Uniform hazard spectra from seismic hazard curves

A seismic hazard curve gives the mean annual frequency with which a ground-motion intensity level is exceeded at a site. For an intensity measure $I$ and a level $i^*$, the annual exceedance rate $\lambda_I(i^*)$ is the expected number of exceedances per year, expressed in $\mathrm{year}^{-1}$. Spectral acceleration at a specified oscillator period is one such intensity measure. A curve relates its amplitude to an exceedance rate; a uniform hazard spectrum relates oscillator period to the amplitude associated with a specified rate (Baker 2008).

Under Poisson occurrence with a constant rate and independent occurrences in time, the probability $P_T$ of at least one exceedance during an investigation time $T$, in years, is (Baker 2008; Cornell 1968)

$$P_T = 1 - \exp\left[-\lambda_I(i^*)T\right].$$

For $0 \le P_T < 1$ and $T>0$, the corresponding annual rate is

$$\lambda_I(i^*) = -\frac{\ln(1-P_T)}{T}.$$

The return period $T_R$, in years, is the reciprocal of a positive annual rate (Baker 2008):

$$T_R = \frac{1}{\lambda_I(i^*)}.$$

This is the mean interval between exceedances and does not imply regularly spaced exceedances (Baker 2008). The annual rate characterizes the curve at the chosen intensity level; the probability also depends on the investigation time. A zero probability gives a zero rate and no finite return period. These conversions retain the Poisson assumption and do not supply a temporal model for dependent occurrences (Baker 2008).

convertAEP() maps AEP and IT to POE and TR. Here AEP denotes the annual rate, POE the probability over IT years, and TR the return period. The oscillator period is a separate quantity, stored as Tn in seconds. Thus the probability interval, the mean interval between exceedances and the oscillator period describe different aspects of the calculation.

For a continuous, strictly decreasing hazard curve, inversion at a rate within its range identifies a unique intensity level. With $I_n$ denoting spectral acceleration at oscillator period $T_n$, the uniform hazard spectral ordinate at return period $T_R$ is

$$S_a^{UH}(T_n;T_R) = \lambda_{I_n}^{-1}\left(\frac{1}{T_R}\right).$$

Each ordinate has the same annual exceedance rate. The ordinates may correspond to different earthquakes, so the spectrum does not represent the spectral shape of a single ground motion or specify the probability of simultaneous exceedance across periods (Baker 2008).

For tabulated curves, evaluateUniformHazard() uses piecewise linear interpolation between the logarithms of return period and intensity. Between consecutive points $(T_{R,j},i_j)$ and $(T_{R,j+1},i_{j+1})$, indexed by $j$, its interpolated intensity is defined by

$$\ln i(T_R) = \ln i_j + \frac{\ln T_R-\ln T_{R,j}}{\ln T_{R,j+1}-\ln T_{R,j}}\left(\ln i_{j+1}-\ln i_j\right).$$

The function uses finite positive points and retains the greatest intensity when points share a return period. Requests beyond the curve’s range fail outside its permitted endpoint margin; requests within that margin take the endpoint ordinate. The argument by identifies each curve, preserving distinctions such as site, oscillator period and supplied curve statistic. The result contains these identifying columns, the requested TR and the corresponding Sa in $\mathrm{g}$, with one row per curve and requested return period.

readHazardWorkbook() obtains annual rates from workbook sheets and computes probabilities for the supplied investigation time. Its p field identifies the supplied curve statistic, whereas POE identifies exceedance probability. Evaluating a curve labelled mean produces the ordinate of that mean curve; inversion and averaging generally do not commute, so this ordinate is not generally the arithmetic mean of the realizations’ ordinates.

References

Baker, Jack W. 2008. An Introduction to Probabilistic Seismic Hazard Analysis (PSHA). Version 1.3.

Cornell, C. Allin. 1968. “Engineering Seismic Risk Analysis.” Bulletin of the Seismological Society of America 58 (5): 1583–606.