- Ground-motion model is:
_{0}(R) is an empirically determined attenuation function from Richter (1958) used for calculation of M_{L}, p is confidence level and v is component direction (v = 0 for horizontal and 1 for vertical). log A_{0}(R) not given here due to lack of space. - Uses three site categories:
- s = 0
- Alluvium. 63% of data.
- s = 1
- Intermediate. 23% of data.
- s = 2
- Basement rock. 8% of data.

- Response parameter is acceleration for 0, 2, 5, 10 and 20% damping.
- Note that do not believe the chosen independent parameters are the best physical characterization of strong shaking but they are based on instrumental and qualitative information available to the engineering community in different parts of the USA and the world.
- Data from free-field stations and basements of tall buildings, which assume are not seriously affected by the surroundings of the recording station. Note that detailed investigations will show that data from basements of tall buildings or adjacent to some other large structure are affected by the structures but do not consider these effects.
- Equation constrained to interval M
_{min}≤ M ≤ M_{max}where M_{min}= -b(T)∕2f(T) and M_{max}= [1 - b(T)]∕2f(T). For M > M_{max}replace f(T)M^{2}by f(T)(M - M_{max})^{2}and for M < M_{min}replace M by M_{min}everywhere to right of log_{10}A_{0}(R). - Use almost same data as Trifunac (1976a). See Section 2.16.
- Use same regression method as Trifunac (1976a). See Section 2.16.
- Note that need to examine extent to which computed spectra are affected by digitization and processing noise. Note that routine band-pass filtering with cut-offs of 0.07 and 25Hz or between 0.125 and 25Hz may not be adequate because digitisation noise does not have constant spectral amplitudes in respective frequency bands and because noise amplitudes depend on total length of record.
- Find approximate noise spectra based on 13 digitisations of a diagonal line processed using the same
technique used to process the accelerograms used for the regression. Linearly interpolate noise spectra
for durations of 15, 30, 60 and 100s to obtain noise spectra for duration of record and then subtract
noise spectrum from record spectrum. Note that since SA(y
_{1}+ y_{2})≠SA(y_{1}) + SA(y_{2}) this subtraction is an approximate method to eliminate noise which, empirically, decreases the distortion by noise of the SA spectra when the signal-to-noise ratio is small. - Note that p is not a probability but for values of p between 0.1 and 0.9 it approximates probability that
SA(T)
_{,p}will not be exceeded given other parameters of the regression. - -g(T)R term represents a correction to average attenuation which is represented by log
_{10}A_{0}(R). - Do not use data filtered at 0.125Hz in regression for T > 8s.
- Due to low signal-to-noise ratio for records from many intermediate and small earthquakes only did regression up to 12s rather than 15s.
- Smooth coefficients using an Ormsby low-pass filter along the log
_{10}T axis. - Only give coefficients for 11 selected periods. Give graphs of coefficients for other periods.
- Note that due to the small size of g(T) a good approximation would be log A
_{0}(R) + R∕1000. - Note that due to digitisation noise, and because subtraction of noise spectra did not eliminate all noise, b(T), c(T) and f(T) still reflect considerable noise content for T > 1 --2s for M ≈ 4.5 and T > 6 --8s for M ≈ 7.5. Hence predicted spectra not accurate for periods greater than these.
- Note that could apply an optimum band-pass filter for each of the accelerograms used so that only selected frequency bands remain with a predetermined signal-to-noise ratio. Do not do this because many data points would have been eliminated from analysis which already has only a marginal number of representative accelerograms. Also note that such correction procedures would require separate extensive and costly analysis.
- Note that low signal-to-noise ratio is less of a problem at short periods.
- Compare predicted spectra with observed spectra and find relatively poor agreement. Note that cannot expect using only magnitude to characterise source will yield satisfactory estimates in all cases, especially for complex earthquake mechanisms. Additional parameters, such as a better distance metric than epicentral distance and inclusion of radiation pattern and direction and velocity of propagating dislocation, could reduce scatter. Note, however, that such parameters could be difficult to predict a priori and hence may be desirable to use equations no more detailed than those proposed so that empirical models do not imply smaller uncertainties than those associated with the input parameters.
- Plot fraction of data points, p
_{a}which are smaller than spectral amplitude predicted for p values between 0.1 and 0.9. Find relationship between p_{a}and p. Note that response spectral amplitudes should be nearly Rayleigh distributed, hence p_{a}(T) = {1 - exp[-exp(α(T)p + β(T)]}^{N(T)}. Find α, β and N by regression and smoothed by eye. N(T) should correspond to the number of peaks of the response of a single-degree-of-freedom system with period T but best-fit values are smaller than the value of N(T) derived from independent considerations.