Dieses Kapitel vertieft sich in die Mechanik seismischer Quellen im Gestein und konzentriert sich auf die Unterscheidung zwischen elastischer und unelastischer Belastung und deren Rolle in der seismischen Wellenstrahlung. Er untersucht, wie Faktoren wie Gesteinsstärke, Spannungszustand und Verformungsrate die Amplitude und Frequenz seismischer Wellen beeinflussen. Der Text diskutiert auch die Erkennung seismischer Ereignisse in einer bestimmten Entfernung und die quantitative seismologische Verarbeitung aufgezeichneter Wellenformen, um Parameter wie Ursprungszeit, Ort, seismische Potenz und abgestrahlte seismische Energie abzuleiten. Darüber hinaus wird die Beziehung zwischen seismischen Ereignissen und geologischen Merkmalen untersucht, wobei hervorgehoben wird, wie verschiedene Gesteinstypen und Stressbedingungen die seismische Energiestrahlung beeinflussen. Das Kapitel schließt mit praktischen Beispielen der seismischen Ereignisanalyse, die die Anwendung dieser Prinzipien in realen Szenarien veranschaulichen.
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Abstract
This chapter starts with the glossary of terms used to describe seismic sources, and a short description of seismic potency, radiated seismic energy, and associated parameters like apparent stress, energy index, and apparent volume. There is a short description of the static solution for a circular crack subjected to a uniform shear strain given by Eshelby that explains the basic source scaling relations, and a section on the circular \(\omega ^{2}\) source model by Brune. While Brune’s model fits the spectra of seismic events recorded in deep high stress mines, it does not work as well in many mines with a weaker inhomogeneous rock mass and/or at intermediate depth and in caving mines. Seismic events in such an environment radiate less energy per unit of deformation at source. Therefore Chap. 2 introduces more general point source models suggested by Beresnev and Atkinson that allow for “softer spectra”, e.g. \(\omega ^{2.5}\) or \(\omega ^{3}\). It also describes the final static deformation and strains induced by the double-couple source, gives the radiation pattern for the near, intermediate, and far fields and derives the expression for the surface of such a source for a given inelastic strain drop.
2.1 Introduction
Mining excavations, whether underground or open cast, induce elastic (reversible) and inelastic (irreversible) strain within the surrounding rock. Elastic strain is defined as a process during which no new micro-defects are nucleated, while all existing micro-defects convect with the mass without growing in size (Krajcinovic & Mastilovic, 1995). The inelastic deformation of brittle rock is mainly due to fracturing and frictional sliding called cataclastic flow. No potential energy, i.e. the energy that could do work, is associated with inelastic strain. The potential energy accumulated during elastic deformation in a given volume of rock may be unloaded due to changes in stress state in this volume, or it may be released gradually due to creep, viscous, or plastic deformation, or it may be released suddenly during the processes of inelastic deformation.
Sudden inelastic deformation associated with fracture and slip radiates seismic waves. The amplitude and frequency of seismic waves radiated from such a source depend, in general, on the strength and state of stress of the rock, the size of the source, and the rate at which the rock is deformed during the fracturing process. The following relations apply here (all other parameters being the same):
1.
The amplitude and frequency increase with an increase in rock strength and stress.
2.
The amplitude and frequency increase with an increase in co-seismic deformation rate at source.
3.
The predominant frequency, i.e. the frequency at which the most energy is radiated, decreases with an increase in source size.1
Seismic sensors detect waves caused by inelastic deformation within a certain distance. Since the attenuation of seismic waves increases with their frequency, the higher the amplitude and the lower the frequency, the longer the distance over which one can receive waveforms at a reasonable signal to noise ratio. Quantitative seismological processing of recorded waveforms can routinely provide information on the following parameters pertaining to the source of the seismic radiation:
Origin time of the event, \(t_{0}\)
Location, \(X=\left (x,y,z\right )\)
Seismic potency, P, in m\(^{3}\), which measures the overall co-seismic inelastic deformation at the source, and its tensor, \(P_{i,j}\)
Seismic energy radiated from the source of the seismic event, E, in Joules
Estimate of stress release at source which scales with the \(E/P\) ratio, in Pa, called apparent stress, \(\sigma _{A}\)
Seismic waveforms do not provide direct information about the absolute stress, but mainly about the strain and stress release at the source. However, sources of seismic events associated with weaker geological features or with a softer or fractured patches of rock yield more slowly under lower driving stress and radiate less seismic energy per unit of inelastic co-seismic deformation, than equivalent sources within strong, solid, and highly stressed rock. Therefore, by comparing the radiated seismic energies of seismic events with similar potencies, one can gain insight into the stresses acting within the part of the rock mass giving rise to these events.
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The bulk of seismicity induced by mining originates close to excavations where the rock mass is fractured, and therefore, the \(E/P\) ratios of these events are low, in some cases between 10 and 100 Pa. These events are also of a “volumetric” nature, i.e. with a relatively high isotropic component of inelastic deformation at source, and generate lower velocity of ground motion. However, the same mine can have very dynamic events that originate away from excavations by rupturing intact rock that deliver \(E/P\) ratios between 10\(^{6}\) and 10\(^{7}\) Pa and ground motion at source above 1 m/s.
Many mine seismic networks record over 1000, and some over 10000, microseismic events per day, each event being recorded by at least 10, and in some cases more than 50 sites. To get reasonable locations, source parameters and source mechanisms every event should be recorded by preferably 10 or more three-component seismic sensors surrounding the source to ensure adequate sampling of the radiation pattern.
In mines, seismic sensors installed underground are grouted in boreholes away from excavations to avoid site effects. The main reason to install sensors at the skin of an excavation is to measure the amplification of ground motion for support design, and these waveforms are excluded from source inversion. However, over the last 10 years or so, some mines have installed low frequency or strong ground motion sensors on surface to recover the seismic potency of larger events and for ground motion studies.
2.2 Seismic Sources: Glossary of Terms
Seismic Source in Rock
A seismic source is a volume of sudden inelastic deformation in rock that radiates perceptible seismic waves. The velocity of that deformation varies from a few tens of cm/s for slow events to a few metres per second for very dynamic events rupturing intact rock. In mines, the average ground motion at source is 0.5 to 1.5 m/s. The volume of inelastic co-seismic deformation, \(V=P/\Delta \epsilon \), for strains, \(\Delta \epsilon \), greater than 10\(^{-3}\) that crack rock, varies from a fraction of a m\(^{3}\) for events with \(\log P\leq -3.0\), to 10\(^{7}\) m\(^{3}\) for events with \(\log P=4\). Strain changes between 10\(^{-3}\) to 10\(^{-4}\) cause minor damage to solid hard rock, and \(V\left (\geq P/10^{-4}\right )\) is correspondingly larger.
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Rupture and Slip
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 can be higher than \(v_{S}\) for super-shear rupture. Rupture speed controls the frequency content of the recorded ground velocity. 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.
Slip follows rupture, very fast at the tip of the rupture and slowing 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. Rise time refers to slip duration at a given point in a source.
Directivity
It is similar to the Doppler effect. Unilateral rupture directivity will produce earthquake source pulses and source spectra that vary with azimuth (Ben-Menahem, 1961). In the time domain, it produces shorter duration, higher amplitude source time functions in the direction of rupture, and longer duration, lower amplitude source time functions in the opposite direction. However, the area under the source time function is the same, regardless of the azimuth, and is proportional to the seismic potency of the event. In the frequency domain, it produces higher spectral amplitudes at higher frequencies in the direction of rupture and a lack of such high frequency signal in the opposite direction. Low frequency amplitudes of source spectra remain unchanged with azimuth. For a circular crack in a purely elastic media with radius r rupturing outwards from the centre with rupture velocity \(v_{r}\), the pulse width of ground velocity, or equivalently the rise time of the far-field displacement pulse, is \(\tau \left (\theta \right )=r/v_{r}-\left (\sin \theta \right )r/v_{\pi }\), where \(\theta \) is the take-off angle, i.e. the angle between the normal to the source plane and the ray leaving the source, and \(v_{\pi }\) is the phase velocity. The full pulse width is the duration between the first break and the second zero crossing, \(\Delta T=r/v_{r} +\left (\sin \theta \right )r/v_{\pi }\). The rise time decreases with increasing \(\theta \), and the full pulse width increases with decreasing \(\theta \). In a real media, the high frequencies are more attenuated more than low frequencies, resulting in a broadening of the pulse width travel time.
Source Time Function
The source time function (STF) defines the deformation \(u\left (t\right )\), velocity \(\dot {u}\left (t\right )\), or acceleration \(\ddot {u}\left (t\right )\) at source versus time. The rise time at a given point at source is the duration of slip at that point, and the average rise time \(\left \langle \tau \right \rangle =\bar {u}/\left \langle \dot {u}\right \rangle \sim 1/f_{0}\), where \(f_{0}\) is the corner frequency. In a crack-like model, the slip duration at a given point is comparable to the overall duration of the rupture, i.e. slip at a given point continues until information is received that the rupture has stopped propagating. The stopping of rupture generates strong healing waves that propagate inwards from the rim of the fault. These waves are of three types: P-, S-, and Rayleigh waves. Soon after the passage of the Rayleigh waves, slip rate inside the fault decreases to zero and the fault heals. In some cases, only a portion of the overall rupture surface is undergoing slip at any given point in time, and therefore, there may be larger transient stress changes in the vicinity of the propagating slip pulse (Heaton, 1990). The initial pulse width is the duration between the first break and the first zero crossing of the seismic signal. For wavelengths much larger than source size and for periods much longer than source duration, the source volume is relatively small and can be approximated as a point concentrated in space with finite potency. Shorter, higher frequency waves are sensitive to the finite extent and detailed variation of the slip process at the source, and they require finite source models. The source time function can also be presented as seismic moment or potency and their derivatives in the time domain, e.g. \(P\left (t\right )\), \(\dot {P}\left (t\right )\), \(\ddot {P}\left (t\right )\).
Double Couple
A model of seismic source caused by shear slip across an internal surface of zero thickness in an isotropic elastic medium for which the equivalent force system consists of two orthogonal couples with the same moment and opposite sign. The corresponding moment or potency tensors have both zero trace (purely deviatoric) and zero determinant. Physically, this is a representation of a shear dislocation source without any volume changes.
Radiation Pattern
A radiation pattern is a geometric description of amplitude and sense of initial motion distributed over the P and S wavefronts on the sphere around the source. If the radius of the sphere is large enough relative to both the size of the source and the dominant wavelength, the radiation pattern represents the far field of a point source. If the radius of the sphere is large compared to the dominant wavelength but comparable with the size of the source, the radiation pattern represents the far field of an extended source. The radiation patterns for the displacement and the velocity are the same since taking the time derivative does not affect the angular distribution. Since radiation from seismic sources reflects the strain distribution near the source, it has a degree of symmetry. The polarity of the initial P-wave pulse leaving the source may be compressional—away from the hypocentre, dilatational—towards the hypocentre, or null at the nodal plane where amplitudes tend to go to zero. For a double-couple point source moving with speed \(v_{r}\) along the OX axis, the angular distribution of displacement is \(u_{P}\left (\theta \right ) = \sin \left (2\theta \right )/\left [1-\left (v_{r}/v_{P}\right )\cos \theta \right ]\), \(u_{S}\left (\theta \right ) = \cos \left (2\theta \right )/\left [1-\left (v_{r}/v_{S}\right )\cos \theta \right ]\), where \(\theta \) is anticlockwise from OX. It has elongated lobes in the direction of rupture, and more so for faster rupture (Ben-Menahem, 1961; Hirasawa & Stauder, 1965), and it is singular for \(v_{r} \geq v_{P}\) and \(v_{r} \geq v_{S}\). For \(v_{r} = 0\), it gives the radiation pattern for a single double-couple non-moving source (Aki & Richards, 2002). A double-couple radiation pattern is symmetric with respect to nodal planes. In mines, tunnels and excavations act as strong scatterers of seismic waves and may modify the radiation patterns of the primary waves. In practice, radiation patterns are observable mainly at lower frequencies.
Source Spectra and Corner Frequency
A point source at which stress is released instantaneously would radiate P- and S-wave displacement pulses that propagate as Dirac delta functions in the retarded time \(t-R/v_{P,S}\). Their spectra would be flat. Seismic source theory predicts a far-field displacement spectrum that is constant at low frequencies and inversely proportional to some power of the frequency at high frequencies (Aki, 1967). The corner frequency of the displacement spectrum, \(f_{0}\), is where the high and low frequency trends intersect. In this sense, it is a model parameter. The predominant frequency, \(f_{E}\), the frequency at which the maximum seismic energy is radiated, i.e. the maximum of the ground motion velocity spectrum, is a more rigorously defined parameter of the seismic spectrum. In the \(\omega ^{-2}\) model, the corner frequency and predominant frequency coincide, \(f_{0}=f_{E}\), but in other spectral models they differ.
When the stress is released over a finite time, the radiated pulses are broadened proportionally. There is a corner in both the P-wave and the S-wave displacement spectra at frequencies proportional to the reciprocal of the time for the stress to be released at the source. Similarly, if stress is released instantaneously, but the source size is made finite, the P and S pulses will be broadened, and their spectra will have corner frequencies. The P- and S-waves propagate through an attenuating medium with frequency dependent velocities. This phenomenon is called dispersion, and it leads to a shift in the respective frequencies.
Maximum Frequency, \({\mathbf {f}}_{max}\)
While the theoretical acceleration spectrum is flat at high frequencies, the observed spectra are characterised by a trend of exponential decay. Hanks (1979) and Hanks (1982) suggested that the acceleration spectrum is flat above the corner frequency only to a second corner frequency, called \(f_{max}\), above which it decays rapidly. The origin of the rapid decay may be the source or path including the site effect or a combination of the above.
Fault Plane Solution or Focal Mechanism
The fault plane solution determines the direction of slip, which is controlled by the orientation of the elastic strain field at the time of rupture, and a possible orientation of the rupture plane. In its simplest form it uses the directions of P-wave motions recorded at a number of stations surrounding the source. Plotting all available directions in the lower hemisphere of a stereographic projection, one can define two orthogonal planes separating compressional and dilatational motion. The axes of maximum shortening and maximum lengthening bisecting the quadrants are known as the P and T axes, respectively. These are the principal strain axes that do not necessarily coincide with the principal stress axes. The P axis lies within the quadrant of dilatational motions, and the T axis lies within the quadrant of compressional motions. Both are orthogonal to the intersection of the two nodal planes where their amplitudes are zero. The axis formed by this intersection is called the B or the null axis. The directions of P-wave motion or moment tensor inversion alone cannot resolve which nodal plane is the rupture plane.
2.3 Seismic Potency and Potency Tensor
A seismic source in rock is a volume of sudden inelastic deformation that radiates perceptible seismic waves. It is always of finite extent in all three dimensions although the thickness of the source is usually much smaller than the other two dimensions. Scalar seismic potency for a physical source is a product of the inelastic strain change and the volume subjected to that strain, \(P = \Delta \epsilon V\). Scalar seismic potency modelled as a single dislocation source is the product of an average slip and source area, \(P = \bar {u}A\) (Ben-Menahem & Singh, 1981; Ben-Zion & Zhu, 2002). Seismic moment \(M= \mu P = \mu \Delta \epsilon V = \Delta \sigma V\), where \(\mu \) is rigidity.
In hard rock, strain changes \(\Delta \epsilon \leq 10^{-6}\) are considered elastic, \(\Delta \epsilon \geq 10^{-4}\) damage inhomogeneous rock and \(\Delta \epsilon \geq 10^{-3}\) crack intact rock (Jeffreys, 1975). Therefore, a seismic event with \(\log P=2.0\), i.e. \(m_{HK}=2.25\), is not just a 100 m\(^{3}\) of volume, but such an event creates approximately \(V = 10^{2}/10^{-3} = 10^{5}\) m\(^{3}\) of rock subjected to cracks, which is not insignificant and frequently leads to damage when its source is close to an excavation.
For small deformations, the total strain at a given point may be written as a sum of elastic \(\varepsilon \) and inelastic contributions, \(\epsilon \). The inelastic strain tensor, \(\epsilon _{ij}\), also called transformational strain, represents the inelastic brittle deformation at the seismic source that resets the reference levels of the elastic stress and strain tensors after the event (Eshelby, 1957). The seismic potency tensor, \(P_{ij}\), and seismic moment tensor, \(M_{ij}\), can then be defined as
where \(c_{ijkl}\) is the tensor of elastic moduli of the rock surrounding the source and the product \(m_{ij}=c_{ijkl}\epsilon _{kl}\) is also called seismic moment density tensor or stress glut (Backus & Mulcahy, 1976). The scalar potency, \(P = \sqrt {2P_{ij}P_{ij}}= \left \Vert P_{ij}\right \Vert \), and its units are m\(^{3}\). Assuming zero net torque and zero net rotation, both tensors are symmetric and have six independent components. The advantage of seismic potency is that it makes no assumptions about material properties, \(c_{ijkl}\), outside the source which are poorly constrained.
The seismic potency tensor can be decomposed into an isotropic component, \(\text{P}_{\text{ISO}}=\frac {1}{3}\delta _{ij}\text{tr}\left (\mathbf {P}\right )\), where \(\text{tr}\left (\mathbf {P}\right )=\sum _{i=1}^{3}P_{ii}\) and \(\delta _{ij}\) is the Kronecker delta, and the deviatoric remainder, \(\text{P}_{\text{DEV}}=P_{ij}-\text{P}_{\text{ISO}}\),
A purely isotropic tensor is characterised by three equal eigenvalues. In fluid mechanics, this represents a fluid at rest, and the magnitude of the eigenvalues stands for hydrostatic pressure. Positive eigenvalues signify an ideal explosion and negative signify implosion. To quantify the strength of the isotropic component, Zhu and Ben-Zion (2013) introduced a dimensionless parameter,
that varies from \(-1\) for implosion to 1 for explosion. Introducing the normalised isotropic tensor, \(I_{ij} = \left (1/\sqrt {3}\right )\delta _{ij}\), and normalised deviatoric tensor, \(D_{ij}\), that satisfies \(D_{ii}=0\) and \(D_{ij}D_{ji}=1\), the potency tensor can be written as
The deviatoric tensor can be decomposed further into a double-couple term, DC, with zero determinant representing shear deformation on a plane, and a remainder non-double-couple term, e.g. a compensated linear vector dipole, CLVD, which is a dipole of magnitude 2 compensated by two unit dipoles (Knopoff & Randall, 1970).
The eigenvalues of the normalised deviatoric tensor are \(\lambda _{1}\geq \lambda _{2}\geq \lambda _{3}\), where the largest one \(\lambda _{1}\) corresponds to the T axis of the deviatoric tensor \(D_{ij}\), the intermediate \(\lambda _{2}\) corresponds to the null axis eigenvector N, and \(\lambda _{3}\) corresponds to the P axis eigenvector P. When \(\lambda _{2}=0\), the deviatoric tensor \(D_{ij}\) is a pure double couple. The condition \(D_{ii}=0\) requires that \(\lambda _{1}+\lambda _{2}+\lambda _{3}=0\) and \(D_{ij}D_{ij}= 1\) that \(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=1\). From these three conditions imposed on the eigenvalues, Zhu and Ben-Zion (2013) show that
While the DC component can be associated with slip on a planar surface, the CLVD component corresponds to compensated uniaxial compression or extension of a volume which requires a more complicated geological setting and may not be well constrained by inversion (Frohlich & Davis, 1999). In addition, the DC-CLVD decomposition is not unique because the CLVD symmetry axis can be aligned with any of the principle axis. Aligning the symmetry CLVD axis with the N axis, Zhu and Ben-Zion (2013) show that
where \(D_{ij}^{\text{DC}}=\left (T_{i}T_{j}-P_{i}P_{j}\right )/\sqrt {2}\) and \(D_{ij}^{\text{CLVD}}==\left (2N_{i}N_{j}-T_{i}T_{j}-P_{i}P_{j}\right )/\sqrt {6}\) are the normalised DC and CLVD tensors, which are orthogonal, and therefore \(D_{ij}^{\text{DC}}D_{ij}^{\text{CLVD}}=0\). The strength of the CLVD component is given by \(\chi =\lambda _{2}/\sqrt {3/2}\), which varies between \(1/2\) and \(-1/2\).
Now \(D_{ij}=\sqrt {1-\chi ^{2}}D_{ij}^{\text{DC}}+\chi D_{ij}^{\text{CLVD}},\) and inserting into Eq. (2.4) Zhu and Ben-Zion (2013) obtained
which has six independent parameters, i.e. three amplitude factors: P, \(\xi \), and \(\chi \), and three angles that specify the orientation of the principal axes of the deviatoric tensor.
Here the isotropic parameter \(\xi \) represents the fractional volumetric source component, and \(\chi \) represents the relative strength of the CLVD component within the deviatoric tensor. Following the Chapman and Leaney (2012) suggestion to use the squared ratios of the scalar moment of each component to the total scalar moment to represent the relative strengths of the ISO, DC, and CLVD components Zhu and Ben-Zion (2013) obtained,
so that \(\Lambda ^{\text{ISO}}+\Lambda ^{\text{DC}}+\Lambda ^{\text{CLVD}}=1\).
Figure 2.1 shows the permissible values of the fractional strength parameters. Note that the maximum CLVD strength in this decomposition is 25 per cent, i.e. at \(\xi =0\) and \(\chi =\pm 1/2\).
Fig. 2.1
Diagram showing permissible values of the fractional source strengths \(\Lambda ^{\text{ISO}}\), \(\Lambda ^{\text{CLVD}}\), and \(\Lambda ^{\text{DC}}\). The pure explosion and implosion sources are indicated by the grey solid and open circles, respectively. The pure DC source is located at the centre. The contours show DC levels of 75% and 50% (Reproduced from Zhu and Ben-Zion, 2013)
A potency tensor representation is a theoretical relation between the real ground motions at a given site and the potency tensor at the source. To invert for the potency tensor, we need the recorded waveforms and the synthetic waveforms, i.e. Green’s functions that describe the impulse response of the rock mass at a given site to a force excitation at the source. Different schemes for moment or potency tensor inversion of mine events are described in Fletcher and McGarr (2005), Malovichko and van Aswegen (2013), Sen et al. (2013), Ma et al. (2018), Willacy et al. (2019).
Sources of mid-size and larger events in underground mines interact with excavations and often display a substantial implosive component (McGarr 1992). Seismic moment or potency tensors derived from waveforms of these events processed using elastodynamic Green’s functions for an unbounded homogeneous medium may lead to an incorrect interpretation. Malovichko (2020) suggested a correction for conventional expressions for seismic point sources based on the volumetric integral of stress-free strain that includes a term that depends on the displacement at the surface of the excavation. He showed that it affects the type of mechanism and orientations of principal axes.
2.4 Radiated Seismic Energy
The energy release during fracture and frictional sliding is due to the transformation of elastic strain into inelastic strain. This transformation may occur at different rates ranging from slow creep-like events to very fast dynamic seismic events with an average velocity of deformation at the source of up to a few metres per second. Slow type events have a long time duration at the source and thus radiate predominantly lower frequency waves, as opposed to dynamic sources of the same size. Since the excitation of seismic energy can be represented in terms of the temporal derivatives of the source function, one may infer that a slower source process implies less seismic radiation. In terms of fracture mechanics, the slower the rupture velocity, the less energy is radiated; the quasi-static rupture would radiate practically no energy.
To assess the physical sources of radiated energy, let us consider the formula for seismic energy E for the single fracture-type source, expressed in terms of source parameters (Kostrov, 1974; Kostrov & Das, 1988; Rivera & Kanamori, 2005),
where \(\gamma _{eff}\)—the effective surface energy, which includes the total loss of mechanical energy, in particular inelastic work, and heat flow from the fracture edge, A—the fracture area with the displacement \(u_{i}\), \(\Delta \sigma _{ij}\)—the difference between the final (at the end of the event) and the initial stress, \(n_{j}\)—the unit vector normal to the fracture plane, \(t_{s}\)—the source duration, \(\dot {\sigma }_{ij}\)—the time derivative of stress or traction rate.
Expression (2.9) allows estimating seismic energy from the stresses and displacements only, and it is not necessary to know the absolute value of stress at source. Therefore, seismic energy depends only on the stress perturbation at source, \(\Delta W\), caused by rupture, and is independent of the pre-stress within the medium. However, it is more likely that high pre-stress will drive higher stress change. The term \(\Delta \sigma _{ij}u_{i}n_{j}\) in Eq. (2.9) cannot be interpreted as the local energy density at a point on the fault plane. It represents the energy released from the tubular volume formed by the integral lines of the vector \(F_{j}\equiv \Delta \sigma _{ij}u_{i}\) passing through a unit area at a point on the fault plane. Thus, this represents the energy coming from the volume of the medium, rather than the fault plane (Rivera & Kanamori, 2005).
The second term, containing the traction rate, \(E_{F}\), strongly depends on how the fracture propagates and how it correlates with slip. The energy due to the radiation of high frequency waves during accelerating and decelerating rupture is called radiation friction or radiation loss. If traction rate and slip are uncorrelated, the third term will vanish.
From dimensional analysis, it follows that the first two terms in this equation vary with the fracture area as \(A^{3/2}\), whereas the fracture work, \(E_{G}=2\gamma _{eff}A\), is proportional to A (Kostrov & Das, 1988). Thus, the relative contribution of the fracture work to seismic energy increases with a decrease in the size of the fracture. Consequently, for sufficiently small fractures, the first term may almost cancel the second term, suppressing the acoustic emission so that “silent” fracture would occur.
In the time domain, the radiated seismic energy of the P- or S-wave is proportional to the integral of the radiation pattern corrected far-field velocity pulse squared \(\dot {u}^{2}(t)\) of duration \(t_{s}\),
where \(\rho \) is rock density, \(v_{S,P}\) is S- or P-wave velocity, and R is the distance from the source. Different \(u(t)\) may have the same time integral over \(t_{s}\) and thus the same potency P, but their time derivatives \(\dot {u}(t)\) may differ, producing different energies E. If \(u(t)\) varies very rapidly, the radiated energy can be very large since \(E\sim \dot {u}^{2}(t)\).
In the far field of seismic observations, the P- and S-wave contributions to the total radiated energy are proportional to the integral of the square of the P and S velocity spectrum. For a reasonable signal to noise ratio in the bandwidth of frequencies available on both sides of the corner frequency, the determination of that integral from waveforms recorded by a seismic network is fairly objective. The high frequency component of seismic radiation needs to be recorded by the seismic system if a meaningful insight into the stress regime at the source is to be gained.
2.5 Apparent Stress, Energy Index, and Apparent Volume
Aki (1966) suggested that the ratio of elastic energy released by the source, W, to seismic moment, M, is independent of the average relative displacement at source, \(\overline {u}\), and the source area A. Brune (1968) and Wyss and Brune (1971) compared \(W = \overline {\sigma }\cdot \overline {u}A = \overline {\sigma }\cdot P\), where \(\overline {\sigma }\) is the average of the initial and final stress, and the radiated seismic energy, \(E=\xi W\), where \(\xi \) is seismic efficiency, and called the product, \(\xi \overline {\sigma }\), apparent stress (Wyss & Brune, 1971),
Apparent stress is the ratio of the observed radiated seismic energy to seismic potency, and therefore, it is a model independent measure of the dynamic stress release in the source region. Apparent stress is proportional to the integral of the square of the velocity spectrum divided by the amplitude of the low frequency asymptote to the displacement spectrum, or, when the acceleration spectrum is considered, apparent stress depends linearly on the product of the comer frequency and the amplitude of the high frequency asymptote to the acceleration spectrum. It is then a more reliable and less model dependent parameter describing an average stress release at the source than the corner frequency cubed dependent static stress drop.
Figure 2.2 left shows the energy, E vs. \(\log P\) for the data set described in Chap. 5 Sect. 1.5, where the green line represents the ordinary least squares fit. The colour here scales with \(\log \sigma _{A}\). Figure 2.2 right shows apparent stress, \(\sigma _{A}\), vs. \(\log P\) for the same data set where colour indicates the time of the event. In this case, there is an increase in seismic energy with increasing potency.
Fig. 2.2
Energy, E, vs. \( \log P\) plot of events with the least square fit (left). Apparent stress, \(\sigma _{A}\), vs. \( \log P\) for the same data set (right)
Although seismic waveforms do not have direct information about absolute stress but merely about the dynamic stress drop at the source, a number of seismological studies and numerous underground observations suggest that a reliable estimate of apparent stress can be used as an indicator of the local level of stress. Gibowicz (1990) considered apparent stress as an independent parameter of stress release in the case when P- and S-wave contributions to the seismic energy are included. (Mendecki, 1993) showed an example where apparent stress associated with seismic events of magnitudes between 1.3 and 1.5 varies from 0.2 to 40 bar, being generally higher at greater depth and within less faulted rock. In general, the apparent stress expresses the amount of radiated seismic energy per unit volume of inelastic co-seismic deformation.
Let us imagine the source of a seismic event associated with a relatively weak geological feature or with a soft patch in the rock mass. Such a source will yield slowly under lower differential stress producing larger seismic potency and radiating less seismic energy, resulting in a low apparent stress event. The opposite applies to a source associated with a strong geological feature or hard patch in the rock mass. In the case of a so-called complex or multiple event, the rapid deformation process at the initial source can push the stresses in the adjacent volume to a level much higher than could normally be maintained by the rock, producing higher apparent stress sub-event(s) that need not be an indication of a generally high ambient stress prior to the event. Although the estimate of apparent stress does not depend on the rupture complexity (Hanks & Thatcher, 1972), the complexity of the event should be tested before meaningful interpretation in terms of stress can be given.
Seismic sources similar in terms of their potency may differ by up to two orders of magnitude in radiated energies, reflecting stress differences at the place and time of their occurrence (Mendecki, 1993). Since radiated seismic energy broadly increases with increasing seismic potency, to gain insight into the stress regime, one would need to compare the radiated seismic energies associated with seismic events of similar seismic potencies. This notion was translated into a practical tool by van Aswegen and Butler (1993) and called the energy index, EI. The energy index of an event is the ratio of the observed radiated seismic energy of that event E, to the average energy \(\bar {E}(P) = 10^{d\log P+c}\) radiated by events of the observed potency P, for a given area of interest,
For \(d = 1.0\), the energy index is proportional to apparent stress. In general, it is advisable to use the logarithm of the ratio since it equally affects changes in the numerator or denominator. It also solves the problem of lack of symmetry, i.e. if E is greater than \(\bar {E}\left (P\right )\), the ratio can take theoretically any value greater than 1, but if E is less than \(\bar {E}\left (P\right )\), the ratio is restricted to the range of 0 to 1. The logged ratio restores the symmetry, i.e. \(\log \left (E/\bar {E}\left (P\right )\right ) = -\log \left (\bar {E}\left (P\right )/E\right )\).
Figure 2.3 left shows the Z-coordinate of events vs. time where colour scales with apparent stress, \(\sigma _{A}\). Here, the lower the Z the deeper the event. Figure 2.3 right shows apparent stress of events vs. Z-coordinate where colour scales with time. The thin black line shows a linear fit to the data, \(\log \sigma _{A} = -0.001Z+5.712\), which for this data set indicates that \(\sigma _{A}\) increases by 11 kPa with every 100 m increase in depth.
Fig. 2.3
Z-coordinate of events vs. time (left) and apparent stress, \(\sigma _{A}\), vs. the Z-coordinate of the event for the same data set (right)
The energy index is a relative measure of stress because it is specific to a given data set, i.e. it is a function of the volume and time from which data are selected. Therefore, as more data becomes available, the \(\log E\) vs. \(\log P\) fit may not be applicable, which makes comparison with the previous data set problematic. However, it is a very useful tool to test for the relative spatial differences in stresses. The energy index of small events is a better test for stress level than larger events, since small ones do not disturb stresses in the area.
Source volume V is the volume of co-seismic inelastic deformation that radiated the recorded seismic waves and can be estimated from \(V=M/\Delta \sigma =\mu P/\Delta \sigma \). Since apparent stress scales positively with stress drop and it does not depend on the corner frequency, Mendecki (1993) defined the apparent volume as
Equation (2.13) shows that for a given seismic potency source volume scales inversely with seismic energy. Apparent volume, like apparent stress, depends on seismic potency and radiated seismic energy, and, because of its scalar nature, can easily be manipulated in the form of cumulative or contour plots, providing insight into the rate and distribution of co-seismic deformation and/or stress transfer in the rock mass. In a cumulative plot, the apparent volume amplifies the influence of soft events that for a given potency radiate less energy, and reduces the impact of fast dynamic events, and therefore it is useful to observe periods of accelerating deformations.
Figure 2.4 shows the cumulative plots of seismic potency and apparent volume vs. time for the same data set. The way the Cum\(V_{A}\) depends on potency and energy makes it more sensitive to softer seismic events and will more likely emphasise the phase of accelerating seismic deformation before larger events, in this case before a \(\log P=2.61\) event, which is not visible on the CumP plot. The main applications of apparent volume are in cumulative and spatial contouring plots.
Fig. 2.4
Cumulative potency (left) and apparent volume (right) for the same data set
For a circular crack of radius r and a uniform strain drop \(\Delta \epsilon \) over the crack surface, \(A=\pi r^{2}\), the displacement profile is given by
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 at the centre, \(u_{max}=24r\Delta \epsilon /\left (7\pi \right )\), see Fig. 2.5.
Fig. 2.5
Eshelby (1957) circular crack displacement profile for \( \log P=2.0\) and \(\varDelta \epsilon =5\cdot 10^{-4}\)
The mean displacement \(\bar {u}\) is the integral of \(u(x)\) given above divided by the area of the crack, \(\pi r^{2}.\) Integration in polar coordinates, \(\left (x,\varphi \right )\), gives \(\intop _{0}^{r}xdx\intop _{0}^{2\pi }d\varphi \sqrt {r^{2}-x^{2}}=\left (2/3\right )\pi r^{3}\), and finally,
For the average strain change at source \(\Delta \epsilon =5\cdot 10^{-4}\), the maximum displacement given by Eq. (2.18) is \(u_{max}=1.5*\bar {u}=0.0052\sqrt [3]{P}\), which is 13% more than \(u_{max}=0.0046\sqrt [3]{P}\) given by McGarr and Fletcher (2003). Figure 2.6 shows source radius, r, and average displacements, \(\bar {u}\), for different \(\log P\) events.
Fig. 2.6
Eshelby source radius (left) and average source displacement (right) vs. \( \log P\) and \(m_{HK}\) for different strain drops
The strain energy in an elastically deformed volume is given by \(\Delta W= {1}/{2}\intop _{V}\sigma _{ij}\epsilon _{ij}dV\), where \(\sigma _{ij}\) is the stress tensor, \(\epsilon _{ij}\) is the strain tensor, and V is the strained volume. The total change in strain energy \(\Delta W\) due to a seismic event is \(\Delta W=W_{0}-W_{1}\), where \(W_{0}\) and \(W_{1}\) are the strain energies before and after the event, respectively. The amount of strain energy released by a seismic event cannot be directly estimated from waveforms since the radiated seismic energy is dependent only on the stress drop and not on the initial stress. This means that two events with the same source displacement can have very different strain energy releases. In the circular crack described above the strain energy, \(\Delta W=\mu \Delta \epsilon \bar {u}A/2 = \Delta \sigma P/2=8\mu r^{3}\Delta \epsilon ^{2}/7\).
2.7 Circular Crack Model by Brune
Near Field
To construct his model, Brune (1970) assumed a shear stress pulse, \(\sigma _{eff}\), applied to a circular crack with a finite radius r instantaneously at \(t=0\), over the entire interior, i.e. it assumes an infinite rupture velocity, and therefore the crack propagation is neglected. The stress pulse \(\sigma _{eff}\) is the effective stress or the dynamic stress drop, and it represents the difference between the initial stress and the dynamic frictional stress. For a complete stress drop, we can assume that the effective stress is equal to stress drop, \(\sigma _{eff}=\Delta \sigma \), or that the effective strain is equal to the dynamic shear strain change (drop), \(\epsilon _{eff}=\Delta \epsilon =\sigma _{eff}/\mu \).
The above assumption is equivalent to applying a sudden uniform stress pulse on the interior surface of the crack. Brune also assumed that the crack surface reflects shear waves during rupture. The stress pulse sends a shear pulse propagating orthogonal to the crack, and the initial time function for this pulse follows directly from the boundary conditions,
where x is the distance from the source, \(v_{S}\) is the S-wave velocity, and \(H\left (t\right )=0\) for \(t<0\), \(H\left (t\right )=1\) for \(t\geq 0\) is the Heaviside unit step function. The displacement created by the stress pulse can be obtained by solving the constitutive equation \(\mu \partial u/\partial x=-\sigma _{eff}\left (x,t\right )\) or \(\partial u/\partial x=-\Delta \epsilon \left (x,t\right )\),
where T is the time required for elastic waves to propagate from the ends of the rupture to the observation point. Therefore, Eqs. (2.23) and (2.22) are for the initial displacement and velocity at a point very close to the centre of the source and neglecting the finite size of the crack. Similar equations were derived by Jeffreys (1962). Assuming \(\Delta \epsilon =3.3\cdot 10^{-4}\), i.e. \(\Delta \sigma =10\) MPa and \(v_{S}=3\) km/s Brune, estimated the initial near-field ground velocity at 1 m/s.
The initial ground velocity starts decaying to zero when the effects of the source edges are felt at the observation point, which Brune models by introducing an exponential decay factor,
where \(\tau \approx r/v_{S}\), where r is the source radius. For \(\Delta \epsilon =2.5\cdot 10^{-4}\) at \(t=\tau \), it gives \(\dot {u}_{NF}\left (t=\tau \right )\simeq 0.8\) m/s. Integrating Eq. (2.22) gives
which for \(\Delta \epsilon =2.5\cdot 10^{-4}\) at \(t=\infty \) gives \(u_{NF}\left (t=\infty \right )\simeq 0.013\) m. The near-field displacement spectrum is
which decays as \(f^{-1}\) at low and as \(f^{-2}\) at high frequencies. Figure 2.7 shows the near-field source time functions for displacement and velocity described by Eqs. (2.23) and (2.22) and the near-field displacement spectra for three different effective strains. The assumptions: \(r=50\) m, \(\Delta \epsilon _{1}=2.5\cdot 10^{-4}\), \(\Delta \epsilon _{2}=5\cdot 10^{-5}\), \(\Delta \epsilon _{3}=1\cdot 10^{-5}\), \(v_{S}=3250\) m/s, \(f_{0S}=0.373v_{S}/r=24.2\) Hz, and \(\tau =1/\left (2\pi f_{0S}\right )=0.0066\) seconds.
Fig. 2.7
Brune near-field displacement and velocity source time functions and displacement spectra described by Eqs. (2.23), (2.22), and (2.24) for selected effective strains
The near field is essentially determined by the motion on one side of the source, while the far field represents the contributions from both sides of the source, i.e. the elastic waves radiated by the opposing side of the source diffract around the source surface and differentiate the far-field spectrum modifying its low frequency part (Keilis-Borok et al., 1960). Brune approximated this effect by multiplying the displacement function by an exponential with decay time of the order of \(r/v_{S}\) and by a factor \(r/R\) to correct for spherical spreading,
where R is the distance, and \(\varLambda _{S}=\sqrt {2/5}=0.632\) is the root mean square radiation pattern of the S-wave, i.e. the mean energy radiation computed by integrating the squared radiation pattern over the surface of the unit sphere (Aki & Richards, 2002). Here, the displacement, \(u_{FF}\left (t\right )\), continues indefinitely, although \(\lim _{t\rightarrow \infty }\left (t\cdot \exp \left (-t/\tau \right )\right )=0\). Velocity and acceleration have jumps at \(t=0\). The Fourier amplitude spectra of the S-wave far-field displacement, velocity, and acceleration are
where \(f_{0}\) is the corner frequency that represents the transition between the flat and sloping parts of the displacement spectrum. It is used to estimate source radius, although this relation is poorly constrained. The displacement spectra decay with \(f^{-2}\) past the corner frequency; therefore, it is a so-called \(\omega ^{-2}\) model. The acceleration spectrum predicts that ground acceleration is flat for arbitrarily high frequencies. The acceleration spectrum usually decays after a high frequency corner identified as \(f_{max}\).
Since the seismic energy scales with velocity squared, the maximum energy is radiated at the predominant frequency, i.e. the frequency where the velocity spectrum is at maximum. In this case, i.e. the \(\omega ^{-2}\) model, the corner frequency and predominant frequency are the same, which is not the case for other models. Figure 2.8 shows the distance corrected far-field S-wave displacement, velocity, and acceleration in time and their spectra. The assumptions are: \(r=50\) m, \(R=500\) m, \(\Delta \epsilon _{1}=2.5\cdot 10^{-4}\), \(\Delta \epsilon _{2}=5\cdot 10^{-5}\), \(\Delta \epsilon _{3}=1\cdot 10^{-5}\), \(v_{S}=3250\) m/s, \(f_{0S}=0.373v_{S}/r=24.2\) Hz, \(\tau =1/\left (2\pi f_{0S}\right )=0.0066\) seconds, and potency \(P=\left (16/7\right )\Delta \epsilon r^{3}\) which for \(\Delta \epsilon _{1}=2.5\cdot 10^{-4}\) gives \(\log P=1.85\). No Q correction was applied at this stage. Note that the theoretical acceleration spectrum extends to infinity, which is not physical; however, the observed acceleration spectra decay above a certain frequency called \(f_{max}\), (Hanks, 1982).
Fig. 2.8
Far-field distance corrected S-wave displacement, velocity, and acceleration in time (top) and spectra (bottom) and for selected effective strains
Brune estimated spherical average corner frequency by comparing the zero frequency limit of the shear wave displacement spectrum given by Eq. (2.28) with Keylis-Borok (1959) equation for the far-field S-wave spectrum at zero frequency, \(\Omega _{FFS}\left (f=0\right )=\Omega _{0S}\),
where \(k=\sqrt {7/\left (16\pi \right )}= 0.373\) is the corner frequency normalised by the wave velocity and source radius, referred as the normalised corner frequency when comparing the results of different source scenarios. If we assume that source radius can also be recovered from the P-wave radiation, then, taking \(r=kv_{P}/f_{0P}\), the ratio of P- to S-wave corner frequency is
Corner frequency is not a physical parameter of a seismic source, and there are considerable variations when computing \(f_{0}\) using different source models. Madariaga (1976) developed a dynamic model, with a stress singularity at the rupture front, and assuming rupture velocity \(v_{r}=0.9v_{S}\) gave an average \(k_{S}=0.21\) for the S-wave and \(k_{P}=0.32\) for P-wave. Brune et al. (1979) considered different source models with different assumptions and suggested an average \(k=1/3\). Kaneko and Shearer (2014) studied a quasi-dynamic circular crack model with a cohesive zone that prevents a stress singularity at the rupture front and, for \(v_{r}=0.9v_{S}\), gave \(k_{S}=0.26\) and \(k_{P}=0.38\).
Keylis-Borok (1959) related the average static stress drop on a circular source to the average slip, \(\Delta \sigma = 7\pi \mu \bar {u}/\left (16r\right )\), that for the S-wave can be written as
Here, any uncertainty in the corner frequency \(f_{0}\) is cubed when computing static stress \(\Delta \sigma \). Aki (1966) and Wyss and Brune (1971) proposed to use the two independent source parameters, seismic potency and radiated seismic energy, to estimate stress release at source, \(\sigma _{A}= E/P\), that does not require an estimate of the source dimension.
If we assume that seismic potency can also be estimated from the zero frequency limit of the P-wave displacement spectrum, \(\Omega _{0P}=\varLambda _{P}\mu P\)/\(\left (4\pi \rho Rv_{P}^{3}\right )\), then the ratio
where \(\varLambda _{P}=\sqrt {4/15}=0.516\). For a Poissonian solid, i.e. \(v_{P}=\sqrt {3}\cdot v_{S}\), the ratio is \(\left (3/2\right )\sqrt {27}\simeq 7.8\). Seismic potency derived from the low frequency limit of the displacement spectra of P- and S-waves should be the same,
The simplest representation of a seismic source is the far-field point source model, i.e. a one moment tensor source with a single source time function radiating from a point. In contrast, an extended seismic source is rupturing and slipping at all points behind the propagating rupture front, and it can be treated as a set of point-like sub-sources.
There are crack-like rupture models, discussed in this section, where the duration of deformation at a given point at source is comparable to the overall duration of rupture, and pulse-like rupture models where the slipping portion is constrained to the vicinity of the propagating rupture tip and the duration of deformation at a point is short relative to the rupture duration (Heaton, 1990). Pulse-like models deliver a spectrum with two corner frequencies: the lower one that scales inversely with the overall duration of rupture, while the higher one is inversely proportional to pulse duration.
The rupture of a fault and the subsequent slip are fast non-linear physical processes involving very large irrecoverable deformations and violation of the continuity of the material. The effect of a seismic source on the rock mass in the immediate vicinity of the fault is to induce motions which cannot be described within the framework of linear elasticity. It is only at a certain distance from the source, where the local displacements are below the plasticity threshold, that one can consider the local wave field as a superposition of the waves radiated by all parts of the source. It is the applicability of the superposition principle which separates the domain of non-linear material behaviour around the source from the rest of the rock mass which resembles the laws of linear elasticity. The terms near field, intermediate field, and far field apply only to the rock mass outside the broken zone at the source. In that sense, a seismic source is always volumetric even when the rupture can be assumed to take place on a 2D surface.
Ideally, seismic source time functions that describe the displacement, velocity, and acceleration of ground motion at source should have the following properties:
The motion must start at a given moment of time, say at \(t=0\), with no motion before that.
The motion must stop after a finite period of time has elapsed.
There must be no change in the direction of the velocity during the motion.
There must be no intermittent motion, i.e. no stop and restart.
There must be no discontinuities or infinite values in the displacement and velocity.
The acceleration time function cannot become infinite but can exhibit discontinuities, i.e. sudden changes in the driving force.
The far-field displacement source pulse of a simple dislocation source is one-sided, and the Fourier transform of such a pulse is always flat at low frequencies and falls off at high frequencies.
If the far-field displacement is a simple one-sided pulse, then the velocity pulse is two-sided, and the acceleration pulse is three-sided with zero area, i.e. zero velocity at end of record.
The far-field displacement due to a point source is controlled by the slip rate, \(u_{FF}(t)=C\cdot \dot {u}(t-r/v_{P,S})\). This statement dictates the following relation between the Fourier transform of the far-field displacement and the final static displacement at the source, \(u_{FF}(\omega ) = C\intop _{-\infty }^{\infty }\dot {u}\left (t\right )\exp \left (-i\omega t\right )dt\), where \(\omega \) is angular frequency and C a constant.
Therefore, the low frequency limit of the far-field displacement spectrum is
since \(u_{NF}\left (t=-\infty \right )=0\) and is related to the scalar seismic potency.
In the frequency domain, the Fourier spectrum of the displacement in the far field should be the same, up to a frequency independent factor, as the spectrum of the velocity at the source. The spectrum of the velocity in the far field should be related to the spectrum of the acceleration at the source, and the spectrum of the acceleration in the far field can be expressed in terms of the spectrum of the acceleration rate at the source. This means that it will be enough to obtain the spectra of the displacement, velocity, acceleration, and acceleration rate at source in order to have all spectra in the near field as well as in the intermediate and far field.
Source Time Functions
In the near field, there is no separation between the P- and S-wave because the arrivals of these waves are delayed by less time than the total seismic source duration. They arrive very close together and interfere to produce a static displacement field; therefore, the near-field term refers to ground motion unrelated to a specific type of wave. The far-field approximation is valid at distances R much larger than the dominant wavelength \(\lambda \), i.e. for \(R/\lambda \gg 1\), and \(\lambda =v/f\), where v is the wave velocity and f the predominant frequency. Seismic waveforms contain a range of frequencies. The high frequency part of the spectrum reflects the details of the far-field ground motion, while the low frequency is determined by the properties of the motion at the source and in the near field. Since in the far field there should be no residual static deformation and because the maximum peak ground velocities, PGV , are usually associated with the lower frequencies of S-waves, one can consider the far field to be at distances where the recorded seismic strains, \(PGV/v_{S}<10^{-6}\), which are assumed elastic for a hard rock. Practically, sites in hard rock that record the maximum ground velocity from the S-wave of less than 3 mm/s can be considered far field. If such low ground velocities are recorded at distances that give a short S-P window, then spectra should be calculated from the S-wave only.
2.8.1 Source Time Functions and Spectra
Beresnev (2019) proposed a simple expression for the displacement source time function that caters for different far-field spectral models,
where \(H\left (t\right )\) is the Heaviside step function, \(\varGamma \left (z+1,t/\tau \right ) = \intop _{t/\tau }^{\infty }x^{z}\exp \left (-x\right )dx\) is the upper incomplete gamma function, and, for a non-negative integer \(\varGamma \left (z+1\right )=z!\), \(u_{\infty }\) is the final displacement at \(t=\infty \) and \(\tau \) is related to the rise time. Parameter \(z=1\) gives the most frequently used omega-square model, \(\omega ^{-2}\), \(z=2\) gives the omega-cube model, \(\omega ^{-3}\), and \(z=1.5\) gives an \(\omega ^{-2.5}\) model. For real positive values of z, the slip velocity and acceleration at source are
Note that the acceleration time function for the \(\omega ^{-2}\) model, i.e. for \(z=1\), exhibits a finite jump at \(t=0\), which means that the acceleration is undefined there. For all values of \(z\geq 1\), the velocity is continuous and equal to zero for \(t=0\). The velocity is maximum at \(t=z\cdot \tau \), where its value is
The velocity spectrum is controlled by two physical parameters: the final displacement and the maximum slip velocity defined by the parameters z and \(\tau \).
Figure 2.9 shows the displacement, velocity, and acceleration source time functions given by Eqs. (2.35), (2.36), and (2.37) for \(z=1\) that defines the \(\omega ^{2}\) model, \(z=1.5\) for the \(\omega ^{2.5}\) model, and \(z=2\), for the \(\omega ^{3}\) model. Assumptions: \(\Delta \epsilon =2.5\cdot 10^{-4}\), \(r=50\) m, give \(P=\left (16/7\right )\Delta \epsilon r^{3}=71.43\) and \(\log P=1.85\), the final max displacement \(u_{\infty }=24r\Delta \epsilon /\left (7\pi \right )=0.0136\) m, and \(v_{S}=3250\) m/s gives \(f_{0S}=0.21v_{S}/r=13.65\) Hz and \(\tau =1/\left (2\pi f_{0S}\right )=0.01166\) seconds. The maximum slip velocities are \(\dot {u}_{1}=0.43\) m/s at \(t_{z1}=0.01166\) for the \(\omega ^{-2}\) model, \(\dot {u}_{1.5}=0.36\) m/s at \(t_{z1.5}=0.0175\) for the \(\omega ^{-2.5}\) model, and \(\dot {u}_{2}=0.32\) m/s at \(t_{z2}=0.02332\) seconds for the \(\omega ^{-3}\) model.
Fig. 2.9
Displacement, velocity, and acceleration source time functions given by Eqs. (2.35), (2.36), and (2.37) for the \(\omega ^{2}\) model (\(z=1\)) in red, the \(\omega ^{2.5}\) model (\(z=1.5\)) in blue, and the \(\omega ^{3}\) model (\(z=2\)) in green
At low frequencies, the displacement spectrum behaves as \(f^{-1}\), the velocity spectrum is flat, and acceleration spectrum is proportional to f. This is true for all three cases irrespective of z. Figure 2.10 shows the displacement, velocity, and acceleration source spectra given by Eqs. (2.39), (2.40), and (2.41) for \(z=1\) that defines the \(\omega ^{2}\) model, \(z=1.5\) being the \(\omega ^{2.5}\) model, and \(z=2\), for the \(\omega ^{3}\) model.
Fig. 2.10
Displacement, velocity, and acceleration source spectra given by Eqs. (2.39), (2.40), and (2.41), for the \(\omega ^{2}\) model (\(z=1\)) in red, the \(\omega ^{2.5}\) model (\(z=1.5\)) in blue, and the \(\omega ^{3}\) model (\(z=2\)) in green
The far-field P- and S-wave displacements radiated by a shear dislocation at an individual source patch area element A at distance R can be expressed as
where \(\dot {u}\left (t_{P}^{\prime }\right )\) and \(\dot {u}\left (t_{S}^{\prime }\right )\) are the displacement rate functions at source for the respective retarded times \(t^{\prime }_{P}=t-R/v_{P}\) and \(t^{\prime }_{S}=t-R/v_{S}\) for P- and S-wave (Aki & Richards, 2002). The term \(1/R\) stands for geometrical spreading with distance for body waves in a homogeneous elastic medium. Taking potency \(P=u_{\infty }A\), the far-field displacement time function for the P- and S-wave is
where \(\Omega _{P,S}=C_{FFP,S}\cdot P/R\), \(C_{FFP}=\varLambda _{P}v_{S}^{2}/\left (4\pi v_{P}^{3}\right )\), \(C_{FFS}=\varLambda _{S}/\left (4\pi v_{S}\right )\), and \(\varLambda _{P}=\sqrt {4/15}=0.516\) and \(\varLambda _{S}=\sqrt {2/5}=0.632\) are the root mean square radiation patterns of P- and S-waves, respectively, and P is seismic potency.
The displacement time function reaches its maximum for
For \(z=1\), it is not zero at the onset of motion, \(t^{\prime }_{P,S}=0\). This indicates a kinematic inconsistency of the \(\omega ^{-2}\) model since a discontinuity in the slip velocity, i.e. a jump from zero to a finite value at the beginning of the motion, would require an infinite driving force. The function behaves properly for any \(z>1\), regardless of how small.
As expected, the slip velocity time function for \(z=1\) is \(\dot {u}_{FFP,S}\left (R,t\!\!\rightarrow \! 0,z=1\right )\)\(=\Omega _{P,S}\)\(\left (1/\tau ^{2}\right )\neq 0\), while for \(z=1+\epsilon \), \(\epsilon >0\), one has
The Dirac delta singularity at \(t_{P,S}^{\prime }=0\) is cancelled for \(z>1\) but survives for \(z=1,\) i.e. for the \(\omega ^{-2}\) model, which is the effect from the discontinuity in the slip velocity for that value of the parameter z.
Figure 2.11 shows the distance corrected far-field time domain displacements, velocities, and accelerations for \(z=1\), \(1.5\), and \(2.0\), with the same assumptions as in Fig. 2.9. For \(z<2\), the power of the term \(\left (t^{\prime }_{P,S}/\tau \right )^{z-2}\) in Eq. (2.45) is negative and goes to infinity for \(t=0\), see Fig. 2.11 right.
Fig. 2.11
Displacement, velocity, and acceleration far-field time functions given by Eqs. (2.43), (2.44), and (2.45) for the \(\omega ^{2}\) model (\(z=1\)) in red, the \(\omega ^{2.5}\) model (\(z=1.5\)) in blue, and the \(\omega ^{3}\) model (\(z=2\)) in green. Dashed lines for P-wave and solid for S-wave
In general, the modelled time functions can have kinks and singularities which can reflect on their behaviour and on the behaviour of their derivatives. Nevertheless, such functions can produce regular and smooth Fourier transforms as long as the original goes sufficiently fast to zero at infinity and provided that its singularities are integrable. In this case, the Fourier transforms of the far-field time functions are smooth.
Far Field Spectra
The expression for the slip velocity at the source defines, up to a constant factor, the displacement time function in the far field, i.e. the Fourier transform of the ground displacement in the far field will be proportional to the complex-valued function,
The modulus of \(F\left (f\right )=\sqrt {[\text{Re}(F)]^{2}+[\text{Im}(F)]^{2}}\) defines the amplitude spectra of the far-field displacement, velocity, and acceleration for a given z. Figure 2.12 top row shows the log-log plots and bottom row log-linear plots of the far-field displacement, velocity, and acceleration spectra.
Fig. 2.12
Top row. Log-log plots of the far-field displacement, velocity, and acceleration spectra given by Eqs. (2.46), (2.47), and (2.50) for the \(\omega ^{2}\) model (\(z=1\)) in red, the \(\omega ^{2.5}\) model (\(z=1.5\)) in blue, and the \(\omega ^{3}\) model (\(z=2\)) in green. Dashed lines for the P-wave and solid for the S-wave. Bottom row. The same as top row but in a linear scale to demonstrate their asymmetry with respect to the corner frequency
where \(\Omega _{P,S}=C_{FFP,S}\cdot P/R\). For \(f=0\), it gives the Keylis-Borok equation for the zero frequency displacement spectral level of S-wave, \(\Omega _{FFS}\left (f=0,z\right )=\Omega _{0S}=\varLambda _{S}P\)/\(\left (4\pi Rv_{S}\right )\). The velocity spectrum is
where \(\eta \) is the exponent in the power law frequency dependence of Q, and \(Q^{-1}\) is the fractional loss of energy per cycle of oscillation due to intrinsic absorption and due to scattering caused by energy redistribution. For \(\eta =0\), the amplitude decay increases exponentially with frequency. Larger Q implies less attenuation, and high frequency waves will attenuate faster than low frequency waves. For \(\eta =1\), the amplitude decay is frequency independent.
For a relatively solid hard rock mass, Q is well above 500, but for a softer and/or fractured rock mass, it can be as low as 25. For strains between \(10^{-3}\) and 10\(^{-6}\), attenuation is strain dependent, and therefore, amplitudes decay more rapidly in the intermediate field of seismic radiation, where strains are larger than 10\(^{-3}\) and where Q may be as low as 10. In general, Q is frequency dependent (e.g. Fedotov & Boldyrev, 1969; Rautian & Khalturin, 1978; Aki, 1980), and therefore, it is larger for P-waves than for S-waves.
To correct for amplitude decay due to attenuation and scattering, one should multiply the observed far-field spectra by \(\exp \left [\pi fR/\left (v_{P,S}Q_{P,S}f^{\eta }\right )\right ]\), where \(R/v_{P,S}\) is the travel time from source to the recording site.
Solid lines in Fig. 2.13 left show the shapes of function (2.51) for \(Q_{S}=200\) and \(\eta =0\), \(\eta =0.1\) and \(0.2\). Dashed lines show the same but for \(Q_{S}=25\), i.e. for a strongly attenuating rock mass. Figure 2.13 centre and right shows a shift in the predominant frequency and decay in spectral amplitudes of the far-field velocity model spectra due to attenuation over distance. Solid lines represent P- and S-wave velocity spectra with no Q attenuation and dashed lines with attenuation for \(Q_{P}=50\) and \(Q_{S}=25\) at distances 500 m (centre) and 2000 m (right). The relatively low values of Q were selected to illustrate the point.
Fig. 2.13
The shape of function 2.51(left). P- and S-wave of the \(\omega ^{-2}\), i.e. \(z=1\) model, far-field unattenuated (solid lines) and Q attenuated (dashed lines) velocity spectra over distances of 500 (centre) and 2000 m (right)
An alternative model for spectral decay at high frequencies is \(a\left (f\right )= A_{0}\exp \left (-\pi f\kappa \right )\),where \(\kappa =R/\left (vQ\right )\), and its frequency dependent modification described below under \(f_{max}\) (Anderson & Hough, 1984 and Hendel et al., 2020).
Site Effect
Site effect may be defined as the modification of the amplitude, frequency, and duration of the incoming wave field due to the specific mechanical properties and geometrical features of the ground surrounding the sensor site. In most cases, the ground motions are amplified at certain frequencies in fractured rock relative to the motion in solid rock. Seismic systems in mines are designed to locate events and to estimate their source parameters. For this reason, sensors are installed at 6 to 10 m in boreholes that penetrate the stress-induced fractures surrounding the excavations to avoid the surface noise and the very site effects that amplify ground motion at the skin of excavations. The further away is the sensor from the skin of an excavation the lower the site effect. Site effect at sites 10 m away from an excavation is negligible.
One way to reduce site effect is by averaging the corrected spectra over the sites accepted for source inversion and then estimating source parameters from the averaged spectrum. This is effective if site effects at different recording sites are at different frequencies and not too strong, which is mostly the case for sensors installed underground in boreholes. Surface sites are more prone to site effect, and to remove it, one needs to estimate the site transfer function and then deconvolve by division in the frequency domain. However, if the site response is coherent, it could easily be enhanced rather than reduced by the conventional averaging of spectra over a number of stations. An effective way to correct spectra for such propagation effects is to employ homomorphic deconvolution in the cepstral domain, e.g. Dysart et al. (1988) and Mendecki (1993). An advantage of this technique is that one needs no prior knowledge about the site effect one is trying to remove. This filtering procedure starts with the displacement amplitude spectrum. A complex function in the frequency domain is then constructed of which the real part is the logarithm of the amplitude spectrum and the imaginary part is zero. Taking the logarithm transforms the product to a sum so that the cepstrum contains the superposition of the source and the spike train. The inverse Fourier transform of this zero phase spectrum is taken, and the resulting cepstrum is filtered by excising peaks or troughs and/or by zeroing portions of the trace after the first or second zero crossings. A forward Fourier transform is then performed on the filtered cepstrum, and the filtered amplitude spectrum is constructed from the antilog of the transformed function.
Empirical Green’s Functions
The removal of path effect can also be done by de-convolving seismograms of the event by seismograms of a suitable small event of similar mechanism located close, preferably within one source dimension, to the hypocentre of the target event e.g. Aki (1967), Bakun and Bufe (1975), Hartzell (1978), Viegas et al. (2010), Abercrombie (2015), and recently by Ross and Ben-Zion (2016) for large data sets. The small event then acts as an empirical Green’s function (EGF), i.e. simulating the impulse response of the medium. Therefore, the source properties of the EGF event cannot be imprinted in the frequency range used to deconvolve. It is a very effective method if such a suitable small event is available. One of the conflicting requirements is that the size, i.e. the source duration, of the candidate EGF event should be considerably smaller than that of the larger event to simulate an impulse response, while the signal to noise ratio should be above the background noise in the frequency range of interest at all recording sites used in the source inversion. That limits the useful frequency bandwidth to only that with signal from both large and small events. Source parameters are estimated by comparing the observed spectral ratio between the target \(\Omega _{1}\left (f\right )\) and EGF event \(\Omega _{2}\left (f\right )\) and a given spectral model,
where \(P_{T}\), \(f_{0T}\) and \(P_{EGF}\), \(f_{0EGF}\) are the potencies and corner frequencies of the target and the EGF event, respectively (Abercrombie & Rice, 2005). If z is fixed and the \(P_{EGF}\) and \(f_{0EGF}\) are known, there are only two parameters to be inverted, \(P_{T}/P_{EGF}\) ratio and \(f_{0T}\). The same can be done to the velocity spectrum to estimate energy. A comprehensive review of the applications of EGFs can be found in Hutchings and Viegas (2012).
Maximum Frequency, \({\mathbf {f}}_{max}\)
While the theoretical acceleration spectrum for the \(\omega ^{-2}\) model is flat at high frequencies, the observed acceleration spectra are characterised by a trend of exponential decay. Hanks (1979) and Hanks (1982) suggested that the acceleration spectrum is flat above the corner frequency only to a second corner frequency, called \(f_{max}\), above which it decays rapidly. The origin of the rapid decay may be the source or path including the site effect or the combination of the above. Cases with \(f_{max}<f_{0}\) would limit the resolution of corner frequency, \(f_{0}\), and therefore impose an artificial minimum source dimension on seismic events.
There are two types of parametric models widely used to shape the observed high frequency acceleration spectra. Boore (1983a) proposed the power law frequency decay model,
where \(\kappa =R/\left (vQ\right )\) is the spectral decay parameter and can be derived from \(\partial \ln a\left (f\right )/\partial f =\) -\(\pi \kappa \), for frequencies above \(f_{max}\). Some studies suggest that that the slope of the acceleration spectrum in log-linear space is not constant but rather curved, so the estimated \(\kappa \) does depend on the chosen frequency band of analysis. Hendel et al. (2020) modified the kappa model to account for the non-linear spectral decay at high frequencies by incorporating a power law frequency dependent Q which they call the zeta model, \(a\left (f\right )=\exp \left [-\pi \zeta f^{1-\eta }\right ]\), where \(\zeta =Rf_{0}/\left (vQ\right )\) and \(\eta \) is a parameter. The \(P\left (f\right )\), \(\kappa \) and the zeta models are to be applied to the \(\omega ^{-2}\) source model in which the acceleration spectrum is flat above the corner frequency. The \(\omega ^{-1.5}\) and \(\omega ^{-3}\) spectra are not flat above corner frequency, see Fig. 2.12 right column, and provide a natural high-cut filtering to be modelled as a source effect.
Figure 2.14 shows S-wave far-field acceleration spectra for the \(\omega ^{-2}\) model (\(z=1\)) in red, the \(\omega ^{-2.5}\) model (\(z=1.5\)) in blue, and the \(\omega ^{-3}\) model (\(z=2\)) in green all attenuated for constant Q (left) and the frequency dependent \(Q\left (f\right )\) (centre). The right figure shows the \(\omega ^{-2}\) model attenuated for constant Q and for \(f_{max}\), and \(\omega ^{-2.5}\) and \(\omega ^{-3}\) models corrected only for a constant Q. It is more difficult to resolve the corner frequency when \(f_{0}/f_{max}=1\) because then source spectra are strongly attenuated for high frequencies past the \(f_{max}\).
Fig. 2.14
S-wave far-field acceleration spectra for the \(\omega ^{-2}\) model (\(z=1\)) in red, the \(\omega ^{-2.5}\) model (\(z=1.5\)) in blue, and the \(\omega ^{-3}\) model (\(z=2\)) in green all corrected for frequency dependent \(Q \left (f \right )\)(left) and for constant Q(centre). The right figure shows the \(\omega ^{-2}\) model corrected for constant Q and for \(f_{max}\) and \(\omega ^{-2.5}\) and \(\omega ^{-3}\) models corrected only for a constant Q
Figure 2.15 shows acceleration waveforms (left column), integrated velocity waveforms (centre column), and the smoothed FFT of the S-wave window (right column) of two events recorded at the same site by three-component accelerometers installed in a borehole in an underground hard rock mine.
Fig. 2.15
Acceleration waveforms (left), integrated velocity waveforms (centre), and smoothed FFT of accelerations (right) of a \( \log P=1.05\) event recorded 17 km away (top row ) and a \( \log P=0.87\) event recorded 243 m away (bottom row)
The first event with \(\log P=1.05\) located 17 km away from the recording site (top row), and the second event with \(\log P=0.87\) located 243 m away (bottom row). The smoothed FFT of noise is marked by dashed lines. The spectrum was smoothed with the Konno and Ohmachi function, \(w\left (f,f_{c}\right )=\left \{ \mbox{sinc}\left [b\log \left (f/f_{c}\right )\right ]\right \} ^{4}\), where \(b=40\) is the coefficient for bandwidth and \(f_{c}\) is the centre frequency (Konno & Ohmachi, 1998). The distant event shows a considerable depletion of high frequencies over the frequency range 1 to 1000 Hz and maintains a constant slope of the acceleration spectrum in log-linear domain up to 800 Hz. The casual fit gives \(\kappa =0.0019\) second and would imply a constant Q over that frequency range.
The second, close event shows very limited depletion in high frequency over the same frequency band, and the spectrum is scalloping up to 500 Hz but overall reasonably linear up to 1000 Hz with \(\kappa =0.0008\) second. Note that although the second event has 8.3 times higher PGA and 14 times higher PGV , its cumulative absolute velocity \(CAV = \intop _{0}^{t_{d}}|a\left (t\right )|dt\) is 2.1 times lower, and the cumulative absolute displacement, \(CAD = \intop _{0}^{t_{d}}|\mbox{v}\left (t\right )|dt\), 1.25 times lower than that of the larger distant event. The reason is the duration of ground motion, \(t_{d}\), which for the distant event is 8 times longer. This illustrates the importance of duration. The first event with lower PGA and PGV but longer duration will consume more deformation capacity of support than the second event.
2.9 Frequency Range \(\log P\) and \(\log E\)
Potency Recovery
The potency and energy range of recorded seismic events is limited by the frequency range of the monitoring system, \(\left (f_{1},f_{2}\right )\), which is mainly determined by the capabilities of the seismic sensors. The following ratio of the far-field displacement spectra quantifies the potency recovery as a function of the ratio of available frequencies at the lower end of the spectrum to corner frequency \(f_{1}/f_{0}\), for different spectral models,
which applies to the P-wave if \(f_{0}=f_{0P}\) and to the S-wave if \(f_{0}=f_{0S}\). Figure 2.16 (left) shows that the \(\omega ^{-2}\) model has the least under-recovery followed by \(\omega ^{-2.5}\) and \(\omega ^{-3}\). It shows that with \(f_{1}\) at the corner frequency the \(\omega ^{-2}\) model recovers 50% of seismic potency, \(\omega ^{-1.5}\) recovers 42%, and \(\omega ^{-3}\) 35%. The \(\omega ^{-2}\) model recovers 80% of seismic potency at \(f_{1}/f_{0}=0.5\), \(\omega ^{-2.5}\) at \(f_{1}/f_{0}=0.44\), and \(\omega ^{-3}\) at \(f_{1}/f_{0}=0.4\). The 80% recovery underestimates \(\log P\) by \(\log \left (0.8P\right ) = \log P - 0.0969\), and the \(m_{HK}\) by 0.0646, and at 50% recovery, i.e. \(f_{1}/f_{0} = 1.0\) the \(\log P\) is underestimated by 0.3 units and the \(m_{HK}\) by 0.2.
Fig. 2.16
Recovery of seismic potency as a function of \(f_{1}/f_{0}\)(left) and the S-wave \( \log P\) at 80% potency recovery as a function of \(f_{1}\) for the \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models and for three selected strain drops (right)
The rate of deformation at seismic sources in softer or in fractured rock is slower than in stronger solid rock, and such sources radiate the predominant portion of their energy at lower frequencies, resulting in a lower corner frequency. Therefore, the recovery of seismic potency will be lower in low strain drop conditions. Potency recovery as a function of the lowest frequency available to calculate spectra, \(f_{1}\), taking into account strain drop, can be derived from \(P=\left (16/7\right )\Delta \epsilon r^{3}\) (Eshelby, 1957) and, for the S-wave, \(r=0.26v_{S}/f_{0}\) (Kaneko & Shearer, 2014),
where \(f_{0}=f_{1}/f_{r}\left (z\right )\) and \(f_{r}\left (z\right )\) is the \(\left (f_{1}/f_{0}\right )\) ratio derived from Eq. (2.55) for the required potency recovery \(\Omega _{FFR}\left (z\right )\),
Figure 2.16 (right) shows \(\Omega _{FFR}\left (z\right )=0.8\), i.e. the 80% potency recovery as a function of \(f_{1}\) for selected strain drops (Eqs. 2.56 and 2.57). Clearly, the recovery of seismic potency increases in harder rock, i.e. higher \(\Delta \epsilon \).
Energy Recovery
The predominant frequency \(f_{E}\), i.e. the frequency at which the maximum energy is radiated, is at the maximum of the velocity power spectrum. Solving the equation \(\partial \dot {\Omega }_{FF}\left (f,z\right )/\partial f = 0\),
Therefore, for the \(\omega ^{-2}\) model, i.e. \(z=1\), the predominant frequency is at the corner frequency, \(f_{E}\left (z=1\right ) = f_{0}\). However, for the \(\omega ^{2.5}\) model, i.e. \(z=1.5\), the \(f_{E}\left (z=1.5\right ) = 0.816f_{0}\), and for \(\omega ^{3}\) model, i.e. \(z=2\), the \(f_{E}\left (z=2\right ) = 0.707f_{0}\).
Seismic radiated energy is proportional to the integral of the velocity power spectrum,
which applies to the P-wave if \(\dot {\Omega }_{FF}=\dot {\Omega }_{FFP}\) and to the S-wave if \(\dot {\Omega }_{FF}=\dot {\Omega }_{FFS}\).
The ratio \(S_{V2}\left (z=1.5,0,\infty \right )/S_{V2}\left (z=1,0,\infty \right )\simeq 0.85\) shows that the \(\omega ^{-2.5}\) model produces 85% of the seismic energy of the \(\omega ^{-2}\) model and the ratio \(S_{V2}\left (z=2,0,\infty \right )/S_{V2}\left (z=1,0,\infty \right )= 0.5\) that the \(\omega ^{-3}\) model produces 50% of the seismic energy of the \(\omega ^{-2}\) model.
The model independent seismic energy can be calculated by \(\tilde {E}_{P,S} = 4\pi \rho v_{P,S}\tilde {S}_{V2P,S}\), where \(\tilde {S}_{V2P,S}\) is the observed, instrument, distance, and Q corrected velocity power spectrum for the P- or S-wave. However, the observed spectrum is limited by the frequency range of the system, and we can recover only the \(\left (f_{1},f_{2}\right )\) part of it. The theoretical energy recovery as a function of z can be defined as
The integral \(2\intop _{f_{1}}^{f_{2}}\left [\dot {\Omega }_{FF}\left (f,z\right )\right ]^{2}df\) does not have a simple analytical solution for all z, but it does if z is either integer or half integer. For \(z=1\), \(z=1.5\), and \(z=2\), the respective integrals give
where \(A=\arctan \left (f_{2}/f_{0}\right )-\arctan \left (f_{1}/f_{0}\right )\), \(c_{1}=1+\left (f_{1}/f_{0}\right )^{2}\), \(c_{2}=1+\left (f_{2}/f_{0}\right )^{2}\). The following equations quantify the portion of the velocity power spectrum lost in the frequency range \(\left (0,f_{1}\right )\) and \(\left (f_{2},\infty \right )\) for \(z=1\), \(z=1.5\), and \(z=2\).
The ratio \(S_{V2}\left (z,f_{0},\infty \right )/S_{V2}\left (z,0,\infty \right )\) shows that the \(\omega ^{-2}\) model gives 82% of the seismic energy above the corner frequency, while the \(\omega ^{-2.5}\) model produces 65% and the \(\omega ^{-3}\) model only 50%. The theoretical energy recovery due to bandwidth limitation is \(S_{V2}\left (z,f_{1},f_{2}\right )/S_{V2}\left (z,0,\infty \right )\). Figure 2.17 shows the recovery of seismic energy given by Eq. (2.60) as a function of \(f_{2}/f_{0}\) for \(f_{1}/f_{0}=0.2\).
Fig. 2.17
Recovery of seismic energy as a function of \(f_{2}/f_{0}\) when \(f_{2}/f_{0}=0.2\)(left) and \(f_{2}/f_{0}=0.5\)(right)
that for the \(\omega ^{-2}\) model, i.e. \(z=1\) gives \(S_{D2}=\left (\pi /2\right )\Omega _{FF}^{2}f_{0}\), for the \(\omega ^{-2.5}\) model, i.e. \(z=1.5 S_{D2}=\left (4/3\right )\Omega _{FF}^{2}f_{0}\), and for the \(\omega ^{-3}\) model, i.e. \(z=2 S_{D2}=\left (3\pi /8\right )\Omega _{FF}^{2}f_{0}\).
Corner frequency can be derived from the ratio of the velocity to displacement power spectra
which for the \(\omega ^{-2}\) model, i.e. \(z=1\) gives \(f_{0}=\sqrt {S_{V2}/S_{D2}}/\left (2\pi \right )\), see also Boore (1983b) and Andrews (1986)). For the \(\omega ^{-2.5}\) model, i.e. \(z=1.5 f_{0}=\sqrt {S_{V2}/S_{D2}}/\left (\pi \sqrt {2}\right )\), and for the \(\omega ^{-3}\) model, i.e. \(z=2 f_{0}=3\sqrt {S_{V2}/S_{D2}}/\left (2\pi \right )\). Inserting \(f_{0}\left (z\right )\) into Eq. (2.63) gives the low frequency displacement spectral plateau, \(\Omega _{PS}\left (z\right )\), as a function of z,
and P- and S-wave potency, \(P_{P}=4\pi Rv_{P}^{3}\Omega _{P}/\left (\varLambda _{P}v_{S}^{2}\right )\) and \(P_{S}=4\pi Rv_{S}\Omega _{S}/\varLambda _{S}\). The following equations quantify the portion of the displacement power spectrum in the frequency range \(\left (f_{1},f_{2}\right )\) for \(z=1\), \(z=1.5\) and \(z=2\),
The following equations quantify the portion of the displacement power spectrum lost in the frequency range \(\left (0,f_{1}\right )\) and \(\left (f_{2},\infty \right )\) for \(z=1\), \(z=1.5\) and \(z=2\).
In practice we estimate the power spectra in the frequency band \(\left (f_{1},f_{2}\right )\), and therefore we underestimate seismic energy and introduce a bias into the corner frequency and other derived source parameters. The corner frequency, \(f_{0}\), corrected for bandwidth limitation as a function of z is
where \(\tilde {f}_{0}\) is the corner frequency estimated within the frequency range \(\left (f_{1},f_{2}\right )\). The following equations give the corrected \(f_{0}\) for \(z=1\), \(z=1.5\), and \(z=2\).
For \(z=1\), i.e. the \(\omega ^{-2}\) model, Eq. (2.69) was derived by Di Bona and Rovelli (1988).
Bandwidth corrections for seismic energy can be carried out by dividing the observed energy, \(\tilde {E}=4\pi \rho v_{P,S}\tilde {S}_{V2P,S}\), estimated for a given frequency range \(\left (f_{1},f_{2}\right )\) by the energy recovery, \(E\left (z\right )=\tilde {E}/\widetilde {ER}\left (z\right )\), given by Eq. (2.60) with Eq. (2.61), for \(z=1\), \(z=1.5\) and \(z=2\), respectively, which gives
where \(\tilde {S}_{V2P,S}\), is the observed, instrument, distance, and Q corrected velocity spectrum. Since the recovery of \(\Omega _{FF}\), and therefore seismic potency, is limited by \(f_{1}\) only, the bandwidth correction can be done by dividing the observed \(\tilde {\Omega }_{FF}\) by the potency recovery given by Eq. (2.55),
First estimates of spectral source parameters of mine events were conducted by Smith et al. (1974), Spottiswoode and McGarr (1975), Gibowicz et al. (1977), and Cichowicz (1981), all using digitised analog waveforms, so it was not conducive to routine application. The introduction of digital seismic systems to mines in the late 80s facilitated a real time quantitative seismology, whereby apart from its timing and location each seismic event is quantified by seismic potency, or seismic moment, their tensors, by radiated seismic energy and the associated predominant frequency (Mendecki, 1993). With the recent introduction of machine learning that automates the phase picking and event classification (e.g. Gal et al., 2021; Zhu et al., 2022), we estimate there are approximately 10\(^{6}\) mine seismic events processed and quantified that way per day, bulk of them using the \(\omega ^{-2}\) model.
However, the \(\omega ^{-2}\) model fits spectra of seismic sources originating in a hard, relatively homogeneous rock well, the harder and more homogeneous the rock the better the fit. In mines with a softer and/or less homogeneous rock mass, the \(\omega ^{-2}\) model tends to overestimate higher the frequencies. The difficulties are that in some mines some seismic events are well described by the \(\omega ^{-2}\) model, while in another part of the same mine they tend to conform to the \(\omega ^{-2.5}\) or even \(\omega ^{-3}\) model, therefore a need for adaptive processing. Since there is a trade-off between Q and the exponent of the spectral decay, it is not recommended to invert Q together with \(\varOmega _{0}\) and corner frequency \(f_{0}\). In mines, most attenuation and scatter occur close to seismic sources due to the mine workings and fractured rock around them, while in crustal earthquakes most inelastic attenuations and site effects are close to surface (Cichowicz et al., 1988 and Cichowicz & Green, 1989).
In the frequency domain, the recorded waveform, \(W\left (f\right )\), is a convolution of the instrument response \(I\left (f\right )\), the source function, \(\Omega \left (f\right )\), the path effect modelled by the attenuation and scattering function \(Q\left (f\right )\), and the site effect \(S\left (f\right )\). To get source, we need to deconvolve the instrument response, path, and site effect, \(\Omega \left (f\right )=W\left (f\right )/\left [I\left (f\right )Q\left (f\right )S\left (f\right )\right ]\). The instrument response for a 4.5 Hz geophone with damping \(b=0.72\) is given by \(I\left (f\right )= \left \Vert f^{2}/\left [20.25+\left (0+6.48i\right )f-f^{2}\right ]\right \Vert \), and for 14 Hz omnidirectional geophone with the same damping used in longer boreholes, \(I\left (f\right )= \left \Vert f^{2}/\left [196+\left (0+20.16i\right )f-f^{2}\right ]\right \Vert \), see Mountfort and Mendecki (1997). Frequently the default damping is between 0.5 and 0.7.
Spectral parameters, namely the zero frequency asymptote \(\Omega _{0}\), and corner frequency \(f_{0}\), are derived from the instrument, distance, and Q corrected spectrum of each recorded waveform and then averaged. A more efficient method is to correct the individual spectra and then average, or stack, each spectral component over all the sites involved. This method tends to average out incoherent site effects, source directivity, and random fluctuations of the high frequency spectrum.
In many cases, most frequently in smaller mines, the distances from seismic sources to the nearest sensors are too short to meet the requirement of being in the far field. Since in the near field the displacement spectra at low frequencies decay as \(f^{-1}\) and at high frequencies as \(f^{-2}\), it may introduce bias to \(\varOmega _{0}\) and to seismic energy, see Eq. (2.24) and Fig. 2.7. If we assume that the intermediate field is where co-seismic strains are of the order 10\(^{-5}\), then for hard rock with \(v_{S}=3000\) m/s seismic sites that record \(PGV\geq 3\) cm/s would be affected. In addition a short distance from source to sensor makes the P-wave spectral window too short for spectral analysis and it should be rejected. Other complications in mines are frequently used single-component sensors, mains electrical noise, and, in some cases, long cables that attenuate the signal before A/D conversion. Below are three examples of events with similar energy but with spectra that fit different models.
Example 1, \(\log E\simeq 7.1\), 4.5 Hz Sensor, \(\omega ^{-2.5}\) Model
Figure 2.18 shows a 1.05 second snap of the 4.4 second buffer waveforms recorded in an underground hard rock caving mine 1.2 km below surface by a three-component 4.5 Hz geophone grouted at the end of an 8 m borehole away from the skin of the excavation.
Fig. 2.18
Velocity waveforms of a mid-size seismic event recorded by 4.5 Hz geophones in an 8 m borehole away from the excavation in an u/g mine (left), and integrated displacement waveforms (right)
The event located 607 m from the site. The vector \(PGV=2.91\cdot 10^{-3}\) m/s, \(PGD=1.19\cdot 10^{-5}\) m, and the cumulative absolute displacement, defined as the integral of the absolute value of a velocity time series, \(CAD = \intop _{0}^{t_{d}}|\mbox{v}\left (t\right )|dt\), calculated for the full 4.4 s buffer, \(CAD=1.62\cdot 10^{-4}\) m. The P-wave window for spectral analysis is taken from just before the P-arrival to the S-arrival, and the S-wave window from just before the S-arrival to \(S_{w}\). Sampling frequency, \(s_{f}=6\) kHz, and after the DC offset removal waveforms are high passed by a 2nd order 1 Hz Butterworth filter run both ways. No site effect deconvolution was needed for this event since it was recorded by sensors embedded in solid rock.
Figure 2.19 shows the velocity and displacement cepstra filtered at the 15th zero crossing and the resulting velocity and displacement spectra.
Fig. 2.19
Velocity and displacement cepstra filtered at 15 zero crossing (top row) and velocity and displacement spectra before, in grey, and after, in black, cepstral filtering
Figure 2.20 shows the distance, Q corrected, and cepstral filtered S-wave velocity and displacement spectra fitted with \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models. The observed spectra deviate from the \(\omega ^{-2}\) model, shown here in red, and fit \(\omega ^{-2.5}\) better shown in blue. Note that seismic energy estimates from one site can be strongly affected by the radiation pattern.
Fig. 2.20
Distance and Q corrected and cepstral filtered S-wave velocity (left) and displacement (right) spectra for \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models
Example 2, \(\log E\simeq 7.1\), 4.5 Hz Sensor, \(\omega ^{-3}\) Model
Figure 2.21 top row shows a 1.2 second snap of the 5.0 second buffer waveforms recorded in an underground hard rock caving mine 850 m below surface by a three-component 4.5 Hz geophone grouted at the end of an 8 m borehole away from the skin of the excavation. The event located 370 m from the site. The vectors \(PGV=4.43\cdot 10^{-3}\) m/s, \(PGD=3.91\cdot 10^{-5}\) m, and the \(CAD=2.64\cdot 10^{-4}\) m, calculated for the full 5.0 second buffer. Sampling frequency, \(s_{f}=6\) kHz, and after DC offset removal, the waveforms are high passed by a 2nd order 1 Hz Butterworth filter run both ways.
Fig. 2.21
Velocity and displacement waveforms of a mid-size seismic event recorded by 4.5 Hz geophones in an 8 m borehole away from the excavation in an u/g mine (top row). The distance, Q corrected, and cepstral filtered S-wave velocity and displacement spectra fitted with \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models (bottom row)
Figure 2.21 bottom row shows the distance, Q corrected, and cepstral filtered S-wave velocity and displacement spectra fitted with \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models.
In this case, the observed spectra fit the \(\omega ^{-3}\) model shown in green. It indicates a lower stress environment at the site of the event due to the shallow depth and a more inhomogeneous and weaker rock mass than in the first example.
Example 3, \(\log E\simeq 7.8\), 4.5 Hz Sensor, \(\omega ^{-2}\) Model
Figure 2.22 top row shows 1.2 second snap of the 5.0 second buffer waveforms recorded in an underground tabular hard rock gold mine 2000 m below surface by a three-component 4.5 Hz geophone grouted at the end of an 8 m borehole away from the skin of the excavation.
Fig. 2.22
Velocity waveforms of a mid-size seismic event recorded by 4.5 Hz geophones in an 8 m borehole away from the excavation in an u/g mine (left), and integrated displacement waveforms (right)
The event located 713 m from the site on a known normal fault. The vectors \(PGV=5.31\cdot 10^{-3}\) m/s, \(PGD=3.59\cdot 10^{-5}\) M, and the \(CAD=4.4\cdot 10^{-4}\) m, calculated for the full 5.0 second buffer. Sampling frequency, \(s_{f}=6\) kHz, and after the DC offset removal waveforms are high passed filtered by a 2nd order 1 Hz Butterworth filter run both ways.
Figure 2.23 bottom row shows the distance, Q corrected, and cepstral filtered S-wave velocity and displacement spectra fitted with \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models. The observed spectra fit the \(\omega ^{-2}\) model, shown in red, which is expected for events associated with fault slip in high stress environment.
Fig. 2.23
The distance, Q corrected, and cepstral filtered S-wave velocity (left) and displacement (right) spectra fitted with \(\omega ^{-2}\), \(\omega ^{-2.5}\), and \(\omega ^{-3}\) models
2.11 Final Static Deformation for Double-Couple Source
2.11.1 Radiation Patterns
Aki and Richards (2002) gave a relatively simple formula for the displacement vector due to a moment or potency tensor corresponding to a point shear dislocation. This equation shows the following:
1.
The far-field displacements attenuate as \(R^{-1}\) and are proportional to particle velocity at the source.
2.
The far-field and the near-field radiation patterns are similar.
3.
There is a final static displacement that attenuates as \(R^{-2}\).
Two coordinate systems are assumed: One is Cartesian, with the source at the origin and the slip in the XY-plane, and the other is spherical polar with the same origin and the polar axis along the Cartesian OZ. The azimuth \(\phi \) is measured from the OX axis of the Cartesian system.
The famous Aki and Richards’s Eq. (4.32) can be expressed in terms of the scalar seismic potency time function \(P\left (t\right )\),
where \(\left (R,\theta ,\phi \right )\) are the spherical polar co-ordinates of the receiver, the integration variable \(\zeta \) has units of time, \(t^{\prime }_{P}=t-R/v_{P}\) and \(t^{\prime }_{S}=t-R/v_{S}\) are the respective retarded times, and the radiation patterns, \(\varLambda _{N}\), \(\varLambda _{IP}\), \(\varLambda _{IS}\), \(\varLambda _{FP}\), and \(\varLambda _{FS}\) are vectors as given by Aki and Richards (2002) in Eq. (4.33) as
where \(\varLambda _{FSV}\) and \(\varLambda _{FSH}\) are the two polarisations of the far-field S-wave. The scalar coefficients \(a, b\), and c and the Cartesian components of the unit vectors \(\hat {r},\hat {\theta }\) and \(\hat {\phi }\) are
The first term with \(1/R^{4}\) in Eq. (2.76) is named near field as it is negligible at distances far away from the source. Madariaga et al. (2019) wrote the first term of Eq. (2.76) in a slightly different form to stress that the near and intermediate terms decay as \(1/R^{2}\) and cannot be separated in the study of seismic radiation. Indeed, applying the mean value theorem of calculus, one can write
which is of order \(1/R^{2}\). The P- and S-waves cannot be separated at very short distances from source, which does not mean that the two types of seismic waves do not exist there. The reason that we cannot separate them is that they arrive almost simultaneously and that their amplitudes are not significantly different.
The next two terms in 2.76 are said to belong to the intermediate field of the seismic radiation. Both the near field and the intermediate field terms are fully determined by the potency time function at the source. The last two terms are dominant in the far field and are determined by the potency rate. They decrease with the distance to the source as \(1/R\) and go to zero when the time goes to infinity, unlike the near-field term and the two intermediate field terms.
A common feature of the individual terms in 2.76 is that in each of them the angular dependence is factored out that makes it possible to visualise those terms as 3D surfaces in polar coordinates,
where the subscript \(\Xi \) stands for \(N,IP,IS,FP\) or FS and the components \(\Lambda _{\Xi R}\), \(\Lambda _{\Xi \theta }\), and \(\Lambda _{\Xi \phi }\) are given in Fig. 2.24 bottom right, see also Eq. (2.77).
Fig. 2.24
Spatial distribution of the radiation of seismic waves for a double-couple source. Table bottom right gives parameters \(\varLambda _{\Xi R}\), \(\varLambda _{\Xi \theta }\), \(\varLambda _{\Xi \phi }\) and the RMS values
The RMS radiation pattern is obtained by averaging \(\left \Vert \Re _{\Xi }\right \Vert ^{2}\) over the unit sphere around the source and then taking the square root of the result,
Figure 2.24 illustrates the spatial distribution or the radiation from a double-couple source in the near, intermediate, and far fields. Note that the vector \(\varLambda _{IP}=4a\hat {r}-2b\hat {\theta }+2c\hat {\phi }\) is neither orthogonal to the wavefront at the point of the receiver to qualify it as a purely P- or pressure wave of longitudinal polarisation, nor it is tangential to the wavefront at the point of the receiver, to qualify it as a purely S- or shear wave of transverse polarisation. The same applies to \(\varLambda _{IS}=-3a\hat {r}+3b\hat {\theta }-3c\hat {\phi }\). Therefore, the intermediate field terms in the classical expression do not describe seismic waves of different polarisation but rather do so for groups of waves which arrive at the receiver almost at the same time. The same can be said for the near-field term.
2.11.2 Final Static Displacement and Induced Strain
Nearly all seismic related damage in underground mines is observed in excavations relatively close to the seismic sources where rock is subjected to large inelastic deformations. Dynamic strains of the order of \(10^{-4}\) or greater are associated with ground velocities over 30 cm/s, which are frequently associated with localised damage to underground infrastructure. Larger strains crack intact rock. It is therefore important to gain insight into the extent of that deformation caused by seismic sources.
Taking \(t\rightarrow \infty \) in Eq. (2.76), i.e. \(u\left (R,\theta ,\phi ;t\rightarrow \infty \right )\), gives the final static displacement field for a shear dislocation by potency P that does not depend on the details of the source time function. This permanent deformation is due to the near-field and the intermediate field terms since in the far field \(\lim _{t\rightarrow \infty }\dot {P}\left (t\right )=0\). The convolution integral, i.e. the near-field term, can be taken exactly in the limit \(t\rightarrow \infty \),
which decays along direction \(\left (\theta ,\phi \right )\) as \(1/R^{2}\), see Aki and Richards (2002) equation (4.34). The maximum static displacement is obtained for \(\theta =\pi /4\) and \(\phi =0\), which for a Poissonian solid gives
The components of the strain tensor in spherical polar coordinates can be expressed as partial derivatives with respect to \(R, \theta \) and \(\phi \) of the displacement vector components relative to the spherical polar system,
The spherical polar components of the displacement vector in Eq. (2.75) are obtained by decomposing \(\Lambda _{N}\), \(\Lambda _{IP}\), and \(\Lambda _{IS}\) as combinations of the spherical polar base vectors \(\hat {r},\hat {\theta }\) and \(\hat {\phi }\) and then grouping the terms. The spherical polar components of the static co-seismic displacement are
which can be written \(\epsilon _{ij}\left (P,\nu \right )=P/\left (4\pi R^{3}\right )e_{ij}\), where the components of the new tensor \(e_{ij}\left (\theta ,\phi ,\nu \right )\) are equal to the respective expressions in the square brackets in Eq. (2.85) and depend on \(\theta , \phi \) and the Poisson ratio, \(\nu \), since
The second order invariant of the strain tensor, \(I_{2}=\text{Tr}\left (\epsilon \cdot \epsilon \right )\), can be used to define a scalar measure for the level of inelastic deformation \(\epsilon _{p}\),
Figure 2.25 shows the shape of the source volume for an event with \(\log P=2.0\) for \(\epsilon _{p}\geq 10^{-6}\) in grey and \(\epsilon _{p}\geq 10^{-4}\) in red, as given by Eq. (2.87) assuming \(v_{S}^{2}/v_{P}^{2}=1/3\), i.e. Poisson ratio \(\nu =0.25\).
Fig. 2.25
Shape of the source volume with \(\epsilon _{p} \geq 10^{-6}\) in grey and \(\epsilon _{p} \geq 10^{-4}\) in red given by Eq. (2.87) in a 3D view (left) and top view (right)
Figure 2.26 left shows the source volume with \(\epsilon _{p}\geq 10^{-4}\) for a seismic event with \(\log P=2.0\) as a function of Poisson ratio, and Fig. 2.26 right shows the Poisson ratio as a function of \(v_{S}/v_{P}\), for reference.
Fig. 2.26
Volume of the source region with strains \(\epsilon _{p} \geq 10^{-4}\) as a function of Poisson ratio (left) and Poisson ratio as a function of \(v_{S}/v_{P}\) for reference (right)
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