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Variability and error in rainfall over a small tropical watershed
Journal of Hydrology, 34 (1977) 161--169 161 cv Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
V A R I A B I L I T Y A N D E R R O R IN R A I N F A L L O V E R A S M A L L T R O P I C A L WATERSHED
E.U. NWA
International Institute of Tropical Agriculture (HTA), Ibadan (Nigeria) (Received September 21, 1976; accepted for publication November 10, 1976)
ABSTRACT Nwa, E.U., 1977. Variability and error in rainfall over a small tropical watershed. J. Hydrol., 34: 161--169. This paper presents an analysis of the variability of rainfall over a small tropical watershed and also examines the errors involved in measuring rainfall over such small areas. Using a gauge density of 8.86 ha/gauge (5-rain-gauge network) as standard, it was shown that a 3-rain-gauge network would be sufficient for the watershed and would keep the average coefficient of variation below 10% and the average error in mean rainfall below 0.8 ram. The average error in mean rainfall and the average coefficient of variation are exponential functions of the mean rainfall and the number of rain gauges or gauge density.
INTRODUCTION Many hydrologic studies require accurate rainfall a m o u n t s over watersheds. T h e d i f f i c u l t y o f o b t a i n i n g a c c u r a t e values has l e d several i n v e s t i g a t o r s t o e x a m i n e t h e v a r i a b i l i t y o f r a i n f a l l o v e r w a t e r s h e d areas a n d t h e e r r o r s i n v o l v e d in t h e e s t i m a t i o n o f m e a n r a i n f a l l a m o u n t s . B u t m o s t o f t h e s t u d i e s d e a l t w i t h large areas. G a n g u l i et al. ( 1 9 5 1 ) u s e d 26 rain gauges t o s t u d y t h e a c c u r a c y o f e s t i m a t ing m e a n r a i n f a l l over an a r e a o f 1 8 , 6 4 8 k m 2 w h i l e N i c k s a n d H a r t m a n {1966), and Linsley and Kohler (1951) investigated the variations of rainfall over areas o f 2 , 9 2 7 a n d 5 7 0 k m 2, r e s p e c t i v e l y . T h e s t u d y o f C u r r y e t al. ( 1 9 6 6 ) was s p e c i f i c a l l y m e a n t t o e x a m i n e t h e e r r o r s r e s u l t i n g f r o m m e a n r a i n f a l l e s t i m a t e s o v e r small areas; t h e y u s e d 22 rain gauges as s t a n d a r d o v e r an a r e a o f 92 k m 2 (414 h a / g a u g e ) . E r r o r s r a n g e d f r o m 10 t o 30% f o r t h e 78 k m 2 / g a u g e n e t w o r k a n d f r o m 3 t o 10% f o r t h e 13 k m 2 / g a u g e n e t w o r k , d e p e n d i n g on t h e mean rainfall amount. One Russian study r e p o r t e d by Toebes and Ouryvaev ( 1 9 7 0 ) i n d i c a t e d t h a t e r r o r s o f o v e r 20% c a n r e s u l t w h e n o n e rain gauge is u s e d t o average m e a n rainfall o v e r o n e k m 2 ( 1 0 0 h a / g a u g e ) w i t h i n a t i m e i n t e r val o f o n e d a y . In t h a t s t u d y t h e e r r o r s i n c r e a s e d as t h e a r e a o v e r w h i c h rainfall a m o u n t s w e r e a v e r a g e d d e c r e a s e d . T h e a b o v e i n v e s t i g a t i o n s p o i n t o u t t h e n e e d , in h y d r o l o g i c s t u d i e s , t o e x a m i n e t h e e r r o r s t o be e x p e c t e d w h e n e s t i m a t i n g rainfall a m o u n t s over small areas.
162 T h e p u r p o s e o f this p a p e r is t h e r e f o r e to investigate the errors involved in d e t e r m i n i n g rainfall a m o u n t s over very small watersheds in the h u m i d tropics and to e x a m i n e the variability o f rainfall over these small areas. PROCEDURE T h e w a t e r s h e d is l o c a t e d in the I n t e r n a t i o n a l I n s t i t u t e o f Tropical Agriculture (IITA) n e a r Ibadan in the western section of Nigeria. T h e fan-shaped w a t e r s h e d has a total relief o f 32 m and an area o f 4 4 . 3 0 ha. Five rain gauges were l o c a t e d in the w a t e r s h e d as s h o w n in F i g . l ; f o u r o f these are weighingt y p e , r e c o r d i n g gauges with 24-h clocks while one is a n o n - r e c o r d i n g gauge. T h e rainfall d a t a f r o m August 8, 1974 to D e c e m b e r 31, 1975, w h e n all five gauges were operating, were used in this analysis. This p r o v i d e d 15 m o n t h s o f r e c o r d f r o m which was o b t a i n e d a total o f 134 rainfall events for the study. Based on the range o f daily rainfall values obtained, the s t o r m s were divided into seven classes. T h e range and m e a n o f rainfall in each class as well as the n u m b e r o f s t o r m s are given in Table I.
x LEGEND -
il,I
', '\, 90
SURFACE + SUBSURFACE MEASURING WEIR RAINFALL
STATION
METEOROLOGICAL
FLOW
No STA] ~©N
RECORDING RAIN GAUGE
/
,
\
,\
'~
NONRECORD~N G RAiN -
==~
INTERMITTENT WATERSHED
GAUGE
STREAM ROAD
x FENCE WATERSHED O
1OO
#=7=:~
BOUNDAR ~'
200 m e t r e s
d
Fig. 1. Rain-gauge and Thiessen network for watershed.
T h r e e m e t h o d s are generally a c c e p t e d for c o m p u t i n g the m e a n rainfall over watersheds. These are the a r i t h m e t i c mean, Thiessen p o l y g o n , and isohyetal m e t h o d s ; it is usually assumed t h a t the a c c u r a c y o f the m e a n rainfall increases in the o r d e r listed above. Because o f the d i f f i c u l t y o f using the isohyetal m e t h o d for every s t o r m , the Thiessen m e t h o d is o f t e n used as the n e x t best a p p r o a c h . The Thiessen m e t h o d was used t o d e t e r m i n e the w a t e r s h e d m e a n rainfall in this study. T o investigate the differences in m e a n rainfall o b t a i n e d by the isohyetal and Thiessen m e t h o d s , a few storms of varying sizes were
163
TABLE I Mean rainfall, number of storms and other variables in various rainfall classes Rainfall class
Rainfall range (ram)
Class mean (ram)
1 2 3 4 5 6 7
0.00-- 4.99 5.00-- 9.99 10.00--14.99 15.00--19.99 20.00--24.99 25.00--37.99 38.00--60.00
2.50 7.50 12.50 17.50 22.50 31.50 49.00
Total
Number of storms in class
Amount of storm in class (mm)
% of total amount
45 28 16 12 11 13 9
107.57 201.92 202.65 211.77 254.60 408.77 399.85
6.0 11.3 11.3 11.9 14.2 22.9 22.4
134
1,787.13
100.0
selected and a n a l y z e d b y b o t h m e t h o d s . Table II shows t h a t the differences are small, lying b e t w e e n 0.1 and 4.3% o f the isohyetal values; the high percentage difference o c c u r r e d at the low rainfall range. In this range a difference o f 0.01 m m can still give over 2% difference. The Thiessen m e t h o d was considered sufficiently a c c u r a t e f o r this analysis. As can be seen in F i g . l , the longest distance b e t w e e n any t w o rain gauges is 760 m. Despite the s h o r t distances b e t w e e n the rainfall stations, a close e x a m i n a t i o n o f the daily d a t a revealed t h a t differences b e t w e e n the high and low c a t c h can be as m u c h as 60% o f the high c a t c h particularly in the l o w e r rainfall range. M o n t h l y p e r c e n t a g e differences are m u c h l o w e r b u t values o f 25% have been observed. T h e isohyets for June 12 and O c t o b e r 1, 1975 are s h o w n in Figs. 2 and 3, respectively, while the d i s t r i b u t i o n for 1 9 7 5 is s h o w n in Fig.4. These figures illustrate the variations in the rainfall a m o u n t s observed in the watershed. In view o f the observed variations in the rainfall a m o u n t s at the various stations it was decided to e x a m i n e the c o e f f i c i e n t o f variation t h a t exists w h e n d i f f e r e n t n u m b e r s o f rain gauges are used to d e t e r m i n e the w a t e r s h e d m e a n rainfall. This will indicate the n u m b e r o f gauges t h a t is a d e q u a t e t o d e t e r m i n e the m e a n rainfall f o r small watersheds in this region. T o d o this, f o u r n e t w o r k s consisting o f 5, 4, 3 and 2 rain gauges were used. The n u m b e r o f rain gauges and the c o r r e s p o n d i n g gauge density for the w a t e r s h e d are given in Table III. F o r each n e t w o r k the c o e f f i c i e n t o f variation was d e t e r m i n e d for each of the 134 rainfall events. Using the w a t e r s h e d mean rainfall, all the events were g r o u p e d into their classes as s h o w n in Table I. T h e a r i t h m e t i c average m e t h o d was used to o b t a i n the m e a n rainfall for the 2-station n e t w o r k since the Thiessen m e t h o d c a n n o t be used for n e t w o r k s o f less than three gauges. An average c o e f f i c i e n t o f variation was o b t a i n e d b y s u m m i n g all the values in
1975
12/09/74 25/10/75 21/10/75 11/07/75 19/06/75 12/06175 01/08/75
Date
0.64 2.79 10.67 23.88 30.99 41.91 50.80 1,443.22
90 0.38 1.78 10.92 20.83 30.23 36.83 44.45 1,413.44
92
Rainfall station number
1,383.03
0.51 1.02 11.18 19.30 27.43 34.29 45.72
108
1,455.44
0.51 3.81 9.65 21.84 34.29 44.45 49.53
228
1,432.04
0.25 3.05 11.18 21.34 30.48 48.26 45.72
247
Comparison of mean rainfall (mm) by Thiessen and isohyetal methods
TABLE II
1,427.31
0.44 2.52 10.71 21.54 30.88 41.18 44.69
Thiessen average
1,425.56
0.46 2.58 10.80 21.50 30.63 41.63 44.61
Isohyet average
1.001
0.957 0.977 0.992 1.002 1.008 0.989 1.002
Thiessen/ isohyet ratio
t65
LEGEN
D
SURFACE + SUBSURFACE ~LOW MEASURING WEIR ~-2 mm ISOHYETS ~" i
METEOROLOGICAL
'~
STATION
RECORDING RAIN GAUGE NONRECOF~DING RAiN GAUGE
--
~
INTERMITTENT
~
WATERSHED
x
x FENCE WATERSHED O
,~'
L
............
STREAM ROAD
100
BOUNDARY
~)0 reel r('s
\
Fig.2. Rainfal| (ram) distribution, June 12, 1975.
LEGEND
',,', S I
SURFACE ~r SUBSURFACE ELCW MEASURING WEIR
t
./*/* mmmISOHYETS
,i,
Y
(
4~
~
~
METEOROLOGICAL
~'
RECORDING RAIN GAUGE
~'
NONRECORDING RAIN GAUGE
f
~NTERMITTENT STREAM WATERSHED ROAD
r~i 7 x
x FENCE WATERSHED
ili
OEEBVOR
STATION
0
100
BOUNDAI~'~
~0O metre5
Fig.3. Rainfall (ram) distribution, October 1, 1075.
each class and dividing by the number of storms in the class. This procedure provided seven coefficients of variation for each network. The class mean rainfall as s h o w n in Table I was plotted against the class average coefficient of variation for each of the four networks. These graphs are given in Fig.5. The n e x t consideration was the determination of the errors in mean rainfall mnounts. The approach used by Curry et al. ( 1 9 6 6 ) was adopted. Five networks consisting of 5, 4, 3, 2 and 1 rain gauges were used. Because of the l o w coefficient of variation obtained above, the Thiessen mean rainfall from the
166
LEGEND .
(¢
SURFACE + SUBSURFACE FLO~", MEASURING WEIR
~00~
,t/ / /'
" x
~ ~
~
@
<\
:;7 ~Y
]'
,
2,
,
#
,/
2?
METEOROLOGrC&L
STALON
RECORDING ~AIN OA:JOE
NC'NqECCRDINS FM,IN GAJOE
tl/
,,
ISOHYETS
NTERMITTEN T STREAM .~_~
!
WA+ERSHED ROA[) FENCE
s/
]
'/~
/
"-. . . . . . . . ResE~vom
~ .
/
WATERSHEC
,
:) :
1[)0
BU NDh*R',
20C rr,etr es 1
Fig.4. Annual rainfall (mm), 1975.
TABLE III Number of rain gauges and corresponding gauge density for the watershed Number of gauges,
N
1
Gauge density,
G (ha/gauge)
2
44.30
3
22.15
4
14.77
5
11.08
8.86
GAUGE DENSITY, G h a / g a u g e
> 'J
5 2c
0
G=886
~
z~
G =1108
~
a
G=I&77
7 w 10
i
o
o L#
0
L
~
~7
~ MEAN
£
2~
RAINFALL
OVER
d8
3'2
3'6
"0
.L
L
WATERSHED, mm
Fig.5. Coefficient of variation for various mean rainfall and rain-gauge network.
5 - s t a t i o n n e t w o r k w a s c o n s i d e r e d t o be t h e true w a t e r s h e d rainfall. T a k i n g o n e o f t h e less d e n s e n e t w o r k s at a t i m e , t h e m e a n rainfall w a s o b t a i n e d f o r e a c h o f t h e 1 3 4 s t o r m e v e n t s . T h e error f r o m e a c h s t o r m e v e n t w a s o b t a i n e d as t h e a b s o l u t e d i f f e r e n c e b e t w e e n t h e m e a n rainfall f r o m e a c h o f t h e less
167
dense networks and the corresponding true rainfall from the 5-station network. Using the true rainfall values, the storm events were divided into their corresponding classes and average errors were obtained for each class by summing all the errors in the class and dividing by the corresponding n u m b e r of storms. As in the coefficient of variation c o m p u t a t i o n , this procedure gave seven average errors for each network. Fig.6 gives the graphs o f class mean rainfall plotted against the corresponding error for each network. 2c GAUGE DENSITY, G ha/gauge A G= 4430 Q G=2215 0 G =IZ~ 77
1E E
•
~
~
g
O(
,~
;
li
16 210 22 218 3'2 MEAN RAINFALL OVER WATERSHEDS, mm
e
36
20
24
418
Fig.6. Average e r r o r for various m e a n rainfal] and rain-gauge n e t w o r k .
R E S U L T S AND DISCUSSION
In his study o f the variability of rainfall in Nigeria, Balogun (1972) observed that areas of high annual rainfall recorded low values of coefficient of variation (Cv) while areas of low rainfall recorded high C v values. Although his result was for annual values over a very large area, it is comparable to t hat shown in Fig.5, for a small watershed, where the average C v increases as the daily watershed rainfall a m o u n t decreases. The C v values for the 3- and 4station networks were quite close for all the mean rainfall values; consequently only one curve was fitted to the data of b ot h networks. This indicates that it does n o t make m uch difference in the mean rainfall w h e t h e r 3 or 4 gauges are used in the watershed. F o r the 5-, 4- and 3-station networks, the average coefficient o f variation is much less than 10% for all the rainfall classes except class 1; in this case the average C v is between 10 and 12%. In spite of the large n u m b e r of storms in class 1, th ey make up only 6% of the total a m o u n t of rainfall (Table I). The above discussion indicates t hat 3 rainfall stations can be used in the watershed w i t h o u t introducing much error in the mean rainfall. To fit the data shown in Fig.5, several types of equations were tried; the three that gave the best fit are given below: lnC v=al+b,lnP+b21nN
(1)
lnC v=a2+b21nP+b31nG
(2)
168
and: In C v = a3 + b 3 P 1/2 + b4 In G
(3)
w h e r e C v is t h e c o e f f i c i e n t o f v a r i a t i o n in p e r c e n t , P is t h e class m e a n rainfall in m m , N is the n u m b e r o f rain gauges, G is the gauge d e n s i t y in ha/gauge, and a and b are coefficients. Eqs. 2 and 3 are the f o r m s used b y Linsley and K o h l e r ( 1 9 5 1 ) a n d C u r r y et al. (1966), respectively, to fit the e r r o r r e l a t i o n s h i p in t h e i r studies. A c o m p u t e r was used t o o b t a i n the c o e f f i c i e n t s a n d the m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s (R 2) f o r the three e q u a t i o n s ; the resulting, respective e q u a t i o n s are given b e l o w : C v = 3 . 7 1 0 p--0.396 N - 0 . 6 8 9 ;
R 2 = 0.82
C v = 1 . 0 9 6 p - 0 . 3 9 6 G0.689;
R 2 = 0.82
and C v = 0 . 9 0 e--0.206 pin 60.689;
R 2 = 0.77
w h e r e e is the N a p i e r i a n c o n s t a n t . T h e results o f the e r r o r analysis are given in Fig. 6. As was t h e case w i t h C v values, t h e average errors f o r the 3- and 4-station n e t w o r k w e r e q u i t e close and t h e r e f o r e o n l y o n e curve was also f i t t e d t o t h e d a t a o f b o t h n e t w o r k s . Alt h o u g h the average e r r o r increased as the m e a n w a t e r s h e d rainfall increased f o r all t h e rainfall classes, it was less t h a n 0.8 m m f o r the 3- a n d 4 - s t a t i o n n e t w o r k . This c o n f i r m s the p r e v i o u s s t a t e m e n t t h a t little e r r o r w o u l d be i n t r o d u c e d in the m e a n w a t e r s h e d rainfall b y using o n l y 3 rain gauges. Eqs. 1 - - 3 also gave t h e b e s t fit in the e r r o r relationship. T h e final f o r m s o f these e q u a t i o n s are given as: E = - - 1 . 8 2 6 p0.557 N-0.758;
R 2 = 0.90
E = - - 4 . 7 0 p0.557 G0.758;
R 2 = 0.90
and E = - - 4 . 5 0 e 0"3085P': G°.755;
R
= 0.92
w h e r e E is t h e average e r r o r in m m , a n d t h e o t h e r variables are as p r e v i o u s l y defined. CONCLUSIONS E x p e r i e n c e s in o t h e r regions c o u l d h a v e led o n e t o c o n c l u d e t h a t o n e rain gauge is s u f f i c i e n t t o m e a s u r e the m e a n rainfall o v e r a small w a t e r s h e d o f this size. T h e results o f this investigation h a v e s h o w n t h a t t h e rain gauge d e n s i t y t h a t can give a c e r t a i n level o f a c c u r a c y in o n e g e o g r a p h i c a l region m a y n o t necessarily give t h e s a m e level o f a c c u r a c y in a n o t h e r g e o g r a p h i c a l region.
169
F o r hydrologic investigations that require accurate rainfall amounts over watersheds, it is i m p o r t a n t to study the variability and the errors involved in measuring rainfall over watersheds in the region of interest. Since the average coefficient of variation is mostly less than 10% and the average error is less than 0.8 m m for the 3-station network, it is concluded that this gauge density (14.77 ha/gauge) is sufficient to measure the mean rainfall over the watershed. F r o m the work of others referred to in this paper and the results of this investigation it can be concluded t hat the error or the coefficient of variation relationship is likely to fit one or m ore equations of the form: E or C v = alpb~N c ,
E or C v = a2PblGd~
or
E o r C v = a 3 e b:d9"2 G d
REFERENCES Balogun, C., 1972. The variability of rainfall in Nigeria. Niger. J. Sci., 6: 87--92. Curry, H.A., Wheaton, R.Z. and Kidder, E.H., 1966. Errors resulting from mean rainfall estimates on small watersheds. Trans. Am. Soc. Agric. Eng., 9: 126--128. Ganguli, M.K., Rangarajan, R. and Panchang, G.M., 1951. Accuracy of mean rainfall estimates Data of Damoda catchment. Irrig. Power J., 8: 278--284. Linsley, R.K. and Kohler, M.A., 1951. Variations in storm rainfall over small areas. Trans. Am. Geophys. Union, 32: 245--250. Nicks, A.D. and Hartman, M.A., 1966. Variation of rainfall over a large gaged area. Trans. Am. Soc. Agric. Eng., 9: 437--439. Toebes, C. and Ouryvaev, V. (Editors), 1970. Representative and Experimental Basins. An International Guide for Research and Practice. UNESCO, Paris, 348 pp.
V A R I A B I L I T Y A N D E R R O R IN R A I N F A L L O V E R A S M A L L T R O P I C A L WATERSHED
E.U. NWA
International Institute of Tropical Agriculture (HTA), Ibadan (Nigeria) (Received September 21, 1976; accepted for publication November 10, 1976)
ABSTRACT Nwa, E.U., 1977. Variability and error in rainfall over a small tropical watershed. J. Hydrol., 34: 161--169. This paper presents an analysis of the variability of rainfall over a small tropical watershed and also examines the errors involved in measuring rainfall over such small areas. Using a gauge density of 8.86 ha/gauge (5-rain-gauge network) as standard, it was shown that a 3-rain-gauge network would be sufficient for the watershed and would keep the average coefficient of variation below 10% and the average error in mean rainfall below 0.8 ram. The average error in mean rainfall and the average coefficient of variation are exponential functions of the mean rainfall and the number of rain gauges or gauge density.
INTRODUCTION Many hydrologic studies require accurate rainfall a m o u n t s over watersheds. T h e d i f f i c u l t y o f o b t a i n i n g a c c u r a t e values has l e d several i n v e s t i g a t o r s t o e x a m i n e t h e v a r i a b i l i t y o f r a i n f a l l o v e r w a t e r s h e d areas a n d t h e e r r o r s i n v o l v e d in t h e e s t i m a t i o n o f m e a n r a i n f a l l a m o u n t s . B u t m o s t o f t h e s t u d i e s d e a l t w i t h large areas. G a n g u l i et al. ( 1 9 5 1 ) u s e d 26 rain gauges t o s t u d y t h e a c c u r a c y o f e s t i m a t ing m e a n r a i n f a l l over an a r e a o f 1 8 , 6 4 8 k m 2 w h i l e N i c k s a n d H a r t m a n {1966), and Linsley and Kohler (1951) investigated the variations of rainfall over areas o f 2 , 9 2 7 a n d 5 7 0 k m 2, r e s p e c t i v e l y . T h e s t u d y o f C u r r y e t al. ( 1 9 6 6 ) was s p e c i f i c a l l y m e a n t t o e x a m i n e t h e e r r o r s r e s u l t i n g f r o m m e a n r a i n f a l l e s t i m a t e s o v e r small areas; t h e y u s e d 22 rain gauges as s t a n d a r d o v e r an a r e a o f 92 k m 2 (414 h a / g a u g e ) . E r r o r s r a n g e d f r o m 10 t o 30% f o r t h e 78 k m 2 / g a u g e n e t w o r k a n d f r o m 3 t o 10% f o r t h e 13 k m 2 / g a u g e n e t w o r k , d e p e n d i n g on t h e mean rainfall amount. One Russian study r e p o r t e d by Toebes and Ouryvaev ( 1 9 7 0 ) i n d i c a t e d t h a t e r r o r s o f o v e r 20% c a n r e s u l t w h e n o n e rain gauge is u s e d t o average m e a n rainfall o v e r o n e k m 2 ( 1 0 0 h a / g a u g e ) w i t h i n a t i m e i n t e r val o f o n e d a y . In t h a t s t u d y t h e e r r o r s i n c r e a s e d as t h e a r e a o v e r w h i c h rainfall a m o u n t s w e r e a v e r a g e d d e c r e a s e d . T h e a b o v e i n v e s t i g a t i o n s p o i n t o u t t h e n e e d , in h y d r o l o g i c s t u d i e s , t o e x a m i n e t h e e r r o r s t o be e x p e c t e d w h e n e s t i m a t i n g rainfall a m o u n t s over small areas.
162 T h e p u r p o s e o f this p a p e r is t h e r e f o r e to investigate the errors involved in d e t e r m i n i n g rainfall a m o u n t s over very small watersheds in the h u m i d tropics and to e x a m i n e the variability o f rainfall over these small areas. PROCEDURE T h e w a t e r s h e d is l o c a t e d in the I n t e r n a t i o n a l I n s t i t u t e o f Tropical Agriculture (IITA) n e a r Ibadan in the western section of Nigeria. T h e fan-shaped w a t e r s h e d has a total relief o f 32 m and an area o f 4 4 . 3 0 ha. Five rain gauges were l o c a t e d in the w a t e r s h e d as s h o w n in F i g . l ; f o u r o f these are weighingt y p e , r e c o r d i n g gauges with 24-h clocks while one is a n o n - r e c o r d i n g gauge. T h e rainfall d a t a f r o m August 8, 1974 to D e c e m b e r 31, 1975, w h e n all five gauges were operating, were used in this analysis. This p r o v i d e d 15 m o n t h s o f r e c o r d f r o m which was o b t a i n e d a total o f 134 rainfall events for the study. Based on the range o f daily rainfall values obtained, the s t o r m s were divided into seven classes. T h e range and m e a n o f rainfall in each class as well as the n u m b e r o f s t o r m s are given in Table I.
x LEGEND -
il,I
', '\, 90
SURFACE + SUBSURFACE MEASURING WEIR RAINFALL
STATION
METEOROLOGICAL
FLOW
No STA] ~©N
RECORDING RAIN GAUGE
/
,
\
,\
'~
NONRECORD~N G RAiN -
==~
INTERMITTENT WATERSHED
GAUGE
STREAM ROAD
x FENCE WATERSHED O
1OO
#=7=:~
BOUNDAR ~'
200 m e t r e s
d
Fig. 1. Rain-gauge and Thiessen network for watershed.
T h r e e m e t h o d s are generally a c c e p t e d for c o m p u t i n g the m e a n rainfall over watersheds. These are the a r i t h m e t i c mean, Thiessen p o l y g o n , and isohyetal m e t h o d s ; it is usually assumed t h a t the a c c u r a c y o f the m e a n rainfall increases in the o r d e r listed above. Because o f the d i f f i c u l t y o f using the isohyetal m e t h o d for every s t o r m , the Thiessen m e t h o d is o f t e n used as the n e x t best a p p r o a c h . The Thiessen m e t h o d was used t o d e t e r m i n e the w a t e r s h e d m e a n rainfall in this study. T o investigate the differences in m e a n rainfall o b t a i n e d by the isohyetal and Thiessen m e t h o d s , a few storms of varying sizes were
163
TABLE I Mean rainfall, number of storms and other variables in various rainfall classes Rainfall class
Rainfall range (ram)
Class mean (ram)
1 2 3 4 5 6 7
0.00-- 4.99 5.00-- 9.99 10.00--14.99 15.00--19.99 20.00--24.99 25.00--37.99 38.00--60.00
2.50 7.50 12.50 17.50 22.50 31.50 49.00
Total
Number of storms in class
Amount of storm in class (mm)
% of total amount
45 28 16 12 11 13 9
107.57 201.92 202.65 211.77 254.60 408.77 399.85
6.0 11.3 11.3 11.9 14.2 22.9 22.4
134
1,787.13
100.0
selected and a n a l y z e d b y b o t h m e t h o d s . Table II shows t h a t the differences are small, lying b e t w e e n 0.1 and 4.3% o f the isohyetal values; the high percentage difference o c c u r r e d at the low rainfall range. In this range a difference o f 0.01 m m can still give over 2% difference. The Thiessen m e t h o d was considered sufficiently a c c u r a t e f o r this analysis. As can be seen in F i g . l , the longest distance b e t w e e n any t w o rain gauges is 760 m. Despite the s h o r t distances b e t w e e n the rainfall stations, a close e x a m i n a t i o n o f the daily d a t a revealed t h a t differences b e t w e e n the high and low c a t c h can be as m u c h as 60% o f the high c a t c h particularly in the l o w e r rainfall range. M o n t h l y p e r c e n t a g e differences are m u c h l o w e r b u t values o f 25% have been observed. T h e isohyets for June 12 and O c t o b e r 1, 1975 are s h o w n in Figs. 2 and 3, respectively, while the d i s t r i b u t i o n for 1 9 7 5 is s h o w n in Fig.4. These figures illustrate the variations in the rainfall a m o u n t s observed in the watershed. In view o f the observed variations in the rainfall a m o u n t s at the various stations it was decided to e x a m i n e the c o e f f i c i e n t o f variation t h a t exists w h e n d i f f e r e n t n u m b e r s o f rain gauges are used to d e t e r m i n e the w a t e r s h e d m e a n rainfall. This will indicate the n u m b e r o f gauges t h a t is a d e q u a t e t o d e t e r m i n e the m e a n rainfall f o r small watersheds in this region. T o d o this, f o u r n e t w o r k s consisting o f 5, 4, 3 and 2 rain gauges were used. The n u m b e r o f rain gauges and the c o r r e s p o n d i n g gauge density for the w a t e r s h e d are given in Table III. F o r each n e t w o r k the c o e f f i c i e n t o f variation was d e t e r m i n e d for each of the 134 rainfall events. Using the w a t e r s h e d mean rainfall, all the events were g r o u p e d into their classes as s h o w n in Table I. T h e a r i t h m e t i c average m e t h o d was used to o b t a i n the m e a n rainfall for the 2-station n e t w o r k since the Thiessen m e t h o d c a n n o t be used for n e t w o r k s o f less than three gauges. An average c o e f f i c i e n t o f variation was o b t a i n e d b y s u m m i n g all the values in
1975
12/09/74 25/10/75 21/10/75 11/07/75 19/06/75 12/06175 01/08/75
Date
0.64 2.79 10.67 23.88 30.99 41.91 50.80 1,443.22
90 0.38 1.78 10.92 20.83 30.23 36.83 44.45 1,413.44
92
Rainfall station number
1,383.03
0.51 1.02 11.18 19.30 27.43 34.29 45.72
108
1,455.44
0.51 3.81 9.65 21.84 34.29 44.45 49.53
228
1,432.04
0.25 3.05 11.18 21.34 30.48 48.26 45.72
247
Comparison of mean rainfall (mm) by Thiessen and isohyetal methods
TABLE II
1,427.31
0.44 2.52 10.71 21.54 30.88 41.18 44.69
Thiessen average
1,425.56
0.46 2.58 10.80 21.50 30.63 41.63 44.61
Isohyet average
1.001
0.957 0.977 0.992 1.002 1.008 0.989 1.002
Thiessen/ isohyet ratio
t65
LEGEN
D
SURFACE + SUBSURFACE ~LOW MEASURING WEIR ~-2 mm ISOHYETS ~" i
METEOROLOGICAL
'~
STATION
RECORDING RAIN GAUGE NONRECOF~DING RAiN GAUGE
--
~
INTERMITTENT
~
WATERSHED
x
x FENCE WATERSHED O
,~'
L
............
STREAM ROAD
100
BOUNDARY
~)0 reel r('s
\
Fig.2. Rainfal| (ram) distribution, June 12, 1975.
LEGEND
',,', S I
SURFACE ~r SUBSURFACE ELCW MEASURING WEIR
t
./*/* mmmISOHYETS
,i,
Y
(
4~
~
~
METEOROLOGICAL
~'
RECORDING RAIN GAUGE
~'
NONRECORDING RAIN GAUGE
f
~NTERMITTENT STREAM WATERSHED ROAD
r~i 7 x
x FENCE WATERSHED
ili
OEEBVOR
STATION
0
100
BOUNDAI~'~
~0O metre5
Fig.3. Rainfall (ram) distribution, October 1, 1075.
each class and dividing by the number of storms in the class. This procedure provided seven coefficients of variation for each network. The class mean rainfall as s h o w n in Table I was plotted against the class average coefficient of variation for each of the four networks. These graphs are given in Fig.5. The n e x t consideration was the determination of the errors in mean rainfall mnounts. The approach used by Curry et al. ( 1 9 6 6 ) was adopted. Five networks consisting of 5, 4, 3, 2 and 1 rain gauges were used. Because of the l o w coefficient of variation obtained above, the Thiessen mean rainfall from the
166
LEGEND .
(¢
SURFACE + SUBSURFACE FLO~", MEASURING WEIR
~00~
,t/ / /'
" x
~ ~
~
@
<\
:;7 ~Y
]'
,
2,
,
#
,/
2?
METEOROLOGrC&L
STALON
RECORDING ~AIN OA:JOE
NC'NqECCRDINS FM,IN GAJOE
tl/
,,
ISOHYETS
NTERMITTEN T STREAM .~_~
!
WA+ERSHED ROA[) FENCE
s/
]
'/~
/
"-. . . . . . . . ResE~vom
~ .
/
WATERSHEC
,
:) :
1[)0
BU NDh*R',
20C rr,etr es 1
Fig.4. Annual rainfall (mm), 1975.
TABLE III Number of rain gauges and corresponding gauge density for the watershed Number of gauges,
N
1
Gauge density,
G (ha/gauge)
2
44.30
3
22.15
4
14.77
5
11.08
8.86
GAUGE DENSITY, G h a / g a u g e
> 'J
5 2c
0
G=886
~
z~
G =1108
~
a
G=I&77
7 w 10
i
o
o L#
0
L
~
~7
~ MEAN
£
2~
RAINFALL
OVER
d8
3'2
3'6
"0
.L
L
WATERSHED, mm
Fig.5. Coefficient of variation for various mean rainfall and rain-gauge network.
5 - s t a t i o n n e t w o r k w a s c o n s i d e r e d t o be t h e true w a t e r s h e d rainfall. T a k i n g o n e o f t h e less d e n s e n e t w o r k s at a t i m e , t h e m e a n rainfall w a s o b t a i n e d f o r e a c h o f t h e 1 3 4 s t o r m e v e n t s . T h e error f r o m e a c h s t o r m e v e n t w a s o b t a i n e d as t h e a b s o l u t e d i f f e r e n c e b e t w e e n t h e m e a n rainfall f r o m e a c h o f t h e less
167
dense networks and the corresponding true rainfall from the 5-station network. Using the true rainfall values, the storm events were divided into their corresponding classes and average errors were obtained for each class by summing all the errors in the class and dividing by the corresponding n u m b e r of storms. As in the coefficient of variation c o m p u t a t i o n , this procedure gave seven average errors for each network. Fig.6 gives the graphs o f class mean rainfall plotted against the corresponding error for each network. 2c GAUGE DENSITY, G ha/gauge A G= 4430 Q G=2215 0 G =IZ~ 77
1E E
•
~
~
g
O(
,~
;
li
16 210 22 218 3'2 MEAN RAINFALL OVER WATERSHEDS, mm
e
36
20
24
418
Fig.6. Average e r r o r for various m e a n rainfal] and rain-gauge n e t w o r k .
R E S U L T S AND DISCUSSION
In his study o f the variability of rainfall in Nigeria, Balogun (1972) observed that areas of high annual rainfall recorded low values of coefficient of variation (Cv) while areas of low rainfall recorded high C v values. Although his result was for annual values over a very large area, it is comparable to t hat shown in Fig.5, for a small watershed, where the average C v increases as the daily watershed rainfall a m o u n t decreases. The C v values for the 3- and 4station networks were quite close for all the mean rainfall values; consequently only one curve was fitted to the data of b ot h networks. This indicates that it does n o t make m uch difference in the mean rainfall w h e t h e r 3 or 4 gauges are used in the watershed. F o r the 5-, 4- and 3-station networks, the average coefficient o f variation is much less than 10% for all the rainfall classes except class 1; in this case the average C v is between 10 and 12%. In spite of the large n u m b e r of storms in class 1, th ey make up only 6% of the total a m o u n t of rainfall (Table I). The above discussion indicates t hat 3 rainfall stations can be used in the watershed w i t h o u t introducing much error in the mean rainfall. To fit the data shown in Fig.5, several types of equations were tried; the three that gave the best fit are given below: lnC v=al+b,lnP+b21nN
(1)
lnC v=a2+b21nP+b31nG
(2)
168
and: In C v = a3 + b 3 P 1/2 + b4 In G
(3)
w h e r e C v is t h e c o e f f i c i e n t o f v a r i a t i o n in p e r c e n t , P is t h e class m e a n rainfall in m m , N is the n u m b e r o f rain gauges, G is the gauge d e n s i t y in ha/gauge, and a and b are coefficients. Eqs. 2 and 3 are the f o r m s used b y Linsley and K o h l e r ( 1 9 5 1 ) a n d C u r r y et al. (1966), respectively, to fit the e r r o r r e l a t i o n s h i p in t h e i r studies. A c o m p u t e r was used t o o b t a i n the c o e f f i c i e n t s a n d the m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s (R 2) f o r the three e q u a t i o n s ; the resulting, respective e q u a t i o n s are given b e l o w : C v = 3 . 7 1 0 p--0.396 N - 0 . 6 8 9 ;
R 2 = 0.82
C v = 1 . 0 9 6 p - 0 . 3 9 6 G0.689;
R 2 = 0.82
and C v = 0 . 9 0 e--0.206 pin 60.689;
R 2 = 0.77
w h e r e e is the N a p i e r i a n c o n s t a n t . T h e results o f the e r r o r analysis are given in Fig. 6. As was t h e case w i t h C v values, t h e average errors f o r the 3- and 4-station n e t w o r k w e r e q u i t e close and t h e r e f o r e o n l y o n e curve was also f i t t e d t o t h e d a t a o f b o t h n e t w o r k s . Alt h o u g h the average e r r o r increased as the m e a n w a t e r s h e d rainfall increased f o r all t h e rainfall classes, it was less t h a n 0.8 m m f o r the 3- a n d 4 - s t a t i o n n e t w o r k . This c o n f i r m s the p r e v i o u s s t a t e m e n t t h a t little e r r o r w o u l d be i n t r o d u c e d in the m e a n w a t e r s h e d rainfall b y using o n l y 3 rain gauges. Eqs. 1 - - 3 also gave t h e b e s t fit in the e r r o r relationship. T h e final f o r m s o f these e q u a t i o n s are given as: E = - - 1 . 8 2 6 p0.557 N-0.758;
R 2 = 0.90
E = - - 4 . 7 0 p0.557 G0.758;
R 2 = 0.90
and E = - - 4 . 5 0 e 0"3085P': G°.755;
R
= 0.92
w h e r e E is t h e average e r r o r in m m , a n d t h e o t h e r variables are as p r e v i o u s l y defined. CONCLUSIONS E x p e r i e n c e s in o t h e r regions c o u l d h a v e led o n e t o c o n c l u d e t h a t o n e rain gauge is s u f f i c i e n t t o m e a s u r e the m e a n rainfall o v e r a small w a t e r s h e d o f this size. T h e results o f this investigation h a v e s h o w n t h a t t h e rain gauge d e n s i t y t h a t can give a c e r t a i n level o f a c c u r a c y in o n e g e o g r a p h i c a l region m a y n o t necessarily give t h e s a m e level o f a c c u r a c y in a n o t h e r g e o g r a p h i c a l region.
169
F o r hydrologic investigations that require accurate rainfall amounts over watersheds, it is i m p o r t a n t to study the variability and the errors involved in measuring rainfall over watersheds in the region of interest. Since the average coefficient of variation is mostly less than 10% and the average error is less than 0.8 m m for the 3-station network, it is concluded that this gauge density (14.77 ha/gauge) is sufficient to measure the mean rainfall over the watershed. F r o m the work of others referred to in this paper and the results of this investigation it can be concluded t hat the error or the coefficient of variation relationship is likely to fit one or m ore equations of the form: E or C v = alpb~N c ,
E or C v = a2PblGd~
or
E o r C v = a 3 e b:d9"2 G d
REFERENCES Balogun, C., 1972. The variability of rainfall in Nigeria. Niger. J. Sci., 6: 87--92. Curry, H.A., Wheaton, R.Z. and Kidder, E.H., 1966. Errors resulting from mean rainfall estimates on small watersheds. Trans. Am. Soc. Agric. Eng., 9: 126--128. Ganguli, M.K., Rangarajan, R. and Panchang, G.M., 1951. Accuracy of mean rainfall estimates Data of Damoda catchment. Irrig. Power J., 8: 278--284. Linsley, R.K. and Kohler, M.A., 1951. Variations in storm rainfall over small areas. Trans. Am. Geophys. Union, 32: 245--250. Nicks, A.D. and Hartman, M.A., 1966. Variation of rainfall over a large gaged area. Trans. Am. Soc. Agric. Eng., 9: 437--439. Toebes, C. and Ouryvaev, V. (Editors), 1970. Representative and Experimental Basins. An International Guide for Research and Practice. UNESCO, Paris, 348 pp.