OPTIMALITY OF EMPIRICAL Z-R RELATIONS

This paper attempts to justify mathematically the two empirical approa ches to the problem of deriving Z-R relations from (Z, R) measurements , namely the power-law regression and the 'probability matching method '. The basic mathematical assumptions that apply in each case are expl icitly identified. In the first case, the appropriate assumption is th at the scatter in the (Z, R) measurements reflects exactly the randomn ess in the connection between Z and R due to a lack of sufficient a pr iori information about either of them. In the second case, the assumpt ion is that the measurements have been classified into categories a pr iori, in a way that allows one to expect a nearly one-to-one correspon dence between Z and R in each catergory, the scatter in the measuremen ts being due to residual noise. The paper then shows how the assmuptio ns naturally lead, in the first case, to a 'conditional-mean' Z-R rela tion of which the power laws are regression-based approximations, and, in the second case, to a 'probability-matched' relation.