The Spatial Assessment of the Current Seismic Hazard State for Hard Rock Underground Mines

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.

Appendices

Appendix 1: Error in exceedance probability resulting from error in M _UL

See Figs. 22, 23, 24, 25.

The truncated cumulative distribution of event magnitude can be written as follows (Gibowicz and Kijko 1994):

$$F\left( M \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{for }}\;M < m_{\text{min}} } \hfill \\ {\frac{{1 - e^{{ - \beta \left( {M - m_{\text{min}}} \right)}} }}{{1 - e^{{ - \beta \left( {{M}_{\text{UL}} - m_{\text{min}} } \right)}} }}} \hfill & {{\text{for }}\;m_{\text{min}} \le M \le M_{\text{UL}} } \hfill \\ 1 \hfill & {{\text{for }}\;M > M_{\text{UL}} } \hfill \\ \end{array} } \right.$$

(35)

where F = the cumulative distribution functions; β = the power law exponent = b·ln(10); m_min = magnitude of completeness; M_UL = upper truncation value of magnitude. The meaning of M_UL is discussed in Sect. 3.1.1; M = magnitude.

For the mean number of events over the time frame under consideration, $\bar{n}$, the exceedance probability of ML can be written as follows (refer to Sect. 3.4 and “Appendix 2”):

$$P\left( {M > {\rm ML}|\bar{n}} \right) = 1 - \left[ {F\left( {\rm ML} \right)} \right]^{{\bar{n}}}$$

(36)

which can be rewritten as:

$$F'_{ \hbox{max} } = \begin{array}{*{20}c} {1 - \left( {\frac{{1 - {e}^{ - \beta \left( \delta \right)} }}{{1 - {e}^{{ - \beta \left( {{{\delta }} + {{\Delta }}} \right)}} }}} \right)^{{\bar{n}}} } & {{\text{for}}\; \delta > 0; \;{{\Delta }} > 0} \\ \end{array}$$

(37)

Where F′_max = the exceedance probability; Δ = M_UL – M; δ = M – m_min; $\bar{n}$ = 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 $\bar{n}$ as a function of δ and the mean number of events of magnitude ML, $\bar{n}_{\text{ML}}$, i.e.:

Fig. 22

Inverse cumulative distribution plots illustrating the definition of the factors in Eq. (37)

$$F'_{ \hbox{max} } = \begin{array}{*{20}c} {1 - \left( {\frac{{1 - {e}^{ - \beta \left( \delta \right)} }}{{1 - {e}^{{ - \beta \left( {{{\delta }} + {{\Delta }}} \right)}} }}} \right)^{{\frac{{\bar{n}_{\text{ML}} }}{{1 - F\left( {\delta ,{{\Delta }}} \right)}}}} } & {{\text{for}}\; \delta > 2; \;{{\Delta }} > 0} \\ \end{array}$$

(38)

Consider the influence of error, Δε, in the estimation of M_UL on the assessment of the exceedance probability for given magnitude M. For an error of Δε in the estimate of M_UL the error in the exceedance probability can be defined as:

$$\varepsilon P = \frac{{F'_{ \hbox{max} } \left( {{{\Delta }} + {{\Delta \varepsilon }}} \right) - F'_{ \hbox{max} } \left( {{\Delta }} \right)}}{{F'_{ \hbox{max} } \left( {{\Delta }} \right)}}$$

(39)

where εP = the error in probability of exceeding ML; Δε = the error in estimation of M_UL.

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 $\bar{n}_{\text{ML}}$. Upper bound estimates for εP can therefore be obtained by assuming lower bound values for b, Δ and $\bar{n}$. Figure 23 shows the upper bound of εP as a function of Δ for different values Δε with conservative lower bound estimates $\bar{n}_{\text{ML}} \to 0$, b = 0.75. The error, εP, decreases with an increase in b-value and an increase in $\bar{n}_{\text{ML}}$ and is therefore larger for smaller hazards, reducing with an increase in hazard.

Fig. 23

εP⁺ and εP⁻ as a function of Δ for different Δε

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):

$$P\left( {M > {\rm ML} |\bar{n}} \right) = 1 - F_{ \hbox{max} } \left( {{\rm ML},\bar{n}} \right) = 1 - e^{{ - \bar{n} \cdot \left( {1 - F\left( {\rm ML} \right)} \right)}}$$

(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.

$$P\left( {M > {\rm ML} |\bar{n}} \right) = 1 - F_{ \hbox{max} } \left( {{\rm ML},\bar{n}} \right) = 1 - \left[ {F\left( {\rm ML} \right)} \right]^{{\bar{n}}}$$

(41)

For the formulation in Eq. (42) to be conservative, the following inequality must be true

$$R = \frac{{1 - e^{{ - \bar{n} \cdot \left( {1 - F\left( {\rm ML} \right)} \right)}} }}{{1 - \left[ {F\left( {\rm ML} \right)} \right]\bar{n}}} \le 1$$

(42)

Equation Eq. (42) holds true for all ${\text{for }}n \ge 0 {\text{ and }}0 \le F\left( {\rm ML} \right) \le 1$, as shown in Fig. 24. From the figure, it can be seen that the difference between the two formulations approaches zero for large $\bar{n}$ 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.

Fig. 24

Ratio between Eqs. (40) and (41) with different n and F(ML) values

To illustrate this further in the context of this paper, consider the scenario with a magnitude of completeness, m_min, of − 2, M_UL = 3, and b = 1. The resulting formulations of F_max(ML) are compared in Fig. 25.

Fig. 25

Comparison between Eqs. (40) and (41) for m_min = − 2, M_UL = 3, b = 1