# 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 over a one-year period. The methodology employed for estimating the AEP is based on the Cornell-McGuire methodology85 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 $T_R=1/\lambda_I (i^\ast) =-T_e/\mathrm{ln}⁡(1-P_{T_e} [I>i^* ])$

### Uncertainty Analysis

The design earthquake obtained by the PSHA methodology incorporates the spatial and temporal variability directly, by means of the total probability theorem. Because some GMPE include in their formulation other source parameters, such as the rake angle of the fault plane, the dip of the fault plane, the faulting type (strike-slip, normal, reverse), the PSHA model implemented in OpenQuake86 also incorporates the epistemic uncertainty in the unknown source parameters through a logical tree of intensity models, recurrences and maximum rupture area sizes. The different branches of the model determine different spectral ordinates $$S_a(T)$$ in rock. Using a simple algorithm, the median and variance of a PDF probability density function of the seismic intensity of the PSHA model are estimated, where the median of the intensities represents the most probable value taken by the recurrence parameters, the hypocentre, and the maximum size of the fault.

## Results

#### Exceedance Probabilities (AEP)

The annual exceedance probability (AEP) curves for PGA are shown in Figure 22. All values were obtained from ground-motion models assuming rock site conditions (NEHRP site class AB). The inverse of the annual exceedance probability is the return period (TR) and the hazard curves in terms of TR are shown in Figure 23.

#### Peak Ground Accelerations (PGA)

Table 11 and Table 12 present the mean and 84th percentile of PGA values expected at GGM for different return periods and site conditions. The maximum design earthquake in rock assumes an AEP of 1:10,000 years and results in $$PGA\approx$$ 0.194 $$\mathrm{g}$$ (mean) and $$PGA\approx$$ 0.268 $$\mathrm{g}$$ (84th percentile). For site class “C,” it reported $$PGA\approx$$ 0.23 $$\mathrm{g}$$ (mean) and $$PGA\approx$$ 0.359 $$\mathrm{g}$$ (84h percentile).

 Peak-Ground Accelerations (PGA) [g] - (mean) NEHRP TR=500 TR=1000 TR=2500 TR=5000 TR=10000 AB 0.021 0.038 0.076 0.124 0.194 B 0.022 0.039 0.083 0.139 0.222 C 0.028 0.048 0.095 0.151 0.230 D 0.035 0.060 0.116 0.183 0.276
 Peak-Ground Accelerations (PGA) [g] - (+84%) NEHRP TR=500 TR=1000 TR=2500 TR=5000 TR=10000 AB 0.028 0.051 0.104 0.171 0.268 B 0.033 0.061 0.128 0.215 0.344 C 0.043 0.075 0.148 0.235 0.359 D 0.055 0.093 0.181 0.286 0.433

#### 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 24 presents the mean UHS spectral ordinates for different exceedance probabilities (return periods) in rock. 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 probabilistic methodology. Figure 25 shows mean values and other percentiles for the 10,000-years uniform hazard spectra. Mean uniform hazard spectra for other site conditions than rock in terms of NEHRP classes are shown in Figure 26. The plots for other percentiles are available in the project’s dashboard PSHA.

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

2. Pagani et al., “OpenQuake Engine.”↩︎