 3-D velocity structure and anisotropy of the upper mantle around the contact zone of the Saxothuringicum and Moldanubicum in the western margin of the Bohemian Massif in central Europe. Spatial variations of P-wave velocities and lateral variations of the particle motion of split shear waves over the region monitor changes of structure and anisotropy within the deep lithosphere and the asthenosphere. A joint interpretation of P-residual spheres and shear-wave splitting results in an anisotropic model of the lithosphere with high velocities plunging divergently from the contact of both tectonic units. Lateral variations of the mean residuals are related to a southward thickening of the lithosphere beneath the Moldanubicum. 相似文献
17.
18.
—Approximate PP plane wave displacement coefficients of reflection and transmission for weak contrast interfaces separating weakly but arbitrarily anisotropic elastic media are presented. The PP reflection coefficient for such an interface has been derived recently by Vavry?uk and P?en?ík (1997). The PP transmission coefficient presented in this paper was derived by the same approach. The coefficients are given as a sum of the coefficient for the weak contrast interface separating two nearby isotropic media and a term depending linearly on contrasts of the so-called weak anisotropy (WA) parameters (parameters specifying deviation of properties of the medium from isotropy), across the interface. While the reflection coefficient depends only on 8 of the complete set of the WA parameters describing P-wave phase velocity in weakly anisotropic media, the transmission coefficient depends on their complete set. The PP reflection coefficient depends on "shear-wave splitting parameter" γ. Tests of accuracy of the approximate formulae are presented on several models. 相似文献
19.
Backus and Crampin derived analytical equations for estimating approximate phase-velocity variations in symmetry planes in weakly anisotropic media, where the coefficients of the equations are linear combinations of the elastic constants. We examine the application of similar equations to group-velocity variations in off-symmetry planes, where the coefficients of the equations are derived numerically. We estimate the accuracy of these equations over a range of anisotropic materials with transverse isotropy with both vertical and horizontal symmetry axes, and with combinations of transverse isotropy yielding orthorhombic symmetry. These modified equations are good approximations for up to 17% shear-wave anisotropy for propagations in symmetry planes for all waves in all symmetry systems examined, but are valid only for lower shear-wave anisotropy (up to 11%) in off-symmetry planes. We also obtain analytical moveout equations for the reflection of qP-, qSH-, and qSV- waves from a single interface for off-symmetry planes in anisotropic symmetry. The moveout equation consists of two terms: a hyperbolic moveout and a residual moveout, where the residual moveout is proportional to the degree of anisotropy and the spread length of the acquisition geometry. Numerical moveout curves are computed for a range of anisotropic materials to verify the analytical moveout equations. 相似文献
20.
I. Lerche 《Pure and Applied Geophysics》1987,125(5):717-751
We demonstrate how multiples, generated at the interfaces of plane parallel beds, modify the propagation characteristics of an originally coherent seismic wave. For waves propagating at an angle to the bedding plane we find that theSV andP-waves couple so that neither is a pure mode. TheSH-wave, while modified in its propagation characteristics by multiples, remains a pure mode. The coupling ofSV-multiples into the quasi-P-mode appears weaker than the coupling ofP-wave multiples into the quasi-SV mode; at least this is so for the two simple cases of (a) density fluctuations only and (b) correlatedV
p
andV
s
fluctuations which conserve Poisson's ratio.We also find that the coupling is sensitive to both the angle of propagation and frequency. In addition there is a cut-off angle forP-wave multiples influencing the quasi-SV mode. Propagation angles larger than the cut-off permit theP-multiples to modify the phase of the quasi-SV mode, but not its effective attenuation. No such cut-off effect is found for SV-multiples influencing the quasi-P mode, whose angle-dependent and frequency-dependent phase distortion and effective attenuation are influenced both byP-wave multiples andSV-multiples.In view of the mathematical complexity of the expressions describing the phase, and effective attenuation of modes when allowance is made forP-andS-wave multiples, we strongly advocate numerical coding of the major mathematical formulae. By so doing a systematic study can be undertaken of the frequency and offset dependence of seismic waves as a function of seismic source input and power spectral behavior of the fluctuations in density and elastic constants of beds. It is our opinion that the full mathematical expressions are too involved to permit an analytic, systematic investigation to be given of the phase and attenuation of seismic waves with any degree of sophistication or generality. 相似文献