## 1. Introduction

[2] Rain is of great interest in several fields. Hydrologists investigate the rainfall process and develop models mainly with the aim of managing water resources, as well as of predicting civilian risks associated to extreme downpour events [*Onof et al.*, 2000]. The meteorological and climatological communities are interested in characterizing precipitation as part of the hydrologic cycle (ECMWF, http://www.ecmwf.int/, accessed May 2011). Last but not least, extensive investigations on rainfall are carried out by researchers involved in wireless telecommunication systems (TLC) because, as well known, hydrometeors cause strong absorption, scattering and depolarization (all contributing to signal impairment) on radiowaves propagating through the atmosphere at frequencies higher than 10 GHz [*Crane and Crane*, 1996].

[3] As extensively documented [*Capsoni and Luini*, 2009; *Bell*, 1987], rain is a process characterized by a marked variability both in space and time: rain events, especially those associated to deep convection, can be bounded to a few square kilometers and last only a few minutes. As a consequence, for TLC applications [*Capsoni et al.*, 2010; *Luini and Capsoni*, 2010], rainfall data should be sampled with high spatial and temporal detail to adequately catch the dynamics even of the most intense rain events, which, although infrequent, are the main cause of the outage experienced by wireless systems.

[4] Rainfall measurements have being routinely carried out worldwide since very long time by employing different instruments and methods.

[5] Raingauges [*Collier*, 1986] and disdrometers [*Thurai et al.*, 2007] can sample the rainfall process with high temporal resolution (typically one minute or even less). Instruments with short integration time are commonly deployed for experimental purposes (e.g., investigation on radiowave propagation in the atmosphere), while for routine hydrologic and meteorological applications, integration time of one hour or longer is typically employed (NCEP, http://data.eol.ucar.edu/codiac/dss/id=21.004, accessed May 2011, and ARPA, http://ita.arpalombardia.it/ita/index.asp, accessed May 2011).

[6] Remote sensing of rainfall from space is achieved by means of radars, radiometers (e.g., Tropical Rainfall Measuring Mission, TRMM [*Kummerow et al.*, 2000]) and imagers (e.g., Meteosat Second Generation (MSG) satellites [*Aminou et al.*, 1999]) onboard of LEO or GEO satellites. Rain measurements are available with a spatial resolution not finer than 2 km × 2 km and with a limited accuracy due to the cumbersome task of removing the effects of ground clutter from measured data. When merged with additional measurements (raingauges, radar or other Earth Observation (EO) products), rainfall data inferred from space-borne instruments can achieve a good accuracy and a better coverage of the Globe, although anyway, with coarse temporal and spatial detail. This is the case for example of the TRMM Multisatellite Precipitation Analysis 3B42 products (TMPA 3B42), obtained by combining orbital and ground-based rainfall data, which consist in rain rate values averaged over 3 h and over pixels of dimension 0.25° × 0.25° for latitudes between ±50° [*Huffman et al.*, 2007].

[7] Weather radars are complex instruments, specifically designed to remotely sense precipitation, which provide almost instantaneous tridimensional pictures of the rain field by scanning the surrounding volume at different elevation angles [*Doviak and Zrnic*, 1993]. Due to the time required to acquire the various azimuth scans, operational weather radars commonly provide a complete picture of the precipitation field every 5 min and with 1 km × 1 km or 2 km × 2 km grid spacing.

[8] Numerical Weather Prediction (NWP) data are the output of meteorological models, which aim at forecasting weather by solving numerically the atmospheric fluid and thermo dynamics equations. Several meteorological models are available nowadays (e.g., see NOAA, http://www.emc.ncep.noaa.gov/modelinfo/, accessed May 2011, and UK MetOffice, http://www.metoffice.gov.uk/, accessed May 2011) and are mainly classified as global or regional, depending on the area covered by the forecasts. These latter models can provide estimates at the fine spatial and temporal resolutions necessary to adequately sample the rainfall process (e.g., 0.5 km × 0.5 km and 30 s), although at the expense of extremely long computation times and prohibitive storage requirements: their use to derive statistically meaningful results, as required for the characterization of the radiowave propagation through the atmosphere and for TLC system planning, is impractical, if even possible. On the other hand, meteorological re-analysis products, such as the ERA40 database generated by the European Centre for Medium-term Weather Forecast (ECMWF), are of specific interest for the research propagation community and for regulatory bodies (International Telecommunication Union – Radiocommunication sector, ITU-R) because of their expected higher accuracy with respect to forecasts, the worldwide coverage and the large number of available years. These features are nowadays a basic requirement by propagation impairment prediction models. Unfortunately ERA40 data are available only with coarse spatial and temporal resolutions, respectively equal to 1.125° × 1.125° (latitude × longitude) and 6 h [*Uppala et al.*, 2005].

[9] As it is clear from the discussion above (which does not claim to be comprehensive), a huge amount of rainfall data is at disposal nowadays. For the design of wireless TLC systems at frequencies above 10 GHz, other than the local complementary cumulative distribution function (CCDF) of point rain rate (a basic input of any model), the knowledge of the spatial correlation of rain (hereinafter referred to as SCoR) is required. In the prediction of rain attenuation statistics, the spatial inhomogeneity of the rainfall rate along the link is taken into account through a path reduction factor [*ITU Radiocommunication Sector* (*ITU-R*), 2009]. In the design of a site diversity scheme [*Goldhirsh et al.*, 1997] or in the optimization procedure of an adaptive onboard common resource-sharing technique [*Paraboni et al.*, 2009], the spatial correlation of rain (directly linked to the probability of simultaneous rain attenuation at more than one station) is the driving parameter. Finally, the trend of the correlation coefficient with distance is strictly related to the inputs of a class of rainfall/rain field models oriented to the generation of synthetic rain fields over areas as large as a continent to be used for simulation purposes [*Bertorelli and Paraboni*, 2005; *Jeannin et al.*, 2009]. For telecommunication applications, SCoR could be, in principle, evaluated on a global scale using rainfall products of meteorological re-analyses or Earth Observation missions, provided that the effects of their coarse spatial and temporal detail is somehow removed. The spatial correlation of rain is also subject of extensive investigations by the hydrological and meteorological communities because SCoR represents a key information for the parameterization of stochastic models devised to synthesize spatially and temporally correlated rainfall data (e.g., refer to*Baigorria and Jones* [2010], and to the extensive literature review therein, for models of this kind), which, in turn, can be used as input to agricultural, hydrological and environmental models for different purposes (simulated crop yields, prediction and management of water resources, climate change studies, etc).

[10] In the literature several authors studied the spatial distribution of precipitation [*Huff*, 1970; *Zawadzki*, 1973], many of them focusing on the correlation of point rainfall as measured by raingauge networks [*Park and Singh*, 1996; *Habib et al.*, 2001]. Some of these works analyzed the spatial variability of rainfall at different time scales [see, e.g., *Baigorria et al.*, 2007; *Fukuchi*, 1988; *Barbaliscia et al.*, 1992], spanning from quasi instantaneous rain rate (e.g., 1-min integration time) to monthly, seasonal or yearly scale [*Huff and Shipp*, 1969]. Other authors showed the dependence of the rainfall correlation coefficient on the dimension of the area which rain rate values are averaged on, using radar data or EO products [*Gebremichael and Krajewski*, 2004]. In *Yang et al.* [2011], the authors preliminary addressed the impact of the spatial or temporal integration on rain field features relevant to the design and performance prediction of satellite communication systems (i.e., probability of rain, spatial correlation of rain and point rain rate statistics), using a restricted set of the same data employed in this work. The comparison of the results presented in all abovementioned works is cumbersome, if even possible, because of the different instruments used to estimate SCoR (raingauges, ground-based radars, space-borne sensors, etc), the different time and space scales considered (e.g., from 1-min to monthly or even yearly integration time), as well as the dimension of the area under analysis, typically limited to a few kilometers for networks of raingauges with high temporal resolution or extending to large regions for products with coarse spatial and/or temporal detail.

[11] In this paper, we aim at providing a comprehensive description of the effects of averaging quasi instantaneous rain rates in time or space (or in time plus space) on the correlation index by taking advantage of the extensive coverage of the NIMROD radar network and of its fine temporal and spatial resolutions. Such analysis is useful to define a means for the de-integration of SCoR derived from rainfall products with long integration time or wide integration area or, furthermore, with both poor spatial and temporal detail, typical of meteorological re-analyses (ERA40) or Earth Observation missions (TMPA 3B42).

[12] The remainder of this paper is organized as follows. Section 2 gives an overview of the rainfall data set employed in sections 3 and 4 to investigate how spatial rainfall statistics modify because of integration in time or in space, respectively. An analytical model for the correlation index of rain *ρ* as a function of the distance between two sites *d* is introduced and general formulations are proposed in order to easily calculate *ρ* from the knowledge of *d* and of the integration time *T* or the integration area *A.* Furthermore, section 5extends the analysis by investigating the impact of joint spatial and temporal integration on rainfall data, which typically characterize products of meteorological re-analyses and of EO missions.