Dieses Kapitel taucht in die komplexe Welt der Bodenbewegung ein und konzentriert sich auf seismische Wellen, vorübergehende Belastungen und Belastungen. Es beginnt mit der Erklärung der Beschaffenheit von Wellen, einschließlich ihrer Art, Eigenschaften und wie sie sich durch verschiedene Medien ausbreiten. Anschließend untersucht der Text die Dynamik der Bodenbewegung, einschließlich der Verstärkung seismischer Wellen an der Haut von Ausgrabungen und der Entwicklung von Bodenbewegungsgleichungen (GMPE) zur Beurteilung seismischer Gefahren. In diesem Kapitel werden auch die Auswirkungen von Bodenbewegungen auf unterirdische Bauwerke diskutiert, einschließlich des Schadenspotenzials und der Entwicklung seismischer Fragilitätskurven. Praktische Anwendungen wie Schadensaufnahme und kumulative CAD-Diagramme werden ebenfalls vorgestellt. Das Kapitel schließt mit einer Diskussion über die Wahrnehmbarkeit von Bodenbewegungen und ihre Auswirkungen auf den Bergbau. Dieser umfassende Überblick bietet wertvolle Einblicke in die Komplexität der Bodenbewegung und ihre Auswirkungen auf unterirdische Bauwerke und macht sie zu einer unverzichtbaren Ressource für Fachleute auf diesem Gebiet.
KI-Generiert
Diese Zusammenfassung des Fachinhalts wurde mit Hilfe von KI generiert.
Abstract
This chapter starts with the general description of transient strains and stresses and the dynamic stress concentration factors for a plane harmonic incident P- and S-wave hitting a tunnel. Then it gives a rudimentary explanation of the ground motion (GM) at seismic source, describes parameters that measure the intensity of GM, and presents a case study on amplification of GM at the skin of an excavation. Understanding, quantifying, and forecasting GM motion in the near and intermediate field of seismic radiation is the most important issue in mine induced seismicity since most damage caused by seismic events is observed in excavations very close to their sources. Since the maximum ground velocity at source is controlled by the strength of the rock mass, small and large events produce similar ground velocity at source, but large events affect a substantial volume of rock, and hence, the probability of hitting a vulnerable structure is considerably higher. In smaller mines, the strong ground velocities and displacements associated with larger events may affect the entire infrastructure.
The next section is dedicated to the ground motion prediction equation (GMPE) and its applications, e.g. plotting the expected GM at strategic locations in real time, seismic fragility curves that show the probability that tunnel support may be damaged under different seismic loads given its remaining deformation capacity, and the GM alert program, called GMAP. The next section describes the real-time version of GMAP, called GMAS, that does not require the GMPE and estimated source parameters to issue an alert.
The next section describes mapping seismic ground motion hazard, which in earthquake seismology is called the probabilistic seismic hazard assessment and its limitations, the major being the uncertainty in the spatial distribution of the expected seismicity. The last section, modelling seismic hazard, describes what is called the deterministic hazard, i.e. kinematic modelling of GM produced by potential seismic sources defined by their expected maximum potency or energy, placed at the most likely locations that can produce the highest intensity of GM motion at a given site. Examples of modelling GM produced by extended and complex sources are given.
6.1 Seismic Waves, Transient Strains, and Stresses
Introduction
A wave is a disturbance that transports energy from its source through the rock without the transport of matter. It is the energy of the wave, not the particles of the medium, that travels through the medium. Wave motion can be transient, periodic, or random. Transient motion is the response of the medium to a sudden pulse-like excitation. Periodic motion is repetitive, recurring in the same form at fixed time increments, e.g. harmonic motion. In random motion, the instantaneous amplitude can be predicted only on a probabilistic basis (e.g. random noise).
The maximum disturbance in each cycle is known as the amplitude of the wave and is determined by its source. The amplitude of a wave is equal to the maximum displacement, velocity, or acceleration of ground motion from the equilibrium position. The number of cycles that pass by a fixed point per second is called frequency, f, and is expressed in hertz [Hz]. The frequency of a wave is determined by its source. The time in seconds required for a complete cycle to be produced or to pass a given point is the period of the wave, T, and it is the reciprocal of its frequency, \(T=1/f\)\(\mbox{or}\)\(f=1/T\). The length corresponding to one complete cycle of the wave is called the wavelength, \(\Lambda \). It can be measured from crest to crest, trough to trough, or between corresponding points on adjacent pulses. Wavelength is affected by both the source and the medium.
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Phase velocity quantifies how fast crests or troughs travel. The wavefront associated with any particular phase advances a distance \(\Lambda \) in time T, and therefore, the phase velocity \(v_{\pi }=\Lambda /T\)\(=\)\(\omega /k\), where \(k=2\pi /\Lambda \) is the wavenumber and \(\omega =2\pi f\) is the circular frequency. Note that for a given frequency, the wavelength increases with increasing wave propagation velocity, \(\Lambda \)\(=\)\(v_{\pi }/f\). In perfectly elastic homogeneous media, all frequencies travel with the same velocity. Attenuating media are dispersive and allow waves of different frequencies to travel at different velocities.
Wave Speed
The speed of any wave depends upon the properties of the medium through which the wave is travelling. Typically, there are two essential types of properties which affect wave speed: inertial and elastic properties. The greater the inertia, or the mass density, of individual particles of the medium, the less responsive they will be to the interactions between neighbouring particles and the slower the wave. If all other factors would be equal, a sound wave would travel faster in a less dense material than in a more dense material. Elastic properties relate to the strength of interaction between particles measured by the tendency of a material to resist deformation upon applied force, which is called stiffness. The stiffer the material the faster the wave. Even though the inertial factor may favour gases, the elastic factor has a greater influence on the wave speed, and therefore: \(v_{solids}\)\(>\)\(v_{liquids}>v_{gases}\). In general, the velocity of a seismic wave increases with rock mass stiffness and decreases with its density, \(v\propto \sqrt {\mbox{stiffness}/\mbox{density}}\).
P-Wave
The dilatational or longitudinal or primary waves, where the elastic medium expands and contracts and in which particles move in the direction of propagation. The wave velocity is
where \(\kappa \) is the bulk modulus, \(\mu \) the shear modulus, \(\rho \) is density, Y is the Young modulus, and \(\nu \) the Poisson ratio. Note the dependence of \(v_{P}\) on both, the bulk and the shear modulus. The reason is that during propagation of dilatation the medium is subjected to a combination of compression and shear. The cross-sectional area of a small cube element normal to the direction of propagation will not be changed, but its dimension along the propagation will be altered. There is thus a change in the shape of the element as well as in its volume, and the resistance of the medium to shear as well as its compressibility comes into play (Kolsky, 1963).
S-Wave
The transverse or shear or secondary waves, where the medium changes in shape, but not in volume. Particles move perpendicular to the direction of propagation, which occurs with velocity
Given a reference plane, the S-wave particle motion can arbitrarily be decomposed into horizontal SH and vertical SV components which, in homogeneous isotropic media, travel with the same speed. The polarisation, i.e. the direction of particle motion relative to the direction of wave propagation, of both P- and S-waves, is linear. In an anisotropic medium, where properties vary with direction, the S-wave splits into a fast and slow component. These split waves propagate with different velocities that cause a time delay and related phase shift. Accordingly, the two split S-wave components superimpose on an elliptical polarisation. The orientation of the main axis and the degree of ellipticity are controlled by the fast and slow velocity directions of the medium with respect to the direction of wave propagation and the degree of anisotropy. An independent propagation of the P- and S-waves is only guaranteed for sufficiently high frequencies where spatial variations in elastic properties occur over much larger distances than the wavelength of the waves involved. Fluids have no shear strength, \(\mu \)\(=\) 0, and thus do not propagate shear waves.
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Ratio \(v_{P}/v_{S}\)
The ratio of the P- to S-wave velocity depends only on the Poisson ratio \(\nu \), see Fig. 6.1, and since \(-1\)\(<\)\(\nu \)\(\leq \)\(0.5\), we have
Thus in a given medium, the P-wave is always faster than the S-wave. The ratio \(v_{P}/v_{S}\) increases with increasing \(\nu \), and for \(\nu \)\(=\)\(0.15\), the \(v_{P}/v_{S}\)\(=\)\(1.558\), for \(\nu \)\(=\)\(0.2\)\(v_{P}/v_{S}\)\(=\)\(1.633\). For a Poissonian solid, where the Poisson ratio \(\nu \)\(=\)\(0.25\), \(v_{P}/v_{S}\)\(=\)\(\sqrt {3}\) and \(v_{S}\)\(=v_{P}/\sqrt {3}\). For most consolidated rocks, the \(v_{P}/v_{S}\) ratio is between 1.5 and 2.0.
Since hydrostatic pressure cannot cause an increase in volume and the volume decrease must remain finite, the bulk modulus \(\kappa \) can only assume positive values. The shear caused by a state of simple shearing has the direction of the stress, and therefore, the shear modulus \(\mu \) must be positive. For an incompressible material, \(\kappa \), and hence, \(\lambda \) (Lamé constant) must become infinite. These conditions ensure that the strain energy is positive, and they impose the following fundamental inequalities for the elastic moduli of isotropic materials: \(\infty \geq \lambda >\frac {2}{3}\mu \), \(\infty >\mu >0\), \(0<Y\leq 3\mu \), \(-1<\nu \leq \frac {1}{2}\), where Y is the Young’s modulus (Table 6.1). Abstract distance between two elastic media can be measured by d\(=\)\(\sqrt {\left [\log \left (\kappa _{1}/\kappa _{2}\right )\right ]^{2}+\left [\log \left (\mu _{1}/\mu _{2}\right )\right ]^{2}}\).
Table 6.1
Relationship between elastic moduli
\(Y=\)
\(2\mu (1+\nu )\)
\(3\kappa (1-2\nu )\)
\(\frac {9\kappa \mu }{3\kappa +\mu }\)
\(\frac {\lambda (1+\nu )(1-2\nu )}{\nu }\)
\(\frac {\mu (3\lambda +2\nu )}{\lambda +\mu }\)
Flow chart depicting a process with interconnected nodes and arrows. The chart begins with a starting point, branching into multiple decision nodes, each leading to different outcomes. Key terms include "Start," "Decision," "Process," and "End." Arrows indicate the flow direction, guiding through various steps and decisions. The layout visually represents a sequence of actions and choices, illustrating a systematic approach to a specific task or problem.
\(\nu =\)
\(\frac {\lambda }{2(\lambda +\mu )}\)
\(\frac {Y}{2\mu }-1\)
\(\frac {3\kappa -2\mu }{2(3\kappa +\mu )}\)
\(\frac {\lambda }{3\kappa -\lambda }\)
\(\frac {3\kappa -Y}{6\kappa }\)
Flow chart depicting a process with interconnected nodes and arrows. The chart begins with a start node, followed by decision points and actions leading to various outcomes. Each node contains text describing specific steps or decisions. Arrows indicate the flow of the process, guiding the viewer through different paths based on conditions or results. The layout is organized to visually represent the sequence and relationship between each step.
\(\mu =\)
\(\frac {Y}{2(1+\nu )}\)
\(\frac {\lambda (1-2\nu )}{2\nu }\)
\(\frac {3(\kappa -\lambda )}{2}\)
\(\frac {3\kappa (1-2\nu )}{2+2\nu )}\)
\(\frac {3\kappa Y}{9\kappa -Y}\)
Flow chart illustrating a process with interconnected nodes and arrows. The chart begins with a central node labeled "Start" and branches into multiple paths, each leading to different outcomes. Key terms include "Decision," "Process," and "End." Arrows indicate the flow direction, guiding through various decision points and actions. The layout emphasizes the sequence and relationship between steps in the process.
\(\kappa =\)
\(\frac {Y}{3(1-2\nu )}\)
\(\frac {2\mu (1+\nu )}{3(1-2\nu )}\)
\(\frac {\mu Y}{3(3\mu -Y)}\)
\(\frac {\lambda (1+\nu )}{3\nu }\)
\(\lambda +\frac {2}{3}\mu \)
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\(\lambda =\)
\(\frac {2\mu \nu }{1-2\nu }\)
\(\frac {\mu (2\mu -Y)}{Y-3\mu }\)
\(\frac {\nu Y}{(1+\nu )(1-2\nu )}\)
\(\frac {3\kappa \nu }{1+\nu }\)
\(\kappa -\frac {2}{3}\mu \)
Flow chart illustrating a process with interconnected nodes and arrows. The chart begins with a central node labeled "Start" and branches into multiple paths, each leading to different decision points and outcomes. Key terms include "Decision," "Process," and "End." Arrows indicate the flow direction, guiding through various steps and choices. The layout emphasizes decision-making pathways and outcomes.
One can reduce the number of parameters to one under certain simplifying assumptions. The most frequent is the so-called Poisson’s relation, \(\lambda =\mu \), from which it follows: \(\nu =0.25\), Y\(=\)\(5\mu /2\), and \(\kappa \)\(=\)\(5\mu /3\). Another simplification is to assume that material is incompressible, and then \(\kappa \)\(=\)\(\lambda \)\(=\)\(\infty \), \(\nu \)\(=\)\(0.5\), and \(\mu \)\(=\)\(Y/3\).
Surface Waves
Seismic surface waves are an example of guided waves that propagate along free surfaces, internal discontinuities, or other waveguides. Seismic surface waves are created by the interference of seismic body waves. There are two major kinds of surface waves: Love waves, which are shear waves trapped near the surface, and Rayleigh waves, which have rock particle motions that are very similar to the motions of water particles in ocean waves.
Love waves are generated when an SH ray hits a reflecting horizon near surface at post-critical angle, and all the energy is trapped within the wave guide (Love, 1911). They propagate by multiple reflections between the top and bottom surfaces of the low speed layer near the surface. Love waves are similar to S-waves with no vertical displacement. They move the ground from side to side horizontally parallel to the Earth’s surface, at right angles to the direction of propagation, and produce horizontal shaking. The Love wave velocity is equal to that of shear waves in the upper layer for very short wavelengths, and to the velocity of shear waves in the lower layer for very long wavelengths.
Rayleigh waves, also known as a “ground roll”, are the result of incident P and SV plane waves interacting at the free surface and traveling parallel to that surface. The particles oscillate in a vertical plane along the direction of propagation. There are two components to their oscillations: vertical up and down motion and the horizontal forward and back motion. At the free surface, the initial vertical motion is up, but the initial horizontal movement is backward. Thus the particle motion at the free surface is elliptical retrograde (i.e. the particle moves opposite to the direction of propagation at the top of its elliptical path), and the vertical displacement is about 1.5 times the horizontal displacement. The penetration into the ground increases with wavelength \(\Lambda \). The vertical amplitude of particle motion decreases exponentially with depth. The horizontal amplitude becomes zero at a depth of \(0.19\cdot \Lambda \), and below that depth, it reverses its direction and the particle motion becomes forward elliptical. The Rayleigh wave velocity, \(v_{R}\), varies from \(0.9\,v_{S}\), for the media with Poisson ratio \(\nu \)\(=\)\(0.1\), to \(0.93\,v_{S}\), for \(\nu \)\(=\)\(0.4\). For a Poissonian solid, \(\nu \)\(=\)\(0.25\), Rayleigh waves travel with a phase velocity \(v_{R}\)\(\simeq \)\(0.92v_{S}\). In mines, surface waves are developed as the wavefront reaches the tabular excavations.
Transient Strains and Stresses
Any function \(f\left (x-v_{\pi }t\right )\) represents a wave propagating in the x direction, and the function \(u(x,t)\)\(=\)\(u_{0}\sin \left (2\pi fx/v_{\pi }-2\pi ft\right )\) represents a simple harmonic wave with an amplitude \(u_{0}\), frequency f propagating in the x direction with velocity \(v_{\pi }\). The ground (particle) velocity at x then is \(\text{v}\)\(=\)\(\partial u/\partial t\)\(=\)\(-2\pi fu_{0}\cos \left (2\pi ft-2\pi fx/v_{\pi }\right )\) and acceleration a\(=\)\(\partial \mbox{v}/\partial t\)\(=\left (2\pi f\right )^{2}u_{0}\sin \left (2\pi ft-2\pi fx/v_{\pi }\right )\). The ground velocity \(\mbox{v}\) is shifted in phase by \(\pi /2\) with respect to u and by \(\pi \) with respect to a. The dynamic transient strain at point x is \(\epsilon _{d}(x,t)\)\(=\)\(\partial u(x,t)/\partial x\)\(=\)\(2\pi f\left (u_{0}/v_{\pi }\right )\cos \left (2\pi ft-2\pi fx/v_{\pi }\right )\), and it shows that the particle velocity and the dynamic strain in a harmonic motion are related by \(\epsilon _{d}\)\(=\)\(\mbox{v}/v_{\pi }\). In summary, for harmonic motion,
Therefore, ground displacement of 1 mm at a frequency of 20 Hz would result in a ground velocity of 0.125 m/s and an acceleration of 15.8 m/s\(^{2}\), and the observed PGV of 10 mm/s associated with a wave propagating in the rock mass at 3300 m/s would cause 3\(\cdot \)10\(^{-6}\) strains, which is close to the upper limit for the elastic behaviour of rock. Larger ground velocities in this medium would result in inelastic deformation. If we assume that hard rock ruptures at shear strain \(\epsilon _{d}\)\(=\)\(5\cdot 10^{-4}\) and that \(v_{\pi }\)\(=\)\(v_{S}\)\(=\) 3250 m/s, then the maximum expected ground velocity at source, \(\mbox{v}\)\(=\)\(1.625\) m/s, see Sect. 6.3. For two waves of the same frequency and different displacement amplitudes, \(u_{1}/u_{2}\)\(=\)\(\mbox{v}_{1}/\mbox{v}_{2}\)\(=\)\(a_{1}/a_{2}\). These results are accurate for the harmonic pulse; otherwise, they are only approximations.
Assuming a linear elastic isotropic material in plane strain, the maximum normal stress is
where Y is the Young’s modulus, \(\nu \) is the Poisson ratio, \(\epsilon _{max}\) is the maximum axial strain, \(PGV_{P}\) is the peak ground velocity in the direction of P-wave propagation, and \(v_{P}\) is the apparent P-wave velocity. A similar expression can be derived for the shear stress due to the propagating S-wave, \(\tau _{max}\)\(=\)\(\pm \rho v_{S}\left |PGV_{S}\right |\), where \(v_{S}\) is the apparent S-wave velocity and \(PGV_{S}\) is the peak ground velocity in the direction of S-wave polarisation. These approximations are more accurate at lower frequencies and quite sensitive to the value of apparent propagation velocity which may be highly variable in the fractured rock surrounding excavations. Indeed, the selection of the appropriate apparent propagation velocity is not simple, and the associated uncertainty limits the accuracy of strain estimates from recorded data. At higher frequencies, there may be additional differential displacements and strains caused by spatial variability of ground motions, i.e. changes in the amplitudes and phases of the motions as well as arrival time perturbations of the waveforms at the various locations in the ground near the surface.
6.2 Static and Dynamic Stresses Around a Circular Tunnel
Underground structures, specifically those embedded in hard rock, are far more resilient to shaking than surface ones. Therefore, with the exception of large events, most damage caused by seismic events is observed in excavations very close to their sources. Since the maximum ground velocity at source is controlled by the strength of the rock mass, small and large events produce similar ground motion at source, but large events affect a substantial volume of rock, and hence, the probability of hitting a vulnerable structure is considerably higher. In smaller mines, the strong ground velocities associated with larger events may affect the entire infrastructure. The maximum ground motion that can be experienced at a given site is controlled by the maximum ground velocity at source, the interaction of radiation from different parts of the source and from different travel paths, and by site effects.
The response of tunnels to seismic motion may be understood in terms of three principal types of inflicted deformation: axial, curvature, and hoop (tangential). Axial and curvature deformations develop when the direction of the incident wave is not exactly normal to the axis of the tunnel. Axial deformations are represented by alternating regions of compressive and tensile strain that travel as a wave-train on the surface of the tunnel along its axis. Curvature deformations also travel along the length of the tunnel trying to bend it in alternating directions. Hoop deformations result from waves of normal or nearly normal incidence to the tunnel axis. Three effects of the hoop deformations can be observed: (1) a distortion of the cross section, (2) dynamic stress concentration near the free surface of the tunnel, and (3) a circulation of seismic wave energy around the tunnel (which is possible only for wavelengths which are shorter than the radius of the tunnel).
An analytical treatment of the dynamic stress field near a tunnel of arbitrary shape and dimensions is not possible, yet valuable intuition can be obtained from case studies formulated for tunnels of simple geometry exposed to plane harmonic seismic waves. A typical example is the solution of the dynamic equations of motion for a plane SH wave at normal incidence to an infinite cylindrical tunnel of circular cross section surrounded by a perfectly elastic material. The same case can be reformulated for incident P-wave or SV-wave. The stationary case of a plane harmonic wave can be used as a stepping stone to the description of the dynamic effect on a tunnel due to a seismic pulse of complex frequency content. This would simply involve a direct and an inverse Fourier transform of the input data. This approach is of practical importance when the effect from an actual seismic wave on an existing tunnel needs to be evaluated. In this case, the velocity seismogram needs to be transformed into the frequency domain, and after computing the effect from each frequency, the obtained stress needs to be transformed back from the frequency domain to the time domain in order to compute the stress concentration factors and the related seismic response spectra.
At the other extreme of reformulating the problem, one can study the stress field around a tunnel due to a static remote loading. At a first glance, this static stress field should have very little in common with the picture of dynamic stress concentration. However, in many cases, the wavelength corresponding to the predominant frequency of a seismic wave can be one or two orders of magnitude greater than the radius of the tunnel, and therefore, the final results for the stress concentration factors in the dynamic and static cases turn out to be similar.
Of practical importance is the case of a plane wave at normal incidence to an infinite circular cylinder. The mathematical formulation of the problem is truly 2D and can be solved exactly. The corresponding static case was solved by Kirsch in the nineteenth century. The 2D-stress field near a circular opening under a constant remote stress \(\sigma _{xx}=\sigma _{1}\), \(\sigma _{yy}=\sigma _{2}\), and \(\sigma _{xy}=\sigma _{12}\) is
where a is the radius of the circular hole (the cross section of the tunnel) and the components of the stress tensor are given in polar coordinates, \(x=r\cos \left (\theta \right )\) and \(y=r\sin \left (\theta \right )\), for \(0\le r<\infty \) and \(0\le \theta \le 2\pi \).
The analytical static solution gives the radial stress component \(\sigma _{r}\), the tangential (or hoop) stress \(\sigma _{\theta }\), and the shear stress \(\sigma _{r\theta }\) at any point outside the hole, \(r\ge a\) and \(0\le \theta \le 2\pi \). The circular hole in the 2D Kirsch case corresponds to the cross section of an infinite tunnel, and this is why the values for the stress components on the free surface \(r=a\) are of special interest. One can verify that the components contributing to the normal stress are zero as required for the boundary of a cavity, \(\sigma _{r}\left (r=a\right )=0\) and \(\sigma _{r\theta }\left (r=a\right )=0\). The hoop stress on the surface of the tunnel, \(r=a\), varies with the polar angle \(\theta \) and can be expressed in terms of the corresponding value of the load \(\sigma _{\theta }^{\infty }\)—this would be the hoop stress if the hole was absent. The special case of the hoop stress at the edge of the hole for \(r=a\) is
The stress concentration factor \(\gamma \) is defined as the ratio of the maximum hoop stress on the cavity wall (\(r=a\)) to the maximum remote stress. For the special case of biaxial loading, \(\sigma _{12}=0\) (or \(\tau _{12}=0\) in Fig. 6.2), \(\sigma _{1}>0\), and \(|\sigma _{2}|\le \sigma _{1}\), the stress concentration factor is reached for \(\theta =\pm \pi /2\) and has the value
The biaxial static concentration factor ranges between 4 and 2. The maximum \(\gamma =4\) is reached for pure shear loading \(\sigma _{2}=-\sigma _{1}\), and the minimum \(\gamma =2\) corresponds to uniform, or hydrostatic, loading \(\sigma _{2}=\sigma _{1}\).
A special case of biaxial loading is \(\sigma _{2}=-\sigma _{1}\nu /(1-\nu )\), where \(\nu \) is the Poisson ratio of the surrounding material. As it turns out, the static stress concentration factor for the above case can be very close to the similar dynamic quantity defined for a long tunnel exposed to a plane monochromatic P-wave.
The general problem of evaluating the stress field at the boundary of an underground tunnel in the presence of a propagating seismic wave is quite complicated but, under some simplifying assumptions, a solution can be found (Mow & Pao, 1971). Usually, it is assumed that the tunnel is an infinitely long cylinder with circular cross section surrounded by a perfectly elastic material. Further, a plane harmonic incident wave is assumed to hit the tunnel at a right angle to the axis. In this formulation, the problem is two dimensional and admits an exact solution for the superposition of the incident and reflected waves. The dimension-less ratio of the maximum resultant tangent stress on the wall of the tunnel to the maximum stress in the incident wave defines the dynamic stress concentration factor. In the case of incident P- and SV-waves, the tangent stress is the hoop \(\sigma _{\theta \theta }\), while for SH incidence \(\sigma _{\theta z}\) is tangent to the boundary of the tunnel. All functions in the exact solutions are expressed in terms of the polar angle \(\theta \) and the radial distance r to the axis of the tunnel. The polar angle is measured from the direction of the incident wave, and the radial variable, r, enters in the final solution through two dimension-less combinations: \(\omega r/v_{P}\) and \(\omega r/v_{S}\), where \(\omega =2\pi f\) is the circular frequency, \(v_{P}\) is the P-wave velocity, and \(v_{S}\) is the S-wave velocity. The exact evaluation of the stress components for any frequency, tunnel radius, and location in the surrounding rock can be computationally quite demanding, but obtaining asymptotic expressions for the dominant frequency of typical seismic waves and for a tunnel radius of just a few metres can be relatively easy. This would require taking the limit \(\omega a/v_{P}\rightarrow 0\) and \(\omega a/v_{S}\rightarrow 0\) in the expression for the tangent stress. The wavelength in the case of circular frequency \(\omega \) and wave speed v is \(\Lambda =2\pi v/\omega \) so \(\omega a/v=2\pi a/\Lambda \) is usually a very small number. Indeed, the wavelength corresponding to the dominant frequency of a seismic event will be two orders of magnitude greater than the radius of the tunnel which justifies the limit \(\omega a/v\propto a/\Lambda \rightarrow 0\). But this limit is also consistent with \(\omega \rightarrow 0\) irrespective of the wave speed and the tunnel radius. The zero, or low, frequency regime is equivalent to the static case, hence, the relationship between the Kirsch solution and the dynamic stress concentration factor for P-wave incidence. The low frequency or quasi-static limits for the three types of plane harmonic waves at normal incidence to a tunnel are
where \(k^{2}\)\(=\)\(\left (v_{P}/v_{S}\right )^{2}\)\(=\)\(\left (2-2\nu \right )/\left (1-2\nu \right )\). For practical purposes, these quasi-static limits can be very useful taking into account the fact that they are only 10% to 15% lower than the respective dynamic counterparts. However, this is not true for high frequencies when significant oscillations are observed in the stress concentration factors.
The properties of the rock surrounding the tunnel are also of importance when computing the stress concentration factors. This is shown in Fig. 6.3 where the dynamic and static stress concentration factors for incident P-wave and SV-wave are plotted as functions of the Poisson ratio of the rock.
Fig. 6.3
Maximum dynamic stress concentration factor for P-wave (left) and SV-wave (right) on a cylindrical cavity
The total stress is a superposition of the static and dynamic stresses. The dynamic stress concentration around the cavity for a P-wave for an isotropic, elastic medium was determined by Mow and Pao (1971), and it depends on \(\nu \) and on the dimension-less frequency, \(\Omega \), of the wave
where f is the frequency of the wave and \(\Lambda \) is the wavelength. Note that for a very long wavelength, i.e. very low frequency, \(\Omega \) tends to zero and only the static stress concentration applies. The full expression for the dynamic stress concentration is rather complex, but its peaks are approximately 10% to 15% greater than the static one and occur at \(\Omega \simeq 0.25\), and therefore, the maximum stress concentration occurs at \(\varLambda /a\)\(\simeq \) 25, i.e. at wavelengths approximately equal to 25 times the cavity radius. For SH-waves in which the particle motion is normal to the plane of the cross section, the dynamic stress concentration factor is 2.1 and corresponds to
and therefore, the maximum shear stress concentration occurs at wavelengths approximately equal to 16 times the cavity radius. For \(v_{\pi }\)\(=\)\(v_{S}\)\(=\) 3000 m/s and the characteristic radius of the cavity of \(a=4\) metres, the frequency of the maximum dynamic stress is at \(f=187.5/8\)\(=\) 47 Hz, and for \(PGV_{S}\)\(=\)\(0.1\) m/s, the maximum dynamic shear stress \(\tau _{max}\)\(=\)\(0.8\) MPa.
Mow and Pao (1971) selected the largest value of the dynamic stress concentration factor over the entire range of frequencies for a given \(\nu \) and plotted it as a function of \(\nu \), for both P- and SV-wave, see Fig. 6.3.
6.3 Ground Motion at Source
Rupture is a propagating pulse that precedes slip at a seismic source. Its speed varies from \(0.6v_{S}\) to \(0.9v_{S}\) for sub-shear rupture and more than \(v_{S}\) for super-shear rupture. Slip follows rupture, and it is very fast at the tip of the rupture and slows dramatically past the rupture front. Slip velocity is the velocity of one side of the source with respect to the other. An average slip velocity varies from a few cm/s to a few m/s. Rupture may be unilateral, propagating in one direction across the source, bilateral, nucleating at the centre of the source and propagating in both directions, or it may be inhomogeneous. Seismic radiation and thus near-field motions strongly depend on rupture velocity: Slow ruptures radiate little seismic energy, while fast ruptures generate higher ground motion amplitudes. If all other factors are equal, the amplitudes of the radiated seismic waves increase with co-seismic deformation rate, rupture speed, rock strength, and with ambient stress.
The near-source ground velocity is equal to half of the slip velocity —the velocity of one side of the source with respect to the other. The ground velocity at source is controlled by the maximum stress at source which, in turn, is limited by the strength of the rock mass.
Consider a small piece of ground attached to an infinite source bounded on a plane by the extension of a rupture propagating with velocity \(v_{r}\rightarrow \infty \) over a small increment of time \(\Delta t\), and away from the source plane by the extension of the propagating S-wave with velocity \(v_{S}\), see Fig. 6.4 left. If the applied effective stress \(\sigma _{eff}\) available to accelerate the two sides of the source is released instantaneously, then the rock mass acceleration, a, and velocity, \(\mbox{v}\), can be derived from a\(=\)\(F/m\), where here force F\(=\)\(\sigma _{eff}\left (v_{r}\Delta t\right )^{2}\) and mass m\(=\)\(\rho \left (v_{r}\Delta t\right )^{2}v_{S}\Delta t\), and therefore
where \(\mu \) = \(\rho v_{S}^{2}\) is rigidity, \(\rho v_{S}\) is the shear wave impedance, and \(\epsilon _{eff}\) is the effective shear strain. If the effective stress \(\sigma _{eff}=\mu \), then the ground velocity would be equal to S-wave velocity, \(\dot {u}=v_{S}\). However, since \(\sigma _{eff}\ll \mu \), the ground velocity \(\dot {u}\ll v_{S}\).
According to the above equations: (1) Rock strength does not limit peak acceleration. For small \(\Delta t\) (at high frequencies, \(1/\Delta t\)), there is practically no limit on peak ground acceleration. A fracture may produce a steep change in ground velocity which results in high acceleration at high frequencies (Andrews et al., 2007). (2) The ground velocity will always be much smaller than the rupture velocity because the effective stress is much smaller than the shear modulus. (3) Rock strength limits ground velocity, which does not depend on frequency. (4) Ground velocity at source does not depend on the size of the event.
For a finite circular source of diameter 2r with instantaneous stress release, the effects of the edges of the crack will abate the ground velocity with time. For a simple taper, \(\exp \left (-v_{S}t/r\right )\), given by Brune (1970), integration of equation (6.11) over the process time, r/\(v_{S}\), (Kanamori, 1972) gives the average ground velocity,
If the effective strain, \(\sigma _{eff}/\mu \), at seismic source is between \(5\cdot 10^{-4}\) and \(10^{-3}\) and \(v_{S}=3250\) m/s, the near-source ground velocity would vary between 1 and 2 m/s. For an inhomogeneous rupture, the effective strain may be considerably different at different parts of the source producing varying dynamic strain drops and varying ground motion. The highest ground motion would be produced by rupture of intact rock (Gay & Ortlepp, 1978; Ortlepp, 1997; van Aswegen, 2008).
The effective stress cannot be measured directly, but different approximations can be made. The best proxy would be the dynamic stress drop; however, it cannot be derived reliably from observations. One option is to assume that \(\sigma _{eff}\) is equal to the bulk shear strength of the rock within the volume of interest, which averages for most hard rocks between 40 and 60 MPa. An intact rock may be considerably stronger. If a reliable database is available, one can, as a lower bound, also use the maximum stress drop derived from recorded waveforms, which in many hard rock mines averages between 2 and 4 MPa. Again the maximum values associated with failure of intact rock may go as high as 40 to 60 MPa.
The near-source ground velocity for a finite source and finite rupture velocity for different source models are quoted in Table 6.2. For \(\sigma _{eff}\)\(=\) 30 MPa, \(\rho \)\(=\) 2700 kg/m\(^{3}\), \(v_{S}\)\(=\) 3250 m/s, assuming the rupture velocity \(v_{r}\) between \(0.5v_{S}\) and \(0.75v_{S}\), the estimates of the near-source ground motion would vary between 1.14 m/s and 1.47 m/s, see Fig. 6.5. In general, the faster the rupture the faster the slip and the higher the near-field ground motion.
Fig. 6.5
Near-source ground velocity for a finite bilateral source and finite rupture velocity (Burridge, 1969)
Models of near-source ground velocity as a function of rupture velocity according to Burridge (1969)\(^{(1)}\), Ida (1973)\(^{(2)}\), and McGarr and Fletcher (2001)\(^{(3)}\), where \(f \left (v_{r} \right )\) is a monotonic function that ranges from 0.11 to 0.4 as rupture velocity increases from \(0.6v_{S}\) to \(0.9v_{S}\)
Equations (6.11) and those in Table 6.2 are applicable when the rate of stress release with an increase in slip velocity is, at all times, less than the shear wave impedance, \(d\sigma _{eff}/d\dot {u}\)\(<\)\(\rho \,v_{S}\), both in Pa\(\cdot \)s/m. Slip rates given by these equations may underestimate strain rates at the edges of the moving source. If the rate of loading exceeds the rate at which energy can be removed by elastic waves, the system is no longer linear. To remove this excess energy, the large strains need to travel faster than small ones—the particle velocity exceeds the shock wave velocity. This is also what happens during super-shear rupture when the crack tip is moving faster than the S-wave velocity (e.g. Weertman, 1969; Burridge, 1973; Savage, 1971; Andrews, 1976; Archuleta, 1984; Spudich & Cranswick, 1984; Dunham, 2007; Lu et al., 2010; Andrews et al., 2007).
6.4 Ground Motion Characteristics
Near-source ground motions can be amplified by rupture forward directivity, which occurs when the rupture direction and slip direction are aligned and move towards the site. When the source ruptures towards the site at a speed close to shear wave velocity, most of the radiated energy arrives there in a short time interval, and the cumulation of these pulses results in a single large low frequency pulse observed at the beginning of the seismogram. Rupture directivity produces a long period pulse in the direction normal to the fault plane. In the case where the rupture propagates away from the site, the arrival of seismic waves is distributed in time. This condition, referred to as backward directivity, is characterised by ground motions with relatively long duration and lower amplitude. Neutral directivity occurs for sites located off to the side of the rupture area where the rupture is neither predominantly towards nor away from the site (Archuleta & Hartzell, 1981; Somerville et al., 1997).
Fling step is a result of a static ground displacement and is generally characterised by a unidirectional velocity pulse and a monotonic step in the displacement time history. Fling step displacements occur in the direction of slip and therefore are not strongly coupled with the rupture directivity pulse. In a strike-slip source, the directivity pulse occurs on the strike-normal component, while the fling step occurs on the strike parallel component. In a dip-slip source, both the fling step and the directivity pulse occur on the strike-normal component.
Secondary Sources
Spall fractures are an example of secondary sources that occur when a high-intensity transient stress wave from the primary source reflects from a free surface. This phenomenon is the result of interference near a free surface between the portion of an oncoming incident compression wave which has not yet been reflected and the portion which has been reflected and transformed into a tensile wave. Usually, the amount of tension increases as the reflected wave moves back inward from the surface. The transient tensile stress resulting from the superposition is at the quarter wavelength depth and twice the maximum stress in the incident pulse. If the medium is not capable of withstanding these induced tensile stresses, it will break, creating a secondary source or sources of seismic radiation at that time and depth. The energy from the secondary sources arrives at the surface later in the strong motion, see Fig. 6.6.
They can be recognised by high frequency content in waveforms recorded by station(s) close to the spall but far from the primary source. Spalling occurs in rocks, soils, liquids and in cohesionless materials.
Figure 6.7 left shows 5.375 seconds of waveforms of the \(\log P=3.0\) event recorded in a mine 1230 metres from source with \(PGV=1.78\) cm/s at frequency approximately 5 Hz. Figure 6.7 right shows a zoomed section with a hidden secondary small high frequency event that was triggered close to the recording site.
Fig. 6.7
Three component velocity waveforms of large event recorded 1230 metres away (left) and a zoomed section showing the hidden secondary event that happened close to the recording site (right)
6.4.1 Engineering Characteristics of Ground Motion
There is a number of parameters that measure the intensity of ground motion and its potential for damage. They are mainly based on measured maximum amplitude, energy, and duration.
Peak Ground Characteristics
Ground motion characteristics include peak ground acceleration, PGA, velocity, PGV , and displacement, PGD. The PGA is most convenient for structural engineers, since the maximum force experienced by a rigid structure of mass m is \(F_{max} = m\cdot PGA\). However, the PGA is a poor parameter for evaluating potential for damage. For example, a large PGA associated with a high frequency pulse may be absorbed by the inertia of the structure with little deformation, since PGD\(\propto \)\(PGA/f^{2}\), where f is frequency. On the other hand, a more moderate acceleration associated with a long duration pulse of low frequency may result in a significant deformation of structures. The PGV , which at source is a half of the slip velocity, is less sensitive to the higher frequencies than PGA, can be measured directly and reliably, and provides a better indication of damage potential.
Duration
The degradation of stiffness and strength of rock are sensitive not only to the amplitude of ground motion but also to its duration and an associated number of load or stress reversals above the elastic regime. There are three different types of duration: bracketed, uniform, and significant.
The bracketed duration measures the duration of the ground motion from the first to the last occurrence of amplitude exceeding a specified threshold. The uniform duration is defined as the sum of the time intervals during which the ground motion is greater than the threshold. A functional way to display it is to construct a histogram and the cumulative graph of the time that the amplitude of ground motion spent above a certain level. Figure 6.8 left shows waveforms with PGA\(=\)\(82.1\) m/s\(^{2}\) produced by an event with \(\log E=6.5\) recorded 93 metres from the source, and Fig. 6.8 right the uniform duration plot. PGA\(\geq \) 10 m/s\(^{2}\) lasted for 0.05 seconds and PGA\(\geq \) 1 m/s\(^{2}\) lasted for 0.09 seconds.
Fig. 6.8
An example of uniform duration associated with a complex event (left) measured on the cumulative graph of the time that the amplitude of ground motion spent above certain level (right)
The significant duration defines ground motion duration as the length of the time interval between the accumulations of two specified levels of ground motion energy at the site. For example, the amount of time in which the central 90% of the integral of the squared velocity or acceleration takes place, \(t_{90}\) (Trifunac & Brady, 1975). As distance increases, the bracketed and uniform durations tend to zero, but, as energy becomes dispersed with distance, the \(t_{90}\) tends to increase. It is useful then to calculate the central 90% of energy, \(E_{90}\)\(=\)\(E\left (t_{95}\right )\)\(-\)\(E\left (t_{5}\right )\) and the power, \(P_{90}\)\(=\)\(E_{90}/t_{90}\).
Arias intensity, \(I_{a}\)\(=\)\(\pi /\left (2g\right )\intop _{0}^{t_{d}}\left [a\left (t\right )\right ]^{2}dt\), where \(I_{a}\) is in m/s, \(a(t)\) is the acceleration time history in units of acceleration due to gravity g, and \(t_{d}\) is the duration of ground motion (Arias, 1970).
The cumulative absolute velocity, CAV\(=\)\(\intop _{0}^{t_{d}}|a\left (t\right )|dt\), and has units of velocity, m/s (EPRI, 1988). In mines, however, the utility of \(I_{a}\) and CAV are limited for two reasons: (1) Mines mostly employ velocity transducers. (2) The highest accelerations recorded by piezoelectric accelerometers are associated with very high frequencies, where there is little displacement and little or no damage potential.
The cumulative absolute displacement, CAD, is defined as the integral of the absolute value of a velocity time series, CAD\(=\)\(\intop _{0}^{t_{d}}|\mbox{v}\left (t\right )|dt\), which has units of displacement. CAD is the area under the absolute velocity time history and is more sensitive to lower frequency ground motion, i.e. to larger displacements.
The gradient of the time integral is equal to the absolute velocity. Since \(\mbox{v}\left (t\right )\)\(=\)\(du/dt\), the integral over velocity can be written as the summation of incremental peak-to-valley and valley-to-peak displacements, regardless of sign, in the displacement time series, CAD\(=\)\(\sum _{j}^{n}|\Delta u_{j}|\), where \(\Delta u_{j}\) is the jth value of incremental displacement in the time series and n is the total number of incremental displacements. CAD can also be interpreted as the area under the plot of ground velocity, \(\mbox{v}\), versus duration, \(t_{d}\left (\geq \mbox{v}\right )\), see Fig. 6.9. Since CAD could be overly influenced by a time series of long duration that contained insignificant amplitudes, one can introduce an integration threshold, \(CAD\left (\geq \mbox{v}_{min}\right )\)\(=\)\(\intop _{0}^{t_{d}}|\mbox{v}\left (t\right )|_{\geq \mbox{v}_{min}}dt\). CAD is a better indicator of damage potential than a single PGV measurement and can replace PGV in the ground motion prediction equation.
Figure 6.10 top row shows the three-component velocity and displacement waveforms of a \(\log P=2.61\) event recorded by 4.5 Hz geophone and at the bottom row X, Y , and Z components of the absolute velocity that, after integration, deliver CAD. The PGV and CAD values are shown in the right margin of each graph.
Fig. 6.10
Velocity waveforms and integrated displacements (top) of a \( \log P=2.61\) (\(m2.66\)) and energy \( \log E=8.65\) event recorded 402 metres from source. X, Y, and Z components of the absolute velocity that after integration delivers CAD(bottom). At the bottom of each graph, there is the date and time of the event and information on sensor and filter applied to processing
The amount of S-wave (or P-wave) energy transmitted by the seismic wave across an area A normal to the direction of propagation during the time interval \(t_{2}-t_{1}\) is \(E_{GM}=\rho Av_{P,S}\intop _{t_{1}}^{t_{2}}\dot {u}^{2}dt\), where \(\rho \) is the rock density, \(v_{S,P}\) is the S- or P-wave velocity, and \(\dot {u}\) is the ground velocity.
6.5 Ground Motion Amplification at the Skin of Excavation
The bulk of the damage caused by seismic events in mines occurs at the source region, i.e. within the volume \(V=P/\Delta \epsilon \), with the strain change of the order of \(10^{-4}\), which for \(\log P=2\) gives \(V=10^{6}\) m\(^{3}\). Complex, cascading sources may create considerably larger volumes of rock destruction. Damage to an excavation away from the source is related to the local geological site conditions that amplify ground motion, and to the capacity of support to sustain loading. Therefore, to select the appropriate support, it is advisable to estimate the expected site response. Site response can be estimated from theoretical models given that sufficient information about the near-surface geology is available, from weak ground motion or from the waveforms of seismic events recorded at the surface of the excavation and at a reference site.
Modelling the site response is very useful in understanding the physics of the problem; however, its success may be limited if the detailed 3D structural and geotechnical information is not accurate. The most reliable estimation of site response in mines, however, is obtained from waves recorded close to or at the skin of the excavation and the same waves recorded downhole deeper in the rock mass which is then taken as the reference site. The fundamental requirement then is that the distance between these two sites is small relative to the distance to the source and that the reference site should be free of site effect. One can then compare the ratio of the recorded ground motion characteristics and compute the spectral ratios of the S-wave group to estimate the frequency dependent site response.
Event1 \(\log P=1.0\) at Distance 55 m
Figure 6.11 shows waveforms of a \(\log P=1.0\) seismic event recorded in an underground hard rock mine 55 metres away from the source by two three-component 14 Hz geophone sets sampled at 6 kHz, one installed at the surface of the 5 m by 6 m tunnel and the other 10 metres downhole.
Fig. 6.11
Three-component velocity waveforms recorded at the end of the 10 metre borehole (top left), at the surface of the u/g excavation (top right), and integrated displacements (bottom row) of a \( \log P=1.0\) event located 55.5 metres away. The grey area indicates the part of the waveform taken for spectral analysis of the event, and the grey lines at the top of the grey area show the part taken for spectral analysis of noise. The black line shows the cumulative energy of ground motion
The recorded average background noise level at the surface of the excavation on the x-component, vertical \(\uparrow \), and y-component, parallel to the axis of the tunnel \(||\), is 16 times higher and on the z-component, orthogonal to the tunnel \(\bot \), 10 times higher than the respective components of the downhole sensor set. The maximum ground velocity at the surface is 1.7 times higher than downhole, with the x-component, \(PGV_{\uparrow }\), amplified 1.72 times, the y-component, \(PGV_{\shortparallel }\), amplified 3.24 times, and the z-component, \(PGV_{\bot }\), 1.24 times. The cumulative absolute displacement, CAD, is also amplified: \(CAD_{\uparrow }\) 2.09 times, \(CAD_{\shortparallel }\) 2.7 times, and \(CAD_{\bot }\) 1.16 times (Table 6.3).
Table 6.3
\( \log P=1.0\) located 55 m away: PGV mm/s, PGD mm, and CAD mm, per component and their ratios
\(PGV_{\uparrow }\)
\(PGV_{\shortparallel }\)
\(PGV_{\bot }\)
\(PGD_{\uparrow }\)
\(PGD_{\shortparallel }\)
\(PGD_{\bot }\)
\(CAD_{\uparrow }\)
\(CAD_{\shortparallel }\)
\(CAD_{\bot }\)
Surface
17.47
28.12
20.86
0.043
0.026
0.122
0.547
0.574
0.736
Borehole
10.17
8.69
16.87
0.049
0.031
0.158
0.261
0.213
0.633
Ratio
1.72
3.24
1.24
0.88
0.84
0.77
2.09
2.7
1.16
Figure 6.12 left shows the spectra of the \(\log P=1.0\) event that includes 0.2 seconds of the pre-P-arrival noise and 0.2 seconds of coda, smoothed with the Konno-Ohmachi function
where \(b=40\) is the coefficient for bandwidth and \(f_{c}\) is the centre frequency (Konno & Ohmachi, 1998). The resulting spectral ratios, Fig. 6.12 right, show site amplifications between 2 and 3 times at 18, 40, and 120 Hz on the x- and y-components. Higher amplifications of 4 to 7 times are recorded at 150 to 250 Hz. Spectral ratios increase again past 300 Hz, most likely due to the interaction of higher frequencies with the tunnel, but at those high frequencies displacements are small, and therefore potential for damage is low.
Site amplification can also be quantified by taking PGV , or CAD, from continuous records every \(\Delta t\) and calculating the ratio between the records at the skin of excavation and in the borehole. While here frequency is disregarded, the method benefits greatly from the large number of measurements, e.g. taking sampling at 6 kHz, there are \(5.184\cdot 10^{8}\) samples per component per day to choose from. The method is also simple, and therefore, it can be implemented online to monitor changes in site effect at a particular site or between different sites as mining progresses.
Figure 6.13 left shows the smoothed probability density kernel of the observed logged ratio of the PGV for each component at the surface and the same measured downhole, taken every \(\varDelta t=0.25\) second from the recorded continuous waveform over the 24 hour period that includes the \(\log P=1.0\) event shown in Fig. 6.11. To avoid spurious amplifications associated with weak ground motion, a threshold of \(0.003\) mm/s was applied to the data before the ratio was taken that reduced the number of ratios from 345600 to 669 on the x-component, to 621 on the y-component, and to 1084 on the z-component. It shows that the most probable amplification of ground motion on the x-component is \(\log \left (PGV_{S}/PGV_{B}\right )\)\(=\)\(0.2\), i.e. \(PGV_{S}\) on the x-component is most likely \(1.6\) times higher than \(PGV_{B}\), on the y-component 2.2 times higher, and on the z-component 2.5 times higher.
Fig. 6.13
Smooth probability density function of the logged ratio of the PGV per component measured at the surface and at 10 metres downhole taken every 0.25 second from the recorded continuous waveform over the 24 hour period that includes the \( \log P=1.0\) event (left). The survival function for the same dataset, i.e. the probability that the amplification ratio is greater or equal to a given value is also shown (right)
Figure 6.13 right shows the survival function, i.e. the probability that the amplification ratio is greater than or equal to a given value. In this case, the probability that the amplification of PGV on the x-component is greater than 2 is almost 50%, and on the y- and z-components it is 70%. The probability that the amplification of PGV on the x- and z-components is greater than 3 is 30%, and on the y-component 38%.
Note that logging the ratio solves the problem of lack of symmetry, i.e. if \(PGV_{S}\) is greater than \(PGV_{B}\), the ratio can take theoretically any value greater than 1, but if \(PGV_{S}\) is less than \(PGV_{B}\), the ratio is restricted to the range of 0 to 1. The logged ratio restores the symmetry, i.e. \(\log \left (PGV_{S}/PGV_{B}\right )\)\(=\)\(-\log \left (PGV_{B}/PGV_{S}\right )\).
Event2 \(\log P=0.2\) at Distance 286 m
Figure 6.14 shows waveforms of a smaller event with \(\log P=0.2\) located 286 metres away and recorded by the same set of sensors.
The recorded average background noise level at the surface of the excavation on the x-component, \(\uparrow \), is 8 times higher, on the y-component \(||\), 11 times higher and on the z-component \(\bot \), 5 times higher than at the respective components of the downhole sensors. The maximum ground velocity at the surface is 4.8 times higher than downhole, with the x-component, \(PGV_{\uparrow }\), amplified 4.2 times, the y-component, \(PGV_{\shortparallel }\), 5 times and the z-component, \(PGV_{\bot }\), 5.5 times. The cumulative absolute displacement, CAD, is amplified 3.7 times, and the energy of ground motion, \(EGM,\) 3.8 times. The PGD is amplified less than PGV but more than in the previous case. The maximum amplification of the PGD is 2.7 times in the direction orthogonal to the tunnel, 1.4 times along the tunnel, and 1.3 times on the vertical component (Table 6.4).
Table 6.4
\( \log P=0.2\) located 286 m away: PGV mm/s, PGD mm, and CAD mm, per component and their ratios
lsg
\(PGV_{\uparrow }\)
\(PGV_{\shortparallel }\)
\(PGV_{\bot }\)
\(PGD_{\uparrow }\)
\(PGD_{\shortparallel }\)
\(PGD_{\bot }\)
\(CAD_{\uparrow }\)
\(CAD_{\shortparallel }\)
\(CAD_{\bot }\)
Surface
1.995
1.809
1.785
0.00199
0.0014
0.00173
0.0515
0.042
0.0443
Borehole
0.469
0.362
0.322
0.00152
0.0010
0.00063
0.0141
0.0108
0.0113
Ratio
4.25
5.0
5.54
1.31
1.41
2.73
3.65
3.87
3.9
Figure 6.15 left shows the smoothed spectra of the \(\log P=0.2\) that also includes 0.2 seconds of the pre-P-arrival noise and 0.2 seconds of coda. The resulting spectral ratios, Fig. 6.15 right, show site amplifications up to 2 times in the frequency range up to 120 Hz on all components. Higher amplifications of 5 to over 20 times are recorded at 150 to 300 Hz, and then again up to 50 times past 300 Hz, most likely due to the interactions of shorter waves with the tunnel. Again, at such high frequencies, displacements are small, and therefore potential for damage is low.
Figure 6.16 left shows the smooth probability density kernel of the observed logged ratio of the PGV for each component at the surface and the same measured downhole.
The ratio is taken every \(\varDelta t=0.25\) second from the recorded continuous waveform over the 24 hour period that includes the \(\log P=0.2\) event shown in Fig. 6.14. The same threshold of 0.003 mm/s was applied to avoid the spurious amplifications associated with weak ground motion, which reduced the number of ratios from 345600 to 500 on the x-component, 401 on the y-component, and 1274 on the z-component.
It shows that the distribution of amplification of ground motion on the x-component peaks at \(\log \left (PGV_{S}/PGV_{B}\right )\)\(=\)\(0.15\), i.e. where \(PGV_{S}\) on the x-component is \(1.4\) times higher than \(PGV_{B}\). On the y-component, it is between 1.6 and 4 times higher and on the z-component 3 times higher. Figure 6.15 right shows the survival function, i.e. the probability that the amplification ratio is greater or equal to a given value. In this case, the probability that the amplification of PGV on any of the three components is greater than 2 is almost 85% and greater than 3 is 70%.
Summary
(1) Larger seismic events produce strong ground motion at frequencies below 15 Hz, and at these frequencies both sets of sensors displace in tandem, and therefore, there is no, or very little, amplification. (2) The PGV is subjected to largest amplification, then CAD and least the PGD. The component of the largest amplification, being parallel, vertical or orthogonal, depends on the direction of the incoming wave.
6.5.1 Ejection Velocity
Ortlepp (1993) presented evidence of rock ejection velocities associated with seismic events in mines of the order of 10 m/s and greater and suggested that they may be due to a rock failure phenomenon different than the classical slip on geological structures. Ground motion can be modified at, or close to, the surface of an excavation during the buckling of slabs in the sidewalls inducing an ejection velocity that can well exceed that of ground motion at source.
Assuming buckling of excavation sidewalls subjected to high levels of compressive stress as a possible mechanism of failure (McGarr, 1997) estimated the ejection velocity, \(v_{ej}\), as
where \(\sigma _{c}\) is the uniaxial compressive stress at failure in Pa, \(\nu \) is the Poisson ratio, Y is the Young modulus in Pa, and \(\rho \) is the rock density in kg/m\(^{3}\) (Fig. 6.17). Indeed, taking reasonable values of \(\sigma _{c}\), Y , \(\nu \), and \(\rho \) for hard rocks, the ejection velocities could be well in excess of 10 m/s. The ejection velocity, \(v_{ej}\), in Eq. (6.14) depends very weakly on the Poisson ratio.
For larger mass, m, such velocities can exert a considerable force, F\(=\)\(mv_{ej}/\Delta t\), on a stiff support system, i.e. for short deformation time \(\Delta t\). However, in case of face bursts driven by sudden loading where rock is shattered to small pieces, the average velocity of ejection is a decreasing function of m, i.e. \(v_{ej}\sim 1/\sqrt {m}\).
If a block of rock is moving with velocity \(v_{ej}\) while being at a level where the gravitational potential energy is \(mgh_{0}\) and eventually hits a wall at a point where its potential energy is \(mgh_{1}\) and is stopped there, the work done, \(\Delta W\), will be numerically equal to the work for accelerating the body from rest to velocity \(v_{ej}\), plus the work for moving the rock in the field of the Earth’s gravity from the point of impact, level \(h_{1}\), to the starting level \(h_{0}\), and therefore,
Ejection from the roof hits the floor. The depth of failure is \(d_{f}\), and \(\rho \) is the density of the rock. The total mass of the ejected rock fragment is \(m_{f}=\rho d_{f}A\), where A is the area of the face of the burden. The mass per unit area of the face is \(m_{A} = \rho d_{f}\). The work done on the floor when it is hit by the ejected rock is \(\Delta W = \frac {1}{2}m_{f}v_{ej}^{2} + m_{f}gh\), in Jules, where h is the height of the tunnel. The work per face-unit area of the ejected rock is, \(\Delta W_{A} = \frac {1}{2}m_{A}v_{ej}^{2} + m_{A}gh\), in Jules/m\(^{2}\).
2.
Ejection from the floor. A block of rock is detached from the floor and pushed up by a dynamic event with an initial velocity \(v_{ej}\). If the ejected rock hits the roof, the work done on the roof when it is hit by the ejected rock is \(\Delta W = \frac {1}{2}m_{f}v_{ej}^{2} - m_{f}gh\), and the work per face-unit area of the ejected rock is \(\Delta W_{A} = \frac {1}{2}m_{A}v_{ej}^{2} - m_{A}gh\). If the ejected rock does not reach as high as the roof, \(\Delta W_{A} = \frac {1}{2}m_{A}v_{ej}^{2}\).
3.
Ejection from the side wall. A block of rock is detached from one of the walls of the tunnel at height \(h_{0}\) and pushed towards the opposite wall by a dynamic event with an initial velocity \(v_{ej}\). The vector of the initial velocity subtends angle \(\theta \) with the horizontal:
(a)
If the ejected rock hits the opposite wall at some height \(h_{1} < h_{0}\), then the work per face-unit area of the ejected rock is \(\Delta W_{A} = \frac {1}{2}m_{A}v_{ej}^{2} -+ m_{A}g \left (h_{0}-h_{1}\right )\).
(b)
If the ejected rock drops on floor some distance d from the wall \(\Delta W_{A} = \frac {1}{2}m_{A}v_{ej}^{2} -+ m_{A}g \left (h_{0}-h_{1}\right )\), then the work per face-unit area of the ejected rock is \(\Delta W_{A} = \frac {1}{2}m_{A}v_{ej}^{2} + m_{A}g h_{0}\). The distance d at which the ejected rock hits the floor is related to the initial velocity by the relation,
which for \(\theta =0\) gives \(v_{ej} = d \sqrt {g/\left (2h_{0}\right )}\).
6.6 Simple GMPE and Its Utility
6.6.1 Introduction
The ground motion prediction equation (GMPE) gives the expected value of a given ground motion parameter, e.g. PGV or CAD, as a function of seismic energy, potency or magnitude, and distance.
There is not much literature on GMPE for mining induced seismicity and even less for underground sites. McGarr et al. (1981) used ground motion data recorded 3 km underground in a South African gold mine to develop the relationship \(\log \left (R\cdot PGV\right )\)\(=\)\(3.95+0.57m_{L}\), where both R is in cm and the peak ground velocity, PGV , is in cm/s, and \(m_{L}\) is local magnitude.
Kaiser and Maloney (1997) proposed a similar equation but with the exponent \(a^{*}\)\(=\)\(0.5\) as a scaling law for support design in rockburst conditions. It is written in the form, PGV\(=\)\(C^{*} \cdot M^{a^{*}}/R\), where M is seismic moment expressed in GNm. The parameter \(C^{*}\) depends on the stress drop environment and based on data from the Creighton Mine, \(C^{*}\)\(=\)\(0.1.\) to \(0.3\) for events with stress drops less than \(2.5\) MPa and with \(C^{*}\)\(=\)\(0.5\) to \(1.0\) for higher stress drop events. However, it is recommended to adjust \(C^{*}\) to a specific dataset at hand. Translating to the seismic potency domain gives \(PGV =\)\(C^{*}P^{1/2}/R\), where PGV is in m/s, P in m\(^{3}\), and R in metres. Taking into account that M\(=\)\(\mu P\) and \(\mu \)\(=\) 30 GPa, the parameter \(C^{*}\)\(=\) (\(1.1\) to \(1.64\)) for events with stress drop less than \(2.5\) MPa and \(C^{*}\)\(=\) (\(2.74\) to \(5.48\)) for higher stress drop events. The relations by McGarr et al. (1981) and by Kaiser and Maloney (1997) do not cater for attenuation and near-source saturation.
McGarr and Fletcher (2005) developed GMPE for PGV and PGA based on the ground motion recorded on surface due to coal mining that generated events with \(m\leq 2.2\), recorded at distances of 500 metres to 10 km.
Mendecki (2008) developed and compared the GMPE-PGV for four underground mines: two gold mines in South Africa, one in Australia, and an iron ore mine in Sweden, all based on data recorded by three -component geophones installed in boreholes drilled from underground excavations.
Atkinson (2015) used the NGA-West 2 database, containing horizontal component response spectra and PGV for events \(3.0 \leq m \leq \)\(6.0\) recorded on surface at distances up to 40 km to develop a GMPE that could be applied to induced seismicity. She concluded that ground motion from small to moderate induced events may be significantly larger than that predicted by most currently used GMPE.
Mendecki et al. (2018) developed GMPE for PGA due to induced seismicity based on 15541 observations recorded by 14 surface sites in the central area of the Main Syncline of the Upper Silesia Coal Basin in Poland.
The development of a GMPE for underground mines is in some respects different to that in earthquake seismology. There are very few accelerometers installed in mines, and in most cases, these are piezoelectric which are not strong ground motion instruments. They deliver high accelerations at high frequencies which are of little interest since at those frequencies there is not much ground velocity and even less ground deformation. Double integration of frequently noisy acceleration waveforms to displacement may also prove difficult. It is only recently that the semiconductor micro-electromechanical systems accelerometers, MEMS, are being used in mines. In addition, underground support designs are based on the expected demand in terms of PGV and deformation. Most mines install a mixture of 4.5 and 14 Hz geophones, and only recently have lower frequency sensors been deployed. For larger events, the higher frequency sensors filter lower frequencies and underestimate the ground motion parameters, and to some degree seismic potency and energy.
Figure 6.18 shows the three-component velocity and integrated displacement waveforms of a \(\log P=2.86\) event recorded at a distance 2252 metres from the source by a 1 Hz sensor and a 4.5 Hz sensor located next to each other in the same borehole. While the shape of the waveforms is similar, the 4.5 Hz sensor recorded significantly lower ground motions. PGV recorded by the 4.5 Hz sensors is 1.8 times lower, PGD is 3.5 times lower, and the cumulative absolute displacement, CAD, is 2.3 times lower. The 14 Hz geophone would record even lower ground motion. For smaller events, the differences are less significant. It is therefore advisable to select PGV and CAD recorded by the same type of sensors while developing GMPE.
Fig. 6.18
Velocity and displacement waveforms of a \( \log P=2.86\) event recorded by 1 Hz (left column) and 4.5 Hz sensors (right column), located at the same site
One can increase the signal range and the travel limits of geophones by overdamping. With normal damping of 0.7, the frequency response is flat to ground velocity above the natural frequency, \(f_{n}\), and is proportional to \(f^{2}\) below, which is caused by a double pole at that frequency, see Fig. 6.19. As the damping increases beyond 1, the poles separate, in such a way that the product of the pole frequencies remains constant. Between these poles, the velocity response is proportional to frequency, effectively making it flat to acceleration over this frequency range. For 4.5 Hz geophones, the maximum damping which can reasonably be achieved is 3.4, which means the acceleration response covers the frequency band from 0.7 Hz to 30 Hz. In this configuration, the ADC voltage clip limit is raised by a factor of 5 to 0.5 m/s, which is then slightly greater than the minimum internal displacement clip limit.
Fig. 6.19
The solid lines show the ground velocity required to produce 2 mm peak internal displacement in a 4.5 Hz geophone, for overdamped and maximally flat responses. The voltage limits for a typical audio ADC are marked by dashed horizontal lines, and pole frequencies by dashed vertical lines, after Mountfort and Mendecki (2019)
Traditionally, the most important ground motion parameter in underground mines was the instrumental PGV which is used for support design. Recently, Mendecki (2018) developed a GMPE for the cumulative absolute displacement, CAD, to be applied to monitor the consumption of the deformation capacity of support due to seismicity. The main interest for mines are the ground motion parameters at distances between 50 and 1000 metres. Larger distances are of interest for surface structures, e.g. tailings dams or processing plants. For the sizes of events in mines, the PGV may drop by two orders of magnitude between 100 and 1000 metres from the source. CAD decays more slowly with distance than PGV , mainly because of the increased duration of waveforms with distance due to scattering. Although we caution that the GMPE is not the best tool to estimate the near-source ground motion (Mendecki, 2016), the geotechnical engineers, lacking other credible data, resort to such extrapolation when considering support specifications. Therefore, there is a need to constrain PGV at source to a physically acceptable level.
Seismic systems in mines are designed to locate events and to estimate their source parameters. For this reason, sensors are installed at least 6 to 10 metres into boreholes to avoid the very site effects that amplify ground motion at the skin of excavations. Since the GMPE derived from such measurements certainly underestimates seismic load, mines conduct separate site amplification measurements at selected locations (Milev & Spottiswood, 2005; Cichowicz, 2008; Mendecki, 2013; Dineva et al., 2016; Mendecki, 2016, 2017).
Unlike crustal seismology, in mines the bulk rock mass properties are changing due to rock extraction, specifically in caving and open pit mines, and, because new strong ground motion data is coming fast, the GMPE needs to be updated at least once a year. For the same reason, the GMPE developed for mines is characterised by large scatter. Moreover, many waveforms of larger events recorded at closer distances are displacement clipped or voltage saturated, which limits the number of observations in the near field. The distances to larger and intermediate size events are also uncertain because of the unknown orientation of sources and the complex nature of larger events.
6.6.2 Simple GMPE for PGV and CAD (Based on Mendecki, 2019)
Ground Motion Prediction Equation
The ground motion prediction equation (GMPE) gives the expected value of a given ground motion parameter, GMP, e.g. peak ground velocity PGV or the cumulative absolute displacement CAD, as a function of seismic potency or energy or magnitude and distance. Its main utility in mines is to predict ground motion at selected sites in the near and intermediate fields of larger events that may occur in the future, on the basis of observations of ground motion caused by smaller events.
We start with the following prediction equation for a ground motion parameter, GMP, caused by an event of potency P at distance R,
where R for larger events should be measured orthogonal to the characteristic rupture plane, \(c_{P}\) is the potency dependence parameter, \(c_{R}\) controls the geometrical attenuation rate, and c is a free parameter (Esteva, 1970; Campbell, 1981). The term \(c_{L}P^{1/3}\) modulates ground motion at small distances, where geometric attenuation is small, and to saturate them at source.
The amplitudes of GMP predicted by the GMPE above are positively correlated with \(c_{P}\) and c and negatively with \(c_{R}\) and \(c_{L}\). The term \(c_{P}\log P\) is consistent with the definition of earthquake magnitude as a logarithmic measure of the amplitude of ground motion. The term \(-c_{R}\log R\) is consistent with the geometric spreading of the seismic wavefront as it propagates away from the source, and it also caters in part for the attenuation due to anelasticity and scattering.
The GMPE at source gives \(GMP\left (R=0\right )\)\(=\)\(cc_{L}^{-c_{R}} P^{c_{P}-c_{R}/3}\), i.e. at source \(\log GMP\) is a linear function of \(\log P\). For \(c_{P}\)\(=\)\(c_{R}/3\) the GMP at source is independent of event size, \(GMP\left (R=0\right )\)\(=\)\(c/ c_{L}^{c_{R}}\). For \(c_{P}\)\(>\)\(c_{R}/3\), it delivers larger GMPs at source for events with larger potencies. The case \(c_{P}\)\(<\)\(c_{R}/3\) predicts that lower potencies generate higher GMPs at source than larger potencies, which is rather unlikely. The case \(c_{R}\)\(=\)\(1.0\) with no attenuation and for \(c_{L}\)\(=\) 0 gives the familiar GMP\(=\)\(cP^{c_{P}}/R\), McGarr et al. (1981); Kaiser and Maloney (1997).
From the GMPE given by Eq. (6.16), we can calculate R\(=\)\(\left (cP^{c_{P}}/GMP\right )^{1/c_{R}} -c_{L}P^{1/3}\), i.e. the distance over which seismic source with potency P generates the ground motion parameter \(\geq GMP\). We can also calculate the minimum potency, or \(\log P\), that delivers a given level of GMP at a distance R. The case \(c_{P}\)\(=\)\(c_{R}/3\) gives \(P\left (R\right )=R^{3}/\left [\left (c/GMP\right )^{1/c_{R}}-c_{L}\right ]^{3}\), but in the general case the solution must be obtained numerically.
There are many forms of GMPE, and most of them developed for predicting surface ground motion resulting from earthquakes, see Douglas (2018) for a review. Some of them are complex and have more than five or even 10 coefficients to cater for magnitude, distance, site effects, source mechanisms (normal, strike slip, or reverse faulting), and, in some cases, even directivity. However, more complex models are more susceptible to the danger of overfitting, i.e. modelling spurious details of the data rather than the data generating process. The inversion procedure for parameters in more complex models should therefore be carried out in two stages to decouple potentially correlated variables, in this case, \(c_{P}\) and \(c_{R}\) (Joyner & Boore, 1993, 1994) or (Abrahamson & Youngs, 1992). However, such a process can only alleviate the problem, and the real physical meaning of these parameters may be lost, a point well made by McGarr and Fletcher (2005).
Simple GMPE-PGV
Equation (6.16) for PGV is \(\overline {PGV}\left (P,R\right )\)\(=\)\(cP^{c_{P}} \)\(\left (R+c_{L}\cdot P^{1/3}\right )^{-c_{R}}\), which at source, for \(R=0\), gives \(\overline {PGV}\left (P,R=0\right )\)\(=\)\(c c_{L}^{-c_{R}} P^{c_{P}-c_{R}/3}\). The average ground velocity at source is \(PGV_{0} = 0.63v_{S}\epsilon _{eff}\). For \(\sigma _{eff}\)\(=\) 25 MPa, \(\rho \)\(=\) 2700 kg/m\(^{3}\), \(v_{S}\)\(=\) 3300 m/s, and the rupture velocity \(v_{r}\)\(=\)\(0.75v_{S}\), the estimates of the near-source ground motion vary between 1.0 m/s and 2.1 m/s.
The effective stress cannot be measured directly, but one can assume that it is equal to the bulk shear strength of the rock within the volume of interest, which for hard rock varies between 10 MPa, for an inhomogeneous rock to 100 MPa for an intact homogeneous hard rock. This, assuming the rigidity of the order of 10 GPa, translates to \(10^{-4} \leq \epsilon _{eff} \leq 10^{-3}\). A more practical proxy for \(\sigma _{eff}\) or \(\epsilon _{eff}\) is the upper limit of the static stress drop, \(\varDelta \sigma \), or strain change, \(\varDelta \epsilon \), derived from waveforms recorded in the area of interest.
Now, from \(\overline {PGV}\left (P,R=0\right )\)\(=\)\(0.63 v_{S} \varDelta \epsilon \)\(=\)\(c c_{L}^{-c_{R}} P^{c_{P}-c_{R}/3}\), we can derive c\(=\)\(0.63v_{S}\varDelta \epsilon c_{L}^{c_{R}} P^{-c_{P}+c_{R}/3}\). For \(c_{P}=c_{R}/3\), parameter \(c= 0.63v_{S}\varDelta \epsilon c_{L}^{c_{R}}\) is independent of potency P, and the GMPE for \(\overline {PGV}\) is
While this expression has four parameters: \(v_{S}\), \(\varDelta \epsilon \), \(c_{L}\), and \(c_{R}\), two of them, \(v_{S}\) and \(\varDelta \epsilon \), are constrained by the type of rock and can be assumed, and the other two, \(c_{L}\) and \(c_{R}\), need to be inverted from data.
Simple GMPE-CAD
Equation (6.16) for CAD is \(\overline {CAD}\left (P,R\right )\)\(= c\cdot P^{c_{P}}\left (R+c_{L}\cdot P^{1/3}\right )^{-c_{R}}\). For a circular crack with a uniform strain change \(\Delta \epsilon \) over the source surface, the displacement profile is given by \(u\left (x\right )\)\(=\)\(24\Delta \epsilon \sqrt {r^{2}-x^{2}}/\left (7\pi \right )\), where x is the radial distance from the centre of the crack and r is the radius of the crack (Eshelby, 1957). The maximum displacement is in the middle of the crack, i.e. at \(x=0\), and therefore, \(u_{max}\)\(=\)\(24r\Delta \epsilon / \left (7\pi \right )\). Integration over the crack length in polar coordinates, \(\left (x,\varphi \right )\), gives the mean displacement at source \(\bar {u}\)\(= 24\varDelta \epsilon /\left (7\pi ^{2}r^{2}\right ) \cdot \intop _{0}^{r}xdx \intop _{0}^{2\pi }d\varphi \sqrt {r^{2}-x^{2}}\), which translates to \(\bar {u}\)\(=\)\(48\varDelta \epsilon /\left (7\pi r^{2}\right ) \cdot \intop _{0}^{r}\sqrt {r^{2}-x^{2}}xdx\), and finally \(\bar {u}\)\(=\)\(16r\Delta \epsilon / \left (7\pi \right )\). This gives seismic potency, P\(=\)\(\bar {u}\pi r^{2}\)\(=\)\(\left (16/7\right ) r^{3} \varDelta \epsilon \), the source radius \(r = \left (7/16\right )^{1/3} \left (P/\varDelta \epsilon \right )^{1/3}\), and
where the constant \(q_{0}\)\(=\)\(\left (24/7\pi \right ) \left (7/16\right )^{1/3}\)\(=\)\(0.828494\). For the average strain change at source \(\varDelta \epsilon =5\cdot 10^{-4}\), the maximum displacement \(u_{max}\)\(=\)\(0.0053 \sqrt [3]{P}\), which is not far from \(u_{max}\)\(=\)\(0.0046 \sqrt [3]{P}\) given by McGarr and Fletcher (2003).
If we assume that the cumulative absolute displacement at the source is equal to the maximum source displacement, i.e. \(CAD_{0}=u_{max}=q_{0}\varDelta ^{2/3}P^{1/3}\), then at source, \(\overline {CAD}\left (P,R=0\right )=c c_{L}^{-c_{R}} P^{c_{P}-c_{R}/3}=CAD_{0}=q_{0} \varDelta \epsilon ^{2/3} P^{1/3}\). For \(c_{P}=\left (1+c_{R}\right )/3\), parameter c\(= q_{0} \Delta \epsilon ^{2/3} c_{L}^{c_{R}}\) is independent of potency P, and, after simple algebra, the GMPE for \(\overline {CAD}\) can be written as
Equation (6.19) has three parameters, \(\varDelta \epsilon \), \(c_{L}\), and \(c_{R}\), and since \(\varDelta \epsilon \) is constrained by the type of rock and can be assumed, the other two, \(c_{L}\) and \(c_{R}\), need to be inverted from data.
6.6.3 SGMPE Example
We analysed ground motion data from 265 seismic events in the potency range \(0.0 \leq \log P \leq 2.79\) recorded at distances between 87 and 996 metres from source that delivered 837 observations of PGV and CAD. The minimum PGV is 0.0218 cm/s, and the maximum 9.9 cm/s. The minimum CAD is 0.0028, cm and the maximum 0.512 cm.
Figure 6.20 shows \(\log \varDelta \epsilon \) vs. \(\log P\) and \(\log f_{0}\) vs. \(\log P\) on the background of different constant strain changes at source. The upper limit of strain change at source is assumed to be \(6\cdot 10^{-4}\).
Fig. 6.20
\( \log \varDelta \epsilon \) vs. \( \log P\)(left) and \( \log f_{0}\) vs. \( \log P\)(right) of events accepted for fitting
where PGV is in m/s, P in m\(^{3}\), and R in metres. The \(c_{R} =1.466\) with the standard errors of \(sd_{cR} =\)\(\pm 0.03576\) and \(c_{L} = 3.5872\) with the standard error \(sd_{cL} =\)\(\pm 0.37495\), and \(\sigma _{\log PGV}\)\(=\)\(0.25622\).
If we assume 5% uncertainty in \(v_{S}\) and 20% uncertainty in \(\varDelta \epsilon \), then the expected peak ground velocity at source varies between \(103.1 \leq PGV_{0} \leq 171.0\) cm/s, irrespective of \(\log P\). Figure 6.22 left shows the data selected for fitting with dots coloured by size range and the fitted model plotted in the middle of each PGV data range. Figure 6.22 right shows extrapolations for larger events.
Fig. 6.22
The GMPE-PGV fit and data (left) and predictions for larger potencies (right)
Figure 6.23 shows the results of residual analysis: \(\log \left (\text{Obs/Pred}\right )\) as a function of \(\log P\) and \(\log \left (\text{Pred}\right )\) vs. \(\log \left (\text{Obs}\right )\).
Fig. 6.23
\( \log \left (PGV_{obs}/PGV_{pred} \right )\) vs. \( \log P\)(left) and \( \log PGV_{pred}\) vs. \( \log PGV_{obs}\) (right )
where CAD is in m, P in m\(^{3}\), and R in metres, see Fig. 6.24. The \(c_{R} = 0.6083\) with the standard errors of \(sd_{cR} =\)\(\pm 0.02395\) and \(c_{L} = 0.3788\) with the standard errors of \(sd_{cL} =\)\(\pm 0.0959\) and \(\sigma _{\log CAD}\)\(=\)\(0.1908\). If, in addition, we assume 20% uncertainty in \(\varDelta \epsilon \), the expected maximum seismic displacement at source for \(\log P=3.0\) varies between 4.5 and 5.89 cm.
Fig. 6.24
The GMPE-CAD fit and data (left) and predictions for larger potencies (right)
Figure 6.24 left shows the data selected for fitting with dots coloured by size range and the fitted model plotted in the middle of each CAD data range. Figure 6.24 right shows extrapolations for larger events.
Figure 6.25 shows the results of residual analysis: \(\log \left (\text{Obs/Pred}\right )\) as a function of \(\log P\) and \(\log \left (\text{Pred}\right )\) vs. \(\log \left (\text{Obs}\right )\).
Fig. 6.25
\( \log \left (CAD_{obs}/CAD{ }_{pred} \right )\) vs. \( \log P\)(left) and \( \log CAD_{pred}\) vs. \( \log CAD_{obs}\) (right )
The velocity of ground motion at the source of a seismic event is mostly independent of its size, i.e. large and small events have similar velocities of ground deformation at source. However, small events affect smaller volumes of rock and radiate waves of higher frequencies that are attenuated faster than lower frequencies. Therefore, smaller events are perceptible over shorter distances.
Humans experience the effect of ground motion as a movement of the ground and as a sound. Large seismic events, e.g. with \(\log P\geq 3.0\) (\(m_{HK}=2.95\)), radiate most of their energy in a frequency band of 5 to 20 Hz and are easily felt as a movement over a distance of a few kilometres. Small events, e.g. with \(\log P=-2.0\) (\(m_{HK}\simeq -0.38\)), which can be seen as a 15 metre size crack, radiate most of their energy at frequencies of 100 Hz and more and can be felt and/or heard underground at 100 metres away.
In general, ground motions lower than 1 mm/s are hardly perceptible as movement, but, by interacting with the environment, they may generate a perceptible noise. This is what happens underground when waves generated by a seismic event interact with the fracture zone around a tunnel and with support elements and generate an audible noise. Ground motion between 1 and 10 mm/s is perceptible as a movement and higher than 10 mm/s becomes unpleasant. Ground motion stronger than 10 cm/s can cause local falls of ground or strain bursts and above 50 cm/s can cause rock failure and damage to underground excavations. Since localised blasts generate PGV at higher frequencies, therefore producing less displacement, they will be felt as movement over shorter distances, but their high frequency content will generate more audible noise. In general, the same PGV generated at lower frequency will be felt as movement over larger distance.
Figure 6.26 illustrates the perceptibility and the expected damage potential to the underground structure by different levels of PGV with superimposed SGMPE for selected \(\log P\).
Fig. 6.26
Perceptibility of ground motion illustrated on the GMPE shown in Fig. 6.22
The basic outcome of ground motion hazard analysis for a given site is a seismic hazard curve that shows the annual rate, or probability, at which a specific ground motion level will be exceeded. This is outside the scope of this chapter. It is expected that CAD, which includes the peak and the duration of ground motion, may be a better indicator of damage potential than the PGV alone, being a single measurement over the whole waveform. Below we present two simple applications: the potential damage inspection plot and the cumulative CAD plot.
From the simple GMPE-PGV given by Eq. (6.17), we can calculate the distance, R, over which a seismic source with potency P generates the velocity of ground motion \(\geq PGV\),
A similar equation can be derived from GMPE-CAD. We can also calculate the minimum potency, or \(\log P\), that delivers a given level of PGV as a function of distance,
Now we can plot \(\log P\) vs. distance R of seismic events, on the background of envelopes of a minimum \(\log P\) that delivers a given level of PGV as a function of distance, for a number of strategic sites.
We analysed the last 60 days of seismic history before a \(\log P=2.61\) event at a mine here referred to as MineD, see the seismic hazard case study described in Chap. 5 and subsection Example in Chap. 3, and Sect. 6.6.8.3 below. The size of the event scales with the radius of source volume, and the colour indicates the distance of the event to the site.
Figure 6.27 left shows events with \(\log P\geq -1.0\) vs. distance and the thresholds of ground motion, \(PGV_{x}\), set as 1, 5, and 10 cm/s. The envelopes of a minimum \(\log P\) stop where, according to the SGMPE, a \(\log P_{max}\) event cannot deliver a given PGV beyond that distance. A seismic event that crosses the calibrated envelope for a given site may trigger damage inspection. Note that the developed GMPE does not take into account site effects, i.e. the amplification of ground motion at the skin of excavations, and therefore all these estimates are in solid rock and most likely underestimated.
Fig. 6.27
897 events with \( \log P \geq -1.0\) that occurred between 08 May and 07 July vs. distances to three sites at the mine. The orange, red, and darker red envelopes indicate a minimum \( \log P\) that delivers \(PGV=1\), \(PGV=5\), \(PGV=10\) cm/s, respectively, as a function of distance
Another useful application is monitoring the consumption of the deformation capacity of the support due to seismicity. Figure 6.28 right shows the cumulative seismic deformation, CumCAD, due to seismic events with \(\log P\geq -1.0\) that exceeded \(PGV=10^{-7}\) m/s, at the same three sites over the same period as in Fig. 6.27. It shows a big jump in CAD at site S1, which was deliberately located at the source of \(\log P=2.61\) event that delivered 4.36 cm of CAD.
Fig. 6.28
Estimates of cumulative co-seismic displacement at the same sites, S1 (left), S2 (centre), and S3 (right), over the same period of time
It shows that Site1 was subjected to a relatively low level of seismic deformation before the main shock. Site 2 accumulated 6.67 cm and 1.52 cm of that was due to the \(\log P=1.24\) event that located very close to this site. Site S3 accumulated 7.12 cm, and the biggest jump was the \(\log P=2.61\) main shock that contributed 0.42 cm. Again, the observed CAD were recorded by sensors in boreholes, and therefore, they do not take into account the amplifying effect of the fracture zone close to excavations and the reaction of the support.
6.6.6 Probability \(\Pr \left ( \geq PGV,R,\varDelta T\right )\)
To calculate the probability that ground motion may exceed a given threshold at distance R from a source, regardless of source location, within the time interval \(\varDelta T\), we need to replace potency in Equation 6.20 for the probability of having an event \(\geq P\) within the time interval \(\Delta T\),
with the potency P derived from the GMPE. The potency as a function of PGV and distance, \(P\left (PGV,R\right )\), can be derived analytically from Equation 6.17 for SGMPE-PGV ,
Figure 6.29 left shows the distance over which a given \(\log P\) generates ground velocities \(PGV\geq \) 15, 10, and 5 cm/s. Figure 6.29 right shows the probabilities of having ground velocity \(PGV\geq \) 15, 10, and 5 cm/s within 180 days, as a function of the distance from the source.
Fig. 6.29
Distance over which a given \( \log P\) generates ground velocity \( \geq PGV\)(left), and \(\Pr \left ( \geq PGV,R,\varDelta T \right )\) as a function of distance from the source for three PGV thresholds (right)
6.6.7 Seismic Fragility Curves and Damage Potential
Fragility curves are functions that describe the conditional probability that the system may be subjected to a different degree of damage over the full range of loads to which that system might be exposed. The probability of damage is a function of both uncertainty in the capacity and uncertainty in the demand, and as demand increases relative to capacity, the probability of damage approaches one.
It is usually assumed that uncertainty in the capacity term follows a log-normal distribution, and therefore, the fragility curve also follows a log-normal distribution. Conforming to the accepted convention, the equation for the fragility function can be written as
where D is an uncertain damage state of the accepted damage classification, d is a particular value of D, \(\Pr \left (D\geq d\big |X=x\right )\) is the conditional probability that \(D\geq d\) is true given \(X=x\) is true, X is the engineering demand parameter, x is a particular value of X, \(F_{d}\left (x\right )\) is a fragility function for damage state d evaluated at x, \(\Phi \) is the standard normal cumulative distribution function, \(\theta _{cd}\) is the median capacity of the structure to resist damage state d, and \(\beta _{cd}\) is the standard deviation of the natural logarithm of the capacity of the structure to resist damage state d (Porter, 2018). In this formulation, only the median and standard deviation of the capacity are required to define the fragility function. For ground motion hazard, Eq. 6.27 can be written as
where the engineering demand parameters GMP can be PGV , PGA, or CAD. The mean capacity of the structure to resist damage \(\overline {GMP}_{c} = GMP_{c} \exp \left (\beta _{c}/2\right )\), and the standard deviation \(\sigma _{GMP_{c}} =\)\(\overline {GMP}_{c} \sqrt {\exp \left (\sigma _{lnGMP_{c}}\right )^{2}-1}\).
The nature of the log-normal fragility function dictates that \(F_{d}\left (GMP=GMP_{c}\right )\)\(=0.5\), and the uncertainty in capacity \(\sigma _{\ln GMP_{c}}\) dictates how steep is the fragility curve. If there is little uncertainty in capacity of the system, the fragility curve will be steep, i.e. there is great degree of certainty that the system will fail at that load. This situation applies to brittle or to well-understood systems. For complex inhomogeneous systems, e.g. the support of u/g tunnels in seismically active mines, the uncertainty in the capacity, \(GMP_{c}\), is larger and the fragility curves are flatter.
Conventionally, the capability of geotechnical structures is evaluated by the design factor of safety, in this case, the ratio of the design capacity to the expected demand, \(F_{S} = GMP_{c}/GMP_{d}\). The demand should be estimated for the maximum expected potency, \(P=P_{max}\) or \(\log P=\log P_{max}\). Obviously, all structures are designed to a factor of safety greater than one to provide an adequate margin of safety, \(M_{S} = GMP_{c} - GMP_{d}\). The SGMPE allows not only to estimate the expected demand in terms of PGV or CAD, but also the expected linear extent of damage, \(R_{d}\left (GMP_{c},P\right )\), in metres in case the demand exceeds the in situ capacity, \(GMP_{d} \geq GMP_{c}\),
which may be of interest to underground tunnels. Equation (6.29) gives \(R_{d}=0\) when the demand is equal to capacity, i.e. \(GMP_{d} = GMP_{c}\), which is the limiting case, without a margin of safety. Any further increase in demand leads to a damaged state, i.e. \(R_{d}>0\). Negative distances, \(R_{d}<0\), indicate a positive margin of safety.
Figure 6.30 shows an example of the fragility curves for four different capacities in terms of \(PGV_{c}\)(left) and \(CAD_{c}\)(right) assuming the structure is at or very close to a source of event with \(\log P = 2.0\), \(2.5\), and \(3.0\).
Fig. 6.30
Fragility curves for four different in situ capacities in terms of \(PGV_{c}\)(left) and \(CAD_{c}\)(right). Distances \(R_{d} \left (PGV_{c}, \log P \right )\) and \(R_{d} \left (CAD_{c}, \log P \right )\) within which the demand exceeds the capacity are calculated for three different \( \log P\) using SGMPE. Dashed lines indicate the range of probabilities for assumed \(\sigma _{\ln GMP_{c}}\)
We assumed \(0.5 \leq \sigma _{\ln PGV_{c}} \leq 0.6\) and \(0.4 \leq \sigma _{\ln CAD_{c}} \leq 0.5\). The PGV graph also shows the corresponding dynamic co-seismic strains, \(\epsilon _{d} = PGV/v_{S}\) as a reference, and the distances \(R_{d}\left (PGV_{c},P\right )\) and \(R_{d}\left (CAD_{c},P\right )\) within which the in situ demand exceeds, \(R_{d}>0\), or is below, \(R_{d}<0\) the capacity of the support. The parameters of the SGMPE are \(v_{S} = 3590\) m/s, \(\varDelta \epsilon =6\cdot 10^{-4}\), and \(c_{RPGV} = 1.466\), \(c_{LPGV} = 3.587\), \(c_{RCAD} = 0.6083\), \(c_{LCAD} = 0.3788\). At source, the \(PGV_{0} = 0.63v_{S}\varDelta \epsilon = 1.357\) m/s is independent of \(\log P\), and \(CAD_{0} = q_{0}\varDelta ^{2/3}P^{1/3}\) is a function of \(\log P\) and gives 0.0274, 0.04, and 0.0589 metres at source for \(\log P = 2.0\), \(2.5\), and \(3.0\), respectively.
The PGV fragility curves in Fig. 6.30 left show the probability of damage for four different in situ capacities, \(PGV_{c} =\) 0.8, 1.0, 1.2, and 1.4 m/s. For the lowest capacity of 0.8 m/s, the distances within which the demand exceeds the capacity are 7.2, 10.6, and 15.6 m for \(\log P=2.0\), \(2.5\), and \(3.0\), respectively. For \(PGV_{c} =1.0\) and \(1.2\), the probabilities of having damage are still over 50%. The minimum capacity to lower the probability of damage to below 50% by these events is 1.4 m/s. The PGV fragility curves are of limited practical value since it is difficult to establish the remaining in situ PGV , or the energy capacity, of the support.
The CAD fragility curves in Fig. 6.30 right are of more practical value because it is easier to assess the remaining in situ deformation capacity of support. The respective factors and margins of safety for CAD are given in Table 6.5.
Table 6.5
Factors and margins of safety for limited CAD fragility curve given in Fig. 6.30
Factors and margins of safety, \(F_{S} | M_{S}\)
\(\log P\)
\(CAD_{c}=3\) cm
\(CAD_{c}=4\) cm
\(CAD_{c}=5\) cm
\(CAD_{c}=6\) cm
2.0
1.097 \(|\) 0.003
1.462 \(|\) 0.013
1.828 \(|\) 0.023
2.193 \(|\) 0.033
2.5
0.747 — \(-\)0.010
0.996 — \(-\)0.000
1.245 — 0.010
1.494 — 0.001
3.0
0.509 — \(-\)0.029
0.679 — \(-\)0.019
0.848 — \(-\)0.009
1.018 — 0.001
6.6.8 Seismic Ground Motion Alert Program—GMAP
6.6.8.1 Introduction
This section is based on Mendecki (2023). Seismic risk is associated with the probability of a loss, which is the product of seismic hazard, the vulnerability of a site, and the exposure. Seismic hazard is the probability of having potentially damaging ground motion at a given site at a future time. The vulnerability of that site is its ability to sustain a certain level of ground velocity and ground displacement. The exposure is defined by the elements at risk which, in the case of safety, is the number of people that could be affected by a potential rockburst.
While the prediction of the time and the location of a potentially damaging seismic event is impossible, one should limit people’s exposure at times of increased seismic activity and/or increased seismic loading by mid-size or larger events close to working places. The reason is very simple, as the rate of seismic activity increases so does the likelihood that one of these events may be larger and damaging.
One of the objectives of seismic monitoring in mines is “To detect strong and unexpected changes in the spatial and/or temporal behaviour of seismic parameters that could lead to rock mass instability and affect working places immediately or in the short term”, see Mendecki (2016) section 1.2. By “strong”, we understand changes exceeding a specified reference level, and the “unexpected” here means spontaneous, not associated with blasts or other controlled mining activities that usually trigger or induce such changes.
After blasting or after larger seismic events, the seismic activity rate is expected to increase, and over the years mine seismologists have developed different methods to monitor and quantify its decay in time and space to allow safe re-entry of personnel (e.g. Spottiswoode, 2000; Turner & Player, 2000; Malek & Leslie, 2006; Woodward et al., 2017; Gospodinov et al., 2022). While some methods use a combination of different seismic parameters, the fundamental one is always the seismic activity rate. One such method, which also includes guidelines for developing a re-entry protocol, is described in Vallejos (2010). Mendecki (2016), section 4.2, described a method to estimate the probability that the seismicity rates in two different time intervals are different by a given factor. Nordstrom et al. (2020) back analysed the utility of 14 different parameters as short term hazard indicators or as early warnings using data from Kiirunavaara mine in Sweden. They concluded that the most successful parameters are accelerated moment, accelerated apparent volume release, and increased activity rate. However, in many cases, the first two parameters are associated with the third one.
The short term hazard associated with aftershock activity quantified in terms of the probability of having an event above a certain size can be estimated by applying a rate function, e.g. the Omori or the stretched exponential, and the size distribution of seismicity for the given area to clusters of events, see Mendecki (2016) section 5.3.
An alert is raised if a given parameter, or a set of parameters, exceeds the imposed reference level(s). The reference level can be estimated by taking an average of a given parameter over times that were outside the influence of blasting and larger events and when there was normal safe working activity in a given area. In the case of seismic activity, it is expected that the coefficient of variation of the data selected to estimate the reference activity will be close to 1.0, i.e. not far from Poissonian. It is important that the reference level takes into account the current vulnerability of the site, or excavation. For example, an area with lower deformation capacity of support and/or a wider span of excavation should have a lower reference level.
Seismic exposure also needs to be defined in space; therefore, we need to be able to delineate the exclusion zone. In small mines, the exclusion zone can be the whole mine, but this is not practical for large mining operations. After an alert has been issued, there is a need to de-alert, and this again includes time and space. After some time, a part of the excluded area may be below the reference level, while others, specifically those close to the sources of larger events or blasts, may not. Typically, the exclusion zone volume increases as the size of the main shock increases. The exclusion time increases with the size of the main shock and decreases as the distance to the main shock increases.
All methods described above are the so-called polygon-based, where the polygon is defined as a seismogenic volume that generates seismicity affecting working places. Therefore, all relevant parameters are derived from the data selected from this polygon. This method has been widely applied in mines for many years. The most difficult, and also subjective, task here is the definition of the polygon. Different polygons select different datasets and therefore will produce different seismic characteristics.
In this section, we describe the polygon-less approach, where one takes into account the influence of all available seismic events, regardless of their location, on a particular working place. The preferred measures of influence are the rates of the following two ground motion parameters: (1) The cumulative absolute displacement, measured over a given minimum ground velocity, CAD\(=\)\(\intop _{0}^{t_{d}}|\mbox{v}\left (t\right )|_{\geq \mbox{v}_{min}}dt\). (2) The ACAD, which is the rate of CAD. The influence of \(CAD_{Rate}\) and \(ACAD_{Rate}\) is moderated by the distance from the seismic source to the place of potential exposure. For example, a \(\log P=2.0\) event generates a similar level of PGV at 200 metres away as a \(\log P=1.0\) at 100 metres, and a \(\log P=2.0\) event generates a similar level of CAD at 200 metres away as a \(\log P=1.0\) at 50 metres.
6.6.8.2 GMAP Algorithm
GMAP is an influence based polygon-less two parameter method where one takes into account the influence of ground motion generated by all available seismic events, regardless of their location, on a particular working place. It is based on the rates of the cumulative absolute deformation, CAD calculated above a given ground velocity threshold, \(CAD_{Rate}\), and on the activity rate of CAD events, \(ACAD_{Rate}\). See Sect. 6.4.1 for more on CAD.
The following few steps describe the procedure to calculate GMAP ratings for each site defined by its coordinates \(\left (x,y,z\right )\), given the alert reference rates, \(CAD_{Rref1}\) and \(ACAD_{Rref1}\), and alarm reference rates, \(CAD_{Rref2}\) and \(ACAD_{Rref2}\). In this case study, alarm reference rates are set as 2 times the alert reference rates.
First define the moving time window, \(\Delta t\), needed to calculate the rates. The window can be constant in time, or it may be defined by a number of events. For all events in the window:
1.
Calculate the threshold seismic strain, \(\epsilon _{s}\)\(=\)\(\text{v}_{min}/v_{S}\), and CAD using the relevant GMPE.
2.
If for a given event seismic strain is greater than or equal to \(\epsilon _{s}\), accumulate CAD to get \(\Sigma CAD\) and calculate its activity, ACAD, defined by \(N_{CAD}\), the number of times CAD is accumulated.
Normalise\(CAD_{Rate}\)and\(ACAD_{Rate}\). They havedifferent units, m/s and 1/s, but can be compared when normalised relative to their individual alert and alarm reference rates.
Take \(\max \left [CAD_{RateN},ACAD_{RateN}\right ]\) to get the GMAP rating.
6.
Repeat for each site and contour and/or move the window by an increment of time, or by one event, to create the time series of GMAP.
Note that any seismic activity or ground motion based alert or re-entry protocol is not a prediction. The method described here is also not a forecast, since it does not state the probabilities of occurrences. GMAP just responds to changes in the selected ground motion parameters.
\(CAD_{Rate}\) will react to small events located close to a given site and to larger events located up to a few hundred metres away. \(CAD_{Rate}\) is also sensitive to an increase in the frequency of mid-size events that are not associated with an overall increase in seismic activity. A single larger event located nearby should issue an instant alert/alarm at that site. \(ACAD_{Rate}\) is sensitive to an increase in the activity rate influencing a given site.
Note that CAD excludes any time dependent aseismic deformation, e.g. bulking which is a function of time and may be delayed. Also the support system prevents part of that deformation from reaching the skin of excavation, so one cannot see or measure it, but this kind of seismic action contributes to the increase in the fracture zone around excavations and may contribute to damage when hit with a larger event.
\(CAD_{Rate}\) and \(ACAD_{Rate}\) are calculated in a moving window. The moving window can be defined by a fixed number of events, in which case its duration varies with activity rate, or by a fixed time window. If after alert, the activity rate unexpectedly drops, the fixed number of events window will drop the GMAP rating very slowly, while with the fixed time window the rating will drop quickly. If the GMAP rating is dominated by one larger event with few aftershocks, then the rating will drop as soon as the large event drops from the moving window.
6.6.8.3 GMAP Example
GMAP Data
We analysed the last 60 days of seismic history before a \(\log P=2.61\) event at a mine here referred to as MineD, see the seismic hazard case study described in Chap. 5 and subsection Example in Chap. 3. Figure 6.31 left shows the convex hull span over all 6073 events with \(\log P\geq -4.5\) available for analysis. The size of the event represents the radius of the source volume taken as a sphere, \(V=P/\Delta \epsilon \), where \(\Delta \epsilon \) is the assumed strain change at the source, in this case \(\Delta \epsilon =10^{-4}\), and the colour indicates the time of the event, from the earliest in blue to the latest in red. In this section, we will test GMAP at site S3, which is the average location of all 6073 events. Figure 6.31 right shows the convex hull span over 3578 events that generated \(PGV \geq 10^{-7}\) m/s at the centre of all 6073 events, shown as S3 in blue.
Fig. 6.31
Convex hull span over all available 6073 events (left) and over 3578 events that generated \(PGV \geq 10^{-7}\) m/s at site S3 (right)
Distances of these events to site S3 range between 12 and 394 m. Tables at the bottom of these figures give the number of events, \(N_{E}\), activity rate/day, the \(\log P\) range, the volume of the convex hull, \(V_{CH}\), and its sphericity index, \(\Psi _{CH}\), the total volume of seismic sources with strain change \(\geq \) 0.0001, \(V\left (\Delta \epsilon \geq 10^{-4}\right )\), and the ratio of the convex hull to the volume of inelastic deformation of events \(V_{CH} / V\left (\Delta \epsilon \geq 10^{-4}\right )\). After selection of the 3578 events, \(V_{CH} / V\left (\Delta \epsilon \geq 10^{-4}\right )\) ratio dropped from 177.2 to 11.67, which indicates that the volume selected for stability is reasonably saturated with co-seismic inelastic deformation. The sphericity index, \(\Psi _{CH}\), here dropped slightly from 0.86 to 0.83, however, while running moving windows through these 3578 events it varied between 0.65 and 0.86.
Figure 6.32 shows the cumulative number of events, CumN, and the cumulative apparent volume, Cum\(V_{A}\) in km\(^{3}\), for all 6073 events. The size of the event represents the radius of the source volume and colour scales with distance to the site. There were 17 mid-size events with \(\log P\geq 0.0\) which gives an activity rate 0.28/day, and the coefficient of variation of all events is \(C_{v}=1.32\), which indicates a degree of time clustering. The vertical red, blue, and green lines indicate times of the three largest events, respectively. The CumN vs. time plot is quite steady with an almost constant activity rate, while Cum\(V_{A}\) shows a bit more structure and an increase in its rate before the main shock. However, the apparent volume does not enter into GMAP.
Fig. 6.32
CumN(left) and Cum\(V_{A}\)(right) vs. time plotted for all 6073 events
The largest event with \(\log P=2.61\) occurred on 07 July and located 130 m from site S3, the second largest with \(\log P=1.24\) on 14 June located 99 m away, and the third largest with \(\log P=1.1\) on 08 June 137 m away. The distance between the largest event and the second largest is 273 m, the largest and the third largest 85 m, and the second and the third largest 211 m. The distance between an event and a given site is taken as the Euclidean distance minus the radius of the event, \(r =\)\(\sqrt [3]{3P/\left (4\pi \Delta \epsilon \right )}\), where the strain change, \(\Delta \epsilon \), is the same used in the development of the SGMPE, in this case 6\(\cdot \)10\(^{-4}\).
Time History of GMAP Parameters
Figure 6.33 top row shows the cumulative CAD and \(N_{CAD}\) vs. time. The vertical red, blue, and green lines on cumulative plots here indicate times of the three largest CAD events, respectively, i.e. events that generated the three largest seismic deformation at site S3. Figure 6.33 bottom row shows normalised \(CAD_{Rate}\) and A\(CAD_{Rate}\) and the resulting GMAP alerts.
Fig. 6.33
Cumulative CAD and \(N_{CAD}\) vs. time, (top row), and time histories of normalised \(CAD_{Rate}\), \(ACAD_{Rate}\), and resulting GMAP alerts (bottom row)
The reference rate \(CAD_{Rref1}\) was set as 0.072 cm/day and \(ACAD_{Rref1}\) as 44 CAD events per day. Alarm levels were set as double the alert levels. Alert plots are based on a fixed number 119 event moving window, which is equal to 2 times the average activity rate per day.
Of the total 6073 seismic events only 3578 classified as CAD events because most of them were too far and/or too small to induce \(PGV \geq \)10\(^{-7}\) m/s at the site. Most of these events located between 20 and 120 metres from site S3. The total CAD over that time was 7.12 cm, and the largest event with \(\log P=2.61\) that was 130 metres away imposed 0.42 cm of CAD. The second largest CAD was 0.09 cm associated with event with a \(\log P=1.24\) located 99 metres away.
The selected dataset is characterised by a reasonably stable activity rate which always makes alerting more difficult. However, the data has a significant number of mid-size events not far from the site of interest, therefore, with the exception of the last three days before the main shock, GMAP alert time history is dominated by \(CAD_{Rate}\) rather than \(ACAD_{Rate}\).
The largest event with \(\log P=2.61\) occurred on 7 July at 15:38:57. The first alarm was on 02 July, then there were seven additional alarms, and the last one on 7 July at 15:30:40. There was also a spike type alarm exactly at the time of the event.
The second largest event with \(\log P=1.24\) and \(CAD=0.1\) cm occurred on 14 June at 04:36:38. There was only one Alarm before the event on 12 June at 09:52:48, and there was a spike type of alarm at the time of event.
The third largest event with \(\log P=1.1\) and \(CAD=0.06\) cm occurred on 8 June at 18:47:01 at distance 137 metres from site S3. At the time of this event GMAP was in a state of alert. There was the alarm issued on 06 June at 13:41:48, then three Alarms on June 07, and the last one on 08 June at 09:23:03.
Figure 6.34 shows the cumulative CAD and \(N_{CAD}\) for the last 68.35 hours before the largest event of \(\log P=2.61\), excluding the main shock.
Fig. 6.34
The last 68.35 hours of CumCAD(left) and Cum\(N_{CAD}\) before the main shock (right)
During this time, there were 238 events giving the activity rate 83.56 events/day. The maximum event was with \(\log P=-0.12\) that occurred 47 hours before the main shock at a distance of 97 metres and generated 0.17 mm of CAD. There was a clear acceleration in the cumulative CAD and \(N_{CAD}\) during the last 6 hours before the main shock. However, it was partially smoothed over by the 119 events moving window. The shorter window would have alerted earlier, but it would also have issued more false alarms.
Comparison with Stability Analysis
Figure 6.35 left shows the time history of GMAP alerts, which is based on \(CAD_{Rate}\) and \(ACAD_{Rate}\). Figure 6.35 right shows a simple version of the stability function for site S3, \(\Psi _{CH} \cdot \text{Median}\left [\log \sigma _{A}\right ] / \lambda \), where \(\sigma _{A}\) is apparent stress, \(\lambda \) the activity rate, and \(\Psi _{CH}\) sphericity of the convex hull span over the events in each moving window.
Both cases are based on the same dataset, the PGV threshold of 10\(^{-7}\) m/s, and the moving window of 119 events. See Sect. 3.5.2 in Chap. 3 for more on the examples of stability analysis. Note that the low level of the stability function indicates a less stable rock mass with higher potential for larger events or a swarm of events. In such cases, GMAP alerts go up. It is clear that these two plots are qualitatively similar, or inversely correlated, i.e. both indicating increase potential for larger ground motion at the site at the same times. This is interesting because with the exception of the activity rate, \(\lambda \), these two plots are based on different parameters. Let us consider the same three events we analysed above:
The largest event with \(\log P=2.61\) on 7 July. From 02 July, GMAP is practically at alarm level, and the stability function is well below the 30 days mean.
The second largest event with \(\log P=1.24\) on 14 June. GMAP is high and stability is low.
The third largest event with \(\log P=1.1\) on 8 June. GMAP is high and stability is low.
6.6.8.4 General Comments
A prediction can be understood as a deterministic binary statement, true or false, about a future event that can be validated or falsified with a single observation. A forecast can be defined as a statement of probability about a future event. An individual forecast can never be validated by a single observation and requires multiple observations to establish a degree of confidence. As mentioned above, GMAP is neither a prediction nor a forecast, since a forecast requires stating the probabilities of occurrences.
The utility of GMAP is not only in issuing ground motion alerts or alarms after a sudden increase in GMAP rating, or in delineating the exclusion zones, but also in guiding control measures to mitigate seismic hazard. If the GMAP rating is systematically increasing or stays high in a given area, then the mine may change the spatial and temporal manner of rock extraction, e.g. change the sequence of blasting, scatter the production blasts, and/or slow down the rate of mining. There is a view that these control measures just delay the inevitable. However, experience shows that scattered rock extraction changes the nature of seismic release by producing more smaller or mid-size events and fewer large ones (van Aswegen & Mendecki, 1999; Handley et al., 2000; Vieira et al., 2001; Mendecki, 2001 Figure 8; Mendecki, 2005; Durrheim et al., 2005). Therefore, if guided by GMAP, the mine applies control measures to manage the seismic response, then the GMAP success or failure rate cannot be tested by the number of larger events that did or did not occur after alerts or alarms. In such cases, it would be more appropriate to measure it by the overall positive changes in the size distribution of seismic events. Paradoxically then, the case where there is no larger event after GMAP alerts would be considered a success. Like any other alerting method, GMAP has limitations:
1.
GMAP may fail to alert for mid-size events located far, typically a few hundred metres from a given site, if there is no increase in seismic activity within that distance. During testing, GMAP never failed on a large event located within a few hundred metres from a given site; however, it may alert or alarm at the time of the event.
2.
It needs to be calibrated, mainly the alert rate reference levels, \(CAD_{Rref1}\) and \(ACAD_{Rref1}\), and the alarm multiplier.
3.
GMAP requires a reasonably accurate GMPE that should be updated at least once a year.
4.
It relies on seismic events being processed before the rating can be updated, and therefore, it depends on the speed and the quality of seismological processing, i.e. it is not a real-time system.
6.7 Seismic Ground Motion Alert System—GMAS
GMAS, like GMAP, is based on the two ground motion parameters, \(CAD_{Rate}\) and \(ACAD_{Rate}\), but it computes these rates in real time from the waveforms provided by seismic sensors at a given location. Unlike GMAP, GMAS does not need a ground motion prediction equation, is not subject to association, and does not rely on the seismological processing that delays the process and introduces uncertainties, and therefore, it gives instantaneously local alerts or alarms when given threshold parameters are exceeded.
The following few steps describe the procedure to calculate GMAS ratings for an area around the GMAS sensor, given the alert reference rates, \(CAD_{Rref1}\) and \(ACAD_{Rref1}\), and alarm reference rates, \(CAD_{Rref2}\) and \(ACAD_{Rref2}\), which in this case study are set as 2 times the alert reference rates.
Define the moving time window, \(\Delta t\), needed to calculate the rates. The window can be constant in time, or it may be defined by a number of events. For all events in the window:
1.
Take a segment, say \(\delta t=0.25\) seconds long, of the continuous data stream as recorded by a seismic site and test if the PGV in this section exceeds a predefined threshold, say \(PGV\geq \)10\(^{-5}\) m/s. If so, declare a ground motion event.
2.
Integrate that segment of the waveforms to get CAD for the GM event.
3.
Accumulate CAD in the window to get \(\Sigma CAD\) and calculate its activity ACAD defined by \(N_{CAD}\), and the number of times CAD is accumulated.
Normalise\(CAD_{Rate}\)and\(ACAD_{Rate}\). They havedifferent units, m/s and 1/s, but can be compared when normalised relative to their individual alert and alarm reference rates.
Take \(\max \left [CAD_{RateN},ACAD_{RateN}\right ]\) to get GMAS rating.
GMAS Example
We analysed 15.36 hours of continuous three-component waveforms recorded at Site10 by 4.5 Hz geophones installed at 8 metres in a borehole and sampled at 6 kHz,. The first sample was at 00:00:09 and the last at 15:21:32. A seismic event with \(\log P=1.1\)\(\left (m=1.65\right )\) and \(\log E=6.4\) occurred at 10:21:43:608 and located 196 metres from the sensor. Figure 6.36 shows the recorded velocity and integrated displacement waveforms. The long tail on the displacement waveforms may indicate a permanent displacement.
Fig. 6.36
Velocity and displacement waveforms of the \( \log P=1.1\) event
Figure 6.37 shows the time history of PGV , CAD, and CumCAD of all 221087 segments of data with the red lines marking the mean values. The mean rate of cumulative CAD was 1.21 cm/day. There were a few bursts of CAD activity in the morning, but the CumCAD rate plot remained structureless, it increased moderately at 07:30, but then remained steady until the main shock. Obviously, the cumulative number of all segments of data here would be just a straight line.
Fig. 6.37
PGV(left),CAD(centre), and CumCAD(right) time histories of all 221087 segments of data
Figure 6.38 shows the CumCAD and Cum\(N_{CAD}\) vs. time where we selected segments of data with \(CAD \geq \overline {CAD} =\)3.5\(\cdot \)10\(^{-6}\) cm and with predominant frequency less than 150 Hz. There are 16751 such GM events, and now the data show more interesting structure, including a significant increase in both, the rate of CumCAD, and the rate of Cum\(N_{CAD}\) approximately 3 hours before the \(\log P=1.1\) event. The largest \(PGV=5.67\) cm/s, and the second largest \(CAD=0.031\) cm occurred at 10:21:43.358. The second largest \(PGV=2.24\) cm/s and the largest \(CAD=0.033\) mm occurred at 10:21:43.608, all of them associated with the \(\log P=1.1\) event.
Fig. 6.38
CumCAD(left) and Cum\(N_{CAD}\)(right) of GM events
Reference alert level for \(CAD_{Rate}\) was set at 0.3042 cm/day, and alarm level was set as 2 times the alert level. Reference alert level for \(ACAD_{Rate}\) was set at 41850 GM events/day and alarm level at double the alert level. The moving window was set at 350 GM events, see Fig. 6.39.
Fig. 6.39
Time histories of \(CAD_{rate}\) and \(ACAD_{rate}\)
Here we assumed the normalised GMAS alert level as 1 and alarm level as 2. Figure 6.40 shows time histories of normalised \(CAD_{Rate}\) and \(ACAD_{Rate}\), in this case dominated by \(ACAD_{Rate}\), and time histories of the final GMAS alerts.
Fig. 6.40
Time histories of normalised \(CAD_{rate}\), \(ACAD_{rate}\)(left) and resulting GMAS alerts (right)
The first GMAS alert was at 07:31:29 followed by an alarm at 07:38:16, and the GMAS stayed there practically till 09:24:54 when it dropped all the way to below the alert level. Then after a short spike above the alert level at 09:50:53, the GMAS rating jumped to alarm level at 10:01:55 and stayed there during the main shock and de-alerted only at 10:50:18.
The \(PGV{\,=\,}\) 5.67 cm/s associated with the main shock was recorded by the sensor embedded at 8 metres in a borehole, and it would be amplified at the surface of the excavation. This level of ground motion may not be damaging, but it would certainly be felt strongly by people close to this site, and they would spontaneously evacuate the area. There was another GMAS alarm at 14:01:17 triggered by a flurry of smaller GM events with maximum \(PGV{\,=\,}0.08\) cm/s and max \(CAD{\,=\,}0.0073\) cm.
The area where the sensor is located is not well covered by the seismic system, and there is limited seismic data available; therefore, GMAP and stability analysis would not alert for this event at that site.
General Comments
Like GMAP, GMAS is not a prediction, and it is also not a forecast since it does not state the probabilities of occurrences. GMAS just responds to changes in the local ground motion parameters, but, unlike GMAP, it works in real time. Like any other alerting method, GMAS has limitations:
1.
GMAS is an inexpensive way to monitor and Alert/Alarm on stronger, potentially damaging ground motion. It does not need a ground motion prediction equation, is not subject to association, and does not rely on seismological processing, and therefore, it works in real time and, in cases of complex seismic events, may give a short notice to evacuate.
2.
GMAS is local and in most cases will Alert/Alarm personnel to frequent mid-size events relatively close to the excavation. Its area of influence will scale positively with the recorded intensity of GM and its frequency.
3.
If run on a number relatively closely spaced sites, it forms a real -time ground motion hazard monitoring system for the area.
4.
One needs to define what constitutes a false alarm. This is an important question because it affects the calibration process. It would certainly be the case if an alarm goes off and people in the area do not feel anything. But it may not be the case when GMAS alarms on GM that make people uncomfortable, and they would evacuate anyway.
5.
GMAS needs to be calibrated, mainly the alert rate reference levels, \(CAD_{Rref1}\) and \(ACAD_{Rref1}\). One can easily calibrate GMAS to Alert/Alarm at the time of a large GM event. However, it is difficult, if not impossible, to eliminate alarms on weaker but perceivable ground motion that may affect a small area. One can reduce the number of such alarms by increasing reference levels and/or the duration of the moving window. However, it is also advisable to define the number and the level of alerts and/or alarms in a given time span before advising personnel to evacuate. This should be a part of the calibration process that then feeds into the re-entry protocol.
6.8 Mapping Seismic Ground Motion Hazard
This section is based on the key note lecture delivered at the Rockburst and Seismicity in Mines Symposium in Santiago, Chile Mendecki, 2017.
The size distribution analysis disregards space; therefore, the given probabilities apply to the whole mine, while in many cases it is quite obvious that seismic hazard varies considerably in space, and these differences should be taken it into account when managing seismic risk. Subdividing space into sub-volumes, fitting a power law to the data extracted from each sub-volume, and calculating their probabilities are not the best strategy to estimate spatial hazard. There are two potential problems with this approach: (1) Seismic activity within these sub-volumes may not be independent, and therefore, it would be inappropriate to fit two different power laws to data. (2) There is a trade-off between the spatial resolution and the amount of data one can extract from these sub-volumes, so there may not be sufficient data for a reliable power law fit.
A better option to delineate seismic hazard in space is to estimate how frequently a given level of ground motion can be reached or exceeded at a given site, X, in future time \(\Delta T\), i.e. \(\Pr \left [\geq GMP\left (X\right ),\Delta T\right ]\), where the ground motion parameter GMP can be PGV , PGA, or CAD. The ground motion hazard incorporates the size distribution analysis, the ground motion prediction equation, GMPE, and the distribution of distances from the relevant seismic events to a given site.
6.8.1 Methodology
The presented methodology to estimate ground motion hazard is based on the principles described in Cornell (1968), McGuire (2004), and Baker et al. (2021), with changes appropriate to accommodate data provided by mine seismic networks. For a given site, \(X=\left (x,y,z\right )\), this methodology consists of the following steps:
1.
For a given seismogenic volume of rock and time span of data, \(\Delta t\), estimate the expected size of the next largest event, \(P_{max}\), and obtain the empirical and theoretical probability density function of the size distribution, \(f\left (P\right )\), and the expected activity rate, \(\lambda \left (\geq P_{min}\right )\)\(=\)\(N\left (\geq P_{min}\right )/\Delta t\), of events above a given potency threshold \(P_{min}\).
2.
Develop a simple GMPE and the respective survival function, \(\Pr \left [\geq GMP\left (X\right );\right .\)\(\left .P,R\right ]\), i.e. the probability that for a given seismic potency P and distance R the recorded ground motion at point X will exceed a given level of GMP.
3.
For given site, define the empirical or theoretical probability density function of the distribution of distances, \(f\left [R\left (X\right )\right ]\), of events with co-seismic strain below a given threshold, say 10\(^{-7}\). This will deliver a different number of events, and therefore, different activity rate for each site. The empirical distribution of distances is frequently based on more recent, shorter data set than the one used to derive the size distribution.
4.
Combine the above information to compute the rate of exceedance,
where the ranges of seismic potency and distances are discretised into \(n_{P}\) and \(n_{R}\) intervals, respectively, and \(f\left (P_{i}\right )\) and \(f\left [R_{j}\left (X\right )\right ]\) are the theoretical and/or the empirical probability density functions. The above equation assumes that the size distribution and the distance distribution are independent.
5.
Compute the probability of observing at least one event in a period of time \(\Delta T\) into the future. If the probability distribution of time between events is close to Poissonian, i.e. independent of time, where the coefficient of variation of the inter-event times is close to one, then the probability of observing at least one event in a period of time \(\Delta T\) is \(\Pr \left [\geq GMP\left (X\right ),\Delta T\right ]\)\(=\) 1 \(-\)\(\exp \left [-\lambda \left [\geq GMP\left (X\right )\right ]\cdot \Delta T\right ]\). In cases where there is a clear trend in seismic activity, one can apply the non-stationary Poisson process with a suitable intensity function.
6.8.2 Example: Data and Size Distribution
The data set starts on 07 September 2007 and ends on 07 July 2013 just after a \(\log P=2.61\)\(\left (m=2.66\right )\) event. It spans 2130 days and includes 2818 events with \(\log P\geq -1.0\)\(\left (m\geq 0.25\right )\), which gives the rate of 1.32 events/day. These events delivered \(\Sigma P = 2526.06\) m\(^{3}\) of seismic potency at the rate of 1.186 m\(^{3}\)/day. The largest event has \(\log P=2.61\)\(\left (m=2.66\right )\), and there are 30 events with \(\log P\geq 1.0\)\(\left (m\geq 1.59\right )\). The mean recurrence interval, \(\bar {t}\left (\log P\geq -1.0\right ) = 0.756\) days with the standard deviation of \(1.125\) days, which gives the coefficient of variation \(C_{v}\left (\log P\geq -1.0\right ) = 1.49\). The mean recurrence interval, \(\bar {t}\left (\log P\geq 1.0\right ) = 73.44\) days with the standard deviation of \(94.76\) days, which gives the coefficient of variation \(C_{v}\left (\log P\geq 1.0\right ) = 1.29\). The black vertical line is at the time of the second largest event with \(\log P = 2.24\) that occurred on 13 June 2012 at the same depth but 500 m away from the main shock.
Figure 6.41 shows the cumulative potency, CumP, and the history of records vs. time where the colour indicates the distance of the event from the main shock of \(\log P=2.61.\)
Figure 6.42 left shows the size distribution of unbinned data where the colour indicates the time of the event, and two UT fits: the black with \(\log P_{max} = 3.04\) and the red assuming that \(\log P_{max} = \log P_{nrb} = 2.809\). The blue straight line is the open-ended fit with \(\beta =0.9415\) shown as a reference.
Fig. 6.42
Size distribution and the probabilities of exceedance
Figure 6.42 right shows the expected ranges of the probabilities of exceedance for the next 90, 180, and 360 days. The black line is for the upper range associated with \(\log P_{max}\), and the red line shows the lower range associated with \(\log P_{nrb}\). The probability of at least one event exceeding the maximum observed \(\log P_{maxo} = 2.608\) within one year, \(\Pr \left (\geq \log P_{maxo};1Y\right )\), lies between 0.066 and 0.111 and \(\Pr \left (\geq \log P=2.0;1Y\right )\) is between 0.449 and 0.475.
The ground motion prediction model should provide a probability distribution instead of a single value of the ground motion parameter. In general, the ground motion prediction model can be described as \(\ln \varphi \)\(= \overline {\ln \varphi }\)\(+\)\(\sigma \), where in this case \(\varphi =PGV\) or \(\varphi =CAD\), \(\ln \varphi \) is a random variable, assumed to be well described by a normal distribution, and \(\overline {\ln \varphi }\) and \(\sigma \) are the predicted mean and standard deviation of \(\ln \varphi \) , respectively. Note that if \(\ln \varphi \) values are normally distributed, the non-logarithmic values are log-normally distributed.
Under such an assumption, the probability of exceeding a given level of ground motion parameter, \(GMP_{x}\), is the survival function of the normal distribution,
where \(\Phi \) is the standard Gaussian cumulative distribution function. One can express it via the probability density function that can easily be evaluated numerically, \(\Pr \left [\geq \varphi ;P,R\right ]\)\(=\)\(\intop _{GMP_{x}}^{\infty }f\left (u\right )du\), where \(f\left (u\right )\) is the probability density function of \(\varphi \) given P and R, which gives
The scatter measured by \(\sigma _{\ln \varphi }\) in data recorded in boreholes in solid rock in mines is expected to be lower than that recorded at the skin of underground excavations. Typical values of \(\sigma _{\ln PGV}\) calculated to date in mines range from 0.4 to 0.65 and \(\sigma _{\ln CAD}\) 0.3 to 0.55. Figure 6.43 shows the PGV and CAD survival functions for \(\log P=2.0\) at distances of 20, 50, and 100 metres for the SGMPE developed in Sect. 6.6.3.
Fig. 6.43
Survival functions for \( \log P=2.0\): \(\Pr \left [ \geq PGV_{x};P,R \right ]\) at \(PGV_{x}=0.2\) m/s, \(\sigma _{\ln PGV}=0.59\)(left), and \(\Pr \left [ \geq CAD_{x};P,R \right ]\) at \(CAD=0.005\) m, \(\sigma _{\ln CAD}=0.4393\)\(\sigma _{ \log PGV}=0.256\)(right), at distances 20, 50, and 100 metres from the seismic source
Figure 6.44 left shows the convex hull span over all 6073 events available for analysis, and Fig. 6.31 right shows the convex hull span over 3578 events that generated \(PGV \geq 10^{-7}\) m/s at the centre of all 6073 events, shown as S3 in blue. The three sites at which we will evaluate probabilities of exceedance are shown as S1, S2, and S3. This is the same data set we used in Sect. 6.6.8.3.
Fig. 6.44
Convex hull span over all available 6073 events (left) and over 3326 events that generated \(PGV \geq 10^{-7}\) m/s at site S3 (right). The three sites are shown as S1, S2, and S3 in blue
To estimate the expected ground motion at a given site, it is necessary to establish the distribution of distances from seismic sources to that site. The location of a seismic event is represented as a point which, in most cases, is assumed to be the rupture initiation. Earthquake seismologists use distance to the epicentre or hypocentre, distance to the closest point on the rupture surface, or distance to the closest point on the surface projection of the rupture.
However, in mines, seismicity follows rock extraction and clusters along geological features; therefore, the assumption of random spatial distribution frequently cannot be made. A better option is to construct an empirical probability distribution of distances to a given site. In most cases, these empirical probabilities cannot be fitted with any reasonable theoretical distribution.
Figure 6.45 shows the empirical probability density function of distances to the same three hypothetical sites S1, S2, and S3 as described in Sect. 6.6.8.3 and marked by blue dots. The log-normal distribution fit to the data defined by a mean and standard deviation is marked by solid red lines.
Fig. 6.45
The empirical probability density marked as blue dots and log-normal fit LGN\( \left (R \right )\) marked by red line for sites S1, S2, and S3
The data set used to derive these distribution ranges from 08 May to 07 July, i.e. the last 60 days before the largest event, and includes 6073 events. The best log-normal distribution fit of distances is for site S3 with 3578 events that exceeded the strain threshold of 10\(^{-7}\), and it has the mode at about 65 m and small standard deviation. The second best is for site S1 with 1106 events that has a mode at about 85 m, but the observed data also has a second mode at 130 m that cannot be reproduced by a log-normal distribution and consequently has larger standard deviation. The distribution for site S2 with 1059 events is the worst of the three, and it moved the observed mode at 100 m to almost 60 m, which would overestimate hazard. The distance between an event and a given site is taken as the Euclidean distance minus the radius of the event, \(r =\)\(\sqrt [3]{3P/\left (4\pi \Delta \epsilon \right )}\), where the strain change, \(\Delta \epsilon \), is the same used in the development of the SGMPE, in this case 6\(\cdot \)10\(^{-4}\).
There is no acceptable model for source to site distances in mining, and therefore we use the empirical distribution in GM hazard calculation. In mines, seismic activity follows rock extraction, and therefore, the empirical distances distribution should be updated at least every 3 months.
6.8.5 Probabilities and Hazard Maps
The basic outcome of ground motion hazard analysis for a given site is the probability of exceedance, i.e. the probability that there will be at least one event that will exceed a given GM parameter in time \(\Delta T\) in the future. All plots below are based on the theoretical size distribution and empirical distances distributions.
Table 6.6 gives probabilities that there will be at least one event every 90 and 360 days with \(PGV \geq 0.05\) m/s or \(CAD \geq 0.0005\) m.
Table 6.6
Probabilities of exceedance of a given PGV and CAD over 90 and 360 days
\(\Pr \left (PGV\geq \right . \)
\(\Pr \left (PGV\geq \right .\)
\(\Pr \left (CAD\geq \right .\)
\(\Pr \left (CAD\geq \right .\)
Site
\(\left . 0.05,90\right )\)
\(\left . 0.05,360\right )\)
\(\left . 0.0005,90\right )\)
\(\left . 0.0005,360\right )\)
S1
0.032
0.1219
0.039
0.1471
S2
0.0851
0.2995
0.0665
0.2407
S3
0.128
0.4217
0.115
0.3866
Figure 6.46 left shows the probabilities of exceedance of PGV over 90, 180, and 360 days for all three sites, and Fig. 6.46 right shows the same for CAD.
Fig. 6.46
Probabilities of observing at least one event greater than a given PGV(left) or CAD(right) within the period of 90, 180, and 360 days
Ground motion hazard can also be presented as a seismic hazard curve that gives the annual rate at which a specific ground motion level may be exceeded. The hazard curve may be superimposed on the probability rating scheme, or likelihood scores, defined by the mine, see Fig. 6.47.
There is a wide spectrum of views in the literature on the utility of PSHA, from suggestions to drop it altogether (e.g. Mulargia et al., 2017) to the more pragmatic, stating that the shortcomings of the method do not invalidate the existence of the hazard curve, which comprises the basic assumption for PSHA (Anderson & Biasi, 2016).
The Nature of the Problem
There is a difference in the nature of ground motion hazard to underground structures due to seismic events in mines and to surface structures due to earthquakes. The main difference is the distances involved. As stated before, the near-source ground motion due to small and larger seismic events is similar, but the hypocentral distances to underground excavations in mines are very small, say from a few metres to a few hundred metres. Earthquakes are usually kilometres away. Therefore, earthquake engineers are mainly concerned with ground motions from larger earthquakes, but these earthquakes are less frequent and their activity rates are less certain.
Recurrence times for large events in mines are also uncertain, and therefore one can expect that the hazard maps for mines can better estimate the potential for less severe damage caused by smaller and medium size events than the infrequent large events.
This is why mines prone to larger events may wish to supplement probabilistic analysis with the deterministic one, i.e. ground motion simulation. Such a simulation involves kinematic modelling of ground motion produced by sources defined by their expected maximum potency, or magnitude, and placed at the most likely locations that can produce the strongest level of peak ground velocity at a given site or sites. For more details, see the next section in this book.
Probabilistic and deterministic methods for hazard assessment have advantages and disadvantages. Probabilistic methods can be viewed as inclusive of all deterministic events with a non-zero probability of occurrence. In this context, a proper deterministic method that models a particular larger event should ensure that that event is realistic, i.e. with a finite probability of occurrence. This points to the complementary nature of deterministic and probabilistic analyses: Deterministic events can be checked with a probabilistic analysis to ensure that the event is probable, and probabilistic analyses can be checked with deterministic events to see that rational, realistic hypotheses of concern have been included in the analyses (McGuire, 2001). However, the deterministic analysis can better account for specifics, i.e. the path and the site effects associated with a given strategic structure.
GMPE and Site Effect
The reliability of a GMPE depends mainly on the selection of the appropriate data, and the main obstacle in selecting a good data set is the limited capability of the sensors employed to record stronger ground motion. This may limit the predicted level of ground motion at closer distances. It is important then for mines to deploy strong ground motion sensors. A more serious limitation though is the unknown amplification of ground motion at the skin of excavations.
Distribution of Distances
There is no acceptable model for source to site distance distribution in mining, and therefore, in this example, we used the empirical distribution that reflects the recent past but does not extrapolate into the future. One option is to model the future spatial distribution of seismicity which for mines is very uncertain, or to update the analysis more frequently.
Size Distribution
We assume that the magnitude distribution of future seismic events will be close to the one derived on the basis of existing data, i.e. that the predicted upper limit of the next record breaking potency, \(P_{max}\), and parameters \(\alpha \) and \(\beta \) of the assumed power law are reasonably accurate. This is not the most risky assumption since in most cases the future seismic rock mass response to mining is reasonably informed by its past size distribution. There are, of course, “black swan” events that considerably exceed the largest observed one, but they are not frequent enough to discourage forecasting. The experience of the author is that the bigger jumps occur more frequently when data follow the open-ended power law closer than the upper truncated one.
A Homogeneous Poisson Process
In some cases, e.g. when rock extraction is interrupted, seismic activity in the time domain is far more clustered than in the volume mined domain. In such a case, it is recommended to perform the analysis in the volume mined domain. Alternatively, one can apply the non-stationary Poisson process with a suitable intensity function.
6.9 Modelling Ground Motion—Deterministic Hazard
6.9.1 Introduction
General Description
Seismic waves in rock, propagating away from the source, interact with interfaces between different rock types, fractured zones, and underground excavations. In the process, they experience reflection, refraction, diffraction, scattering, and inelastic attenuation. Reflection, refraction, and diffraction are all manifestations of boundary behaviour of waves and are associated with some significant change of the wave vector leading to the bending of the wave path. Path bending can be observed only for waves in two or three dimensions. Reflection is a sudden change in the direction of wave propagation. It occurs when the wavefront reaches the interface between two different media. Refraction is caused by the change in speed when the wave travels through the interface between two different media. This change of wave speed leads to a change in the wavelength and in the direction of propagation. Diffraction is observed when a wave reaches some obstacle (an edge or an opening) of size co-measurable with the wavelength. It is a complex physical phenomenon involving not only bending of the wave path but also changes in the wave intensity due to interference. The said changes in the wave intensity can take the form of alternating minima and maxima when the peak value in a diffractive maximum is significantly larger than the intensity of the incident wave.
As an outgoing wave travels away from the source, its amplitude decreases due to geometrical spreading, attenuation, and scattering. Local effects include constructive and destructive interference of scattered waves, trapping in lower velocity layers and site effects. The combined effect from all these factors is that the wave field becomes more complex as the wavefront progresses.
The deterministic hazard, as it is discussed here, involves kinematic modelling of ground motion produced by specific seismic sources, i.e. sources, defined by their expected maximum potency or energy and placed at most likely locations that can produce the strongest level of peak ground velocity, PGV , or cumulative absolute displacement, CAD, at a given site. The likely locations of these sources may be inferred from the past data or, preferably, they can be determined by numerical modelling of the induced shear stresses on geological structures. In a kinematic model, the source process is defined by the spatial and temporal distribution of the slip vector, by the local slip velocity function, and by the rupture velocity, without taking into consideration the forces and stresses acting at the source.
There are three steps in the kinematic modelling process. Firstly, one needs to build a model of the mine including the geometry of any existing or planned excavations, geological structures, and rock mass properties such as rock density, wave velocities, and attenuation factors. Secondly, one needs to prescribe the location, the shape and the size of the source as well as the details of the slip rate that will generate seismic waves. The simplest example of a kinematic source model is one of zero size called a point source. A point source is defined by its location and the source time function, i.e. the local displacement time history. The seismic wave field created by a point source is relatively easy to reconstruct, but it can be compared with actual observations only at stations sufficiently far from the source. Since a true point source, that is one of zero volume, cannot exist, the modelling of the near and intermediate field effects requires the preparation of extended sources as input. The third element of the deterministic modelling of strong ground motion is the formulation of the initial and boundary conditions for the equations of motion.
Equations of Motion
The equations of motion for elastic materials are the mathematical expression of Newton’s second law: The rate of change in the momentum of a body is equal to the net applied force. The change of the momentum for the material in a small volume dV is \(\rho \ddot {\mathbf {u}} dV\), where \(\ddot {\mathbf {u}}\)\(=\)\(\partial ^{2}\mathbf {u}\left (\mbox{x},t\right )/\partial t^{2}\) and \(\mathbf {u}(\mathbf {x},t)\) is the displacement of the centre of mass. The net applied force comes from two contributions. One is due to interactions with the world outside the volume element. It is called body force. An example of body force is the weight or the force due to gravity. The other contribution to the net force applied on the volume element dV is a measure of the response of the surrounding medium to local deformations. The complete information for all response forces is contained in a single mathematical object called the stress tensor. For perfect elastic materials, the components of the stress tensor are linear functions of the components of the tensor of deformation which for small deformation is equal to the strain tensor. The forces due to the local deformation act on the surface dV , and their resultant is expressed as a combination of the spatial derivatives of the stress tensor \(\nabla \cdot \sigma \). The components of the stress tensor are functions of the gradients of the displacement \(\nabla \mathbf {u}\). These are the constitutive relations, and they can be either linear or nonlinear thus determining the overall properties of the equations of motion. The complete 3D equations of motion for a linear elastic solid are
where \(c_{ijkl}\) are constant coefficients. If the material is homogeneous, \(c_{ijkl}\) are symmetric: \(c_{ijkl}\)\(=\)\(c_{klij}\)\(=\)\(c_{jikl}\)\(=\)\(c_{ijlk}\), which reduces the 81 components of the \(c_{ijkl}\) to 21 independent components. If the material is also isotropic, the number of independent components of \(c_{ijkl}\) is further reduced to just two Lame’s parameters \(\lambda \) and \(\mu \). The equations of motion can now be written as the nine simultaneous equations: The first three are expressing Newton’s second law
In the above, \(\partial _{tt}\) is the operator of taking the double derivative with respect to time, \(\partial _{x}\), \(\partial _{y}\), \(\partial _{z}\) are the operators of taking the first order spatial derivatives, \(u_{x},u_{y},u_{z}\) are the three components of the displacement vector, \(\sigma _{xx}\), …, \(\sigma _{yz}\) are the nine components of the stress tensor, and \(f_{x}\), \(f_{y}\), \(f_{z}\) are the three components of the applied body force. The stress tensor is symmetric, so only six of its components are independent.
The spatial derivatives are a measure of the change in the function when moving from a given point to one of its neighbours. The governing differential equations of elastodynamics are the mathematical expression of the fact that a disturbance at one point in a continuum propagates by means of interactions between nearest neighbours. The result is a wave field which fills the volume of the elastic solid. The numerical methods for solving differential equations are simple in the case of a first order problems, that is, when only first order derivatives are present. The equations of elastodynamics are of second order, but they can easily be transformed into a system of first order differential equations by introducing the velocities \(\mbox{v}_{j}\left (x,y,z,t\right )\)\(=\)\(\partial u_{j}\left (x,y,z,t\right )/\partial t\) as new unknown functions and differentiating the constitutive equations with respect to time,
These equations are meaningful only for a particular domain in space and in time, and therefore, the components of the source function, \(f_{x}\), \(f_{y}\), and \(f_{z}\), need to be defined within the same domains. All differential equations have infinitely many solutions, and one needs to impose additional conditions on the components of the velocity and the stress to ensure the uniqueness of the elastic wave radiated by a given source.
Space-Time Grid
The space-time grid is a set of discrete points separated in space by \(\varDelta x\), \(\varDelta y\), and \(\varDelta z\) and in time by \(\varDelta t\). In most applications, the grid is regular: \(\varDelta x\)\(=\)\(\varDelta y\)\(=\)\(\varDelta z\)\(=\)h. In a simple grid, all functions are approximated at the same grid points. In a partly staggered grid, displacement or particle velocity components are located at one set of grid points, whereas the stress tensor components are assigned to another set. A staggered grid of step h is just two regular grids of the same step but shifted by \(h/2\) relative to each other in the three spatial directions. This procedure puts extra nodes at the middle of the ribs and the faces of the h-cubic cells and has the advantage that, when the components of the velocities and the stress are allocated to particular nodes, the spatial derivatives are approximated by central differences of step \(h/2\), hence more accurately. The staggered distribution of quantities in space is related through the equations of motion to the staggered distribution of quantities in time.
The partial derivatives in the equations of motion need to be replaced by approximations which are related to the definition of derivatives, e.g. the first order partial derivative with respect to x for the function \(u_{z}\left (x,y,z,t\right )\) at the point \(\left (x_{0},y_{0},z_{0},t_{0}\right )\) can be approximated by the central difference
and for the time derivative by the forward difference at \(({\mathbf {x}}_{0},t_{0})=(x_{0},y_{0},z_{0},t_{0})\),
$$\displaystyle \begin{aligned} \partial_{t}\sigma_{xz}\left({\mathbf{x}}_{0},t_{0}\right)\simeq\left[\sigma_{xz}\left({\mathbf{x}}_{0},t_{0}+\varDelta t\right)-\sigma_{xz}\left({\mathbf{x}}_{0},t_{0}\right)\right]/\varDelta t. \end{aligned}$$
One and the same partial derivative of a given function can be approximated by infinitely many different finite difference expressions which can be obtained by combining different Taylor expansions of the same function. A finite system of equations for the desired approximation can be obtained from the truncated Taylor expansions. The lowest order term which was neglected in the truncation procedure defines the order of accuracy for the particular finite difference approximation. There are two meanings of “order” when the term is used to describe a finite differences approximation: One can approximate a derivative of a certain order, for instance, all derivatives in the nine differential equations for the velocity and the stress are of first order, and then one can describe a particular finite differences approximation as being of a given order of accuracy meaning the lowest order term in the truncated Taylor series which was neglected.
It can be shown that the forward difference is a first order approximation because the truncation error is proportional to \(\varDelta t\), and the central difference is a second order approximation because the truncation error is proportional to \(h^{2}\). One example of a fourth order approximation formula is
The choice of h and \(\Delta t\) depends on the properties of the material and of the source time function. A good rule is h\(=\)\(v_{S}/\left (kf_{max}\right )\), where \(v_{S}\) is the velocity of the slower wave, in this case, S-wave \(f_{max}\) is the maximum frequency to be resolved and k represents the number of evaluations per wavelength, usually \(k=6\). The time step \(\Delta t\) depends on the shortest time taken by the P-wave to cross the distance h between two neighbouring nodes, and it is limited by the stability condition which, for the staggered grid, reads: \(\varDelta t\)\(<\)\(0.5h/v_{P}\). The finite difference representation of the partial derivatives applies only to points of the interior for which all nearest neighbours are within the model and for moments of time for which the time derivative can be approximated. This excludes the beginning, i.e. \(t=t_{0}\).
The numerical solution of the dynamical equations of motion on a staggered grid is obtained by first imposing the initial conditions, that is, by giving the initial values of the velocity and the stress components at the respective grid nodes, and then solving for the nodal values at the next moment of time, evaluating the finite difference expressions on the right hand sides of the equations. In this way, the numerical solution is woven like a 3D carpet one time step at a time. There are two types of nodal values which require special attention: the nodes on the boundaries and the nodes on the seismic source. The boundaries and respectively the boundary nodes can be of two types: free surfaces and virtual boundaries. Free surfaces are characterised by zero normal stress, and the corresponding nodal boundary values must reflect this fact. Virtual surfaces are not related to any existing feature in the model domain. Their role is just to separate a finite volume out of the much bigger material world so that the numerical model would be of a reasonable size and could be solved. It is very difficult to impose boundary conditions on a virtual boundary because it must stay “invisible” for every incidence of waves on it. The nodes on the source are similar to the boundary nodes in the sense that the values of the physical fields at these points are not obtained by solving the equations but are taken from tabulated time functions. This is how a source point works: At the beginning, the source function is zero, and the source node behaves just like any other node in the model. That is, it obtains a new value after each iteration according to the solution of the finite differences equations. But when the time stepping procedure reaches the moment when the source function becomes non-zero, at that time step the velocity component in the source node takes its value from the source function. This value is not what the solution of the model would have given, and the difference acts as a perturbation of the wave field. In other words, it becomes a point source with a short, pulse-like velocity function. The time sequence of such pulses constitutes a point-like source which, together with others of the same type, represents the extended kinematic source model.
6.9.2 Implementation and Examples
This section is based on Mendecki and Lötter (2011). A numerical solution of the 3D wave equation can be achieved by finite difference modelling. In principle, the finite difference method does not have restrictions on the type of constitutive equation, boundary conditions, curved interfaces, and different source types and allows general material variability. The implementations of finite difference schemes for solving non-linear and non-isotropic problems, though, can be complicated and of little practical value.
As a first step, partial derivatives in Equation 6.33 are replaced by numerical first, second, or fourth order estimates obtained from Taylor expansions truncated after the desired number of terms. The decision on where on a finite difference grid to place spatially dependent material properties, \(\rho \), \(\lambda \), and \(\mu \), and physical quantities, \(v_{i}\), \(\sigma _{ij}\), constitutes the choice between various staggered grid schemes. A fourth order in space scheme is described in Graves (1996). However, a staggered grid causes instability problems when the medium possesses high contrast discontinuities. Saenger et al. (2000) and Saenger and Bohlen (2004) proposed a new rotated staggered grid where all medium parameters are assigned to the centre of each elementary cell (Fig. 6.49). This modified finite differences scheme provides a more accurate way of including in the model underground excavations and interfaces between different materials. To reduce the influence of nonphysical wave reflections of any virtual boundaries, it is important to apply either absorbing boundary conditions (e.g. Clayton & Engquist, 1977) or an attenuating shell around the model that quenches the waves and minimises reflection.
Fig. 6.49
Elementary 2D cell for velocities and stresses in a rotated staggered grid (left), and its 3D view (right)
The wave field from an extended source is modelled as the superposition of the wave fields from multiple point sources in a finite difference scheme. The seismic potency of a finite source is the sum of the potencies of the constituent point sources, and it is expected that these point sources will have similar principal axes but may vary in magnitude, as not all parts of the source will experience the same net displacement. Figure 6.50 demonstrates such a simple scheme with colours proportional to the final displacement experienced by each point. At the edge of this elliptical source, displacement tends towards zero (blue), while maximum displacement occurs at an off-centre point on the principal axis of the source ellipse.
Fig. 6.50
An extended source built from a distribution of point sources
When building a kinematic model of an extended source, the consistency of the set of source parameters and time histories for each point of the fracturing fault is very important. Two features of seismic sources that enforce some constraints on the possible distributions of these parameters are self-similarity of the final slip distribution and the expected characteristics of the high frequency part of the displacement spectrum as seen in the far field. These consistency constraints ensure smooth and monotone slip histories for all parts of the source, while remaining physically admissible.
To provide explicit constraints on the parameters controlling the rupture, the kinematic k-square earthquake source model of Herrero and Bernard (1994) can be used. The idea of the k-squared model and its generalisations is to prescribe the Fourier transform of the displacement in the 2D wavenumber space and then to obtain the spatial distribution in real space by performing the inverse Fourier transformation. One implementation of the k-square model proposes a final displacement at \((x,y)\) on the fault as
where \(D\left (k_{x},k_{y}\right )=\exp \left [i\Phi \left (k_{x},k_{y}\right )\right ]/\sqrt {1+\left [\left (k_{x}/k_{c}\right )^{2}+\left (k_{y}/k_{c}\right )^{2}\right ]^{2}}\), with the function \(\Phi (k_{x},k_{y})\) in the Fourier transform being a random phase and \(k_{c}\)\(\sim \) 1/(source size), is the corner wavenumber which, like the corner frequency in the Brune model, demarkates the low-wavenumber asymptote from the high-wavenumber behaviour in the Fourier transform of the slip distribution. Fourier transform decays as an inverse square. An extended source can then be constructed by distributing its total seismic potency over the sub-sources proportional to the modelled permanent displacement at their positions. For sub-source \((i,j)\) at point \((x_{i},y_{j})\) on the fault, we thus obtain \(P_{0}^{ij}=u(x_{i},y_{j})\triangle A_{ij}=u_{ij}h^{2}\) . Similarly, rise times over different parts of the source can be chosen proportionally to the final slip, \(T_{i}\)\(=\)\(T^{max}u\left (\xi _{i},u_{max}\right )\), where \(T^{max}\) is a chosen rise time corresponding to the sub-source with maximum final displacement.
The modelled spatial distribution of the final displacement on the source cannot ensure on its own the correct high frequency behaviour of the far-field displacement spectra. A method of specifying the slip velocity time functions for the sub-sources needs to be added to the kinematic source model. Beresnev and Atkinson (1997) have given an example of a slip velocity time function which leads to a far-field displacement spectrum adhering to \(\omega ^{-2}\) decay. The parametrisation of their source time function \(\nu \left (t;\tau ,\zeta \right )=\)\(u^{\infty }\left (\frac {2}{\tau }\right )^{2}t\cdot \exp \left (-2t/\tau \right )\) will reproduce the rise time for the sub-sources \(T_{ij}\) when \(\tau =\frac {1}{2}T_{ij}\) with \(T_{ij}\leqq T^{max}\).
Every sub-source in an extended kinematic source model needs to be assigned its own initiation time \(t_{0}^{ij}\). In solving a finite difference model, time is measured in time steps \(\Delta t\) starting from zero. The velocity time function of sub-source \((i,j)\) is zero before the initiation time \(t_{0}^{ij}\) after which it follows the respective velocity time function, for instance the Bereznev expression. The choice of initiation times \(t_{ij}^{0}\) for the sub-sources is equivalent to specifying the propagation of the rupture front over the fault. It is an important element of seismic source modelling because the spatial distribution of the initiation times is equivalent to the complete rupture scenario. At this stage, one can model acceleration of rupture, sub- or super-shear rupture velocity, and the stopping phase. For instance, one can make the simple assumption that rupture speed is faster when parallel to slip and slower when orthogonal to slip. In this case, one obtains an extended source in the shape of an ellipse, with eccentricity determined by the ratio of these two orthogonal rupture velocities. This is the scenario illustrated by Fig. 6.50.
Alternatively, one can choose a spatial distribution of the rupture velocity \(v_{ij}^{rupt}\) and compute the initiation time \(t_{0}^{ij}\) for a sub-source as \(\sum (h/v_{lm}^{rupt})\) where the sum is taken over the fault nodes on the shortest path from the hypocentre to sub-source \((i,j)\). Now, the ground motion experienced at a site is controlled by the maximum velocity of deformation at the seismic source, i.e. slip velocity, by the interaction of radiation from different sub-sources from different travel paths and by site effects.
6.9.2.1 Example 1: Extended Sources in Heterogeneous Media
Displacement on a fault originates at the focus, i.e. at the hypocentre, and propagates towards its edges, in a manner resembling the extended source described above. As the focus is not necessarily in the centre of the fault, much of the propagation is unidirectional.
Traditionally, it has been assumed that the upper bound for rupture propagation velocity is the Rayleigh wave velocity (Broberg, 1996). However, field observation has shown several examples where the rupture velocity exceeds shear wave velocity, \(v_{r}>v_{S}\), Dunham and Archuleta (2004).
To examine the extreme ground motions that can be caused at points placed close to the source in the direction of rupture propagation, we modelled two similar extended sources based on the same k-square slip distribution, but with different rupture scenarios.
In the first case, we took a sub-shear rupture speed \(v_{r}=0.9v_{S}\) , while in the second case we let the rupture propagate with super-shear velocity \(v_{r}=\sqrt {2}v_{S}\). Also present in this model was a tabular stope that caused reflections and partly obstructed the waves from directly traveling to the upper part of the rock mass. The displacement follows the rupture front from the focus towards its edges. As the focus is not necessarily in the centre of the fault, much of the propagation is unidirectional. From an inspection of the wave field in Figs. 6.51 and 6.52, for which corresponding frames refer to the same points in time, it can be seen that rupture progresses faster in the super-shear case and that a Mach cone evolves (see Fig. 6.52 frame 4).
Fig. 6.51
Snapshots in section of time steps in a sub-shear rupture process, chronologically from left to right, top to bottom
6.9.2.2 Example 2: Complex Sources in Heterogeneous Media
We model a complex seismic source conceptualised by David Ortlepp and described in Ortlepp (1984) and Ortlepp (1997) page 63 caption (e), see Fig. 6.53 left. It is a system of two rectangular extended faults. A kinematic source model was created for each of the faults following the methodology outlined in the previous section. Rupture initiates at the fault which is further away from the stope. The wave radiated by this fault triggers rupture on the second fault, the one that interacts with the stope. Here we assume that the second rupture is induced by the first one almost immediately.
Fig. 6.53
Strike section through a stope showing a double source mechanism conceptualised by Ortlepp (first) . Rupture history of the Ortlepp source, with initiation time per point obtained from radial distance from the focus (second). Slip history of the Ortlepp source, with final displacements determined by a k-square distribution (third). Synthetic sensor placement around the fault-stope corner (forth)
Due to the available freedom in placing faults and sub-sources in a finite difference grid, we are now able to independently choose both the orientations and the velocity of rupture, \(v_{r}\), on these faults. In particular, we can chose for the initiating fault a sub-shear rupture velocity (\(v_{r}<v_{S}\)) and a super-shear rupture velocity (\(v_{r}>v_{S}\)) for the second fault. The rupture and slip histories of such a scenario are illustrated in Fig. 6.53.
One can see that rupture and slip do not need to be in the same direction at all source points. For our model, we introduce a fault with a dip of \({60^{\circ }}\), touching a horizontal stope. Some points of interest are marked around it. It is at these points where we record the modelled ground motion. In particular, as the rupture of the fault starts below and progresses upwards, we are especially interested in comparing ground motions of the footwall and the hanging wall.
After performing a sufficient number of iterations with the described kinematic source model, we investigated the recorded velocity seismograms at sensors 4, 7, and 9 (Fig. 6.54). It was found that particle velocities close to the fault exceed 10 m/s as one can see on the synthetics recorded at sensor 9. Both, sensors 4 and 7, which are opposite to each other on the footwall and hanging wall of the fault, experience about 2 m/s velocities, although a higher frequency content is observed at sensor 7 due to the interaction of the stope with the wave field.
Fig. 6.54
Synthetic seismograms recorded around the fault. Sensor 4: Velocities near fault and footwall corner (left), Sensor 7: Velocities just ahead of the stope (centre). Sensor 9: Velocities close to the fault (right)
Sensors 12 and 15 are placed opposite to each other on the hanging wall—sensor 12, and the footwall—sensor 15, of the stope. The corresponding synthetic velocity seismograms are shown in Fig. 6.55.
Fig. 6.55
Synthetic seismograms (as absolute velocity) comparing the hanging wall and footwall ground motions of sensors 12 and 15
Clearly the footwall sensor records ground motion earlier than the hanging wall sensor. This is not so much due to the shorter straight line distance to source, but because of the longer path the elastic waves need to travel around the stope which is modelled as a reflector. Evidently, the synthetic velocity seismograms recorded at sensor 12 have significantly lower amplitudes. This difference in the recorded velocity amplitudes from two stations so close to each other would not have been seen if the effect from the underground excavation was not a property of the numerical model.
Figure 6.56 shows successive snapshots of the seismic wave field from a two-fault Ortlepp source. In the first few frames, the radiation from the low-potency initial fault is visible, but upon initiation of the second, high-potency, fault, its contribution to the wave field becomes dominant. When the rupture on the second fault reaches the stope, interaction, reflection, and constructive interference lead to high ground motions.
Fig. 6.56
Snapshots of a 2D section of the 3D wave field induced by the synthetic Ortlepp shear event
6.9.2.3 Example 3: Surface Ground Motions Induced by Mining Events
In this model, the event is 1.5 km below surface, at an epicentral distance of 5 km from a major city centre. The event is modelled as a rupture of an elliptical reverse fault with axes of 300 m and 100 m. The fault strikes at \({0^{\circ }}\), dips at \({45^{\circ }}\), and slips at a rake of \({90^{\circ }}\). The maximum slip velocity is chosen to be 2 m/s, and the average displacement over the whole source is set as \(\overline {u}=0.2\) m. We assumed \(\log P=3.5\) and a predominant frequency of 3 Hz.
The model domain is filled with a mostly homogeneous hard rock. There is a soil layer three-grid spacings thick just under the free surface (the grid is of spacing \(h=\)25m). For the hard rock, we choose \(\rho =2700\) kg/\(\text{m}^{3}\), \(v_{P}=5500\) m/s, and \(v_{S}=3500\) m/s, whereas for the 50 m soil layer, which is constructed over the top three-grid points in the rock below the air in the finite difference grid, we choose \(\rho =2000\) kg/\(\text{m}^{3}\), \(v_{P}=4000\) m/s, and \(v_{S}=2000\) m/s. To model the free surface effect and air above it, we choose \(\rho =2000\) kg/\(\text{m}^{3}\), \(v_{P}=300\) m/s, and \(v_{S}=0\) m/s with high density to avoid stability problems during the kinematic modelling phase on the finite difference grid.
We have computed waveforms in three points of interest indicated at Fig. 6.57. Forward modelling allows us to track velocities and dynamic stresses of the wave field not only at the points of interest, but in the entire model domain. Thus we can visualise the wave field on multiple planes of interest in subsequent time snapshots. This representation is shown in Fig. 6.58. With reference to Fig. 6.57, we have recorded synthetic seismograms at sensors 1, 2, and 3, and these are shown in Fig. 6.59.
Fig. 6.57
Points of interest at which seismograms are recorded. Sensor 1 is just below the urban area of interest, sensor 2 at epicentre, and sensor 3 is halfway between the hypocentre and urban city centre
Snapshots of velocity fields during subsequent stages of the kinematic model run of an underground event and observed surface ground motions. The intersection of the two vertical sections represents our point of interest, the area below the urban area where damage could potentially occur
We show only the vertical component (blue) and the horizontal component (green) because our source is symmetric relative to the plane of the sensors. The stronger horizontal ground motions at sensor 1 (Fig. 6.59 left) which are actually further from the hypocentre are expected to have a more damaging effect.
The predominant frequency at sensor 1 was calculated by computing a power spectrum over the two non-trivial components of the observed waveform. The predominant frequency of the ripple recorded by sensor 2 at about \(t=\) 2s is 10 Hz (estimated from the average over eight consecutive full periods).
While potentially a numerical effect, this can be compared to the expected horizontal S-wave resonance (20 Hz) and the expected vertical S-wave resonance (10 Hz). Both horizontal and vertical ground motions exceeding 15 mm/s are observed on the surface and in the area of interest. The predominant frequency of this ground motion is 5.8 Hz. At sensor 2, a significant vertical particle velocity is recorded. After integrating to displacement, this leads to an upward permanent displacement of the surface equal to about 1 mm. It is to be expected of reverse faulting but unlikely to be particularly damaging to surface structures. At sensor 3, the initial arrivals and their immediate coda are clear of reflected waves, as this synthetic recording is underground, halfway between the hypocentre and the surface.
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Abrahamson, N. A., & Youngs, R. R. (1992). A stable algorithm for regression analyses using the random effects model. Bulletin of the Seismological Society of America, 82(1), 505–510.CrossRef
Anderson, J. G., & Biasi, G. P. (2016). What is the basic assumption for probabilistic seismic hazard assessment? Seismological Research Letters, 87(2A), 323–326. http://doi.org/10.1785/0220150232.CrossRef
Andrews, D. J. (1976). Rupture velocity of plane strain shear cracks. Journal of Geophysical Research, 81(32), 5679–5687.CrossRef
Andrews, D. J., Hanks, T. C., & Whitney, J. W. (2007). Physical limits on ground motion at Yucca Mountain. Bulletin of the Seismological Society of America, 97(6), 1771–1792.CrossRef
Archuleta, R. J. (1984). A faulting model for the 1979 Imperial Valley earthquake. Journal of Geophysical Research, 89(B6), 4559–4585.CrossRef
Archuleta, R. J., & Hartzell, S. H. (1981). Effects of fault finiteness on near-source ground motion. Bulletin of the Seismological Society of America, 71(4), 939–957.CrossRef
Arias, A. (1970). A measure of earthquake intensity. In R. J. Hansen (Ed.), Seismic Design for Nuclear Power Plants (pp. 438–483). MIT Press.
Atkinson, G. M. (2015). Ground motion prediction equation for small to moderate events at short hypocentral distances, with application to induced seismicity hazards. Bulletin of the Seismological Society of America, 105(2A), 981–992. http://doi.org/10.1785/0120140142.CrossRef
Baker, J. W., Bradley, B. A., & Stafford, P. J. (2021). Seismic hazard and risk analysis. Cambridge University Press.CrossRef
Beresnev, I., & Atkinson, G. M. (1997). Modeling finite-fault radiation from the omega-n spectrum. Bulletin of the Seismological Society of America, 87(1), 67–84.CrossRef
Brune, J. N. (1970). Tectonic stress and the spectra of seismic shear waves from earthquakes. Journal of Geophysical Research, 75(26), 4997–5009.CrossRef
Burridge, R. (1969). The numerical solution of certain integral equations with non-integrable kernels arising in the theory of crack propagation and elastic wave diffraction. Philosophical Transactions of the Royal Society of London, A 265, 353–381.
Burridge, R. (1973). Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohesion. Geophysical Journal of the Royal Astronomical Society, 35(4), 439–455.CrossRef
Campbell, K. W. (1981). Near-source attenuation of peak horizontal acceleration. Bulletin of the Seismological Society of America, 71(6), 2039–2070.
Cichowicz, A. (2008). Near-field pulse-type motion of small events in deep gold mines: Observations, response spectra and drift spectra. In 14th World Conference on Earthquake Engineering, Beijing, China.
Clayton, R., & Engquist, B. (1977). Absorbing boundary conditions for acoustic and elastic wave equations. Bulletin of the Seismological Society of America, 67(6), 1529–1540.CrossRef
Cornell, C. A. (1968). Engineering seismic risk analysis. Bulletin of the Seismological Society of America, 58(5), 1583–1606.CrossRef
Dineva, S., Mihaylov, D., Hansen-Haug, J., Nystrom, A., & Woldemedhin, B. (2016). Local seismic systems for study of the effect of seismic waves on rock mass and ground support in Swedish underground mines Zinkgruvan, Garpenberg, Kiruna). In Ground Support 2016, Lulea, Sweden (pp. 1–11).
Douglas, J. (2018). Ground motion prediction equations 1964–2018. Review. University of Strathclyde, Glasgow.
Dunham, E. M. (2007). Conditions governing the occurrence of supershear ruptures under slip-weakening friction. Journal of Geophysical Research, 112(B07302), 1–24. http://doi.org/10.1029/2006JB004717.
Dunham, E. M., & Archuleta, R. J. (2004). Evidence for a supershear transient during the 2002 Denali fault earthquake. Bulletin of the Seismological Society of America, 94(6B), S256–S26.CrossRef
Durrheim, R. J., Spottiswoode, S. M., Roberts, M. K. C., & Brink, A. v. Z. (2005). Comparative seismology of the Witwatersrand basin and Bushveld Complex and emerging technologies to manage the risk of rockbursting. The Journal of The South African Institute of Mining and Metallurgy, 105, 409–416.
EPRI (1988). A criterion for determining exceedance of the operating basis earthquake. Tech. Rep. EPRI NP-5930, Electrical Power Research Institute, Palo Alto, California.
Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 241(1226), 376–396.
Esteva, L. (1970). Seismic risk and seismic design decisions. In R. J. Hansen (Ed.), Seismic risk and seismic design criteria for nuclear power plants (pp. 142–182). MIT Press.
Gay, N. C., & Ortlepp, W. D. (1978). Anatomy of a mining-induced fault zone. Geological Society of America Bulletin, 90, 47–58.CrossRef
Gospodinov, D., Dineva, S., & Dahner-Lindkvist, C. (2022). On the applicability of the RETAS model for forecasting aftershock probability in underground mines, Kiirunavaara Mine, Sweden. Journal of Mine Seismology, Springer, 4(26). http://doi.org/10.1007/s10950-022-10108-6.
Graves, R. W. (1996). Simulating seismic wave propagation in 3D elastic media using staggered grid finite differences. Bulletin of the Seismological Society of America, 86(4), 1091–1106.CrossRef
Handley, M. F., de Lange, J. A. J., Essrich, F., & Banning, J. A. (2000). A review of the sequential grid mining method employed at Elandsrand Gold Mine. The Journal of The Southern African Institute of Mining and Metallurgy, 100(3), 157–168.
Herrero, A., & Bernard, P. (1994). A kinematic self-similar rupture process for earthquakes. Bulletin of the Seismological Society of America, 84(4), 1216–1228.CrossRef
Ida, Y. (1973). The maximum acceleration of seismic ground motion. Bulletin of the Seismological Society of America, 63(3), 959–968.
Joyner, W. B., & Boore, D. M. (1993). Methods for regression analysis of strong motion data. Bulletin of the Seismological Society of America, 83(2), 469–487.CrossRef
Joyner, W. B., & Boore, D. M. (1994). Methods for regression analysis of strong motion data - Errata. Bulletin of the Seismological Society of America, 84(3), 955–956.CrossRef
Kaiser, P. K., & Maloney, S. M. (1997). Scaling laws for the design of rock support. Pure and Applied Geophysics, 150(3-4), 415–434.CrossRef
Kanamori, H. (1972). Determination of effective tectonic stress associated with earthquake faulting. The Tottori earthquake of 1943. Physics of the Earth and Planetary Interiors, 5, 426–434.CrossRef
Kirsch, E. G. (1898). Die theorie der elastizitat und die bedurfnisse der festigkeitslehre. Zeitschrift des Vereines deutscher Ingenieure, 42, 797–807.
Kolsky, H. (1963). Stress waves in solids (212 p.). Dover Publications.
Konno, K., & Ohmachi, T. (1998). Ground motion characteristics estimated from spectral ratio between horizontal and vertical components of microtremors. Bulletin of the Seismological Society of America, 88(1), 228–241.CrossRef
Love, A. E. H. (1911). Some problems of geodynamics. Cambridge University Press.
Lu, X., Rosakis, A. J., & Lapusta, N. (2010). Rupture modes in laboratory earthquakes: Effect of fault prestress and nucleation conditions. Journal of Geophysical Research, 115(B12302), 1–25. http://doi.org/10.1029/2009JB006833.
Malek, F., & Leslie, I. (2006). Using seismic data for rockburst re-entry protocol at Inco’s Copper Cliff North Mine. In 41st U.S. Symposium on Rock Mechanics, Golden, Colorado, ARMA-06-1163.
McGarr, A. (1997). A mechanism for high-wall rock velocities in rockbursts. Pure and Applied Geophysics, 150(3-4), 381–391.CrossRef
McGarr, A., & Fletcher, J. B. (2001). A method for mapping apparent stress and energy radiation applied to the 1994 Northridge Earthquake Fault Zone-Revisited. Geophysical Research Letters, 28(18), 3529–3532. http://doi.org/10.1029/2001GL013094.CrossRef
McGarr, A., & Fletcher, J. B. (2003). Maximum slip in earthquake fault zones, apparent stress, and stick-slip friction. Bulletin of the Seismological Society of America, 93(6), 2355–2362.CrossRef
McGarr, A., & Fletcher, J. B. (2005). Development of ground-motion prediction equations relevant to shallow mining induced seismicity in the Trail Mountain area. Bulletin of the Seismological Society of America, 95(1), 31–47. http://doi.org/10.1785/0120040046.CrossRef
McGarr, A., Green, R. W. E., & Spottiswoode, S. M. (1981). Strong ground motion of mine tremors: Some implications for near-source ground motion parameters. Bulletin of the Seismological Society of America, 71(1), 295–319.CrossRef
McGuire, R. K. (2001). Deterministic vs probabilistic earthquake hazards risks. Soil Dynamics and Earthquake Engineering, 21, 377–384.CrossRef
McGuire, R. K. (2004). Seismic hazard and risk analysis. Earthquake Engineering Research Institute.
Mendecki, A. J. (2001). Data-driven understanding of seismic rock mass response to mining: Keynote Address. In G. van Aswegen, R. J. Durrheim, & W. D. Ortlepp (Eds.), Proceedings 5th International Symposium on Rockbursts and Seismicity in Mines, Johannesburg, South Africa (pp. 1–9). South African Institute of Mining and Metallurgy.
Mendecki, A. J. (2005). Persistence of seismic rock mass response to mining. In Y. Potvin, & M. R. Hudyma (Eds.), Proceedings 6th International Symposium on Rockburst and Seismicity in Mines, Perth, Australia (pp. 97–105), Australian Centre for Geomechanics.
Mendecki, A. J. (2008). Forecasting seismic hazard in mines. In Y. Potvin, J. Carter, A. Diskin, & R. Jeffrey (Eds.), Proceedings 1st Southern Hemisphere International Rock Mechanics Symposium, Perth, Australia (pp. 55–69). Australian Centre for Geomechanics.
Mendecki, A. J. (2013). Characteristics of seismic hazard in mines: Keynote Lecture. In A. Malovichko & D. A. Malovichko (Eds.), Proceedings 8th International Symposium on Rockbursts and Seismicity in Mines, St Petersburg-Moscow, Russia (pp. 275–292). ISBN:978-5-903258-28-4.
Mendecki, A. J. (2016). Mine seismology reference book: seismic hazard (1st edn.). Institute of Mine Seismology. ISBN:978-0-9942943-0-2. www.imseismology.org/msrb/.
Mendecki, A. J. (2017). Mapping seismic ground motion hazard: Keynote lecture. In J. A. Vallejos (Ed.), Proceedings 9th International Symposium on Rockbursts and Seismicity in Mines, Santiago, Chile.
Mendecki, A. J. (2018). Ground motion prediction equations for DMLZ. Technical PTFI-REP-GMPE-201801-AJMv1, Institute of Miner Seismology.
Mendecki, A. J., & Lötter, E. C. (2011). Modelling seismic hazard for mines. In Australian Earthquake Engineering Society 2011 Conference, Barossa Valley.
Mendecki, M. J., Duda, A., & Idziak, A. (2018). Ground-motion prediction equation and site effect characterization for the central area of the main syncline, Upper Silesia Coal Basin, Poland. Open Geoscience, 10(1), 474–483. http://doi.org/10.1515/geo-2018-0037.
Milev, A. M., & Spottiswood, S. M. (2005). Strong ground motion and site response in deep South African mines. The Journal of The South African Institute of Mining and Metallurgy, 105, 1–10.
Mountfort, I. P., & Mendecki, A. J. (2019). Measuring ground motion with geophones. Acta Geophysica (in submission).
Mow, C.-C., & Pao, Y.-H. (1971). The diffraction of elastic waves and dynamic stress concentration. R-482-pr, United States Air Force.
Mulargia, F., Stark, P. B., & Geller, R. J. (2017). Why is probabilistic seismic hazard analysis (PSHA) still used? Physics of the Earth and Planetary Interiors, 264, 63–75.CrossRef
Nordstrom, E., Dineva, S., & Nordlund, E. (2020). Back analysis of short-term seismic hazard indicators of larger seismic events in deep underground mines (Lkab, Kiirunavaara mine, Sweden). Pure and Applied Geophysics, 177, 763–785, http://doi.org/10.1007/s00024-019-02352-8.CrossRef
Ortlepp, W. D. (1984). Rockbusts in South Africa gold mines: A phenomenological view. In N. C. Gay, & E. H. Wainwright (Eds.), Proceedings 1th International Congress on Rockbursts and Seismicity in Mines, Symposium Series 6. The South African Institute of Mines and Metallurgy.
Ortlepp, W. D. (1993). High ground displacement velocities associated with rockburst damage. In R. P. Young (Ed.), Proceedings 3rd International Symposium on Rockbursts and Seismicity in Mines, Kingston, Ontario, Canada (pp. 101–106). Balkema, Rotterdam.
Ortlepp, W. D. (1997). Rock fracture and rockbursts - An illustrative study. Monograph Series M9 (126 p.). South African Institute of Mining and Metallurgy.
Porter, K. (2018). A beginners guide to fragility, vulnerability, and risk. University of Colorado Boulder.
Saenger, E., & Bohlen, T. (2004). Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid. Geophysics, 69, 583–591.CrossRef
Savage, J. C. (1971). Radiation from supersonic faulting. Bulletin of the Seismological Society of America, 61(4), 1009–1012.CrossRef
Somerville, P. G., Smith, N. F., Graves, R. W., & Abrahamson, N. A. (1997). Modifications of empirical strong ground motion attenuation relations to include the amplitude and duration effects of rupture directivity. Seismological Research Letters, 68(1), 199–222.CrossRef
Spottiswoode, S. M. (2000). Aftershocks and foreshocks of mine seismic events. In 3rd International Workshop on the Application of Geophysics to Rock and Soil Engineering, Melbourne (pp. 1–6).
Spudich, P., & Cranswick, E. (1984). Direct observation of rupture propagation during the 1979 Imperial Valley earthquake using a short baseline accelerometer array. Bulletin of the Seismological Society of America, 74(6), 2083–2114.CrossRef
Trifunac, M. D., & Brady, A. G. (1975). A study on the duration of strong earthquake ground motion. Bulletin of the Seismological Society of America, 65(3), 581–626.
Turner, M. H., & Player, J. R. (2000). Seismicity at Big Bell Mine. In Proceedings Massmin 2000 (pp. 791–797). The Australasian Institute of Mining and Metallurgy.
Vallejos, J. A. (2010). Analysis of seismicity in mines and development of re-entry protocols, PhD thesis, Queen’s University Kingston, Ontario, Canada.
van Aswegen, G. (2008). Ortlepp Shears - dynamic brittle shears of South African gold mines. In Y. Potvin (Ed.), Proceedings 1st Southern Hemisphere International Rock Mechanics Symposium, Perth (pp. 111–120).
van Aswegen, G., & Mendecki, A. J. (1999). Mine layout, geological features and seismic hazard. Final report gap 303, Safety in Mines Research Advisory Committee, South Africa (pp. 1–91).
Vieira, F. M. C. C., Diering, D. H., & Durrheim, R. J. (2001). Methods to mine the ultra-deep tabular gold-bearing reefs of the Witwatersrand Basin, South Africa. In W. A. Hustrulid, & R. L. Bullock (Eds.), Underground Mining Methods: Engineering Fundamentals and International Case Studies (pp. 691–704), Society for Mining, Metallurgy and Exploration, Inc (SME).
Weertman, J. (1969). Dislocation motion on an interface with friction that is dependent on sliding velocity. Journal of Geophysical Research, 74(27), 6617–6622.CrossRef
Woodward, K., Wesseloo, J., & Potvin, Y. (2017). Temporal delineation and quantification of short term clustered mining seismicity. Pure and Applied Geophysics, 174, 2581–2599. http://doi.org/10.1007/s00024-017-1570-6.CrossRef
