Seismic Rays

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Inverse Problem ◽  
Seismic Tomography ◽  
Seismic Waves ◽  
Seismic Station ◽  
Three Dimensional ◽  
Arrival Times ◽  
The Earth ◽  
Straight Lines ◽  
Ray Path ◽  
Uniform Composition

This chapter focuses on the underlying mathematics of seismic rays. Seismic waves caused by earthquakes and explosions are used in seismic tomography to create computer-generated, three-dimensional images of Earth's interior. If the Earth had a uniform composition and density, seismic rays would travel in straight lines. However, it is broadly layered, causing seismic rays to be refracted and reflected across boundaries. In order to calculate the speed along the wave's ray path, the time it takes for a seismic wave to arrive at a seismic station from an earthquake needs to be determined. Arrival times of different seismic waves allow scientists to define slower or faster regions deep in the Earth. The chapter first presents the relevant equations for seismic rays before discussing how rays are propagated in a spherical Earth. The Wiechert-Herglotz inverse problem is considered, along with the properties of X in a horizontally stratified Earth.

Linearization of the eikonal equation

Geophysics ◽  
1994 ◽  
Vol 59 (10) ◽  
pp. 1631-1632 ◽  
Author(s):  
David F. Aldridge
Keyword(s):  
Inverse Problem ◽  
Perturbation Theory ◽  
Path Integral ◽  
Seismic Waves ◽  
Eikonal Equation ◽  
Nonlinear Inverse Problem ◽  
Nonlinear Functional ◽  
Arrival Times ◽  
Traveltime Tomography ◽  
Fermat's Principle

Seismic traveltime tomography is a nonlinear inverse problem wherein an unknown slowness model is inferred from the observed arrival times of seismic waves. Nonlinearity arises because the raypath connecting a given source and receiver depends on the slowness. Specifically, if L(s) designates a raypath through the slowness model s between two fixed endpoints, then the path integral for traveltime [Formula: see text] is a nonlinear functional of s because it does not, in general, satisfy the superposition condition (i.e., [Formula: see text] where [Formula: see text] and [Formula: see text] are two different slowness models). The tomographic inverse problem can be solved after linearizing the traveltime expression about a known slowness model [Formula: see text]. This linearized expression is usually obtained by appealing to Fermat’s principle (e.g., Nolet, 1987). Alternately, the required relation can be rigorously derived via ray‐perturbation theory (Snieder and Sambridge, 1992). The purpose of this note is to present a straightforward derivation of the same result by linearizing the eikonal equation for traveltimes. Wenzel (1988) adopts this approach, but his method of proof cannot be generalized to heterogeneous 3-D media. A full 3-D treatment is given here. The proof is remarkably simple, and thus it is quite possible that others have discovered it previously.

2-D random media with ellipsoidal autocorrelation functions

Geophysics ◽  
1993 ◽  
Vol 58 (9) ◽  
pp. 1359-1372 ◽  
Author(s):  
L. T. Ikelle ◽  
S. K. Yung ◽  
F. Daube
Keyword(s):  
Aspect Ratio ◽  
Seismic Data ◽  
Random Media ◽  
Large Scale ◽  
Seismic Waves ◽  
Homogeneous Medium ◽  
Small Scale ◽  
Arrival Times ◽  
The Earth ◽  
Pulse Arrival

The integration of surface seismic data with borehole seismic data and well‐log data requires a model of the earth which can explain all these measurements. We have chosen a model that consists of large and small scale inhomogeneities: the large scale inhomogeneities are the mean characteristics of the earth while the small scale inhomogeneities are fluctuations from these mean values. In this paper, we consider a two‐dimensional (2-D) model where the large scale inhomogeneities are represented by a homogeneous medium and small scale inhomogeneities are randomly distributed inside the homogeneous medium. The random distribution is characterized by an ellipsoidal autocorrelation function in the medium properties. The ellipsoidal autocorrelation function allows the parameterization of small scale inhomogeneities by two independent autocorrelation lengths a and b in the horizontal and the vertical Cartesian directions, respectively. Thus we can describe media in which the inhomogeneities are isotropic (a = b), or elongated in a direction parallel to either of the two Cartesian directions (a > b, a < b), or even taken to infinite extent in either dimension (e.g., a = infinity, b = finite: a 1-D medium) by the appropriate choice of the autocorrelation lengths. We also examine the response of seismic waves to this form of inhomogeneity. To do this in an accurate way, we used the finite‐difference technique to simulate seismic waves. Special care is taken to minimize errors due to grid dispersion and grid anisotropy. The source‐receiver configuration consists of receivers distributed along a quarter of a circle centered at the source point, so that the angle between the source‐receiver direction and the vertical Cartesian direction varies from 0 to 90 degrees. Pulse broadening, coda, and anisotropy (transverse isotropy) due to small scale inhomogeneities are clearly apparent in the synthetic seismograms. These properties can be recast as functions of the aspect ratio [Formula: see text] of the medium, especially the anisotropy and coda. For media with zero aspect ratio (1-D media), the coda energy is dominant at large angles. The coda energy gradually becomes uniformly distributed with respect to angle as the aspect ratio increases to unity. Our numerical results also suggest that, for small values of aspect ratio, the anisotropic behavior (i.e., the variations of pulse arrival times with angle) of the 2-D random media is similar to that of a 1-D random medium. The arrival times agree with the effective medium theory. As the aspect ratio increases to unity, the variations of pulse arrival times with angle gradually become isotropic. To retain the anisotropic behavior beyond the geometrical critical angle, we have used a low‐frequency pulse with a nonzero dc component.

Array studies of seismic waves arriving between P and PP in the distance range 90° to 115°

1972 ◽  
Vol 62 (1) ◽  
pp. 385-400 ◽  
Author(s):  
C. Wright
Keyword(s):  
Local Structure ◽  
Seismic Waves ◽  
Statistical Study ◽  
Detailed Examination ◽  
Coherent Signals ◽  
Time Curve ◽  
Arrival Times ◽  
Asymmetric Reflection ◽  
The Earth ◽  
Lateral Variations

Abstract Arrival times, slownesses and azimuths for coherent signals arriving between P and 60 secafter PP have been measured for 12 earthquakes recorded at the Yellowknife array at distances between 90° and 115°. The slowness (dT/dΔ) and azimuth values for P indicate that corrections to dT/dΔ for local structure beneath the array are small and can be neglected. A statistical study of arrivals from 10 events at distances less than 103° did not demonstrate conclusively the existence of PdP waves, and revealed a pattern of slowness values for precursors to PP similar to that observed by Wright and Muirhead (1969) at a distance of 106.0°. Further, a more detailed examination of three events at distances of 93.1°, 105.5° and 114.5° showed the presence of precursors with slowness values of about 10 sec/deg. These results required the development of an asymmetric reflection hypothesis in which the large amplitudes of these waves are produced by cusps in the travel-time curve near 20° and lateral variations in the uppermost regions of the Earth.

Core Phases Observed with AlpArray

2020 ◽  
Author(s):  
On Ki Angel Ling ◽  
Simon Stähler ◽  
Domenico Giardini ◽  
Kasra Hosseini ◽  
The AlpArray Working Group
Keyword(s):  
Large Scale ◽  
Seismic Waves ◽  
Incidence Angle ◽  
Body Wave ◽  
Sampling Rate ◽  
European Alps ◽  
S Wave ◽  
Arrival Times ◽  
The Core ◽  
Ray Path

&lt;p&gt;In most seismic tomographic models, the first P and/or S wave data generated by regional and teleseismic events are used to conduct tomographic inversion. Despite the abundance and precise measurement of the first body wave arrival times, the non-uniform distribution of their ray path leads to a lower resolution in the mantle below 1000km in depth. Curiously, there are particularly few ray paths sampling the lowermost mantle below dense seismic arrays, due to the limited incidence angle range of P and S waves. Previous studies have demonstrated the importance of core phases, resulting from reflection and/or conversion of seismic waves at the core discontinuities, in seismic tomography by improving the ray path coverage and constraining the structures in the lower mantle. Therefore, adding core-grazing phases (Pdiff, Sdiff) as well as core phases (e.g. PKP, PKIKP, SKS) in tomography could deliver high-resolution tomographic images of deep mantle structures in poorly resolved regions and may even reveal undiscovered features.&lt;/p&gt;&lt;p&gt;To increase the topographic resolution in the Alpine region, the AlpArray Initiative deployed about 250 temporary stations alongside the local permanent stations in the European Alps forming a greater AlpArray seismic network. This large-scale network provides a dense sampling rate and high-quality seismic data across the region, which gives us a unique opportunity to observe core phases coming from all directions in such a large aperture. We investigate the visibility of core phases observed with AlpArray and find that it is uniquely suited to observe high order core phases (P&amp;#8217;P&amp;#8217;, PcPPcPPKP, PKPPKPPKP) from sources in Alaska, Japan, and Sumatra in a distance range of 60-110 degrees. We show some array processing methods to improve the resolution of seismic observation and examine the waveforms in different frequency ranges. We find significant deviations in core phase amplitudes from predictions which are most likely linked to other structures directly above the core mantle boundary and can serve to test tomographic models in this depth region. The insight gained from this modelling is used to discuss the usability of core phases in future tomographic studies.&lt;/p&gt;

Isolation Performance of Intelligent Seismic Isolation System Using Air Bearing

2009 ◽  
Author(s):  
Satoshi Fujita ◽  
Keisuke Minagawa ◽  
Mitsuru Miyazaki ◽  
Go Tanaka ◽  
Toshio Omi ◽  
...  
Keyword(s):  
Seismic Isolation ◽  
Seismic Waves ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Air Bearings ◽  
Natural Period ◽  
Isolation System ◽  
Vertical Excitation ◽  
Long Period ◽  
Isolation Performance

This paper describes three-dimensional isolation performance of seismic isolation system using air bearings. Long period seismic waves having predominant period of from a few seconds to a few ten seconds have recently been observed in various earthquakes. Also resonances of high-rise buildings and sloshing of petroleum tanks in consequence of long period seismic waves have been reported. Therefore the isolation systems having very long natural period or no natural period are required. In a previous paper [1], we proposed an isolation system having no natural period by using air bearings. Additionally we have already reported an introduction of the system, and have investigated horizontal motion during earthquake in the previous paper. It was confirmed by horizontal vibration experiment and simulation in the previous paper that the proposed system had good performance of isolation. However vertical motion should be investigated, because vertical motion varies horizontal frictional force. Therefore this paper describes investigation regarding vertical motion of the proposed system by experiment. At first, a vertical excitation test of the system is carried out so as to investigate vertical dynamic property. Then a three-dimensional vibration test using seismic waves is carried out so as to investigate performance of isolation against three-dimensional seismic waves.

SEISMIC RESPONSE OF POWER TRANSMISSION TOWER-LINE SYSTEM UNDER MULTI-COMPONENT MULTI-SUPPORT EXCITATIONS

2012 ◽  
Vol 06 (04) ◽  
pp. 1250025 ◽  
Author(s):  
TIAN LI ◽  
LI HONGNAN ◽  
LIU GUOHUAN
Keyword(s):  
Seismic Waves ◽  
Power Transmission ◽  
Incident Angle ◽  
Ground Motions ◽  
Three Dimensional ◽  
Transmission Tower ◽  
Line System ◽  
Time Stepping ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

The effect of multi-component multi-support excitations on the response of power transmission tower-line system is analyzed in this paper, using three-dimensional finite element time-stepping analysis of a transmission tower-line system based on an actual project. Multi-component multi-support earthquake input waves are generated based on the Code for Design of Seismic of Electrical Installations. Geometric non-linearity was considered in the analysis. An extensive parametric study was conducted to investigate the behavior of the transmission tower-line system under multi-component multi-support seismic excitations. The parameters include single-component multi-support ground motions, multi-component multi-support ground motions, the correlations among the three-component of multi-component multi-support ground motions, the spatial correlation of multi-component multi-support ground motions, the incident angle of multi-component multi-support seismic waves, the ratio of the peak values of the three-component of multi-component multi-support ground motions, and site condition with apparent wave velocity of multi-component multi-support ground motions.

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