Handout (379.2 kB)
The scheme can well explain many ideal situations proposed in literature, such as 1) the number-controlled situation with b=1, 2) the size-controlled situation with b=1.63, 3) the scaled-number(Nt/Dc)-controlled situation with b=1.5 and 4) the inverse number-size situation with b=1.87. This scheme completes 5) the independent number-size situation with b=1.63, and 6) the number-controlled but inversely related to size situation with b<1. Different from the previous studies explaining b by the shape parameter (mu) of the gamma function which is a less physically meaningful parameter, this scheme suggests that b is independent to any absolute value of DSD moments and it should be involved with impacts of dominant micro-physical rain process through the correlation between Nt and Dm and the competition of their variability. For the coefficient a, the scheme predicts a small variation of a when b is close to 1 for the number-controlled situation, and a large variation of a when b is close to 1.63 for the independent number-size situation. These characteristics are consistent to the 69 power law Z–R relationships quoted by Battan (1973).
Two-year DSD data collected in the south of France is used to illustrate the impacts of Nt and Dm on the variation of Z-R relationships according to this scheme.

