Radio Science

Estimating the spatial cumulative distribution of rain from precipitation amounts

Authors


Abstract

[1] This contribution describes SPET (Spatial P(R) Estimation Technique), a methodology aimed at estimating the spatial rain rate complementary cumulative distribution function, PS(R), from Numerical Weather Prediction (NWP) rain precipitation data. SPET has been calibrated making use of a large database of rain maps derived from the S-band weather radar sited in Spino d'Adda, Italy, while its performance has been assessed against an independent data set derived from the NIMROD C-band radar network. Results indicate that SPET performance improves as both the observation area and reference time interval increase, which adds confidence to its use to estimate PS(R) starting from global rainfall data produced by meteorological re-analyses. SPET also proved to correctly predict the fractional rainy area, showing an RMS of the relative error that falls approximately between 15% and 20% for area and time interval values typical of NWP re-analysis data. As a final step, the proposed technique has been used to predict long-term rainfall statistics collected by rain gauges worldwide, receiving as input rainfall data extracted from 10 years of ERA40 products (ergodicity of the rain field). Very good performances have been obtained, only limited by the poor quality of the input data.

1. Introduction

[2] Modern satellite telecommunication systems are intended to provide users with advanced services, both in the broadcasting domain (e.g., parallel information streams allowing the real-time selection between Standard or High Definition TV depending on the local channel condition [European Telecommunications Standards Institute, 2006]) as well as in the interactive domain (e.g., full duplex links delivering Internet via satellite (iDirect, http://www.idirect.net, accessed on June 2011)). In this scenario, increased data rates and link reliability represent key requirements, for the fulfillment of which satellite network operators are presently planning the employment of frequencies in the Ka and Q/V bands. The main drawback associated to such bands resides in the propagation impairments caused by the atmosphere, whose detrimental effects become stronger and stronger as frequency increases [Riva, 2004]. Among all the atmospheric impairments, those related to precipitation definitely play the prevailing role and have the largest impact on the overall performance of such kind of systems [Capsoni et al., 2009]. As a result, knowledge of the meteorological conditions (in particular, of precipitation) across the whole satellite service area (which can extend to a continental region) is fundamental for the correct planning of the system [Luini et al., 2011].

[3] In the last decades, the complexity and accuracy of Numerical Weather Prediction (NWP) models considerably increased. Thanks to their global coverage and statistical stability associated to long observation periods, products obtained from NWP data re-analyses, made available nowadays by the European Centre for Medium-Range Weather Forecasts (ECMWF) (http://www.ecmwf.int/, accessed on June 2011), are the main input to current International Telecommunication Union Radiocommunication Sector (ITU-R) recommendations devoted to the estimation of the atmospheric impairment statistics. As an example, recommendation P.840–4 [ITU-R, 2009], focused on the attenuation due to clouds and fog, receives as input statistics of total liquid water content (reduced to 0°C temperature), which are derived from the ERA40 database provided by the ECMWF [Uppala et al., 2005]. The main limitation of such data is the coarse spatial and temporal detail (1.125° × 1.125° latitude/longitude grid and 6 h, respectively, for the ERA40 database), which are absolutely unsatisfactory for some applications. This is the case, for instance, of an advanced satellite system with an on-board reconfigurable antenna that allows to selectively radiate more power toward the areas more severely affected by atmospheric fades [Paraboni et al., 2009]. As it is easily arguable, the effectiveness of the power allocation algorithm driving the antenna reconfiguration is strictly related to the accuracy with which the actual meteorological situation (i.e., attenuation level) at ground is known over the entire service area. Since small scale rain structures cannot be adequately resolved by the coarse spatial and temporal detail of NWP re-analysis data, suitable methods are necessary to derive more precise information on the precipitation field from meteorological cumulative quantities.

[4] This contribution refines (in terms of accuracy and adaptability) and further consolidates (in terms of performance assessment) SPET (Spatial P(R) Estimation Technique). First introduced by Capsoni and Luini [2009], the method aims at estimating the small-scale rain rate spatial distribution in terms of complementary cumulative distribution function (hereinafter indicated as PS(R)), from rain data produced by meteorological re-analyses. Specifically, input data to SPET are the total rain amount, Mt, and convective rain amount, Mc, accumulated in a reference time period (e.g., 3, 6 h) within a given area (e.g., 1.125° × 1.125° latitude/longitude grid, roughly 100 × 100 km2 at mid latitudes), on a snapshot basis. Specifically, section 2 briefly presents SPET by describing its rationale and its calibration, while section 3.1 is devoted to assessing the performance of the proposed methodology in estimating PS(R), as well as the fractional rainy area, against an extensive independent weather radar data set (NIMROD network). In section 3.2, taking advantage of the quasi-ergodicity property of the rainfall process, 10 years of ERA40 rainfall products are used as input to SPET to evaluate its ability in estimating long-term rain gauge derived P(R)s collected worldwide and gathered into the DBSG3 global database provided by the ITU-R. Finally section 4 draws some conclusions.

2. SPET: The PS(R) Estimation Technique

[5] The methodology proposed here for the estimation of PS(R) requires as a first step the assumption of an analytical model of the P(R), whose parameters can be adjusted according to the Mt and β = Mc/Mt values associated to a given time interval and a given area. In the literature several models of the P(R) have been proposed so far, mainly with the aim of fitting point rain rate distributions measured by rain gauges in time (hereinafter named PT(R)). Exponential, Weibull, Moupfouma [Moupfouma and Dereffye, 1982] and lognormal distributions are some examples. Each one of them has its advantages and limitations: for instance, the lognormal distribution is a common choice thanks to its simplicity and mathematic treatability [Sauvageot, 1994; Féral et al., 2003; Panagopoulos and Kanellopoulos, 2003]; however, its adaptability to measured PT(R)s is not always satisfactory, especially when high rain rates are concerned and, moreover, it is suitable for fitting distributions conditioned to having rain and not unconditioned ones. In this work, we prefer to make reference to the following model of the P(R):

display math

where R is the rain rate (mm/h) exceeded with probability P, while P0, n, Rlow and Ra are coefficients that allow to tune the expression in (1) so as to reproduce at best the reference P(R). In particular, Ra represents the maximum expected rain intensity, i.e., the limit of validity of (1), and n is the shape parameter that decreases with the increase of the convective rain amount Mc. Finally, Rlow and P0 are the parameters that concur to identify the probability of having rain P(R = 0).

[6] This model, first introduced by Capsoni et al. [2006], presents a high degree of flexibility: thanks to its analytical form purposely proposed, it can well reproduce both part of the P(R)s mainly due to low/moderate rain rate (stratiform regime) and the one where intense (convective regime) rain rate values prevail.

[7] The flexibility and the accuracy of the model in (1) emerged when it was used to fit the rain gauge derived PT(R)s measured in different climates worldwide and gathered in the DBSG3 database provided by the ITU-R (DBSG3, http://saruman.estec.esa.nl/dbsg3/, accessed on 2011). Very positive results were obtained (root mean square value of the relative error ranging from 5% and 10%) and the fitting procedure allowed to identify the following power law relationships:

display math

[8] The Mt value contributes to the identification of the coefficients of (1) through its imposition as the integral of the P(R), i.e.:

display math

where p(R) is the rain rate probability density function and γ(a,x) is the incomplete gamma function, defined as:

display math

[9] As a result, by inverting (4), P0 can be calculated as a function of the other fitting coefficients and of Mt:

display math

[10] For the applications envisaged in this study, the optimum coefficients a, b, c, d, e, f, g, h and i in (2) must be re-identified because the β values relative to a short time period (e.g., 3, 6 h) vary within a much larger interval than the average yearly values (βy): while the latter basically give a hint of the balance between stratiform events and the convective activity in a year (e.g., average βy = 0.23 as extracted from the ERA40 database for Milan, Italy), T-hour β values, which span the full range 0–1, provide the specific information on the precipitation type associated to each short time period. In other words, using in the coefficient identification process only the data coming from the pool of yearly PT(R)s would result in a model with reduced sensitivity to the input precipitation data.

[11] Therefore, to properly calibrate the SPET based on (1), a large database of radar derived rain fields with high spatial and temporal detail has been employed, consisting of several rain events collected by the S-band meteorological radar sited in Spino d'Adda, Italy (latitude = 45.4°N and longitude = 9.5°E). More specifically, the considered database includes approximately 22000 pseudo CAPPI (Constant Altitude Plane Position Indicator) radar images, extracted from rain events (in the period from 1988 to 1992) which have proven to be fully representative of the local yearly rainfall statistics [Capsoni et al., 2008]. The maximum operational range of the radar considered for this study is 40 km in order to avoid the inclusion of clutter pixels due to the surrounding mountains. The spatial and temporal resolutions of the radar scans are respectively 0.5 × 0.5 km2 and 77 s.

[12] The key idea consists in deriving from the available rain maps all the information required to calibrate the model, i.e., on one side, the average rainfall quantities analogous to Mt and β provided by NWP re-analyses, and, on the other side, the associated reference PS(R) against which the model's predictions shall be compared. To this aim, consecutive radar images have been grouped into 3-h time slots (typical of global NWP re-analysis data) and the overall PS(R) has been calculated as PS(R) = P(R > R*), i.e., the probability that the rain rate exceeds a given threshold. Afterwards, the average rain rate values, both total and convective, have been derived as follows:

display math

where N is the total number of rain rate samples (pixels) in all the maps pertaining to a 3-h time interval and Rsc is the threshold used to discern between stratiform and convective rainfall. It is worth mentioning that although the β produced by NWP re-analyses relies on other additional meteorological information than the rain intensity (e.g., occurrence of wind updrafts and downdrafts), when such information is not available, as in the case of radar data, the discrimination between stratiform and convective precipitation based on a fixed threshold appears a simple yet effective choice [Capsoni et al., 2006]. Figure 1 shows the monthly mean values of β extracted from the ERA40 database and calculated from the Spino d'Adda radar images using (6) for Rsc = 6, 8 and 10 mm/h.

Figure 1.

Monthly mean β values extracted from the ERA40 database and from the radar images using (6) for Rth = 6, 8 and 10 mm/h.

[13] Based on such results, Rsc = 8 mm/h has been chosen as the most appropriate threshold to apply to the radar data in order to derive a suitable estimate of β.

[14] According to the procedure outlined above, 126 3-h time periods have been extracted from the Spino d'Adda radar database, and characterized in terms of PS(R), Mt and β.

[15] The power law coefficients in (2) have been derived through a global optimization procedure which minimized the root mean square value of the following error figure:

display math

where Re(P) and Rm(P) are the rain rates respectively extracted from the 126 PS(R) estimated by the model and derived from radar measurements at the same probability level P. As a result of the optimization, we obtained:

display math

in which inline image = β for β ≥ 0.001 and inline image = 0.001 for β < 0.001, necessary conditions to prevent Rlow from tending to infinity. The joint application of (5) and (8) in (1) allows the estimate the PS(R) relative to a given time period and reference area which the input Mt and β values refer to. It is worth pointing out that, with respect to Capsoni and Luini [2009], where the Rlow coefficient was fixed to 1 mm/h, in this work its dependence on β has been elucidated as well, leading to an increased adaptability of the analytical P(R) and, consequently, to an increased performance of SPET, as will be shown in the next section.

3. Assessment of SPET

3.1. NIMROD Radar Data

[16] The NIMROD network, managed by the UK Meteorological Office (Met Office), consists of 19 C-band weather radars deployed throughout the British Isles. The detailed positions of the radar equipments is indicated as asterisks in Figure 2, which also depicts the overall coverage of the network.

Figure 2.

Full NIMROD network coverage and the 200 × 200 km2 area considered in this study (square defined by black solid line).

[17] Each radar completes a sequence of azimuth scans at different elevations (between four to eight) every 5 min. Afterwards, each radar transmits the acquired data in full resolution polar format to Radarnet IV (the Met Office centralized processing system at Exeter), where basic processing, quality controls and typical correction procedures are applied [Harrison et al., 1998]. Finally, concurrent data from all the available radars are merged to produce composite images of the estimated rain rate at ground level over the whole Great Britain, every 5 min and with a spatial resolution of 1 × 1 km2.

[18] In this work, a full year (2009) of NIMROD 1 × 1 km2 composite rain maps, freely available on the Web for research purposes [Met Office, 2003], has been selected. The whole database consists of 99695 rain maps, which corresponds to a data availability of 94.84%. As indicated by the black solid line in Figure 2, a square area of dimension 200 × 200 km2 has been extracted, the rationale being: (1) Focus on the area where weather radars are denser and, consequently, data availability and accuracy are increased. (2) Consider a region over land, where the characterization of the rainfall process is of major interest for propagation-oriented applications. (3) Take into account a size typical of reanalyzed NWP data (e.g., ERA40), which are the main target of the proposed estimation technique.

[19] With the aim of evaluating the sensitivity of SPET performance to the integration time and/or area of the input data, the NIMROD maps have been processed in order to obtain PS(R), Mt and β relative to different time intervals (T = 1, 2, 3, 4, 5, 6 h) and to different dimensions of the selected area (A = D2, with D = 100, 140, 200 km).

[20] Figures 3 and 4 respectively show, for both radar data sets, Eε and RMSε, the overall (i.e., taking into account all the PS(R) of the database associated to a given T and D couple) mean and root mean square value of ε(P), calculated by selecting P values higher than 10−4 and lower than PS(1 mm/h), with the aim of considering statistically stable results (P = 10−4 corresponds to 12 samples (pixels) for T = 1 h and D = 100 km).

Figure 3.

Trend with D and T of Eε, the overall mean value of the error figure in (7): NIMROD (solid lines) and Spino d'Adda data (dashed lines).

Figure 4.

Trend with D and T of RMSε, the overall root mean square of the error figure in (7): NIMROD (solid lines) and Spino d'Adda data (dashed lines).

[21] For completeness, Table 1 reports NP, the number of PS(R)s on which Eε and RMSε in Figures 3 and 4 have been evaluated, thus providing a hint about the statistical meaningfulness of the results. It is worth noticing that the number of T-hour intervals in a year obviously decreases as T increases, while the dependence of NP on D is bound to the condition that only PS(R) with R values associated to P ≥ 10−4 are considered in the tests: in fact, the probability to include more rainy pixels tends to increase with the observation area.

Table 1. Number of PS(R)s – NIMROD and Spino d'Adda Dataa
 T = 1T = 2T = 3T = 4T = 5T = 6
  • a

    D expressed in kilometers and T expressed in hours.

D = 100 (NIMROD)317417061194936807685
D = 140 (NIMROD)3628191213411044889756
D = 200 (NIMROD)44842326161112491055878
D = 80 (Spino)3381811261068581

[22] The performance results achieved against the NIMROD data set are in line with those obtained against the Spino d'Adda rain maps (in this case, D = 80 km). Indeed, the results from the two databases present a similar trend with T, and the better error figures associated to the Spino d'Adda database are ascribable to the fact that its rain maps were used to calibrate the model and, also, to the fact that it consists of separate rain events, i.e., with a limited number of maps with very poor rain coverage, which are the ones more difficult to reproduce.

[23] Moreover, the performance analysis points out a key behavior: the accuracy achieved by SPET increases for larger D and T values. This adds confidence to its use as a PS(R) estimation technique starting from global reanalyzed NWP data, typically characterized by wide integration area and long integration time. D has the highest impact, while the reduction of RMSε tends to be negligible after 4 h. These results were somehow expected because of the inherent statistical nature of the prediction model: with the increase in the area (and/or in the time) the resulting PS(R) becomes more and more long-term shaped and the model in (1) is more suitable to reproduce it.

[24] As a final remark on the accuracy of SPET in estimating PS(R), it is worth underlining that the RMSε values in Figure 4 are lower than those reported respectively by Capsoni et al. [2011] for the NIMROD database and by Capsoni and Luini [2009] for the Spino d'Adda data set, which confirms the benefit of introducing also the dependence of the coefficient Rlow on β.

[25] An attractive result of SPET is the estimate of the fraction of the area affected by rain in the reference time interval. This quantity, defined as fractional rainy area η(Rth) = A(Rth)/A [Eltahir and Bras, 1993] (A is the total observation area and A(Rth) is the area where rain intensity is higher than Rth mm/h), is of key importance for the description of land surface hydrology in climate models as well as in propagation related meteorological models. The method presented by Eltahir and Bras [1993] permits to estimate η(Rth) starting from the total cumulated rain rate Mt; however, the main assumption of the method is that the spatial distribution of the rainfall rate over the area is always a realization of the same PS(R). As it is clearly shown in the present work, this assumption is questionable because rain events strongly differ from season to season (and sometimes from hour to hour), and the above assumption may lead to marked errors in the estimation of the fractional rainy area.

[26] To evaluate the ability of SPET in estimating η, let us introduce the following error figure:

display math

where, for each time interval, ηe(Rth) and ηm(Rth) are the fractional rainy area respectively estimated through SPET, by evaluating the expression in (1) for R = Rth, and derived from radar data.

[27] Figures 5 and 6 show Eϕ and RMSϕ, the average and the root mean square values of ϕ(Rth), respectively. The performance results are depicted considering all the time slots derived from the NIMROD and Spino d'Adda data sets, although in the former case Rth = 1 mm/h, while in the latter one Rth = 0.5 mm/h. In fact, the limited observation area (D = 80 km) and the receiver characteristics of the Spino d'Adda radar allow an increased sensitivity to rain rate if compared to NIMROD rain maps.

Figure 5.

Trend with D and T of Eϕ, the overall mean value of the error figure in (9): NIMROD (solid lines) and Spino d'Adda data (dashed line).

Figure 6.

Trend with D and T of RMSϕ, the overall root mean square of the error figure in (9): NIMROD (solid lines) and Spino d'Adda data (dashed line).

[28] Figure 5 indicates that SPET tends to slightly underestimate the fractional rainy area, while Figure 6 confirms again the dependence of the model's performance on D and T. In the overall, SPET provides a satisfactory estimation of η for values of integration area and time interval typical of NWP re-analysis data: for D ≥ 140 km and T ≥ 3 h, RMSϕ falls approximately between 15% and 20%. Although the fractional rainy area for Rth = 0 mm/h is a quantity of key interest both for hydrology and propagation related applications and, in principle, it could be indeed estimated by the proposed model, no reference data are available (the direct measure of this quantity is extremely critical) to assess the accuracy of the model estimation and thus no attempt to derive η(0) has been proposed here.

3.2. Rain Gauge Derived Rainfall Statistics

[29] As documented in the literature, weather precipitation is a quasi-ergodic process, which, in practice, implies that the long-term first-order statistics collected by a rain guage at one site, PT(R), is equivalent to the one obtained by cumulating the information on the precipitation affecting the surrounding area (of lateral dimension approximately between 100 km and 300 km), PS(R) [Eltahir and Bras, 1993; Goldhirsh, 1990]. The main aim of this section is to further increase confidence in the application of the model to NWP-like data. The key idea is to use time series of Mt and β extracted from the ERA40 database (6-h time resolution and 1.125° × 1.125° spatial resolution) as input to SPET, and to sum them to obtain the long-term PS(R) statistics, which, based on the quasi-ergodicity property previously mentioned, should represent a reliable estimate of PT(R) for the site. In fact, the mean yearly rainfall statistics relative to site j can be calculated by means of SPET as follows:

display math

where M is the total number of 6-h intervals in the period 1991–2000, Pi(RMti(j),βi(j)) is the P(R) estimated by SPET for the i-th time interval, using as input Mti(j) and βi(j), which are calculated, for each interval i, by bilinear interpolation of the values associated to the four ERA40 pixels surrounding the site (as recommended by ECMWF). Figure 7 provides a sketch of the full procedure for the estimation of PTj(R) starting from ERA40 data.

Figure 7.

Sketch of the procedure for the estimation of PTj(R) starting from ERA40 data.

[30] The accuracy in reproducing PT(R) through (10) has been tested against the rainfall statistics gathered in the DBSG3 database. Only multiple-year (at least 10) PT(R)s have been used after aggregation to obtain long-term curves, for the quasi-ergodicity property to hold.

[31] Figures 8 and 9 show the results relative to Toledo (latitude = 39.88°N and longitude = −4.88°E, 15 years) and Castellon (latitude = 39.95°N and longitude = −0.07°E, 15 years), which represent one of the best and one of the worst predictions of the model, respectively. The line with circles identifies the data extracted from the DBSG3 database, the solid line is the PT(R) estimated by SPET. For convenience, also the PT(R) calculated after the method currently recommended by ITU-R for the prediction of long-term PT(R) [ITU-R, 2007] has been added to the figures (dashed line). The estimated accuracy, in terms of RMSε, Eε and σε is indicated in the figure legend.

Figure 8.

Results relative to Toledo (latitude = 39.88°N and longitude = −4.88°E, 15 years): DBSG3 data (line with circles), PT(R) estimated through SPET (solid line) and PT(R) derived from ITU-R recommendation P.837–5 (dashed line).

Figure 9.

Results relative to Castellon (latitude = 39.95°N and longitude = −0.07°E, 15 years): DBSG3 data (line with circles), PT(R) estimated through SPET (solid line) and PT(R) derived from ITU-R recommendation P.837–5 (dashed line).

[32] From the results reported in Figures 8 and 9 the performance of SPET in reproducing the local PT(R) appears to be tightly dependent on the accuracy of the input total amount of water accumulated in an average year. The figure title reports such a quantity as provided by the different sources: Mt(ERA40) is the mean yearly accumulation obtained from the ERA40 database (input to SPET), Mt(ITU-R) is extracted from the meteorological database which the ITU-R recommendation relies on, and Mt(GHCN) is the long-term mean yearly amount of water collected by the nearest rain gauge pertaining to the GHCN (Global Historical Climatology Network) database [Vose et al., 2007], which, if relative to many years of observation, as it is usually the case, can be assumed as the most accurate estimate of the local Mt.

[33] Mt(ERA40) differ from Mt(ITU-R), as the latter is the result of the calibration of the original ERA40 database by means of additional rain gauge–based measurements (Global Precipitation Climatology Centre (GPCC) database, http://gpcc.dwd.de, accessed on June 2011), procedure that was conceived to correct the definite overestimation of Mt shown by ERA40 data against different reference rainfall data sets, such as the GHCN database and TRMM (Tropical Rainfall Measuring Mission) 3B42 product [European Space Agency (ESA), 2007]. SPET appears to correctly estimate the shape of the measured P(R) in both cases, as demonstrated by the extremely low σε values, but its overall performance is strongly reduced because of the Mt values in input; for Toledo, Mt(ERA40) is close to Mt(GHCN) and RSMε is low (9.4%), while the opposite is true for Castellon, for which RMS = 55.9%. Unfortunately, it is not possible to take advantage of the abovementioned calibration procedure using the calibrated ERA40 data as input to SPET, because the prediction method operates on 6-h intervals, while the calibration was performed on yearly basis [ESA, 2007].

[34] Figures 10 and 11 extend the assessment analysis by depicting the performance of SPET in estimating PT(R) on the selected subset of the DSBG3 database. The resulting 57 experiments are relative to sites whose latitude spans approximately from −23°N to 46°N. Figure 10 confirms the tendency of SPET to underestimate the local PT(R) more than the ITU-R model (average values of Eε are −23.9% and −7.1% respectively), as a direct consequence of the marked underestimation of the mean yearly Mt(ERA40): the average value of the relative difference between Mt(ERA40) and Mt(GHCN) in each site, ΔMt(ERA40), is −7.1%, while the one between Mt(ITU-R) and Mt(GHCN), ΔMt(ITU-R), is 9.8%. Although SPET shows a much higher negative bias than the ITU-R model, still the overall RMSε of the former is only slightly higher the one of the latter.

Figure 10.

Eε for all the DBSG3 experiments with at least 10 years of measurement: SPET (line with circles) and ITU-R model (line with squares).

Figure 11.

RMSε for all the DBSG3 experiments with at least 10 years of measurement: SPET (line with circles) and ITU-R model (line with squares).

[35] An exhaustive analysis of the results seems to indicate that the underestimation of Mt in the ERA40 data is associated to an “error” in the convectivity ratio β, or, in other words, that the underestimation on the stratiform and convective rainfall amounts is different. This fact may be explained considering that convective events are typically more limited both in space and time than stratiform ones and consequently, NWP models may be less accurate in identifying and resolving the former than the latter, which leads to underestimated β values. This tendency seems to be confirmed by the results in Figures 12 and 13, which focus only on the European sites (i.e., a temperate region) included in the DBSG3 database (again only sites with at least 10 years of measurement). In this case, the average value of ΔMt(ERA40) is −26.2%, but the average RMSε of SPET is lower than that in Figure 11 (even lower than the one of the ITU-R model). The occurrence of convective events in Europe is fairly limited with respect to equatorial and tropical regions, consequently the error on β is expected to be reduced and the estimates provided by SPET more accurate.

Figure 12.

Eε for the DBSG3 experiments in Europe with at least 10 years of measurement: SPET (line with circles) and ITU-R model (line with squares).

Figure 13.

RMSε for the DBSG3 experiments in Europe with at least 10 years of measurement: SPET (line with circles) and ITU-R model (line with squares).

4. Conclusion

[36] This contribution describes in detail SPET (Spatial P(R) Estimation Technique), a methodology aimed at estimating the small-scale rain rate spatial distribution in terms of complementary cumulative distribution function, PS(R). Input data to SPET are the total rain amount, Mt, and the total convective rain amount, Mc, accumulated in a reference time period within a given area, on a snapshot basis as provided by NWP re-analysis rainfall data.

[37] SPET has been developed using a large database of rain fields collected by the S-band meteorological radar sited in Spino d'Adda (Italy) and consisting of more than 20000 radar images relative to several rain events occurred in a 4-year period.

[38] The performance of SPET in estimating PS(R) has been assessed against an independent rain field data set derived from the NIMROD C-band radar network over Great Britain. Results indicate a slight overestimation of the PS(R) and a clear dependence of SPET performance on the observation area A and the time interval T which the input Mt and β values refer to. Specifically, SPET performance improves for larger observation areas and longer time intervals (up to 4 h) because, as A or T increases, the resulting PS(R) becomes more and more long-term shaped and the model is more suitable to reproduce it (RMS of the error figure: ≈20% for A = 1002 km2 and T = 1 h, ≈15% for A = 2002 km2 and T = 6 h). This outcome adds confidence to its use as a PS(R) estimation technique starting from global NWP re-analysis data. The same radar data set has been used also to evaluate the fractional rainy area, which is directly derivable from the estimated PS(R). Results indicate again a dependence of the performance on A and T (RMS of the error figure: ≈38% for A = 1002 km2 and T = 1 h, ≈15% for A = 2002 km2 and T = 6 h), but definitely prove the usefulness of SPET for the prediction of η, showing an RMS of the error that falls approximately between 15% and 20% for values of area and time interval typical of NWP re-analysis data (D ≥ 140 km and T ≥ 3 h).

[39] As a final step, in order to further increase confidence in the application of SPET to NWP re-analysis data, according to the quasi-ergodicity principle of the rain field, the model has been used to estimate long-term rainfall statistics collected by rain gauges in different sites spread worldwide, using as input time series of Mt and β (1991–2000) extracted from the ERA40 database. Its performance, evaluated against the global DBSG3 database provided by the ITU-R, has shown to be tightly bound to the accuracy of the input rainfall data. Notwithstanding the marked underestimation of Mt (and, likely, of β) characterizing precipitation products included in the ERA40 database, the obtained overall RMS of the error turns out to be slightly higher (lower) than that achieved by recommendation ITU-R P.837–5 on a global (European) basis.

Acknowledgments

[40] This work has been partially supported by ESA contract 22216. The authors would like to thank Antonio Martellucci from the European Space Agency for granting access to ERA40 data, the British Atmospheric Data Center (BADC) for providing NIMROD data and Adrian Townsend from the University of Bath for his kind help in processing NIMROD 1 × 1 km2 composite images.

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