Corresponding author: L. Luini, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci, 32, Milano MI 20133, Italy. (email@example.com)
 Nowadays a huge amount of data is available for the statistical characterization of rainfall worldwide, although unfortunately not always with the adequate spatial and temporal resolution required for the very high demanding telecommunication applications. On the basis of the NIMROD radar network composite rain maps, first, this paper investigates separately the impact of space or time integration on the spatial correlation of rainfall ρ, a key parameter for most Propagation Impairment Mitigation Techniques (PIMTs), as well as for many prediction models such as time-space rain field generators. Analytical formulations are proposed to model the average trend ofρ with the distance d between two sites as a function of the integration time T or the integration area A, which, in turn, can be used to de-integrate the spatial correlation information estimated respectively from networks of raingauges with long integration time or from radar data with coarse spatial resolution. As an example, the last part of the paper compares the spatial rain decorrelation trends estimated by a database of radar maps collected in Northern Italy with the ones de-integrated from products of meteorological re-analyses (ERA40) or Earth Observation missions (TMPA 3B42).
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 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].
 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.
 Rainfall measurements have being routinely carried out worldwide since very long time by employing different instruments and methods.
 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).
 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].
 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.
 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].
 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 toBaigorria and Jones , 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).
 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. , 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.
 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).
 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.
2. The NIMROD Weather Radar Network
 The NIMROD network, managed by the UK Meteorological Office (MetOffice), consists of 19 C-band weather radars deployed throughout the British Isles: the position of the radar sites is indicated as asterisks inFigure 1, which also depicts the overall coverage of the network.
 Each radar performs in 5-min time a series of azimuth scans at different elevations. Afterwards, the acquired data are centralized to Radarnet IV, the MetOffice processing system at Exeter, in full resolution polar format. All basic processing, quality controls and correction procedures, such as the elimination of ground clutter and the conversion of radar reflectivityZ (mm6/m3) into rain rate R (mm/h), are carried out on polar data [Harrison et al., 1998]. The conversion of raw data to a Cartesian grid is achieved by combining the various scans of multiple radars available at each pixel: the resulting composite rain maps represent a reliable estimate of the rain rate at ground level over the Great Britain, have 1 km × 1 km grid spacing and are available every 5 min.
 In this work, a full year (2009) of NIMROD 1 km × 1 km composite rain maps, freely available on the Web for research purposes [UK Meteorological Office, 2010], have been selected to investigate the effects of temporal and spatial integration on rainfall. The whole database consists of 99695 maps, which corresponds to an availability of 94.84%. Moreover, rain composites, whose original dimension is 2175 km × 1725 km, have been limited to a rectangular area of dimension 1000 km × 600 km, identified by the black solid line in Figure 1, the rationale being: (1) consider a region mostly over land, where the characterization of rainfall is of major interest for propagation-oriented applications; and (2) focus on the area where weather radars are denser and, consequently, data availability and accuracy are increased.
3. Temporal Integration
 Time integration of rainfall is the common process which raingauges and disdrometers rely on. Although with different procedures depending on the specific instrument, they all basically provide as output the rain rate (mm/h) by dividing the amount of rain accumulated (mm) by the length of the integration time T (h). A too short integration time (e.g., 1 s) yields inaccurate rain rate values, because of the too limited sample volume; on the contrary, a long integration time (e.g., 1 h) filters out the most intense rain rates, because their duration is typically limited to a few minutes. A good compromise, at least for TLC applications, is achieved with T= 1 min, as recommended by ITU-R [ITU Radiocommunication Sector (ITU-R), 2007].
 The effect of time integration on rainfall is assessed by processing NIMROD radar data at different integration times, namely T = 0.5, 1, 2, 4 and 6 h. Consecutive rain maps are sorted so as to extract time series relative to all pixels and, afterwards, rain rates included in a T-minute interval are integrated numerically as follows:
where t and τare the temporal indexes of the un-integrated and integrated rainfall intensity, respectively. As it is clear from(1), the temporal integration process simply corresponds to averaging the rain rates extracted from the NT subsequent rain maps pertaining to an interval of T minutes.
where E[•] and σ[•] are the mean and standard deviation operators, while, according to (1), RT(i1, τ) and RT(i2, τ) are the T-minute integrated rain rate time series, respectively relative to pixelsi1 and i2.
 In order to derive a complete picture of SCoR over the selected radar area, the expression in (2) should be calculated for each couple of pixels. However, due to the extremely large amount of possible combinations C between all the pixels on the map (being n = 1000 ⋅ 600 = 16 × 105 the pixels, C = n(n + 1)/2 = 1.8 × 1011), which would be impractical to handle, a subset of pixels has been selected so as to obtain a manageable number of combinations, while covering at best the whole region under investigation. The chosen pixel grid, which consists of 1519 pixels (i.e., C = 1154440 combinations), is shown in Figure 2: the distance between adjacent pixels follows partially a geometric progression (central part of the map) and partially a uniform sampling (image borders). This approach has the advantage of selecting several couples of pixels at short distance, which are necessary to analyze in detail the spatial correlation of precipitation where the variation is expected to be particularly marked (approximately d < 50 km) [Gebremichael and Krajewski, 2004].
Figure 3shows the correlogram (un-integrated data): each point identifies a couple of pixels, regardless of their position on the map, and the black dashed line represents the average trend ofρ. The correlation index steeply decreases for d values between 0 and 200 km and afterwards slowly tends to 0 for d > 400 km. As a matter of fact, the rainfall process can be considered to be almost uncorrelated (ρ < 0.1) after approximately 100 km. A similar steep decorrelation trend has been obtained in Yang et al.  as well, where the authors calculated SCoR over an area of 256 km × 256 km using one month (January 2008) of NIMROD radar data with 1 km × 1 km grid spacing. The results obtained here can be compared only qualitatively with those reported in Yang et al.  because of the different amount of data used to derive SCoR in the two works: the marked differences are likely due to seasonal (selection of a full year versus a single month) and regional effects (selection of a small area versus the whole UK).
 An advantage in calculating the spatial correlation using the time series approach based on (2) relies in the chance to appreciate the dependence of ρ on the selected sites: Figure 3 evidences the spread of the ρ values around the average trend (average standard deviation of ρ over all separation distances equal to 0.02): for d > 200 km, some points are still correlated, while others even show slight anticorrelation (ρ < 0). Such variability is partially related to the number of samples used to calculate ρ (more years of radar data would contribute to reduce the spread), but it is manly associated to topographic and regional effects characterizing UK: in other words, the stationarity of rain field [Bell, 1987], inherently assumed by introducing a dependence of ρ only on the site separation distance, is not verified throughout the whole UK.
 In order to assess whether the selected subset of pixels used to calculate ρ by means of (2) is representative of the full rain map, we have also applied a spectral approach to estimate the rainfall spatial correlation (spatial covariance estimated as the inverse Fourier transform of the spatial periodogram [Proakis and Manolakis, 1996]; details in Appendix A). The results are reported in Figure 4, which depicts the spatial correlation derived from full NIMROD maps (un-integrated data), and inFigure 5, which shows the trend of correlation with distance as estimated using the time series approach (spatial technique in the label) and the spectral approach (spectral technique in the label). As is clear from Figure 5, the two techniques provide very similar decorrelation trends (root mean square of the difference between the curves equal to 0.0168), which proves that the chosen subsample of pixels is representative of the full rainfall map. Moreover, Figure 4allows to appreciate that the pattern of SCoR over UK is slightly anisotropic as it tends to be more elongated in the North-South direction: ford = 10 km, SCoR has an almost isotropic pattern, with ρ = 0.40 and ρ= 0.39 for the North-South and East-West baselines (approximately a relative difference of 2.5%), respectively, while ford = 100 km, the anistropy is more evident with ρ = 0.12 and ρ= 0.09 for the North-South and East-West baselines (approximately a relative difference of 30%), respectively.
Figure 6 quantifies the effects of time integration on the spatial correlation of rain by depicting the average trend of ρT with d for the selected integration times. In fact, as the integration time becomes longer, intense rain events are more and more filtered out and, consequently, the rain rate range associated to the time series progressively reduces. This effect is well evidenced in Figure 6: as an example, the ratio between the un-integrated and the 6-h integrated curves lies between 3 and 5 for 50 km <d < 400 km.
 The average trend of ρT with d can be modeled by a double exponential law which, as shown also in Bertorelli and Paraboni , allows to properly take into account both the fast (d approximately shorter than 50 km) and slow (d approximately larger than 50 km) decrease:
where a(T), b(T) and c(T) are regression coefficients that guarantee the best fit between ρT(d) calculated from radar data and (3). Making reference to Figure 7 (T = 2 h), the fitting accuracy of (3) can be evaluated by considering the scatterplot between the data used for the regression (circles) and those predicted by (3) (black solid line): the slope mf and intercept qf of the linear regression model are equal to 1.004 and −0.0058, respectively, while the coefficient of determination is R2 = 0.9952. Similar satisfactory values are found for all the Tvalues considered (including un-integrated data): the lowest and highestR2 values are obtained respectively for T = 0.5 h (R2 = 0.9922, mf = 1.0589 and qf = −0.0061) and T = 4 h (R2 = 0.9961, mf = 1.0094 and qf = −0.0051).
 The analytical expressions for the coefficients in (3) show a very regular behavior with T:
where T is expressed in minutes.
 From equations (3) to (6), and T between 1 and 360 min, we can define a conversion factor CFT(d) aimed at de-integrating second-order statistics obtainable from rainfall data with integration timeT typically longer than 1 min (ρT(d)m in (8)):
 In (8), (d) represents the estimate of ρ1(d) achieved through CFT(d).
 Although CFT(d) is likely to be area dependent, its applicability to data collected in regions characterized by temperate climate (e.g., Europe) is expected to hold with good approximation, as it is the case for the scaling coefficients used to de-integrate in time the first order statistics of rain rate [Emiliani et al., 2009; ITU Radiocommunication Sector (ITU-R), 2007]. The specific characteristics of the site (or area) are, in fact, already embedded in ρT(d)m. Indeed, the variability of CFT(d) from region to region is likely to be quite limited, being CFT(d) a scaling function that operates on second order statistics; for this reason, when no data at high resolution are available and information on the local SCoR is a key requirement, the applicability CFT(d) outside temperate regions can provide a first rough estimate of ρ1(d) for the site of interest.
4. Spatial Integration
 Radars and other EO instruments for the remote sensing of precipitation are intrinsically based on the concept of spatial integration because the data acquisition process takes advantage of the backscattering properties of hydrometeors (weather radars) or of the electromagnetic emission generated by precipitating clouds (radiometers). In both cases, rain intensity is estimated on almost instantaneous basis by receiving signals from a volume associated to the antenna field of view. As a result, a spatial averaging effect is inherently applied and the acquired rain rate values depend on the features of the instrument (e.g., −3 dB antenna beam width) and on its distance from the sampled volume: the larger the distance, the more marked the averaging effect.
 To evaluate the impact of using spatially averaged rain rates, NIMROD data have been numerically integrated over square pixels with lateral dimension S = 4, 8, 25, 50 and 100 km. To this aim, a new pixel lattice with S km × S km grid spacing has been superimposed to each NIMROD composite rain map at fine resolution, which corresponds to grouping rain rate values in adjacent pixels of dimension A = S2 and then processing them as follows:
where i and jare the spatial indexes of the un-integrated and integrated rainfall intensity, respectively. The right end side of(9) points out that the spatial integration process basically corresponds to averaging the NS rain rates included in a bigger S-kilometer wide pixel.
 Second order statistics can be studied by resorting again to the correlation index between two pixels ρS(j1, j2) obtained after spatial integration, defined as:
 Differently from the case of temporal integration, an obvious consequence of the spatial integration process is the progressive reduction in the number of available pixels as S increases: as an example, each NIMROD rain map consists of 1000⋅600 = 6⋅105 pixels for S = 1 km and (1000/100)⋅(600/100) = 60 pixels for S = 100 km. Thus, the approach presented in section 3 for the smart pixel selection has been applied also to the lattice resulting from the spatial integration, but only for S = 4 and 8 km, with the aim of maintaining approximately the same number of combinations C = 1154440 already considered when dealing with time integration; for S = 25, 50 and 100 km, all pixels have been taken into account, leading respectively to C = 461280, 28920 and 1830 combinations, in order to maximize the statistical meaningfulness of the results. Another evident consequence of the spatial integration process is the progressive increase with S in the minimum distance dmin among pixels (reference are the centers of the pixels) such that, obviously, dmin = S. Both the effects mentioned above can be appreciated in Figure 8, which shows the average trend of ρS with d for different S values: the reduced number of available pixels explains the less regular trend of ρ50km(d) and ρ100km(d) (particularly noticeable for d > 300 km), while the increased dmin is clearly visible in the lack of samples for distances shorter than 50 km and 100 km respectively.
 Notwithstanding this, the results presented in Figure 8 give the chance to appreciate the dependence of SCoR on S: the use of rainfall data with coarse spatial resolution, typical of global products of meteorological re-analyses and Earth Observation missions (S = 25, 50 and 100 km), definitely impacts on the correlation between two sites (e.g., for d = 50 km, ρ(50 km) ≈ 0.2 and ρ25km(50 km) ≈ 0.4); it is also interesting to notice that even the integration over a small area has a non negligible effect on ρ, especially for short distances (e.g., for d = 10 km, ρ(10 km) ≈ 0.5 and ρ8km(10 km) ≈ 0.7).
 The double exponential law in (3) appears to be an adequate model also for the average trend of ρS with the distance:
where g(S), h(S) and p(S) are regression coefficients that guarantee the best fit between ρS(d) calculated from radar data and (11). The law proposed in (11) provides a very good fit to ρS(d) for every selected value of S: in this case 0.9864 < mf < 1.0259 and −0.0052 < qf < 0.0005, while the lowest and highest R2 values are obtained respectively for S = 100 km (R2 = 0.9924) and T = 4 h (R2 = 0.9961). The dependence of the regression coefficients on S is well represented by the following analytical formulations:
where S ranges between 1 and 100 km.
 Similarly to the methodology proposed in section 3, equations (11), (12), (13) and (14) can be used to define a conversion factor CFS(d) aimed at de-integrating second-order statistics obtainable from rainfall data integrated over pixels with lateral dimensionS typically larger than 1 km (ρS(d)m in (16)):
 Based on the same considerations presented before for CFT(d), the conversion factor CFS(d) may reveal useful in order to derive a first estimate of ρ1(d) from spatially integrated rainfall data in temperate regions and, to a lesser extent, also in regions with a different climate.
5. Temporal and Spatial Integration
 This section investigates the impact of the joint space-time integration on the spatial correlation of rain. The reason is to provide useful hints for the use of rainfall databases with poor spatial and temporal resolution as input to electromagnetic wave propagation oriented applications. Specifically, in this section we will make reference to rainfall data extracted from the ERA40 and the TMPA 3B42 databases. The former, made available by the ECMWF on global basis, are the output of a meteorological re-analysis project aimed at assimilating long-term historical observational data (approximately 40 years) using a single consistent analysis scheme [Uppala et al., 2005]. The latter is obtained as a product of the Tropical Rainfall Measuring Mission (TRMM) by combining orbital data (collected by space-borne sensors such as the Special Sensor Microwave/Imager, SSM/I, and the Precipitation Radar, PR) and ground-based rainfall data (included in the Global Precipitation Climatology Project, GPCP [Adler et al., 2003]). These databases contain average rainfall data with a temporal detail of 6 h and over pixels of dimension 1.125° × 1.125° (whole Globe) for the ERA40 database, and of 3 h and over pixels of dimension 0.25° × 0.25° (only latitudes between ±50°) for the TMPA 3B42 data set. In order to mimic the effect of joint space-time integration, NIMROD composite rain maps have been processed as follows:
where t and τare respectively the temporal indexes of the un-integrated and temporally integrated rainfall intensity;NT is the number of RS(j, t) samples, defined in (9), included in a time interval of T hours.
 The analysis proposed hereafter follows the procedures outlined in sections 3 and 4but it is not as extensive because the joint space-time integration is a too complex process, which makes cumbersome, if even possible, to define simple analytical formulations such as(3) and (11). As an alternative, results are derived for values of integration time and area specific of the two abovementioned databases. Figure 9 depicts the average trend with distance of the correlation coefficient for spatially plus temporally integrated data (the definition of the latter, ρST follows (10), where RS is replaced by RST), for T = 6 h and S = 100 km (roughly the ERA40 pixel at midlatitudes), and T = 3 h and S = 25 km (roughly the TMPA 3B42 pixel throughout its coverage region, i.e., ±50° latitude). For comparison, Figure 9also depicts the results for un-integrated data and for integration withS = 100 km. As an example, at 100 km distance, ρ(100 km) ≈ 0.12, ρ100km(100 km) ≈ 0.55 and ρ100km6h(100 km) ≈ 0.7. Note also how rainfall data integrated over a relatively small area (A = 525 km2), when further integrated in time (T = 3 h), provide an increase in the correlation index as from data averaged over an area about 16 times larger (A = 10000 km2).
 The results in Figure 9 allow us to define conversion factors similar to those in (7) and (15) relative to the ERA40 or TMPA 3B42 databases:
whose trend with d is shown in Figure 10, together with analytical fitting expressions and the associated regression accuracy.
 The applicability of (18) and (19)to low resolution rainfall data relative to regions other than UK has been preliminary assessed against the data derived from the S-band weather radar installed for research purposes at the experimental station of Spino d'Adda, a few kilometers far from Milan in the Padana Valley (Italy). In particular, we have considered 4920 maps of rain intensity at ground level (collected during several events in the years 1998, 1999, 2000 and 2006), resulting from the composition of radar scans at different elevation angles. Data in the original polar format have been processed and specific procedures applied (e.g., use of image filters and of the Doppler information to identify static obstacles) in order to improve the rain rate estimation from reflectivity, made particularly critical due to the high mountains surrounding the Padana Valley. The time interval between consecutive rain maps is 15 min, rain rate estimates are mapped on a regular 1 km × 1 km Cartesian grid and the maximum operational distance from the radar is 150 km.
Figure 11 depicts the trend of ρ with dfor un-integrated data derived from the Spino d'Adda radar maps and from the application of(19) and (18) to rainfall data extracted respectively from the TMPA 3B42 products (RTRMM) and the ERA40 database (RERA40). For completeness, also the decorrelation trends obtained from RTRMM and RERA40 samples have been added to Figure 11. These trends have been calculated using the same procedure outlined in section 3 for NIMROD data: first the value of the correlation coefficient was derived using RTRMM and RERA40 time series as input to (2) for each couple of pixels and afterwards the correlogram reported in Figure 11 was calculated by averaging ρ values associated to pixels at the same distance. Considering that Spino d'Adda radar maps define a circle with radius equal to 150 km, we have identified overlapping areas using ERA40 and TMPA 34B2 data, of latitude/longitude dimension 3.375° × 3.375° (3 × 3 ERA40 pixels) and 3° × 3° (12 × 12 TMPA 3B42 pixels), respectively.
 The correlation coefficient estimated from TMPA 3B42 data is very satisfactory (root mean square of the estimation error equal to 0.0211), while the one obtained from ERA40 data is well acceptable (root mean square of the estimation error equal to 0.0483). The difference is likely due to the coarser resolution and more limited accuracy of the ERA40 data with respect to TMPA 3B42 data. Although not directly noticeable in Figure 11, it is worth pointing out that the rainfall decorrelation trend with distance is slightly steeper over the Padana Valley than over the British Isles, which reflects the different climate in the two regions (prevalent stratiform events in Northern Europe, higher convective activity in the Mediterranean area).
 In this paper, 1 km × 1 km composite rain maps collected by the NIMROD radar network covering Great Britain have been processed to assess the impact of the time integration T or the area integration A = S2 on the spatial rainfall statistics. The main goal was to quantify the enhancement in the correlation (at equal distance between the sites) associated with the rain rate averaging as T or S increases: indeed the correlation coefficient ρrelative to un-integrated data is smaller than 0.1 ford > 100 km, while temporally or spatially integrated data still show a marked correlation (e.g., ρ6h(100 km) ≈ 0.5 and ρ100km(100 km) ≈ 0.55).
 Simple double exponential laws have been introduced to model the average trend of ρ with the distance between two sites d, as a function of A or T. The proposed models show a satisfactory fitting accuracy for all T values (0.5, 1, 2, 4 and 6 h) and S values (4, 8, 25, 50 and 100 km) considered in this work, with the coefficient of determination R2associated to the regression ranging between 0.9922 and 0.9961. Such models are afterwards used to define conversion coefficients, whose applicability is hoped to hold with good approximation in temperate regions outside UK. Their aim is to de-integrate the spatial correlation information estimated from networks of raingauges with long integration time or from radar data with coarse spatial resolution.
 As a final step, NIMROD maps have been also processed to mimic rainfall data included in the ERA40 and TMPA 3B42 databases (joint space-time integration) in order to provide useful hints for the use of global large databases of rainfall with limited spatiotemporal resolution (meteorological and Earth Observation databases) to estimate the high-resolution spatial correlation of rain, which turns out to be a key information for the design of advanced telecommunication systems.
 Future extensions of the work include the further validation of the proposed conversion coefficients against additional rainfall databases and the investigation of the monthly variation of the correlation functions.
 The spatial correlation of rain can be calculated from radar derived rain maps using the Fourier analysis outlined in this section. The method inherently assumes that the rain field is stationary, i.e., that the covariance C between two points of coordinates [x1, y1] and [x2, y2] depends only on their separation distance, i.e., that C([x1, y1], [x2, y2]) = C(|x2 − x1|, |y2 − y1|) = C(dx, dy).
 The Discrete Fourier Transform (DFT) Fk(u, v) of the k-th rain maprk(x, y), consisting of N × M pixels, is given by (N = 600 and M = 1000 for NIMROD 1 km × 1 km composite images):
 The average periodogram, an estimate of the power spectral density (PSD), is obtained as (Wis the total number of available rain maps, e.g., 99695 in this work considering un-integrated data):
from which, according to the Wiener–Khinchin's theorem [Proakis and Manolakis, 1996], the average covariance of the rain field can be obtained by applying the inverse DFT:
 The correlation coefficient map (see Figure 4) is calculated by normalizing the covariance with respect to its maximum value:
from which the decorrelation trend depicted in Figure 5 is finally obtained by averaging ρ values corresponding to the same separation distance .
 The authors would like to thank the UK Meteorological Office and the British Atmospheric Data Center (BADC) for the provision of NIMROD radar data; Adrian Townsend from the University of Bath for kindly providing the software code to extract NIMROD 1 km × 1 km composite images; and Stefano Tebaldini from Politecnico di Milano for his useful suggestions.