Pulse distortion and Hilbert transformation in multiply reflected and refracted body waves

1975 ◽  
Vol 65 (1) ◽  
pp. 55-70 ◽  
Author(s):  
George L. Choy ◽  
Paul G. Richards
Keyword(s):  
Phase Shift ◽  
Travel Time ◽  
Body Waves ◽  
Wave Form ◽  
Distorted Wave ◽  
Arrival Times ◽  
The Earth ◽  
Frequency Independent ◽  
Wave Forms ◽  
First Arrival

abstract Many seismic body waves are associated with rays which are not minimum travel-time paths. Such arrivals contain pulse deformation due to a phase shift in each frequency component. For sufficiently high frequencies, the phase shift each time a ray touches an internal caustic is π/2 and frequency-independent. The distorting effect of a frequency-independent phase shift is successfully observed in seismograms from events in several regions. The data examined are long-period (T > 9 sec). They include deep earthquakes (depth > 500 km), in which a series of well-separated S phases (S, sS, SS and sSS) are available. These show that the wave form of SS, which has been distorted in propagation through the Earth, can be derived from the wave form of sS, which is not distorted. Shallow events, in which multiple S phases overlap, also exhibit behavior predicted by phase distortion. Rays supercritically reflected or refracted at a discontinuity in the Earth also suffer a constant phase shift, which in general can have any value. An important case is SKKS: its undistorted wave form resembles that of SKS, which has a minimum travel-time path. Without exception, all the distorted wave forms bear little or no resemblance to the original wave form. That is, neither the first arrival of energy nor the subsequent relative position of peaks and troughs on a distorted wave form appear at the ray theoretical times. Thus, T-Δ curves constructed by choosing arrival times to correspond to the first arrival of energy may be biased. Similarly, doubt is cast on differential travel times chosen from first motions, or from averaging several points on what appear to be corresponding peaks and troughs of two wave forms. Some of the rays most important to seismology, in which the distortion phenomenon occurs, include P and S (where d2T/dΔ2 > 0), PKPAB, PP, SS, and SKKS. Removal of phase distortion in the data is computationally straightforward. By exploiting the resulting wave forms to full advantage in correctly picking arrival times, we may hope to improve velocity models of the Earth. It is shown that matched filtering to obtain differential travel times is appropriate for certain pairs of body waves if they are phase-corrected.

The Alaskan earthquake of July 22, 1937*

1940 ◽  
Vol 30 (4) ◽  
pp. 353-376
Author(s):  
John N. Adkins
Keyword(s):  
Travel Time ◽  
Epicentral Distance ◽  
Time Interval ◽  
Time Curve ◽  
Arrival Times ◽  
The Earth ◽  
Geographical Latitude ◽  
The World ◽  
First Motion ◽  
First Arrival

Summary The study of the Alaskan earthquake of July 22, 1937, is based on the examination of original seismograms and photographic copies from seismological observatories throughout the world. The arrival times of P at 71 stations were used in locating the epicenter. By Geiger's method and the use of Jeffreys' travel times, the position of the epicenter was found to be: geographical latitude, 64.67±.04° N, longitude, 146.58±.12° W, and the time of occurrence to be 17h 9m 30.0±.25s, U.T. The epicenter lies in the Yukon-Tanana upland in central Alaska, which is not a region of frequent major earthquakes. The disagreement caused by the apparently early arrivals at College and Sitka was reduced by replacing the standard travel-time curve of P by a linear travel-time curve in the interval of epicentral distance 0° to 16° and by interpreting the first arrival at College as P. It was possible to determine the direction of the first motion of P for 51 stations. The observed distribution of first motion and the geological trends in the region of the epicenter are consistent with the earthquake's having been caused by movement along a fault with strike between N 30° E and N 37° E, and dip between 64° and 71° to the southeast, in which the southeast side of the fault was displaced relatively northeastward with the line of movement pitching between 12° and 16° northeast. A wave designated F (for “false S”) was found to precede S on the records by 20 to 55 seconds, depending on the epicentral distance. The wave is longitudinal in type and the arrival times define a linear travel-time curve. It is suggested that this wave may be a longitudinal surface wave, of the type proposed by Nakano, produced at the surface of the earth by the arrival of a transverse wave which has been reflected at a surface of discontinuity within the earth. The records show two impulses near the time when S is expected. The average time interval between the two impulses is 11.3 sec. The first, called S1, has a plane of vibration intermediate in direction between the plane of propagation and the normal thereto. The second impulse, called S2, is nearly pure SH movement. The writer wishes to express his indebtedness to Professor Perry Byerly for invaluable suggestions and criticism during the course of the investigation.

scholarly journals An analysis of the first-arrival times picked on the DSS and wide-angle seismic section recorded in Italy since 1968

2009 ◽  
Vol 47 (6) ◽  
Author(s):  
L. De Luca ◽  
R. De Franco ◽  
G. Biella ◽  
A. Corsi ◽  
R. Tondi
Keyword(s):  
Travel Time ◽  
Velocity Model ◽  
P Wave ◽  
Wide Angle ◽  
Seismic Section ◽  
Arrival Times ◽  
First Arrival ◽  
Time Offset ◽  
Refraction Data ◽  
1D Velocity Model

We performed an analysis of refraction data recorded in Italy since 1968 in the frame of the numerous deep seismic sounding and wide-angle reflection/refraction projects. The aims of this study are to construct a parametric database including the recording geometric information relative to each profile, the phase pickings and the results of some kinematic analyses performed on the data, and to define a reference 1D velocity model for the Italian territory from all the available refraction data. As concerns the first goal, for each seismic section we picked the P-wave first-arrival-times, evaluated the uncertainties of the arrival-times pickings and determined from each travel time-offset curve the 1D velocity model. The study was performed on 419 seismic sections. Picking was carried out manually by an algorithm which includes the computation of three picking functions and the picking- error estimation. For each of the travel time-offset curves a 1D velocity model has been calculated. Actually, the 1D velocity-depth functions were estimated in three different ways which assume: a constant velocitygradient model, a varying velocity-gradient model and a layered model. As regards the second objective of this work, a mean 1D velocity model for the Italian crust was defined and compared with those used for earthquake hypocentre locations and seismic tomographic studies by different institutions operating in the Italian area, to assess the significance of the model obtained. This model can be used in future works as input for a next joint tomographic inversion of active and passive seismic data.

scholarly journals Travel times, apparent velocities and amplitudes of body waves

1968 ◽  
Vol 58 (1) ◽  
pp. 339-366
Author(s):  
Bruce R. Julian ◽  
Don L. Anderson
Keyword(s):  
Surface Wave ◽  
Travel Time ◽  
Velocity Gradient ◽  
Body Wave ◽  
Travel Times ◽  
S Waves ◽  
The Earth ◽  
Geometric Spreading ◽  
First Arrival ◽  
Wave Models

abstract Surface wave studies have shown that the transition region of the upper mantle, Bullen's Region C, is not spread uniformly over some 600 km but contains two relatively thin zones in which the velocity gradient is extremely high. In addition to these transition regions which start at depths near 350 and 650 km, there is another region of high velocity gradient which terminates the lowvelocity zone near 160 km. Theoretical body wave travel time and amplitude calculations for the surface wave model CIT11GB predict two prominent regions of triplication in the travel-time curves between about 15° and 40° for both P and S waves, with large amplitude later arrivals. These large later arivals provide an explanation for the scatter of travel time data in this region, as well as the varied interpretations of the “20° discontinuity.” Travel times, apparent velocities and amplitudes of P waves are calculated for the Earth models of Gutenberg, Lehmann, Jeffreys and Lukk and Nersesov. These quantities are calculated for both P and S waves for model CIT11GB. Although the first arrival travel times are similar for all the models except that of Lukk and Nersesov, the times of the later arrivals differ greatly. The neglect of later arrivals is one reason for the discrepancies among the body wave models and between the surface wave and body wave models. The amplitude calculations take into account both geometric spreading and anelasticity. Geometric spreading produces large variations in the amplitude with distance, and is an extremely sensitive function of the model parameters, providing a potentially powerful tool for studying details of the Earth's structure. The effect of attenuation on the amplitudes varies much less with distance than does the geometric spreading effect. Its main effect is to reduce the amplitude at higher frequencies, particularly for S waves, which may accunt for their observed low frequency character. Data along a profile to the northeast of the Nevada Test Site clearly show a later branch similar to the one predicted for model CIT11GB, beginning at about 12° with very large amplitudes and becoming a first arrival at about 18°. Strong later arrivals occur in the entire distance range of the data shown, 1112°. to 21°. Two models are presented which fit these data. They differ only slightly and confirm the existence of discontinuities near 400 and 600 kilometers. A method is described for predicting the effect on travel times of small changes in the Earth structure.

The Earth’s Correlation Wavefield: Proof of Concept, Origin and Applications

2020 ◽  
Author(s):  
Hrvoje Tkalčić ◽  
Sheng Wang ◽  
Thanh Son Pham
Keyword(s):  
Body Wave ◽  
The Body ◽  
Body Waves ◽  
Proof Of Concept ◽  
Arrival Times ◽  
Multiple Receivers ◽  
The Earth ◽  
Motion Time ◽  
Global Seismology ◽  
Waveform Distortions

<p>We have recently shown that all features in the earthquake-coda correlogram can be explained by the similarity of seismic phases that have a common slowness for the analysed receiver pair. This includes both the features that have their equivalents in the conventional traveltime stacks, but also those that were previously unexplained. Consequently, the information contained in the correlograms – cross-correlated ground-motion time-series in a two-dimensional representation – can be used to constrain Earth’s internal structure, however, that requires a proof of concept and further investigation into the origin of the correlation wavefield. We thus first decompose relevant correlogram features into discrete constituents with respect to their arrival times and we uniquely identify contributing seismic phases to each constituent. This confirms that the correlation wavefield does not arise due to the reconstruction of body waves between the two receivers (a.k.a. Green’s function) – instead, it is dominated by the interaction of various body waves, and its features are characterised by complex sensitivity kernels.</p><p>We demonstrate that the event locations relative to the receivers alter the similarities between the body waves, and may result in significant waveform distortions and inaccuracies in arrival-time predictions. We further show that the nature of source-mechanism and energy-release dynamics are the key influencers responsible for individual correlograms equal in quality to a stack of hundreds of correlograms. In other words, a single seismic event that meets a set of criteria in the presence of multiple receivers can completely `illuminate’ the Earth’s interior. Quantitative kernel-decomposition and identification of body-wave pairs that contribute to a given feature in the correlogram, along with informed choices of seismic events, thus makes the correlation-wavefield tomography and other applications fully feasible. This has the potential to change the course of global seismology in the coming decades.</p>

Tomography without rays

1993 ◽  
Vol 83 (2) ◽  
pp. 509-528 ◽  
Author(s):  
Charles J. Ammon ◽  
John E. Vidale
Keyword(s):  
Simulated Annealing ◽  
Travel Time ◽  
Velocity Structure ◽  
Random Noise ◽  
Time Inversion ◽  
Arrival Times ◽  
Inversion Technique ◽  
Travel Time Inversion ◽  
First Arrival ◽  
Inversion Procedure

Abstract We present two new techniques for the inversion of first-arrival times to estimate velocity structure. These travel-time inversion techniques are unique in that they do not require the calculation of ray paths. First-arrival times are calculated using a finite-difference scheme that iteratively solves the eikonal equations for the position of the wavefront. The first inversion technique is a direct extension of linearized waveform inversion schemes. The nonlinear relationship between the observed first-arrival times and the model slowness is linearized using a Taylor series expansion and a solution is found by iteration. For a series of two-dimensional numerical tests, with and without random noise, this travel-time inversion procedure accurately reconstructed the synthetic test models. This iterative inversion procedure converges quite rapidly and remains stable with further iteration. The second inversion technique is an application of simulated annealing to travel-time topography. The annealing algorithm is a randomized search through model space that can be shown to converge to a global minimum in well-posed problems. Our tests of simulated annealing travel-time topography indicate that, in the presence of less than ideal ray coverage, significant artifacts may be introduced into the solution. The linearized inversion scheme outperforms the nonlinear simulated annealing approach and is our choice for travel-time inversion problems. Both techniques are applicable to a variety of seismic problems including earthquake travel-time tomography, reflection, refraction/wide-angle reflection, borehole, and surface-wave phase-velocity tomography.

Neubestimmung der natürlichen irdischen Tritiumzerfallsrate und die Frage der Herkunft des „natürlichen“ Tritium

1959 ◽  
Vol 14 (4) ◽  
pp. 334-342 ◽  
Author(s):  
F. Begemann
Keyword(s):  
Phase Shift ◽  
Decay Rate ◽  
Production Rate ◽  
Residence Time ◽  
Cosmic Radiation ◽  
Sunspot Cycle ◽  
Energy Component ◽  
Tritium Content ◽  
The Earth ◽  
Snow Samples

The terrestrial decay rate of “natural” tritium has been re-determined from measurements of the tritium content of old snow samples from Greenland. The finding by CRAIG and BEGEMANN and LIBBY has been confirmed that the tritium decay rate is about 10 times higher than was anticipated previously.Two mechanisms to explain the discrepancy are discussed,a) production by the low energy component of the cosmic radiation andb) the accretion of solar tritium by the earth, as suggested by FELD and ARNOLD.It is shown that in case all the tritium is produced by cosmic radiation the tropospheric production rate may be expected to vary in antiphase with the sunspot cycle, whereas in case of accretion of solar tritium by the earth the variation should be in phase with the sunspot cycle. In both cases a phase shift between the stratospheric production rate and the amount of tropospheric tritium is to be expected because of the residence time of tritium in the stratosphere. A measurement of the phase shift should allow to determine this residence time.The data obtained on the Greenland samples appear to show such a variation of the production rate. The results can be explained best by assuming that all the tritium is produced by cosmic radiation. This result, however, is only preliminary. More systematic measurements are required to decide between the two possibilities.

Elastic displacements in the near-field of a propagating fault

1969 ◽  
Vol 59 (2) ◽  
pp. 865-908
Author(s):  
N. A. Haskell
Keyword(s):  
Fault Plane ◽  
Near Field ◽  
Strong Motion ◽  
Dislocation Motion ◽  
Displacement Discontinuity ◽  
Wave Form ◽  
Strong Motion Station ◽  
Ramp Function ◽  
Wave Forms ◽  
Parkfield Earthquake

abstract Displacement, particle velocity, and acceleration wave forms in the near field of a propagating fault have been computed by numerical integration of the Green's function integrals for an infinite medium. The displacement discontinuity (dislocation) on the fault plane is assumed to have the form of a unilaterally propagating finite ramp function in time. The calculated wave forms in the vicinity of the fault plane are quite similar to those observed at the strong motion station nearest the fault plane at the Parkfield earthquake. The comparison suggests that the propagating ramp time function is roughly representative of the main features of the dislocation motion on the fault plane, but that the actual motion has somewhat more high frequency complexity. Calculated amplitudes indicate that the average final dislocation on the fault at the Parkfield earthquake was more than an order of magnitude greater than the offsets observed on the visible surface trace. Computer generated wave form plots are presented for a variety of locations with respect to the fault plane and for two different assumptions on the relation between fault length and ramp function duration.

Some numerical solutions for Lamb's problem

1974 ◽  
Vol 64 (2) ◽  
pp. 473-491
Author(s):  
Harold M. Mooney
Keyword(s):  
Rayleigh Wave ◽  
Poisson’S Ratio ◽  
Pulse Width ◽  
Numerical Solutions ◽  
Pulse Height ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Form ◽  
Lamb’S Problem ◽  
Wave Forms

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

scholarly journals The wave form of atmospherics at night

1940 ◽  
Vol 176 (965) ◽  
pp. 180-202 ◽  
Keyword(s):  
Wave Form ◽  
Time Interval ◽  
Time Separation ◽  
The Earth ◽  
Lightning Channel ◽  
Reflexion Coefficient ◽  
Primary Pulse

The wave form of all atmospherics received at night from sources within 2000 km. can be accurately described as a ground pulse followed by a series of sky pulses produced by successive reflexions between the ionosphere and the earth, thirty such reflexions being frequently recorded. The time separation between the peaks of these pulses is determined by the distance travelled and the height of the layer. The primary pulse emitted by the source is usually a single complete oscillation of period ranging from 50 to 400//sec. A t distances greater than 500 km. the ground pulse and the first sky pulse merge owing to the shortness of the time interval between them . Differences of amplitude, form and phase between pulses can arise from differences in angle of emission from the parent lightning channel. The height of the reflecting layer can be determined within ± 1 km. It ranged from 85-5 to 90-5 km. during two winter months, with a mean of 88-0 km. The distances of the sources as found by analysis of the pulse series were corroborated by independent location with cathode-ray direction-finders. The reflexion coefficient of the layer for the pulses of longer period exceeded 0-80. The velocity of the ground pulse where it can be tested is within 0.7 % of that of light.

scholarly journals Sensor-based localization of epidemic sources on human mobility networks

2021 ◽  
Vol 17 (1) ◽  
pp. e1008545
Author(s):  
Jun Li ◽  
Juliane Manitz ◽  
Enrico Bertuzzo ◽  
Eric D. Kolaczyk
Keyword(s):  
Mixture Model ◽  
Human Mobility ◽  
Gaussian Mixture ◽  
Local Conditions ◽  
Arrival Times ◽  
Modes Of Transmission ◽  
Kwazulu Natal ◽  
First Arrival ◽  
Control Of Epidemics ◽  
The Impact

We investigate the source detection problem in epidemiology, which is one of the most important issues for control of epidemics. Mathematically, we reformulate the problem as one of identifying the relevant component in a multivariate Gaussian mixture model. Focusing on the study of cholera and diseases with similar modes of transmission, we calibrate the parameters of our mixture model using human mobility networks within a stochastic, spatially explicit epidemiological model for waterborne disease. Furthermore, we adopt a Bayesian perspective, so that prior information on source location can be incorporated (e.g., reflecting the impact of local conditions). Posterior-based inference is performed, which permits estimates in the form of either individual locations or regions. Importantly, our estimator only requires first-arrival times of the epidemic by putative observers, typically located only at a small proportion of nodes. The proposed method is demonstrated within the context of the 2000-2002 cholera outbreak in the KwaZulu-Natal province of South Africa.

Sign in / Sign up

Export Citation Format

Share Document