Some Effects of the Stochastic Structure of Rain on Remote Sensing Observations. Part 1: Drop Size Distribution as a Probability Density and the Natural Variability of Rain

It is shown in recent work that adoption of a stochastic viewpoint on the concept of the drop size distribution (DSD) leads to a useful change of perspective regarding the interpretation of radar measurements of rain. As we will see, this ultimately leads to a constructive definition of ``steadiness'' of rain.

We begin by interpreting the normalized part of the DSD as a probability density function. This then allows the decoupling of number concentration from the distribution of drop diameters so that an expression for the relative dispersion of the rainfall rate can be derived. We then address the following questions: What exactly is so natural about ``natural variability'' and Is there a standard of ``perfect steadiness'' with respect to which natural variability can be defined?

In response, we first discuss artificial monodisperse rain. It is argued that even though all rain is random and therefore fluctuations are inevitable, statistical stationarity of the drop count time series must certainly hold for the ideal steady rain. However, statistical stationarity does not preclude pulsations or ``patchiness'' of rain. Therefore we suggest that another condition must be satisfied for the ideal steady rain, that is, the drop counts per unit time must be Poisson-distributed.

We then return to the drop size distribution and modify the ``steadiness'' constraints to include all drop sizes. Simple observational examples are considered next and are used to compute natural variability and its implications on radar reflectivity. In part 2, these results are applied to a systematic investigation of Z-R relations and their variability.

Finally, the notion of the pair correlation function is introduced and applied to quantify deviations from the stationary Poisson series. We compare the local character of the pair correlation function with the cumulative measure of natural variability: clustering index. Particular attention is paid the resolution dependence of both measures because this is crucial for proper comparison of measurements taken with instruments of vastly different resolution, e.g., radars and raingauges and for the detection and characterization of natural variability of precipitation.