Treatment of and sensitivity to epistemic uncertainty in seismic risk assessment of infrastructures

Seismic risk assessment of an infrastructure, intended as a system of systems including buildings, lifelines and critical facilities, is typically affected by several sources of uncertainty, classified as aleatory or epistemic. Momentum to this work came from the need not only to properly take into account all these uncertainties, but also to provide the confidence in the estimate and quantify the contribution of the employed models and parameters to the total uncertainty. After a brief overview about treatment of and sensitivity to epistemic uncertainty, this paper focuses on some critical issues, advocates the use of parallel models arranged in a so-called logic tree, and demonstrates the applicability of (modified) ANOVA to evaluate sensitivity of the total variance in the risk to each component of the input epistemic uncertainty, with reference to a “synthetic” city composed of buildings and a water network. Results show how the methodology can give the analyst a clear indication on which models or parameters are the most influential and thus deserve increased “knowledge” in order to reduce the total epistemic uncertainty in the problem. INTRODUCTION Modern societies heavily rely on their infrastructure to produce and distribute the continuous flow of essential goods and services they need (PCCIP, 1997). From a systemtheoretic point of view, the infrastructure is a system of systems (SOS), a super-system including a number of spatially distributed systems (i.e., buildings, lifelines and critical facilities). The best practice for the seismic risk assessment of an infrastructure should include not only taking into account all the relevant uncertainties affecting the problem, but also providing the confidence in the estimated output and quantifying the contribution of each input model to the total output uncertainty (i.e., sensitivity). Uncertainty affects seismic risk to an infrastructure in the following aspects: • Cause or hazard: Regional seismicity (event magnitude and location, local seismic intensities at vulnerable components’ sites). • Physical damage: Fragility of vulnerable components as a function of local seismic intensities (fragility functions). • Functional consequences: Network flow analysis. • Impact (e.g., estimation of injured, fatalities, displaced population, economic loss). The above uncertainties can be classified as aleatory or epistemic, and both characterize the seismic input as well as the physical system. For instance, the geometry of seismic sources, their activity rate, the maximum magnitude of earthquakes they can generate, are all examples of quantities affected by epistemic uncertainty and entering into the seismic hazard evaluation, while the damage state of a component given a 13 International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019 2 value of the intensity measure is an example of aleatory uncertainty affecting the system. This paper focuses on epistemic uncertainty, that is, uncertainty in the choice of a model among different candidate models, called Type I in the following, or uncertainty in the parameters of a chosen model, denoted as Type II, or both. Published literature to date features many works addressing the treatment of and the sensitivity to epistemic uncertainty (of the output) in seismic risk assessment (e.g., Helton and Oberkampf, 2004). However, most of these works give only a partial view of the problem, dealing with either the treatment of uncertainties (e.g., Celic and Ellingwood, 2010, and Rokneddin et al., 2015) or the computation of confidence bounds (e.g., Rubinstein and Kroese, 2016), or the sensitivity (e.g., Celarec et al., 2012). Further, the available works focus on seismic hazard only, or seismic risk assessment of structural systems. The goals of this paper are i) to summarize relevant approaches in the current literature, discussing some critical issues, and ii) to propose and demonstrate a methodology that overcomes those issues, with reference to the seismic risk assessment of an infrastructure. The next two sections include a brief overview about treatment of and sensitivity to epistemic uncertainty in seismic risk assessment, while Section 3 presents an application to a “synthetic” city, composed of buildings and a water network, derived from that in Franchin and Cavalieri (2015). Results are presented in terms of distribution of the mean annual frequency (MAF) of exceedance curves of displaced population due to epistemic uncertainty, and contribution of different components of epistemic uncertainty to total output uncertainty. 1. TREATMENT OF EPISTEMIC UNCERTAINTY 1.1. Possible approaches As already said, uncertainty in the problem is partly aleatory and partly epistemic. It is useful to recall the possible approaches to the treatment of the epistemic component, which vary depending on its type: 1. Epistemic uncertainty of Type I: parallel models Θ are considered in each step of the analysis, arranged in what is often called a logic tree, and distinct simulations are run for each different combination of branches, thus yielding multiple results (e.g., MAF of exceedance curves of a performance metric). Weights, summing up to 1, are attached to branches to reflect subjective degrees of belief of the analyst in each model. This is common practice in probabilistic seismic hazard analysis (PSHA). A typical uncertainty in model form considered in PSHA is represented by the ground motion prediction equation (GMPE). The outcome is usually expressed in terms of mean hazard curve over the logic tree, obtained as a weighted average of curves from each branch (Bommer and Scherbaum, 2008). Often, upper and lower fractile curves or, alternatively, a confidence interval around the mean curve are computed based on the same data (set of curves from the tree) to quantify the effect of epistemic uncertainty on the results. 2. Epistemic uncertainty of Type II: each model parameter θ is modelled with a random variable, whose distribution describes its epistemic uncertainty. • a) These variables (e.g., the maximum magnitude M! "#) can be arranged in a hierarchical model, together with aleatory uncertainty (e.g., the magnitude M). In this case the risk analysis yields a single result, incorporating the effect of both aleatory and epistemic uncertainty (e.g., Franchin and Cavalieri, 2015). • b) Alternatively, and with a higher associated computational effort, the risk analysis can be repeated for discrete values of each parameter θ (e.g., 16, 50 and 84 fractiles). This approach practically leads to arrange parameters in 13 International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019 3 a logic tree, as done for Type I uncertainty. 3. Epistemic uncertainty of both Types I and II: • a) One possibility is to adopt approach 1 for Type I and approach 2a for Type II uncertainties. This comparatively cheaper approach should be followed only as a way to carry out the expectation over all sources of uncertainty, presenting the results as the mean over the logic tree. It should not be used to compute confidence intervals or fractiles, because they would refer only to part of the total epistemic uncertainty. • b) The second approach, involving approach 1 for Type I and approach 2b for Type II uncertainties, consists in building an expanded logic tree combining both Type I and II uncertainties. Since they come from a probability distribution, both discrete values of the model parameters and their weights attached to tree branches could be assigned, for instance, according to Miller and Rice (1983). It should be clear from the above that the best practice for the treatment of Type II epistemic uncertainty would be to adopt approaches 2b or 3b. The problem with these approaches, however, is that, in rigour, they can only be applied when all parameters are statistically independent, otherwise variation in one parameter changes the (conditional) distribution of the others and neglecting this makes the results dependent on the ordering of branches. This occurs in PSHA, for instance, with reference to the parameters of the GutenbergRichter recurrence relationship (a and b values, and M! "# ), for which independent sequential branches are adopted, neglecting the correlation between a and b (Bommer and Scherbaum, 2008). To the best knowledge of the authors, this issue remains unsolved, and possible solutions are proposed in Section 2.2. 1.2. Quantification of output uncertainty due to the epistemic component In a probabilistic framework where a logic tree is used to deal with epistemic uncertainty (approaches 1, 2b and 3b), risk is estimated through a chain of modules, intended as groups of parallel choices (i.e., alternative models or model parameter values). The logic tree results in a number N of MAF of exceedance (λ) curves, denoted λ-curves in the following, for a performance metric of interest. Each λ-curve is related to one out of N simulations (e.g., Monte Carlo), each encompassing multiple runs. Propagation of epistemic uncertainty through the logic tree results in a distribution f! (x) of the output X (either λ for a fixed performance metric value or performance metric values for a fixed value of λ). As a minimum, f! (x) can be summarized through its mean μ! and variance σ! Computing the mean is normally termed harvesting the logic tree in PSHA practice. Finally, variability in X is also often expressed through weighted fractiles, as an alternative to σ! , and the confidence interval around the mean curve is also used (but of course this is related to σ! but not alternative to it). For a tree with NΘ models and Nθ parameters, μ! can be estimated as the weighted average of X: