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A Student’s Guide to Point Sources in Homogeneous Media
Abstract
The analysis of point sources in homogeneous media is presented in an introductory but rigorous manner. Stokes solution to the elastic wave equation with a concentrated body force is extended to include couples and moment tensor sources. The single couple and the double couple without moment are discussed in detail because of their importance in earthquake source studies. To simplify the treatment, the transformation properties of tensors under rotations are used throughout. This guide is intended to complement that written by Jost and Herrmann (1989) on moment tensors. A FORTRAN 77 program is provided to plot P, SV, SH radiation patterns as well as S-wave polarizations on a lower hemisphere focal sphere for an arbitrary symmetric moment tensor.
... The point source, defined as a body force f acting at a point, has been adapted as an AE excitation source in seismic displacements, crack formation and fracture, and concentrated vertical step force [12][13][14][15][16]. In the elastic field, the point source has been treated as a concentrated time harmonic source [17]. ...
... In the elastic field, the point source has been treated as a concentrated time harmonic source [17]. Although the AE generated by the point source is important for characterizing real signals observed in practical NDT investigations, theoretical modeling has been limited to spherical geometries with an infinite domain, given that a homogeneous solution is obtainable from the three-dimensional wave equation in spherical coordinates [12,13,17]. Most theoretical works on elastic wave propagation in cylindrical coordinates have focused on situations with or without external perturbations [18][19][20][21][22]. ...
... From Equations (13) and (15), ...
In this paper, the displacement fields responsible for acoustic emission (AE), excited from a point source in a transversely isotropic cylinder, are derived by solving the Navier-Lamé (NL) equation. The point source as an internal defect is represented by a spatiotemporal concentrated force. The introduction of three potentials correlated with the point source to displacement field vector decouples the coupled NL equation in cylindrical coordinates. Under these conditions, we solve the radial, tangential, and axial displacement fields. Analytical simulations of AE were carried out at several point source locations. Our results demonstrate that analytical modeling is a powerful tool for characterizing AE features generated from an internal defect source.
... This derivative offers a practical and general way to obtain the displacement field due to a couple, but one can actually start with the two displacements indicated in (2), introduce a number of approximations and then take the limit. This approach was used by Lay and Wallace (1995) to determine the far field, but it is much simpler to use the derivative (see Pujol and Herrmann, 1990). ...
... This source does not generate SH waves. For the equations used to generate the plots see also Pujol and Herrmann (1990). Fig. 3. P , S, SV and SH wave radiation patterns for the double couple obtained by superposition of the two single couples of Fig. 1. ...
... From Pujol (2003). For the equations used to generate the plots see also Pujol and Herrmann (1990). The only difference between this P radiation pattern and that in Fig. 2 is a factor of 2. Fig. 4. Geometry for Gauss' theorem in the presence of a surface of discontinuity (Σ). ...
As noted in the Introduction to the classic paper by Burridge and Knopoff (1964) (this issue), this paper played a critical role in the development of the theory of the earthquake source. Because the mathematical development presupposes a level of knowledge beyond that of the seismologist or seismology student not actively engaged in theoretical studies, I will discuss a number of basic results that will help understand the paper as well as to put it into a broader perspective. I hope this tutorial will entice those untheoretically oriented to read the Burridge and Knopoff paper, so that they can appreciate the essential elegance (as opposed to a brute-force approach) of the arguments used to derive the expression for the body force equivalent. To make the tutorial self-contained a few excerpts from the Introduction are repeated here.
For a homogeneous isotropic elastic medium the equation that governs the propagation of seismic waves can be written as \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ {\rho}{\ddot{u}}(\mathbf{x},t)=({\lambda}+2{\mu})\mathrm{grad\ div}{\ }\mathbf{u}(\mathbf{x},t)-{\mu}{\ }\mathrm{curl\ curl}{\ }\mathbf{u}(\mathbf{x},t)+\mathbf{f}(\mathbf{x},t)\] \end{document}(1) where u is the vector that describes the displacement of the particles in the medium, x = ( x 1, x 2, x 3) denotes the observation point, such as a seismic station, the double dots indicate second derivative with respect to time, λ and μ are the Lame's parameters, ρ is density, and f is the body force per unit volume that is the source of the waves ( e.g. , Aki and Richards, 1980).
Although solving Equation 1 for a given f is a difficult problem, solutions for simple forces have been known since 1849. The problem that seismologists faced when they began to study the generation of waves by an earthquake was to find the body force to be used in Equation 1. As noted below, the relation between faults and earthquakes was well established before a model for …
... The final slip in the fault plane has always been assumed to be the result of a seismic source without nets of force, torque, and volume change. The radiations of P and S waves for shear slip faults are concluded with remarkably straight-forward formulas (Aki and Richards 1980;Herrmann 1989;Pujol and Herrmann 1990). In addition to shear faulting, a possible expansive and/or compressive process in a direction normal to the fault plane during rupture has been reported and proposed. ...
... Such a linear dipole will contribute P waves carrying the same strength and traveling in two opposite directions with a velocity of a, whereas the shear couple contributes S waves with a velocity of b. The displacements of P and S waves are proportional to the time derivatives of moment functions (Aki and Richards 1980;Pujol and Herrmann 1990) of the linear dipole and the shear couple, both are with arm in the radiate direction at the source. ...
... where $ r, q, and j are three unit column vectors. In Cartesian coordinates of north, east, and down (15) (16) (17) u P , u S , u SV , and u SH are the same as those expressions derived by Pujol and Herrmann (1990) for a generous body force representing a seismic source as a linear combination of couples. These are also presented for an arbitrary orientation of a double couple (Aki and Richards 1980). ...
A shear slip fault, an equivalence of a double couple source, has often been assumed to be a kinematic source model in ground motion simulation. Estimation of seismic moment based on the shear slip model indicates the size of an earthquake. However, if the dislocation of the hanging wall relative to the footwall includes not only a shear slip tangent to the fault plane but also expansion and compression normal to the fault plane, the radiating seismic waves will feature differences from those out of the shear slip fault. Taking account of the effects resulting from expansion and compression to a fault plane, we can resolve the tension and pressure axes as well as the fault plane solution more exactly from ground motions than previously, and can evaluate how far a fault zone opens or contracts during a developing rupture. In addition to a tensile angle and Poisson's ratio for the medium, a tensile fault with five degrees of freedom has been extended from the shear slip fault with only three degrees of freedom, strike, dip, and slip.
... where m is the moment vector composed of six components of moment tensor M, G p3 , m is the spatial derivative oG p3 =ox m and G p3 is the vertical component of the amplitude at the receiver measured along the x m -axis produced by the point force at the source directed along the x p -axis. For surface monitoring (Fig. 1), we get the following form of components G k (Pettitt, 1998;Pujol & Herrmann, 1990): ...
We study the sensor layouts of surface monitoring systems for injection areas in hydraulic fracturing experiments, which maximize the accuracy of retrieved moment tensors. The moment tensors (MTs) are determined using noisy synthetic P-wave amplitudes of microearthquakes with varying mechanisms. An optimum sensor layout is searched by minimizing a difference between the true and inverted MTs. First, we tested the efficiency of the circular networks. We focused on one-circle and two-circle layouts of various sizes and numbers of sensors on each circle and with their centre in the epicentral region. The results show that the optimal layout for events distributed within a circular epicentral area is characterized by: (1) one sensor fixed in the centre of the area; (2) about 1/10 of the total number of sensors gathered near the centre; and (3) the rest of sensors evenly distributed on a circle surrounding the area with the take-off angle of 135°. In homogeneous media, this angle corresponds to the radius of the circle comparable with depth of microearthquakes. Second, we compared the efficiency of the circular layout with other layouts: regular-grid, star-shaped and uniform focal-sphere layouts. Taking 25 sensors and 81 sensors as examples, we show that the two-circle layout is always the best, which means that we can achieve more accurate MTs using the circular layout than with other layouts, if the number of sensors in the experiment is fixed. This contradicts a common opinion that networks, which cover the target area, work equally well irrespective of their layout.
... To infer the conditions responsible for generating seismic waves, a formal mathematical description of the source is needed. Some of the first earthquake models treated a slipping fault as a point in space (point-source models; Nakano, 1923;Pujol and Herrmann, 1990); provided that observations are made at hypocentral distances much larger than the fault dimensions, this approximation has remained valid. Pointsource models show the equivalency between shear fracture dislocation (i.e., across a fault plane) and the double couple body forces that exist on either side of the dislocation surface (Burridge and Knopoff, 1964). ...
Dynamic rupture models are physics-based simulations that couple fracture mechanics to wave propagation and are used to explain specific earthquake observations or to generate a suite of predictions to understand the influence of frictional, geometrical, stress, and material parameters. These simulations can model single earthquakes or multiple earthquake cycles. The objective of this article is to provide a self-contained and practical guide for students starting in the field of earthquake dynamics. Senior researchers who are interested in learning the first-order constraints and general approaches to dynamic rupture problems will also benefit. We believe this guide is timely given the recent growth of computational resources and the range of sophisticated modeling software that are now available. We start with a succinct discussion of the essential physics of earthquake rupture propagation and walk the reader through the main concepts in dynamic rupture model design. We briefly touch on fully dynamic earthquake cycle models but leave the details of this topic for other publications. We also highlight examples throughout that demonstrate the use of dynamic rupture models to investigate various aspects of the faulting process.
... In this work we will use three kinds of sources [11,5] a. Dipole without a moment. This type of source is equivalent to fault shift. ...
A lot of seismic phenomena with small magnitude has been taking place in the Ladoga lake region. One of them is the earthquake which was recorded on the 31$^\text{th}$ of July, 2010 followed by an earthquake swarm near Nikonovsky cape. There is only one station in the region that is why it is difficult to process the data. In this work we propose a new method for obtaining main event focal mechanism and constructing the synthetic seismograms for this event. The data of swarm events is used to test the method of constructing synthetic seismograms and the data of close events to determine the parameters of the media. Since the amplitude of the wave P is close to zero, the location of the nodal plane can be submitted. That is the reason why we can find the focal mechanism for the earthquake using data of the single seismostation. %use data of the single seismostation \textcolor[rgb]{1.00,0.00,1.00}{to work out} the focal mechanism for the earthquake. Resulting fault plane has the same inclination as known tectonics line which the earthquake source is placed on. Our proposed method is based on Green function method with using several kinds of sources and temporal $\delta$-shaped sequences.
... Given that the fracture surface is parallel to the plane, x 1 Ox 2 , of the coordinate system, and the crack tip reaches the origin of the coordinate system, O, the observation point A can be shown in figure 4. In order to study the near-field seismic displacements at point A, three mutually perpend icular unit vectors-R, Φ, and Θ-are introduced, and among them, vector R is oriented from point O towards point A, vector Φ is the unit vector tangential to the latitudinal line at point A, and vector Θ is the unit vector tangential to the longitudinal line at point A. Their tensor expressions are given in equation (13) (Pujol and Herrmann 1990). Based on the relationship between the vibrating direction at point A and the propagating direction of the seismic wave, vector R is parallel to the propagating direction of the seismic wave, while vectors Φ and Θ are perpend icular to the propagating direction of the seismic wave, and Φ always keeps its horizontal direction and Θ its vertical direction. ...
To study the near-field seismic impact of coal fractures in stress concentration zones, we established a source generation model based on finite dislocation source theory and dynamic fracture mechanics, derived an analytical expression for near-field seismic displacements caused by coal fractures in the zone and numerically computed the resultant near-field seismic displacements within the coal mass. The results show that (1) the larger difference between the vertical and horizontal normal stresses in the stress concentration zone leads to a greater fracture speed, which thereby causes a stronger seismic impact; (2) the P-wave component in the near-field seismic displacements mainly impacts on the middle of the roadway, while the SH- and SV wave components mainly affect the junctions between the roadway and both the roof and the floor, and the damage caused by the SH- and SV waves within the coal mass is more significant than that caused by the P-waves; and (3) the effective way to mitigate the seismic impact induced by coal fractures in stress concentration zones is to reduce the difference between the vertical and horizontal normal stresses as far as possible. It is hoped that this study will provide a better understanding of the seismic impacts induced by coal fractures in stress concentration zones and thus help engineers to discover ways to prevent roadway failure.
... The single couple is co-planar to the double couple but is oriented in the opposite direction with a moment that is mutually orthogonal to the force direction. In the same medium, a pair of opposite forces is considered in the adjacent two portions under the transient role condition, the combination of their forces is zero, and they have the same force magnitude and direction, as shown in Figures 3 and 4. The total displacement is the sum of the displacement produced by each force (Pujol and Herrmann, 1990): ...
Microseismic monitoring is an effective means of forecasting instabilities in ruptured rock masses during deep mining. Based on the complex geological conditions of a mine in Southwest China, a digital 24-channel microseismic monitoring system was established to monitor microseismic events during deep mining. In the monitoring period, the focal mechanism solutions of roof collapse accidents were analysed using a double couple model. The results show that the focal mechanism solutions of the double couple model can be used to explain roof ruptures and instabilities. Two seismic source events were located at the F4 active fault, and the other seismic source event was located at the rock stratums demarcation line between little eighth orebody and C1b stratum. It was concluded that the active microseismic events were caused by deep mining. The method can be used to forecast roof collapse.
... ( Pujol and Herrmann, 1990) where u k is the displacement field recorded at sensor k, G ki,j are the Green's functions describing the wave propagation between source and receiver (the comma indicates the partial derivative at the source with respect to the coordinates after the comma), and M ij is the moment tensor. The summation convention applies. ...
A moment tensor inversion (MTI) code originally developed to compute source mechanisms from mining-induced seismicity data is now being used in the laboratory in a civil engineering research environment. Quantitative seismology methods designed for geological environments are being tested with the aim of developing techniques to assess and monitor fracture processes in structural concrete members such as bridge girders. In this paper, we highlight aspects of the MTI_Toolbox program that make it applicable to performing inversions on acoustic emission (AE) data recorded by networks of uniaxial sensors. The influence of the configuration of a seismic network on the conditioning of the least-squares system and subsequent moment tensor results for a real, 3-D network are compared to a hypothetical 2-D version of the same network. This comparative analysis is undertaken for different cases: for networks consisting entirely of triaxial or uniaxial sensors; for both P and S-waves, and for P-waves only. The aim is to guide the optimal design of sensor configurations where only uniaxial sensors can be installed. Finally, the findings of recent laboratory experiments where the MTI_Toolbox has been applied to a concrete beam test are presented and discussed.
... [16] We represent the geometry of an earthquake source mechanism with a unit potency tensorP ij [e.g., Ben-Zion, 2008]. The far-field P wave radiation pattern of a general P ij can be written [e.g., Pujol and Herrmann, 1990;Aki and Richards, 2002] as ...
[1] We analyze spatio-temporal patterns in rotation angles of double-couple–constrained mechanisms of aftershocks of the 1992 Landers earthquake. The rotation angles provide information on the distribution of source geometries in different regions of space and time with respect to the mainshock focal mechanism. The results indicate that the mechanisms of the early aftershocks are more scattered and less aligned with the mainshock than those of the long-term events. This is most pronounced around the northern end of the Landers rupture, least pronounced around the central section, and intermediate around the southern end of the rupture. The relatively large scatter and misalignment of the mean rotation angles of the early focal mechanisms around the edges of the Landers rupture suggest possible volumetric earthquake strain in these regions. The results may reflect isotropic source terms produced by dynamic generation of rock damage. Synthetic tests indicate that the observed differences in the rotation distributions of the early and long-term events around the end regions of the Landers rupture can result from neglecting in the inversion process isotropic components that are 0.03–0.15 of the total event moments.
... Additionally, a simple method is needed to compare focal mechanism solutions obtained by different methods. For accessible discussion of earthquake focal mechanisms see Jost & Herrmann (1989) and Pujol & Herrmann (1990). Methods for determining focal mechanism are discussed by Ekström et al. (2005, and references therein) and by Snoke (2003). ...
We derive new, simplified formulae for evaluating the 3-D angle of earthquake double-couple (DC) rotation. The complexity of the derived equations depends on both accuracy requirements for angle evaluation and the completeness of desired solutions. The solutions are simpler than my previously proposed algorithm based on the quaternion representation designed in 1991. We discuss advantages and disadvantages of both approaches. These new expressions can be written in a few lines of computer code and used to compare both DC solutions obtained by different methods and variations of earthquake focal mechanisms in space and time.
We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type.
Dynamic rupture models are physics-based simulations that couple fracture mechanics to wave propagation and are used to explain specific earthquake observations or to generate a suite of predictions to understand the influence of frictional, geometrical, stress and material parameters. These simulations can model single earthquakes or multiple earthquake cycles. The objective of this paper is to provide a self-contained and practical guide for students starting in the field of earthquake dynamics. Senior researchers who are interested in learning the first order constraints and general approaches to dynamic rupture problems will also benefit. We believe this guide is timely given the recent growth of computational resources and the range of sophisticated modeling software that are now available. We start with a succinct discussion of the essential physics of earthquake rupture propagation and walk the reader through critical concepts in dynamic rupture model design. We briefly touch on fully dynamic earthquake cycle models, but leave the details of this topic for other publications. We also highlight examples throughout that demonstrate the use of dynamic rupture models to investigate various aspects of the faulting process.
This work discusses the dynamic development of hydraulic fractures, their evolution and the resulting seismicity during fluid injection in a coupled numerical model. The model describes coupling between a solid that can fracture dynamically and a compressible fluid that can push back at the rock and open fractures. With a series of numerical simulations it is shown how the fracture pattern and seismicity change depending on changes in depth, injection rate, Young’s modulus and breaking strength. Simulations indicate that the Young’s modulus has the largest influence on the fracture dynamics and also the related seismicity. Simulations of rocks with a Young’s modulus smaller than 10 GPa show dominant mode I failure and a growth of fracture aperture with a decrease in Young’s modulus. Simulations of rocks with a Young’s modulus higher than 10 GPa show fractures with a constant aperture and fracture growth that is mainly governed by a growth in crack length and an increasing amount of mode II failure. This change in fracture geometry evolution has an effect on the observed seismicity. Rocks with a Young’s modulus of 10 GPa have the smallest moment magnitude while both decrease and increase of Young’s modulus value contribute to a growth of the seismic moment magnitude. The signal is further altered by non-linear change in dip and tensile angle depending on the Young’s modulus value.
It is proposed that two distinct failure regimes are observed in the simulations. Below 10 GPa a fracture propagates through growth in aperture, this causes the fracture tip to be under constant extension. For rocks above 10 GPa, the aperture is small and the fracture is under compression. In this case fracture growth is driven by stress intensification at the crack tip, which causes fracture opening to have greater proportion of mode II compared to mode I. To suppliment the observations made from numerical simulations, laboratory experiments with air injection into vertically orientated Hele-Shaw cell were carried out. Strain analysis of the recorded experiments showed stress regimes that are very similar to the ones observed during numerical simulations with soft rocks. In both cases negative strain fields could be observed in front of the fracture tip. This indicates that fracture propagation for soft materials is driven by tensile failure and walls being pushed apart. Further analysis on fracture propagation mechanisms and solid media response were carried out.
These results are applicable to the prediction of fracture dynamics and seismicity during fluid injection, especially since we see a transition from one failure regime to another at around 10 GPa, a Young’s modulus that lies in the middle of possible values for natural shale rocks.
Quantitative characterization of fracture mechanisms in concrete is a challenging problem that has not been convincingly solved to date. Several monitoring approaches have been proposed to capture and characterize different fracture mechanisms and warn against potential sudden structural failure. However, due to the complex nature of concrete, often only qualitative approaches are applied, which are unable to accurately characterize fracture sources in concrete due to intricate and simultaneously occurring fracture processes and significant measurement noise. In this study, the authors further investigate the feasibility of employing a monitoring approach that combines the seismology-based method of moment tensor inversion (MTI) with the acoustic emission (AE) technique for the quantitative characterization of flexural and shear cracks in reinforced concrete beams. To evaluate this approach in a realistic large-scale setting, the authors conducted two laboratory experiments on two separate normal-weight reinforced concrete beams of the same overall dimensions: 305 mm×610 mm×4.88 m (12 in.×24 in.×16 ft). Such large-scale experiments are scarce and only very few studies have been conducted to date. One beam was designed to fail in flexure and the second one in shear. According to the proposed approach, AE signals recorded from concrete cracking were inverted using a MTI code that the authors modified to visualize stereographic projections appropriate for structural testing, in order to study the sources of fracture and infer their nature. Furthermore, the authors employed a new high-fidelity point-contact sensor that measures actual displacements, which is the input required for a MTI. The results show that the cracks produced during loading of the flexure beam are dominantly tensile (81%) while the cracks from the shear beam were dominated by shearing (73%). These findings demonstrate the potential of the MTI method for quantitative structural-health monitoring (SHM) of large-scale reinforced concrete beams.
In einem unter Last stehenden Bauteil bildet sich ein globales Spannungsfeld aus. Dieses ist u. a. von den Materialkennwerten, der Geometrie des Bauteils, dessen Auflagerbedingungen und der Art der Lasteinwirkung abhängig. Übersteigt das Spannungsniveau im Bauteil lokal die Festigkeit des Materials, kommt es zum Versagen. Es entsteht z. B. ein Riss. Dabei wird Energie freigesetzt, wovon ein Teil für die bleibende Veränderung des Materials und die Erzeugung von Wärme aufgewendet wird. Ein geringer Teil der Energie wird in Form eines seismischen oder akustischen Wellenfeldes freigesetzt [Lawn, 1993]. Bei Mikrorissen in spröden Werkstoffen oder Gefügeänderungen in Metallen spricht man dabei von "Schallemissionen".
Dank der Fortschritte auf den Gebieten der digitalen Messwerterfassung und -verarbeitung hat sich die Schallemissionsanalyse (SEA) zu einem leistungsfähigen Messverfahren entwickelt [Drouillard, 1996]. Vor allem die Möglichkeit digitalisierte transiente Signale von einer großen Zahl von Schallereignissen speichern zu können, erschloss der SEA eine Reihe von Anwendungen. Die SEA ist ein passives Verfahren. Sie erlaubt es, den Rissfortschritt auch in undurchsichtigen Materialien mit Hilfe weniger stationärer Sensoren zu untersuchen. Im Rahmen einer signalbasierten oder quantitativen SEA [Scruby, 1985, Sachse & Kim 1987] kann im Anschluss an die Datenerfassung eine umfassende Aufbereitung, Bearbeitung und Analyse der Ereignisse erfolgen. Auf der Basis transienter Signale, die die Ausbreitung des Wellenfeldes im Raum beschreiben, können z. B. Bruchherde lokalisiert und Bruchmechanismen an Hand des Modells des Momententensors analysiert werden [z. B. Ohtsu et al. 1991, Landis & Shah, 1993, Maji & Sahu 1994, Grosse, 1996, Zang et al. 1998, Köppel & Vogel 2000, Manthei et al. 2001, Finck et al. 2003].
Im Rahmen dieser Arbeit werden verschiedene Auswerteverfahren der SEA beschrieben und angewendet. Im theoretischen Teil werden gängige Bruchmodelle und die Annahme einer Punktquelle hinsichtlich der Abstrahlung seismischer Energie diskutiert. Einen Schwerpunkt bildet hierbei u. a. der Einflusses des seismischen Nahfeldes auf die SEA. Auf Basis dieser theoretischen Untersuchungen wurden in einer Reihe von Experimenten unterschiedliche Schädigungsmechanismen aktiviert und mit Hilfe von verschiedenen Methoden untersucht. Einen Schwerpunkt bilden dabei relative [Dahm, 1993] und hybride Verfahren [Andersen, 2001] zur Inversion auf den Momententensor.
This study used data from a three-month continuous reservoir monitoring experiment in Peace River, Alberta, Canada to measure spatial and temporal variations in near-surface S-wave velocity (VS) and attenuation (QS) in the weathering layer. The permanently buried sources generate a strong refracted S-wave that was recorded on buried 3C receivers. A method to perform receiver-side up-down separation and extract primary and ghost S-wavefields is presented. These wavefields are then used to measure near-surface VS and QS for the near-surface layer above the buried receivers, which is the top 12 m in this case. The measured V S values range between 180 and 220 m/s and QS values between 8 and 22. Maps of VS and QS show a robust correlation (low VS with low QS) and a clear spatial variation that can be associated with soil type. Both properties increase slowly over a three-month period in one section of the survey area while remaining constant in another. The cumulative increase in VS and QS is 10%.
This study presents the application of an automated moment tensor analysis procedure on microcracks which were generated during a triaxial compression test of a cylindrical rock salt specimen (diameter 150 mm, length 300 mm). The acoustic emission signals were detected in a frequency range between 20 kHz and 1 MHz using 12 acoustic emission sensors mounted on the surface of the specimen cylinder. The moment tensor analysis was applied to about 30,000 events, which were precisely lo- cated using at least 16 P- and S-wave arrival times. For more than 40 per cent of these events, i.e. approximately 12,500 events, stable moment tensor solutions could be evaluated using the first motion of the P-wave radiation patterns. Most of the evaluated events showed significant isotropic source components which is in good agreement with dilatation of rock during com- pressional loading. The majority of the events were caused by tensile opening which is lead- ing to dilatation of the rock. The tension (T) axes which are normal to the crack plane of these tensile microcracks, were predominantly orientated in radial direction of the cylindrical specimen, perpendicular to the maximum principal stress, which is in axial direction. The direction of the tensile opening calculated by the moment tensor evaluation coincides very well with the direction of the minimum principal stress. The applied collapsing method discovers cellular structures with a cell size in the range of a few centimetres. However, it seems that the events occur only in zones where the cell inter- faces are favourably orientated in the stress field. These events are attributed to cracking at grain interfaces which occurs in rock salt under very slow creep loading above the dilatancy boundary.
This study presents the application of an automated moment tensor analysis procedure on microcracks, which were generated during a triaxial compression test of a cylindrical rock salt specimen (diameter 150 mm, length 300 mm). The acoustic emission signals were detected in a frequency range between 20 kHz and I MHz using 12 acoustic emission sensors mounted on the surface of the specimen cylinder. The moment tensor analysis was applied to about 30,000 events, which were precisely located using at least 16 P- and S-wave arrival times. For more than 40% of these events, approximately 12,500 events, stable moment tensor solutions could be evaluated using the first motion of the P-wave radiation patterns. Most of the evaluated events showed significant isotropic source components, which is in good agreement with dilatation of the rock during compressional loading. The majority of the events were caused by tensile opening leading to dilatation of the rock. The tension (7) axes, which are normal to the crack plane of these tensile microcracks, were predominantly oriented radially in the cylindrical specimen, perpendicular to the maximum principal stress, which is oriented axially. The direction of the tensile opening calculated by the moment tensor evaluation coincides very well with the direction of the minimum principal stress. The applied collapsing method discovers cellular structures with a cell size in the range of a few centimetres. However, it seems that the events occur only in zones where the cell interfaces are favourably orientated in the stress field. These events are attributed to cracking at grain interfaces which occurs in rock salt under very slow creep loading above the dilatancy boundary.
Numerical modeling of seismic waves in transversely isotropic (TI) media is often restricted to special cases where the plane of isotropy coincides with a coordinate plane of the model medium. We remove this special limitation by developing a scheme in which symmetry axes of individual component TI media are oriented arbitrarily with respect to the coordinate axes of the composite model. In these general TI media, the elastic constants for each homogeneous anisotropic region are a 6 X 6 matrix of nonzero elements calculated by an arbitrary rotation. Then, 3-D modeling can readily deal with the coupling of the three components of wave motion. However, required computer memory and execution time may exceed practical limits. Therefore, we implement a finite-element modeling process for TI media in which elastic properties vary only in two dimensions but component media have planes of isotropy in arbitrary directions. We compute three components of particle motion since the latter are coupled together in these media. The computational load is about twice that of the special cases where the planes of isotropy coincide with the coordinate planes. Three-component synthetic profiles corresponding to two sample models clearly illustrate the behavior of seismic waves in anisotropic media, including shear-wave splitting and coupling between the in-line and cross-line motion.
This paper addresses two problems: the random rotation of double-couple (DC) earthquake sources and the display of earthquake focal mechanisms. We consider several equivalent representations for DC sources and their properties and provide mathematical expressions for their mutual transformation. Obviously, a 3-D rotation of any object is more intricate than a 2-D rotation. Any rotation of a DC source is further complicated by its symmetry properties. Applying statistical tests to a DC distribution often requires one to compare it to a completely (or uniform) random DC pattern. We review several methods for obtaining random distribution of DC orientation; some of these seemingly natural techniques yield an incorrect result.
The DC random rotation problem is closely connected to displays of focal mechanisms. In such displays, a strike or an azimuth of a focal mechanism can be neglected; hence, we are confronted with mapping a 2-D distribution onto a flat surface. We review different methods for such displays and discuss more specifically how to project a random focal mechanism distribution on a flat 2-D display with uniform probability density. Such displays can be used to analyse earthquake patterns statistically in various tectonic regions.
A splitting test was performed to generate a well-defined crack surface in a concrete cube. Using acoustic emission technique
the formation of micro cracks was observed and correlated with the stress distribution during the test. Subsequent to the
experiment the topography of the crack surface was evaluated, using a process of stepwise grinding and digitalization of photographs
of cross-sections of the cube. Inversion of the amplitudes of the spatial displacement distribution yields a system of equivalent
moments for selected acoustic events, the so-called moment tensor. Analysis of the moment tensor allows a fracture mechanics
based interpretation of failure and gives information about the stress field within the specimen. A comprehensive 3D visualization
combines all this information and allows for a broad understanding of the failure process.
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