Shear Wave Amplitude Research Articles

Numerical Simulation Study on the Influence of Fracture on Borehole Wave Modes: Insights from Fracture Width, Filling Condition, and Acoustic Frequency.

Unconventional reservoirs, such as shale and tight formations, have become increasingly vital contributors to oil and gas production. In these reservoirs, fractures serve as crucial spaces for fluid migration and storage, making their precise assessment essential. Array acoustic logging stands out as a pivotal method for evaluating fractures. To investigate the impact of fracture width, fracture-filling conditions, and acoustic frequency on compressional and shear waves, a three-dimensional variable mesh finite difference program was employed for acoustic logging numerical simulation. Firstly, numerical models representing fractured formations with varying fracture widths and distinct fluid-filling conditions were established, and array acoustic logging numerical simulations were conducted at different frequencies. Subsequently, the waveform data were processed to extract acoustic characteristic parameters, such as velocities and amplitude attenuations of compressional and shear waves. Finally, a quantitative analysis was conducted to examine the variation patterns of characteristic parameters of refracted compressional and shear waves in relation to fracture properties. The research results indicate that amplitude attenuation information derived from borehole wave modes is particularly sensitive to the changes in fracture properties. As fracture width increased, we observed a significant amplitude attenuation in both compressional and shear waves, proportional to the logarithm of the attenuation coefficients. Furthermore, when the fracture width was constant, gas-filled fractures exhibited more prominent amplitude attenuation than water-filled fractures, with shear wave attenuation being more sensitive to the filling material. Moreover, from a quantitative perspective, the analysis revealed that the attenuation coefficients of refracted compressional and shear waves exhibited an exponential variation with gas saturation. Notably, once fracture width and filling conditions were established, the amplitudes of compressional and shear waves at the dominant frequency of 40 kHz were significantly reduced compared to those at 8 kHz, accompanied by increased attenuation. Subsequent quantitative analysis revealed that, when the product of fracture width and dominant frequency remains constant, the corresponding attenuation coefficient ratios approach 1. This indicates that the attenuation process of acoustic propagation in fractured media follows the principle of acoustic similarity. The findings of this study provide reference for further research on fracture property evaluation methods based on array acoustic logging data.

Impact of Randomized Soil Properties and Rock Motion Intensities on Ground Motion

Seismic site response is inevitably influenced by natural variability of soil properties and anticipated earthquake intensity. This study presents the influence of variability in shear wave velocity (Vs) and amplitude of input rock motion on seismic site response analysis. Monte Carlo simulations were employed to randomize the Vs profile for different scenarios. A series of 1-D equivalent linear (EQL) seismic site response analyses were conducted by combining the randomized Vs profile with different levels of rock motion intensities. The results of the analyses are presented in terms of surface spectral acceleration, amplification factors (AFs), and peak ground acceleration (PGA). The mean and standard deviation of these parameters are thoroughly discussed for a wide range of randomized Vs profile, number of Vs randomizations, and intensities of input rock motions. The results demonstrate that both the median PGA and its standard deviations across different number of Vs profile realization exhibit a slight variation. As few as twenty Vs profile realizations are sufficient to compute reliable response parameters. Both rock motion intensity and standard deviation of Vs variability cause significant variation in computed surface parameters. However, the variability in the number of records used to conduct site response has no significant impact on ground response if the records closely match the target spectrum. Incorporating the multiple sources of variabilities can reduce uncertainty when conducting ground response simulations.

Focused Shear Wave Beam Propagation in Tissue-Mimicking Phantoms.

Ultrasound transient elastography (TE) technologies for liver stiffness measurement (LSM) utilize vibration of small, flat pistons, which generate shear waves that lack directivity. The most common cause for LSM failure in practice is insufficient shear wave signal at the needed depths. We propose to increase shear wave amplitude by focusing the waves into a directional beam. Here, we demonstrate the generation and propagation of focused shear wave beams (fSWBs) in gelatin. Directional fSWBs are generated by vibration at 200-400 Hz of a concave piston embedded near the surface of gelatin phantoms and measured with high-frame-rate ultrasound imaging. Five phantoms with a range of stiffnesses are employed. Shear wave speeds assessed by fSWBs are compared with those by radiation-force-based methods (2D SWE). fSWB amplitudes are compared to predictions using an analytical model. fSWB-derived shear wave speeds are in good agreement with 2D SWE. The amplitudes of fSWBs are localized to the LSM region and are significantly greater than unfocused shear waves. Overall agreement with theory is observed, with some discrepancies in the theoretical source condition. Focusing shear waves can increase the signal in the LSM region for TE. Challenges for translation include coupling piston vibration with the patient skin and increased attenuation in vivo compared to the phantoms employed here. Fibrosis is the most predictive measure of patient outcome in non-alcoholic fatty liver disease. Increased shear wave amplitude in the LSM region can reduce fibrosis assessment failure rates by TE, thus reducing the need for invasive methods like biopsy.

Resonance, Velocity, Dispersion, and Attenuation of Ultrasound-Induced Shear Wave Propagation in Blood Clot In Vitro Models.

Improve the characterization of mechanical properties of blood clots. Parameters derived from shear wave (SW) velocity and SW amplitude spectra were determined for gel phantoms and in vitro blood clots. Homogeneous phantoms and phantoms with gel or blood clot inclusions of different diameters and mechanical properties were analyzed. SW amplitude spectra were used to observe resonant peaks. Parameters derived from those resonant peaks were related to mimicked blood clot properties. Three regions of interest were tested to analyze where resonances occurred the most. For blood experiments, 20 samples from different pigs were analyzed over time during a 110-minute coagulation period using the Young modulus, SW frequency dispersion, and SW attenuation. The mechanical resonance was manifested by an increase in the number of SW spectral peaks as the inclusion diameter was reduced (P < .001). In blood clot inclusions, the Young modulus increased over time during coagulation (P < .001). Descriptive spectral parameters (frequency peak, bandwidth, and distance between resonant peaks) were linearly correlated with clot elasticity values (P < .001) with R2 = .77 for the frequency peak, .60 for the bandwidth, and .48 for the distance between peaks. The SW dispersion and SW attenuation reflecting the viscous behavior of blood clots decreased over time (P < .001), mainly in the early stage of coagulation (first minutes). The confined soft inclusion configuration favored SW mechanical resonances potentially challenging the computation of spectral-based parameters, such as the SW attenuation. The impact of resonances can be reduced by properly selecting the region of interest for data analysis.

Laser Vibrometry of Shear Waves in a Layer of a Gel-Like Medium

Abstract—A laser scanning vibrometer was used to measure the amplitudes and phases of the vibrational velocity of shear waves excited by a one-dimensional source in the form of a narrow rectangular bar in a gellike medium. The vibrations of 26 plates reflecting the laser beam and located inside an optically transparentphantom along a segment with a length of 84.5 mm at a distance of 20 mm from the source were measured.The angular distributions of the amplitude and phase of shear waves at discrete frequencies from 59 to 500 Hz were measured in continuous mode. In pulsed mode, the vibrator excited a pulse in the medium with a durationof 1.5 periods of the 300 Hz frequency. The amplitudes and phases of shear waves were calculated by fastFourier transform of the time profile of the vibration velocity of the plates with a duration of 50 ms. The angular amplitude distributions measured in the pulsed and continuous modes are qualitatively the same. At all frequencies, the distributions are symmetrical with respect to the vertical axis. The maximum oscillation amplitude is observed at angles close to ±45°. The velocity of shear waves, calculated from the measured phase distributions, increases from 2 to 2.5 m/s with a change in frequency from 50 to 500 Hz. It is shown that this velocity behavior is well described by a relaxation model of the medium with one relaxation time equal to 0.3 ms. Shear wave attenuation depends on frequency and exceeds 1 cm-1 for waves with frequencies above250 Hz. The maximum attenuation per wavelength is observed near the relaxation frequency of the medium in the 300–400 Hz range. The results can be used to optimize devices for measuring the elasticity of soft tissues.

Rapid MR elastography of the liver for subsecond stiffness sampling.

Depicting the stiffness of biological soft tissues, MR elastography (MRE) has a wide range of diagnostic applications. The purpose of this study was to improve the temporal resolution of 2D hepatic MRE in order to provide more rapid feedback on the quality of the wavefield and ensure better temporal sampling of respiration-induced stiffness changes. We developed a rapid MRE sequence that uses 2D segmented gradient-echo spiral readout to encode 40 Hz harmonic vibrations and generate stiffness maps within 625 ms. We demonstrate the use of this technique as a rapid test for shear wave amplitudes and overall MRE image quality and as a method for monitoring respiration-induced stiffness changes in the liver in comparison to 3D MRE and ultrasound-based time-harmonic elastography. Subsecond MRE allowed monitoring of increasing shear wave amplitudes in the liver with increasing levels of external stimulation within a single breath-hold. Furthermore, the technique detected respiration-induced changes in liver stiffness with peak values (1.83 ± 0.22 m/s) at end-inspiration, followed by softer values during forced abdominal pressure (1.60 ± 0.22 m/s) and end-expiration (1.49 ± 0.22 m/s). The effects of inspiration and expiration were confirmed by time-harmonic elastography. Our results suggest that subsecond MRE of the liver is useful for checking MRE driver settings and monitoring breathing-induced changes in liver stiffness in near real time.

Shear wave transmissivity and wave-induced vorticity diffusion through conversion scattering to slow-shear diffusion wave

In the poroelasticity theory of de la Cruz and Spanos, fluid shearing within the viscous boundary layer, i.e. fluid vorticity, manifests as an independent process, namely the diffusive slow shear wave. Quantifying its impact on elastic wave propagation remains a research challenge. We analyse the transmissivity of a horizontally polarized shear wave travelling across a stack of porous fluid-saturated layers. The fluid shearing developed at each contact is captured, in a macroscopic sense, through conversion scattering into this diffusive slow shear wave. Generalizing the reflectivity method for elastic waves to the poroelastic case, we develop semi-analytical results for an arbitrary number of layers. We find that the conversion scattering into this diffusion wave reduces the shear wave amplitude, and this reduction accumulates with the number of layers. In the limit when the layer thickness corresponds to the pore diameter, the resulting wave quality factor is close to the predictions of previously reported attempts to capture the fluid vorticity. However, our approach is distinctively different since it is purely based on a wave-theoretic analysis of interacting waves, acknowledging vorticity diffusion as an independent process, thereby avoiding the problems arising when incorporating a pore-scale phenomenon in a macroscopic wave propagation theory.

Abdominal MR elastography with multiple driver arrays: performance and repeatability.

To evaluate the performance and repeatability assessing liver, spleen, and kidney stiffness with magnetic resonance elastography (MRE), using arrays of pneumatic passive drivers. An array of four flexible, pneumatically activated passive drivers for abdominal MRE were developed and tested in this study. Multiple MRE acquisitions were performed prospectively in a series of eleven volunteers, with activation of all combinations of the four drivers, individually and simultaneously. MRE exams were repeated three times to study within-day and between-day test-retest repeatability. Semi-quantitative evaluation of wave propagation and penetration, and quantitative assessment of tissue stiffness was conducted for liver, spleen, and kidneys. When driver location and amplitude were sufficient to achieve necessary shear wave illumination in any given region of interest, the results showed excellent test-retest repeatability in abdominal organ stiffness with both single and multiple driver configurations. The results confirmed that multiple driver arrays provided suitable shear wave illumination over a larger region of the abdomen, allowing more reliable stiffness measurements in multiple organs. MRE assessment of the spleen was found to be prone to effects of excessive shear wave amplitude, however. A multiple driver array provides shear wave illumination over a larger region of the abdomen than obtained with a single driver, for MRE assessment of multiple abdominal organs, providing excellent test-retest repeatability in stiffness measurements. However, careful tuning of the location and amplitude of each driver is essential to achieve consistent results.

Quantifying the Impact of Imaging Through Body Walls on Shear Wave Elasticity Measurements.

In the context of ultrasonic hepatic shear wave elasticity imaging (SWEI), measurement success has been determined to increase when using elevated acoustic output pressures. As SWEI sequences consist of two distinct operations (pushing and tracking), acquisition failures could be attributed to (i) insufficient acoustic radiation force generation resulting in inadequate shear wave amplitude and/or (ii) distorted ultrasonic tissue motion tracking. In the study described here, an opposing window experimental setup that isolated body wall effects separately between the push and track SWEI operations was implemented. A commonly employed commercial track configuration was used, harmonic multiple-track-location SWEI. The effects of imaging through body walls on the pushing and tracking operations of SWEI as a function of mechanical index (MI), spanning 5 different push beam MIs and 10 track beam MIs, were independently assessed using porcine body walls. Shear wave speed yield was found to increase with both increasing push and track MI. Although not consistent across all samples, measurements in a subset of body walls were found to be signal limited during tracking and to increase yield by up to 35% when increasing electronic signal-to-noise ratio by increasing harmonic track transmit pressure.

Study on the steering capability of a meander-line coil EMAT

Electromagnetic acoustic transducers (EMATs) are well-established as a means of ultrasonic wave generation and reception without the use of a mechanical coupling. When comprising a bias magnetic field and a meander-line coil (MLC), these waves propagate at an angle normal to the emission surface. With the appropriate frequency, the propagation pathway of these ultrasonic waves can be steered to a particular angle. This paper presents the methodology used to find the steering limit of an MLC EMAT and the results from simulations and experimental validations on aluminium. The results show that the maximum shear wave amplitude occurred at around 30°, the steering limit was approximately 50° and the simulations were validated by the experimental set-up to a satisfactory degree.

Surface excitation of focused shear wave beams in soft elastic media: Experiment

Shear waves are employed in medical imaging to reveal variations in the viscoelastic properties of soft tissues, which are useful biomarkers for pathologies such as breast lesions and liver disease. Shear wave excitation methods that employ acoustic radiation force or surface vibration with a small piston have limitations associated with imaging depth and shear wave amplitude. We introduce a new method for surface excitation of shear waves that employs longitudinal motion of a concave-shaped piston source to generate a focused shear wave beam, thereby increasing shear wave amplitude and penetration depth. Focused shear waves are excited in a gel phantom in the frequency range of 200 Hz to 400 Hz using a concave piston with 40 mm radius of curvature and 50 mm diameter. The wave field comprises both transverse and longitudinal displacement components, which are polarized perpendicular and parallel to the propagation direction, respectively. Transversely and longitudinally polarized wave fields are measured using ultrasound speckle tracking. The longitudinally polarized shear wave is of interest because it is measured in elastography applications. Preliminary comparisons exhibit good agreement between measured beam patterns and those predicted by an analytical theory for shear wave beam propagation due to surface excitation with a piston source.

Particle image velocimetry for estimating shear wave elasticity imaging (SWEI) in the esophagus dysplasia

In this study, a 128-channel high-frequency pulse generator/receiver system was used to emit multi-angle wave propagation, and the generator plane excitation was used to generate a focused beam, which induced the phenomenon of particle displacement in the esophagus phantom recorded. Shear wave electrography is generated by using a micro-ultrasound probe (ARFI), which excites the shear waves with the output acoustic energy. In addition, we perform the method of PIV proper orthogonal decomposition to better qualify and quantify the differences between the PIV and Doppler, especially for analysis block into PIV for particle tracking, in which the external environment, with a Doppler similarity index of 82%, and PIV mainly uses the Doppler algorithm for analysis and obtains 97 frames, and fixed 0.33 ms time difference to output pictures. Each pass interrogation area is set to 128 in sequence, the distance is defined in calibration, in which the low filter contrast of image-based validation is set to 0.004, and the bright filter objects are set to 0.5. Through the analysis results of PIV, each particle's speed and movement are critical for the hospital to think. The relative position of the lesion and the effect of ARFI on the lesion are to be understood by understanding the direction of movement of the contrast agent. Overall, the shear wave and Doppler output amplitudes provided the best image quality and PIV results for esophagus dysplasia.

Influencing Factors of Shear Wave Radiation of a Dipole Source in a Fluid-Filled Borehole

In shear wave far detection logging, dipole-source radiation is the main factor influencing the amplitude of the reflected shear waves. In this paper, a method is derived with the far-field asymptotic solution to calculate the dipole-source radiation of shear waves in a fluid-filled borehole. Then the dipole-source radiation of the shear waves is simulated under both low and high frequencies. In addition, the influences of formation elastic parameters on the dipole-source radiation of the shear waves are analyzed and the variations of the radiation characteristics of the shear wave with source main frequency and borehole radius are compared. Results show that the density and compressional wave velocity of the formation have little effect on the dipole-source radiation of the shear waves. However, the shear wave velocity not only affects the shear wave amplitude radiated to the formation by the dipole source (radiation performance), but also affects the energy distribution of the shear wave at different locations in space (radiation direction). The dipole source has better radiation performance and radiation coverage at low frequency and the optimal excitation frequency in different formations is very close, which is good for the application of this technology under different circumstances. At low frequency, the borehole has little influence on the dipole-source radiation, no matter how large the borehole radius is. However, at high frequency, the borehole modulation of the dipole-source radiation cannot be ignored, especially at large borehole radius.

Using laboratory data to understand how pore aspect ratio influences elastic parameters and amplitude variation with offset

Pore geometry is an important parameter in reservoir characterization that affects the permeability of reservoirs and can also be a controlling factor on the impact of pressure and saturation on reservoirs elastic properties. We have used selective laser sintering 3D printing technology to build physical models to experimentally investigate the impacts of pore aspect ratio on compressional (P-) and shear wave (S-wave) velocities and amplitude variation with offset (AVO). We printed six models to study the effects of the pore aspect ratio of prolate and oblate pore structures on elastic properties and AVO signatures. We found that the P-wave velocity is reduced by decreasing the pore aspect ratio (flatter pore structure), whereas the S-wave velocity is less sensitive to the pore aspect ratio. This effect is reduced when the samples are water saturated. We developed new experimental and processing techniques to extract realistic AVO signatures from our experimental data and found that the pore aspect ratio has similar effects on AVO as fluid compressibility. This indicated that not considering the pore aspect ratio in AVO analysis can lead to misleading interpretations. We also found that these effects are reduced in water-saturated samples.

Features of modeling of stress and strain waves in an anisotropic medium on the example of a wooden element

This article describes the hypotheses of the occurrence, propagation, and modification of stress and strain waves caused by external loads in isotropic and anisotropic infinite and finite elastic media. A model of an infinite elastic medium experiencing a point external impulse is presented. The model demonstrates the propagation of a longitudinal plane wave. Compaction and rarefaction of the medium are observed in the plane with wave propagation. A graph of changes in the amplitude of a longitudinal plane wave is presented in the same coordinate system. The problem is posed of expanding the numerical model of a finite elastic medium in the form of an anisotropic wooden rod experiencing a plane external impulse. The model should demonstrate the propagation of longitudinal and transverse waves and describe the volumetric deformation of an anisotropic material. Compaction and rarefaction of the medium are shown in the plane, coinciding with the direction of wave propagation. A graph of the change in the shear wave amplitude is presented in the same coordinate system. The combination of these two graphsreveals the difference in wave propagation velocities and the combination of amplitudes. The model will make it possible to identify the presence of Rayleigh waves and to describe the reflection of waves from the boundary of the medium.

Effects of Wave Coherence on Longshore Variability of Nearshore Wave Processes

AbstractCoherent waves are pairs of waves having identical frequency and waveform with constant phase difference leading to a stationary wave interference in the wavefield of phase‐resolving wave models. The resulting stationary interference due to the presence of coherent water waves results in the longshore variation of the wave height in longshore uniform topography. Consequently, this variation affects the radiation stress throughout the domain and leads to the longshore variation in mean water level, mean current field, shear wave amplitude, wave setup, and wave runup. Here, we have developed a new method of wave energy spectrum discretization for the phase‐resolving wave models like FUNWAVE‐TVD, which either eliminates the coherent waves at the offshore boundary condition or limits them to a controlled number. Then, effects of the coherent waves on some nearshore wave processes are probed using both a flat bottom case as well as a longshore uniform sloping beach. The results shed light on the effects of wave coherence on the nearshore wave processes and demonstrate the usefulness of the newly developed wavemaker as a tool for the scientific community to further assess the hydrodynamics in nearshore environments.

Generic uniqueness and stability for the mixed ray transform

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1 + 1 1+1 and 2 + 2 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of q S qS -polarized elastic waves provide partial information about the mixed ray transform of 2 + 2 2+2 tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the \text{Dirichlet-to-Neumann} map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

Shear wave generation by mode conversion in picosecond ultrasonics: Impact of grain orientation and material properties

AbstractPicosecond ultrasonics has seen wide application in the investigation of interfaces in semiconductors and 3D imaging in biological systems. Shear ultrasonic waves are important for elastic constant and grain orientation measurement in ceramic materials. In this study, we investigate the impact of grain orientation and material's elastic properties on the generation efficiency of shear waves by mode conversion at a transducer thin film/substrate interface. The solution of acoustic wave equations suggests that crystal grain orientation has a strong impact on the generated shear wave amplitude. This dependence is found to be closely related to the magnitude of longitudinal component of wave displacements. The applicability of analytical model is validated by the experimental results from time‐domain Brillouin scattering. Moreover, material properties determine acoustic wave amplitudes based on the acoustic mismatch model and, particularly, large elastic anisotropy defined by Zener ratio favors strong shear waves. In the light of this analysis, several recommendations on suitable grain orientations and film materials are made to facilitate shear wave detection.

An application of WKBJ theory for triad interactions of internal gravity waves in varying background flows

AbstractMotivated by the question of whether and how wave–wave interactions should be implemented into atmospheric gravity‐wave parametrizations, the modulation of triadic gravity‐wave interactions by a slowly varying and vertically sheared mean flow is considered for a non‐rotating Boussinesq fluid with constant stratification. An analysis using a multiple‐scale WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) expansion identifies two distinct scaling regimes, a linear off‐resonance regime, and a nonlinear near‐resonance regime. Simplifying the near‐resonance interaction equations allows for the construction of a parametrization for the triadic energy exchange which has been implemented into a one‐dimensional WKBJ ray‐tracing code. Theory and numerical implementation are validated for test cases where two wave trains generate a third wave train while spectrally passing through resonance. In various settings, of interacting vertical wavenumbers, mean‐flow shear, and initial wave amplitudes, the WKBJ simulations are generally in good agreement with wave‐resolving simulations. Both stronger mean‐flow shear and smaller wave amplitudes suppress the energy exchange among a resonantly interacting triad. Experiments with mean‐flow shear as strong as in the vicinity of atmospheric jets suggest that internal gravity‐wave dynamics are dominated in such regions by wave modulation. However, triadic gravity‐wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.

Effect of Random 3D Correlated Velocity Perturbations on Numerical Modeling of Ground Motion from the Source Physics Experiment

ABSTRACTExplosions are traditionally discriminated from earthquakes, using the relative amplitude of compressional and shear waves at regional and teleseismic distances known as the P/S discriminant. Pyle and Walter (2019) showed this technique to be less robust at shorter distances, in detecting small-magnitude earthquakes and low-yield explosions. The disparity is largely due to ground motion from small, shallow sources being significantly impacted by near-surface structural complexities. To understand the implications of wave propagation effects in generation of shear motion and P/S ratio during underground chemical explosions, we performed simulations of the Source Physics Experiment (SPE) chemical explosions using 1D and 3D velocity models of the Yucca Flat basin. All simulations used isotropic point sources in the frequency range 0–5 Hz. We isolate the effect of large-scale geological structure and small-scale variability at shallow depth (&amp;lt;5 km), using a regional 3D geologic framework model (GFM) and the GFM-R model derived from the GFM, by adding correlated stochastic velocity perturbations. A parametric study of effects of small-scale velocity variations on wave propagation, computed using a reference 1D velocity model with stochastic perturbations, shows that the correlation length and depth of stochastic perturbations significantly impact wave scattering, near-surface wave conversions, and shear-wave generation. Comparisons of recorded and simulated waveforms for the SPE-5 explosion, using 3D velocity models, demonstrate that the shallow structure of the Yucca Flat basin contributes to generation of observed shear motion. The inclusion of 3D wave scattering, simulated by small-scale velocity perturbations in the 3D model, improves the fit between the simulated and recorded waveforms. In addition, a relatively low intrinsic attenuation, combined with small-scale velocity variations in our models, can confirm the observed wave trapping and its effect on duration of coda waves and the spatial variation of P/S ratio at basin sites.