Inter‐annual variability,risk and confidence intervals associated with propagation statistics. Part II: parameterization and applications

This set of two companion papers aims at providing a model for the inter‐annual variability of earth‐space propagation statistics and for the inherent risk and CIs. In part I, it was proposed to model the yearly variance σ² of empirical complementary CDFs so that where is the variance of estimation, the inter‐annual climatic variance and p the long‐term probability. Particularly, an analytical formulation of was derived and parameterized from synthetic rain attenuation data. Considering the statistical framework developed in part I, this part II is specifically devoted to the parameterization of the variance of estimation from experimental data of rain attenuation and rainfall rate. Then, a methodology to model and parameterize worldwide the inter‐annual climatic variance is presented. The model of yearly variance of the empirical complementary CDFs is finally compared against yearly experimental variances derived from data collected worldwide. The knowledge of this variability is very useful for system design as it allows the risk on a required availability and associated with a given propagation margin to be quantified. Copyright © 2013 John Wiley & Sons, Ltd.     

A mathematical theory of de‐integrating long‐time integrated rainfall statistics. Part II: from 1 day to 1 minute

We have developed, tested and discussed a theory for de‐integrating the probability distribution (PD) of the daily rain rate to the PD of the rain rate integrated in 1 min, through many simple steps, with a good or very good precision, for a large range of probabilities and in many sites. The theory can also estimate the number of rain events (in the sense of rainstorms), N_R, their average duration and the (conditional) PD of daily rainy time. The theory needs only three inputs, measured on site: the daily rain rate PD, the number of rainy days, N_D (only for finding N_R) and the PD of the rain rate integrated in two consecutive and disjoint couples of days. The theory contains two complementary parts, both successfully tested: the first deals mainly with duration of daily rainy time and rain events, and the second deals with the main issue, namely de‐integrating daily rain rate PDs in 1‐min PDs. We have tested the theory on duration of rainy time in Spino d'Adda, Gera Lario, Fucino and Prague and subsequently with real (blind) field tests in Milan, Lugano, many sites in the USA, and Canada. The sites tested belong to very different climatic regions. Nevertheless, the predictions are generally very close to the experimental data. The theory, and its powerful predictions, can be useful for several research communities: radio propagation, agriculture, climatology, hydrology and applied meteorology. For all disciplines and applications, seasonal studies or even monthly studies could be pursued because the data banks available can be very large, even for restricted sub‐data banks. The theory will estimate its parameters on on‐site seasonal, or monthly, measurements. Future developments could deal with de‐integrating large‐area and long‐time integrated rain rate, observed by means of meteorological satellites, for obtaining ‘point’ 1‐min rain rate PDs concerning a small homogeneous area. We have tested the theory to sites with very different climates and for latitudes between 65°N and 28°N. Therefore, we think that the theory can be applied globally in this latitude range because its parameters are derived from local measurements and, perhaps, down to the tropics. Because of the simplicity of the theory, and its use of few local measurements, it may be applied also to equatorial sites. This, however, is only a conjecture because we have not tested the theory directly there. Copyright © 2013 John Wiley & Sons, Ltd.