Fitting the stochastic ω−2 source model to observed response spectra in western north America: Trade-offs between Δσ and κ

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We subtracted the logarithms of the response-spectral ordinates found in the empirical analysis of Joyner and Boore (1988, Table 2) from the logarithms of the stochastic-model calculations to form residuals. We computed residuals for M = 5.5, 6.5, and, 7.5 at 10 of the 12 periods considered by Joyner and Boore (0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0, and 2.0 sec; their periods of 3.0 and 4.0 sec were excluded because the quality of the empirical fits are questionable for those periods). A distance of 20 km was used. This distance is the horizontal distance r 0 in the Joyner and Boore equations. Joyner and Boore used a point source model as the basis for their regression equations and determined perioddependent pseudo-depths h that gave slant distances r = ~/r02 + h 2 . The stochastic model is also a point source model, and for consistency the slant distances used in the calculations were the same as used in the evaluation of the Joyner and Boore equations (in other words, r 0 was fixed at 20 km, but r depended on oscillator period, ranging from 20.54 km for T = 2.0 sec to 22.97 km for T = 0.1 sec).