Report - PEER - University of California, Berkeley

of events with [X1>x 1 , X 2 >x 2 , …] and P[C|X], is the conditional probability of Cgiven an event with parameters X. With this information the “total probabilitytheorem” states that λC = ∫ P [ C | X] | dλ(X)| . The differential | d λ(X)| is the meanannual frequency density (or absolute value of the partial derivative of λ(X)) timesdx 1 dx 2 …. In the following sections we seek various ways to estimate P[C|X] underthe assumption that the seismologist has sole responsibility for λ(X) and that this awell studied and commonly practiced problem of engineering seismology. Forsimplicity and concreteness we shall assume below that X = [M, R], the magnitudeand distance of the earthquake.2.2 Option A: Direct Estimation of P[C|X=x] and λ CEstimating P[C|X] is a joint responsibility of the seismologist and structural engineer.A direct way of estimating P[C|X] is for the seismologist to prepare a sample of n’(more strictly, a random sample of equally likely) accelerograms. The structuralengineer must then analyze his structure for each record and count the number ofobservations, r, of the event C, e.g., collapse. His estimate of P[C|X] is then simplyr/n’. This process must be repeated for m well selected sets of the parameters, X i , i =1, …m, for a total of n = n’m records. Then the estimate of λ Cis λC≈ ∑ P[ C | Xi] ∆λ(Xi) in which ∆λ(X i ) is approximately the annual frequency ofevents with characteristics X i . (The set of m sets of characteristics should beeffectively exhaustive and exclusive.)In practice one must have a sample size n’ large enough to estimate each of the mP[C|X]’s adequately. For comparative purposes suppose that this condition can besatisfied by estimating the median MIDR to within a standard error of 10%. Then thenecessary sample size is about (0.8/0.1) 2 , or more than 50, given that the coefficientof variation (COV) of the MIDR of a typical frame in near failure regime is at least0.8. (This is conservative as, given only {M, R}, the standard deviation of the naturallog of the peak response of a simple linear oscillator is 0.7 or more.) Assuming thatm = 10 to 20 in order to cover adequately the range of say X = {M,R}, the totalrequired sample size is of order 1000. In advanced application this number can bereduced by a factor of 2 or more by using “smart” Monte Carlo, or by, for example, aresponse surface analysis 2 or regression of MIDR on X. Of course once theseanalyses have been completed they can be used to find λ C for many different events,C, such as other failure modes or other values of MIDR or economic losses. Examplesof U. S. researchers using such methods include Ang, Wen, and Beck and their coworkers.2 Note that this would be in effect a structure-specific MIDR “attenuation law”.41