# Hazard disaggregation

## Methodology

The PSHA combines the probabilities of all earthquake scenarios with predictions of resulting ground motion intensity, as described in previous sections. To identify the contributing events for a given ground motion level and return period, the seismic hazard disaggregation computes the relative contribution of earthquake parameters magnitude and distance to the total hazard. The full disaggregation methodology can be found in Bazzurro et. al.50 Conceptually, the contribution of a target magnitude to the total hazard can be obtained from the Bayes’ theorem from the total AEP and the marginal AEP from:

$\bar{m}=\mathrm{mode}\left[ \dfrac{\lambda[I>i^\ast|M=m_1]}{\lambda_I(i^\ast)},...,\dfrac{\lambda[I>i^\ast|M=m_N]}{\lambda_I(i^\ast)}\right]$ where $$\lambda[I>i^\ast|M=m_u]$$ is the marginal AEP for events of magnitude $$\bar{m}$$ is the most frequent (mode) value from all possible magnitudes $${m_k}$$

$\lambda[I>i^\ast|M=m_u]\simeq \sum_{N_S}\nu_o^{(s)}\sum_{N_R}\sum_{N_M}\Phi_C^{(s)} (\varepsilon_{j,k}^\ast)f_M^{(s)}(\bar{m})\,\Delta m f_{R|M}^{(s)}(r_k|m_u)\Delta r$ and $$\lambda[I>i^\ast]$$ is the total AEP for all sources,

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

Likewise, the contribution of a target distance to the total hazard can be obtained from the most frequent value from all possible marginal AEPs for $${r_v}$$

$\bar{r}=\mathrm{mode}\left[ \dfrac{\lambda[I>i^\ast|R=r_1]}{\lambda_I(i^\ast)},...\dfrac{\lambda[I>i^\ast|R=r_N]}{\lambda_I(i^\ast)} \right]$

with the marginal AEP given by:

$\lambda[I>i^\ast|R=r_v]\simeq \sum_{N_S}\nu_o^{(s)} \sum_{N_R}\sum_{N_M}\Phi_C^{(s)} (\varepsilon_{j,k}^\ast)f_M^{(s)}(m_j)\,\Delta m f_{R|M}^{(s)}(r_v|m_j)\Delta r$ With this procedure, the fractional contribution of different subsets of the events to the total hazard is computed. The disaggregation by magnitude and distance bins allows the dominant scenario earthquake (M-R bins) to be identified. When several mechanisms contribute to the total hazard, the disaggregation procedure can be performed for a family of sources (for example, all sources related with subduction interface mechanisms) and identify the M-R bins for each mechanism. The disaggregation by sources allows the dominant seismic source to be identified.

A given intensity level $$i^*$$, represents an annual exceedance probability and all M-R bins shall be defined for all annual exceedance probabilities of the intensities at the project site. On the other hand, the ground-motion models are defined for spectral ordinates $$Sa(T_n)$$ representing seismic intensities of an SDOF oscillator with fundamental period $$T_n$$, and therefore, there will be a pair of modal $$M-R$$ values controlling the hazard input for each spectral ordinate $$Sa(T_n)$$.

## Results

The disaggregation analysis was performed for fundamental periods $$T_n$$ ranging between $$T_n=0$$ (PGA) and $$T_n=4.0\,\mathrm{s}$$, and for annual exceedance probabilities of 1/500,1/1,000,1/2,500, 1/5,000 and 1/10,000 years.

M-R bins associated with fundamental periods of $$T_n=0$$ (PGA) represent characteristic design events that control the seismic hazard in rigid structures, such as low rise embankments or buried structures. Figure 21 shows the M-R contributions for an event with an AEP of 1/10,000 years and $$T_n=0\,\mathrm{s}$$, for rock site conditions.

M-R bins associated with fundamental periods near $$T_n=0.5\,\mathrm{s}$$ represent characteristic design events that control the seismic hazard in flexible structures, such as tall waste rock dumps, elevated structures, or soft soil strata. Figure 22 shows the M-R contributions for an event with an AEP of 1/10,000 years and $$T_n=0.5\,\mathrm{s}$$, for rock site conditions.

In the case of deep strata of very soft soils where potentially liquefiable layers exist, the design characteristic events will be associated with M-R bins with fundamental periods close to $$Tn=1.5\,\mathrm{s}$$. Figure 23 shows the M-R contributions for an event with an AEP of 1/10,000 years and $$T_n=1.5\,\mathrm{s}$$, for rock site conditions.

For rigid and PGA-controlled structures ($$T_n<<0.5\,\mathrm{s}$$), the hazard disaggregation reported bins with $$M_w \sim 4.75$$ (Figure 21). Flexible structures $$(T_n>0.5\,\mathrm{s})$$ reported events with $$M_w \sim 5.10-5.75$$ (Figure 23) as main contributors.

For all AEPs and spectral ordinates, nearby sources (25 - 35 km) have a preponderant contributions on the site’s seismic hazard.

Table 6 presents a summary of deaggreggation pairs $$M_w-R$$ for the different natural periods.

 Hazard Deaggreggation - Modal Values NEHRP Tn TR Mm Rm [km] AB 0 500 4.75 63 AB 0 1,000 4.75 38 AB 0 2,500 4.75 30 AB 0 5,000 4.75 30 AB 0 10,000 4.75 22

1. “Disaggregation of Seismic Hazard,” Bulletin of the Seismological Society of America 89 (1999): 501–20.↩︎