Statistical modeling of ground motion relations for seismic hazard analysis

Abstract

We introduce a new approach for ground motion relations (GMR) in the probabilistic seismic hazard analysis (PSHA), being influenced by the extreme value theory of mathematical statistics. Therein, we understand a GMR as a random function. We derive mathematically the principle of area-equivalence; wherein two alternative GMRs have an equivalent influence on the hazard if these GMRs have equivalent area functions. This includes local biases. An interpretation of the difference between these GMRs (an actual and a modeled one) as a random component leads to a general overestimation of residual variance and hazard. Beside this, we discuss important aspects of classical approaches and discover discrepancies with the state of the art of stochastics and statistics (model selection and significance, test of distribution assumptions, extreme value statistics). We criticize especially the assumption of logarithmic normally distributed residuals of maxima like the peak ground acceleration (PGA). The natural distribution of its individual random component (equivalent to exp(ε 0 ) of Joyner and Boore 1993) is the generalized extreme value. We show by numerical researches that the actual distribution can be hidden and a wrong distribution assumption can influence the PSHA negatively as the negligence of area equivalence does. Finally, we suggest an estimation concept for GMRs of PSHA with a regression-free variance estimation of the individual random component. We demonstrate the advantages of event-specific GMRs by analyzing data sets from the PEER strong motion database and estimate event-specific GMRs. Therein, the majority of the best models base on an anisotropic point source approach. The residual variance of logarithmized PGA is significantly smaller than in previous models. We validate the estimations for the event with the largest sample by empirical area functions, which indicate the appropriate modeling of the GMR by an anisotropic point source model. The constructed distances like the Joyner-Boore distance do not work well for event-specific GMRs. We discover also a strong relation between magnitude and the squared expectation of the PGAs being integrated in the geo-space for the event-specific GMRs. One of our secondary contributions is the simple modeling of anisotropy for a point source model. ground motion relation, probabilistic seismic hazard analysis, area-equivalence, regression analysis, extreme value statistics, model selection, statistical test, random function * *

Estimation of the upper bound of seismic hazard curve by using the generalised extreme value distribution

“…The results demonstrated that a superior fit to the data and, in turn, more accurate acceleration estimates are obtained under the latter assumption. Similar considerations were expressed by Raschke (2013), who criticized the log-normal assumption and noted that the GEVD is the natural distribution for residuals of maxima such as PGA. Huyse et al (2010) applied the peaks over threshold method to analyse both the raw PGA data and the logarithmic residuals of PGA, and concluded that the generalised Pareto distribution (GPD), with the negative shape parameter, provided a model that was more accurate for the tail fractions of both the studied datasets.…”

A novel model and estimation method for the individual random component of earthquake ground-motion relations

“…The latter is a result of extreme value statistics (cf. Schlather, 2002, theorems 1 and 2), as was mentioned already for PSHA by Raschke (2013a).…”

“…However, this leads to an overestimation of the dispersion measures V(ε a ) and σ a =√V(ln(ε a )) and, consequently, results in a global overestimation of the seismic hazard (cf. Raschke, 2013a). I particularly emphasise that a global overestimation of V(ε a ) and σ a cannot compensate for the local biases of Fig.…”

“…The ground-motion relation (GMR) is an important element of PSHA that describes the relation between the local ground-motion intensity and the event parameters, such as the magnitude. Another term for ground-motion relation is ground-motion prediction equation; however, in this paper, the term GMR (Atkinson, 2006;Raschke, 2013a) is preferred. The reasons are that GMR includes more elements than one equation and regression analysis does not have to be applied directly.…”

“…Accordingly, an alternative estimation for V(ε a ) and σ a was needed and was developed, as described in this paper. Raschke (2013a) has suggested estimating the parameters of ε a by determining the sample of the random difference ξ by:…”