Calculation Of Phase Velocities Research Articles

Inversion of Functional Graded Materials Elastic Properties from Ultrasonic Lamb Wave Phase Velocity Data Using Genetic Algorithm

An inverse method based on genetic algorithm (GA) is presented to determine the elastic properties of functional graded materials (FGMs) plate from lamb wave phase velocity data. The Legendre polynomial expansion method is used as a forward solver to calculate the phase velocity dispersion curves of lamb wave in FGMs plate, which has a distribution function to express the inhomogeneity of materials’ elastic properties. The properties of FGMs plate can be characterized by minimizing the standard deviations between the actual and calculated phase velocities of lamb waves. By using GA, the elastic parameters of FGMs plates with three different distribution functions are inversed, and the convergence and deviations of the inversion are also discussed. The investigation shows that this inverse method is efficient and accurate for determining the materials parameters of FGMs and the deviation can be controlled below 5 %.

Calculation of Rayleigh-wave phase velocities due to models with a high-velocity surface layer

Rayleigh-wave phase velocities have been utilized to determine shear (S)-wave velocities in near-surface geophysics since early 1980s. One of the key steps is to calculate theoretical dispersion curves of an earth model. When the S-wave velocity of the surface layer is higher than some of the layers below, however, the Rayleigh-wave phase velocity in a high-frequency range calculated by existing algorithms approaches the lowest S-wave velocity among the layers above the half-space, rather than a value related to the S-wave velocity of the surface layer. According to our numerical modeling results based on wave equation, trends of the Rayleigh-wave dispersive energy approach about a 91% of the S-wave velocity of the surface layer at a high-frequency range when its wavelength is much shorter than the thickness of the surface layer, which cannot be fitted by a dispersion curve calculated by existing algorithms. We propose a method to calculate Rayleigh-wave phase velocities of models with a high-velocity surface layer by considering its penetration depth. We build a substituted model that only contains the layer with the lowest S-wave velocity among the layers above the half-space and the layers above it. We use the substituted model to replace the original model to calculate phase velocities when the Rayleigh-wave wavelength is not long enough to penetrate the lowest S-wave velocity layer. Several synthetic models are used to verify fitness between the dispersion curve calculated by our proposed method and the trend of the highest dispersive energy. Examples of inversion also demonstrate high accuracy of using our method as the forward calculation method during the inversions.

Wave Height and Wave Velocity Measurements in the Vicinity of the Break Point in Laboratory Plunging Waves

Results are presented of laboratory experiments undertaken to study the dynamics of wave propagation and transformation within the surf zone. The study involved measuring the external flow characteristics of regular plunging waves propagating along a 20 m long flume fitted with a 1:20 plane slope. To achieve this, monochromatic waves of frequency 0.4 Hz and a deep water wave height of 12 cm were generated by a servo-controlled piston-type wave maker. A set of calibrated parallel-wire capacitive wave gauges were employed to measure statistics of the free surface elevation along the slope in order to get an insight into the wave breaking behavior. To characterize the wave field, free surface elevation measurements were made in the vicinity of the break point. The measured time series data were analyzed at each flume position to obtain statistics of the mean water level, wave height, and wave velocity along the flume. Results show that as the wave propagates from deep water towards shallow water, there is an increase in the wave height, reaching a maximum height of about 21.5 cm at the break point, and then decreases sharply thereafter. Wave phase velocity calculations at different flume positions were made from the measured time series. Cross correlation techniques were used to determine the phase difference between the reference wave near the generator and the wave at various points along the flume. The local wave velocity was obtained by taking the phase difference between two points spaced 0.2 m apart. A comparison was made between the measured wave phase velocity, its linear shallow water ((gh)) approximation and the roller model concept wave velocity (1.3(gh)), at various points along the flume. The measured wave velocity c was found to lie in the range (gh)<c<1.3(gh) for most of the positions except near the break point. After the break point, the measured wave velocity is up to 38% higher than the theoretical value predicted using the roller model concept. Also noted is the variability of the phase speed in the breaking region. The present experiments of quantifying the mean macroscopic properties of breaking waves are a necessary prerequisite for more detailed experiments involving internal fluid velocity measurements that will follow.

Near-Surface Shear-Wave Velocity Structure of the Chiayi Area, Taiwan

This paper uses microtremor‐array measurements to obtain an overall view of S ‐wave velocity structure in the Chiayi area, Taiwan. The Rayleigh‐wave dispersion curves are calculated using the frequency‐wavenumber (![Graphic][1] ) spectrum method and then the S ‐wave velocity structure of the area is estimated by surface‐wave inversion. According to the inversion results, the S ‐wave velocities decrease from east to west. If the S ‐wave velocity of the bedrock is assumed to be 1500 m/s, the depths of the alluvium are between 560 and 1400 m gradually increasing from east to west. In order to understand the variations of the shallow S ‐wave velocity in the area, we sketch 2D and 3D maps using the imaging techniques based on interpolation algorithms. From the 3D results, the thickness of the sediments having an S ‐wave velocity in the range 270–1500 m/s increases from east to west; whereas the thickness of the layers having an S ‐wave velocity in the range 1500–2370 m/s decreases from east to west. The results are also in good agreement with the geological and geophysical information of the Chiayi area. Therefore, microtremor‐array measurement provides a good alternate to estimate shallow S ‐wave velocity structure. Online Material: Figures showing observed and calculated phase velocities, f ‐ k spectra, dispersion curves, inverted velocity models, sensitivity kernels, and V S contour maps and electric resistivity maps. [1]: /embed/inline-graphic-1.gif

Laser optoacoustic method of local porosity measurement of particles reinforced composites

In the present work we have realized experimentally the laser optoacoustic method for the measurements of local volume fraction of air pores (porosity P) of isotropic composite materials. It is based on measurements of phase velocities of thermooptically excited longitudinal ultrasonic waves in composite samples in the frequency range 0.5÷50 MHz. The local porosity (lateral resolution 1÷2 mm) is determined using the dependence of phase velocity of longitudinal acoustic wave on porosity for porous metals and theoretical calculation of phase velocity in a composite with the two-phase medium model. A number of aluminum alloy (silumin) matrix composite samples reinforced by SiC particles of a different mass concentration with the mean particle size of 14 mm was investigated. The samples were disks with the diameter d = 40 mm and the porosity in the center and in the periphery area of each sample was determined. The increase of mass concentration of the filler SiC leads to the growth of P. The results coincide within relative inaccuracy of 2–3% with the gravimetrical measurements of average <P> value, the porosity in the center of each sample was slightly higher than in the periphery.

Using Monte Carlo Method for Calculating Phonon Images

In this paper a Monte Carlo method has been used to calculate the phonon images of a number of semiconductor cubic crystals from III-V and II-VI groups (higher symmetry) and orthorhombic crystals (lower symmetry). An algorithms a MATLAB codes has been prepared for the calculation of phase velocities, slowness surfaces and group velocities and the formation of the phonon images which required a large number of points in the reduced Brillion zone, these points has been transformed to the group velocity space by solving the Christoffel equation for each of these points, a projection in a given direction is obtained, each incident pluses is transformed to gray level to form the final image. A comparison between the Monte Carlo and systematic methods has been done for the calculated images.

Propagation Characteristics of Surface Acoustic Waves in AlN/128° Y–X LiNbO3 Structures

In this study, propagation characteristics of surface acoustic waves in a layered piezoelectric structure (AlN/128° Y–X LiNbO3) are investigated. The phase velocity and coupling coefficient of the layered structure with interdigital transducers (IDTs) and/or thin metal films deposited at various interfaces are calculated numerically. The effects of the polarity of 128° Y–X LiNbO3 on SAW characteristics are illustrated. The calculated phase velocity and coupling coefficient of SAWs in the IDT/ZnO/128° Y–X LiNbO3 structure are also demonstrated for comparison. The simulation results obtained in this study can be applied to the design of SAW devices using AlN/128° Y–X LiNbO3 structures.

Evidence of the theoretically predicted seismo-magnetic conversion

SUMMARY Seismo-electromagnetic phenomena in porous media arise from seismic wave-induced fluid motion in the pore space, which perturbs the equilibrium of the electric double layer. This paper describes with details the original experimental apparatus built within the ultra-shielded chamber of the Low Noise Underground Laboratory of Rustrel (France). We measured seismo-magnetic conversions in moist sand using two induction magnetometers, and a pneumatic seismic source to generate the seismic wave propagation. We ensured to avoid the magnetometer vibrations, which could induce strong disturbances from induction origin. Interpretation of the data is improved by an analytical description of phase velocities for fast (Pf) and slow (Ps) longitudinal modes, transverse mode (S) as well as the extensional mode due to the cylindrical geometry of the sample. The purpose of this paper is to provide elements to measure correctly coseismic seismomagnetic fields and to specify their amplitude. The seismic arrivals recorded in the sample showing a 1200–1300 m s−1 velocity have been associated to P and extensional waves. The measured seismo-magnetic arrivals show a velocity of about 800 m s−1 close to the calculated phase velocity of S waves. Therefore, we show that the seismo-magnetic field is associated to the transverse part of the propagation, as theoretically predicted by Pride (1994), but never measured up to now. Moreover, the combined experimental and analytical approaches lead us to the conclusion that the measured seismo-magnetic field is probably about 0.035 nT for a 1 m s−2 seismic source acceleration (0.1 g).

An Objective Study of 500-hPa Moving Troughs in the Southern Hemisphere

Abstract The location and movement of 500-hPa troughs using an automated method are studied with data from a 24-yr period with the objective of determining the trough formation and dissipation regions in the Southern Hemisphere. To identify the 500-hPa mobile troughs, an objective method that uses the Eulerian centripetal acceleration (ECA) is developed. On average, 868 troughs per year were identified by the method, with an increase in trend during the period studied. The troughs have an average lifetime of 4.3 days, being longer (shorter) in subtropical (high) latitudes. The average calculated phase velocity was 13.6 m s−1, being higher (lower) in middle (high) latitudes. The troughs are normally found in the 60°–40°S latitudinal band, with maximum occurrence at 50°S. The longitudinal distribution of trough genesis has three maximum regions: over the Drake Strait and the South Atlantic Ocean, over the Indian Ocean around 50°S, and over the southwestern Pacific Ocean between 150°E and 150°W. The trough dissipation regions are less concentrated than the genesis regions and also show three maxima: over the west of the Andes, south of the African continent, and south of Australia. The seasonal variation in the trough dissipation shows that the 30°–40°S band is more active during winter than in summer. The difference between the genesis and dissipation location is that formation occurs more in middle and high latitudes, while dissipation is more common in the 40°–30°S latitude belt.

Using Ambient Seismic Noise to Determine Short-Period Phase Velocities and Shallow Shear Velocities in Young Oceanic Lithosphere

Using 10 broadband ocean bottom seismometers from the 11-month-long deployment of the Gravity Lineations Intraplate Melting Petrologic and Seismologic Expedition (GLIMPSE) passive seismic experiment located in the south central Pacific, we have estimated the seismic impulse responses from ambient seismic noise for 45 station-to-station paths. The raw impulse responses show moveout with station-to-station distance, and there is a trend of decreasing signal-to-noise ratios as the station-to-station distance increases. The decrease in signal-to-noise ratio is expected as a smaller range of azimuths of propagating surface waves will contribute constructively to the cross-correlated signal with increasing distance, although scattering may also play a role in the coherence of seismic noise at periods less than 16 sec. From these station-to-station paths, we determined group velocities for the fundamental mode Rayleigh waves of a 2–16-sec period and the second mode Rayleigh wave of a 3.5–7-sec period. We calculate phase velocities for the fundamental mode and second mode Rayleigh wave over the same period range as the group velocities by applying a time variable filter to the noise correlation function and carefully unwrapping the phase spectrum of the resulting filtered impulse responses. Within this period range, there is a transition from waves at short periods, whose energy is mostly in the water column, to waves sensitive to crustal and upper mantle structure. The phase velocities for the second mode, which have peak sensitivity in the lower crust and shallow mantle, show evidence for azimuthal anisotropy. The average phase velocities of the station-to-station paths in the east–west direction are 2% faster than the north–south paths at the 4–7-sec period, consistent with the fast directions determined from SKS wave splitting measurements of N100°E. By incorporating the short-period fundamental and higher mode phase velocities from ambient seismic noise with longer period (16–100 sec) fundamental mode Rayleigh-wave phase velocities determined from teleseismic events, we inverted for the average crustal and upper mantle shear velocity structure and water column depth and velocity. The predicted phase velocities are extremely sensitive to the water column compressional velocity. We determined the average water column velocity to be 1466±3 m/sec, in contrast to the average of 1500 m/sec estimated from shipboard measurements weighted according to the Rayleigh-wave sensitivity kernel. The difference may be due to the dispersive effects of scattering by bathymetry or by the thin variable thickness sediment layer. The inversion also produces a Vp/Vs ratio of 1.88 for the crust. This method can provide useful information about the shallow seismic structure of the oceanic crust and uppermost mantle and is an important complement to longer period studies.

Optimization Design of a Lamb Wave Device for Density Sensing of Nonviscous Liquid

A Lamb wave device composed of a piezoelectric plate loaded with a nonviscous liquid layer is presented. The relation between the Lamb wave phase velocity and the liquid density can be used for liquid density sensing. In this paper, utilizing the partial wave theory, the concept of effective permittivity is introduced to analyze the Lamb wave's excitation and the phase velocity calculation under a certain liquid density. The interface between the Lamb wave device and the liquid layer is metallized to eliminate the influence of liquid electrical properties when sensing liquid density. Based on the theory model, the phase difference measurement method is adopted to study the device's sensitivity to liquid density. In order to achieve high sensitivity to liquid density with sufficient excitation efficiency of Lamb wave, the optimal parameters of the Lamb wave device including plate thickness and cut orientation are obtained by numerical calculation. The experimental results are found to be in agreement with the theoretical simulations, verifying the validity of the theory model and the practicability of the optimization design.

Effect of soil texture on the propagation and attenuation of acoustic wave at unsaturated conditions

A central issue in the successful application of acoustic wave method to detect subsurface hydrological properties is a better understanding of the influence of soil texture on the propagation and attenuation of acoustic wave as moisture content is varied, which was numerically investigated in the present study. Our earlier studies have demonstrated the existence of three different modes of acoustic wave in an elastic porous medium containing two immiscible, viscous, compressible fluids. Based on the dispersion equation obtained in the Lo-Sposito-Majer (LSM) model, the phase velocity and attenuation coefficient of the P1 and P2 waves which respectively propagate the fastest and second fastest were determined as a function of water saturation for 11 soil texture classes. The slowest wave (P3) was not characterized in this study since it does not travel far, due to very high attenuation. To provide a more general result, the calculated phase velocity and attenuation coefficient for different soil textures were normalized by those computed for sand. The normalization leads the resulting dimensionless parameters to be frequency independent throughout the whole range (up to 500 Hz) with Darcy's law remaining valid for the description of each fluid flow under wave excitation. The normalized phase velocity of the P1 wave was shown to have a substantially constant value at higher water saturations, but in the lower saturation range it first increases to reach a certain maximum value for different soil types and then decreases. The physical parameter controlling this phenomenon is the ratio of two effective non-wetting fluid storativity factors. Numerical results reveal that the normalized attenuation coefficient of the P1 wave is sensitive to soil texture and water saturation. Sand and loamy sand have the highest and second highest attenuation coefficients for the P1 wave, respectively. The magnitude of the normalized phase velocity of the P2 wave is found to be, with very few exceptions at nearly full saturations, linearly associated with the intrinsic permeability, while relating to the normalized attenuation coefficient of the P2 wave in an opposite manner. These results provide a quantitative clue for acoustic wave method to explore the physical properties in the shallow subsurface.

Scattering attenuation, dispersion and reflection ofSHwaves in two-dimensional elastic media with densely distributed cracks

We compute the synthetic seismograms of multiply scattered SH waves in 2-D elastic media with densely distributed parallel cracks. We assume randomly distributed cracks in a rectangular-bounded region, which simulate a cracked zone. The crack surfaces are assumed to be stress-free. When the incident wavelength is longer than the crack size, the delay in the arrival of the primary wave is observed at stations beyond the cracked zone and the amplitude of the primary wave is amplified in the cracked zone in the synthetic seismograms. This is because the cracked zone behaves as a low velocity and soft material to the incident long-wavelength wave due to the crack distribution. When the half-wavelength of the incident wave is shorter than the crack length, the scattered waves are clearly observed in the synthetic seismograms and the amplitude of the primary wave is largely attenuated beyond the cracked zone. The calculated attenuation coefficient Q−1 of the primary wave is directly proportional to the crack density in the range of νa2≤ 0.1, where ν and a are the number density and half the length of cracks, respectively. This is consistent with that obtained by a stochastic analysis based on Foldy's approximation. A periodic distribution of cracks in a zone is considered as an utterly different model in order to investigate the effect of spatial distributions on the attenuation and dispersion of seismic waves. When cracks are distributed densely, the values of Q−1 for the periodic crack distribution appear to differ from those for the random distribution of cracks in the low wavenumber range. This suggests that the effect of multiple interactions among densely distributed cracks depends on not only the density but also the spatial distribution of cracks at low wavenumbers. The calculated phase velocity of the primary wave is consistent with that from the stochastic analysis in the range of νa2≤ 0.1 and does not depend on the spatial distribution of cracks. This suggests that the multiple crack interactions have a smaller effect to the phase velocity. Therefore, the crack density can be estimated from the values of the phase velocity for the cases of densely distributed cracks even if the effect of the multiple crack interactions is not considered. We can clearly observe the reflected waves in the synthetic seismograms. The elastic constant of a single anisotropic layer equivalent to the cracked zone is derived from the crack density at the long-wavelength limit. The reflection coefficients calculated from the synthetic seismograms are consistent with those of the anisotropic layer calculated from its elastic constants and thickness in low wavenumber range. This means that a fracture zone distributed parallel cracks is considered as an anisotropic layer for long incident wavelengths. Therefore, the elastic constants, crack density and the thickness of the fracture zone can be estimated from the frequency dependence of the reflection coefficients for long incident wavelengths. On the contrary the wavenumber dependence of the reflection coefficients cannot be explained theoretically in the high wavenumber range.

Propagation of shear elastic waves in composites with a random set of spherical inclusions (effective field approach)

The work is dedicated to the problem of plane monochromatic shear wave propagation through elastic matrix composite materials with a homogeneous random set of spherical inclusions. The effective field method (EFM) and quasi-crystalline approximation are used for the calculation of phase velocity and attenuation factor of the mean wave field propagating through the composite. The version of the method developed in the work allows us to obtain the dispersion equation for the wave vector of the mean wave field that serves for all frequencies of the incident field, properties and volume concentrations of the inclusions. The long- and short-wave asymptotic solutions of the dispersion equation are found in closed analytical forms. Numerical solutions of this equation are constructed in a wide region of frequencies that covers the long-, middle- and short-wave regions of the propagating waves. The phase velocities and attenuation factors of the mean wave field in the composites are analyzed for various elastic properties, density and volume concentrations of the inclusions. Comparisons of the predictions of the method with some numerical computation of the effective parameters of matrix composites are presented; possible errors in predictions of the velocities and attenuation factors of the mean wave field in the composites are indicated and discussed.

Study on electromechanical coupling coefficients of surface acoustic waves in layered systems

The aim of this paper is to utilize the effective permittivity approach to calculate the electromechanical coupling coefficients of surface acoustic waves in layered systems. The effective permittivity offers a direct relation to the voltage‐induced charge density at the interface, and can be used to calculate the phase velocity dispersion and the electromechanical coupling coefficients of a layered piezoelectric medium exactly. In this paper, the formulation based on the matrix method for calculating the effective permittivity of a layered piezoelectric medium with three types of electrode arrangements is given first. Then, based on this formulation, we calculated phase velocity dispersions of the generalized SAW mode and the HVPSAW mode in layered systems of ZnO/Diamond and LiNbO3/Diamond. In addition, the electromechanical coupling coefficients of surface acoustic waves in the aforementioned layered systems with three different electrical boundary conditions were analyzed and discussed.

A first detailed look at the Greenland lithosphere and upper mantle, using Rayleigh wave tomography

We present the results of a surface wave study carried out across Greenland as part of the ‘GLATIS’ (Greenland Lithosphere Analysed Teleseismically on the Ice Sheet) project. Rayleigh wave phase velocity dispersion curves were estimated for 45 two‐station paths across Greenland, using data from large teleseismic earthquakes. The individual dispersion curves show characteristics broadly consistent with those of continental shields worldwide, but with significant differences across the Greenland landmass. Reliable phase velocity measurements were made over a period range of 25–160 s, providing constraint on mantle structure to a depth of ∼300 km. An isotropic tomographic inversion was used to combine the phase velocity information from the dispersion curves, in order to calculate phase velocity maps for Greenland at several different periods. The greatest lateral variation in phase velocity is observed at intermediate periods (∼50–80 s), where a high‐velocity anomaly is resolved beneath central‐southwestern Greenland, and a low‐velocity anomaly is resolved beneath southeastern Greenland. The results of the phase velocity inversion were used to construct localized dispersion curves for node points along two parallel north–south profiles in southern Greenland. These curves were inverted to obtain models of shear wave velocity structure as a function of depth, again with the assumption of isotropic structure. A similar inversion was carried out for two two‐station dispersion curves in northern Greenland, where the resolution of the phase velocity maps is relatively low. The models show a high‐velocity ‘lid’ structure overlying a zone of lower velocity, beneath which the velocity gradually increases with depth. The ‘lid’ structure is interpreted as the continental lithosphere. Within the lithosphere, the shear wave velocity is ∼4–12 per cent above global reference models, with the highest velocities beneath central‐southwestern Greenland. However, the assumption of isotropic structure means that the maximum velocity perturbation may be overestimated by a few per cent. The lithospheric thickness varies from ∼100 km close to the southeast coast of Greenland to ∼180 km beneath central‐southern Greenland.

An accurate and logically correct way to verify the numerical dispersion relations of FDTD and ADI‐FDTD methods

AbstractIt is well known that the numerical dispersion relations of all kinds of finite‐difference time‐domain (FDTD) methods, including the conventional FDTD and alternating‐direction implicit (ADI) FDTD methods, are derived from the assumption of plane wave. In the past, however, disregarding the above fact, a point source that creates a cylindrical wave was used to numerically validate the numerical dispersion relations of the FDTD‐related methods. Strictly speaking, using the point source for the validation is illogical and also incorrect. This is due to the fact that the numerical dispersion relation of the cylindrical wave differs from that of the plane wave. It is demonstrated in this paper that the phase velocity calculated from the point source strongly depends on the positions of the observation points, and that the calculated phase velocity never exactly matches the theoretical prediction, even though the phase velocity of the cylindrical wave gets closer to that of the plane wave when the observation points are located far away from the point source (that is, when the cylindrical wave can be approximately treated as the plane wave). On the contrary, numerical tests indicate that, no matter where the observation points are located, the phase velocity obtained using plane‐wave excitation agrees extremely well with the theoretical values. We can therefore conclude that to more accurately and logically verify the numerical dispersion relations of any kinds of FDTD‐related algorithms, plane‐wave excitation has to be employed. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 427–431, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11400

Rapid Inspection of Composite Skin-Honeycomb Core Structures with Ultrasonic Guided Waves

Guided wave inspection of composite skin-honeycomb core structures is an efficient and sensitive alternative to other common inspection methods. This paper shows that sweeping experimentally through the dispersion curves is an effective way to experimentally locate guided wave modes sensitive to skin-core delamination. Composite skin-Nomex honeycomb core specimens were developed with simulated delaminated areas. The delaminated areas were detected with guided waves and confirmed with conventional ultrasonic testing methods. Calculated phase velocity dispersion curves are given to define the practical phase velocity and frequency ranges. Example wave structures in this range are given to illustrate the change in sensitivity as frequency is swept for a given mode.

A Comparison of ESAC and FK Methods of Estimating Phase Velocity Using Arbitrarily Shaped Microtremor Arrays

Focusing on a comparison of the accuracy of the extended spatial autocorrelation (ESAC) method and the frequency–wavenumber spectrum (FK) method, we carried out short-period microtremor measurements of arbitrarily shaped array configurations at a site with a well-known velocity structure at shallow depths. Using both techniques, we measured phase velocities for frequencies between 2.5 Hz and 13.5 Hz and compared the results with the theoretical Rayleigh-wave-dispersion characteristics of both the fundamental and the first higher modes calculated from the PS logging data. Next, we tried to estimate the S -wave velocity structure to a depth of 43 m by fitting the theoretically calculated phase velocities to the experimental data, taking into consideration both the higher mode contributions as well as the fundamental mode. The theoretical dispersion characteristics were successfully fitted to the results analyzed by the ESAC method but not by the FK method. The estimated S -wave velocity structure from the ESAC method results was in good agreement with the PS logging profile. We also confirmed, using the model, that the S -wave transfer function due to vertical incidence coincided with that from the PS logging data. As a result of the study, we conclude that the ESAC method gives more accurate results than the FK method in determining the Rayleigh-wave phase velocity from records of short-period microtremors using arbitrarily shaped array configurations. The ESAC method can also provide a better estimate of the S -wave velocity structure and site amplification than the FK method.

Group velocity along general direction in a general anisotropic medium

An attempt has been made to study the three-dimensional wave propagation in a general anisotropic medium. A procedure is presented to solve the inverse problem of finding the group velocity in a given direction of ray travel, without using numerical differentiation. The phase direction, derived from the given ray direction, is used to calculate phase velocity. Then, such a phase velocity and phase direction are used to calculate group velocity. Analytical expressions are derived for the directional derivatives of phase velocity which are used to calculate group velocity. Algebraic expressions are derived in a computation convenient manner. Newton's method (without numerical differentiation) for solving a system of two non-linear simultaneous equations is the only numerical method used. The ray directions along which a wave (or energy) can travel with more than one group velocity may be difficult to treat. Variations of phase velocity and group velocity with ray direction in three dimensions are plotted for a hypothetical model of general anisotropic medium.