Array measurements of higher mode Rayleigh wave dispersion: an approach utilizing source parameters

A perturbation technique described in a previous paper has been applied to the tape helix model of a traveling wave tube (TWT). The perturbation technique employs two solutions; the unperturbed solution is that for a helix in a conducting sleeve, suspended in vacuum, and the perturbed solution is with the above configuration, but with the introduction of dielectric support rods. Usually the perturbed solution is not known and is approximated by the homogeneous dielectric solution. Then in three-dimensional (3-D) fashion, the perturbation technique weights the dielectric with an energy-like term that is only evaluated within the discrete dielectric. Therefore, the model accounts for the distribution of dielectric material. Extending the perturbation to the tape helix, the current density is accurately portrayed in space. In all cases, the perturbed tape helix solution has a lower least square error in normalized phase velocity than the unperturbed tape helix solution. It had a lower error than the perturbed sheath helix solution of the previous paper, which in turn had a lower least square error than the Naval Research Laboratory's (NRL's) Small Signal Gain (SSG) Program. An initial discussion of the tape helix is presented and the calculated dispersion relation given. This solution incorporates a conducting backwall or shield with a homogeneous dielectric between it and the helix, using n=-6 to n=+6 space harmonics. Then the perturbation theory is applied, first for uniform support rods, and then to include notched rods. The resultant simulations produce phase velocity and coupling impedance data. Deviation from theory to experiment is reported by comparing the square root of the average sum of the squares difference, divided by an average phase velocity, between calculated phase velocities and a second order least squares fit of the measured data. Experimental data on the dispersion properties of four helices came from Northrop. The average error in phase velocity for four cases was 0.95%. For uniform rods the perturbation theory lowers the phase velocity, but doesn't significantly alter the dispersion. However, for notched rods, the perturbation theory raises the phase velocity and flattens the dispersion, in agreement with experiment. Computations can be performed in 1/2 min for a ten-frequency-step problem on a Pentium II processor.

A note on linear dispersion and shoaling properties in extended Boussinesq equations

Physical 1D-average reference models of the Earth offer valuable summaries of the radial variations in rock properties and a reference for geophysical studies. PREM, in particular, has been used widely for >40 years and comprises Vp, Vs, density, radial anisotropy and attenuation profiles, while also fitting the Earth’s mass and moment of inertia. Many of PREM’s features have proven remarkably accurate, despite the limited amount of data used to construct it, but some features are inconsistent with now available data. Also, the upper mantle structure differs so much between Earth’s different tectonic environments that a global average is not quite representative of any of them.  The recent growth in seismic station coverage yields very dense data sampling, globally and over different tectonic environments. Here, we use a large global dataset to construct ten 1D, multi-parameter, reference models of the upper mantle, for the globe and for 9 basic tectonic types: cratons; stable platforms; Phanerozoic continents with normal (<46.5 km) and thick (>46.5 km) crust; rifts and continental hotspots; old oceans; intermediate oceans; young oceans; backarcs.The dataset comprises Love and Rayleigh-wave phase velocities, measured using waveform inversion and all available data since 1990s; surface heat flow measurements; topography/bathymetry. With tomography-based tectonic regionalization, we identify areas within each tectonic environment and compute average dispersion curves in the 20-30 to 310 s period range, which constrain shear velocity and anisotropy in the entire upper mantle.We then use computational-petrology-based inversion to calculate 1D physical models for the globe and the 9 basic tectonic types. Our non-linear gradient search converges to true best-fitting models. The main unknowns in the inversion are the depth of the lithosphere-asthenosphere boundary (LAB); the geotherm from the LAB down to 400 km depth; radial anisotropy (0-800 km). The steady-state geotherm in the lithosphere is computed from the LAB depth and the radiogenic heat production and thermal conductivity profiles by solving the conductive heat transfer equation. Rock composition and the geotherm determine the density, seismic velocities and attenuation down to 400 km. Seismic velocities in the crust, transition zone (410-660 km) and shallow lower mantle can vary to fit the data. Density below 410 km and all parameters in the core and most of the lower mantle are from PREM. Like PREM, our reference models honour the Earth's mass and moment of inertia.Small phase-velocity errors and relative data-synthetic misfits (<~0.1%) are necessary to resolve radial trade-offs in the upper-mantle structure. We achieved this by obtaining very accurate dispersion curves and by meticulously tuning the inversion, its parameterisation and regularisation.The best-fitting models have slightly depleted lithospheric mantle and fertile asthenosphere for most tectonic types. In Archean and Proterozoic continents, the mantle lithosphere is more depleted. No other compositional heterogeneities are required to fit the data. Isotropic-average seismic velocities decrease monotonically from the Moho to the LAB. The geotherms follow the mantle adiabatic temperature gradient in the asthenosphere. Our results provide useful, accurate new reference models for global and regional seismic imaging and other geophysical studies. 

In this paper, based on quadratic interpolation in time domain, an improved first-order absorbing boundary condition (ABC) for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is developed. The ABC is derived in two half-time steps iteration form which is very suitable for the ADI-FDTD method. The reflection of the ABC caused by both the truncated error and the phase velocity error is analyzed in detail. The phase velocity estimation and nonuniform cell are adopted to improve the performance of the ABC. Numerical results show that efficient improvement on the absorbing performance is achieved based on these methods.

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.

Use of peaks and troughs in the horizontal-to-vertical spectral ratio of ambient noise for Rayleigh-wave dispersion curve picking

A two‐dimensional (2D) locally one‐dimensional finite‐difference time‐domain (LOD‐FDTD) method with low numerical dispersion is introduced by virtue of a parameter‐optimized compact fourth‐order scheme. The numerical dispersion error and anisotropy of numerical phase velocity of this new method are proven to be significantly reduced compared with the original second‐order method. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 2783–2787, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23842

In this paper, an accurate numerical dispersion relationship is developed for 3-D ADI FDTD with artificial anisotropy. The numerical dispersion relation with accurate mathematical model helps to calculate the anisotropic parameters, which are used to control the error of the numerical phase velocity. Numerical example shows that the proposed method helps to reduce the numerical dispersion significantly. The effects of anisotropic parameters on the numerical dispersion with different mesh resolution, courant number and cell dimension ratio, are shown for three-dimensional anisotropic ADI-FDTD. A comparison has been made for anisotropic ADI FTDD and conventional ADI FDTD in terms of accuracy and CPU time for the calculation of resonant frequencies of rectangular cavity.

The guided wave has been widely employed to characterize thin plates and layered media. The dispersion curves of phase and group velocities are essential for the quantitative application of guided waves. The technique using leaky Lamb wave (LLW) is one of the excellent methods to obtain dispersion curves. In the present work, a fully automated system for the measurement of backward radiation of LLW has been developed. The specimen moves in two dimensional planes as well as in angular rotation. The signals of backward radiation of LLW were measured from an elastic plate in which specific modes of Lamb wave were strongly generated. Phase velocity of the corresponding modes was determined from the incident angle. The generated Lamb waves propagated forward and backward with the leakage of energy into water. Backward radiated LLW was detected by the same transducer and its frequency components were analyzed to extract the related information to the dispersion curves. The dispersion curves of phase velocity were measured by varying the incident angle. Moving the specimen in the linear direction of LLW propagation, group velocity was determined by measuring the transit time shift in the rf waveform.

The nonstandard finite-difference time-domain (NS-FDTD) method, using a rectangular parallelepipeds structured grid, has been proposed to overcome the dispersion and anisotropic errors of the FDTD method. However, the numerical dispersion and the stability condition have not been examined. Furthermore, the method has been defined only in the isotropic grids. This paper investigates the numerical dispersion and the stability condition of the three-dimensional NS-FDTD method for isotropic and nonisotropic grids. The method is compared with the FDTD method. As a result, this method demonstrates highly accurate characteristics and high Courant stability condition.

A relatively simple method is discussed for estimating the phase velocity and direction of an unknown plane-wave signal, propagating across two-dimensional array in a dispersive waveguide. The finite Fourier transform is applied to the output signal of each sensor, and the phases of the smoothed frequency components calculated. The phases of the components are linearly regressed on sensor positions to produce estimates of the wave slowness components. Neither the phase nor group velocity dispersion curves need to be known, except for upper and lower estimates of the range. By smoothing over frequency and transforming, estimates are obtained of the direction and phase velocity of the signal. The precision of estimates depends on the signal-to-noise ratio, the square root of the number of sensors, and the number of wavelengths that fall in the array. Each frequency is treated separately, so that the direction and phase velocity are obtained as a function of frequency. Since the individual frequency components of the signal are separated, the signals observed on a large array can be combined, even though the dispersion is appreciable over the array. For good signal-to-noise power ratios, the dispersion curves are a direct result from the analysis.

Approximately 2 yr (2010–2011) of continuous seismic records from a subset of the Antarctic component of the Polar Earth Observing Network (POLENET-ANET) seismic network deployed in West Antarctica are used to compute the nine components of the correlation tensor between each pair of stations in the network. Rayleigh wave velocity information from the vertical and radial components was extracted in the form of group and phase velocity dispersion curves, whereas the transverse component provided complimentary Love wave velocity information. The multicomponent Rayleigh wave measurements (ZZ, RR, ZR, and RZ) were averaged and used to infer the measurement uncertainties. The Rayleigh and Love wave group and phase velocities were then regionalized in space using a 2D deterministic tomography. A transect that spans the West Antarctic rift system was extracted from the tomography at individual periods between 7 and 60 s for the four types of surface wave velocities (i.e., Rayleigh and Love phase and group velocities). A transdimensional Bayesian joint inversion algorithm was used to invert these four datasets for a 1D model of isotropic shear-wave velocity versus depth at each point along the transect. In this way, surface wave dispersion curves from multicomponent noise correlations were used to build a 2D isotropic shear-wave velocity model down to ∼55 km depth. In this model, the top of the large low-velocity zone beneath Marie Byrd Land was imaged (up to a 5% decrease in velocity at ∼50 km depth), which provides further evidence for a mantle hot spot beneath the crust that supports the high topography in this region. We also observed a large velocity contrast in the lower crust beneath Marie Byrd Land at a depth where previous long-period seismicity has been observed. This strong contrast occurs more shallow than in previous crustal models, which compared to our model identify a deeper Moho (∼5–10 km deeper) beneath Marie Byrd Land. This new model has implications for interpreting earthquake locations in this region and perhaps necessitates that we revisit past hypocenter estimation studies using updated velocity models for the region.

The local measurement of dispersion curves of intermediate-period surface waves is particularly difficult because of the long wavelengths involved. We suggest an improved procedure for measuring dispersion curves using small-aperture broad-band arrays. The method is based on the hypotheses of plane incoming waves and that averaging over a set of events with a good backazimuth distribution will suppress the effects of diffraction outside the array. None of the elements of the processing are new in themselves, but each step is optimized so we can obtain a reliable dispersion curve with a well-defined uncertainty. The method is based on the inversion for the slowness vector at each event and frequency using time delays between pairs of stations, where the time delays Δt are obtained by frequency-domain Wiener filtering. The interstation distance projected on to the slowness vector (D) is then calculated. The final dispersion curve is found by, at each frequency, calculating the inverse of the slope of the best-fitting line of all (D, Δt) points. To test the algorithm, it is applied to synthetic seismograms of fundamental mode Rayleigh waves in different configurations: (1) the sum of several incident waves; (2) an array located next to or above a crustal thickening; and (3) added white noise, using regular and irregular backazimuth distributions. In each case, a circular array of 23 km diameter and composed of six stations is used. The algorithm is stable over a large range of wavelengths (between half and a tenth of the array size), depending on the configuration. The situations of several, simultaneously incoming waves or neighbouring heterogeneities are well handled and the inferred dispersion curve corresponds to that of the underlying medium. Above a strong lateral heterogeneity, the inferred dispersion curve corresponds to that of the underlying medium up to wavelengths of eight times the array size in the configuration considered, but further work is needed to better understand the limits under which the obtained dispersion curve is not biased. In the case of 5 per cent spectral amplitude white noise, the dispersion curve is also stable for wavelengths up to approximately eight times the array size, but this limit depends of course strongly on the noise level. The method is finally applied to data from two arrays in the French Alps located 50 km apart. It is possible to measure the dispersion curves up to wavelengths approximately ten times bigger than the array diameter. The difference in the dispersion curves is compatible with a crustal and lithospheric thickening under the Alps. However, the observed errors are large, which result in severe limits on the interpretation in terms of lithospheric structure. Longer recording periods may help to reduce the errors. Otherwise the algorithm is likely to be of use mainly in areas where the lateral variations outside the array are smaller than those of the French Alps.

We propose a programmed technique to analyze the dispersion data set of Love surface waves for the determination of two-dimensional (2-D) velocity structure (horizontally polarized shear-wave – ) of the subsurface. We assume synthetic data emulating the common-shot gather for which the weighted preconditioned linear radon transform (WPLRT), single-station (SS) method and two-station (TS) method are used in coordination. In the first step, the WPLRT provides a phase velocity dispersion curve, which is inverted to determine the one-dimensional (1-D) velocity structure representative of the average structure underneath the common-shot gather. Then the SS method provides tens of group velocity dispersion curves proportional to the number of geophones where each curve is inverted to establish the 1-D velocity structure descriptive of the average structure between the source and receiver. In the third step, the TS method yields hundreds of dispersion curves (both phase and group velocity) proportional to the number of inter-stations. Each set of phase and group velocity curve is jointly inverted to determine the 1-D velocity structure expressive of the average structure between two receivers. Each step in the proposed technique involves several phase and group velocity dispersion curves, which are difficult to hand pick. Instead, a programmed scheme, which saves us a lot of processing time, is developed to select the related dispersion curves. The area covered by the common-shot gather is discretized into a set of grid points. The path dependent dispersion curves obtained by the SS and TS methods are converted into individual dispersion curves at these grid points by solving properly defined linear systems through the travel times. The 1-D velocity-depth profiles obtained by the inversion of dispersion curves at the grid points are combined to build the 2-D cross section beneath the area under consideration.

Ultrasound shear wave elastography (SWE) is a technique used to measure mechanical properties to evaluate healthy and pathological soft tissues. SWE typically employs an acoustic radiation force (ARF) to generate laterally propagating shear waves that are tracked in the spatiotemporal domains, and algorithms are used to estimate the wave velocity. The tissue viscoelasticity is often examined through analyzing the shear wave phase velocity dispersion curves, which is the variation of phase velocity with frequency or wavelength. A number of available methods to estimate dispersion exist, which can differ in resolution and variance. Moreover, most of these techniques reconstruct dispersion curves for a limited frequency band. In this work, we propose a novel method used for dispersion curve calculation. Our unique approach uses a generalized Stockwell transformation combined with a slant frequency-wavenumber analysis (GST-SFK). We tested the GST-SFK method on numerical phantom data generated using a finite-difference-based method in tissue-mimicking viscoelastic media. In addition, we evaluated the method on numerical shear wave motion data with different amounts of white Gaussian noise added. Additionally, we performed tests on data from custom-made tissue-mimicking viscoelastic phantom experiments, ex vivo porcine liver measurements, and in vivo liver tissue experiments. We compared results from our method with two other techniques used for estimating shear wave phase velocity: the two-dimensional Fourier transform (2D-FT) and the eigenvector (EV) method. Tests carried out revealed that the GST-SFK method provides dispersion curve estimates with lower errors over a wider frequency band in comparison to the 2D-FT and EV methods. In addition, the GST-SFK provides expanded bandwidth by a factor of two or more to be used for phase velocity estimation, which is meaningful for a tissue dispersion analysis in vivo.