Rock Mechanics and Rock Engineering
https://doi.org/10.1007/s00603-018-1430-4
ORIGINAL PAPER
The Spatial Assessment of the Current Seismic Hazard State for Hard
Rock Underground Mines
Johan Wesseloo1
Received: 5 June 2016 / Accepted: 2 February 2018
© Springer-Verlag GmbH Austria, part of Springer Nature 2018
Abstract
Mining-induced seismic hazard assessment is an important component in the management of safety and financial risk in
mines. As the seismic hazard is a response to the mining activity, it is non-stationary and variable both in space and time.
This paper presents an approach for implementing a probabilistic seismic hazard assessment to assess the current hazard
state of a mine. Each of the components of the probabilistic seismic hazard assessment is considered within the context of
hard rock underground mines. The focus of this paper is the assessment of the in-mine hazard distribution and does not consider the hazard to nearby public or structures. A rating system and methodologies to present hazard maps, for the purpose
of communicating to different stakeholders in the mine, i.e. mine managers, technical personnel and the work force, are
developed. The approach allows one to update the assessment with relative ease and within short time periods as new data
become available, enabling the monitoring of the spatial and temporal change in the seismic hazard.
Keywords Induced seismicity · Seismic hazard · Seismic hazard assessment · Spatial assessment · Hard rock underground
mines
1 Introduction
The management of seismic hazard in any seismically active
mine is an extremely important task due to the safety risks,
as well as the direct and indirect financial risks. The level
of mine-induced seismicity is expected to increase as mines
reach greater depths. In Australia, the deepest mines are
currently operating at depths of about 1600 m. In Canada,
depths of about 3000 m are reached, whilst in South Africa
operating levels are now nearing 4000 m.
Wesseloo (2013) argues that, due to the stochastic nature
of mining-induced seismic events, seismic hazard should
be assessed probabilistically. Probabilistic evaluations also
allow one to quantitatively integrate hazard over space and
over time, which is not possible with, for example, qualitative seismic hazard evaluation (e.g. Hudyma and Potvin
2004, 2010; Kaiser et al. 2005; McGaughey et al. 2007).
Probabilistic seismic hazard assessment in crustal seismology and earthquake engineering has a long history, and
* Johan Wesseloo
johan.wesseloo@uwa.edu.au
1
Australian Centre for Geomechanics, The University
of Western Australia, Crawley, Australia
the principles and general approaches are well established
and understood. Many similarities exist between crustal seismicity and mining-induced seismicity, and many concepts
and theories can be borrowed from crustal seismology, as is
evident from Gibowicz and Kijko (1994). Some of the components of probabilistic seismic hazard assessment applied
to mines are discussed in the Canadian Rockburst Research
Handbook (Mining Research Directorate 1996).
In contrast to crustal seismicity, mining-induced seismicity is a consequence of human activity and is directly
related to the mining activity (Fig. 1). Mining engineers can
also influence the seismic hazard with, for example, mining method, mining sequence and preconditioning. Another
difference between crustal seismicity and mining-induced
seismicity is the fact that the mining activity changes the
geotechnical conditions which give rise to the seismicity.
Over time, the mining-induced stress concentration and rock
competency change, resulting in a complex spatio-temporal
seismic hazard regime. For example, a highly stressed volume of rock may, in a later stage of mining, be completely
de-stressed, changing from a volume with high seismic hazard to a volume with insignificant hazard.
The temporal change in seismicity must also be considered in studies of seismicity induced by human activity
13
Vol.:(0123456789)
J. Wesseloo
Fig. 1 Drop in the rates of seismic activity during 2010 Christmas
break in a gold mine in South Africa (Mendecki and Lotter 2011)
that are not related to mining. Convertito et al. (2012), for
example, propose a technique for time-dependent probabilistic seismic hazard analysis to be used in geothermal fields
to monitor the effects of ongoing field operations. Lasocki
(2005) recognises the importance of accounting for the transient nature of mining-induced seismicity and suggests that
probabilistic characteristics of future seismic source zones
could be predicted based on the characteristics of historical
source zones. It should be noted that Lasocki (2005) considered a scenario where the hazard assessment is performed at
a scale and distance greater than the seismic source zones.
For such a scenario, the overall seismic hazard is a result
of a superimposition of the contributions of the individual
sources, and reasonable results may be achieved. In contrast, for the assessment of the in-mine seismic hazard, sufficiently reliable prediction of the probabilistic characteristics
of future seismic sources is currently not possible for many
mining scenarios.
This paper presents a method for the probabilistic assessment of the in-mine spatial distribution of the current seismic hazard in hard rock mines and considers different ways
to present and communicate the hazard.
2 The Current Seismic Hazard State in Mines
The hazard assessment method proposed here entails the
quantification of the current hazard state. This is conceptually different from other hazard assessment approaches and
therefore requires further clarification.
The word “current” refers to the fact that this hazard
assessment is not an assessment of the hazard for a specific
future time period, but an assessment of the hazard to which
the mine is currently subjected. “Hazard state” refers to the
fact that the hazard assessment is a quantification of the
overall spatial variation of higher and lower hazard throughout the mine and does not concern itself with short-term
13
fluctuations as opposed to, for example, the approach of
Rebuli and van Aswegen (2013).
For our purposes, the current hazard state can be defined
as follows:
The current seismic hazard state is the event size that
can be expected to occur with a confidence of P, with a
continuation of recent seismic conditions at the timescale
under consideration.
It should be noted that the phrase “with a continuation
of recent seismic conditions at the timescale under consideration” is not an assumption. The transient nature of
mining-induced seismicity makes the addition of the phrase
a necessary part of the definition. Within the mining environment, we can safely assume that current conditions will
not continue into the future for any substantial period. This
does imply, though, that the assessment is valid for as long
as the future seismic conditions, over the timescale of interest, remain reasonably constant.
3 Calculating the Current Seismic Hazard
State
A summary of the different components involved in the
probabilistic seismic hazard assessment in crustal seismology is given by Kijko (2011):
• Parameterisation of seismic source zones.
• Temporal and magnitude distribution of seismicity for
each source zone.
• Calculation of the ground motion prediction equation and
its uncertainty.
• Integration of uncertainty in earthquake location, mag-
nitude and strong ground motion relationship to obtain
the probability of exceeding a specified ground motion
parameter at least once.
The definition of seismic source zones is somewhat subjective; there are some examples where the interpretation of
the seismic response and hazard has been distorted because
of an incorrect definition of the zones. Wesseloo (2013)
mentions an occasion where events that occurred near a
major fault in the hanging wall, and events near the same
fault’s inferred location in the footwall, were assumed to be
from a single source. The rock mass responses in these two
areas were markedly different, and the hazard in the footwall
was subsequently overestimated.
The identification of seismic source zones is important,
and a careful evaluation of possible zones will enhance the
seismic hazard assessment. The approach taken here is to
develop a robust method that will require minimal user input
and be insensitive to user input. A grid-based, rather than a
zone-based, approach is followed.
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
Two important components to the hazard assessment,
namely, the spatial evaluation of the frequency–magnitude
distribution and event rate density, are quantified using the
grid-based approach. The variation of the b-value is evaluated using the method discussed in Wesseloo (2014), whilst
the recent historical seismic density rate is obtained using
the method discussed in Wesseloo et al. (2014).
The last component mentioned by Kijko (2011) is the
integration of the strong ground motion relationships and
their uncertainties into an overall estimate of the probability
of exceeding a specified strong ground motion.
3.1 The Upper Limit of Magnitude, MUL
3.1.1 The Meaning of MUL in Mining
The term “Mmax” is used to define the region-characteristic
maximum possible event magnitude or as the upper limit of
event magnitude for a given region (Kijko and Singh 2011)
and defines the upper asymptote of the frequency–magnitude
distribution. In the mining industry, the term “Mmax” is also
used for three other concepts, namely, the largest recorded
event in a dataset, the fitted frequency–magnitude relationship at N = 1 and the distribution of the largest expected
magnitude. To avoid confusion, we will refer to this value
as the upper limit of magnitude, MUL.
In crustal seismology, the value of M UL is generally
assumed to be constant for a particular seismic source zone.
However, in the mining environment the value of MUL is
not a constant and is influenced by a number of factors, for
example, rock mass conditions, mining-induced stress state,
the mining sequence and mining layout. In addition, MUL
is expected to increase with the extraction ratio (Mendecki
2012). For these reasons, we define MUL as the upper limit
of the next largest event.
The influence of each of the mentioned factors on MUL
is currently unknown, resulting in an unknown reliability
in the assessment of MUL. As MUL forms an upper limit of
the frequency–magnitude distribution, the probability of
exceeding MUL is, per definition, zero. When assessing the
probability of exceeding a specified magnitude P[M > ML],
under-estimation of MUL leads to larger errors than overestimation of MUL by the same amount (“Appendix 1”), i.e.:
]|
]|
| [
| [
|𝜀P M > ML|MUL − Δ𝜀 | = 𝜀P− > 𝜀P+ = |𝜀P M > ML|MUL + Δ𝜀 |
|
|
|
|
(1)
where MUL = the upper limit of magnitude; M = magnitude;
ML = magnitude under consideration; εP = the error in
probability of exceeding ML; Δε = the error in estimation of
MUL; εP+, εP−= the error due to overestimation and underestimation of MUL, respectively.
The value of εP+ is always conservative, whilst εP− is
always optimistic. For these reasons, a conservative estimate
of MUL is necessary.
3.1.2 Statistical Methods for the Estimation of MUL
Several statistical methods for the estimation of MUL from
seismic records in crustal seismology are discussed in detail
by Kijko and Singh (2011). These include an approximate
and exact solution to the Kijko–Sellevoll (K–S) method, and
Bayesian adaptations to the Tate–Pisarenko (T–P) and K–S
methods. The exact solution to the K–S method and the two
Bayesian methods is computationally intensive and produces
results similar to the approximate and non-Bayesian solutions and is therefore excluded from routine calculations.
Kijko and Singh (2011) also propose formulations for the
assessment of the standard deviation of MUL as a function of
the resolution of the magnitude for which a value of 0.1 was
used. Lasocki and Urban (2011) quantified the bias and variance of the K–S method’s estimates for MUL which we use
to apply bias and variance corrections to the K–S method
estimates.
To ensure conservatism in the estimation of MUL, the
expected value plus the standard deviation is used as the
estimate of MUL. The maximum value obtained from the
following, discussed in detail by Kijko and Singh (2011),
is used: Tate–Pisarenko (T–P), Kijko–Sellevoll, order statistics, Robson–Whitlock, Robson–Whitlock–Cooke and
Cooke 1980.
The statistical methods assume that the magnitude values are reliably recorded and that the shape of the inverse
cumulative distribution of magnitude is a result of the statistical distribution only. This condition is, however, not
always met as many in-mine seismic systems under-record
the seismic moment of larger events due to sensor frequency
range limitations and, as a result, exhibit a nonlinear (on
the log-linear scale) frequency–magnitude distribution, as
shown in Fig. 2a. Figure 2a shows the frequency–magnitude
distribution (inverse cumulative distribution of magnitude)
of recorded data from a mine network with only 50 Hz sensors. The downward curvature of the distribution is not a
result of the statistical distribution of magnitude but of the
under-recording of the moment by the sensors. Also shown
in the figure are two theoretical lines—the straight Gutenberg–Richter relationship assumed as the true distribution
of magnitude, and a theoretical assessment of the effect of
under-recording using analytical formulations (Boore 1986;
Di Bona and Rovelli 1988; Mendecki 2013b; Morkel and
Wesseloo 2017). The effect of different lower frequency limits of the sensors is illustrated in Fig. 2b.
For databases subjected to under-recording, optimistic values of MUL will result from the use of the statistical
methods.
13
J. Wesseloo
8
Magnitude of largest recorded event
7
Theorecal
“true” distribuon
6
5
4
3
2
1
0
range applicable
to mining
-1
Theorecal “recorded”
distribuon (thick line), and,
distribuon of recoded data
(points)
-2
0
1
2
Log(L);
3
4
5
L = Largest dimension of causive acvity
McGarr et al. 2002
Australian and Canadian Mines
Fig. 3 Maximum magnitude (Richter scale) and maximum dimension
of causative activity after McGarr et al. (2002) with the addition of
four mine sites from Australia and Canada
(a)
mines were obtained from government seismological institutions. The largest dimension of the mine at the time of the
occurrence of the events was obtained from mine survey data.
Although the mine site data are limited, they fit the general trend for induced seismicity. Estimating an upper bound
to this data provides an empirical relation to estimate MUL in
the absence of reliably recorded large event data. The relation shown in Fig. 3 can be written as follows:
Theorecal
“true” distribuon
(2)
where MUL = the upper limit of magnitude; L = the largest
spatial dimension of the human influence.
It should be noted that the largest dimensions of mines L
may vary between, say 500 and 5000 m (log(L) of 2.7 to 3.7)
with associated values of MUL ranging between about 4 and
6. The database includes only two points below log(L) of 3,
both of which have relatively low magnitudes. This highlights the importance for expanding the empirical database
used to populate Fig. 3.
MUL = 1.85 ⋅ log (L) − 0.5
100 Hz
Theorecal
“recorded”
distribuons
50
30
15
4.5
1
(b)
Fig. 2 Effect of sensor frequency limits on the frequency–magnitude
distribution (inverse cumulative distribution of moment magnitude)
of recorded seismic datasets (Morkel and Wesseloo 2017)
3.1.3 Empirical Methods for the Estimation of MUL
McGarr et al. (2002) compiled a dataset of large induced earthquakes with the largest associated dimension of human influence for each of them. Their data are presented in Fig. 3 with
the addition of four data points from hard rock metalliferous
mines in Australia and Canada. The event magnitudes for these
13
3.2 Frequency–Magnitude Distribution
Several formulations of the frequency–magnitude distribution which include an upper limit value for magnitude
exist. Utsu (1999) provides a review of the relationships and
expresses these in the following generalised form:
(3)
where 𝜙(M) is an increasing function of M. The well-known
truncated Gutenberg–Richter relationship frequently used in
crustal seismology can be written in this form with:
log(n) = A − b ⋅ M − 𝜙(M)
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
𝜙(M) =
0
for M ≤ MUL
A − b ⋅ M for M > MUL
(4)
This paper limits itself to the use of the truncated Gutenberg–Richter relationship as this is the formulation most
commonly used in mining seismology (e.g. Gibowicz and
Kijko 1994; Kijko and Funk 1994; Malovichko 2017; Mendecki 2012, 2013a; Mendecki 2008). The hazard assessment
framework discussed here is not dependent on the particular formulation of the frequency–magnitude distribution and
can easily be extended to include any other formulation.
When evaluating the frequency–magnitude distribution for the purpose of hazard assessment, it is important
to ensure that a clean database is used. The characteristics
of blasts are different from those of seismic events, and
several discrimination techniques have been proposed for
the removal of blasts from mine databases (e.g. Dong et al.
2016; Malovichko 2012; Vallejos and McKinnon 2013). The
removal of orepass noise on an individual event basis has
not yet satisfactorily been addressed. Currently, the most
practical approach to addressing this problem is to limit data
for hazard assessment in the vicinity of orepasses to periods when orepasses are not in use (i.e. apply a time filter).
For the purpose of hazard assessment, however, it is not
necessary to separate orepass noise from other events on an
individual basis since statistical approaches can be applied.
Such approaches aim to obtain the statistical characteristics of underlying superimposed populations. Woodward
and Tierney (2017) propose a method for simultaneously
inverting the parameters of the frequency–magnitude relationships for both the orepass noise and seismic events. The
parameters for the frequency–magnitude distribution of the
events obtained this way can be used for hazard assessment.
3.3 The Considered Timescale
When performing the hazard assessment, a data time period
on which to base the assessment must first be chosen. This
time period must be defined with reference to the mining
cycle at different timescales of interest. The selected data
period constitutes the “recent history” that defines the hazard
state, and the hazard assessment is applicable to any timescale for which the mean seismic rate obtained from the data
period is applicable.
Lasocki (2008) mentions that earthquake processes are permanent and are controlled by factors that are constant in time
and, as a result, are assumed to be Poissonian (stationary and
memoryless), with the event rate being governed by the Poisson distribution. Mining-induced seismicity, on the other hand,
is anthropomorphic and is spatially and temporally strongly
correlated to mining activity and, as a result, is temporally
often regarded as a non-Poissonian process. Lasocki (2008)
shows that the reported non-Poissonian character of mininginduced seismicity is due to a non-stationary process, rather
than it being caused by the inter-relations of event occurrence.
He also noted that the process tends towards being stationary
when the time period is reduced. Mining-induced seismicity
can therefore be described by a Poisson distribution with the
rate parameter varying over time. Lehmann and Romano (2005)
show that:
∑ (
∑
)
P(N|𝛬, ΔT) =
P ni |𝜆i , Δti ; with N =
ni
(5)
i
i
where N, Λ, ΔT = number of events, mean event rate and
time period for overall time period; ni, λi, Δti = number of
events, mean event rate and time period for sub-period i.
This implies that for mining-induced seismicity the mean
event rate Λ over the time period ΔT adequately describes
the temporal seismic process at scale ΔT, irrespective of
the individual values of λi over the shorter periods, Δti
(Fig. 4). Its implication in terms of hazard assessment can
be described as follows. The seismic hazard assessed for
period ΔT, based on the overall mean seismic rate, Λ, is
equal to the combined hazard from each of the sub-periods
with associated rate and time length parameters. In other
words, the assessment of hazard over a longer period with
mean seismic rate, Λ, is independent of the short-term variations of seismic rate, λi.
3.4 Probabilistic Seismic Hazard Assessment
of the Current Hazard State
The truncated Gutenberg–Richter formulation can be written
as a cumulative and inverse cumulative distribution function,
as follows (Kijko and Singh 2011):
⎧0
for M < mmin
⎪
−𝛽 (M−mmin )
F(M) = ⎨ 1−e−𝛽 (M −m ) for mmin ≤ M ≤ MUL
UL
min
1−e
⎪1
for M > MUL
⎩
(6)
ΔT
Cumulave number of seismic event
{
Δt1
Δt2
Δt3
Δt4
Δt5
5
4
3
Λ
N
2
1
Time
Fig. 4 Representative mean seismic rate over different time periods
13
J. Wesseloo
F � (M) = 1 − F(M)
(7)
where F, F′ = the cumulative and inverse cumulative
distribution functions, respectively; β = the power law
exponent = b·ln(10); mmin = magnitude of completeness;
MUL = upper truncation value of magnitude. The meaning
of MUL is discussed in Sect. 3.1.1; M = magnitude.
The cumulative probability function of the largest event
in a set of n events is given as follows (Gibowicz and Kijko
1994):
n
(8)
The probability of at least one event out of a total of n
events having a magnitude exceeding ML is given by:
Fmax (M, n) = F(M)
P(M > ML|n) = 1 − Fmax (ML, n) = 1 − [F(ML)]n
(9)
In crustal seismology, for the purpose of forecasting
future hazard for a specific time period, the uncertainty in
the number of events, n, is taken into account as follows
(Gibowicz and Kijko 1994):
Fmax (M|Δt) =
∞
∑
P(n|Δt)[F(M)]n
n=0
(10)
where P(n|Δt) = the probability of experiencing n events in
a future time interval Δt.
Generally, P(n|Δt) is defined as follows for a stationary
Poisson’s process:
P(n|Δt) =
(𝜆 ⋅ Δt)n ⋅ e−𝜆⋅Δt
n!
(11)
where λ = mean event rate applicable over period of length
Δt; n = number of events within a time interval Δt.
This leads to the following relationship for the probability
of at least one event exceeding ML (Gibowicz and Kijko
1994):
P(M > ML|Δt) = 1 − Fmax (ML|Δt) = 1 − e−𝜆⋅Δt⋅(1−F(ML))
(12)
Equations (11) and (12) can be rewritten so that the applicable time period is implicit as the period length over which
the mean number of events, n , is applicable:
P(n|̄n) =
n̄ n ⋅ e−̄n
n!
(13)
P(M > ML|̄n) = 1 − Fmax (ML|̄n) = 1 − e−̄n⋅(1−F(ML)) (14)
where n̄ = the mean number of events experienced within a
time period Δt.
As the mining-induced seismicity is a result of the rock
mass’ response to mining and is influenced by the mining
activity, it is transient over space and time. Reliable forecasting of P(n|Δt) for any specified future period based on
historical data is problematic and, for many mining methods
13
and environments, not currently possible. This component
is not important for our purposes as the method proposed
here is not an attempt to forecast future hazard for a specific
future period, but a method to quantify (or monitor) the current hazard state.
For our purposes, we ignore the uncertainty in the number of events for a future time period and assume that n = n̄ .
Thus, for the assessment of the current hazard state, we
assume:
P(M > ML) = 1 − [F(ML)]n̄
(15)
where n̄ = the mean number of events applicable to time
period Δt.
It should be noted that, since 1 − F(ML)n̄ ≥ 1−
e−̄n⋅(1−F(ML)) for n̄ ≥ 0 and 0 ≤ F(ML) ≤ 1, the assumption
that n = n̄ is conservative for all permissible conditions. The
difference between the two formulations (Eqs. 14 and 15)
approaches zero for large n̄ and large probabilities, whilst
the largest difference between the two formulations occurs
for small probabilities and low seismic rate and is inconsequential (“Appendix 2”).
Assuming that all other statistical characteristics of
sources of seismicity remain the same, the current hazard
state is applicable to any future period for which the mean
seismic rate obtained from the data period is applicable.
For the purpose of calculating the current hazard state, n̄
is calculated from the mean event rate density obtained from
the grid-based approach for the data period under consideration (Sect. 3.5).
3.5 The Grid‑Based Approach
This section briefly presents the grid-based approach discussed in detail in Wesseloo et al. (2014) and Wesseloo
(2014). The approach consists of the discretisation of a volume of interest into a fine regular grid. To each gridpoint,
representative seismic parameter values are assigned, based
on the events in the neighbourhood of the gridpoint. For
the assessment of the seismic hazard state in space for the
period under review, the b-value and the event density rate
are evaluated.
What is regarded as the “the neighbourhood” depends on
the resolution of the seismic system and is defined with a
maximum search distance, Rmax, when evaluating the parameters for each gridpoint.
Different approaches are needed to obtain the b-value and
the event rate density. These two approaches are discussed
separately in the following paragraphs.
3.5.1 Event Rate Density
For obtaining the event density for the period under consideration, each event that occurred within the period of interest
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
carries a unit intensity. The event density is calculated with
a variable smoothing function where the kernel bandwidth
is linked to the event source size, and the event density at
each gridpoint is obtained from the sum of all the values
of all the events registered to that gridpoint. This can be
expressed as follows:
∑
All gridpoints
Fi =
( )
K 𝜃i,j
(16)
(17)
j
𝜃i,j =
sj − si
hi
(
)3
1 − 𝜃 2 if 𝜃 < 1
K(𝜃) =
0
if 𝜃 ≥ 1
1:1
(18)
(19)
where pj = event density at gridpoint j; s = location vector
of the event and gridpoints; K(θ) = kernel function; hi = the
influence zone of each event, i.
F is a correction factor that ensures that no errors are
introduced due to the discretisation of the volume space, i.e.
that the following condition is met:
∑
all ridpoints
pj ⋅ Vj = N
(20)
j
where Vj = the volume of grid cell j; N = total number of
events used in the evaluation of event density distribution.
In this calculation, the influence zone for each event, hi, is
based on the event size with added ceiling and floor values
(Fig. 5). A lower cut-off value equal to the grid spacing is
imposed to ensure stability of the method for coarser grid
discretisation. This lower limit ensures that each of the more
numerous small events are associated with at least one gridpoint. A limiting ceiling value is also introduced for numerical efficiency and stability. The results are not sensitive to
the ceiling value. In this study, the source radius as defined
by Brune (1970) is used to define the source size.
The mean event density rate is obtained as the event density divided by the time length for the period of interest.
3.5.2 b‑Value
The grid-based approach for the spatial assessment of the
b-value is performed as follows:
• Events with magnitudes much smaller than the esti-
mate of the overall sensitivity based on the whole data-
Fig. 5 Definition of the event influence zone with floor and ceiling
values
Δ
mmin = system sensivity
i
( )
1
⋅ K 𝜃i,j
Fi
Data excluded
from analysis
∑
All events
pj =
b
1
Fig. 6 Illustration of frequency–magnitude distribution showing the
range of data excluded from the b-value calculations
set (mmin − Δ) are excluded from the analysis (Fig. 6).
This is done to speed up the calculations by excluding
very small events that do not contribute to obtaining
the b-value. A Δ of 1 magnitude unit yields satisfactory
results.
• For each gridpoint, the mmin and b-value are obtained
from the closest Nb events. The search distance R is limited to a value Rmax. The algorithm for automatically
obtaining the mmin and b-value is outside the scope of
this paper, but is discussed in detail in Wesseloo (2014).
The data included in the calculation of the b-value for
every gridpoint are illustrated in Fig. 7.
13
J. Wesseloo
3.6 Probabilistic Strong Ground Motion
on Excavation
Δ
overall mmin
It is common practice to use the strong ground motion as a
proxy for seismic hazard at the excavation, and, in mining,
it is common to use the peak particle velocity (ppv) (Potvin
and Wesseloo 2013). Milev and Spottiswoode (2005) have
made a large number of ppv measurements at the surface of
excavations at the TauTona mine, Kloof mine and Mponeng
mine using a custom-designed surface-mounted instrument
Nb events within
distance R
local mmin
Data excluded
from analysis
Nb
(a)
called the peak velocity detector (PVD). “Theoretical” ppv
values calculated from the seismic monitoring systems projected at excavation locations were then compared to ppv
values measured at the surface of excavations from the
PVD instrument. The ratio between the ppv experienced at
the surface of an excavation and the theoretical body wave
ppv are often called the “site amplification factor”. The site
amplification factor varied between about 1 and 25 for each
of the three mine sites, suggesting that the site effect may
vary significantly, even within a mine. The site amplification
factor is likely to be the cumulative effect of different factors
which include the radiation pattern, the complex interaction
of the body waves with the geology and excavation and the
effect of surface waves (Potvin and Wesseloo 2013).
Depending on the seismic source mechanism and orientation, different radiation patterns will result in the strong
ground motion varying in different directions from the
source. The radiation pattern for a double-couple shear
mechanism in three dimensions is illustrated in Fig. 8. Colour plots of the theoretical body wave intensity in the vicinity of excavations are shown in Fig. 9. The first two plots in
Fig. 9 show the variation for two different slip orientations,
whilst the third plot shows the result when the three-dimensional radiation pattern is ignored.
The body waves are reflected and refracted, and s- and
p-wave conversion occurs as they travel through the rock
mass resulting in complex wave forms interacting with the
excavations. When the body waves interact with excavations, surface waves are generated and propagate along the
surface of an excavation. Durrheim (2012) explains that the
fractured zone typically present around excavations at depth
creates a contrast in velocity, which contributes to “trap
seismic energy as the low velocity surface layer enhances
- grid point
- events excluded, mag < mmin-Δ
- Nb events within distance R used in b-value calculaon
- events excluded, enough events found closer to grid point
- events too far from grid point, not in the neighbourhood
(b)
Fig. 7 Frequency–magnitude distribution (a) and a special distribution of events (b) illustrating what data are included and excluded in
the determination of the local b-value associated with a gridpoint
13
Fig. 8 Relative 3D distribution of s-wave (a) and p-wave amplitude
(b) for a double-couple shear mechanism (Potvin and Wesseloo 2013)
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
Fig. 9 Illustration of the influence of the radiation pattern on the wave intensity in three dimensions; a, b illustrate the difference in ppv′ of diffirence in the slip direction, and c assuming uniform radiation (Potvin and Wesseloo 2013)
the formation of surface waves such as Raleigh and Love
waves”. Body waves are also reflected off the excavation
boundary, and superimposition of the complex waveforms
occurs. The complex interaction of the body waves with
excavations and geological contacts is illustrated by the
experimental work of Daehnke (1997). Figure 10 shows the
isochromatic fringes from the photoelastic experiment performed by Daehnke (1997). Local amplification or reduction
in ppv is also different for various excavation-to-wavelength
ratios (Wang and Cai 2015).
The effect of radiation pattern and the reflection and
refraction of the seismic waves can be assessed through
numerical modelling for a single event (e.g. Hildyard et al.
1995; Hildyard 2007; Mendecki and Lotter 2011; Wang and
Cai 2015), but cannot, with current technology, practically
be performed as part of a probabilistic process.
The relationship between the body wave ppv and the
actual ppv at the surface of an excavation can be expressed
as follows:
(
)
log (ppv) = log ppv� + 𝛿(i, j)
(21)
where ppv = the ppv at the boundary of the excavation;
ppv′ = body wave ppv without any wave–excavation interaction; δ(i, j) = the logarithm of the amplification factor for the
specific location i and event j.
The site amplification is poorly understood, and for this
reason the body wave ppv in the rock mass (denoted here as
ppv′) is used as a proxy for vibration intensity. For dynamic
ground support design, ppv′ is often used as an input to
empirical (Duan 2016; Duan et al. 2015; Heal 2010; Heal
et al. 2006), and analytical methods (Mining Research Directorate 1996) with an amplification factor applied separately
as part of the design process. For these reasons, we currently
restrict ourselves to the evaluation of ppv′.
Hazard maps based on ppv′ would be more smeared than
a hazard map based on the actual ppv because the localised amplification or reduction in the body wave ppv due to
the factors mentioned above is not taken into account. As
technology improves, both the assessment of ppv′ and the
design methodologies’ reliance on these values need to be
improved.
3.6.1 Strong Ground Motion Relationship
For the strong ground motion relationship, the following
generic formulation is used for which the coefficients should
be obtained through back-analysis from recorded data:
(
)
(
)
log ppv� = c0 + cM ⋅ M + cR ⋅ log R + cRs ⋅ Rs
+ cSSD ⋅ log (SSD) + 𝜀
(22)
where ppv′ = peak particle velocity of the body wave; M = is
the magnitude; ε = an error function; R = the epicentral
distance to the site under consideration; Rs = saturation distance; SSD = seismic source parameter, static stress drop,
which quantifies the difference between the mean shear
stress before and after an event.
Neglecting to quantify δ(i,j) of Eq. (21) for every combination of location i and event j results in a dispersion of the
actual values about the regression (Lasocki 2005), and this
is quantified by the error function ε.
This proposed equation includes a saturation distance,
Rs, resulting in the saturation of the relationship in the near
field. The saturation distance is expected to be proportional
to the source size, which for practical purposes can be written as a function of the magnitude. In lieu of data from the
near field, formulating the saturation distance as a function
of the magnitude is also important to ensure stable results
when obtaining site-specific coefficients for the above formulation. Mendecki (2008, 2013a) formulates a strong
ground motion relationship using log(potency) as magnitude
scale and uses a saturation distance proportional to the cube
root of potency.
13
J. Wesseloo
1
1
; c = 2; cSSD = ; cRs = 0; c0 = −4.32 for R in metres, and
2 R
2
𝜀(P) = qnormal(P, 𝜇 = 0, 𝜎 = 0.405), with M = MR + 1.15
cM =
(23)
where SSD = the static stress drop in Pa; P = probability of non-exceedance i.e., reliability; qnormal = inverse
cumulative normal distribution; μ = mean of error function; σ = standard deviation of error function; MR = Richter
magnitude.
To obtain the relationship recommended by the CRH for
use in design, let P = 0.9 with ε(P = 0.9) = 0.519.
Potvin and Wesseloo (2013) extend the relationship used
in the CRH to include near-field saturation. The generic
formulation can be reduced to that proposed by Potvin and
Wesseloo (2013) using the following in addition to the coefficients mentioned above:
(a)
cRs = 1; Rs = Ro = D ⋅ 10
MR +1.5
3
1.14
;D= √
3
SSD ⋅
1
106
(24)
where Rs = the radius of saturation in metres; Ro = source
radius in metres; SSD = the static stress drop in Pa;
MR = Richter magnitude.
3.6.2 Calculation Process for Probabilistic Assessment
of ppv′
(b)
Detonaon point
Clamped non-cohesive
parng plane
Open slot
Araldite B
Makrolon
(c)
Fig. 10 (a, b) Photoelastic model showing the reflection and refraction of stress wave when; a interacting with a long narrow opening, and b when interacting with a similar opening bounded by two
discontinuous planes above and below the opening (after Daehnke
1997). a “stope” in homogeneous medium, b “stope” bounded by
non-cohesive parting planes, c definition sketch
The generic formulation can be used to express the wellknown far field relationship proposed by the Canadian Rockburst Handbook (CRH) (Kaiser and Maloney 1997; Mining
Research Directorate 1996), using the following coefficient
values:
13
From the strong ground motion relationship (Eq. 22), the
probability of the ppv′ exceeding a specified value, PPV′,
for a given magnitude, ML, and a given distance, R, can
be obtained. For each grid cell, k, the event rate density, n,
and the power law exponent, β, are obtained through the
grid-based approach [Fig. 11 a(i)]. The probability density
function of the largest event for each grid cell, the derivative
of Eq. (8), is given by:
(
)
k
M|̄nk , 𝛽 k
(
) dFmax
k
k k
(25)
fmax M|̄n , 𝛽 =
dM
The distance between the possible event locations and
the target point of interest is taken as the shortest linear distance, i.e.:
√
(
)2 (
)2 (
)2
(26)
Ri (x, y, z) =
xi − x + yi − y + zi − z
For every target point, i, and every grid cell, k, the probability of ppv′ exceeding the value of PPV′ is given by:
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
]k
[
P ppv� ≥ PPV� i =
zk+ yk+ x+k MUL
� � � �
( [
])
k
(m|̄nk , 𝛽 k ) ⋅ P ppv� > PPV� |M, Ri (x, y, z) dMdxdydz
fxk (x)fyk (y)fzk (z) ⋅ fmax
(27)
zk− yk− x−k m−
where m = magnitude; m− = the smallest magnitude that
could result in any ppv-value of interest; MUL = overall
MUL for the whole mine; x−k , yk−,zk− = minimum x, y and z
boundaries for grid cell k; x+k , yk+, zk+ = maximum x, y and
z boundaries for grid cell k; nk, βk = event density rate and
power law exponent for grid cell k; fxk , fyk , fxk = probability density function of event x, y and z location in cell k;
k
fmax
= probability[ density function ]of the magnitude of
the largest event; P ppv′ > PPV′ |m, r = the strong ground
motion relationship (Eq. 22).
Using the discretised volume and assuming a uniform
seismic density distribution throughout each grid cell, the
discretised formulations of Eq (27) can be expressed as follows with reference to Fig. 11.
J
L Y
∑
∑
[
]k ∑
P ppv� ≥ PPV� i =
pMkj ⋅ p𝜀y
y
j
l
(28)
[
] ΔM ⋅ Δ𝜀
�
�
⋅ P ppv > PPV |Mkj , Rikl , 𝜀y ⋅
L
where J, L, Y = total number of bins in the magnitude, grid
cell and epsilon, respectively; ΔM, Δε = bin widths for M
and ε discretisation.
For every grid cell, k, the probability distribution of the
largest magnitude is known, and for every magnitude bin, j,
the probability of Mkj can be assessed [Fig. 11 b(iii)]. The
distance Rikl is known for every combination of target point,
i, and sub-cell, l, of any grid cell, j [Fig. 11b(i)]. For any
given Rikl and Mkj, the strong ground motion relationship
can be evaluated and the ppv′ calculated for every ε bin, y
(Fig. 11 b(iv)). The ppv′ijkly and its associated probability are
calculated for every combination, ijkly, from which the full
ppv′ probability curve can be obtained. With the evaluation
of Eq. (28) the exceedance probability of PPV′ at any target
point, j, resulting from any grid cell, k, can be obtained.
The probability of ppv′ exceeding a prescribed PPV′ at
target point, i, due to the hazard at every grid cell, k, can then
be calculated as follows:
∏(
[
]
[
]k )
P ppv� ≥ PPV� i = 1 −
1 − P ppv� ≥ PPV� i
(29)
k
In many design approaches, the 90 percentile value of ε
is used (Heal 2010; Heal et al. 2006; Kaiser and Maloney
1997; Mining Research Directorate 1996). In industry, a
constant value for ε is often used (generally 90 or 95 percentile value). For the approach presented here, however,
the full distribution of ε is used as the propagation of the
uncertainty quantified by ε, is an important component in a
quantitative risk approach.
The probabilistic evaluation of the ppv′ on the excavation
target point is computationally intensive. The number of calculations scales linearly with both the number of excavation
points and the number of gridpoints. In three-dimensional
space, the gridpoints scale quadratically with the decreased
grid spacing. The high resolution required for a full hazard map requires a large number of calculations. In order to
facilitate understanding, the intricacies of code optimisation
are ignored in the previous explanation of the calculation
process. Several code optimisation techniques can be used
to reduce the number of loops and calculations required. The
calculation algorithm introduced in an mXrap app (Wesseloo and Harris 2015) was optimised to the point where
the map shown in Sect. 4.3.2 could be produced in about
20 s on a standard laptop computer. A detailed discussion
of the code optimisation is outside the scope of this paper.
3.6.3 Overall Strong Ground Motion Hazard Associated
with Different Areas of the Mine
Section 3.6.2 leads to a spatial distribution of exceedance
probabilities at different locations in the mine. It is, however,
important to also assess the exceedance probabilities for sections of infrastructure in different locations and for different
lengths along the infrastructure. Consider, for example, the
conceptual mining layout for a longhole stoping sub-level
(Fig. 12). A dynamic failure in either the decline, the ore
drive or any of the stope drives will each differently impact
the economic and safety risk of the mine. If, for example,
rockburst damage occurs anywhere along the section of the
decline shown in Fig. 12, no access to the ore from that
sub-level or any sub-level below is possible for the period
necessary for rehabilitation. To quantify this risk, the seismic hazard over the total shown decline length needs to be
evaluated. Quantifying the risk of partial ore loss of different
quantities will require the evaluation of the seismic hazard
over the different lengths of different drives. Similarly, the
exposure of personnel to the seismic hazard differs for the
different types and lengths of infrastructure.
13
J. Wesseloo
Primary stope drive
i)
i)
Secondary stope drive
Fig. 12 Conceptual mining layout for a primary/secondary longhole
stoping sub-level (Joughin 2017)
ii)
Annualisation (Secon 4.1)
The target points on the excavation are, however, at arbitrary spacing and cannot be assumed to be independent.
Two target points far apart may be assumed to be independent, whilst this may not be true for target points close to
each other. Since the calculations are performed on target
points of arbitrary spacing, an adjustment must be applied
to account for the size of an independent unit, SIU (Fig. 13).
Gilbert (1986) and Dowding and Gilbert (1988) investigated the dynamic stability of rock slopes and the effect of
wave propagation on the stability of slopes. They concluded
that, to excite a block, wavelengths greater than four times
the block size are required. In a similar vein, we link the SIU
to the wavelength with a factor of cλ, i.e.:
iii)
iii)
p[Mk] probability distribution of largest magnitude
for given k and bk (Equation 25)
iv)
iv) Volume
normalisation
(Section 4.2)
SIU = 𝜆o ⋅ c𝜆
v) Spatial
distribution
of seismic hazard
(Section 4.3.1)
v) Display of probabilistic strong ground
moon hazard map (Section 4.3.2)
(a)
(b)
where SIU = the size of an independent unit; λo = wavelength at the event corner frequency; cλ = a defined fraction
of the wavelength.
From the Brune model (Brune 1970), one can obtain an
estimate of the wavelength at the corner frequency of an
event, as follows:
√
3
2 ⋅ 𝜋 3 7 10 2 ⋅(Mw +6)
(31)
⋅
⋅
𝜆o =
2.34
16
Δ𝜎
Fig. 11 Conceptual illustration of the components of the probabilistic
strong ground motion calculation
The joint probabilities for nt independent points along the
infrastructure can be calculated as follows:
P[a ≥ A] = 1 −
nt
∏
(
)
1 − P[a ≥ A]i
(30)
i=1
Fig. 13 Definition sketch of excavation lengths used in calculated
joint probabilities for length Γ
13
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
where λ o = wavelength at the event comer frequency;
Mw = moment magnitude (Hanks and Kanamori 1979);
Δσ = static stress drop.
For current purposes, cλ is assumed to be 14 , according to the results of Gilbert (1986) and Dowding and Gilbert (1988), and a conservative value for Δσ = 10 MPa is
assumed. For other magnitude scales, empirical site-specific
relationships between corner frequency, magnitude and
static stress drop can be obtained through regression of site
data.
For the purpose of calculating the joint probabilities, the
approach described in Sect. 3.6.2 is adjusted in the following manner. For every target point, i, and every gridpoint,
k, the probability of exceeding PPV′ is obtained for every
magnitude bin, j, as follows:
P[ppv� ≥ PPV� |Mj ]ki =
L Y
∑
∑
l
pMkj ⋅ p𝜀y ⋅
y
[
] ΔM ⋅ Δ𝜀
P ppv� > PPV� |Mkj , Rikl , 𝜀y ⋅
L
(32)
The joint PPV′ exceedance probability for a length of
mine infrastructure, Γ, is calculated as follows (Fig. 13):
[
]
P ppv� ⩾ PPV� 𝛤
))
(
(
nt
Li
∏
∏ ∏(
)
=1−
1 − P[ppv� ⩾ PPV� |Mj ]ki SIUj
i=1
j
k
(33)
where Li = the length of mine infrastructure associated with
calculation point, i; ­SIUj = the size of the independent unit
for magnitude j; Γ = length of mine infrastructure for which
the joint probability is calculated; nt = the number of target
points within length Γ.
4 Expressing Seismic Hazard
The spatial distribution of probabilities presents a difficulty
in communicating the true nature of the hazard. The contour/isosurfaces of probabilities can easily be interpreted
in a relative sense: “hot” and “cold” volumes showing
higher and lower relative spatial probability of occurrence.
The spatial probability distribution in units of probability/
unit time/unit volume, however, leads to small numbers that
carry little or no meaning for engineers, management and
the workforce.
For this reason, we need to normalise values for both time
and volume. These normalisations are necessary to enable
us to directly compare hazard levels of different duration
contained within different sized volumes. They also provide
a consistent system for communicating the hazard, without
which, the management of the seismic hazard is impeded.
4.1 Annualisation of Hazard
For the purpose of this study, we normalise the hazard probability to equivalent annualised values. This normalisation
leads to a “yearly probability”, but it is important to note
that this should not be interpreted as the probability for a
physical year (future or historical). It is the probability value
appropriate to the timescale for which the mean seismic rate
is applicable, expressed in equivalent annualised terms.
Annualisation also allows one to calculate the associated
risk and evaluate it against corporate accepted annualised
risk levels.
To enable direct comparison between hazards of different
durations, it is necessary to normalise the calculated probability to the same equivalent timescale. This normalisation
can be performed as follows:
(
) Tn
PTn = 1 − 1 − PTe Te
(34)
where Tn, Te = the normalised time frame and the original
timeframe, respectively; PTn, PTe = the equivalent probabilities expressed for timeframes Tn and Te, respectively.
A hazard with a weekly probability of 1% can
be expressed with equivalent annualised values as
1 − (1 − 0.01)52 = 40%, whilst a hazard with a biennial
probability of 50% can be expressed in equivalent annualised
values as 1 − (1 − 0.5)0.5 = 30%.
These calculations assume that both the hazards are present over a long time period. If, for example, a hazard with
a weekly probability of 1% is only present for 1 week in
the year, the yearly hazard would also be 1%. This sometimes leads to misunderstanding in the application of hazard
assessments in industry, where a short-lived but repeating
hazard is sometimes evaluated in isolation.
Seismic hazard in a mine is transient in space and time
and although the seismic hazard at a specific location might
be short lived, the hazard is of a repeating nature. For example, the hazard associated with the mining of a single stope
might only be present during the time it takes to mine that
stope, but, a similar hazard might occur due to the mining
13
J. Wesseloo
1
(a)
Stopes
Historic seismicity
Seismicity during
“current “ month
(b)
of the stopes is mined for a month during which a seismic
response is induced in the indicated annulus around it. The
seismicity in this annulus ceases when mining in this stope
is complete. During the following month, the next stope is
mined with the associated induced seismicity limited to its
surrounding annulus. The argument can be further simplified
by assuming a constant b = 1 over the whole volume and the
whole year, and by assuming that the total number of events
occurring in the annulus of each of the stopes is the same at
1000 events > ML-2.
For each stope, the probability of exceeding ML2 is 3.92%,
and if evaluated in isolation, may be regarded as acceptable.
Cumulating the number of events for all 12 stopes, however,
results in the total probability of exceeding ML2 of 38%.
In the year, the company is exposed to the total hazard of
P[ML > 2] = 38%, even though each stope only has an individual monthly hazard of 3.9%. If exposing the company to
the yearly hazard of 38% is not acceptable, by implication, it
is not acceptable to expose the company to the hazard associated with every one of those stopes individually.
Normalisation can be done to any timescale. Short time
periods, however, lead to small numbers, which are in our
experience often misinterpreted as small hazards and should
be avoided. We propose the use of 1 year (annualisation),
as this corresponds with the practice in other branches of
engineering, financial risk management and corporate governance (Jonkman et al. 2003; Stacey et al. 2007; Terbrugge
et al. 2006; Wesseloo and Read 2009).
4.2 Normalisation to Meaningful Volume
(c)
Fig. 14 Conceptual mine layout for illustrative example, consisting of
12 stopes with an annulus of seismicity induced by the stoping activity. a Month 1, b month 6, c month 12
of the next stope. Both long-term and repeating hazards can
be normalised temporally using Eq. (34).
To illustrate this concept, consider a simple fictitious
mining scenario that consists of 12 stopes (Fig. 14). Each
13
The same principles also apply to the spatial representation
of hazard. Before true comparison between the hazards for
two different volume sizes is possible, the hazard must be
expressed in values normalised to the same representative
volume scale. With the grid-based approach, the hazards in
each grid are directly comparable as calculations are performed on a uniform grid. Representing this distribution
through space therefore leads to a true spatial comparison.
As the mine is exposed to the total hazard, in addition to
the representation of the hazard through the whole mining
volume, the total hazard for the whole mine or large mining
block under consideration must also be provided.
4.3 Displaying Spatial Distribution of Seismic
Hazard
4.3.1 Displaying Hazard Through the Rock Mass Volume
The display of the distribution of the hazard at any particular
time through space is possible with contours and isosurfaces,
which gives a qualitative indication of “hot” and “cold”
areas. Quantitatively, the meaning of the contour/isosurface
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
Fig. 15 Elevation view and cross section view of Tasmania mine with isosurfaces of hazard rating
Fig. 16 Seismic hazard map showing the probability of exceeding
ML2 within 50 m
Fig. 17 Strong ground motion probability map showing the probability of exceeding ppv′ = 1 m/s
13
J. Wesseloo
values is obscure with values in probability per year per ­m3.
For this reason, we need to introduce a characteristic volume
for normalisation of the volumetric component.
One way to express the hazard spatially through the use
of contours or isosurfaces is to plot a hazard rating. We can
define a hazard rating as the magnitude with an annualised
exceedance probability, P, in a characteristic volume, i.e. the
magnitude with an exceedance probability P/year/characteristic volume. Hudyma and Potvin (2004) propose a seismic hazard rating with which many mines in Australia and
Canada are familiar. The hazard rating values resulting from
P = 15%, with the volume of a sphere having a 50 m radius,
are similar to the hazard rating values of the Seismic Hazard
Scale proposed by Hudyma and Potvin (2004) and were used
for the definition of the hazard rating scale used here. Once
the probabilistic hazard has been calculated on a grid basis,
it is a simple extension to calculate the hazard rating and
display it using contours or isosurfaces, as shown in Fig. 15.
4.3.2 Displaying Hazard on Excavations
For any point in space, one can easily calculate the probability of exceeding a specified magnitude, ML, within a
specified distance, R, i.e. P[m ≥ ML|r ≤ R]. This is computationally efficient compared to mapping probabilistic
strong ground motion evaluations and the computations
Fig. 18 Conceptual layout of parallel mine excavations with associated probability of exceedance values (Sect. 3.6.2) (a) and spatially cumulative exceedance probability values (Sect. 3.6.3) with subsequent zonation according to the specified risk classes (Sect. 4.3.2) (b)
13
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
among very small probabilities, with a well-calibrated range
limited to between 20 and 80%. Expressing hazards should
therefore be done in a way that utilises values in the wellcalibrated range of human cognitive discrimination.
For this purpose, the target calculation points are grouped
into zones of similar hazard levels. The total exceedance
probabilities for each of these zones are calculated with the
method discussed in Sect. 3.6.3.
The zonation is performed as follows: target points are
sor ted wit h increasing hazard as def ined by
P[ppv� ≥ PPV� |Mj ]i (Eq 29), i.e. target point 1 having the
lowest
hazard] (Fig. 18a). For every target point, n t,
[
P ppv� ≥ PPV� 1…n (Eq. 33) is calculated, leading to a spat
Fig. 19 Strong ground motion probability map showing the total
probability of exceeding a ppv′ = 0.6 m/s in different zones according
to likelihood classes used in mine site risk matrices
scale linearly with the number of points evaluated. A map
of P[m ≥ ML|r ≤ R], like that shown in Fig. 16, can therefore be generated efficiently and for large problems. For the
example shown in Fig. 16, the annualised probability of
experiencing an event of ≥ ML2 within a distance of 50 m
from the mine excavations is evaluated and expressed as a
contour map.
A probabilistic hazard map of ppv′ can be obtained, as
discussed in Sect. 3.6.2. This results in a probabilistic ppv′
map as shown in Fig. 17. The example shown in Fig. 17
plots a map of the probability of exceeding a ppv′ of 1 m/s
due to any possible event located anywhere in the rock mass
surrounding the mine.
The probability at each point on the surface is quite small
(< 0.2% anywhere in the mine), with only a small section of
the mine reaching those levels. Although the hazard distributions and the hazard values are correct, the small values
associated with zones on the map are subject to misinterpretation. The interpretation of such contours is generally that
there is total probability of 0.1–0.2% of exceeding the design
PPV′ in the areas of the highest contour value. There is, however, a 0.1–0.2% chance of exceeding the ppv′ everywhere
within that zone—the risk that needs to be communicated to
stakeholders should be the integration of that risk over the
whole zone. Moreover, Vick (2002) discusses the experimental work by Fischhoff et al. (1977) and Hogarth (1975)
and concludes that people have little ability to distinguish
tially cumulative exceedance probability map (Fig. 18b).
The zonation is based on these cumulative exceedance probability values.
A risk class is assigned to each target point, n, according
to which class its cumulative exceedance probability value
belongs. Risk classes can be defined according to the company risk matrix commonly used on mines leading to the
hazard being expressed in terms familiar to all the stakeholders. The results of the zonation process for Tasmania mine
are shown in Fig. 19.
5 Discussion and Conclusion
The previous sections provided a methodology for assessing
the current seismic hazard state in mines and for monitoring the changes in the hazard state on an ongoing basis.
Due to the fact that many of the sources of seismic hazard
in mines are of a transient nature, the continuous tracking
of the change in the seismic hazard state provides valuable
information for the management of the seismic hazard. Not
all sources of seismic hazard are transient, and, as a result,
careful consideration must be given to the data period used
for the hazard assessment. Faults, dykes and lithological
contact may deform co-seismically as they respond to the
mining-induced stress changes. Such sources are stationary
and often react to the mining activity on a longer timescale.
Depending on the mining method, sequence and layouts,
the stress change and associated change in seismic hazard
from such sources can occur in relatively short time frames.
For pillars and stationary abutments, the associated seismic
hazard may increase as loading increases and decrease after
complete yielding occurs. Generally, the change in seismic
hazard associated with pillars and stationary abutments
occurs on a longer timescale. In the vicinity of active mine
workings, temporary pillars and abutments result in transient
sources. As a result, transient high hazard sources superimpose on stationary sources creating a complex seismic
hazard regime changing with mining activity.
13
J. Wesseloo
Fig. 20 Probabilistic strong
ground motion hazard maps
showing the migration of inmine seismic hazard over time.
Plots of annualised likelihood
of exceeding ppv′ exceeding 0.6 m/s based on a hazard
period of 3 months
13
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
Fig. 21 Plots of annualised
likelihood of exceeding ppv′
exceeding 0.6 m/s based on different data periods
a
a
2011-02-01 | 24 Months
2011-02-01 | 12 Months
b
c
e
d
a) Dissipated hazard sources
shown due to long me periods.
b) Recent mining acvity in this
area.
c) Strong reacon to mining of the
abutment in this area.
d) Recent mining of stopes in new
area.
e) Hazard around unmined pillar.
No mining occurring near this
area.
2011-02-01 | 6 Months
Limiting the hazard assessment period to short time periods gives more weight to transient sources and results in
a better representation of spatial change in seismic hazard
associated with the mining activity on a shorter timescale.
Extending the hazard data period to long periods will result
in a spatial smoothing of the transient hazard and give a
better definition of the stationary hazard sources. Seismic
hazard assessment, therefore, needs to be performed with
reference to the mining cycle at different timescales of interest. As the method can be updated within a relatively short
calculation period, a frequent update of the hazard assessment will provide indication of areas where the hazard is
fairly stationary or increasing/decreasing rapidly.
13
Acknowledgements My sincere appreciation to Dr Lindsay Linzer and
William Joughin for their comments on the original manuscript and to
Gerhard Morkel for fruitful discussions during the writing of this paper.
I thank the mXrap Consortium (https​://mxrap​.com/the-mxrap​-conso​
rtium​) for their financial support of this work. The support of Paul
Harris with the implementation of this work into efficient algorithms
is greatly appreciated. This work was not done in isolation and I would
like to acknowledge the contribution of my colleagues at the Australian
Centre for Geomechanics, through fruitful discussions, suggestions and
challenging questions.
∆
∆
Log scale
The migration of the seismic hazard zones over time is
illustrated in Fig. 20. For this figure, the data period is kept
constant at 3 months and the likelihood zones (similar to
Fig. 19) of ppv′ exceeding 0.3 m/s on the ore drives, footwall
drives and the declines are shown; only the ore drives are
visible from the viewpoint shown in the figure.
Figure 21 plots the hazard zones shown in Fig. 20 using
different data periods. Using 24- and 12-month data periods
includes data from transient sources around mine workings
that have dissipated. This results in overestimation of the
hazard in these areas. Six month and three month periods
show similar results and show the strong reaction to the mining of the abutment in the upper area and the stoping of new
areas lower down. A high seismic hazard is also shown in
an unmined pillar, whilst no mining activity is occurring
near that area.
In Australia, the dynamic demand, for which ground support is designed, is often specified in terms of a magnitude
occurring at a specified distance (e.g. ML2 at 50 m) or in
terms of a design ppv′ that the support is required to withstand. The probabilistic hazard assessment procedure presented here allows for the assessment of seismic hazard in
terms of these design demand values. Using this method,
areas where ground support is under-designed can be highlighted and appropriate action taken.
The method allows for the seismic hazard to be evaluated
within the general risk management framework of the mine
with the hazard reflected in the company’s risk matrix.
UL
J. Wesseloo
Mmin
MUL
ML
Fig. 22 Inverse cumulative distribution plots illustrating the definition
of the factors in Eq. (37)
12
0.5
0.25
Δ
εP+
→ 0, b = 0.75
Δ
Δ
Δ
Appendix 1: Error in exceedance probability
resulting from error in MUL
See Figs. 22, 23, 24, 25.
The truncated cumulative distribution of event magnitude
can be written as follows (Gibowicz and Kijko 1994):
Δ
Δ
Fig. 23 εP+ and εP− as a function of Δ for different Δε
13
The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines
0.01
0.1
CDF of largest event size
R
1
10
n = 100
F(ML)
Fig. 24 Ratio between Eqs. (40) and (41) with different n and F(ML)
values
(35)
where F = the cumulative distribution functions; β = the
power law exponent = b·ln(10); mmin = magnitude of completeness; MUL = upper truncation value of magnitude. The
meaning of MUL is discussed in Sect. 3.1.1; M = magnitude.
For the mean number of events over the time frame under
consideration, n̄ , the exceedance probability of ML can be
written as follows (refer to Sect. 3.4 and “Appendix 2”):
P(M > ML|̄n) = 1 − [F(ML)]n̄
which can be rewritten as:
)n̄
(
−𝛽(𝛿)
�
Fmax
= 1 − 1−e−𝛽(𝛿+Δ)
for 𝛿 > 0; Δ > 0
1−e
�
= 1−
Fmax
(
1−e−𝛽(𝛿)
1−e−𝛽(𝛿+Δ)
)
n = 100
0.1
0.01
ML - mmin
(37)
Fig. 25 Comparison between Eqs. (40) and (41) for mmin = − 2,
MUL = 3, b = 1
Consider the influence of error, Δε, in the estimation of
MUL on the assessment of the exceedance probability for
given magnitude M. For an error of Δε in the estimate of
MUL the error in the exceedance probability can be defined
as:
𝜀P =
for 𝛿 > 2; Δ > 0
1
10
(36)
Where F′max = the exceedance probability; Δ = MUL – M;
δ = M – mmin; n̄ = the mean number of events over the time
frame under consideration.
These factors are illustrated in Fig. 22. This formulation
is independent of the actual value of ML which is implicitly
defined by factors Δ and δ. This formulation can further be
made independent of δ by writing n̄ as a function of δ and the
mean number of events of magnitude ML, n̄ ML , i.e.:
n̄ ML
1−F(𝛿,Δ)
Inverse CDF of largest event
⎧0
for M < mmin
⎪
−𝛽 (M−mmin )
F(M) = ⎨ 1−e−𝛽 (M −m ) for mmin ≤ M ≤ MUL
UL
min
1−e
⎪1
for M > MUL
⎩
ML - mmin
(38)
�
�
Fmax
(Δ + Δ𝜀) − Fmax
(Δ)
�
Fmax
(Δ)
(39)
13
J. Wesseloo
where εP = the error in probability of exceeding ML;
Δε = the error in estimation of MUL.
With εP+ and εP− denoting εP for positive and negative
values of Δε, respectively.
The error value εP decreases with increases in b, Δ and
n̄ ML. Upper bound estimates for εP can therefore be obtained
by assuming lower bound values for b, Δ and n̄ . Figure 23
shows the upper bound of εP as a function of Δ for different values Δε with conservative lower bound estimates
n̄ ML → 0, b = 0.75. The error, εP, decreases with an increase
in b-value and an increase in n̄ ML and is therefore larger for
smaller hazards, reducing with an increase in hazard.
Appendix 2: The Effect of Excluding
the Uncertainty in the Number of Events
Described by the Poisson Distribution
Two formulations for the exceeding probability of at least
one event exceeding ML are compared here. The first is
a formulation used in forecasting future seismicity and
includes the uncertainty of the number of events in the future
period Δt, which can be written as (Eq. 14):
P(M > ML|̄n) = 1 − Fmax (ML, n̄ ) = 1 − e−̄n⋅(1−F(ML))
(40)
The second formulation used in this work for the assessment of the current hazard state ignores the uncertainty
related to the number of events and assumes a mean number
of events to occur.
(41)
For the formulation in Eq. (42) to be conservative, the
following inequality must be true
P(M > ML|̄n) = 1 − Fmax (ML, n̄ ) = 1 − [F(ML)]n̄
R=
1 − e−̄n⋅(1−F(ML))
≤1
1 − [F(ML)]̄n
(42)
E qu a t i o n E q . ( 4 2 ) h o l d s t r u e fo r a l l
for n ≥ 0 and 0 ≤ F(ML) ≤ 1, as shown in Fig. 24. From
the figure, it can be seen that the difference between the two
formulations approaches zero for large n̄ and large values of
F(ML), whilst the largest difference between the two formulations occurs for small values of F(ML) and low seismic
rate. A small amount of conservatism will therefore result
in areas of high hazard, and lower accuracy with higher conservatism in areas of low hazard.
To illustrate this further in the context of this paper, consider the scenario with a magnitude of completeness, mmin,
of − 2, MUL = 3, and b = 1. The resulting formulations of
Fmax(ML) are compared in Fig. 25.
13
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