Spurious power-law relations among rainfall and radar parameters

In previous work, the authors examined statistical physics of rain from the point of view of the modern theory of random processes. In particular, the importance of statistical stationarity (homogeneity) was emphasized in order to attribute a clear physical meaning to the notions of drop size distribution and relations between the radar re ectivity factor, Z, and rainfall rate, R, used in radar meteorology. In this work we return to the case of the simplest rain model, namely uncorrelated raindrops having a prescribed drop size distribution. As shown in previous work, in such rain linear relations are anticipated among the various rainfall parameters. Taking the direct approach of Monte Carlo simulations and using the techniques typical of rainfall studies over the last several decades, we then sample from a collection of rain events in which the drop occurrences are uncorrelated but each event has the same prescribed drop size distribution. Surprisingly, it is found that apparently realistic but spurious nonlinear power-law relations still appear among rainfall parameters even though the rain is not only statistically homogeneous but purely random as well. We show that this occurs largely because of an inadequate number of drops in each sample. The drop samples typically observed, while variable in size, are often too small to sufŽ ciently represent the size distribution or its moments. That is, each sample of rain drops, is one random, partial realization of the probability density function of diameter, that still yields functionally related pairs of variables such as Z and R. We show, however, that because of the inadequate number of drops, n, in each sample the least-square-error power-law Ž ts yield spurious exponents that depend upon n. Thus, even though resulting Ž ts may adequately describe the deŽ cient observations, they are spurious and physically meaningless. An inspection of the literature reveals that nearly all reported Z–R and other rainfall parameter relations over the last several decades are likely to be spurious because n was too small by factors of hundreds to thousands. It is clear that such Ž tted relations are unlikely to re ect the actual physical properties of the rain and are simply artefacts of inadequate sampling and, to a lesser extent, Ž tting procedures. In particular the relations of Marshall and Palmer are probably artefacts.