Historically the peak over threshold method has been used to derive average recurrence intervals, while the annual maxima method has been used to derive annual exceedance probabilities.
Conversion between these can be done using the Langbein’s formula (Langbein, 1949) and its inverse:
\mathrm{AEP} = 1 - \exp(-1/\mathrm{ARI})\mathrm{ARI} = -1/\ln(1 - \mathrm{AEP})These formulas are approximately equal to the commonly-used equation \mathrm{ARI} = 1/\mathrm{AEP} above an ARI of 10 years.