## 1. Introduction

[2] Currently, the ITU-R provides Rec. ITU-R P.837-2 for calculating the average annual, one-minute averaged, rainrate distribution for any point on Earth. This may be used to calculate the average annual rain-fade distribution experienced by a terrestrial link. However, telecommunications operators are increasing interested in the second order statistics of rain-fade. Such statistics include the temporal power spectrum, rain-fade duration statistics, rain-fade slope statistics and rain-fade covariance for pairs of links, or the same link at different times. These statistics are important for the design of fading countermeasures, such as time/path diversity, and for interference coordination. The second order statistics of rain-fade depend directly upon the second order statistics of rainrate variation.

[3] Although there exists a large number of pulse models describing rainrate variation, including the model by *Capsoni et al.* [1987] designed for propagation studies, these models do not contain information on the high resolution stochastic variation of rainrate in time and space. The Synthetic Storm model of *Matricciani* [1996] does provide high resolution, temporal, rain-fade variation but relies of simultaneous raingauge measurements. *Veneziano et al.* [1996] have postulated that log rainrate, while raining, may be modeled as a stationary, Gaussian stochastic process. Furthermore, theoretical models of rain as a passive tracer in a turbulent, two-dimensional flow [*Kraichnan and Montgomery*, 1980; *Lovejoy and Schertzer*, 1992] predict that the temporal and spatial spectral density of log rainrate follows a segmented power law form. These fluid dynamical models often lead to spectral density power laws with exponents expressed as ratios of small integers. Over ranges of scales where the spectral density follows a simple power law, there is no special scale and the random variable exhibits self-similarity. Models of rainrate variation that assume power law spectra are often termed “fractal models.” *Crane and Shieh* [1989] identify one-dimensional (1-D) spatial log rainrate spectra with two power law segments. Below the scale of energy injection, typically the size of a front, the power spectrum has the form |*f*|^{−3}, where *f* is spatial or temporal frequency, reflecting a direct entropy cascade toward larger wave numbers. Above the scale of energy injection an inverse energy cascade toward smaller wave numbers leads to spectra of the form |*f*|^{−5/3}. Based on the observation of eight storms, Veneziano et al. suggest that spatial or temporal 1-D log rainrate has a spectral density function (s.d.f.) with four power law segments, see Figure 1. The corner frequencies are associated with the scale of convective cells and cell clusters. A log-log plot of such a spectral density is segmented-linear with the gradient of each segment equal to the power law exponent.

[4] Assuming that log rainrate statistics exhibit self-similarity allows the low resolution available data to be used to develop models of rainrate variation at scales of interest to the radio propagation community. *Paulson and Gibbins* [2000] have verified the self-similarity of log rainrate variation at a point over scales from 10 s to one day. Other authors, such as *Venugopal et al.* [1999], have used courser meteorological radar data to investigate spatial statistical scaling and symmetries between temporal and spatial variation. In this paper, radar data are used to verify self-similarity of log rainrate variation in both time and space. Extrapolating this model to higher resolution than the scales measured by the radar, a three-dimensional model for the second order statistics of log rainrate variation is proposed. From this model, complete descriptions of the first order and second order statistics of rainrate and specific attenuation variation within rain events is developed. In section 4, this model is extended to predict the second order statistics of the rain attenuation experienced by pairs of radio links within a rain event. This model is verified using data from an experimental 38 GHz link.