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1.
The dispersion relation for Rayleigh waves in a layer on a half-space is modified by introducing the quadratic wave number instead of the phase velocity. The implicit function theorem is then used to derive analytical formulae for the group velocity and for the phase- and group velocity partial derivatives with respect to the parameters of the medium. As two examples, the method is applied to the interpretation of dispersion curves for short-period Rayleigh waves observed in a sedimentary basin of the West Carpathians, and in the Moravo-Silesian region, Czech Republic. 相似文献
2.
I. Lerche 《Pure and Applied Geophysics》1986,124(6):1021-1037
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. 相似文献
3.
Two well-known methodologies have been used, for the first time, to derive a detailed and reliable lithospheric model of the Adriatic Plate, consistent with the thus far available data: non linear inversion of phase and group velocity dispersion curve obtained from surface-wave tomography, and body-waves tomography. To carry out the body-waves tomography, about 16500 P phases and 2000 S phases from 1219 seismic events, 73 seismic stations located at the border of the investigated region and a vertically heterogeneous starting model derived from the nonlinear inversion of average dispersion measurements, are used. The 3-D velocity model of the upper lithosphere, obtained from the combined analysis, shows a rather clear structural anomaly on the northeastern side, where the crust is thicker, and an uprising of the top of the lid in the northern part of the plate with a very thin, if any, transition zone from crust to mantle.Acknowledgement We are grateful to Bruno Alessandrini and Luisa Filippi (National Seismic Service, Rome – Italy) for providing the computer code SIMULPS and for their very useful advice on the local earthquakes tomography method. This work has been supported by the Italian project MURST-COFIN prot. 2001048324-001 (2001). 相似文献
4.
Practical VTI approximations: a systematic anatomy 总被引：3，自引：0，他引：3
Transverse isotropy (TI) with a vertical symmetry axis (VTI) often provides an appropriate earth model for prestack imaging of steep-dip reflection seismic data. Exact P-wave and SV-wave phase velocities in VTI media are described by complicated equations requiring four independent parameters. Estimating appropriate multiparameter earth models can be difficult and time-consuming, so it is often useful to replace the exact VTI equations with simpler approximations requiring fewer parameters. The accuracy limits of different previously published VTI approximations are not always clear, nor is it always obvious how these different approximations relate to each other. Here I present a systematic framework for deriving a variety of useful VTI approximations. I develop first a sequence of well-defined approximations to the exact P-wave and SV-wave phase velocities. In doing so, I show how the useful but physically questionable heuristic of setting shear velocities identically to zero can be replaced with a more precise and quantifiable approximation. The key here to deriving accurate approximations is to replace the stiffness a13 with an appropriate factorization in terms of velocity parameters. Two different specific parameter choices lead to the P-wave approximations of Alkhalifah (Geophysics 63 (1998) 623) and Schoenberg and de Hoop (Geophysics 65 (2000) 919), but there are actually an infinite number of reasonable parametrizations possible. Further approximations then lead to a variety of other useful phase velocity expressions, including those of Thomsen (Geophysics 51 (1986) 1954), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Harlan (Stanford Exploration Project Report 89 (1995) 145), and Stopin (Stopin, A., 2001. Comparison of v(θ) equations in TI medium. 9th International Workshop on Seismic Anisotropy). Each P-wave phase velocity approximation derived this way can be paired naturally with a corresponding SV-wave approximation. Each P-wave or SV-wave phase velocity approximation can then be converted into an equivalent dispersion relation in terms of horizontal and vertical slownesses. A simple heuristic substitution also allows each phase velocity approximation to be converted into an explicit group velocity approximation. From these, in turn, travel time or moveout approximations can also be derived. The group velocity and travel time approximations derived this way include ones previously used by Byun et al. (Geophysics 54 (1989) 1564), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Tsvankin and Thomsen (Geophysics 59 (1994) 1290), Harlan (89 (1995) 145), and Zhang and Uren (Zhang, F. and Uren, N., 2001. Approximate explicit ray velocity functions and travel times for P-waves in TI media. 71st Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, 106–109). 相似文献
5.
The Selection of Field Acquisition Parameters for Dispersion Images from Multichannel Surface Wave Data 总被引：3，自引：0，他引：3
The accuracy and resolution of surface wave dispersion results depend on the parameters used for acquiring data in the field. The optimized field parameters for acquiring multichannel analysis of surface wave (MASW) dispersion images can be determined if preliminary information on the phase velocity range and interface depth is available. In a case study on a fill slope in Hong Kong, the optimal acquisition parameters were first determined from a preliminary seismic survey prior to a MASW survey. Field tests using different sets of receiver distances and array lengths showed that the most consistent and useful dispersion images were obtained from the optimal acquisition parameters predicted. The inverted S-wave velocities from the dispersion curve obtained at the optimal offset distance range also agreed with those obtained by using direct refraction survey. 相似文献
6.
Oldřich Novotný L. V. Burlako T. A. Proskuryakova 《Studia Geophysica et Geodaetica》1996,40(2):167-177
Summary Dispersion relations for Love and Rayleigh waves in a layer on a half-space are modified by introducing quadratic slownesses instead of velocities. The advantages of this approach are demonstrated on analytical formulae for computing the group velocity. 相似文献
7.
The derivation of P and S velocities at the core-mantle boundary (CMB) from long-period diffracted waves by the use of the simple ray-theoretical formulav
CMB=r
c
/p (v
CMB=velocity at the CMB;r
c
=core radius;p=ray parameter) yields apparent velocity values which differ from the true velocities. Using a dominant period of about 20 sec for calculating theoretical seismograms, we found a linear relation between the apparent velocity and the average velocity in a transition zone at the base of the mantle with fixed velocity on top.The ray parameters determined from long-period earthquake data are found to be 4.540±0.035 and 8.427±0.072 sec/deg for Pdiff and Sdiff, respectively. These values yield apparent velocities of 13.378±0.103 for P and 7.207±0.062 km/sec for S waves. By means of the theoretical relation between apparent and average velocity and under the assumption of linear variation of velocity with depth, one can invert the apparent velocities into true CMB velocities of 13.736±0.170 and 7.320±0.124 km/sec. These results imply positive velocity gradients at the base of the mantle and hence no significant departures from adiabaticity and homogeneity.Contribution No. 211 of the Geophysical Institute, University of Karlsruhe. 相似文献
8.
9.
Summary The partial derivatives of the dispersion curves have been calculated by means of a method founded on the theorem of implicit functions, which was described in[1]. The partial derivatives of the dispersion curves of Love waves are studied in detail for a model of a single-layered medium. Analytical formulae for the partial derivatives are used for deriving the limits of these partial derivatives when the period approaches zero or the critical period. 相似文献
10.
Anatoly A. Nikitin Boris D. Plyushchenkov Arkady Yu. Segal 《Geophysical Prospecting》2016,64(5):1335-1349
We derived the velocity and attenuation of a generalized Stoneley wave being a symmetric trapped mode of a layer filled with a Newtonian fluid and embedded into either a poroelastic or a purely elastic rock. The dispersion relation corresponding to a linearized Navier–Stokes equation in a fracture coupling to either Biot or elasticity equations in the rock via proper boundary conditions was rigorously derived. A cubic equation for wavenumber was found that provides a rather precise analytical approximation of the full dispersion relation, in the frequency range of 10?3 Hz to 103 Hz and for layer width of less than 10 cm and fluid viscosity below 0.1 Pa· s [100 cP]. We compared our results to earlier results addressing viscous fluid in either porous rocks with a rigid matrix or in a purely elastic rock, and our formulae are found to better match the numerical solution, especially regarding attenuation. The computed attenuation was used to demonstrate detectability of fracture tip reflections at wellbore, for a range of fracture lengths and apertures, pulse frequencies, and fluid viscosity. 相似文献
11.
M. DOBR
KA 《Geophysical Prospecting》1988,36(3):318-331
The WKB-method is used for the derivation of both the complex dispersion relation and displacement functions for Love channel-waves that propagate in a coal seam of varying thickness. The constant Q-model is used to describe the anelastic friction. With numerical solutions of the absorption-dispersion relation, the influence of thickness changes on the phase velocity and absorption coefficient of Love seam-waves is analysed at various frequencies. It is shown that the changes in the seam thickness can be optimally detected around the average Airy-phase frequency. An equivalence is pointed out between the wave guide structures: homogeneous with varying seam thickness and horizontally inhomogeneous with constant seam thickness. 相似文献
12.
S.C. Mohapatra 《地球物理与天体物理流体动力学》2014,108(6):615-638
Expansion formulae associated with the interaction of oblique surface gravity waves with a floating flexible plate in the presence of a submerged horizontal flexible structure are derived using Green’s integral theorem in water of finite and infinite water depths. The associated Green’s functions are derived using the fundamental solution associated with the reduced wave equation. The integral forms of the Green’s functions and the velocity potentials are advantageous over the eigenfunction expansion method in situation when the roots of the dispersion relation coalesce. As an application of the expansion formulae, diffraction of oblique waves by a finite floating elastic plate in the presence of a submerged horizontal flexible membrane is investigated in water of finite depth. The accuracy of the numerical computation is demonstrated by analysing the convergence of the complex amplitude of the reflected waves and the energy relation. Effect of the submerged membrane on the diffraction of surface waves is studied by analysing the reflection and transmission coefficients for various parametric values. Further, the derivation of long wave equation under shallow water approximation is derived in a direct manner in the appendix. The concept and methodology can be easily extended to deal with acoustic wave interaction with flexible structures and related problems of mathematical physics and engineering. 相似文献
13.
Steven D. London 《地球物理与天体物理流体动力学》2013,107(4):317-333
We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane. 相似文献
14.
O'Leary González Leonardo Alvarez Mariangela Guidarelli Giuliano F. Panza 《Pure and Applied Geophysics》2007,164(10):1985-2007
An overview of the crust and upper mantle structure of Central America and the Caribbean region is presented as a result of
the processing of more than 200 seismograms recorded by digital broadband stations from SSSN and GSN seismic networks. Group
velocity dispersion curves are obtained in the period range from 10s to 40s by FTAN analysis of the fundamental mode of the
Rayleigh waves; the error of these measurements varies from 0.06 and 0.09 km/s. From the dispersion curve, seven tomographic
maps at different periods and with average spatial resolution of 500 km are obtained. Using the logical combinatorial classification
techniques, eight main groups of dispersion curves are determined from the tomographic maps and eleven main regions, each
one characterized by one kind of dispersion curves, are identified. The average dispersion curves obtained for each region
are extended to 150s by adding data from a larger-scale tomographic study (Vdovin et al., 1999) and inverted using a nonlinear procedure. A set of models of the S-wave velocity vs. depth in the crust and upper
mantle is found as a result of the inversion process. In six regions we identify a typically oceanic crust and upper mantle
structure, while in the other two the models are consistent with the presence of a continental structure. Two regions, located
over the major geological zones of the accretionary crust of the Caribbean region, are characterized by a peculiar crust and
upper mantle structure, indicating the presence of lithospheric roots reaching, at least, about 200 km of depth. 相似文献
15.
Stability and grid dispersion in the P-SV 4th-order in space, 2nd-order in time, displacement-stress staggered-grid finite-difference scheme is investigated in the case of a homogeneous unbounded medium. All results, however, also apply to the velocity-stress and displacement- velocity-stress finite-difference schemes.
Independent stability conditions for the P and S waves are obtained by exact separation of equations for the two types of waves.
Since the S-wave group velocity can differ from the actual velocity as much as 5% for the sampling ratio 1/5, commonly used in numerical modelling, the sampling of the minimum S wavelength by 6 grid spacings (with the velocity difference not larger than 2.5%) is recommended.
Grid dispersion is strongest for a wave propagating in a direction of a coordinate axis and weakest for a wave propagating along a plane diagonal.
Grid dispersion in the 4
th
-order scheme for the sampling ratios s = 1/5 and s = 1/6 is smaller than grid dispersion in the 2
nd
-order scheme for s = 1/10 and s = 1/12, respectively. 相似文献
16.
T. K. De 《Pure and Applied Geophysics》1980,118(2):1170-1178
Rayleigh's principle and the concept of the local wave number have been utilised for the approximate determination of the dispersion of Love waves propagating in a laterally heterogeneous layer lying over a homogeneous half-space. The shear wave velocity and the rigidity in the surface layer have been assumed to decrease with the increase of the lateral distance from the origin. The range of validity of the dispersion equation obtained by this method has been examined critically. It was found that: (a) for existence of Love waves the minimum value of shear wave velocity in the layer must be less than that in the matter below, and (b) the phase velocity of Love waves decreases with the increase of the lateral distance from the origin. 相似文献
17.
The elastic and anelastic structure of the lithosphere and asthenosphere of the Iberian Peninsula is derived by means of tomographic
techniques applied to local phase and group velocities and local attenuation coefficients of Rayleigh wave fundamental mode.
The database consists of surface wavetrains recorded at the broadband stations located in the Iberian Peninsula on the occasion
of the ILIHA project. Path-averaged phase and group velocities and attenuation coefficients were previously obtained by standard
filtering techniques of surface wavetrains and, subsequently, local dispersion curves were computed according to the Yanovskaya-Ditmar
formulation. First, a principal component analysis (PCA) and the average linkage (AL) clustering algorithm are applied to
these local values in order to classify the Iberian Peninsula in several rather homogeneous domains from the viewpoint of
the similarity of the corresponding local dispersion curves, without previous seismotectonic constraints. Second, averaged
phase and group velocities and attenuation coefficients representing each homogeneous region are used to derive the respective
elastic and anelastic models of the lithosphere and asthenosphere. This purpose is achieved by using the uncoupled causal
inversion of phase and group velocities and attenuation coefficients. The main features of the homogeneous regions are discussed
by taking as reference the Hercynic, Alpine and Neogene domains of the Iberian Peninsula, and two questions affecting the
reliability of the elastic-anelastic models are revised. First, the coherence of the shear-velocity and Qβ−1 models obtained by causal uncoupled inversion for each region is analysed. Second, the influence of the causal phase and
group velocities on the shear-velocity models is evaluated by comparing elastic and anelastic models derived from causal uncoupled
inversion with those deduced from non-causal inversion. 相似文献
18.
The idea of Gerstner's trochoidal waves is used to reconstruct a model of short, finite-amplitude progressive waves on frontal surfaces of the Margules Type. Stable waves associated with a negative (westward) group velocity occur in the model. A wave train can be maintained through the formation of new waves in the rear of the existing waves as a result of the westward energy transport. 相似文献
19.
Atsushi Kubokawa 《地球物理与天体物理流体动力学》2013,107(2):119-137
Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [?π/L b , π/L b ], where L b is the periodic length of the topography. When the topographic?β?is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude. 相似文献
20.
《水文科学杂志》2013,58(1)
Abstract Abstract Accurate application of the longitudinal dispersion model requires that specially designed experimental studies are performed in the river reach under consideration. Such studies are usually very expensive, so in order to quantify the longitudinal dispersion coefficient, as an alternative approach, various researchers have proposed numerous empirical formulae based on hydraulic and morphometric characteristics. The results are presented of the application of artificial neural networks as a parameter estimation technique. Five different cases were considered with the network trained for different arrangements of input nodes, such as channel depth, channel width, cross-sectionally averaged water velocity, shear velocity and sinuosity index. In the case where the sinuosity index is included as an input node, the results turned out to be better than those presented by other authors. 相似文献