Rainfall Rate Modeling for Various Rainfall Types in South Africa

A.A. Alonge, Student Member, IEEE and T.J. Afullo, Senior Member, SAIEE School of Electrical, Electronics and Computer Engineering,

University of KwaZulu-Natal, Durban, South Africa

{210546050, afullot}@ukzn.ac.za

Abstract—This paper models the rainfall rate in Durban (29o 97’S, 30o95’E), South Africa for four different rainfall classifications: drizzle, widespread, shower and thunderstorm. Twelve-months data obtained in 2009 via the Joss_Waldvogel disdrometer, gave a total of 729 independent rain events and 5,542,147 raindrops. The measured rain rate probability functions were compared with the Moupfouma model using 1-minute integration time. The maximum likelihood estimators were adopted for the parameters of two other statistical models, the lognormal and gamma models. The results show that gamma model had the closest performance for the various rain classifications under consideration.

Index Term—Integration time, rainfall classifications, rainfall rate distribution, rainfall rate.

I. INTRODUCTION The rapid development and success recorded in advanced

computing has changed the scope of services offered by communication networks with more emphasis placed on data-centric services, trailing behind video and multimedia services. With increasing demand for high speed data and multimedia services in Africa, most especially in South Africa, there is a need to adequately provide optimized wireless services. Precipitation is a major problem encountered in the deployment of microwave and millimetric wave communication systems [1], [2]. Optimized link services must be sustained irrespective of the hostile atmospheric conditions such as rainfall. Rainfall, a form of hydrometeor has been identified as a major setback in signal propagation. It can intermittently, or even abruptly disrupt transmission links during a downpour, resulting in extreme fading in wireless communication systems [2].

Globally, several rainfall rate and rain dropsize distribution (DSD) models have been developed, as a solution to solving the inherent localized problems which are intrinsic and distinct to rainfall and radiometeorological features of that region [2] [7]. Over the years, rainfall microstructural behaviour have been examined by various researchers; of particular interest are two important rainfall parameters: rain rate and rain dropsize. Statistical analysis using Probability Density Functions (PDFs) and Cumulative Distribution Functions (CDFs) have also contributed to research in the area of microwave link planning and budgeting.

This paper attempts to analyze one-minute rain data collected using the Joss_Waldvogel RD–80 disdrometer during

the period of January to December 2009 at the University of KwaZulu-Natal, Durban, South Africa.

II. REVIEW OF RAINFALL RESEARCH IN SOUTH AFRICA In Africa, one of the known works on rainfall was undertaken at Ile-Ife in Nigeria by Ajayi and Ofoche [8]. In their work, it was suggested that the ITU-R one-minute integration time was the most suitable for microwave link designs. Segal [9] and later Olsen et al.[10], also suggested a two-parameter power-law function which has been exhaustively used in the conversion of higher rain rate to 1-minute integration time in several locations. The power-law function is given as:

T / (1) where a and b are the conversion parameters of the rain rates PT and PT at one-minute and τ-minute respectively exceeded at equal probability (PT%).

The conversion factor was defined by Segal as: ρτ P R1 PRτ P (2) where and are the rainfall rate exceeded, with equal probability, P, at different integration times, one-minute and τ-minute.

Later, Moupfouma [4] developed an exponential relationship for the estimation of the cumulative distribution of rain rate in the tropics and equatorial regions. This was extended in [5] to accommodate new parameters for temperate and tropical regions. ITU-R 837-5 [6] also presented approximate rain rate conversion equations for countries such as China, Brazil and Korea with minimal errors. However, the region of Africa has been left out mainly because of insufficient long-term experimental data and scarcity of deployed equipments. This led to the adoption of regional classification of rainfall as indicated in the ITU-R designations in [6]. In [11], Emiliani et al. also examined the various methodologies required for the classification of rainfall rate conversion techniques.

Interestingly, ITU-R classifications have their limitations with regards to microwave link planning and budgeting. Fashuyi et al. [12] and Owolawi et al. [13] reported classification discrepancies in the ITU-R rainfall designations

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for South Africa. In addition, Fashuyi et al. [12] used the approach in [8]–[10] to obtain the rain rate conversion parameters for Durban for conversion of higher integration time and also reclassified ITU-R designations for South Africa. By comparing 5-year hourly rain rate distributions with measurements at a similar region using equiprobable rain rate exceedances, values of a = 9.228 and b = 0.8207 for Durban were obtained. These power-law parameters were then assumed for eleven other cities in South Africa in the estimation of rain attenuation. Odedina et al. [14] used the R0.01 values obtained from the work in [12] to determine the fade margins in five climatic locations in South Africa. More recently, Owolawi et al. [15], [16] investigated the validity of the various statistical models for South Africa. In [15], the Moupfouma tropical model was investigated for the South African subtropical region, with parameters a, b and u being obtained. In [16], by employing a hybrid method for rain rate conversion, rainfall rate probability functions using Weibull, lognormal and gamma distribution models with five-minute rainfall data were examined. The method adopted a second-order polynomial function with three regression coefficients a, b and c for rain rate conversion from five-minutes to one-minute as given by:

P P [mm/h] (3)

where the values of a = 0.0014, b = 1.2021 and c = –0.3543 were obtained for Durban from his work under equiprobable rain rate exceedences.

In all the research related to rainfall rate in South Africa [12]–[16], the apparent lack of one-minute data to determine rainfall attenuation parameters is a challenge. Moreover, no detailed research has been embarked upon in South Africa to determine the behaviour of these parameters under four different rainfall classification: drizzle, widespread, shower and thunderstorm. Hence, it is imperative to establish a new understanding by examining the influence of rainfall parameters under these classifications. In this paper, we make our comparison based on the probability density function (PDF) for the different rainfall types.

III. DATA COLLECTION AND MEASUREMENTS FROM DISDROMETER

The measurements for this research work were undertaken at the Howard campus site of the University of KwaZulu-Natal, Durban. The one-minute rain data was measured using the Joss-Waldvogel RD-80 impact disdrometer installed at the School of Electrical, Electronics and Computer Engineering. The disdrometer has a sampling time of one-minute and measures in real-time, quantities such as rain rate, rain dropsize (with 20 dedicated bins with range from 0.359 mm to 5.373 mm), rain accumulation and rain reflectivity. The measurements were taken over a period of one year between January and December 2009, although a few instances of equipment outages were experienced.

The one-minute samples obtained from the RD-80 disdrometer were classified under four rain events based on

the maximum rain rate recorded in a single rain event. A total of 729 rain events, which corresponds to 40,102 rain rate samples were obtained as summarized in the Table I. It should be noted that the highest rainfall rate recorded was 117.15 mm/h, while the lowest rainfall rate was 0.003 mm/h. In filtering and processing the collected data, some factors were taken into consideration as follows:

1. Rain rate samples with total sum of drops less than 10 were discarded.

2. A minimum duration of five minutes was assumed as the interval between two independent rain events.

The rain samples were then separated into four different rainfall types namely: drizzle (0–5 mm/h), widespread (5–10 mm/h), shower (10–40 mm/h) and thunderstorm (>40 mm/h). In each event, the highest value of rainfall rate was considered and then grouped according to the listed rainfall type. Fig. 1 shows the resolution rainfall rate and total number of drops per unit time recorded for the period of measurement.

TABLE I SUMMARY OF RAIN EVENTS RECORDED IN 2009

Rain Event Drizzle

(0-5mm/h) Widespread (5-10mm/h)

Shower (10-40mm/h)

Thunderstorm (>40 mm/h)

Number of events

638

44

37

10

Total number of raindrops

(mm-3)

2,561,596

952,119

1,686,265

342,167

Rain accumulation

(mm)

5564.41

3452.76

9221.04

4730.4

Figure 1. Resolution of rainfall parameters in 2009: rain rate per unit time above and rain drops per unit time below.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

10-4

10-2

100

102

104

Duration of rain event (minutes)

Rai

n ra

te (m

m/h

)

Resolution of rain rate variation in 2009

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

100

101

102

103

104

Duration of rain events (minutes)

Tot

al n

umbe

r of

rai

n dr

ops

Resolution of rain drop variation in 2009

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Fig. 2. The cumulative distributions for various rainfall types in 2009.

TABLE II

PERCENTAGE OF TIME EXCEEDED AT VARIOUS TIMES FOR DIFFERENT RAINFALL TYPES

The rainfall rate cumulative distribution was obtained for

each rainfall types as shown in Fig. 2. The percentage of time rain rate exceeded at 1%, 0.1%, 0.01% and 0.001% for each rain event are summarized in Table II. The thunderstorm rainfall type had the highest value of rain rate exceeded at 0.01% (R0.01), while the drizzle rainfall type had the lowest at the same probability.

IV. PROBABILITY DENSITY FUNCTION ANALYSIS FOR RAINFALL EVENTS IN DURBAN

The probability density function (PDF) is an important tool in determining the characteristics of a parameter and other related parameters that contributes to its shape function. The various rain events can desirably be examined by considering the influence of their measured PDFs with other existing empirical and statistical models. The importance of this is to obtain fitted values for such models by using regression techniques to estimate suitable valid values for them. Three rain rate models were used to analyze the collected data: Moupfouma, gamma and lognormal rain rate PDF models. Shape and size parameters for the two latter models were optimized using the method of Maximum Likelihood (ML) as utilized in [16]. A. Moupfouma Rain Rate Probability Distribution Function

Moupfouma [4] in his work on tropical locations derived an exponential law for rain rate probability given by:

f r arb u br exp –ur r mm/h 0 0 r mm/h 0 (4)

where the parameters a, b and u depend on the climatic zone under consideration. In practice, F ∞ 1, u is assumed as positive, and b is also taken as positive.

Since South Africa is classified as a subtropical region [12], [14] and Durban is a coastal city, we employed parameters of u closer to our location to calculate a and b, with R0.01 being 65.3 mm/h for the year 2009 (please see Moupfouma et al. [5], [17] and Ajayi et al. [2] for these equations). B. Gamma Rain Rate Probability Distribution Function

The gamma distribution is often being used as a good estimation for medium and high rain rates. The PDF of a two-parameter gamma function is given as: f x xα_1 exp ‐ x ββα Г α for γ x ∞ (5)

The optimized values α and β for the Maximum Likelihood method were defined by Thom [18] and given by:

α 1 1 4A34A (6)

and,

β Nα xjNJ 1 (7)

N is taken as the number of samples of rainfall rate and xj is the rainfall rate value at j=1 to N. A ln 1N xjN

j 1 – 1N ln xjNj 1 (8)

The ML estimates for α and β are given in Table III.

C. Lognormal Rain Rate Probability Distribution Function The lognormal PDF is a distribution which is used for the

estimation of low precipitation rates as discussed in [5], [17]. The PDF is defined as follows: f x 1xσ√2π exp – ln x ‐ µ 22σ2 0 x ∞ (9)

The ML estimators ̂ and are defined in [19]–[20] by (10) and (11): µ exp 1N In xiN

i 1 (10)

0.01

0.1

1

10

0 10 20 30 40 50 60 70 80 90

Perc

enta

ge o

f Tim

e fo

r whi

ch R

ain

Rate

is E

xcee

ded

(%)

Rain Rate (mm/h)

Drizzle Rain Type

Widespread Rain TypeShower Rain Type

Thunderstorm Rain Type

Time for which Rain rate is exceeded (%)

1

0.1

0.01

0.001

Drizzle

(mm/h)

2.43 4.24 4.92 4.98 Widespread 6.23 9.15 9.94 9.98

Shower 13.36 23.13 28.66 29.97 Thunderstorm 44 74.25 79.78 83.90

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TABLE III MOUPFOUMA RAIN RATE PARAMETERS AND METHOD OF LIKELIHOOD ESTIMATORS

FOR PROBABILITY DENSITY FUNCTION IN DURBAN, SOUTH AFRICA

MODELS

EVENT

PARAMETERS

aMOUPFOUMA

a b u

Drizzle Widespread

Shower Thunderstorm

0.019 0.017 0.0099 0.0120

3.24 2.15 1.16 0.64

0.025 0.025 0.025 0.025

GENERAL 0.102 0.7161 0.025

GAMMA

Drizzle Widespread

Shower Thunderstorm

0.35 0.38 0.43 0.32

1.46 0.56 0.29

0.087

GENERAL 0.32 0.48

µ σ

LOGNORMAL

Drizzle Widespread

Shower Thunderstorm

0.03 0.11 0.30 0.32

11.61 11.51 11.03 19.89

GENERAL 1.93 5.43 aThe equations for the parameters a and b of Moupfouma model were obtained from Moupfouma et al. [5], [17] and Ajayi et al. [2].

σ exp 1 1 ln µ 11

where N represents the number of rain samples in an event and x is the value of the rainfall rate at i = 1 to N. Their estimates are given in Table III.

D. Model Testing and Error Analysis The models in this paper were tested using two goodness-

of-fit tests namely: chi-squared statistic and root mean square (RMS) error tests. The use of chi-square statistical test in this paper is used to establish how well the theoretical PDF of the rain rate models fits that of the measured PDF. The chi-square χ statistical test value is given as: χ , ,, 12

where Xmodel,j and Xact,j represent the values of the expected frequency and actual (measured) frequency.

The chosen level of significance for (N–1) degree-of-freedom is 1% (0.01). The chosen model must, therefore, have a χ value lower than its threshold at the (N-1)th row on the chi-squared table. On the other hand, the RMS test value is given by:

RMS =1

Xmodel,i– Xact,i2

N

i=1

13

where Xmodel,j and Xact,j retain the same definitions as given in (12). The value of N is taken as the maximum number of tested frequencies. In adopting the RMS error tests, it is assumed that the model with the lowest value will be the best fit to actual data. By examining the average value of the RMS errors for a specified model, its performance can be easily compared with another model. This approach was used in the determination of the preferred model.

VI. RESULTS AND DISCUSSION

The results are presented in Figures 3–7. The Moupfouma PDF model was first considered with the measured PDF for all the durations of rain events in 2009 using the function parameters in Table III. Fig. 3 shows the corresponding generated graph while Table IV shows the respective model testing indices. The results show that the Moupfouma parameters for Durban slightly over-estimate the measured PDF for rainfall types below 1 mm/h and fits the measured PDF at fairly higher values of rainfall rate - which agrees with Moupfouma’s observation. A closer look at the graphical results indicate that the gamma model has the best fit with the measured data. This is further buttressed by the error tests results where the gamma model has a χ2 test value of 2.7589 (threshold value of 54.776) and RMS error value of 0.1288. Therefore, the gamma model performs best when compared with other models including the known Moupfouma tropical model for all the rainfall durations in 2009.

The other set of results show the respective performances of gamma, lognormal and Moupfouma PDFs under different rainfall types; Figures 4–7 shows the generated graphical results while Table IV shows the model fitting tests for each rainfall types. In Fig. 4, it was noticed that the statistical modeling at drizzle rain (0–5 mm/h) is roughly fitted by the gamma model. Even though it appears that the gamma model over-estimates the measured data, the error tests [Table IV] clearly indicate it as the best fit with χ2 of 12.95 (threshold value of 64.95) and RMS error value of 0.2644. For regime of

Fig. 3. Comparison of the Moupfouma model PDF with the measured PDF and other models for all rain events in Durban in 2009.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.1 1 10

Prob

abili

ty D

ensi

ty F

unct

ion

Rain rate (mm/h)

Measured

Moupfouma modelGamma modelLognormal model

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Fig. 4. Comparison of the various models PDF with the measured PDF for drizzle rain type.

Fig. 5. Comparison of the various models PDF with the measured PDF for widespread rain type.

Fig. 6. Comparison of the various models PDF with the measured PDF for shower rain type.

Fig. 7. Comparison of the various models PDF with the measured PDF for thunderstorm rain type.

widespread rain type (5–10 mm/h) in Fig. 5, the gamma model is again a better fit when compared with the lognormal and Moupfouma model. Again, as given by the error results, gamma model has the lowest χ2 value of 4.1553 with an RMS value of 1.0181 (threshold value of 36.19). The shower rain type (10–40 mm/h) appears to have close values of χ2 for both gamma, lognormal and Moupfouma models with 0.0131%

absolute deviation. It is shown in fig. 6 that the lognormal model is the preferred fit to the measured data with a χ2 of 17.73 (threshold value of 34.805) and RMS of 0.3375. Finally, for thunderstorm rain type (40–120 mm/h) in Fig. 7, we had fewer degree of freedoms for the χ2 test value. From the error results, we concluded that the gamma model still performed better with a χ2 test value of 1.0525 (threshold value of 11.345) and RMS value of 0.2697.

In general, it was noticed that at higher rainfall rate range i.e. for shower and thunderstorm rain types, the gamma model decays rapidly. Meanwhile, at lower rainfall rate range, the lognormal model appears to give a better approximation to the measurements. This has been explained by Moupfouma in [4] and [5]. Also, the Moupfouma model appears to improve with increasing R0.01 as the the rainfall range increases. Broadly speaking, the results from the error tests in Table IV also indicate that the gamma model has the lowest average RMS value, while the χ2 statistic value also confirms the gamma model as a good fit of the measured PDF.

In order to compensate for the deviations of both gamma and lognormal functions. The models were refitted by least

TABLE IV COMPARISON BETWEEN MEASUREMENT AND STATISTICAL MODELS FOR

PROBABILITY DENSITY FUNCTION IN DURBAN

MODELS

aCHI

OTHER

COMPARISON

GENERAL

bχ 2

RMS Error Moupfouma 2.7589 0.2740

Gamma d 2.0864 0.1288 Lognormal 41.2269 0.3208

GAMMA

cχ 2

RMS Error

Drizzle d12.9564 0.2644 Widespread d4.1553 1.0181

Shower 19.0362 2.4226 Thunderstorm d1.0525 0.2697

LOGNORMAL

cχ 2

RMS Errror

Drizzle 53.4229 0.8097 Widespread 45.2152 1.1242

Shower d17.73 0.3375 Thunderstorm 5.7639 0.2529

MOUPFOUMA χ 2

RMS Errror

Drizzle 1979.87 80.565 Widespread 27.22 1.049

Shower 37.59 34.35 Thunderstorm 7.25 0.232

aThe χ 2 threshold for all sampled models are taken as 1% (0.01). bThe degree of freedom for the three models is taken as 33 and the statistic threshold is 54.776. cThe degree of freedom for drizzle, widespread, shower and thunderstorm events are 41, 18, 18 and 3 with statistic threshold of 64.95, 36.19, 34.805 and 11.345 respectively. dRecommended χ 2 values that meet the selected fitted model criteria.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.1 1

Prob

abili

ty d

ensi

ty fu

ncti

on

Rain Rate (mm/h)

Measured

Gamma Model

Lognormal Model

Moupfouma Model

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.1 1 10

Prob

abili

ty D

ensi

ty F

unct

ion

Rain Rate (mm/h)

MeasuredGamma ModelLognormal ModelMoupfouma Model

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.1 1 10

Prob

abili

ty d

ensi

ty fu

ncti

on

Rain Rate (mm/h)

MeasuredGamma ModelLognormal modelMoupfouma Model

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.1 1 10

Prob

abili

ty d

ensi

ty fu

ncti

on

Rain Rate (mm/h)

MeasuredGamma modelLognormal modelMoupfouma Model

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TABLE V REFITTED MODEL PARAMETERS FOR GAMMA AND LOGNORMAL PROBABILITY

DENSITY FUNCTION MODELS

MODEL

RAIN TYPE

α

β

k

R2

Gamma

Drizzle 0.0004 0.9816 -0.2074 0.9869

Widespread 0.0004 0.6267 0.1268 0.9884 Shower -0.0065 0.3605 0.2175 0.9858

Thunderstorm -1.1611 2.1659 0.1408 0.9524

Lognormal

Drizzle -0.4783 3.5021 -0.0196 0.9938 Widespread -0.4380 3.1687 0.1206 0.9766

Shower -0.3736 2.2022 0.0983 0.9756 Thunderstorm -10.432 5.8418 0.0743 0.8444

square regression techniques, these two statistical models can be enhanced by adjusting their probability functions with fitted coefficients relative to the measured PDF. A polynomial function of second-order was derived to achieve this: . α β (14) where α, β and k represent the polynomial coefficients of the fitting; f(rmodel) being the PDF of the statistical model under consideration and f(rmeas.) is the measured PDF of the rainfall rate samples.

The results in Table V shows the fitted values of α, β and k for the different models and rain types. Moupfouma empirical model was, however, not considered in the fitting procedure.

VI. CONCLUSION

In this study, the scope of this work is limited to the PDF analysis of rainfall rate using Maximum Likelihood (ML) estimators for Durban, South Africa. The analysis of the 12-month disdrometer data has given an indication of the behaviour of rainfall rate under various rain types. From the error tests, the gamma model appears as the best fit for Durban with an average χ2 and RMS values of 9.3001 and 0.9937 respectively. The gamma model is well known as an appropriate distribution for the estimation of rainfall at high rain rates with South Africa being a subtropical region. This explains, as well as confirm, the earlier work in [16] suggesting gamma model as the preferable model for modeling rainfall rates in South Africa.

On the other hand, the Moupfouma empirical model compares well with the measured data has shown that it over-estimates and under-estimates at some points of the measured PDF with χ2 statistic of 2.7589. This could be due to its limitation at accurately resolving rain rates below 2 mm/h as stipulated by the model. The gamma model also performs better compared to Moupfouma’s and therefore, provides a better fit considering the overview of all rain events in 2009. Thus, the performance of our tested models are in the following ascending order: lognormal model, Moupfouma model and gamma model.

The use of appropriate rain rate distribution is essential in microwave and millimetric wave link design as it is useful in rain attenuation prediction in terrestrial and satellite systems. It is therefore imperative that adjustments to these statistical models be considered when modeling for locations in South Africa.

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