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Uncertainty Analysis and Expert Judgment in Seismic Hazard Analysis

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Abstract

The large uncertainty associated with the prediction of future earthquakes is usually regarded as the main reason for increased hazard estimates which have resulted from some recent large scale probabilistic seismic hazard analysis studies (e.g. the PEGASOS study in Switzerland and the Yucca Mountain study in the USA). It is frequently overlooked that such increased hazard estimates are characteristic for a single specific method of probabilistic seismic hazard analysis (PSHA): the traditional (Cornell–McGuire) PSHA method which has found its highest level of sophistication in the SSHAC probability method. Based on a review of the SSHAC probability model and its application in the PEGASOS project, it is shown that the surprising results of recent PSHA studies can be explained to a large extent by the uncertainty model used in traditional PSHA, which deviates from the state of the art in mathematics and risk analysis. This uncertainty model, the Ang–Tang uncertainty model, mixes concepts of decision theory with probabilistic hazard assessment methods leading to an overestimation of uncertainty in comparison to empirical evidence. Although expert knowledge can be a valuable source of scientific information, its incorporation into the SSHAC probability method does not resolve the issue of inflating uncertainties in PSHA results. Other, more data driven, PSHA approaches in use in some European countries are less vulnerable to this effect. The most valuable alternative to traditional PSHA is the direct probabilistic scenario-based approach, which is closely linked with emerging neo-deterministic methods based on waveform modelling.

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Correspondence to Jens-Uwe Klügel.

Appendix 1: Terminology

Appendix 1: Terminology

1.1 Entropy and Information

The basics of information theory were developed by Shannon (1948). It has found wide application in system science, data analysis and statistical inference.

For a probability distribution of a random variable with discrete values defined over x ∈ X, the information of the probability distribution is defined (Cooke, 1991) as:

$$ I(p) = \sum\limits_{i = 1}^{n} {p_{i} } \ln \,(p_{i} ) $$
(21)

In the original definition by Shannon (1948) the logarithm was related to base 2, but this difference is not important. The use of the natural logarithm serves the same purpose. This information measure is related to a corresponding entropy measure by:

$$ H(p) = - I(p) $$
(22)

The concept of information (entropy) can be formally extended to continuous probability density functions f x (x):

$$ I(f) = \int\limits_{a}^{b} {f_{x} (x)} \ln (f_{x} (x))\,{\text{d}}x $$
(23)

The disadvantage of the measure defined by Eq. 3 consists in the possible infinite numbers which I(f) can obtain. Therefore, it is common practice (Held, 2008) to use relative information by comparing two different probability density functions (Kullback, 1959):

$$ I(f_{1} ,f_{2} ) = \int\limits_{R} {f_{1} (x)} \ln \left( {{\frac{{f_{1} (x)}}{{f_{2} (x)}}}} \right)\,{\text{d}}x $$
(24)

Probability distributions derived from a given dataset that maximize entropy are called maximum entropy distributions. According to Eq. 2 maximum entropy distributions contain a minimum of information. That’s why they are called non-informative probability distributions. On the other hand, they can be developed with a very limited amount of information on the object of investigation (e.g. for zero events). The Gutenberg–Richter relationship (including the truncated case) belongs to the class of maximum entropy distributions (Berril and Davis, 1980).

Data analysis as used in modern risk analysis is based on the preferred use of informative probability distributions. The “degree of informativeness” of a probability distribution can be established by computing information measures. Usually the best “parametric fit” for a model is established by ranking possible alternatives based on their performance according to their scores with respect to a suite of information criteria (Vose, 2008).

1.2 Probability of Frequency Approach

The probability of frequency approach is a concept of engineering risk analysis developed and broadly used in the nuclear industry (Kaplan and Garrick, 1981) and used in many other engineering fields. It is also regarded as the classical approach to risk analysis (Aven, 2005). In this concept the term probability is used for the subjective probability (knowledge-based, epistemic) and the concept of frequency (aleatory variability) is used for the objective probability based on relative frequency.

Risk analysis is dealing with prediction of future events (representing a risk for the analyzed object) based on induction. Because critical events (like component failures or plant trips) are very rare events, their prediction is difficult. Using statistical inference techniques (or expert judgement), subjective probability distributions are assigned to the calculated frequencies of rare events. These subjective probabilities can be related to confidence levels and confidence intervals. E.g., a 90% confidence interval for an estimated failure frequency is interpreted according to this concept as a 90% probability that the estimated confidence interval will include the true value of the estimated frequency of a rare event. The mean value of the subjective probability distribution (epistemic) assigned to the estimated frequency is typically used as the relevant parameter used in risk analysis.

1.3 Neo-Deterministic Seismic Hazard Analysis

Neo-deterministic seismic hazard analysis is a physics based approach to seismic hazard analysis (SHA) which (in comparison to traditional deterministic seismic hazard analysis) replaces the use of empirical ground motion prediction equations (GMPEs) by waveform modeling. The term modeling SHA is used as a synonym. The use of the word “deterministic” should not be confused with non-probabilistic regulatory approaches, which represent, at best, crude simplifications of modern deterministic seismic hazard analysis. An example of an actual application of the neo-deterministic method is given in Peresan et al. (2009).

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Klügel, JU. Uncertainty Analysis and Expert Judgment in Seismic Hazard Analysis. Pure Appl. Geophys. 168, 27–53 (2011). https://doi.org/10.1007/s00024-010-0155-4

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