Demanda del proyecto

The risk to earthquake action of a given portfolio of buildings is commonly described through a loss exceedance curve that specifies the frequency, usually expressed annually, with which specific values of loss will be exceeded. When using a single Ground Motion Prediction Equation (GMPE), this annual frequency, or rate, can be computed based on the application of the total probability theorem, as follows:

𝛾(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚} = ∑^𝑁_𝑛=1𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}). 𝛾_𝑛 ( 5.3 )

92 Chapter 5 l given the seismic event, or rupture, Rupn , using GMPEm, 𝛾_𝑛 is the annual rate of occurence of Rupn, and N is the number of different (assumed independent) possible earthquake ruptures determined by an earthquake rupture forecast (ERF) (Pagani, et al., 2014). In the present case, the seismological model developed by Vilanova & Fonseca (2007) has been used for the purposes of building the ERF, as described in Chapter 4.

The probability of exceedance of loss l given the occurrence of a Rupn is an uncertain variable, expressed by the following equation:

𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) = ∫ 𝑃(𝐿 > 𝑙|_{𝐼 𝐼}). 𝑓(𝐼|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}). 𝑑𝐼 ( 5.4 ) In which I represents the spatial distribution of seismic intensity – Sa(T1) in the present case - across the portfolio of interest, 𝑓(𝐼|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) is the probability density function of I determined by GMPEm, conditioned on 𝑅𝑢𝑝_𝑛, and 𝑃(𝐿 > 𝑙|_𝐼) is the probability of exceedance of loss l given the spatial distribution of seismic intensity I.

Equation 5.4 considers the fact that, given a 𝑅𝑢𝑝_𝑛, the distribution of intensity across a spatially distributed portfolio of assets is uncertain. Generally, it is not practical to solve the aforementioned equation in its closed form. Thus, given an earthquake rupture, the characterization of 𝑓(𝐼|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) in the dI domain is herein performed through the generation of a number of ground motion fields (Pagani, et al., 2014) of Sa(T1) that incorporates different simulations of spatially correlated ground shaking values at the location of the collection of assets.

Several studies have addressed the issue of generating spatially correlated ground motion fields of spectral ordinates (e.g. Weatherill et. al. (2015), Silva (2016)) and such matter will not be herein addressed in detail. However, provided that a number of random fields J is large enough to adequately reflect 𝑓(𝐼|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}), then Equation 5.4 is numerically solved as presented in Equation 5.5:

𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) = ∑^𝐽_𝑗=1𝑃 (𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝐹𝑗,𝐺𝑀𝑃𝐸𝑚}).¹_𝐽 ( 5.5 )

Where 𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝐹,𝐺𝑀𝑃𝐸𝑚}) is the deterministic value of probability of exceedance of loss l given the ground motion field j generated for the 𝑅𝑢𝑝_𝑛 and 1/J is one equiprobable realization of 𝑓(𝐼|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚})𝑑𝐼.

Equation 5.5 embodies what is commonly referred as a model mixture (Surajit &

Lindsay, 2005) (i.e. sum of probability densities), according to which, if one considers

Chapter 5 93 𝑃 (𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝐹𝑗,𝐺𝑀𝑃𝐸𝑚}) as the mean of a normal random variable with zero variance, the probability density function of 𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) is completely defined by a Gaussian mixture (GM) model parameterized by:

𝜇^𝐺𝑀_{𝑃(𝐿>𝑙|}

𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚) = {𝑃 (𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝐹𝑗,𝐺𝑀𝑃𝐸𝑚})} ; 𝑗 = 1 … 𝐽 ( 5.6 ) 𝜔^𝐺𝑀_{𝑃(𝐿>𝑙|}

𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚)= {¹_𝐽} ; 𝑗 = 1 … 𝐽 ( 5.7 ) In which 𝜇^𝐺𝑀_{𝑃(𝐿>𝑙|}

𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚) and 𝜔^𝐺𝑀_{𝑃(𝐿>𝑙|𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚)} are respectively the mixture mean and weight vectors from all component densities.

Equation 5.3 is thus a weighted sum of (assumed) independent random variables, based on which its mean and variance are established by the following Equations 5.8 and 5.9.

𝜇_{𝛾(𝐿>𝑙)𝐺𝑀𝑃𝐸𝑚} = ∑^𝑁_𝑛=1𝛾_𝑛. 𝜇_{𝑃(𝐿>𝑙|}

𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚) ( 5.8 ) 𝜎²_{𝛾(𝐿>𝑙)𝐺𝑀𝑃𝐸𝑚} = ∑^𝑁_𝑛=1𝛾_𝑛². 𝜎²

𝑃(𝐿>𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) ( 5.9 ) Where 𝜇_{𝑃(𝐿>𝑙|𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚)} and 𝜎²_{𝑃(𝐿>𝑙|𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚)} are the mean and variance of 𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}), respectively.

Despite the fact that 𝜇_{𝑃(𝐿>𝑙|𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚)} and 𝜎²_{𝑃(𝐿>𝑙|𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚)} can explicitly be determined, the body of the probabilistic distribution of 𝛾(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚}is not known a priori. Therefore, its probability density is herein empirically established through the numerical simulation of a sufficiently large number, R, of realizations (denoted as 𝛾^𝑟(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚}), with 𝑟 = 1 … 𝑅 = 2000), as follows:

𝛾^𝑟(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚} = ∑^𝑁_𝑛=1𝑃^𝑟(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}). 𝛾_𝑛 ( 5.10 ) In which 𝑃^𝑟(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) are (uncorrelated) random realizations of 𝑃(𝐿 > 𝑙|_{𝑅𝑢𝑝𝑛,𝐺𝑀𝑃𝐸𝑚}) from each of the N corresponding distributions.

5.3.1.1 Epistemic uncertainty and loss estimation results

A logic tree approach is used in this study in order to consider the epistemic uncertainty associated with the choice of GMPEs to use in conjunction with the

94 Chapter 5 aforementioned seismological model, and, as presented in Chapter 4, the models developed by Atkinson & Boore (2006) and Akkar & Bommer (2010) are considered, with 0.70 and 0.30 logic tree weights, respectively.

A fundamental issue in deciding how to treat the epistemic uncertainty in hazard analysis is the interpretation of what the weights on the logic tree branches represent (Abrahamson & Bommer, 2005). As stated by Vick (2002), one interpretation favours the assumption that the weights are frequency-based probabilities of the alternative models being correct, whereas the alternative view is one according to which logic tree branches represent our relative confidence in the alternative models. As established in Chapter 4, the distributions of a given IMi conditioned on Sa(T1)=a used as target for record selection take into account the contribution of all the scenarios defined by 3D disaggregation on Magnitude, Distance and GMPE, which implies a frequency-based interpretation of logic tree weights assigned to different GMPEs. Therefore, the probabilistic distribution of loss exceedance rate determined by the contribution of all the considered GMPEs is herein determined by the following equation:

𝛾(𝐿 > 𝑙) = ∑^𝑁_𝑚=1^{𝐺𝑀𝑃𝐸} 𝛾(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚}. 𝑃(𝐺𝑀𝑃𝐸_𝑚) ( 5.11 ) Where 𝑃(𝐺𝑀𝑃𝐸_𝑚) is the logic tree weight assigned to GMPEm, and NGMPE is the total number of GMPEs considered.

Similarly to Equation 5.5, Equation 5.11 represents the weighted sum of probability densities. However, since the distribution of 𝛾(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚} is defined numerically, the probabilistic density function of 𝜇_{𝛾(𝐿>𝑙)} is herein computed through the simulation of NGMPE sets of 𝑄^𝑚 realizations of 𝛾^𝑟(𝐿 > 𝑙)_{𝐺𝑀𝑃𝐸𝑚}, in which 𝑄^𝑚= _{𝑚𝑖𝑛{𝑃(𝐺𝑀𝑃𝐸}^{𝑃(𝐺𝑀𝑃𝐸𝑚)}

𝑚)}× 𝑅 and 𝑚𝑖𝑛{𝑃(𝐺𝑀𝑃𝐸_𝑚)} is the minimum of the logic tree weights assigned to each of the considered GMPEm.

As highlighted by Abrahamson & Bommer (2005), if one considers that logic tree weights are, on the other hand, measures of the relative merit of each GMPE, then the mean value of 𝛾(𝐿 > 𝑙) as defined in Equation 5.11 does not correspond to the expected value in its strict statistical sense. The Author acknowledges the importance of this matter;

however, such discussion is considered beyond the scope of this study.

Chapter 5 95