An Elementary Mathematical Model for the Interpretation of Precipitation Probability Forecasts

Abstract

A mathematical model is constructed for interpreting precipitation probability forecasts which have areal connotations. The paper is chiefly concerned with a simple discrete form of the model, in which a forecast area is represented by N rain gages scattered throughout it, and a precipitation event is identified with the observing of more than a trace of precipitation in a particular subset of the N gages. The basic parameters in the model are: (a) conditional probabilities of the events consisting of the selection of particular subsets of the N gages by precipitation, given that a certain proportion r of the gages are selected; and (b) a probability distribution for r. The probabilities in (a) are assumed to be peculiar to the particular forecast area and can easily be estimated from historical data. The probabilities in (b) (or at least the mean value of the distribution) are intended to be the objects of estimation in each forecast. It is apparently customary for forecasters to arrive at a forecast precipitation probability by calculating the theoretical mean value of r and assuming that this is equal to a point probability valid uniformly at each point of the forecast area. As explained within the model, this forecast method actually gives the arithmetic mean of the “point probabilities” of the events that the Jth gage receives more than a trace for J = 1, 2, …, N. Questions concerning biases in published verification data and possible lack of randomness in sequences of precipitation verifications within probability categories are raised as suggestions for further study concerning the operational significance of precipitation probabilities.