Generalized Ray Theory Research Articles

Revisiting Piezoelectric Sensor Calibration Methods Using Elastodynamic Body Waves

The application of absolutely calibrated piezoelectric (PZT) sensors is increasingly used to help interpret the information carried by radiated elastic waves of laboratory/in situs acoustic emissions (AEs) in nondestructive evaluation. In this paper, we present the methodology based on the finite element method (FEM) to characterize PZT sensors. The FEM-based modelling tool is used to numerically compute the true Green’s function between a ball impact source and an array of PZT sensors to map active source to theoretical ground motion. Physical-based boundary conditions are adopted to better constrain the problem of body wave propagation, reflection and transmission in/on the elastic medium. The modelling methodology is first validated against the reference approach (generalized ray theory) and is then extended down to 1 kHz where body wave reflection and transmission along different types of boundaries are explored. We find the Green’s functions calculated using physical-based boundaries have distinct differences between commonly employed idealized boundary conditions, especially around the anti-resonant and resonant frequencies. Unlike traditional methods that use singular ball drops, we find that each ball drop is only partially reliable over specific frequency bands. We demonstrate, by adding spectral constraints, that the individual instrumental responses are accurately cropped and linked together over 1 kHz to 1 MHz after which they overlap with little amplitude shift. This study finds that ball impacts with a broad range of diameters as well as the corresponding valid frequency bandwidth, are necessary to characterize broadband PZT sensors from 1 kHz to 1 MHz.

Broadband Calibration of Acoustic Emission and Ultrasonic Sensors from Generalized Ray Theory and Finite Element Models

This paper describes a calibration method for acoustic emission and ultrasonic sensors that is effective from 1 kHz to 1 MHz. The method combines generalized ray theory and finite element analysis to model wave propagation at higher and lower frequencies, respectively. A ball impact is used as a calibration source, a thick aluminum plate is used as the test block, and hot glue is used as the couplant. We demonstrate this method on five commercial piezoelectric sensors: Physical Acoustics (PAC) R15a, PAC WSa, Panametrics V101, Panametrics V103, and Valpey-Fisher Pinducer. Our calibration results show that reflections and other wave phases can be more clearly identified with the less-resonant Panametrics sensors. The PAC sensors have the greatest sensitivity and are able to detect surface normal displacements at least down to 1 pm amplitude in the 100s of kHz frequency band. Aperture effect is minimized by the small size of the Pinducer. Our method focuses on the amplitude response of the sensors (phase is ignored) and extends the calibration to a frequency band that is lower than typical analyses. Low frequency information is useful for determining the seismic moment of a seismic source (analogous to the magnitude of an earthquake) and can increase the amount of information acquired in a single recording.

Generalized transformation design: Metrics, speeds, and diffusion

We show that a unified and maximally generalized approach to spatial transformation design is possible, one that encompasses all second order waves, rays, and diffusion processes in anisotropic media. Until the final step, it is unnecessary to specify the physical process for which a specific transformation design is to be implemented. The principal approximation is the neglect of wave impedance, an attribute that plays no role in ray propagation, and is therefore irrelevant for pure ray devices; another constraint is that for waves the spatial variation in material parameters needs to be sufficiently small compared with the wavelength. The key link between our general formulation and a specific implementation is how the spatial metric relates to the speed of disturbance in a given medium, whether it is electromagnetic, acoustic, or diffusive. Notably, we show that our generalized ray theory, in allowing for anisotropic indexes (speeds), generates the same predictions as does a wave theory, and the results are closely related to those for diffusion processes.

On mode conversion, reflection, and transmission of magnetoacoustic waves from above in an isothermal stratified atmosphere

We use the exact solutions for magnetoacoustic waves in a two dimensional isothermal atmosphere with uniform inclined magnetic field to calculate the wave reflection, transmission, and conversion of slow and fast waves incident from above ($z=\infty$). This is relevant to the question of whether waves excited by flares in the solar atmosphere can penetrate the Alfv\'en/acoustic equipartition layer (which we identify as the canopy) to reach the photosphere with sufficient energy to create sunquakes. It is found that slow waves above the acoustic cutoff frequency efficiently penetrate (transmit) as acoustic (fast) waves if directed at a small attack angle to the magnetic field, with the rest converting to magnetic (slow) waves, in accord with Generalized Ray Theory. This may help explain the compact nature of seismic sources of sunquakes identified using seismic holography. The incident slow waves can also efficiently transmit at low frequency in inclined field due to the reduction in acoustic cutoff frequency (ramp effect). Incident fast (magnetic) "waves" from infinity with specified nonzero horizontal wavenumber are necessarily evanescent, but can carry energy to the equipartition level by tunnelling. It is found that this can then efficiently convert to acoustic (fast) energy that can again reach the photosphere as a travelling wave. Overall, there appear to be ample avenues for substantial compressive wave energy to penetrate the canopy and impact the photosphere.

Low‐velocity zones along the San Jacinto Fault, Southern California, from body waves recorded in dense linear arrays

AbstractWe derive high‐resolution information on low‐velocity fault zone (FZ) structures along the San Jacinto Fault Zone (SJFZ), Southern California, using waveforms of local earthquakes that are recorded at multiple linear cross‐fault arrays. We observe clear across‐fault delays of direct P and S waves, indicating damage zones at different segments of the SJFZ. We then compute synthetic traveltimes and waveforms using generalized ray theory and perform forward modeling to constrain the FZ parameters. At the southern section near the trifurcation area, the low‐velocity zone (LVZ) of the Clark branch has a width of ~200 m, 30–45% reduction in Vp, and ~50% reduction in Vs. From array data across the Anza seismic gap, we find a LVZ with ~200 m width and ~50% reduction in both Vp and Vs, nearly as prominent as that on the southern section. We only find prominent LVZs beneath three out of the five arrays, indicating along‐strike variations of the fault damage. FZ‐reflected phases are considerably less clear than those observed above the rupture zone of the 1992 Landers earthquake shortly after the event. This may reflect partially healed LVZs with less sharp boundaries at the SJFZ, given the relatively long lapse time from the last large surface‐rupturing event. Alternatively, the lack of observed FZ‐reflected phases could be partially due to the relatively small aperture of the arrays. Nevertheless, the clear signatures of damage zones at Anza and other locations indicate very slow healing process, at least in the top few kilometers of the crust.

Pulsed acoustic field radiation in a laterally bounded layered fluid

The acoustic field radiation from an impulsive source located in a discretely layered and laterally bounded fluid is investigated with the aid of the generalized-ray theory. The main ingredient of the presented approach is the Cagniard–de Hoop technique in combination with the method of images that provide exact space–time expressions for the activated acoustic wavefield components. Illustrative numerical examples are presented.

A Fast and Accurate Scheme for Travel-Time Computations and Ray-Path Determinations of Global Seismic Phases in a Laterally Heterogeneous Model

The pioneering work of Jeffreys and Bullen (1940; JB Tables) established the first truly global seismological travel‐time tables, which have fulfilled a vital and important role in performing earthquake locations and seismic tomography. Such information has enriched our knowledge about the inner structure of the Earth and its dynamic processes. The JB travel‐time tables, which are based on 1D velocity model (radial dependence only), have gradually improved over the years through a series of modifications, for example, the PREM model (Dziewonski and Anderson, 1981), the IASP91 model (Kennett and Engdahl, 1991) and the AK135 model (Kennett et al. , 1995). The major source of error in travel‐time prediction with these tables is due to from unincorporated lateral heterogeneities in the true velocity structure of the Earth. For traditional purposes of earthquake location, the global travel‐time tables (i.e., ISAP91 or AK135) have been generalized by a ray‐parameter method, such as a tau‐p function integral method of Buland and Chapman (1983). Some modified versions of the tau‐p function integral method are available, for example, the TauP Toolkit software (Crotwell et al. , 1999) and TTBox software (Knapmeyer, 2004), which offer improvements in computational efficiency and flexibility of usage. For purposes of global travel‐time tomography, there exist a variety of schemes to predict the travel‐times for a couple of the important global phases. They include perturbation theory (Bijwaard and Spakman, 1999), dynamic ray tracing (Dahlen et al. , 2000), the pseudo‐bending method (Zhao, 2001) nd generalized ray theory (Ballard et al. , 2009), in which most of the global phases are not predicted and accommodated, for example, later phases with the minimax time‐path (see Ray Tracing for Phases with a Stationary Minimax Time Path section for detailed …

SGRAPH (SeismoGRAPHer): Seismic waveform analysis and integrated tools in seismology

Although numerous seismological programs are currently available, most of them suffer from the inability to manipulate different data formats and the lack of embedded seismological tools. SeismoGRAPHer, or simply SGRAPH, is a new system for maintaining and analyzing seismic waveform data in a stand-alone, Windows-based application that manipulates a wide range of data formats. SGRAPH was intended to be a tool sufficient for performing basic waveform analysis and solving advanced seismological problems. The graphical user interface (GUI) utilities and the Windows functionalities, such as dialog boxes, menus, and toolbars, simplify the user interaction with the data. SGRAPH supports common data formats, such as SAC, SEED, GSE, ASCII, and Nanometrics Y-format, and provides the ability to solve many seismological problems with built-in inversion tools. Loaded traces are maintained, processed, plotted, and saved as SAC, ASCII, or PS (post script) file formats. SGRAPH includes Generalized Ray Theory (GRT), genetic algorithm (GA), least-square fitting, auto-picking, fast Fourier transforms (FFT), and many additional tools. This program provides rapid estimation of earthquake source parameters, location, attenuation, and focal mechanisms. Advanced waveform modeling techniques are provided for crustal structure and focal mechanism estimation. SGRAPH has been employed in the Egyptian National Seismic Network (ENSN) as a tool assisting with routine work and data analysis. More than 30 users have been using previous versions of SGRAPH in their research for more than 3 years. The main features of this application are ease of use, speed, small disk space requirements, and the absence of third-party developed components. Because of its architectural structure, SGRAPH can be interfaced with newly developed methods or applications in seismology. A complete setup file, including the SGRAPH package with the online user guide, is available.

Some useful approximations to generalized ray theory for regional distance seismograms

Seismograms recorded at regional distances (2°–12°) are quite complicated due to the waveguide nature of the crust. Generalized ray theory can be used to model the body waves in this distance range but a very large number of rays is required. Here I present a series of approximations to streamline generalized ray theory for the waveguide problem. If a layer over a half-space is used for the structure, then the de Hoop contour for a given ray is most strongly dependent on the fastest velocity of any leg of the ray. This results in analytic approximations to locate the contour. Each ray has two body wave arrivals (a headwave and a reflected arrival) so the displacement response of the ray need only be evaluated at a few points in time about the two arrival times and interpolated in between. A change in structure (increasing crustal thickness or Pn velocity) most strongly affects the relative timing of the headwave and the reflected arrival, so it is possible to ‘stretch’ or ‘squeeze’ the waveform of a representative model to simulate a whole suite of models. Also discussed is the applicability of a single layer over a half-space structure for modelling the observed regional distance waveforms for shallow earthquakes. At periods greater than a few seconds crustal layering can be replaced by a single layer having the appropriate average velocities. Lateral variations in crustal thickness with scale lengths of less than about 100 km can also be modelled with a simple horizontal layer of appropriate average thickness.

Near surface velocity and QS structure of the Quaternary sediment in Bohai basin, China

Heavily populated by Beijing and Tianjin cities, Bohai basin is a seismically active Cenozoic basin suffering from huge lost by devastating earthquakes, such as Tangshan earthquake. The attenuation (QP and QS) of the surficial Quaternary sediment has not been studied at natural seismic frequency (1–10 Hz), which is crucial to earthquake hazards study. Borehole seismic records of micro earthquake provide us a good way to study the velocity and attenuation of the surficial structure (0–500 m). We found that there are two pulses well separated with simple waveforms on borehole seismic records from the 2006 MW4.9 Wen’an earthquake sequence. Then we performed waveform modeling with generalized ray theory (GRT) to confirm that the two pulses are direct wave and surface reflected wave, and found that the average νP and νS of the top 300 m in this region are about 1.8 km/s and 0.42 km/s, leading to high νP/νS ratio of 4.3. We also modeled surface reflected wave with propagating matrix method to constrain QS and the near surface velocity structure. Our modeling indicates that QS is at least 30, or probably up to 100, much larger than the typically assumed extremely low Q (∼10), but consistent with QS modeling in Mississippi embayment. Also, the velocity gradient just beneath the free surface (0–50 m) is very large and velocity increases gradually at larger depth. Our modeling demonstrates the importance of borehole seismic records in resolving shallow velocity and attenuation structure, and hence may help in earthquake hazard simulation.

A two‐dimensional hybrid method for modeling seismic wave propagation in anisotropic media

A hybrid method is developed for calculating synthetic seismograms for seismic waves propagating in two‐dimensional localized heterogeneous anisotropic media. The hybrid method is a combination of analytic and numerical methods, with the numerical method (finite difference, or FD) applied in the heterogeneous anisotropic region only and the analytic method (generalized ray theory, or GRT) outside. The GRT solutions from a seismic source are interfaced with the FD calculation at the boundaries of the anisotropic region; and seismic responses at the Earth's surface are obtained from the output of FD calculation. Synthetic experiments on both isotropic and anisotropic models demonstrate the validity of the new method in simulating seismic wave propagation in complex media. We applied the hybrid method to study the SKS wave propagation in the upper mantle. We calculated the synthetics for a series of laterally varying anisotropic models, including a layer with varying anisotropy, two‐layer anisotropy, separated anisotropic body and those may be related to subduction zones. For these laterally varying models, SKS waveforms exhibit higher sensitivity to the spatial variation of anisotropy than the shear wave splitting parameters do. The results of this study demonstrate the potential of the proposed hybrid method for high‐resolution imaging the laterally varying anisotropic structures from seismological observations, when the tectonic structures can be approximated as two‐dimensional.

High-resolution structures of the Landers fault zone inferred from aftershock waveform data

SUMMARY High-frequency body waves recorded by a temporary seismic array across the surface rupture trace of the 1992 Landers, California, earthquake were used to determine fault-zone structures down to the seismogenic depth. We first developed a technique to use generalized ray theory to compute synthetic seismograms for arbitrarily oriented tabular low-velocity fault-zone models. We then generated synthetic waveform record sections of a linear array across a vertical fault zone. They show that both arrival times and waveforms of P and S waves vary systematically across the fault due to transmissions and reflections from boundaries of the low-velocity fault zone. The waveform characteristics and arrival-time patterns in the record sections allow us to locate the boundaries of the fault zone and to determine its P- and S-wave velocities independently as well as its depth extent. Therefore, the trade-off between the fault-zone width and velocities can be avoided. Applying the method to the Landers waveform data reveals a low-velocity zone with a width of 270‐360 m and a 35‐60 per cent reduction in P and S velocities relative to the host rock. The analysis suggests that the low-velocity zone extends to a depth of ∼7 km. The western boundary of the low-velocity zone coincides with the observed main surface rupture trace.

Generalized ray theory for an Epstein profile

The generalized ray method as extended by Heyman & Felsen is used for investigating an Epstein monotonic transition zone. Such an Epstein profile gives rise to normal turning rays, reversed turning rays and space-domain caustics. These wave species can now all be investigated using a complex integration contour in the forming of the synthetic seismogram. We present numerical results of the method and compare it to Chapman's (WKBJ seismogram) method. The waveforms generally agree well although a DC value difference is sometimes present in the seismograms. During the analysis we show that at the dark side of the caustic the arrival can be interpreted as a contribution due to tunnelling rays.

Multifold phase space path integral synthetic seismograms

SUMMARY We consider a medium consisting of homogeneous layers separated by curved interfaces. In order to evaluate the response of a single generalized ray, the source and receiver wavefields are expanded in a series of plane waves. The coupling of these plane waves at each point of the surfaces of material discontinuity is determined by means of a Kirchhoff integral using generalized reflection and transmission coefficients. The resulting integral, called the multifold phase space path integral (PSPI) consists of a series of integrals over ray parameters and over interfaces touched by the generalized ray on its way from the source to the receiver. This approach is a generalization of the multifold configuration space path integral (CSPI) to which it reduces by successive application of the stationary phase point method over the ray-parameter integrals. The PSPI like the CSPI automatically includes diffractions from corners. In addition classical head waves are included, although for curved interfaces the head waves are only approximate. 2-D synthetic seismograms are converted to equivalent approximate point-source responses by assuming cylindrical symmetry about source and/or receiver. The waveforms and amplitude of PSPI synthetic seismograms compare well with those computed by generalized ray theory for a 1-D model, and with finite difference synthetics for a 2-D model.

Waveform modelling of local earthquakes in southwest Japan

AbstractThe crustal structure beneath southwest Japan is investigated through waveform modellings of four local crustal earthquakes by using the generalized ray theory. High-quality seismograms recorded by the Hi-net seismic network are used. The focal mechanism solutions of the events used are estimated by using the firstP-wave polarity data. Twenty-nine paths are used to characterize the crustal waveguide in SW Japan. Optimal 1-D velocity models for different paths are derived individually by waveform modelling. Synthetic tests are conducted to study the effects of different model parameters and to confirm the focal mechanism solutions. Modelling the waveforms individually is proven applicable for the heterogeneous regions. The resulted synthetics match the observed seismograms well. Considerable variations of the Conrad and Moho depths are found in the study area. The Conrad depth ranges from 14.8 to 19.7 km, and the Moho depth ranges from 28.4 to 33.4 km. Considerable variations in the crustal structure are found for the eastern portion of the study area where Biwa Lake lies. The effects of Biwa Lake on different paths are examined in detail.

On the lack of seismic discontinuities within the inner core

Recent work has suggested that seismic discontinuities might exist within the inner core (IC). Such boundaries could be caused by changes in the strength and orientation of anisotropy, by variations in the fraction of partial melt or by a solid—solid phase transition involving iron. In this work, we search for evidence of IC discontinuities with reflection seismology techniques that use precritical PKiKP waveforms as empirical source pulses. Because any phase reflected from an IC discontinuity will have a small amplitude, we develop a technique combining source and receiver stacks in the search. We perform resolution tests where we compare the data with synthetic waveform stacks created using generalized ray theory and appropriate attenuation operators. Our results show that the IC lacks a significant (impedance jump >3 per cent), sharp (thickness <3–4 km), global discontinuity to depths of 800 km below the boundary with the outer core.

A method for calculating the axisymmetric response of a two-layered half-space under dynamic concentrated loading

The transient response of a linearly elastic structure made of a layer overlaying a half-space, subjected to a normal point force on its free surface, is investigated. Both welded contact and frictionless sliding are considered at the interface. This paper presents a method to calculate the response, for a wide range of loading durations ( T), on the axis of symmetry of the configuration (in the layer and the half-space). Equations of the boundary–value problem are manipulated in an integral transform domain and the Cagniard–de Hoop method is used. The final form of the exact analytical solution is a sum of contributions corresponding to the rays of the generalized ray theory; little computational effort need be developed for evaluating each contribution. While this theory has only been used to obtain early-time responses, long-time responses—up to 30 times the transit time of P-waves in the layer—have been calculated for this study. This work was conducted to help characterize the stress transmitted in the human lung (half-space) when the thoracic wall (layer) is subjected to a non-penetrating impact. Depending on T, multiple reflections of waves in the layer or typical low-frequency response are observed. The influence of the contact condition with respect to T is elucidated.

On the transmission of transient elastodynamic waves at a frictionless contact interface; application to a weakly coupled bimaterial

Abstract A method is presented for a detailed quantitative analysis of the elastic waves interaction with a frictionless sliding interface between two linearly elastic, isotropic and homogeneous half-spaces (media 1 and 2). Frictionless contact (zero shear stresses and discontinuity of tangential displacements at the interface) is relevant, for instance, when media are separated by a thin fluid layer. The required transmission/reflection coefficients of plane waves at the interface are derived in closed form. The elastodynamic problem consists in calculating exact time domain Green’s functions (for displacements and stresses) in medium 2, when a point force acts inside medium 1. They are obtained in two steps: (1) the generalized ray theory yields solutions in a Laplace–Fourier transform domain for each transmitted wave (PP, PS, SP and SS); (2) the Cagniard–de Hoop method is used to return in the time-space domain. Numerical results are relevant to the propagation of an impact wave in an idealized model of the thorax; media 1 and 2 represent the thoracic wall (“stiff”) and the lung (“compliant”). The wave field transmitted at the frictionless interface is compared to that transmitted at a welded interface. It is found that (i) S-waves amplitudes are negligible; (ii) the P-wave field is only slightly changed; (iii) these effects are characteristic of the weak acoustic coupling.

Ridge‐like lower mantle structure beneath South Africa

Recent (ScS‐S) results from probing the deep structure beneath southern Africa display strong delays of up to 10 s at distances beyond 90°. Such delays could be explained by long‐period tomographic models containing smooth (weak) features with the addition of rough (strong) D″ structure (3–9% drops in shear velocities). However, these structures cannot explain the (SKS‐S) differentials sampling the same region. To explain the (SKS‐S) and (ScS‐S) data sets simultaneously requires instead a large‐scale ridge‐like structure with a relatively uniform 3% reduction in shear velocity. The structure is about 1000 km wide and extends at least 1200 km above D″. It is orientated roughly NW‐SE and leans toward the east at latitudes from 15° to about 30°. It proves difficult to explain such sharp features with thermal effects alone and, thus, the importance of high‐resolution waveform modeling to establish their existence. To derive the above results, we developed a special algorithm by matching simulated synthetics to observed broadband waveforms. This is achieved by computing the various arrivals separately using generalized ray theory for a reference model and allowing the arrivals to shift in relative times to match the data. Tomographic models can then be constructed, or existing tomographic models can be altered, to match these data, and new 2‐D synthetics can be constructed as well to better fit the waveform data. These updated synthetics can again be decomposed and reassembled, and the process can be repeated. This algorithm is applied to a combination of analog and digital data along a corridor from South America, producing the high‐resolution 2‐D model described above.

An SH hybrid method and shear velocity structures in the lowermost mantle beneath the central Pacific and South Atlantic Oceans

An SH hybrid method is developed for calculating synthetic seismograms involving two‐dimensional localized heterogeneous structures. The hybrid method is a combination of analytic and numerical methods, with the numerical method (finite difference) applied in the heterogeneous region only and analytical methods applied outside the region. Generalized ray theory solutions from a seismic source are used to initiate the finite difference calculation, and seismic responses at the Earth's surface are obtained from the finite difference output by applying the Kirchhoff theory. We apply the hybrid method and study SH wave propagation near the bottom of the mantle. The shear velocity structures and the interaction of SH waves with these velocity structures at the base of the mantle beneath the central Pacific and South Atlantic Oceans are of particular importance. The observed SH waves sampling these two regions of the core‐mantle boundary show very different characteristics across the epicentral distance range of 83°–108°. The SH waves sampling the core‐mantle boundary beneath the central Pacific show a linearly increasing delay of 4 s from 98° to 108° and discernible ScS phases up to an epicentral distance of 102°. The SH waves propagating through the base of the mantle beneath the South Atlantic Ocean, on the other hand, exhibit a linearly increasing delay of 10 s from 98° to 108°, discernible multiple ScS phases with a same slowness as the direct SH waves up to 108°, and rapid variations of waveforms across small epicentral distances. Synthetic tests indicate that while the observations sampling the central Pacific can be explained by a negative shear velocity gradient of 3% (relative to the preliminary reference Earth model) at the bottom 300 km of the mantle, those sampling the South Atlantic Ocean require a 300‐km‐thick bottom boundary layer with a larger negative velocity gradient (up to 10%) and steeply dipping edges. The negative velocity gradient at the base of mantle beneath the South Atlantic Ocean can be best explained by partial melt driven by a compositional change produced in the early Earth's history and a vertical thermal gradient within the layer, while that beneath the central Pacific may reasonably be attributed to pure thermal effects within a thermal boundary layer.