performance evaluation of models for the conversion ...

Performance evaluation of models for the conversion of the rainfall statistics from long to short integration time ESA Workshop on Radiowave Propagation, Noordwijk, NL; 12/2008

PERFORMANCE EVALUATION OF MODELS FOR THE CONVERSION OF RAINFALL STATISTICS FROM LONG TO SHORT INTEGRATION TIME Luis Emiliani(1), Lorenzo Luini(2), Carlo Capsoni(2), Carlo Riva(2) (1)

Email: [email protected] Department of Electronics and Information (DEI) – Politecnico di Milano, Milano, Italy Email: [email protected]

(2)

ABSTRACT The 1-minute integrated complementary cumulative distribution function (CCDF) of rainfall, P(R)1, is a necessary piece of information in the process of rain attenuation estimation for microwave link design. Whenever local 1-minute integrated rainfall data are not available, the conversion of rain rate CCDFs from any integration time T, P(R)T, to P(R)1 is a viable option. This paper presents a comparative performance evaluation of some models for the conversion of the rainfall statistics from long to short integration time, including some methods found in the mainstream literature and a new physically based methodology. The models’ assessment is performed with the aid of an extensive database of rainfall statistics collected from various locations in the world, in different climatic zones. The performance comparison highlights the benefits of using a physical approach to rainfall statistics conversion, as it provides more accurate and stable predictions, in terms of root mean square of the prediction error variable. INTRODUCTION In today's communications networks panorama, microwave links continue to play an important role. When used as a last mile solution, microwave systems enable point-to-multipoint connectivity with minimum infrastructure deployment and with minimum disruption to other services. When used as part of the backbone of the network, microwave links provide a costeffective means of reaching sites in impervious geographic locations. As widely known, microwave communication systems at frequencies above 10 GHz are heavily affected by tropospheric propagation impairments. Among the various performance-affecting phenomena of interest, rain induced attenuation is the most relevant one [1], as the magnitude of the fade exceeds tens of dB at frequencies in the Ka (20-30 GHz) and Q/V (40-50 GHz) bands, which are in use for both terrestrial services, such as Broadband Wireless Access (BWA), Local Multipoint Distribution Service (LMDS) and Multipoint Video Distribution Systems (MVDS), as well as for satellite networks, part of the Fixed and Broadcast Satellite Service (FSS, BSS). In spite of the problems arising from the strong attenuation phenomena, high frequency bands are becoming more and more attractive as they provide the bandwidth required to offer complex multimedia applications and because they are less congested, when compared to the Ku band. This paper deals with the conversion of rainfall statistics, specifically of the Complementary Cumulative Distribution Function (CCDF), from an integration time T (henceforth referred to as P(R)T) to a 1-minute integrated distribution (hereinafter referred to as P(R)1). This latter integration interval is recommended by the ITU-R (International Telecommunications Union – Radiocommunication sector) because it permits to measure the rain rate temporal variation with adequate accuracy. However, considering that the number of data sources fulfilling this condition is relatively small for various regions of the world (such as countries in South America and Africa), the conversion of P(R)T (typically ranging from 5 to 60 minutes), to P(R)1 through adequate methods represents a valid alternative. Indeed, this latter solution should be preferred whenever local measurements are available, as it allows the preservation of the peculiarities of the rainfall process as much as possible and, thus, it improves the CCDF estimation accuracy [2]. DESCRIPTION OF THE RAINFALL DATABASE As stated in the previous section, the aim of this paper is to conduct a comparative study of some models aimed at the conversion of P(R)T to P(R)1. To this aim, a database of rainfall statistics with different integration times (including T = 1 minute) and collected in various climatic regions of the world has been assembled. The data used for our modeling and testing activity have been derived from various sources, including:

L. Emiliani, L. Luini, C. Capsoni and C. Riva.


• 1-minute integrated rain rate time series, from which CCDFs with longer integration time can be obtained1; • CCDFs available from ITU contributions; • CCDFs available from scientific reports and papers. The rainfall CCDFs contained in the database are sampled according to the probability values defined by recommendation ITU-R P.311 [3] and classified following the Köppen climatic region system, as per [4]. Although any climatic categorization is subject to uncertainty in the vicinity of the boundaries between the regions and to the existence of microclimates that could affect some sites (such as those located in major urban centers [5]), it can at least provide an idea of the meteorological peculiarities common to all the rainfall statistics pertaining to the same zone. The sites where rainfall statistics are available are indicated by the red dots depicted in Fig. 1, whereas Table 1 and Table 2 respectively specify the number of sites pertaining to the different Köppen climate zones and the number of measurements at our disposal for different integration times.

45° N

0°

45° S

Fig. 1. Description of the sites and type of data available in the database of rainfall CCDFs Table 1. Number of sites classified according to the Köppen climate zones Cold zone Temperate zone Tropical zone 4 9 10 Number of sites Table 2. Number of measurements available per integration time T = 1 and 5 min T = 1 and 10 min T = 1 and 20 min T = 1 and 30 min Number of measurements

21

22

11

15

T = 1 and 60 min 17

MODELS USED IN THE TESTING ACTIVITY Various methodologies have been proposed through the years as possible solutions to the problem of converting CCDFs from various integration times to P(R)1. These methods can be classified as physical, analytical or empirical based on their functional principles and, as global or regional, based on their scope of applicability. Physical models address the conversion of statistics based on the physical processes involved in the formation and development of rain and in the evolution of a rain event in time. Empirical conversion methodologies provide simple analytical laws (normally regression-based) expressing the relationship between equiprobable rain rate values. This group of methodologies is perhaps the most common found in the literature, as the functions used to model the relationship are relatively simple to adapt when the 1-minute and T-minute integrated distributions are available. In between these two extremes of modelling complexity, we find analytical models. This approach assumes that the rain rate CCDFs at any integration time T can be represented by a given function whose general shape remains constant but whose parameters vary depending on the integration time.

1

In this study, the following integration times have been considered: 1, 5, 10, 20, 30 and 60 minutes.



This paper will address three methodologies proposed in the literature and belonging either to the physical or to the empirical family. Among the numerous models proposed, these models have been selected because, in spite of their regional nature (in fact, they are all based on conversion coefficients to be determined by means of measured data), their simple formulation (i.e. at most two coefficients) allows to easily extend their applicability to more than one climatic region of the world through the introduction of meaningful average coefficients (a detailed explanation of the selection process is given in [6]). Additionally, the empirical methods selected in this paper have been proposed, at different moments in time, as possible ITU-R recommended approaches for rainfall statistics conversion. Finally, a new physical conversion method inherently applicable on a global basis will be presented in this contribution. The model, an extension of the one proposed in [2] to include multiple integration times, will be tested and its performance will be compared with the one provided by the other selected methods. The Power Law (PL) approach The CCDF conversion through the power law approach is illustrated in (1):

R1 ( P ) = a RT ( P ) b

(1) where R1(P) and RT(P) are the rain rate values, respectively extracted from P(R)1 and P(R)T, relative to the same probability value P, whereas a and b are the model’s coefficients. Table 3 presents the regional and global coefficients determined for the PL approach by averaging the values of a and b obtained from the database described above. Further details regarding the process of obtaining the parameters can be found in [6]. Table 3. Average coefficients for the PL rainfall conversion methodology Global Temperate (C) Cold (D) Tropical (A) a a a b a b a b 5 min to 1 min 0.924 1.044 0.953 0.895 0.895 1.047 0.924 1.044 10 min to 1 min 0.829 1.097 0.903 0.730 0.730 1.086 0.829 1.097 20 min to 1 min 0.736 1.169 0.674 0.784 0.784 1.161 0.736 1.169 30 min to 1 min 0.583 1.265 0.571 0.528 0.528 1.205 0.583 1.265 60 min to 1 min 0.509 1.394 0.442 0.507 0.507 1.375 0.509 1.394 It is worth mentioning that the current ITU-R method [7] implements the PL approach as well, but proposes a different set of average coefficients a and b which are valid only for integration times in the 5 to 30 minute interval and which were obtained from a database of measurements collected in 11 sites (Korea, the Czech Republic, China and Brazil) [8]. In the testing activity both sets of coefficients for the PL approach will be considered, the new ones listed in Table 3 and those proposed by the recommendation ITU-R P.837-5. The Conversion Factor (CF) approach Another common empirical method to convert distributions is the use of a conversion factor, defined as the ratio between equiprobable rain rates extracted from P(R)1 and P(R)T. The conversion factor is modeled as a function of the probability of exceeding the given rain rates and approximated by means of either a power law or a mixed law with power and exponential components, but in this work only the former model, thanks to its simplicity, has been evaluated and extended with global and regional coefficients. This approach will henceforth be known as CF-PL.

CF ( P ) = a P b , 0 < P < 1 R1 ( P ) = CF ( P ) R T ( P )

(2)

The approach of (2) was initially proposed and analyzed in [9]. Table 4 presents the regional and global coefficients determined for the CF-PL approach using the selected sites in the database. Further details regarding the process of obtaining the parameters can be found in [6]. Table 4. Average coefficients for the CF-PL rainfall conversion methodology Global Temperate (C) Cold (D) Tropical (A) a b a b a b a b 5 min to 1 min 0.910 -0.021 0.925 -0.021 0.825 -0.031 0.934 -0.016 10 min to 1 min 0.813 -0.044 0.850 -0.043 0.736 -0.053 0.819 -0.040 20 min to 1 min 0.716 -0.073 0.646 -0.085 0.608 -0.087 0.867 -0.049 30 min to 1 min 0.588 -0.108 0.558 -0.114 0.537 -0.111 0.628 -0.104 60 min to 1 min 0.521 -0.155 0.424 -0.171 0.360 -0.191 0.646 -0.132



The Lavergnat-Golé (LG) model The Lavergnat-Golé model (henceforth LG) was originally proposed in [10]. Although this method is devised starting from physical considerations, i.e. by modeling the rainfall inter-drop time (time interval separating two consecutive rain drops) as a renewal process, nevertheless it relies on an empirical parameter, α, originally adapted to the data available to the authors in [10] (Gometz-la-Ville, France). The conversion of cumulative distributions from an integration time T to the target integration time of 1 minute is a particular application of the general model, and is achieved by means of a conversion factor defined as the ratio of the two integration times:

CF = 1 T R1 = RT CF α and P (R1 )1 = CF α P (RT )T

(3)

A key difference in this approach is the use of a conversion factor operating both on the rain rate as well as on the probability values, as indicated in (3). Moreover, in contrast with the PL and the CF-PL methods, the LG model involves a single, sitespecific conversion parameter that does not depend on the integration time. Therefore, in this exercise, α has been determined by maximizing the conversion performance over all integration times, as opposed to the previous cases, where a fit was performed for each integration time separately. Global and regional coefficients obtained by averaging the local values of α are shown in Table 5. Table 5. Average coefficients for the LG rainfall conversion methodology Mean parameter α

Global

Temperate (C)

Cold (D)

Tropical (A)

0.163

0.185

0.1627

0.143

As a comparison, the value recommended for the temperate climate in [10] is 0.115. The EXCELL conversion model A new physical method has been included in this comparative study. The new methodology, an extension of the one proposed in [2], takes advantage of the EXCELL (Exponential CELL) model [11] for the simulation of the actual counting process of a rain gauge operating at a given integration time. Its rationale can be summarized as follows. The meteorological environment of the location of interest is described by means of an ensemble of synthetic rain structures whose probability of occurrence depend on P(R)1: the EXCELL model, in fact, defines an analytical relationship between the probability of occurrence of the exponentially shaped rain cells and the third order derivative of the local P(R)1. Afterwards, the rain cells identified by the EXCELL model are classified as of the stratiform or of the convective type, according to the procedure outlined in [12], which is based on the assumption that, when considering stratiform rain, rainfall peaks hardly exceed 10 mm/h. The core procedure of the conversion methodology concerns the simulation of the rain field interaction with a rain gauge operating at a given integration time T. To this aim, the synthetic rain cells are moved on the scene at an equivalent cell velocity v in order to take into account the space-time evolution of the whole rain field. In particular:

vconv = v600 k1 (T )

(4)

where v600 is the wind velocity (relative to the isobar 600 hPa) provided on a global basis by the ECMWF (European Centre for Medium-Range Weather Forecast) on a 1.5° x 1.5° geographical grid (ERA-15 database) [13], whereas k1(T) is a reduction factor (> 1) that is dependent on the integration time T of the rain gauge. Instead, cells contributing to stratiform rain translate at velocity vstrat:

vstrat = k 2 (T ) vconv

(5)

which, as expected, is a fraction k2(T) (< 1) of vconv. In this paper, the velocity reduction factors, k1(T) and k2(T), have been determined by exploiting reference rainfall statistics with integration times of 1 minute and of T minutes (with T ranging from 5 to 60 minutes), measured at 6 locations spread worldwide: Montreal (Canada) and Prague (Czech Republic) in the cold region; Spino d’Adda (Italy) and a site in Florida (USA) in the temperate region; Ji-Paranà (Brazil) and Chorillos (Colombia) in the tropical region. The following expressions accurately describe the trend of the reduction factors with T:

k1 (T ) = 0.2 T 0.498 + 0.82 and k 2 (T ) = −0.017 T 0.746 + 0.69 for 5 < T < 60 minutes

(6) For the purpose of estimating P(R)1 from P(R)Τ, which is the actual goal of the conversion method, the procedure outlined in this section is reversed using an optimization procedure based on a Genetic Algorithm [14].



Model evaluation The performance of the conversion methods has been assessed by means of the relative error variable ε(P,T) defined in (7) and compared by calculating ȥ, the root mean square (RMS) value of ε(P,T):

ε (P, T ) = 100

Re (P, T )1 − Rm (P )1 [%] Rm (P )1

(7)

where Rm(P)1 and Re(P,T)1 are the rain rate values respectively relative to the measured P(R)1 and to the P(R)1 estimated from the measured T-minute integrated rainfall CCDF; both rain rates correspond to the same probability level P. ȥ was calculated considering all the points available on the CCDFs, without resorting to interpolation. Table 6 and Table 7 present the results of the model performance evaluation, respectively conditioned to the climatic zone and the integration time. Empirical models – PL, CF-PL and LG – have been applied using the global coefficients listed respectively in Table 3, Table 4 and Table 5: the regional results reported in Table 6 and Table 7 refer to the use of such coefficients to rainfall data collected in those regions. For convenience, also the performance of the currently recommended ITU-R model [7] is reported in the tables. Table 6. Results of the testing activity, per climatic region and over all sites, considering all integration times

ITU PL CF-PL LG EXCELL

Global RMS Mean 18.05 6.01 20.85 3.35 17.96 5.42 15.14 2.93 13.9 -4.47

Tropical climate Temperate climate RMS Mean RMS Mean 24.04 11.97 9.82 0.74 26.91 10.38 11.31 -3.28 19.43 5.13 12.83 2.39 16.98 6.36 12.90 -1.16 15.28 -6.54 10.61 -4.26

Cold climate RMS Mean 12.80 1.87 16.36 -2.14 21.47 11.19 14.30 2.08 16.42 -0.46

Table 7. Results of the global coefficients testing activity, for different integration times

ITU PL CF-PL LG EXCELL

5 to 1 min RMS mean 8.47 4.75 6.75 0.17 8.93 5.13 9.76 6.15 9.47 -1.46

10 to 1 min RMS mean 16.32 8.68 13.13 0.93 11.67 1.31 12.82 4.99 11.72 -3.42

20 to 1 min RMS Mean 17.72 -1.13 17.52 0.85 19.19 4.85 13.72 1.91 15.03 -5.55

30 to 1 min RMS mean 27.77 9.20 24.63 4.97 19.33 5.10 18.17 2.76 15.46 -2.12

60 to 1 min RMS Mean 36.20 11.74 29.41 12.78 21.74 -4.32 19.96 -10.75

As shown in Table 6 and Table 7, the EXCELL conversion model performs best overall (over all regions, over all integration times), followed by the LG method and by the CF-PL approach. Considering Table 6, the EXCELL model outperforms the others both in the tropical and temperate Köppen climatic regions, while the LG method gives better results in the cold region: specifically, it is worth highlighting the better stability shown by physical models with respect to the others, in terms of variation of ȥ with the climatic regions studied. These results appear to confirm the statements in [2], i.e. that physical models should be generally preferred to empirical ones. As indicated in Table 7, the ITU model has been defined only for conversions from T = 5, 10, 20 and 30 minutes to 1 minute: the global results shown in Table 6 actually mask the reduced applicability range of the model and might give a misleading result if it is not properly contextualized. The ITU model should be compared directly with the PL method, as they are both based on the same formulation (see (1)): although in this paper the PL method has been extended to T = 60 minutes and reparameterized according to a more comprehensive rainfall database (with respect to the one used to obtain the coefficients of the ITU model), on average, such kind of approach performs worse than all the other methods. This result suggests that a revision of the conversion method currently recommended by ITU, not in terms of coefficients but rather in terms of approach, would be of benefit. The results reported in Table 7 clearly indicate that, as expected, for all models the conversion performance degrades as T increases: as the integration time increases, the site-specific meteorological peculiarities are more marked. As a consequence, the average coefficients of empirical methods are less effective, whereas, when physical models are concerned, they appear to be too simple to properly represent the actual complexity of the rainfall process.



As a general conclusion, it is clear that conversions employing physical models should be regarded as the best solution, whereas the use of a conversion factor modeled via a power law (CF-PL) represents a valuable alternative solution. In any case, the performance degradation with increasing integration time should be kept in mind. Summary and conclusions The main objective of this paper was to compare methodologies for the conversion of CCDFs of rainfall from a given integration time T to 1-minute. To do so, we have selected from the mainstream literature three methods, based on empirical parameters, according to their simplicity and their suitability to be extended to global use. However, although models of such nature offer a simple means to approach the problem, their applicability is limited to the original domain of the data from which their empirical coefficients have been obtained (sites with similar climatology, topography, orography, urbanization levels, etc). In order to extend their applicability, in this paper, average empirical coefficients have been derived from a comprehensive set of rainfall statistics collected with different integration time and in different climatic regions. Moreover, a novel physically based approach to the rainfall statistics conversion has been introduced. The new methodology relies on the EXCELL model and it is inherently applicable worldwide, as it does not include empirical parameters and receives as input local data. The performance of the four conversion methods has been evaluated and compared: as expected, physical models (EXCELL and LG) have proven to yield the best results, regardless of the integration time of the known distribution and of the climate of the location of interest. Moreover, considering that the EXCELL model performs the conversion using as input global maps of wind speed, it is clear that it is the only one that can be made truly global. In fact, although the LG model provides results comparable to those of the EXCELL model, it inevitably presents the drawback of using an empirical parameter for the conversions, which makes it dependant on the source database used to calculate it. Even though its sensitivity to changes in its parameter is relatively small (as shown in [6]), it can not be considered truly global until an expression for α as a function of climatic parameters is proposed. When empirical conversion methods are concerned, the formulation lying at the basis of the recommendation ITU-R P.837-5 (the PL approach) turned out to be less adequate to global applicability with respect to the CF-PL approach: this result suggests that a revision of the recommendation ITU-R P.837-5 would be of benefit. As a further conclusion, either when empirical or physical methods are concerned, prediction results degrade with increasing integration time: worst results are obtained when the data source is P(R)60. Considering that data with 60-minute integration time is more readily available from weather and sewage management agencies, the use of physical models should be even more recommended over the simplicity of empirical proposals. ACKNOWLEDGEMENTS The authors greatly appreciate the collaboration received from Drs. O. Fiser, J. Restrepo, F. Barbaliscia, Gustavo Rendon and Gustavo Munoz in assembling the measurement database. REFERENCES [1] E. Matricciani, C. Riva, “Evaluation of the feasibility of satellite systems design in the 10–100 GHz frequency range,” Int. J. Satell. Commun., Vol. 16, 237–247, 1998. [2] C. Capsoni, L. Luini, “1-min rain rate statistics predictions from 1-hour rain rate statistics measurements,” IEEE Transactions on Antennas and Propagation, Vol. 56, No. 3, March 2008. [3] ITU-R Recommendation P.311-12, “Acquisition, presentation and analysis of data in studies of tropospheric propagation,” Geneva, 2007. [4] M. Peel, B. Finlayson, T. McMahon, “Updated world map of the Köeppen-Geiger climate classification,” Hydrol. Earth Syst. Sci. Discuss., 4, pp. 439–473, 2007. [5] A. Ramachandra Rao, “Stochastic Analysis of annual rainfall affected by urbanization,” Journal of Applied Meteorology, Vol. 19, Jan. 1980. Pp 41-52. [6] L. Emiliani, L. Luini, C. Capsoni, “Analysis And Parameterization Of Methodologies For The Conversion Of Rain Rate Cumulative Distributions From Various Integration Times To One Minute,” accepted for publication in the IEEE Antennas and Propagation Magazine.



[7] ITU-R Recommendation P.837-5, “Characteristics Of Precipitation For Propagation Modelling”, Geneva, 2007. [8] M. Jung, I. Han, M. Choi, J. Lee, J. Pack, “Study on the Empirical Prediction of 1-min Rain Rate Distribution from Various Integration Time Data,” Korea-Japan Microwave Conference (KJMW), Nov. 2007. pp 89 – 92. [9] B. Segal, “The influence of raingage integration time on measured rainfall-intensity distribution functions,” J. Atmospheric and Oceanic Technology. Vol. 3, pp. 662-671. 1986. [10] J. Lavergnat, P. Golé, “A Stochastic Raindrop Time Distribution Model,” AMS Journal of Applied Meteorology, Vol. 37, Aug. 1998. pp.805-818. [11] C. Capsoni, F. Fedi, C. Magistroni, A. Paraboni, A. Pawlina, “Data and theory for a new model of the horizontal structure of rain cells for propagation applications,” Radio Science, Volume 22, Number 3, Pages: 395-404, May-June 1987. [12] C. Capsoni, L. Luini, A. Paraboni, C. Riva, “Stratiform and convective rain discrimination deduced from local P(R),” IEEE Transactions on Antennas and Propagation, pp. 3566-3569, Vol: 54, Issue: 11, Nov. 2006. [13] J.K. Gibson, P. Kållberg, S. Uppala, A. Hernandez, A. Nomura, E. Serrano, “ERA -15 Description (Version 2),” ECMWF ReAnalysis Project Report Series, Jan. 1999. [14] D. E. Goldberg, “Genetic Algorithms in Search, Optimzation & Machine Learning,” Addison-Wesley, 1989.
