Explorations on the Hershfield Factor

The Hershfield factor (H) essentially constitutes a multiplier aiming to correct the error between fixed time interval maxima (F-maxima) and sliding maxima (S-maxima) as a direct consequence of temporal discretization of hydrometeorological time series. Rainfall is typically recorded as an accumulated value in fixed non-overlapping time intervals, e.g., in daily intervals, and thus the annual maximum value expresses the maximum value of these fixed recordings over a year period. Yet if measurements at a finer time scale are available, e.g., hourly, then the annual daily S-maximum, i.e. the annual maximum value resulting by sliding a 24-hour time interval starting at any hour of the year, in general, is different than the F-maximum value. The H factor attempts to correct for this error. Multiplying the F-maximum, which can be considered as a random variable, with the H factor, theoretically should result in the S-maximum random variable. This implies that the location and scale characteristics of the S-maximum distribution are explicitly related to the value of H and to the characteristics of the F-maximum random variable, while its shape characteristics will be exactly the same as those of the F-maximum distribution. This study further explores the validity of this well-accepted assumption. In order to verify or discard this assumption we perform an unprecedentedly large empirical analysis based on thousands of hourly rainfall records across the USA.