# Probabilistic hazard assessment

## Methodology

A Probabilistic Seismic Hazard Assessment (PSHA) quantifies the contributions of all sources for all possible magnitudes within the affected area. In the same way as a Deterministic Seismic Hazard Assessment (DSHA), the PSHA stages identify all earthquake sources capable of producing significant ground-motion at the site and enable the selection of a minimum number of independent ground-motion models that capture the expected range of all possible ground-motions in the target region. The fundamental difference is the inclusion of random uncertainty in the calculation of the spatial and temporal distribution of earthquakes, as well as the (aleatory) uncertainty associated with the ground-motion models.

In a probabilistic framework, ground-motion intensities (PGA, Sa) are expressed in terms of annual exceedance probability (AEP). The Annual exceedance probability (AEP) is the rate of exceedance of a specific level of seismic intensity due to the potential occurrence of earthquakes from that source during a year. The methodology employed for estimating the AEP is based on the Cornell-McGuire methodology49 which combines earthquake occurrence models, seismic source zone models, magnitude-recurrence relationships, and ground-motion intensity models through the total probability theorem.

Assuming the occurrence of earthquakes follows a Poisson process (i.e. that average time between events is known, but the exact timing of events is random), the probability that a certain strong motion intensity $$I$$ exceeds a certain value $$i^\ast$$ during a time interval $$T_e$$ is given by $$P_{T_e}\left[I>i^\ast\right]=1-\mathrm{exp}(-\lambda_I (i^\ast)\,T_e )$$ where $$\lambda_I (i^\ast)$$ is the annual rate of exceedance and $$T_e$$ is the exposure time. For a given source, the annual rate of exceedance of an intensity target level $$i^\ast$$ $$\lambda_I(i^\ast )$$ can be expressed by the Total Probability Theorem (TPT):

$\lambda_I^{(s)}(i^\ast)\simeq \nu_o^{(s)}\sum_{N_R}\sum_{N_M}P[I>i^\ast|m_j,r_k]^{(s)} f_M^{(s)} (m_j)\,\Delta m\,f_{R|M}^{(s)} (r_k |m_j) \Delta r$

where $$M_{max}^{(s)}$$ is the MCE of the source (s) and $$\nu_o=P[M>M_o^{(s)}]$$ is the annual frequency of occurrence of earthquakes with magnitude greater than a minimum value $$M_o$$. Using the TPT the annual exceedance rate of earthquakes with magnitude greater than $$m^\ast$$ can be expressed in terms of an asymptotically truncated Gutenberg-Richter magnitude-frequency distribution. The discrete mass probability for magnitudes $$P[M=m_j]^{(s)}=f_M^{(s)}(m_j)\,dM$$ can be obtained from the first derivative of the magnitudes cumulative-density function $$F_M^{(s)} (m_j)$$ valid for $$M_{max}^{(s)}<m^\ast<M_o^{(s)}$$: $\lambda^{(s)}[M>m^\ast]=\lambda_M^{(s)}(m^\ast)=\nu_o^{(s)}(1-F_M^{(s)}(m^\ast))\simeq \nu_o^{(s)}\dfrac{\mathrm{e}^{-\beta\left(m^\ast-M_o^{(s)}\right)}}{1-\mathrm{e}^{-β \left(M_{max}^{(s)}-M_o^{(s)} \right) }}$ The discrete mass probability function for distances $$f_{R⁄M}^{(s)} (r^\ast |m^\ast ) dr^\ast=P[R=r^\ast |m^\ast ]^{(s)}$$ can be obtained numerically from the histogram of an array of epicentres inside the boundaries of each source.

The Return Period $$T_R$$ is defined as the inverse of the total AEP $T_R=1/\lambda_I (i^\ast) =-T_e/\mathrm{ln}⁡(1-P_{T_e} [I>i^* ])$

## Results

#### Exceedance Probabilities (AEP)

The annual exceedance probability (AEP) curves for PGA are shown in Figure 15. All values were obtained from ground-motion models assuming rock site conditions (NEHRP site class AB). Figure: 15: Annual exceedance probability (AEP) curves for PGA in [g]

Assuming an exposure period of 50 years, the probability of exceedence (POE) of a given intensity level for PGA are shown in Figure 16 Figure: 16: Total exceedance probability (POE) curves for PGA in [g]

The inverse of the annual exceedence probability is the return period (TR) and the hazard curves in terms of TR are shown in Figure 17 Figure: 17: Return Periods for PGA in [g]

#### Peak Ground Accelerations (PGA)

Table 5 presents 84th percentiles of PGA values expected at Simandou for different return periods. The maximum design earthquake in rock assumed an AEP of 1:10,000 years and results in $$PGA\approx$$ 0.101 $$\mathrm{g}$$. For site class “C,” a 1:10,000-year AEP event reported $$PGA\approx$$ 0.146 $$\mathrm{g}$$ (84%).

 Peak-Ground Accelerations (PGA) [g] - (+84%) NEHRP TR=500 TR=1000 TR=2500 TR=5000 TR=10000 AB 0.011 0.019 0.038 0.063 0.101 B 0.013 0.022 0.045 0.077 0.126 C 0.018 0.030 0.058 0.093 0.146 D 0.023 0.039 0.073 0.116 0.180

For site class “C,” a 1:10,000-year AEP event reported $$PGA\approx$$ 0.146 $$\mathrm{g}$$ (84%).

#### Uniform Hazard Spectrum (UHS)

The Uniform Hazard Spectrum (UHS) is the set of spectral ordinates (for all structural periods of interest) that have the same probability of exceedance. Figure 18 presents the uniform-Hazard spectral ordinates for different exceedence probabilities in rock. Figure: 18: Uniform Hazard Spectra for different exceedence probabilities in [g], NERHP AB

Spectral ordinates at AEP=1:10,000 yr represents the maximum design earthquakes $$S_a(T_n)$$ for rock at the project site, obtained from a probabiistic methodology. Figure 19 shows median values an other quantiles for the 10,000-years uniform hazard spectra. Figure: 19: Uniform Hazard Spectra ordinates for 1:10,000 years AEP in [g]

Uniform hazard spectra for other site conditions than rock are shown in Figure 20 (84%) Figure: 20: Scenario-based spectral ordinates for different site conditions in [g]

The proposed numerical model combines the uncertainties of all stages defining the seismic design parameters for different service levels and confidence intervals. The combination of very long structural periods $$(T_n>3 sec)$$ and very low exceedance probabilities $$(AEP > 1:5000)$$ presents a spurious numerical attenuation as can be seen in the two figures above. The numerical noise at long periods arises from the inteprolation error of the few values reported by the GM models at those ordinates combined with very low exceedance probabilities, and should therefore be ignored as they are physically meaningless. As a general rule, the existence of very flexible structures, such as high tailings dams subjected to moderate to high intensity earthquakes founded on very deep strata of soft soils, requires different methodologies to estimate the dynamic response over long periods, and is beyond the scope of the present study.

1. Robin K. McGuire, Seismic Hazard and Risk Analysis (Earthquake Engineering Research Institute, 2004).↩︎