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Hydrological Sciences - Journal - des Sciences Hydrologiques, 27, 3,9/1982

Deriving rainfall intensity-duration-frequency relationships and estimates for regions with inadequate data

LEKAN OYEBANDE Department of Geography, University of Lagos, Lagos, Nigeria

ABSTRACT Type 1 extreme-value distribution (Gumbel) was applied to the annual extreme rainfall data sets generated by 11 rainfall zones to estimate the parameters and hence the intensity-duration-frequency (IDF) rainfall. The chi-square test confirmed the appropriateness of the fitted distribution. Gumbel graphical plots and the computed confidence limits also showed that the Gumbel EV-1 function fits well the empirical distribution.

Méthode en vue d'obtenir les relations et les estimations de 1'intensitê-durêe-fréquence de la pluie pour les régions sans données suffisantes RESUME La distribution des valeurs extrêmes (Gumbel) du type 1 a été appliquée aux séries de données de la pluies maximales annuelles relevées dans 11 zones pluvieuses en vue de déterminer les paramètres et partant l'intensité-durée-fréquence de la pluie. Le test chi-square (X2) a montré que la distribution choisie est valable. Les graphiques de Gumbel et les calculs d'intervalle de confiance ont montré également que la fonction EV-1 de Gumbel correspond bien à la distribution empirique.

THE DATA PROBLEM

In an earlier study (Oyebande, 1980) which employed an empirical approach it was observed that for about 60% of the stations for which short-duration rainfall records were available, reliable intensity-duration-frequency (IDF) estimates could be obtained for a return period of up to about lO years only. It is well known that a small sample may define a frequency distribution which differs greatly from the population frequency distribution. Thus, it is often emphasized that at least 25-30 years of records are needed to obtain estimates of some practical value for both short and longer durations.

The frequency distribution of the 35 stations used in the above study is shown in Table 1. Records of only 18 stations exceed 20 years in length and of those only three are longer than 25 years. Thus, more than 90% of the available records cannot be used with confidence to obtain IDF estimates for return periods of up to 50 years or more.

However, such estimates are often required for estimation of design floods for engineering purposes, especially from inadequately gauged or ungauged basins. Bell (1969) opined that short-duration

353

354 Lekan Oyebande

Table 1 Distribution of recording stations

Record length (years) No. of stations

5-10 8 11-15 8 16-20 11 21-25 15 26-30 3

35

rainfalls of less than 2 or 3 h are of special importance in flood estimation and that for engineering purposes it is generally necessary to estimate values with return periods of at least 50 years. To make matters worse, the distribution and coverage over the country of the stations is also inadequate. Eighteen stations have at least 20 years of record, giving a density of one station to more than 50 000 km . In order to overcome the two-fold problem of inadequate record length and coverage, regional analysis is used in which data for individual stations are lumped together or compounded to yield larger regional data samples. The present paper thus has two objectives. The first is to define detailed regions, or rather zones according to certain criteria. The other is to use the longer records generated for the zones to obtain IDF relationships and estimates which can be applied to the respective zones with greater confidence. This second step involves fitting a double exponential function to the zonal data sets.

RAINFALL ZONES

A map of Nigeria showing four principal rainfall regions with two sub-regions was produced by Oyebande (1980). A further study of the IDF regime, however, indicated further differentiation into zones. The map proposed in the present study defines 10 principal rainfall zones with sub-zones (Fig.l). The zones were designated according to the following climatic and topographic characteristics.

(a) Rainfall IDF regime characterized by three indices namely, the ratio of: the 10-year 10-min fall to that of the 10-year 30-min fall (A); 2-year 15-min fall to 2-year 60-min fall (B); and 25-year 1-h fall to 25-year 24-h fall (C).

(b) Topographical characteristic as expressed by the altitude of the station, in metres above the mean sea level.

(c) The average rainfall pattern or regime at the station as characterized by mean annual rainfall and mean number of rainy days per year.

Table 1 shows the range of the criteria variables used for delimiting the zones while the location and extent of each zone is shown in Fig.l. There are no recording raingauges in zone V, the hilly eastern area adjacent to the border with the Cameroun Republic. To delineate this region, it was assumed that the dominant geographic factor is of orographic origin. This orographic

Rainfall intensity-duration-frequency relationships 355

Fig. 1 Rainfall zones of Nigeria.

influence is also reflected in the high annual rainfall experienced in the zone (Table 2).

The length of records in station years varies from 20 to 115, with six zones having at least 45 years and only two having less than 30 years. The substantial increase in the length of records strengthens the case for applying mathematical curve fitting which one might not be willing to apply to individual stations.

However, as Table 3 indicates, all the data sets for each zone come from relatively short periods between 1948 and 1978. It was found necessary to test their long term representativeness. This was accomplished by the use of variance heteorogeneity test devised by Hartley (1950). The method is rather simple and convenient but requires a special probability table showing percentile values of "largest F ratio", F m a x =

Sm a x / S m ^ n in a set of k mean squares each

based on n degrees of freedom. The null hypothesis tested is 2 2 2 2

Hg : rjj_ = 0"2 = ... O, = 0 , vs E-, : the four variances are not equal. Annual maximum observation-day rainfall series were used and

representative stations with long-period records (1915-1976) were selected, one from each zone. The long period of 62 years was divided into four sub-periods according to information available from studies of fluctuations in annual rainfall values and trends in rainfall regime for different parts of the country (Obasi et al.,

356 Lekan Oyebande

Table 2 Characteristics of rainfall zones

Zone

1 II III IVa IVb V VI Vila Vl lb VIM IX X

A

(%)

41-46 50-54 52-54 48-49 50-51 — 40-44 59-67 59-63 46-48 45-53 65-69

B (%)

47-49 50-55 56-57 54-55 52-54 — 50-52 59-67 63-67 50-56 58-59 52-73

C (%)

43-49 50-65 52-57 54-60 54-57 — 50-52 53-66 56-61 56-66 74-83 68-72

Altitude (m)

80 80 225-305 113-307 150-190 460-2409

63-260 119-351 645-1285 460-750 350-415 325-520

Annual mean rainfall (mm)

2150 1550-2900 1217-1600 1224-1800 900-1150

1400-3670 1190-1320 710-1070

1281-1400 840-1085 651-776 525-600

Annual mean no. of raindays

170 125-190 106-151 94-109 75-90

_ 89-101 58-85

108-126 50-82 62-64 49-61

A = 100 X (10-year, 10-min. fall/10-year 30-min. fall); B = 100 X (2-year, 15-min. fall/2-year 60-min. fall); C = 100 X (25-year, 1-h fall/25-year 24-h fall).

1980; Okulaja, 1980). According to these studies, after 1918 there

were progressively persistent or drier dry seasons which culminated

in the general drought of 1946. Then there was a return to average

wet conditions from 1952 till 1971, the year the sahelian drought

(1971-1973) began. However, the only significant cyles revealed by

Table 3 Derived zonal rainfall records

Zone Station names No. of stations

Length of period (station years)

Period

I Port, Harcourt, Calabar, Umudike

II Warri, Lagos, Ikeja, Oshodi, Benin, Aero & NI FOR

III Oshogbo, Ondo, Nora, Ibadan

IVa Enugu, llorin, Bida, Makurdi

IVb lbi,Yola

V

VI Lokoja, Minna

Vila Mokwa, Yelwa, Sokoto Vl lb Kaduna, Jos Aero, Lamingo

VIII Bauchi, Zaria, Kano, Gusau, Samaru

IX Potiskum, Maiduguri, Aero & Water Works

X Nguru, Katsina

3

6

4 4

2

2

3

3

45

115

51 82

22

34

35 60

61

38

20

1951-78

1948-78

1956-75

1956-78

1956-73

1956-75

1956-76

1952-78

1951-75

1956-74

1960-75

35


the method of spectral analysis employed for Nigeria as a whole was one of 2-3 years. It would have sufficed to test 1947-1978 against the earlier period(s), but each of the two periods was further divided into two for a more detailed test. The results of the analysis are shown in Table 4. The null hypothesis which assumes homoscedasticity is acceptable for all the zones except zones IVb and VIlb. However, in each case it was the earliest period (1915-1930) that was associated with significantly low variance. These departures were checked by a more robust test by Bartlett. B = 7.33 for Jos and 13.31 for Yola. Since Xo 95 = 7.81 for three degrees of freedom, we conclude that the variances for Jos are equal. The same conclusion cannot be reached for Yola since Xo.99 = H-3 for three degrees of freedom. It is strongly suspected however that there was a shift in the gauge location to a significantly different site after 1930 or thereabout, and that the shift accounted for the significant difference, rather than climatic change.

Table 4 Results of variance homogeneity test on annual series of observation-day maximum rainfall

Zone

1 II III IVa IVb V VI Vila VI lb VII I IX X

Station

Calabar Lagos Ondo Offa Yola

Minna Sokoto Jos Kano Wladuguri Hadejia

1915-X

4.74 4,63 3.34 2.60 2.33

3.14 2.47 2.69 2.80 2.38 2.10

-1930: s2

0.82 3.31 1.72 0.86 0.12

1.01 0.66 0.15 0.53 0.43 0.43

N

15 16 16 15 14

13 14 9

15 14 13

1931-X

5.08 5.25 3.30 2.91 2.99

2.96 2.38 2.78 2.72 2.46 2.70

-1946: s2

0.86 2.49 0.98 2.07 0.47

0.86 0.39 0,75 0.36 0.76 0.98

N

16 15 16 16 16

16 16 16 16 15 16

1947-X

4.98 4.55 3.24 3.30 2.83

3.04 2.99 2.64 3.11 2.81 2.61

-1959: s2

1.25 2.37 0.93 2.10 0.87

0.80 1.18 1.02 1.38 0.96 0.70

N

13 13 13 13 13

13 12 13 13 13 13

1960-X

4.32 5.35 3.15 3.62 2.91

3.54 2.27 2.71 2.40 2.62 2.67

-1976: s2

1.28 3.50 1.07 0.95 096

1.32 0.80 0.51 0.50 0.81 0.83

N

15 17 17 17 15

17 17 17 16 17 12

F*

1.561 1.477 1.849 3.38 8.00*

1.65 3.02 6.80» 3.83 2.23 1.93

* F m a x significant at a = 0.01 : for or = 0.05, k = 4 and n (= N — 1 ) = 16, the critical region is F m a x > 3.80; for a = 0.01, k = 4 and n = 16, the critical region is F m a x > 5.20; fo ra = 0.05, k = 4 and n = 15, the critical region is F m a x > 4.01.

REGIONAL ANALYSIS

A number of investigators such as Bell (1969) , Natural Environment Research Council (1975) and Baghirathan & Shaw (1978) have found the regional analysis useful. The scheme assumes that the annual maximum rainfall events are independent in time and space. It also assumes that a sample of falls in a region or zone is sufficiently varied to represent the population of a long period. The present knowledge of the storm characteristics in Nigeria indicates that the convective showers which account for most of the intense falls are highly localized not only in the northern - zones, but also in the coastal region which is under the monsoonal influence during much of the year. The spread of the rainfall stations together with the effects of localization should ensure that the assumption of independence is satisfied.

The second assumption, however, is not so easily verified, and

358 Lekan Oyebande

implies that the exact value of the effective length of the

compounded data sets remains unknown. Nevertheless, it is important

to provide estimates of the effective length of the single regional

sample generated to enable one to obtain a measure of the

reliability of the IDF estimates obtained from the samples. Table 3

(column 4) shows the nN observations derived from n-point records,

where N is the average length of each point record. nN will be the

effective length of the regional data sample if the n data points

are mutually uncorrelated. However, if they are mutually correlated,

as is likely, the pooling together of n sets of N observations

can be expected to yield only as much information as n eN, where

n_ < n. The term n e is called effective number of points (stations)

in the region or zone.

Yevjevich (1972, p. 245) has shown that

n e = n/(l + r) (n - 1) (1)

where

r = 2E. . £. .,_ r-j/n(n - 1) (2) 3=1 1=3+1 13

and where r. • is the product-moment correlation between variables of

stations i and j. Table 5 shows the average interstation

Table 5 Estimates of effective record length by regions

Region nN 0.2-h Duration: 1-h Duration: 24-h Duration:

I I I * III lVa IVb VI Vi la* Vl lb VIM* IX X

45 115 51 82 22 34 35 60 61 38 20

r

0.471 0.26 0.233 0.220 0.150 0.366 0.348 0.363 0.196 0.280 0.40

neN

23 50 30 49 19 25 21 35 34 24 14

100neN/nl\l

51 43 59 60 87 74 59 58 56 64 71

r

0.380 0.229 0.211 0.205 0.150 0.138 0.294 0.327 0.196 0.280 0.40

neN

26 54 31 51 19 30 22 37 34 24 14

100neN/nN

57 47 65 62 87 88 62 62 56 64 71

r

0.171 0.229 0.205 0.158 0.145 0.138 0.294 0.123 0.190 0.255 0.33

neN

33 54 32 56 19 30 22 48 34 25 15

100nel\l/nN

74 47 62 68 87 88 62 80 56 66 75

'Concurrent series are quite short in certain cases and tend to yield rather high correlation coefficients.

correlation r, and the estimates of the effective length of the regional sample neN as well as the percentage of the gross length that the latter represents.

The above steps used to estimate neN were next applied to annual series of maximum observation-day rainfall which yield much longer regional totals ranging from 86 to 270 observations. Both applications show that the effective length of record obtained by pooling together n-point records exceeds 60% of nN for at least eight of the regions for durations equal to or exceeding 1 h (Table 6) .

In addition to the substantial increase in record length, the IDF determinations from a regional sample for several stations are more reliable than those calculated from one of the stations because the


Table 6 Estimates of effective length of observation-day annual maximum rainfall

Region Stations Period nN r neN 100neN/nN

1

II

III

IVa

IVb VI Vila

Vl lb

VIII

IX X

Aba, Calabar, Forcados Ikot Ekpene, Owerri, Umudike Benin, Ebute Metta, lju, Kwale, Lagos Warri Abeokuta, Agbor, Ibadan, Ondo, Oyo Ado Ekiti, Enugu, Keffi, Makurdi, Ogoja, Okene Ibi, Yola Lokoja, Minna Birnin Kebbi, Kotangora, Sokoto, Zuru Kaduna, Kafanchan, Jos, Pankshin, Wamba Azare, Bauchi, Hadejia, Kafinsoli, Kano Samaru Biu, Maiduguri, Potiskum Daura, Katsina, IMguru

1932-66

1931-75

1931-71

1935-65 1916-31 1918-77

1926-71

1939-68

1941-68 1940-74 1947-75

210

270

205

CD

C

D

CO

C

O

120

184

150

135 105 87

0.166

0.147

0.105

0.128 0.116 0.055

0.081

0.179

0.158 0.311 0.091

115

155

144

113 77

114

148

87

91 65 74

55

57

70

61 90 95

80

58

67 62 85

spottiness of intense falls implies that some falls will be recorded at some stations and not at others during some years.

Fitting Gumbel EV-1 distribution

A double exponential distribution was fitted to the annual maximum series to obtain IDF relationships and estimates. The distribution also known as the Fisher-Tippet distribution was introduced for extreme values by R.A.Fisher & L.H.C.Tippet. It is, however, widely known in hydrology as Gumbel Extreme Value Type 1 (EV-1 distribution) for it was Gumbel (1941) who proposed its use for annual floods. In its cumulative form (which is mathematically simpler than the f(x) form), Gumbel EV-1 is defined as

F(x) = exp {-exp[-a(x - g)]} (3)

where a, 3 are the scale and location parameters respectively. Equation (1) can be written in terms of a reduced variate

y = a(x - B) (4)

to obtain

F(x) = exp [-exp (-y)] (5)

and, s ince F(x) = 1 - (1/T ) , i t can be shown t h a t

360 Lekan Oyebande

y = - In (In T r / T r - 1) (6)

where T is the return period in years. However, for estimating y when T = 1, equation (6) breaks down, so that the following approximation by Schulz (1973, p. 424)

y = In T r - l/2Tr - 1/24T* - 1/8T^ (7)

gave more satisfactory values of y for T = 1 and was used instead of (6) .

The estimates of a and 3 could be obtained by the method of moments, but as Yevjevich (1972, p. 180) has shown, the use of the method of moments for the estimation of parameters of extremely skewed distributions may represent a significant loss of valuable hydrological information. The Gumbel EV-1 is sufficiently skewed to warrant the use of the maximum likelihood method to obtain the final estimates of a and B. The initial estimates provided by the method of moments are

1/a = 0.779697 s and 8 = x - 0.577216/â (8)

where s and x are sample standard deviation and mean respectively. Then using the initial estimates the final estimates were obtained by iterative computations by the maximum likelihood method until the corrections become negligible. The values of a and 3 for each zone and duration are shown in Table 7. The IDF estimates, x, were then

Table 7 Estimates of parameters of equation x = j3 + (1/â)y for zones

Zone 0.2 h 0.4 h 1h 3 h 6 h 12 h 24 h

I

II

III

IVa

IVb

VI

Vila

Vl lb

VIII

IX

X

23.52 118.94 24.83

113.16 19.57

108.78 21.42

111.83 20.15

108.40 32.95

115.60 19.40 99.99 23.28 99.96 18.04 95.65 25.43 98.23 21.53

103.66

19.19 97.43 20.97 85.74 14.99 78.59 16.24 83.38 15.87 78.66 21.76 81.22 15.62 67.07 17.95 72.72 14.64 69.79 21.92 73.30 17.65 69.82

12.10 60.13 14.36 55.06 11.32 43.63 12.65 49.13 12.70 46.29 13.49 47.27

9.21 36.25

9.83 39.78 10.87 40.67 11.53 40.32

9.35 35.75

6.63 26.25

6.29 23.58

5.59 17.43 4.69

20.03 6.30

18.02 6.03

19.80 3.52

14.21 3.87

15.72 4.76

15.92 4.59

15.85 3.86

12.78

3.74 14.48 3.88

13.30 2.94 9.55 3.04

10.56 3.56

10.28 3.13

10.61 1.60 7.53 3.13 8.57 2.70 8.46 2.74 8.65 2.09 6.73

1.87 7.63 2.16 7.03 1.49 4.85 1.57 5.61 1.84 5.20 2.26 5.14 0.88 3.99 1.14 4.49 1.53 4.47 1.43 4.39 1.08 3.43

0.98 4.03 1.26 3.81 0.85 2.61 0.89 2.83 0.92 2.64 0.57 2.81 0.39 2.05 0.68 2.41 0.78 2.28 0.76 2.23 0.57 1.76

The first row in each zone contains the value of a; the second row in each zone contains the value of /3.


obtained by the relationship in equation (4). The standard error (SE) of estimate was calculated using the formula derived by the Natural Environment Research Council (1975) as

SE(x(Tr)) = (a2/N) (1.11 + 0.52y + Côly2)*5 (9)

where x(Tr) is the estimate with T-year return period and y is the reduced variate corresponding to Tr- 1/a2 is substituted for a2. The IDF estimates, x, are not shown, but can be obtained by substituting the values of the statistics B and 1/ct (Table 7) in the equation x = 0 + (1/oOy, where y is given by equation (6). The value of the associated standard error can then be calculated as twice SE(x(Tr)) in equation (9).

A computer programme was written to perform the above and subsequent functions including the chi-square test of goodness of fit given by

X2 = E* = 1 (0± - Ei)2/Ei

where k is the number of class intervals, 0j_ is observed frequency and E^ the expected frequency in the class interval. Much emphasis was placed on the design and procedure of the test for it was considered important to ascertain, as much as possible, the appropriateness of the Gumbel EV-1 for fitting the annual maximum series. Gumbel (1943) has shown that the chi-square test would almost certainly give contradictory results which could make it impossible to draw a conclusion about acceptance or rejection of a hypothesis when arbitrary startingpoints and lengths of intervals are used. Gumbel and Yevjevich (1972, pp. 224-228) suggest the choice of class intervals or cells such that all have equal probabilities p^ = 1/k or equal absolute frequencies, N j_, in order to overcome much of the arbitrariness that accompanies the construction of class intervals. Five class intervals were used in the study for all the zones except zones II and IVa for which 10 class intervals were used because N > 80. The computed probability was then used to compute the reduced variate y from equation (5) and equation (4) was then used to determine the class intervals in terms of extreme rainfall series, x.

The level of significance was obtained from tabulated values of the chi-square distribution corresponding to k - h - 1 degrees of freedom, where h is the number of parameters in the Gumbel EV-1 function and represents additional constraints, so that the degree of freedom in this case is k - 3. It is either 2 or 7 depending on whether five or ten intervals were used.

The EV-1 distribution was also applied to the data for representative stations for each zone. The derived parameters 1/a and (3 are shown in Table 8. A close comparison of Tables 7 and 8 reveals a number of interesting features, including the station to station variation with the same zone and how the regional approach smooths out the variations.

362 Lekan Oyebande

Table8 Estimates of parameters of the equation x = |3 + (1/â)y for individual stations

Station Zone 0.2 h 0.4 h 1h 3 h 6 h 12 h 24 h

Port Harcourt*

Warri*

Lagos*

Ikeja*

Oshogbo

Enugu*

Wlakurdi

llorin*

Yola

Lokoja

Sokoto

Jos Aero*

Kaduna*

Kano

Potiskum

Nguru

I

II

II

II

III

IVa

IVa

IVa

IVb

VI

Vila

Vl lb

Vl lb

VIII

IX

X

26.43 121.03 29.33

120.78 23.55

105.69 23.32

109.99 26.47

109.59 17.95

121.05 27.13

113.81 18.85

105.81 16.89

102.12 20.54

101.17 14.89 90.11 23.63

101.64 19.82 99.79 16.61 90.64 17.55 90.76 19.16

106.34

15.22 100.13 22.65 92.47 22.17 80.18 18.35 86.64 12.89 81.64 14.52 92.27 20.54 86.25 12.19 80.68 13.12 73.83 19.23 82.31 12.80 60.26 18.92 70.32 15.04 73.17 17.05 66.62 13.54 71.00 18.25 70.46

14.00 61.35 14.57 59.91 16.66 55.97 10.99 53.59

7.51 44.60 13.30 53.34 11.61 50.08 10.30 39.74 12.91 44.53 10.58 47.22

7.74 32.92

9.66 39.41 10.00 39.95 14.13 39.87 10.57 41.43 10.16 35.11

6.45 27.31

4.85 27.53

6.82 25.06

4.69 22.27

5.27 18.55 4.20

22.00 4.54

20.38 3.73

18.25 6.27

16.96 4.90

18.95 3.05

12.75 3.14

14.85 4.50

16.25 6.05

15.37 4.74

15.92 3.89

11.97

3.70 15.53 3.44

15.06 3.93

15.03 2.94

12.24 2.73

10.25 2.36

11.85 2.57

11.35 3.15 8.75 3.58 9.69 2.46 9.93 1.33 6.80 1.84 7.98 2.34 9.04 3.41 8.12 2.52 8.47 2.18 6.30

1.61 8.10 1.68 7.90 1.97 8.04 1.62 6.63 1.25 4.90 1.21 6.12 1.39 5.88 1.56 4.93 1.85 4.90 1.38 4.92 0.85 3.76 1.04 4.21 1.24 4.67 1.78 4.13 1.29 4.28 1.15 3.25

0.85 4.29 1.04 4.52 1.11 4.12 1.00 3.58 0.76 2.62 0.62 3.26 0.73 2.97 0.95 2.38 0.94 2.49 0.68 2.57 0.39 2.00 0.63 2.28 0.72 2.43 0.92 2.11 0.70 2.17 0.61 1.66

*Record length of 20-29 years. Other records are 14-19 years.

DISCUSSION OF RESULTS

The results of the chi-square test of goodness of fit showed that in 95% of the cases the null hypothesis that the extreme rainfall series have the Gumbel EV-1 distribution is acceptable at the 5% level of significance. The few cases in which the fitting was not good were either for 0.2 or 0.4 h durations. This may not be surprising for as Hershfield (1962) noted such high intensity-rainfall-frequency data are not usually accurate and could be subject to large errors due mainly to sampling deficiencies.

The estimates of 1/a, g and IDF values should therefore prove reliable and useful. For shorter durations such as 0.2 and 0.4 h, the value of 1/S is higher in zones VI and IX than in other zones, and appreciably lower in zones III, Vila and VIII. However, for durations exceeding 1 h zones I and II show consistently higher values of both 1/S and S than other zones, and zones Vila, VIII and IX are also associated with the lowest values of 3-


iiifi : ^ i

tsnoH h3d UJUJ anoH a 3d ^ ^ NI TivjNiva iinoH a^d

364 Lekan Oyebande

î S I 1 s ! § £ g ? N! TWJNÎVy


t i

366 Lekan Oyebande

The plots of the rainfall intensity vs. return period are shown in Fig.2 for each of the regions or zones. The plots permit a comparison of the observed data with the expected (fitted by EV-1) values, particular for the longer return periods. The 2SE confidence limits are also sketched in the plots and used to determine possible outliers where such are suspected to be significant.

Outliers are defined by Tomlinson (1980, p. 9) as values that plot inconsistently with the rest of the data. In the present study a value that plots outside the confidence interval defined by twice the standard error of estimate (2SE1 is considered as an outlier.

On the whole the fitting of the EV-1 appears satisfactory for at least three reasons:

(a) the EV-1 distribution seems to fit the whole series well for all the regions;

(b) the chi-square value of goodness of fit is not abnormally high, or highly significant (except for the 0.2 h duration for zones III and IV);

(c) the fitting appears to produce a good estimate of the 100-year value as evidenced by zone II and as inferred from zones III and IVa.

The duration of 0.2 h (12 min) produces some outliers (according to the analytical definition above) in several regions for return periods greater than 25 years, but the possible deviation is rather small in virtually all cases. Zone II provides a good example (and some confident assurance) for determining the reliability of the estimates for longer return periods, for though T = 115, all the observed data points fall within the confidence limits, even for the 0.2 h duration.

According to Tomlinson (1980) there are three basic reasons why outliers occur. The values may be incorrect observations; they can represent a rare occurrence; or they may represent an occurrence which has resulted from a different phenomenon from that which produced the other values in the series.

Most of the outliers observed in Fig.2 appear to be due to the first reason, especially since many of them occur in the middle of the plot. It appears, however, that the outliers in zones I, Vllb and X may be due to the second reason. In the case of the 0.2 h plot for zone I for instance, the two largest values may actually represent some 100-year rainfall occurring in a 45-year series.

There are several ways of dealing with outliers. They may be excluded while the parameters are recalculated. On the other hand we may include them and select a more appropriate distribution, fitting technique or plotting position. The deviation of the outliers observed in the present study does not seem to justify doing any of the above.

It can be seen from the plots in Fig.2 that the plots and estimates are generally good for durations of 1 h and longer. The same applies to 0.4 h durations which are not shown in the plots.

CONCLUSION

It is interesting that statistical evidence supports the appropriateness of the Gumbel EV-1 distribution for fitting annual


extreme rainfall series. The results obtained should serve to meet the need for rainfall intensity-duration-frequency relationships and estimates in various parts of Nigeria, both for short and longer recurrence intervals. The need has been particularly great because many large basins remain inadequately gauged while some are hardly gauged at all. The use of the results of this study to provide design floods could be done with greater confidence and somewhat calculable risk for each zone.

It is hoped that as more data become available this objective analytical estimation could also be applied to individual station series with confidence. The results could then be used to construct isarithmic or isopleth maps which not only provide the magnitude of extreme falls of known duration and frequency at points required but also provide a total view of the statistical surface of the extreme rainfall in the country for each duration and frequency.

REFERENCES

Baghirathan, V.R. & Shaw, E.M. (1978) Rainfall depth-duration-frequency studies for Sri Lanka. J. Hydrol. 37, 223-239.

Bell, F.C. (1969) Generalized rainfall-duration-frequency relationship. J. Hydraul. Div. ASCE 95 (HYI), 311-327.

Gumbel, E.J. (1941) The return period of flood flows. Ann. Math. Statist. 12(2), 163-190.

Gumbel, E.J. (1943) On the reliability of the chi-square test. Ann. Math. Statist. 13, 306-317.

Hartley, H.O. (1950) The maximum F-ratio test as a short-cut test for heteorogeneity of variance. Biometrika 37, 308-312.

Hershfield, D.M. (1962) Extreme rainfall relationships. J. Hydraul. Div. ASCE (HY6), 73-92.

Natural Environment Research Council (1975) Flood Studies Report, vols I and II. Natural Environment Research Council, London.

Obasi, G.O.P., Mulero, M.A.A. & Omolayo, S.A. (1980) Fluctuations in annual rainfall values in Nigeria. In: The West African Monsoon Experiment (WAMEX) (Symp. Proc, ed. by D.O.Adefolalu) , 368-455.

Okulaja, F.O. (1980) Trends in rainfall regime along the Nigerian coastline during the pre-sahelian drought era. In: The West African Monsoon Experiment (WAMEX) (Symp. Proc, ed. by D.O.Adefolalu), 488-503.

Oyebande, L. (1980) Rainfall intensity-duration-frequency curves and maps for Nigeria. Occasional Pap. Series, no. 2, Department of Geography, Univ. of Lagos (in press).

Riggs, H.C. (1968) Some Statistical Tools in Hydrology. Techniques of Water Resources Investigations, US Geological Survey, Book 4.

Schulz, E.F. (1973) Problems in Applied Hydrology. Water Resources Publications, Fort Collins, Colorado, USA.

Tomlinson, A.I. (1980) The frequency of high intensity rainfall in New Zealand. Pt. 1, Water and Soil Tech. Publ. no. 19, National Water and Soil Conservation Organization, Wellington, New Zealand.

Yevjevich, V. (1972) Probability and Statistics in Hydrology. Water Resources Publications, Fort Collins, Colorado, USA.

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