A mathematical theory of de-integrating long-time integrated rainfall and its application for predicting 1-min rain rate statistics.

To date, the methods devised for converting long-term experimental probability distribution (pd), P(rho T) (rho(T)), of the rain rate rho(T) integrated in T min (T > > 1 min) to 1-min pd, P(R)(R), of the instantaneous rain rate R, are based on flawed T-min data and, as such, are not based on fully reliable first principles. P rho(T) (rho(T)) is not only an upward translated version of P(R)(R) but also rotated clockwise and distorted. The current methods do not correct these errors. We propose and discuss a mathematical theory, which corrects these errors and thus de-integrates T-min experimental pds into the corresponding 1-min pd, the input required by all rain attenuation prediction methods. The theory is based on simple first principles whose parameters are calibrated by means of a large and reliable rain-rate data bank recorded in Spino d'Adda, a site held as an experimental laboratory and used for exploratory data analysis. We show that P(R)(R) is modelled by four distinct functions in four disjoint ranges, and that this modelling is physically meaningful. We have tested the theory up to integration times of 12 h, with a large experimental data bank of 1-min rain-rate time series recorded in Gera Lario, Fucino, Rome, Prague, and Montreal, besides Spino d'Adda. Defined the fraction of rainy time in an average year, P(o) (%), we have found that: (a) the modelling is very good up to 6 h; (b) in the range from about P(o) to 0.001%, the error values are constant, with average error set at about - 3% and RMS error less than 8% for T <= 120 min, less than about 9% for 120 < T <= 360 min. We have also applied the theory to rain-rate time series provided by meteorological agencies with integration time T = 60 min (blind test) with excellent result.