Recommended Interim Procedure. - Continued

DOE-STD-1024-92

adjust it to reflect the mean hazard. This factor is found as follows. It is the same

value for all sites.

Find a composite 85 percentile/median factor by taking the

Step (3)

(geometric) average of these ratios from the EPRI and LLNL-4GX results above.

Recall that these ratios consistently have a ratio of 2 in the higher frequencies;

therefore the geometric average will be 2 times the lower (EPRI) 85

percentile/median ratio. The EPRI ratio (for PGA) is 3.0 to 3.5 for amplitudes of

0.2g or less. Let us use the higher value, 3.5, for all amplitudes. The

representative, composite factor for 85 percentile/median hazard is, therefore, 2

x 3.5 or 5. Note that the simple average would be very similar ((3.5 + 2 x 3.5)/2 =

5.2). This step is the key one in the proposed procedure. It is a major decision to

base this "uncertainty factor" on the results for the high frequency end of the

spectrum. The decision is based on the concerns already expressed about the

extreme values at the low frequency end, plus the generally greater consistency

within and between studies at the high frequency end. It is hoped that future

studies will permit us to improve upon this step.

Find the corresponding mean/median hazard factor. For

Step (4)

this step we assume an underlying lognormal distribution for simplicity (LLNL has

verified that their uncertainty distributions over hazard are well represented by a

lognormal distribution). It is easily shown5 that for a lognormal distribution a ratio

of 85 percentile/median of x implies a mean/median ratio of exp {(In x)2} or here

exp {(In 5)2} = 3.6.

Step (5)

Find the PGA multiplier corresponding to this hazard

mean/median multiplier. It is well known (based on theory and observation) that

hazard curves plot approximately linearly on log-log paper, at least over the range

of interest here, i.e., hazard ratios of an order of magnitude or less. This implies H

is proportional to y-b, in which H is hazard, y is ground motion level and b is the

slope on log-log paper. It follows that for a mean/median hazard ratio of x (e.g.,

the ratio 3.6 above), the corresponding ratio of ground motion values is (x)1/b, e.g.,

for a mean/median hazard ratio of 3.6 and a b of 3.5, the ratio of corresponding

PGA values is (3.6)1/3.5 = 1.44. This factor should be used to adjust upward the

"composite" PGA found in Step 2 above; the lower PGA value is associated with

the median estimate of the hazard and the larger value is associated with the

mean estimate of the hazard. The slopes of hazard curves in the EUS fall in a

relatively narrow range for a given hazard level. This fact has been confirmed by

several investigators. R.P. Kennedy suggests that in the 10-3 to 10-5 hazard range

Because 85 percentile = (median) exp { InH} and mean = (median) exp { InH} in

which InH is the standard deviation of the log hazard. Solving the first equation for InH in

terms of the ratio 85 percentile/median and substituting into the second equation yields

the desired mean/median ratio.

D-4