A Fuzzy Model for Predicting the Group and Phase Velocities of Circumferential Waves Based on Subtractive Clustering

Phase and group velocities of mantle Love and Rayleigh waves obtained from strain seismograph records of the Chilean earthquake are presented. The velocities of mantle Rayleigh waves of period from 300 to 550 seconds agree with those predicted from periods of free spheroidal oscillation of the earth and do not show a flattening of the group velocity curve for periods greater than 380 seconds. Group velocities for mantle Rayleigh waves reach a maximum of 7.8 km/sec at a period of about 1000 sec. Study of initial phases of Rayleigh waves indicates a difference of phase of π between the azimuth to Isabella and the azimuths to Ñaña and Ogdensburg. Determinations of phase and group velocities of Love waves have been extended to periods of 700 seconds. The phase velocity data of Satô [1958] has been corrected for the polar phase shift. The correct curve has been identified from the numerous possible curves which result from a 2π ambiguity in the phase correlation made by Satô. Values of phase velocities are presented for periods in the range of 60 to 700 seconds. The group and phase velocities of both Love waves and Rayleigh waves agree well with those predicted for the Gutenberg-Bullen A model of the earth. It is verified that analysis of seismograms in terms of progressive wave trains is equivalent to analysis in terms of standing waves. In the presence of absorption, as for the earth, the analysis in terms of progressive wave trains has many advantages. Material supplementary to this article has been deposited with the ADI Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D.C. A copy may be secured by citing the document number 6816 and remitting $1.75 for 35-mm microfilm. Advance payment is required. Make check or money order payable to: Chief, Photoduplieation Service, Library of Congress.

The behavior of phase and group velocities of waves occurring in the ionosphere, which is considered collision-free, is the focus of this article. The phase velocities of the waves (ordinary, polarized waves) occurring in the ionosphere are larger than the speed of light, whereas the group velocities are less than the speed of light, according to the findings. The phase velocities are compatible with changes in electron density under acceptable conditions, however, the group velocities show an antisymmetric variation with the electron density. In the northern hemisphere, the extra-ordinary wave takes negative values, while in the southern hemisphere, it takes positive values.

The behavior of phase and group velocities of a left-polarized wave in the F-region of the ionosphere is studied in this paper. Despite the fact that the magnitudes of a left polarized wave's phase and group velocities in the F-region of the ionosphere are almost identical at low latitudes, they are schematically asymmetric under acceptable conditions. Under the same conditions, the group velocity changes in the same way as the electron density in this region; however, the phase velocity changes in the opposite direction. According to the findings, the left-polarized wave's group velocity and electron density have a linear connection. The phase velocity, on the other hand, cannot be said to be the same. Keywords: Ionosphere, The polarized wave, Group-phase velocity, Equatorial anomaly

AWESoMe 1.1: A code for the calculation of phase and group velocities of acoustic waves in homogeneous solids

Transversely isotropic media with a vertical symmetry axis (VTI) is the model of the subsurface suitable for processing seismic reflection data surface of sedimentary basins formed by shale. The propagation of P waves in VTI media is characterized by four independent elastic parameters and complex algebraic equations for the phase and group velocities. Therefore, there is a need to obtain approximations accurate to the phase and group velocities in VTI homogeneous media. Several authors have described approaches to the phase and group velocities with only three parameters, in homogeneous horizontally layered VTI media through hypotheses as weak anisotropy of the medium and anellipticity wavefront. In this work, we have used rational approximants in shifted-hiperbola approximation and obtained rational anelliptitic approximations of the phase and group velocities in homogeneous horizontally stratified VTI media. We have verified the accuracy of the approximations, compared with other approximations in the literature. As an application, we converted the group velocity approximations in nonhyperbolic moveout approximations and performed parameter estimation by means of semblancebased velocity analysis. The results show the validity of anelliptic rational approximations in inverse processes. Introduction Due to the limitations of isotropic models in more complex lithologies, such as sedimentary basins formed by shales, the seismic reflection survey is considered as a model of subsurface anisotropic media, especially the VTI media. In homogeneous media TIV, the wavefront of the SH phase velocity is elliptical and has exact equation that depends only two elastic parameters. However, the Pand SV-waves have: strongly anelliptical wavefront for both phase velocity and group velocity; algebraically complicated exact equations for the phase velocity; and are characterized by five independent elastic moduli tensors ( ). Moreover, even in TIV media, it is difficult to explain exact equations for the group velocity. Another remarkable feature of anisotropic media is the nonhyperbolic behavior of moveout curve. Thus, it is necessary to obtain approximations for the phase and group velocities and thus for moveout curves, which have precision and are practical to perform the steps of seismic data processing. Thomsen (1986), using the physical characteristics of the elastic parameters and the properties in the vertical direction, introduced a parameter of elastic moduli which facilitates the study of the effects of wave propagation in homogeneous anisotropic media VTI. Alkhalifah and Tsvankin (1995) found that only three of these parameters influence the propagation of P-waves in TIV media. Authors such as Muir and Dellinger (1985), Thomsen (1986), Dellinger (1993), Alkhalifah and Tsvankin (1995), Alkhalifah (1998), Fomel (2004), Psensic (2013), among others, have shown approaches the phase velocity in homogeneous VTI media, which depend explicitly only three elastic parameters. Dellinger, and Muir (1985) and Dellinger (1993) showed the anelliptical approximation of the phase velocities, using the properties of the elliptical anisotropy. Assuming analogy in the form of approximations obtained elliptical approximation of group velocity and consequently the moveout approximations for TI media. Fowler (2003) defined the elliptical component of the phase velocity, and through convenient parameterization of elastic parameters obtained anelliptical approximation for phase velocity in VTI media, equivalent to those obtained by other authors. However, using heuristics pure, converted them to approaches: by dispersion relations, group velocities and time equations. Fomel (2004) inspired by the anelliptical approximation (Dellinger et al,1993) used the shifted-hyperbola approximation (Malovichko 1978; Sword 1987; de Bazelaire 1988; Castle 1994) and Stoltstretch correction (Stolt 1978; Fomel and Vaillant 2001 ) to obtain, separately, the acoustic phase velocity approximation (Alkhalifah, 1998) of the P-wave in VTI media. However, by analogy form, obtained group velocity approximation and non-hyperbolic traveltime, very accurate. In this work, we obtained anelliptical approximation for the phase velocity of the compressional wave in vertical media such as rational approximating the shiftedhyperbola approximation (Fomel, 2004). Using the conversion technique by similarity of form (Dellinger, 1993), we obtained anelliptical approximations to group velocity towards these; we obtain new nonhyperbolic moveout approximations. To prove the accuracy of such approximations, we calculated the relative errors of these compared to other approximations. We also conducted semblance-based velocity analysis, to show the robustness of rational approximations of traveltime in estimating parameters. Phase velocity in VTI media The wave propagation in VTI media is characterized by the independent elastic parameters, density-normalized:

<p>The surface wave phase and group velocities are estimated by dividing the epicentral distance by phase and group travel times respectively in all the available methods, this is based on the assumptions that (1) surface waves originate at the epicentre and (2) the travel time of the particular group or phase of the surface wave is equal to its arrival time to the station minus the origin time of the causative earthquake; However, both assumptions are wrong since surface waves generate at some horizontal distance away from the epicentre. We calculated the actual horizontal distance from the focus at which they generate and assessed the errors caused in the estimation of group and phase velocities by the aforementioned assumptions in a simple isotropic single layered homogeneous half space crustal model using the example of the fundamental mode Love wave. We took the receiver locations in the epicentral distance range of 100-1000 km, as used in the regional surface wave analysis, varied the source depth from 0 to 35 Km with a step size of 5 km and did the forward modelling to calculate the arrival time of Love wave phases at each receiver location. The phase and group velocities are then estimated using the above assumptions and are compared with the actual values of the velocities given by Love wave dispersion equation. We observed that the velocities are underestimated and the errors are found to be; decreasing linearly with focal depth, decreasing inversely with the epicentral distance and increasing parabolically with the time period. We also derived empirical formulas using MATLAB curve fitting toolbox that will give percentage errors for any realistic combination of epicentral distance, time period and depths of earthquake and thickness of layer in this model. The errors are found to be more than 5% for all epicentral distances lesser than 500 km, for all focal depths and time periods indicating that it is not safe to do regional surface wave analysis for epicentral distances lesser than 500 km without incurring significant errors. To the best of our knowledge, the study is first of its kind in assessing such errors.</p>

The paper deals with studying trajectories of motion of individual liquid particles in a two-layer hydrodynamic system with a finite layer thickness as well as analyzing phase and group velocities of internal waves in the system. The problem is modeled for an inviscid incompressible fluid under action of the gravity and surface tension forces in a dimensionless form. Solutions of the problem are sought in the form of progressive waves using the multi-scale method. The solutions are expanded in terms of the nonlinearity coefficient. Dependence of the dispersion ratio of the wavenumber is investigated for different values of the surface tension coefficient and the ratio of the layer densities. Formulas are obtained for the group and phase velocities for internal gravity-capillary waves as well as in the limiting case for capillary waves. A comparison of the values of the phase and group velocities of internal waves for different values of the wave number is carried out. It is proved that with an increase in the wave number, the group velocity begins to outstrip the phase velocity, and their equality occurs at the minimum phase velocity. It is shown that the trajectories are ellipses in which the horizontal semi axes are larger than the vertical ones. Formulas are obtained for the semi axes of elliptic trajectories for each of the layers. The character of the change in the semi axes of elliptical trajectories is analyzed depending on the distance from the interface between two liquid layers as well as on the values of the wave number. It is proved that the semi axes of ellipses decrease unevenly with increasing distance from the boundary. The asymmetry of the particle trajectories of each of the layers is shown for the case when the thickness of the lower layer differs from the thickness of the lower layer. The study of the kinematic characteristics of the particle motion makes it possible to simulate real physical wave processes in the World Ocean. The results are also relevant for creating a theoretical basis for experiments.

Theoretical displacement seismograms are computed for the study of group and phase velocities of Love type surface waves in a dipping layer overlying an elastic body using the solution derived in a previous paper by the present authors. The result does not show that the group velocity in a dipping structure corresponds to the theoretical group velocity of horizontal structure having the thickness equal to an arithmetic average of depths between the source and receiver. The Sato's formula is found to explain the group velocity obtained by the analysis of theoretical seismograms for the short period waves in the cases of dip angles smaller than 2.0°, his formula being derived under the assumption that the phase velocity depends only upon the thickness at an observation station. The analyzed phase velocity agrees well with the phase velocity for horizontal layer with the thickness at an observation point for the short period waves when the dip angle is small.

Determination of the group and phase velocities from time–frequency representation of Wigner–Ville

The effect of dispersion on acoustic wave sensors is considered. The discussion is focused upon layer guided surface acoustic waves (Love waves), which obtain their high mass sensitivity for the first Love wave mode by optimizing the guiding layer thickness, d, such that d∼λl/4; the wavelength in the layer is given by λl=f/vl where f is the operating frequency and vl is the shear acoustic speed of the guiding layer. We show that this optimization of guiding layer thickness corresponds to strong dispersion so that the phase and group velocities can be quite different. From the definition of the phase velocity mass sensitivity, we show that it can be determined from either the slope of the curve of phase velocity with normalized guiding layer thickness, z=d/λl, or from the phase and group velocities measured for a given guiding layer thickness. Experimental data for a poly(methylmethacrylate) polymer guiding layer on 36° XY Lithium Tantalate is presented. Measurements of phase velocity and group velocity determined by a network analyzer were obtained for systematically increasing guiding layer thicknesses; a pulse transit experiment was also used to provide independent confirmation of the group velocity data. Two independent estimates of the mass sensitivity are obtained for z=d/λl<0.22 from (i) the slope of the phase velocity curve and (ii) the measurements of the group and phase velocity. These two estimates are shown to be consistent and we, therefore, conclude that it is possible to determine the mass sensitivity for a Love wave device with a given guiding layer thickness from measurements of the phase and group velocities. Moreover, we argue that the formula using group velocity to determine phase velocity mass sensitivity can be extended to a wide range of other acoustic wave sensors. In addition, we suggest that variations in the group velocity due to deposited mass may be a more sensitive parameter than variations in the phase velocity.

Measurement of vertical phase and group velocities of atmospheric gravity waves by VHF radar

Rayleigh and Love wave phase velocities for great circle paths from ordinary seismograms of World Wide Standard Seismographic Network /WWSSN/ stations

Experimental measurements of the phase and group velocities of body waves in a transversely isotropic medium

The anisotropy of graphite/epoxy composite plates was studied using measurements of phase and group velocities of Lamb waves. Three types of composite structures were used: unidirectional, two-directional with orthogonal fibers, and quasi-isotropic. Two experimental methods were developed. For single mode, the phase and group velocities and the angle of deviation between them were measured using angle variable contact ultrasonic transducers which were situated on the surface of the plates. For multimodes, a new broadband single transducer immersion technique was used to examine the dispersion behavior for the phase velocity as propagation direction was varied relative to the fibers. The experimental data for an So mode was compared with calculations of the angular dependence of the phase and group velocities of bulk waves in the graphite/epoxy composite material. The advantage of the Lamb wave technique over bulk waves is the ability to measure the in-plane anisotropic properties of thin composite plates such as those used in actual aircraft applications.

Rayleigh wave phase and group velocities and attenuation coefficients in the period range 18 to 120 s have been determined for three regions along nine paths across the western United States using a two‐station technique. The attenuation coefficients were found to increase from east to west between the Rocky Mountains and the Pacific coast. Rayleigh wave group and phase velocities were inverted, using a differential procedure, to obtain shear wave velocity models for three regions. These velocity models were then used in an inversion process where Qβ−1 as a function of depth was obtained from observations of Rayleigh wave attenuation. The inversion results show that Qβ values in the upper mantle of the western United States are lowest for the coastal regions and westernmost Basin and Range and highest for the Rocky Mountains and western Great Plains. Intermediate Qβ values appear to occur in the upper mantle beneath the Basin and Range province and the Columbia and Colorado plateaus, although uncertainties in the data prevent a clear separation between that region and coastal regions. Low values of Qβ occur in the upper crust, higher values in the lower crust, and highest values in the uppermost 20 km of the mantle of all these regions. These overlie an upper mantle low‐Q zone in which Qβ values decrease from east to west, lowest values (15–20) being similar to those observed above descending slabs in the western Pacific. The regional pattern of upper mantle shear wave Q values is similar to that observed for upper crustal shear wave Q values inferred from surface wave studies and for Lg coda Q values. The patterns of upper crustal and upper mantle Q variations correlate with the temporal sequence of tectonic and magmatic activity in the western United States, lowest Q values in both depth ranges occurring where tectonic activity has been most recent. Partial melting and/or enhanced dislocation motion in the mantle and variations in the volume of interstitial hydrothermal fluids in cracks in the upper crust may both be ultimately due to processes which occurred during plate consumption and interaction near the western margin of the United States.