5. Time Distribution and Seismic Hazard

Both are simulated by the stretched exponential distribution, \(f\left (t\right ) = \left (q/\tau \right ) \left (t/\tau \right )^{q-1} \exp \left [-\left (t/\tau \right )^{q}\right ]\), with \(\tau =1.0\). The first one with \(q = 1.0\) that gives the exponential distribution of the inter-event times with \(C_{v}=1\), \(C_{v2}=0.71\), and \(P_{v}=0.6\). The second one with \(q=0.35\) which delivers a thick tail power law type distribution in time, therefore strongly clustered with \(C_{v}=3.02\), \(C_{v2}=0.95\), and \(P_{v}=0.83\).

The empirical survival function, \(\Pr \left (\geq t\right )\), for the data set that simulates the exponential distribution is linear for the entire domain of inter-event times, and probabilities of outliers are low. However, the clustered data set with \(C_{v}=3.02\) delivers a highly non-linear survival function with much higher probabilities of extreme inter-event times.

The Poisson distribution can be used to forecast the number of random events in a future time, \(\Delta T\). There are two random variables that arise from a Poisson process: (1) the number of events in a given time interval, which is a discrete variable, (2) the time until the occurrence of the first event, which is a continuous variable. Having a single event in the time interval \(\Delta t\), the probability of finding this event in the sub-interval \(\Delta T = t_{2} - t_{1}\) is \(\Delta T/\Delta t\). Having n events in \(\Delta t\), the activity rate is \(\lambda _{t} = n/\Delta t\), the recurrence time \(\bar {t} = \Delta t / n\), and the expected number of events in \(\Delta T\) is \(\Lambda \left (\Delta T\right ) = n \left (\Delta T/\Delta t\right ) = \lambda _{t}\Delta T\). The probability that exactly \(N < n\) events can be found in the interval \(\Delta T\) is given by the binomial distribution, The mean of the binomial distribution is \(n\Delta T/\Delta t\), and the variance is \(n\Delta T/\Delta t \left (1-\Delta T/\Delta t\right )\), i.e. the variance is less than the mean. For large n and small \(\Delta T/\Delta t\), the binomial distribution can be approximated by Poisson distribution, with the mean value and the variance equal to \(\lambda _{t} \Delta T = n\Delta T/\Delta t\) and the coefficient of variation \(C_{v} = 1/\left (\lambda _{t}\Delta T\right )\). Note that \(\Pr \left (N=1,\bar {t}\right ) = 0.37\).

The probability that there will be no events in \(\Delta T\) is \(\Pr \left (N=0,\Delta T\right ) = \exp \left (-n\Delta T/\Delta t\right )\), and the probability that there will be at least one event is \(F\left (\Delta T\right ) = \Pr \left (N\geq 1,\Delta T\right ) = 1 - \exp \left (-n\Delta T/\Delta t\right )\), which is the exponential distribution with the mean \(1/\lambda _{t}\) and the variance \(1/\lambda _{t}^{2}\), and therefore, the coefficient of variation \(C_{v} = 1\). If \(n \left (\Delta T/\Delta t\right ) < 0.1\), then \(\Pr \left (N\geq 1,\Delta T\right ) \simeq n \left (\Delta T/\Delta t\right )\). Note that \(\Pr \left (N\geq 1,\bar {t}\right ) = 0.63\), see Fig. 5.3.

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The exponential distribution describes the length of time between events, or the inter-event time distribution, and it does not depend on time. The assumption of stationary and independent increments means that at any point the process probabilistically restarts itself, i.e. it has no memory. The probability density function of an exponential distribution is \(f\left (\Delta T\right ) = \lambda \exp \left (-\lambda \Delta T\right )\), and the probability that an event will occur in time interval \(\left (0,1/\lambda \right )\) is \(\Pr \left (0,1/\lambda \right ) = \intop _{0}^{1/\lambda }\lambda \exp \left (-\lambda \Delta T\right )d\left (\Delta T\right ) = 0.632\), and the probability that it will occur in the time interval \(\left (1/\lambda ,\infty \right )\) is \(\Pr \left (1/\lambda ,\infty \right ) = 0.368\), see Fig. 5.3.

If a set of n random variables, \(X_{1}\), \(X_{2}\), ..., \(X_{n}\), where \(X_{1} = P_{1}\), \(X_{2} = P_{2}\), ... \(X_{n} = P_{n}\) is one possible realisation of each of them, is drawn from the same distribution function \(F\left (P\right ) = \Pr \left (\leq P\right )\), then the distribution of the maximum value \(P_{max} = \max \left \{ P_{1},P_{2},...,P_{n}\right \} \) is given by \(\Pr \left (\leq P\right ) = \Pr \left (P_{1}\leq P,P_{2}\leq P,...,P_{n}\leq P\right ) = \prod _{j=1}^{n}\Pr \left (\leq P_{j}\right ) = \left [F\left (P\right )\right ]^{n}\).

If the occurrence of larger events in time \(\Delta t\) is ruled by a homogeneous Poisson process with activity rate \(\lambda _{t} = n/\Delta t\), then the probability of having exactly N events within time \(\Delta T\) is where \(\Lambda \left (\Delta T\right ) = \lambda _{t}\Delta T\) is the expected number of events in \(\Delta T\). The probability that all \(P_{j}\) within \(\Delta T\) are smaller than or equal to some large P is which gives \(\Pr \left (\leq P,\Delta T\right ) = \exp \left \{ -\Lambda \left (\Delta T\right )\left [1-F\left (P\right )\right ]\right \} \), or \(\Pr \left (\leq P,\Delta T\right ) = \exp \left [-\Lambda \left (\Delta T\right )\Pr \left (\geq P\right )\right ]\), and the probability of having at least one event \(\geq P\) within \(\Delta T\), where \(\Pr \left (\geq P\right ) = N\left (\geq P\right ) / n\). For the UT power law, the Eq. (5.7) gives Epstein and Lomnitz (1966) showed that the OE power law, \(N\left (\geq P\right ) = \alpha P^{-\beta }\), gives \(\Pr \left (\leq P,\Delta T\right )=\exp \left [-\alpha \left (\Delta T/\Delta t\right )\exp \left (-\beta m\right )\right ]\), where \(m = \ln P\), which is the first Gumbel distribution of the extreme values (see also Shakal & Willis, 1972 or Kijko, 1982). Note that replacing \(\Delta T\) with the recurrence time \(\bar {t}\left (\geq P\right ) = \Delta t / N\left (\geq P\right )\) gives \(\Pr \left [\geq P,\bar {t}\left (\geq P\right )\right ] = 0.63\), regardless of P. The mean recurrence time can also be presented as where \(\Pr \left (\geq P,\Delta T\right )\) is the probability of exceedance, and \(\Delta T\) here is called the exposure time. Most seismic building codes are based on the following performance levels that define the ability of a structure to sustain its main functions during and after earthquakes of different strengths. (1) Operational Limit: 50% probability of exceedance over 50 years, which is considered frequent, that according to Eq. (5.9) gives \(\bar {t}\left (\geq P\right ) = 72\) years, (2) Immediate Occupancy: 20% in 50 years (occasional) that gives \(\bar {t}\left (\geq P\right ) = 225\) years, (3) Life Safety: 10% in 50 years (rare) that gives \(\bar {t}\left (\geq P\right ) = 475\) years, and (4) Collapse Prevention: 2% in 50 years (very rare) that gives \(\bar {t}\left (\geq P\right ) = 2475\) years. In general, one expects better structural performances for frequent events of lower intensities, and one can accept higher damage for very rare large events. Building codes also adjust seismic input level to the importance of a given structure, e.g. by increasing the exposure time \(\Delta T\) in Eq. (5.9) that leads to an increase of the mean recurrence time. For many structures in mines, the exposure times \(\Delta T\) are shorter. For example, 50% probability of exceedance over 10 years would give \(\bar {t}\left (\geq P\right ) = 14.4\) years, 20% in 10 years would give \(\bar {t}\left (\geq P\right ) = 44.8\) years, and 10% in 10 years would give \(\bar {t}\left (\geq P\right ) = 95\) years.

For a Poisson process, the probability of having N occurrences over a time window equal to the recurrence time, \(\Delta T = \bar {t}\), is \(\Pr \left (N\right ) = \left (1/N!\right )\exp \left (-1\right )\), which gives a probability of 36.8% that such an event will occur once, 18.4% that it will occur twice and 6.1% for three times.

Equation (5.9) can also be presented in terms of potency as a function of recurrence times, which for the upper truncated power law, \(N\left (\geq P\right ) = \alpha \left (P^{-\beta }-P_{max}^{-\beta }\right )\), gives and for the OET distribution \(\overline {t}\left (\geq P\right ) = \Delta t/ \left [N\left (\geq P_{min}\right )\Pr \left (\geq P\right )_{OET}\right ]\).

These equations can be used to calculate the expected potencies over the selected recurrence times and to estimate their probabilities (Eq. 5.7) as a function of \(\Delta T\), as mining progresses, to monitor longer term hazard. In this case, \(P_{max}\) should be taken as the latest estimate of the maximum expected potency. Figure 5.4 left shows the probabilities of exceedance versus \(\log E\) for MineC over 90, 180, and 360 days given by Eq. (5.8), and Fig. 5.4 right shows the probabilities of exceedance of \(\log E\) with the recurrence times of 1, 2, 3, and 5 years for the same data set calculated with Eq. (5.10).

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Note that the term “the mean recurrence time” or “the return period” as it is frequently called may be misleading unless it also gives the standard deviation around the mean. In practice, the Poisson model is justified if this standard deviation is not far from the mean.

In mines where the volume of rock extraction is not a linear function of time seismic hazard can be expressed in terms of the volume of rock extracted, or volume mined \(V_{m}\). Then, the probability that there will be at least one event \(\geq P\) while extracting an additional volume of rock \(\Delta V_{m}\) is where \(\Lambda \left (\Delta V_{m}\right ) = n \left (\Delta V_{m}/V_{m}\right ) = \lambda _{m}\Delta V_{m}\) is the expected number of events in \(\Delta V_{m}\). Since \(\bar {V}_{m}\left (\geq P\right ) = V_{m} / N\left (\geq P\right )\), the Eq. (5.11) gives the ratio \(\bar {V}_{m}\left (\geq P\right ) / \Delta V_{m} = -1 / \ln \left [1-\Pr \left (\geq P,\Delta V_{m}\right )\right ]\). For example, if \(\Pr \left (\geq P,\Delta V_{m}\right ) = 0.05\), we can mine on average 19.5 times \(\Delta V_{m}\) to generate seismic event \(\geq P\). For the UT power law, the Eq. (5.11) gives which shows that if \(\beta \) is constant and \(\alpha \) is proportional to \(V_{m}\), then seismic hazard can be controlled by the rate of mining.

The rock mass dynamics in mines is driven mainly by the transient deformation of nearby excavations in response to extraction of stressed rock and by the near-field deformation of seismic sources, including pillar and abutment failures, and by blasting. To a lesser degree, it is also influenced by waves from remote sources of seismic radiation and by tidal forces. The step loading caused by rock extraction and by seismic events induces large displacements, plastic instability, splitting, buckling, and bursting of rock close to excavations. The energy imparted into the rock mass during step loading is partially converted into strain energy and partly dissipated through friction, local plastic deformation, and strain energy of radiated waves.

After a step loading, one can observe two partly competing stress-related processes occurring in the affected rock mass:

(1) An excitation phase that elevates the average stress level in volumes of rock where loading is faster than the ability to dissipate the excess of strain energy. It is generally short-lived, and it can be observed by monitoring changes in seismically inferred stress and strain by high-resolution networks.

(2) A relaxation phase that cascades the system down from its excited state through intermediate phases toward a non-equilibrium steady state. It is facilitated by different forms of inelastic deformation—seismic and aseismic. It modifies the stress pattern by reducing its elevated levels and moving it further away from excavations. It generates the bulk of the seismic activity after loading, and it lasts much longer than the excitation phase.

Seismic rock mass response to blasting is driven strongly by the stress level in rock surrounding the blast and by the volume of rock blasted. In the case of a preconditioning blast, when there is little or no rock extraction, the stress level plays the major role. In most cases, rock extraction by blasting induces and triggers seismic events immediately and in close proximity to the blast, which helps to set up a proper re-entry protocol, i.e. the exclusion zone and re-entry time.

Aftershock sequences after larger seismic events in mines or after major production blasts are not as well developed and do not last as long as after tectonic earthquakes of similar size. The main reason is faster relaxation facilitated by the presence of openings and extensive fracturing. If the immediate aftershocks are subdued and locate close to the main shock, there is less potential for a large event. As the immediate aftershocks extend further away from the main shock, it is more likely they may trigger larger event, see Fig. 5.20.

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However, rock mass convergence associated with large rock extractions or with larger pillar removals may produce elevated seismic response for days or even weeks with sporadic elevated levels of seismic activity away from the place of extraction. Locations and orientations of larger events during aftershock sequences are controlled by geological structures rather than by the elevated stress level.

The seismic activity after step loading tends to cluster in space and time. In some cases, the rate of activity follows typical aftershock sequences that can be described by the Omori law or the stretched exponential relaxation function. However, in some cases, the response is more complex and cannot be described by a simple intensity function and therefore needs to be treated with non-parametric statistics.

The fundamental premise here is that as the rate of seismic activity increases so does the likelihood that one of these events may be larger and damaging. In many cases, specifically after production or development blasts, the recorded activity is complex and does not follow the typical aftershock sequences that could be described by the Omori law or the stretched exponential relaxation function, and therefore needs to be treated by non-parametric statistics. One way to check if the elevated seismic activity has returned to an acceptable level is to test the null hypothesis of no change. The acceptable level of seismic activity can be defined a priori, or it can be taken as an average activity rate before step loading.

To detect changes in seismic activity rates in two different time intervals, \(\Delta t_{1}\) and \(\Delta t_{2}\), within the same volume of rock, one can count the respective number of recorded events above a certain potency. If the time intervals are equal and relatively long and the observed number of events is significantly different, then a statement can be made about the relative change. However, the associated uncertainty increases as the time intervals get shorter and as the difference in the event counts becomes smaller. The situation is even more difficult if the time intervals are not equal.

Under normal conditions, the event counts can be considered as outcomes of a Poisson process, and therefore, their occurrence can be very irregular. To measure the seismicity rate change, we need the probability density function of the activity rate, \(\lambda \). This can be derived by normalising the Poissonian density function (see Eq. 5.5) so that the integral over \(\lambda \) from 0 to \(\infty \) is unity, which gives

The probability that the seismicity rate in two different time intervals, with densities \(f_{1}\) corresponding to time interval \(\Delta t_{1}\) and \(f_{2}\) to \(\Delta t_{2}\), increased by more than k times is \(\Pr \left [\left (\lambda _{2}/\lambda _{1}\right )>k\right ] = \intop _{0}^{\infty } d\lambda _{1} f_{1}\left (\lambda _{1}\right ) \intop _{k\lambda _{1}}^{\infty } d\lambda _{2} f_{2}\left (\lambda _{2}\right )\), which, taking \(\lambda _{1} = N_{1}\left (\geq \log P\right )/\Delta t_{1}\) and \(\lambda _{2} = N_{2}\left (\geq \log P\right )/\Delta t_{2}\), gives

where \(\varGamma \left (N_{2}+1,kx\Delta t_{2}/\Delta t_{1}\right ) = \intop _{kx\Delta t_{2}/\Delta t_{1}}^{\infty }\exp \left (-t\right )t^{N_{2}+1}dt\) is the upper incomplete Gamma function, and the ratio \(\Delta t_{2}/\Delta t_{1}\) caters for the unequal time intervals (Marsan, 2003).

In 1893 Fusakichi Omori published the 11 page long note “On aftershocks” (Omori, 1893), which was the abstract of the paper published a year later (Omori, 1894), where he described the activity of aftershocks at any time x after the main shock as \(y = k\left (x+h\right )^{-1}\), where k and h are constant, which “represents a rectangular hyperbola and implies that the activity varies nearly inversely with time”. Hirano (1924) and Utsu (1957) modified Omori’s equation by introducing the exponent p, and today the Omori law is presented as

where \(k, c\), and p are parameters and t is time. The total number of events \(N_{t} = N\left (0,\infty \right )\) is derived from \(N_{t}=\int _{0}^{\infty }k\left (t+c\right )^{-p}dt\) that gives \(N_{t} = k c^{1-p} / \left (p-1\right )\) for \(p>1\) and \(N_{t} = \infty \) for \(p \leq 1\) (Fig. 5.22).

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The parameter k depends on the magnitude of the main shock and for given c and p scales with the total number of events. The parameter c is the offset time that accounts for the incompleteness of the data set immediately after the main shock—the better the resolution of the seismic network, the smaller the c. Setting \(c > 0\) also prevents the infinite N at \(t=0\), and its physical meaning, however, is debatable.

When plotting the observed number of events within a certain interval of time versus time on a log-log scale, the data points should tend to a straight line whose slope is an estimate of p. The parameter p controls the rate of decay of aftershocks, and, for slowly decaying sequences with \(p\leq 1\), the Omori law predicts that the total number of events becomes unbounded with increasing time. The parameter p measures the overall relaxation time of the process following the major event. The lower the p-value, the slower the relaxation, which is characteristic of stiffer systems; the opposite would apply to softer systems. As a consequence, the p-value would be expected to correlate negatively with the Gutenberg-Richter b-value, e.g. (Wang, 1994). Note that p and c are positively and almost linearly correlated, which can frustrate the estimation process. In general, the Omori law is an open-ended power law, \(N\left (\geq t\right ) \sim t^{-p}\), has infinite moments and therefore the property of scale invariance, i.e. the relative change of \(N\left (\geq st\right ) / N\left (\geq t\right ) = s^{-p}\) is independent of t, and therefore it lacks a characteristic scale—there is no characteristic time. To be physical, the power law description of a finite system must be bounded; otherwise, aftershocks would continue forever.

The number of events in a time interval \(N\left (t_{1},t_{2}\right )\) after the step loading can be calculated by \(N(t_{1},t_{2})=\int _{t_{1}}^{t_{2}}k/\left (t+c\right )^{p}dt\), which gives

and the ratio

To get the probability distribution, we need to normalise parameter k so \(\int _{0}^{T}k\left (t+c\right )^{-p}dt =1\), where T is the assumed end of the aftershock sequence, which gives

and the cumulative distribution function, \(F\left (t\right ) = \Pr \left (\leq t\right ) = \intop _{0}^{t}f\left (u\right )du\), which gives

and \(\Pr \left (\leq t\right )=\left (t/T\right )^{1-p}\) for \(c = 0\) and \(p \neq 1\). The probability of having an event during the time interval \(\left (t_{1},t_{2}\right )\) is

For \(p \leq 1\), the end of the aftershock sequence T must be finite; otherwise, the integral diverges, but for \(p > 1\) there is a finite number of aftershocks as \(T \rightarrow \infty \) and the respective equations are as follows:

The maximum likelihood (ML) method of estimating parameters of the Omori relation is similar to the method of calculating the \(\beta \)-value of the power law size distribution. However, the results are not always stable, because p and c are almost linearly correlated. One can simplify calculations by assuming that parameter \(c=0\), i.e. simple Omori (SO). Then to get the probability density function, we need to normalise parameter k so \(\intop _{t_{1}}^{t_{2}}kt^{-p} = 1\), which gives \(k = \left (1-p\right ) / \left (t_{2}^{1-p}-t_{1}^{1-p}\right )\). The ML function then is \(L\left (t,p\right ) = \prod _{j=1}^{N}\left (1-p\right ) t_{j}^{-p} / \left (t_{2}^{1-p}-t_{1}^{1-p}\right )\), where N is the number of events in the time period \(t_{2} - t_{1}\). The parameter p can be calculated from \(\partial \ln L\left (p\right )/\partial p = 0\), which gives

where \(t_{dj}\) is the time of seismic events. Equation (5.26) needs to be solved numerically for p. The parameter k can then be calculated from

The standard deviation of p can be obtained from the second derivative of the log likelihood function, \(\left [-\partial ^{2}\ln L\left (p\right )/\partial p^{2}\right ]^{-1/2}\), which gives

The single most important factor influencing estimates of \(s_{d}(p)\) is the number of observations, and for small N, the estimates are unreliable. For \(p=1\), taking \(\partial \ln L(p=1,k)/\partial k = 0\) gives \(k = N / \left (\ln t_{2}\ln t_{1}\right )\).

An alternative to the Omori law is the stretched exponential function introduced by von Rudolf Kohlrausch (1854) to interpret charge relaxation in a Leiden jar, which was the first capacitor. The stretched exponential is also known as the Kohlrausch-Williams-Watts (KWW) function, after Williams and Watts (1970) used it to characterise the dielectric relaxation rates in polymers. The stretched exponential function has a fatter tail than the exponential, but less so than a pure power law distribution, offering a compromise between the two descriptions. It can therefore be used to account for both a limited scaling regime and a cross over to non-scaling. While the Omori law is purely empirical, the parameters of the stretched exponential have a clear physical meaning, i.e. the relaxation time of the exponential decay process and the total number of events.

The function is defined as \(\exp \left [-\left (t/\tau \right )^{q}\right ]\) for \(t \geq 0\), where \(\tau > 0\) is the relaxation time and \(0<q<1\) is called the stretching or the shape exponent. The origin of the stretched exponential is assumed to be the result of a competition between two or more exponential processes. The reciprocal of q is called a heterogeneity parameter, \(h = 1/q\), which is the number of generations in a multiplicative process, and it gives an insight into the heterogeneity in complex systems. The larger the values of relaxation time \(\tau \), the slower the relaxation processes. The smaller the value of the shape exponent \(q < 1\), the quicker the decay for short times and slower for long times.

To apply the stretched exponential to aftershock sequences, it is convenient to start at time \(t_{1}\) after \(t_{0}\) when the step loading was applied (Kisslinger, 1993). Then the number of events yet to occur in a sequence at time \(t_{1}\) is

where \(N\left (t_{0},T\right )\) is the total number of events that will eventually occur and \(T - t_{0}\) is the duration of the process (Fig. 5.23). At time \(t_{1} = \tau \), the number of events still to occur is \(N\left (\tau ,T\right ) = N\left (t_{0},T\right )/e\), regardless of q, so 63% of the work is done during the relaxation time. The number of events that had occurred to time \(t_{1}\) is \(N(t_{0},t_{1}) = N\left (t_{0},T\right ) - N(t_{1},T)\), which gives a prediction of the total number of events at time \(t_{1}\) as

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The expected number of events to occur in the interval \(\left (t_{1},t_{1}+\Delta T\right )\) is

The cumulative distribution function is

and the survival function, \(S\left (t\right ) = \exp \left [-\left (t/\tau \right )^{q}\right ]\). The probability density function is

The probability of having an event in the time interval \(\left (t,t+\Delta T\right )\) is \(\Pr \left (t,t+\Delta T\right ) = F\left (t+\Delta T\right ) - F\left (t\right ) = \exp \left [-\left (t/\tau \right )^{q}\right ] - \exp \left [-\left (\left (t+\Delta T\right )/\tau \right )^{q}\right ]\), and the expected number of events in this time interval is \(N\left (t,t+\Delta T\right ) = N\left (t_{0},T\right ) \Pr \left (t,t+\Delta T\right )\). For \(q = 1\), the stretched exponential becomes a simple exponential distribution and for \(q < 1\) decays slower than exponential for large times (thick tail). For \(q > 1\), it decays faster than exponential and is known as the Weibull distribution. For \(q = 2\), it is also known as the Rayleigh distribution.

All moments of the stretched exponential distribution are finite, the mean, or the expected value is \(\tau \varGamma \left (1+1/q\right ) = \left (\tau /q\right ) \varGamma \left (1/q\right )\), and variance \(\tau ^{2} \varGamma \left (1+2/q\right ) - \tau ^{2} \varGamma \left (1+1/q\right )^{2}\). The gamma function, \(\varGamma \left (t\right ) = \intop _{0}^{\infty }x^{t-1}e^{-x}dx\), reduces to \(\varGamma \left (n\right ) = \left (n-1\right )!\) for positive integers and \(\varGamma \left (t+1\right ) = t\varGamma \left (t\right )\). The median is \(F\left (t\right ) = \exp \left [-\left (t_{0.5}/\tau \right )^{q}\right ] = 0.5\), which gives \(t_{0.5} = \tau \left (\ln 2\right )^{1/q}\).

The function \(t\cdot f\left (t\right ) = q \left (t/\tau \right )^{q} \exp \left [-\left (t/\tau \right )^{q}\right ]\) has a maximum at the relaxation time \(\tau \), regardless of q, and the \(\tau \cdot f\left (\tau \right ) = q/e\) depends only on q, see Fig. 5.24. These two properties of \(t\cdot f\left (t\right )\) may be utilised while fitting the data.

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The maximum likelihood function is

and \(\ln L\left (t_{j};\tau ,q\right ) = \sum _{j=1}^{n} \ln f\left (t_{j}\right ) = n\ln \left (q\right ) + q\sum _{j=1}^{n}\ln \left (t_{j}\right ) - \sum _{j=1}^{n}\ln \left (t_{j}\right ) - n q \ln \left (\tau \right ) - \sum _{j=1}^{n}\left (t_{j}/\tau \right )^{q}\). Taking \(\partial \ln L\left (t_{j};\tau ,q\right ) / \partial \tau = 0\) and \(\partial \ln L\left (t_{j};\tau ,q\right ) / \)\(\partial q = 0\) gives

respectively, where all sums are \(\sum _{j=1}^{n}\). Inserting \(\tau \) given by Eq. (5.34) into Eq. (5.35) gives

which depends only on q but does not have an analytical solution and needs to be solved numerically, and then one can solve for \(\tau \) using Eq. (5.34). The standard deviation of q and \(\tau \) can be obtained from the second derivative of the log likelihood function, \(\left [-\partial ^{2}\ln L\left (q,\tau \right )/\partial q^{2}\right ]^{-1/2}\) and \(\left [-\partial ^{2}\ln L\left (q,\tau \right )/\partial \tau ^{2}\right ]^{-1/2}\), which gives

The hazard function is defined as \(h\left (t_{1}\right ) = f\left (t_{1}\right ) / \left [1-F\left (t_{1}\right )\right ] = f\left (t_{1}\right ) / S\left (t_{1}\right )\), where \(f\left (t_{1}\right )\) is the probability density function, \(F\left (t_{1}\right ) = \Pr \left (\leq t_{1}\right )\) is the cumulative distribution function, i.e. the probability that the event will happen before time \(t_{1}\), and \(S\left (t_{1}\right ) = \Pr \left (\geq t_{1}\right ) = 1 - F\left (t_{1}\right )\) is the survival function, i.e. the probability that the event will not happen until time \(t_{1}\).

If the distribution of the recurrence times is a stretched exponential with \(f\left (t_{1}\right ) = \left (q/\tau \right ) \left (t_{1}/\tau \right )^{q-1} \exp \left [-\left (t_{1}/\tau \right )^{q}\right ]\) and the cumulative distribution function \(F\left (t_{1}\right ) = 1-\exp \left [-\left (t_{1}/\tau \right )^{q}\right ]\), then the hazard function becomes

where \(t_{1}\) is the time since the last event. For systems characterised by \(q=1\), the probability of having another event of similar size is independent of time, i.e. the stretched exponential distribution is reduced to the exponential one, and since the system has no memory, events occur randomly, see Fig. 5.25 (left).

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For \(q>1\), the probability of having another event of that size is increasing as the time since the last event increases. Interestingly, for \(q<1\), the probability of having another event of that size decreases with an increasing time since the last event.

For the intermediate and the long term hazard, one needs to consider the conditional probability that such an event will occur during \(\Delta T\), if the time since the last event of that size is \(t_{1}\), \(\Pr (\geq P;t_{1}\leq t\leq t_{1}+\Delta T\mid t_{1}<t) = \int _{t_{1}}^{t_{1}+\Delta T}f(t)dt/\int _{t_{1}}^{\infty }f(t)dt\), which, assuming the stretched exponential distribution, gives

see Fig. 5.25 (right). The waiting time corresponding to a probability of 0.5 gives the median waiting time,

The respective probability density function is the derivative \(d\left [F\left (\Delta T|t_{1}\right )\right ]d\left (\Delta T\right )\), which gives \(f\left (\Delta T|t_{1}\right ) = f\left (t_{1}+\Delta T\right ) / S\left (t_{1}\right )\), where \(S\left (t_{1}\right ) = 1 - F\left (t_{1}\right )\), and for the stretched exponential,

The expected time to the next event then is \(\left \langle \Delta T\right \rangle = \left [1/S\left (t_{1}\right )\right ] \intop _{0}^{\infty }\Delta Tf\left (t_{1}+\Delta T\right )d\)\(\left (\Delta T\right )\), which for the stretched exponential gives

where \(\varGamma \left [\left (1/q\right ),\left (t_{1}/\tau \right )^{q}\right ]\) is the upper incomplete gamma function.

The Shannon information entropy measures the amount of uncertainty in a given distribution, which is a measure of unpredictability. Similar to the power law distribution, the continuous information entropy of the stretched exponential distribution is defined as \(H\left [f\left (t\right )\right ] = \intop _{_{0}}^{T} f\left (t\right ) \log \left [1/f\left (t\right )\right ] dt\), where \(f\left (t\right )\) is the probability density function, \(f\left (t\right ) = \left (q/\tau \right ) \left (t/\tau \right )^{q-1} \exp \left [-\left (t/\tau \right )^{q}\right ]\), where \(\tau > 0\) is the relaxation time and \(0 < q < 1\) is the shape exponent and t is time. The \(\log \left [1/f\left (t\right )\right ]\) is the information content that an event will occur at time t with probability \(f\left (t\right )\), and, if \(f\left (t\right )\) is high, then knowledge that event occurred gives very little information, since it had a high probability of occurrence to start with. After integration, the information entropy for the stretched exponential distribution can be expressed as

where \(\gamma =\) 0.57721 is the Euler-Mascheroni constant. For \(q = 1\), i.e. for the exponential distribution \(H = \ln \left (\tau \right ) + 1\), which is independent of q, and for \(\tau = 1\), \(H = 1.\)

The uncertainty is low for low q-values, and it reaches its maximum of \(\gamma +\ln \left (\tau /\gamma \right )\) at \(q = \gamma \), regardless of the relaxation time, and then slowly decays. For a given q, the uncertainty, or the unpredictability, increases with an increase in relaxation time, i.e. systems that are slow to relax are less predictable, see Fig. 5.26. Note that the uncertainty here refers to the forecasting ability of the \(f\left (t\right )\) and not to the uncertainty or the error in q.

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In a non-stationary, also called a non-homogeneous, Poisson process, the underlying probability distribution and the intensity of the occurrence of events, \(I\left (t\right )\), vary with time. If \(\Lambda \left (t_{1},t_{1}+\Delta T\right ) = \intop _{t_{1}}^{t_{1}+\Delta T}I\left (t\right )dt\) is the expected number of events in the time interval \(\left (t_{1},t_{1}+\Delta T\right )\), then, according to the Poisson distribution, the probability of having exactly N events in that time interval is \(\Pr \left [N;\left (t_{1},t_{1}+\Delta T\right )\right ] = \left [\Lambda \left (t_{1},t_{1}+\Delta T\right )\right ]^{N} \exp \left [-\Lambda \left (t_{1},t_{1}+\Delta T\right )\right ] / N!\). The expected total number of events is \(\intop _{t_{0}}^{T}I\left (t\right )dt\), where \(t_{0}\) is the beginning and T is the end of the process. The probability of having zero events during that time is \(\Pr \left [N=0;\left (t_{1},t_{1}+\Delta T\right )\right ] = \exp \left [-\Lambda \left (t_{1},t_{1}+\Delta T\right )\right ]\), and the probability of having at least one event is

The non-stationary Poisson process can be applied to model the expected time and size distributions of seismic activity after step loading, e.g. after production blasts, pillar failures, or after larger events. If \(I\left (t,P\right )\) is the intensity function of the underlying process defined in such a way that \(\Lambda \left [\left (t_{1,}t_{1}+\Delta T\right );\left (P_{1},P_{2}\right )\right ] = \intop _{t_{1}}^{t_{1}+\Delta T}\intop _{P_{1}}^{P_{2}}I\left (t,P\right )dPdt\) is the expected number of events in the time interval \(\left (t_{1},t_{1}+\Delta T\right )\) and within the potency range \(\left (P_{1},P_{2}\right )\), then the probability of having exactly N events in this time interval and in this potency range is

If within that time interval the time and size distributions are Independent, then \(I\left (t,P\right ) = I\left (t\right ) I\left (P\right )\) and \(\intop _{t_{1}}^{t_{1}+\Delta T} I\left (t\right ) dt = \Lambda \left (t_{1},t_{1}+\Delta T\right )\) is the expected number of events in this time interval that can be modelled by the Omori law or by the stretched exponential function. The integral \(\intop _{P_{1}}^{P_{2}} I\left (P\right ) dP = \Pr \left (P_{1},P_{2}\right )\) gives the probability of having an event in this potency range, and \(I\left (P\right )\) is the probability density function of the underlying size distribution, e.g. the upper truncated power law. Taking \(N = 0\) gives \(\Pr \left [N=0,\left (t_{1},t_{1}+\Delta T\right );\left (P_{1},P_{2}\right )\right ] = \exp \left [-\Lambda \left (t_{1},t_{1}+\Delta T\right )\Pr \left (P_{1},P_{2}\right )\right ]\), which is the probability that the first event in this potency range will occur after time \(t_{1}+\Delta T\). Therefore, the interval probability that at least one event will occur in potency range \(\left (P_{1},P_{2}\right )\) in this time interval is \(\Pr \left [N\geq 1;\left (t_{1},t_{1}+\Delta T\right );\left (P_{1},P_{2}\right )\right ] = 1 - \exp \left [-N\left (t_{1},t_{1}+\Delta T\right )\Pr \left (P_{1},P_{2}\right )\right ]\), which for \(P_{1} = P\) and \(P_{2} = P_{max}\) can be written as

The probability \(\Pr \left (\geq P\right )\) can be given by the UT power law \(\Pr \left (\geq P\right ) = \left (P^{-\beta }-P_{max}^{-\beta }\right ) / \left (P_{min}^{-\beta }-P_{max}^{-\beta }\right )\), where \(\beta \) and \(P_{max}\) are parameters to be derived from data over the period \(\Delta t\) before the step loading.

Assuming the simple Omori relaxation function, i,e. \(c=0\), the number of events after the step loading is \(N\left (t_{1},t_{1}+\Delta T\right ) = k \left (t_{2}^{1-p}-t_{1}^{1-p}\right ) /\left (1-p\right )\) for \(p\neq 1\) and \(N\left (t_{1},t_{1}+\Delta T\right ) = \ln \left (t_{2}/t_{1}\right )\) for \(p=1\). For the stretched exponential relaxation function, the number of events after the step loading is given by Eq. (5.31). The \(N\left (t_{0},T\right )\) in Eq. (5.31) can be estimated in an on-line scheme as the aftershock data is being acquired, \(N\left (t_{0},T;t_{1}\right ) = N\left (t_{0},t_{1}\right ) / \left (1-\exp \left [-\left ((t_{1}-t_{0})/\tau \right )^{q}\right ]\right )\), where \(N\left (t_{0},t_{1}\right )\) is the observed number of events at time of forecast, \(t_{1} > t_{0}\), and \(\tau \) and q are estimated at the time \(t_{1}\) by the maximum likelihood method described above. This process is then repeated as the aftershock sequence progresses to firm up these estimates.

The total number of events to be produced by the sequence, \(N\left (t_{0},T\right )\), can also be estimated by the productivity law of aftershocks, \(N\left (t_{0},T\right ) = \alpha _{m} P_{m}^{\beta }\), where \(P_{m}\) is the known potency of the main shock, \(\alpha _{m}\) is the scaling factor to be calibrated, and \(\beta \) is the power law exponent.

Parameters k and p for the Omori relation and \(\tau \), q for the stretched exponential can be inverted on-line for the current ongoing sequence by the maximum likelihood method. They can also be taken a priori as average values from previous sequences in the area or, taking a Bayesian-like approach, a weighted average of the a priori and the on-line values (Reasenberg & Jones, 1989).

If after the step loading rock extraction is suspended, the relaxation process would end naturally at time \(t\left [N\left (t_{0},T\right )\right ]\). If, however, production continues or is restarted long before the relaxation process is completed, the stresses may accumulate and hazard may increase. In principle, mining could restart when seismic hazard dips below a predefined acceptable level and then stays there for a given period of time. This exclusion time would scale positively with seismic hazard before loading and the intensity of sudden loading which scales positively with the stress level in the area.

Here we will discuss the aftershock activity after the main shock (MS) with \(\log P=2.61\) that occurred on 07 July 2013, see description of the data up to the MS in Sect. 5.5. Figure 5.27 left shows the cumulative number of events with \(\log P\geq -2.0\) from 8 May to 23 July, where size of the event scales with the source volume and the colour indicates the distance of the event from the MS. The three vertical lines in red, blue, and green mark the times of the three largest events. The rate of events before the MS was steady with no increase in the rate or acceleration of activity before the MS. The first aftershock with \(\log P\geq -2.0\) was recorded 3.95, and the second 6.4 seconds after the MS. Within the first 16 hours after the main shock, there were 165 events that give the activity rate just over 10 events/hour, and within the first 171 hours, the rate of activity dropped to 1.4 events/hour.

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Figure 5.27 right shows the distances from the MS during the same time with colour scaling with the logarithm of apparent stress, \(\log \sigma _{A} = \log \left (E/P\right )\). The bulk of the immediate aftershocks spread 670 metres from the MS. The largest of the immediate aftershock with \(\log P=0.15\) occurred 1.5 minutes after the MS and located 235 metres away. The largest event in the aftershock sequence with \(\log P=1.13\) occurred 162 hours after the MS and located 670 metres away.

Figure 5.28 left shows a 15 event moving window of the coefficient of variation of inter-event times, \(C_{v}\left (t\right )\), seismic diffusivity \(d_{s} = \left \langle d^{2}\right \rangle /\left \langle t\right \rangle \), and the seismicity rate change, where the colour scales with the mean distance of events in each moving window to the MS, that vary from 165 m in red to 391 m in blue. For details, see Sect. 5.6.2 in this chapter. The high \(C_{v}\left (t\right )\) indicates clustering of events in time. Seismic diffusivity, \(\left \langle d^{2}\right \rangle /\left \langle t\right \rangle \), increases due to an increase in the rate of seismic activity, i.e. decrease in the mean time between events, \(\left \langle t\right \rangle \), and/or increase in the spatial extent of events, i.e. increase in the mean distance from the MS or the mean inter-event distance \(\left \langle d^{2}\right \rangle \). Usually, \(d_{s}\) increases when a given system is overloaded and needs to relax. The seismicity rate change gives the probability that there is an increase or a decrease in the rate of events. Probability less than 0.5 means that there was a decrease in the event rate and above 0.5 that there was an increase. Parameter k specifies the magnitude of the change.

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The \(C_{v}\left (t\right )\) before the MS ranged between 0.79 and 1.4, and it spiked to 3.81 at the time of the MS indicating high clustering, but dropped to below 1.27 within 8 hours. The reference diffusivity started to increase 70 hours before MS and jumped to 3661.32 just after that and it took 63 hours to drop to pre-MS level.

Figure 5.28 right shows the probability that the rate of seismicity increased by at least 2 times with respect to the background level before the MS. The seismicity rate change jumped twice before the MS but reached a maximum 0.61 which is low. It spiked to 1.0 at the time of MS and stayed there for 17 hours dropping below the significant 0.75 level after 25 hours.

Figure 5.29 left shows the diffusivity vs. time during 147.7 hours before and 353.8 after an event with \(\log P=2.61\) shown here as a red circle. In this figure, colour scales with distance to the MS, and the size of the event here represents the radius of the source volume taken as a sphere, \(V=P/\Delta \epsilon \), where \(\Delta \epsilon \) is the assumed strain change at the source.

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There was a steady rate of seismic activity of \(0.68\) events per hour before and a typical burst and a power law decay of aftershock activity after the main shock. Figure 5.29 right shows \(\log \left \langle d\left (t\right )^{2}\right \rangle \) vs. \(\log t\) scaling for 100 events before and for the first 240 events after the main shock. It shows a normal diffusion before with \(\gamma =1.04\) and a sub-diffusive process with \(\gamma =0.91\) after the MS.

Figure 5.30 first row shows the cumulative number of events 150 hours before and 200 after the MS with the stretched exponential (SE) fit to the first 16 and 176 hours of aftershock activity in red. Figure 5.30 second row shows the same but with the simple Omori fit (with \(c=0\)) in blue. In both cases, the forecast at the time \(t_{f}\) for the next 48 hours is shown in green. All statistics quoted on the right of the plots relate to the aftershocks at the time of forecast.

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The SE fitted data well and provided a reasonable forecast at 16 and 171 hours after the MS. The SO fitted data at 16 hours well but failed in forecast and at 171 hours did not fit data well but provided a reasonable forecast. The coefficient of variation of time differences between events, \(C_{v}\left (t\right )\), increased from 1.75 in the first fit to 2.47 in the second fit due to the additional 76 intermittent events, but it shows that in both cases there is a temporal clustering and interdependence of events.

Figure 5.30 bottom row shows the interval probabilities, \(\Pr \!\left [\!\geq \!\log \! P;\!\!\left (t_{f},t_{f}{+}\Delta T\right )\!\right ]\), of having at least one event not smaller than a given \(\log P\) within \(\Delta T = 48\) hours after \(t_{f}\), calculated assuming SE, shown in red and SO in blue. The Poissonian probabilities during any 48 hours before the MS are shown in black. In general the SE provided better fit to data.

At 64 hours after the MS, the SE probabilities are considerably higher than before the MS, indicating that the relaxation process is still in progress and seismic hazard is high, and therefore, people should not enter the area affected by aftershock activity. For example \(\Pr \left [\log P\geq 0.0;64h\right ]\) is 0.4, while before the MS it was 0.265 and \(\Pr \left [\log P\geq 1.0;64h\right ]\) is 0.055, and before the MS it was 0.034. At 224 hours after the MS, the relaxation process is mainly completed and seismic hazard dropped. For example, \(\Pr \left [\log P\geq 0.0;240h\right ]\) is 0.079, while before the MS it was 0.265, and \(\Pr \left [\log P\geq 1.0;240h\right ]\) is 0.009, and before the MS it was 0.034.

Note that after the main shock mining was suspended and this is reflected in the interval probabilities, therefore here we compare seismic hazard before the MS driven by production with the interval probabilities after the MS with no production. Obviously, when production will resume, seismic hazard will increase, and therefore, the re-entry into the aftershock area needs to be done with caution.

This example describes aftershock sequence following a large \(\log P=5.47 \left (m_{HK}=4.6\right )\) and \(\log E=10.6\), mining triggered reverse slip event driven by tectonic sub-horizontal stress. The event was initiated off the bottom edge of a small mine and propagated over one kilometre away from the mine along sub-horizontal geological structure. Waveforms from the mine seismic system, from the close by mines, and regional stations were used to evaluate source parameters and mechanism of the main shock (MS) and aftershocks.

Figure 5.31 left shows three-component velocity waveforms recorded by 14 Hz sensors at distance 280 metres, Fig. 5.31 middle shows waveforms recorded by 4.5 Hz sensors at distance 4384 metres, and Fig. 5.31 right shows 200 seconds of waveforms recorded by broadband sensor at a distance of 92 km from the mine.

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The moment tensor solution by the USGS, as well as location and mechanisms of aftershocks, indicates that the event represents a rupture along a plane with strike 240\(^{\circ }\) and dip 24\(^{\circ }\), in the mine’s coordinate system, which extends to the South-South-West of the mine. Analysis of waveforms indicates the two main episodes of rupture, the main shock and 0.7 seconds later the sub-event, that could well be treated as the first aftershock, see Fig. 5.31 left. This sub-event was most likely associated with the rupture of the fault segment about 950 m to the South-South-West from the primary main shock.

The spherical equivalent of the MS is shown in Fig. 5.32 by a large light grey circle that scales with the radius of the source volume with strain change \(\Delta \epsilon =\)10\(^{-4}\), \(r=\left [3P/\left (4\pi \Delta \epsilon \right )\right ]^{1/3}=856\) metres, where P is potency. However, the real shape of this event was rather closer to a flat ellipsoid span over the ellipse shown in Fig. 5.32 in blue. The event triggered 3670 aftershocks with \(\log P\geq -3.0\) within 156 days which are shown in Fig. 5.33. Most of the aftershocks located away from the mine are associated with the sub-horizontal planar structure approximated by ellipse. The largest aftershock with \(\log P=2.66\), shown in Fig. 5.32 in red, occurred within the mine 54 seconds after and 581 metres from the MS. The second largest aftershocks with \(\log P=2.54\) shown in Fig. 5.32 in orange occurred outside the mine 38 days after and 258 metres from the MS. The third largest aftershocks, with \(\log P=1.37\) occurred within the mine 15.6 hours after and 578 metres from the MS.

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Figure 5.33 left shows the cumulative number of events with \(\log P\geq -3.0\) during 1338.7 hours before and 3730.5 after the MS shown here as the large red circle. Colour here scales with distance to the MS, and the size of the event here represents the radius of the source volume taken as a sphere, \(V=P/\Delta \epsilon \), where \(\Delta \epsilon \) is the assumed strain change at the source, in this case \(10^{-2}\). There was a steady rate of seismic activity, \(\lambda _{B}=N\left (\log P\geq -3.0\right ) / \Delta t_{B} = 0.484\) per hour before the MS. Figure 5.33 right shows distances of seismic events to the MS over the same period of time where colour scales with the logarithm of apparent stress, \(\log \sigma _{A} = \log \left (E/P\right )\). The spatial distribution of events before the MS is mostly concentrated at the mine, at distances of 300 to 500 metres from the future MS, with only few events at distances over 500 metres. From the very beginning of the aftershock sequence, the spatial distribution of aftershocks extended to over 2000 metres from the MS. Approximately 2900 hours after the MS, one can see seismic activity at 200 metres from the MS associated with the rehab of mining operations.

Figure 5.34 top row shows the cumulative number of events with the stretched exponential (SE) fit at the time of forecast \(t_{f}=756\) and \(t_{f}=3057\) hours after the MS, shown in red, and its extrapolation into the next 48 hours in green. The simple Omori (SO), i.e. with \(c=0\), was also tested for these data sets and gave very similar results. All statistics quoted on the right of the plots relate to the aftershocks at the time of forecast. The first aftershock with \(\log P\geq -3.0\) was recorded 8 seconds after the MS.

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At \(t_{f}=756\) hours, the rate of aftershocks was \(\lambda _{A} = 2.84\), the background activity rate \(\lambda _{B}=0.48\). The coefficient of variation of time differences between events \(C_{v}\left (t\right ) = 1.6\), which indicates a degree of temporal clustering and interdependence of events. At that time, the mean distance of aftershocks to the MS was 857 metres, extending well beyond the boundary of the mine, which suggests the influence of local tectonics influenced by years of mining in the area.

Figure 5.34 bottom row shows the interval probabilities, \(\Pr \left [\geq \log P;\right .\)\(\left .\left (t_{f},t_{f}+\Delta T\right )\right ]\), of having at least one event not smaller than a given \(\log P\) within \(\Delta T = 48\) hours after \(t_{f}\), calculated assuming SE, shown in red, and SO in blue, which are practically the same.

The Poissonian probabilities during any 48 hours before the MS are shown in dark grey which, at \(t_{f}=756\), are considerably lower which implies that the relaxation process is still in progress and seismic hazard is high. For example, the SE probability, \(\Pr \left (\log P\geq 1.0;t_{f}+\Delta T=48\text{h}\right ) = 0.054\) as opposed to \(0.022\) for any 48 hours before the MS, which is 2.45 times higher. Note that the SE relaxation time, \(\tau \), at this stage is large, but it will get smaller as the relaxation process continues.

The forecast at \(t_{f}=3057\) hours gives a lower hourly rate of aftershocks of 1.13, and the activity rate over the last 1000 aftershocks is 0.51, which is very close to the premain shock background. The coefficient of variation of time differences between events increased to \(C_{v}\left (t\right ) = 2.29\) due to the intermittency of the recent aftershocks, which indicates a strong degree temporal clustering and interdependence of events. The spatial distribution of aftershocks extended even further beyond the boundary of the mine, which confirms the previous premise of tectonic or other mining influence. Figure 5.34 bottom right shows the interval probabilities, \(\Pr \left [\geq \log P;\left (t_{f},t_{f}+\Delta T\right )\right ]\), of having a seismic event not smaller than a given \(\log P\) within \(\Delta T = 48\) hours after \(t_{f}=3057\) hours, calculated assuming SE, shown in red, and SO in blue, which are practically the same and equal to the probabilities at any 48 hours before the MS.

At 756 hours, after the MS, the SE probabilities are considerably higher than before the MS, indicating that the relaxation process is still in progress and seismic hazard is high. For example, \(\Pr \left [\log P\geq 0.0;804h\right ]\) is 0.3, while before the MS it was 0.14 and \(\Pr \left [\log P\geq 1.0;64h\right ]\) is 0.053, and before the MS it was 0.022.

At 3105 hours after the MS, the relaxation process progressed to the stage that seismic hazard is at the same level it was before the MS.

Note that after the main shock mining was suspended, and this is reflected in the interval probabilities coming back to the pre-shock level rather quickly, and therefore, here we compare seismic hazard before the MS driven by external loading and production with the interval probabilities after the MS with no production. Obviously, when production will resume, seismic hazard will increase, and therefore the re-entry into the aftershock area needs to be done with caution.

McGarr (1976b) showed that if after extracting a volume of rock \(V_{m}\), the altered stress and strain field can readjust to an equilibrium state through seismic movements only, the sum of seismic potency released within a given period of time would be proportional to the excavation closure, and in the long term \(\sum P = \sum V_{m}\). Then, assuming the seismic deformation will follow the open-ended power law with parameters \(\alpha \), \(\beta \), one can estimate the maximum possible event size \(P_{max}\) in the seismic sequence, \(P_{max} = \left [\left (1-\beta \right )V_{m}/\left (\alpha \beta \right )\right ]^{1/\left (1-\beta \right )}\). Similar equations were derived by Wyss (1973), Smith (1976), McGarr (1976a), and Molnar (1979) to estimate the expected maximum magnitude of earthquakes, and by McGarr (1984) for mines. See also section Balance of the Effective Volume Mined and \(P_{max}\)in Chap. 4.

Mendecki (2001) studied seismic rock mass response to extraction blasting in a long wall with four 30 m panels at a depth of 3000 m at Tau-Tona mine. The area was covered by a high-resolution microseismic network of five three-component accelerometers that provided enough data to gain insight into the excitation and relaxation processes associated with a sudden transient closure after production blasts. During five days of observations, 15 panels were blasted—minimum 3, maximum 6 panels per day, and no blasts on Sunday. In each stope, there were two rows of 0.9 m long blasting holes separated by 0.5 m vertically and horizontally. They were blasted in pairs top and bottom, with about 200 ms delay. Thus it took approximately 12 to 15 seconds to blast a panel, with face advance by about 0.8 m. A few thousand events were recorded over the five days of observations, but only 1086 with well-developed P- and S-wave signatures were processed and used in analysis. The larger seismic events occurred during Day1 and Day3 after the production blast of at least three contiguous panels. It was concluded that blasting a number of contiguous panels degrades stiffness of the surrounding rock more and for a longer period of time than if the same number of panels were blasted non-contiguously. The loss of stiffness makes the system more vulnerable to larger events.

In another study, Mendecki (2005) analysed the seismic rock mass response to production by blasting at Mponeng mine. The rate of mining \(V_{m}/\Delta t\) ranged from 3 to 33 m\(^{3}\)/day and did not correlate with seismic hazard. Although the highest rate of mining was where non-contiguous panels were blasted, the seismic hazard remained low and larger events occurred during the periods of contiguous blasting.

Martinsson and Torrman (2020) developed a Bayesian hierarchical model that captured the dynamic relationships between seismic activity and production rate, depth, and size of the orebodies. Testing on data from the LKAB Malmberget mine, they inferred that the seismic half-life, defined as the amount of time required for the activity to fall halfway to its steady state value, ranges from weeks up to 2 months for the cases considered. They also found that the effect of the weekly production on the induced seismicity depends exponentially on the average production size in the orebody. The seismic exposure term reveals a dependency on depth, and, for cases considered, an increase in depth by 100 m doubles the seismic activity.

de Beer (2022) tested correlations between production rates, taking into account ring and drawbell blasting and mucking, and seismic potency response in a block cave mine. He found that the rate of mucking is a long term driver of seismic potency release both above and below the undercut. Birch et al. (2024) conducted a similar analysis, fitting a simplified version of a generalised linear model to tons bogged and blasted in weekly intervals over two years at Oyu Tolgoi block cave mine. They found that during this period, which included the start of drawbell development and production bogging, weekly tonnes bogged had much stronger influence on seismicity than blasting.

Most of these papers focus on the seismic rock mass response to production and production rate. Here, we explore the influence of the proximity of group blasting on the intensity of seismic response as measured by the number of larger, potentially damaging seismic events during and after blasting.

The intensity of the seismic rock mass response to a single extraction, undercut, or preconditioning blast scales positively with the stress level at the blasting site and the surrounding area and with the volume of rock extracted. In the case of multiple blasts, it also depends on the number and the proximity of these blasts, i.e. the time differences and the distances between them. For a given stress condition and volumes of rock blasted, more clustered blasts result in a more intense seismic response.

Let us denote the volume of the first blast in a sequence of two blasts by \(V_{b1}\) and ask what should be the time delay, \(\Delta t_{21}\), and the space separation, \(\Delta d_{21}\), to the second blast to avoid superposition of seismic activity. First, let us define the scaled volume of the first blast as

where the numerator is the volume of rock blasted during the first blast, \(V_{b1}\), and the dimensionless denominator quantifies the proximity in time and in space between the first and the subsequent blast. The proximity in time is quantified by the ratio \(\Delta t_{21}/t_{\alpha 1}\), where \(t_{\alpha 1}\) is the characteristic seismic relaxation time, and the proximity in space is measured by the ratio \(\Delta d/d_{\alpha 1}\), where \(d_{\alpha 1}\) is the characteristic size, or the radius, of the seismic relaxation zone. Both \(t_{\alpha 1}\) and \(d_{\alpha 1}\) are functions of \(V_{b1}\). The best proxies for \(t_{\alpha 1}\) and \(d_{\alpha 1}\) are the calibrated exclusion or re-entry time, \(t_{r}\), and the characteristic size of the exclusion zone, \(d_{e}\). Alternatively, they can be scaled appropriately. For a given \(\Delta t_{21}\) and \(\Delta d_{21}\), the larger \(t_{\alpha 1}\) and/or \(d_{\alpha 1}\), the larger \(V_{sb1}\). Since the seismic rock mass response to blasting scales positively with the stress level, so would \(t_{\alpha 1}\) and \(d_{\alpha 1}\), and therefore, Eq. (5.46) indirectly caters for the stress level.

If the time difference between the first and the second blast \(\Delta t_{21}\) is equal to \(t_{\alpha 1}\) and the distance between them, \(\Delta d_{21}\), to \(d_{\alpha 1}\), then

Now, we can define the proximity index of the first blast to the second blast as the ratio,

If \(\Delta t_{21} = t_{\alpha 1}\) and \(\Delta d_{21} = d_{\alpha 1}\), then \(I_{pb1}=1\). For a given \(t_{\alpha 1}\) and \(d_{\alpha 1}\), the index \(I_{pb1}\) measures the deviation from the desired proximity of the second blast from the first one in time and/or in space. The case \(I_{pb1}\leq 1\) means the desirable outcome or better, and \(I_{pb1}>1\) means that, for a given \(V_{b1}\), the second blast may be too close in space and/or in time. Since \(t_{\alpha 1}\) and \(d_{\alpha 1}\) scale positively with \(V_{b1}\), the larger this volume the longer the relaxation time and the larger the relaxation zone. In principle both, \(t_{\alpha }\) and \(d_{\alpha }\) would scale with the cube root of the volume of rock blasted.

Equation (5.48) allows testing whether the preferred time difference and distance between two planned blasts are below the acceptable level of \(I_{pb}\) that produces a manageable seismic response.

In practice, mines frequently group more than two blasts to optimise the required exclusion time. The most practical way to test the proximity is to apply the above formulas to each two consecutive blasts. However, while all pairs of consecutive blasts may be reasonably separated in time and space, there may be some non-consecutive blasts missed by this scheme. For example, the case of three blasts carried out within a small \(\Delta t\) where the first and the second blasts are separated enough in space but the third one is very close in space to the first one. Here, if measured in relation to the second blast, the third one would give reasonable space separation resulting in low \(V_{sb}\) and \(I_{pb}\), while it would not be the case if the first blast would be measured in relation to the third one.

To detect such cases, one can consider all combinations of two blasts in a given blasting sequence. For \(n_{b}\) blasts, there are C\(_{2}^{n_{b}} = n\left (n-1\right )/2\) combinations of two blasts, and for each, we can calculate \(I_{pb}\). Then, we can sort \(I_{pb}\) from the largest to the smallest and compare with the sorted values of consecutive blasts to see if these two lists are the same.

The example below is based on the best blasting data available to the author for publication.