The purpose of this analysis is to show how the distribution of rainfall measurements at a particular location would be more variable than the rainfall characteristics of storms that pass over that location. Suppose that all rainstorms are circular of radius R and the rainfall intensidty is uniform and constant. Let b be the perpendicular distance bween the rainguage location and the vector that passes through the center of the storm in the direction of the storm's movement.
The probability distribution of the variable b could be assumed to be uniform. As shown in the diagram below the rain guage would measure the rainfall on a chord of the circle of the storm where the chord length is equal to Rcos(sin-1(b/R)).