Optimality of empirical Z-R relations

This paper attempts to justify mathematically the two empirical approaches 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 explicitly identified. In the first case, the appropriate assumption is that the scatter in the (Z, R) measurements reflects exactly the randomness in the connection between Z and R due to a lack of sufficient a priori information about either of them. In the second case, the assumption is that the measurements have been classified into categories a priori, in a way that allows one to expect a nearly one-to-one correspondence between Z and R in each catergory, the scatter in the measurements being due to residual noise. The paper then shows how the assmuptions naturally lead, in the first case, to a 'conditional-mean' Z-R relation of which the power laws are regression-based approximations, and, in the second case, to a 'probability-matched' relation.