円形アレーを用いた微動探査における位相速度の直接同定法

We studied the effect of body waves on short period vertical ground motion based on both a numerical simulation and field array observations. The vertical components generated by the vertical point force within a close distance are targeted throughout the study. Firstly, we carried out a numerical simulation and confirmed that the effect of body waves on the full-wave field was generally limited to a close distance from the source with frequency dependence. In a frequency range lower than 5 Hz and at around the 1st predominant frequency (5.8 Hz), the effect appears beyond a far distance; in a range of about 6 to 10 Hz, it still remains up to 10 m or more; in a range higher than 10 Hz, it almost vanishes. Secondly, we analyzed the observed array data excited by a 30-kg sandbag falling 10, 20, and 40 m from the array center. The array configuration was a circle, 1 m in radius, consisting of seven sensors placed in the center and around the circumference at equal intervals. To obtain a better understanding of the results of the array analyses, simulated array data, based on the Green's functions for the full-wave field and for the Rayleigh wave, was produced and analyzed in the same manner as the observed field array data. The phase velocity results were calculated with three different methods, namely, the spatial autocorrelation (SPAC) method, a method based on the spatial 1st Fourier coeffcients in the azimuthal direction with respect to the cross spectral amplitude between the center and one of circumferential stations, and a technique which combined these two methods. In the case of an offset of 10 m (Case 1), the detected phase velocities from the observed data and the simulated data on the full-wave field were much lower than the theoretical phase velocity or that from the Rayleigh wave array data in a frequency range of 5 to 10 Hz, suggesting the strong effect of body waves. In the case of an offset of 20 m (Case 2), the estimated phase velocities from the observed data and the two other simulated array data coincided with the theoretical phase velocity, suggesting the weak effect of body waves. We conclude that in order to properly estimate an underground structure from the array measurement data at a close distance from the source, the analytical technique based on the full-wave Green's function should be applied. Taking a look at the choice of analytical methods for the array data, it can be pointed out that for our interest the two methods which use the spatial 1st Fourier coeffcients produce higher accuracy compared with the SPAC method in a relatively low frequency range. In addition, the phase velocity results were estimated from the CCA (centerless circular array) method by Cho et al. (2004). The CCA method provided similar accuracy compared with the above two methods based on the spatial 1st Fourier coeffcients. The CCA method seems to be applicable to highly directional wave fields located nearby the source.

Estimating near surface shear wave velocity using the SPAC method at a site exhibiting low to high impedance contrast

Dispersion curves of phase velocities of Rayleigh waves are determined by the method of frequency-time analysis in a range of periods of 10–200 s from data of 43 interstation traces in Central Asia. Because the joint use of phase and group velocities significantly decreases the uncertainty in the determination of S wave velocity structures, the same traces were used for calculating group velocities from tomographic reconstructions obtained in [Yanovskaya and Kozhevnikov, 2003, 2006] and determining average velocity structures along these traces. The velocity structures were calculated by the Monte Carlo and linear inversion methods, which gave consistent results. Using velocity values obtained at fixed depths by the 2-D tomography method, lateral variations in velocities at these depths were estimated, which allowed us to construct smoothed vertical velocity structures at some points in the region. The resulting structures were used as initial approximations for constructing local velocity structures solely from previously obtained local dispersion curves of group velocities in the area (32°–56°N, 80°–120°E). Based on these structures, we mapped the lateral distribution of velocity variations at upper mantle depths of 75–400 km and along three vertical profiles. The inferred velocity variations are in good agreement with data on the tectonics of the region.

Lateral heterogeneities and microtremors: Limitations of HVSR and SPAC based studies for site response

An analytical framework is developed to explore the dispersion behavior of Rayleigh waves in a functionally graded piezoelectric (FGP) plate interfaced with a dry sandy foundation, accounting for nonlocal elasticity effects and interfacial imperfections. The governing equations are formulated using nonlocal elasticity theory and solved analytically via harmonic wave analysis approach. Interface effects are modeled using normal and tangential stiffness parameters, allowing for partial bonding at the interface. Numerical simulations are performed considering both electrically open and electrically short surface conditions. The influence of key parameters including material gradation, nonlocal scale effects, interfacial stiffness, and substrate properties on the phase velocity of Rayleigh waves is examined in detail. Parametric analysis reveals that variations in the plate thickness and nonlocal parameter considerably affect the phase velocity and dispersion behavior of Rayleigh waves. The 2D and 3D plots of the displacement and electric potential fields are generated using Mathematica , illustrating the spatial variation and confinement of wave energy within the FGP layer. The results not only validate classical trends under limiting conditions but also reveal important coupled effects that are absent in traditional local models. These findings contribute to a deeper understanding of wave mechanics in smart-layered structures and offer valuable guidance for the design and optimization of advanced piezoelectric sensors, actuators, and SAW devices.

Phase velocities of Rayleigh waves in Finland have been measured by the Fourier analysis method, using records of three stations which are on almost the same great circle as several Greek earthquakes. Phase velocities of Love waves of two of these earthquakes have also been determined. The wave groups were divided by numerical filtering into two frequency channels, which were analyzed separately. The resulting phase velocity curves were then smoothed in order to remove rapid oscillations, and mean values were calculated. The phase velocities were found to be slightly lower than those found for the Canadian Shield (Brune and Dorman, 1963), the maximum deviation for Rayleigh waves being about 0.07 km/sec at a period of 28 seconds and for Love waves about 0.15 km/sec at a period of 32 seconds. At periods shorter than 40 seconds the phase velocities of Rayleigh waves were found to be equal in Northern and Southern Finland within the accuracy of measurement.

Phase velocities of Love waves from five major earthquakes are measured over six great circle paths in the period range of 50 to 400 seconds. For two of the great circle paths the phase velocities of Rayleigh waves are also obtained. The digitized seismograph traces are Fourier analyzed, and the phase spectra are used in determining the phase velocities. Where the great circle paths are close, the phase velocities over these paths are found to be in very good agreement with each other indicating that the measured velocities are accurate and reliable. Phase velocities of Love waves over paths that lie far from each other are different, and this difference is consistent and much greater than the experimental error. From this it is concluded that there are lateral variations in the structure of the earth's mantle. One interpretation of this variation is that the mantle under the continents is different from that under the oceans, since the path with the highest phase velocities is almost completely oceanic. This interpretation, however, is not unique and variations under the oceans and continents are also possible. Group velocities are computed from the phase velocities and are also directly measured from the seismograms. The group-velocity curve of Love waves has a plateau between periods of 100 and 300 seconds with a shallow minimum at about 290 seconds. The sources of error in both Fourier analysis and direct time domain methods of phase velocity measurement are discussed.

Purpose – The purpose of this paper is to illustrate the propagation of Rayleigh waves in an anisotropic inhomogeneous layer placed over an isotropic gravitational viscoelastic half space of third order. Design/methodology/approach – It is considered that the mass density and the elastic coefficients of the layer are space dependent. Dispersion properties of waves are derived with the simple mathematical techniques. Graphs are plotted between phase velocity ‘k’ and wave number ‘c’ for different values of inhomogeneity parameters for a particular model and the effects of inhomogeneity and gravity are studied. Findings – The wave analysis indicates that the phase velocity of Rayleigh waves is affected quite remarkably by the presence of inhomogeneity, gravity and strain rates of strain parameters in the half space. The effects of inhomogeneity and depth on the phase velocity are also shown in corresponding figures. Originality/value – The results presented in this study may be attractive and useful for mathematicians, seismologists and geologists.

Summary In recent years, distributed acoustic sensing (DAS) has enabled the observation of strain over tens of kilometres at metre-level intervals by using optical fibre as a sensor. This study presents an analytical solution for the cross-spectrum of ambient noise with DAS data acquired from arbitrarily shaped and/or multiple fibre-optic cables, with the aim of estimating subsurface S-wave velocity structures using the spatial autocorrelation (SPAC) method. Our formulation accounts for both isotropic and anisotropic wave incidence. The analytical cross-spectrum depends on the angles between the horizontal direction connecting the two measurement points and the axial strain directions at the two points. This study demonstrates that both Rayleigh and Love waves contribute to the cross-spectrum, and that their contributions vary in a complex manner depending on the cable geometry, seismic velocity structure, interstation distance between observation points, and source amplitudes. By using this analytical solution, an integrated analysis combining the SPAC method and the ambient noise tomography method is applicable to DAS data acquired from arbitrarily shaped and/or multiple cables. In addition, the analytical expression considering anisotropic wave incidence will be useful for correcting travel-time anomalies caused by source heterogeneity. The application of our formulation to DAS data from winding or multiple cables will facilitate high-resolution and precise imaging of the three-dimensional structure.

S-wave velocity structure can reflect the differences of physical properties of Earth media, and is the important basis of identifying the structural characteristics of the low velocity zone. This article attempts to use spatial autocorrelation method (SPAC method) to extract Rayleigh-wave phase velocity dispersion curves from the vertical component of the seismic stations' microtremors, and through the inversion of dispersion curve, acquires the S-wave velocity structure. Taking 8 broadband seismic stations' actual measurement from China National Seismic Network for example, we acquired the one-dimensional S-wave velocity structure of 30 km depth range using this method in the Capital area near Beijing. The inversion result shows that the crystalline basement of this region is about 2 km deep; there is S wave low-velocity layer at 5 similar to 8 km and 12 similar to 16 km depth; the depth 8 km and 20 km stands for the speed interface of S-wave that has a big speed difference. This result has good consistency with seismology and artificial seismic exploration results, which illustrates that estimating the crustal S-wave velocity structure using spatial autocorrelation method is feasible and effective.

A feasibility study was made concerning the use of the ellipticity of the Rayleigh wave particle motion for determining earth structures. Variational parameters were computed empirically for both the ellipticity and phase velocity of Rayleigh waves in the period range T = 10-50 seconds. It was found that, in general, the ellipticity and phase velocity are about equally sensitive to structural perturbations, but that near-surface low-velocity sedimentary layers influence the ellipticity much more strongly than they do the phase velocity. Anelasticity has a minor effect on the ellipticity, whereas the presence of interfering waves can have a significant influence. A test of the independence between ellipticity and phase velocity indicated that in our period range ellipticity does contribute independent information, and thus provides an additional constraint toward uniqueness. Using data from LASA, both ellipticity and Rayleigh- and Love-wave phase velocities were measured and the results interpreted in terms of a crustal structure. The ellipticity data proved useful when combined with the phase velocity and some structures that fit the phase velocity data could be rejected on the basis of ellipticity.

According to a Sino-U. S. joint project, eleven broadband digital PASSCAL seismometers had been deployed inside the Tibetan Plateau, of which 7 stations were on the profile from Lhasa to Golmud and other 4 stations situated at Maxin, Yushu, Xigatze and Linzhi. Dispersions and phase velocities of the Rayleigh surface waves (10s–120s) were obtained on five paths distributed in the different blocks of Tibetan Plateau. Inversions of the S-wave velocity structures in Songpan-Ganzi block, Qiang-Tang block, Lhasa block and the faulted rift zone were obtained from the dispersion data. The results show that significant lateral variation of the S-wave velocity structures among the different blocks exists. The path from Wenquan to Xigatze (abbreviated as Wndo-Xiga) passes through the rift-zone of Yadong-Anduo. The phase velocities of Rayleigh waves from 10s to 100s on this path are significantly higher than that on other paths. The calculated mean crustal velocity on this path is 3.8 km/s, much greater than that on other paths, where mean crustal velocities of 3.4–3.5 km/s are usually observed. Low velocity zones with different thicknesses and velocities are observed in the middle-lower crust for different paths. Songpan-Ganzi block, located in the northern part of Tibetan Plateau is characterized by a thinner crust of 65 km thick and a prominent low velocity zone in the upper mantle. The low velocity zone with a velocity of 4.2 km/s is located at a depth form 115 km to 175 km. While in other blocks, no low velocity zone in the upper mantle is observed. The value of Sn in Songpan-Ganzi is calculated to be 4.5 km/s, while those in Qiang-Tang and Lhasa blocks are about 4.6 km/s.

Phase velocities of Rayleigh waves across the East Pacific Rise

As a World Cultural Heritage Site, the Yungang Grottoes face the risk of geological disasters caused by underground goafs. To accurately detect the surrounding void zones of the grottoes and analyze their distribution and structural characteristics, this study uniquely applied the passive-source surface wave spatial autocorrelation (SPAC) method. Four high-density linear arrays were deployed in the Yungang Grottoes area to collect microtremor signals. By extracting the Rayleigh wave dispersion curve and combining it with a non-linear least-squares inversion technique incorporating a damping regularization term, a shallow shear wave velocity profile was established, successfully identifying potential low-velocity anomalies. During the inversion process, the optimized damping factor effectively suppressed the oscillation effects under complex geological conditions, significantly improving the stability and accuracy of the imaging. The results showed that 15 typical low-velocity anomalies with wave velocities below 1,200 m/s were identified, and these anomalies were highly consistent with the locations of goaf areas in historical mining data. The imaging results revealed that the goaf exhibits multiple layers and a multi-center distribution, with the L1 and L4 survey lines being the main areas of goaf activity. With a horizontal resolution of 10–20 m and a maximum detection depth of 320 m, the method presented differences in the integrity and fragmentation of underground rock masses. The SPAC method demonstrated advantages such as high resolution, non-destructive testing capability, and imaging stability in the detection of abandoned mine areas within cultural heritage sites. By optimizing the inversion regularization parameters, this study significantly improves imaging accuracy in complex geological environments, providing effective technical support for the stability assessment of the Yungang Grottoes and for the prevention and control of geological disaster risks at cultural heritage sites.

Effect of magneto-elasticity, hydrostatic stress and gravity on Rayleigh waves in a hydrostatic stressed magneto-elastic crystalline medium over a gravitating half-space with sliding contact