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Spatial and temporal characteristics of high-intensive rainfall in northern Tunisia
Journal of Hydrology, 87 (1986) 285-298 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
285
[1] S P A T I A L A N D T E M P O R A L CHARACTERISTICS OF H I G H - I N T E N S I V E R A I N F A L L IN N O R T H E R N T U N I S I A
RONNY BERNDTSSON and JANUSZ NIEMCZYNOWICZ
Department of Water Resources Engineering, Lund Institute of Science and Technology, University of Lund, Lund (Sweden) (Received February 25, 1986; accepted for publication May 20, 1986)
ABSTRACT Berndtsson, R. and Niemczynowicz, J., 1986. Spatial and temporal characteristics of high-intensive rainfall in n o r t h e r n Tunisia. J. Hydrol., 87: 285-298. When dealing with u r b a n design, it is essential to have knowledge of the magnitude of the spatial variability of short high-intensive rainstorms. Most of the studies of high-intensive rainfall, conducted during recent years, concern regions with a humid climate. Still, the largest temporal and spatial variations in rainfall are to be found in arid and semi-arid regions. This paper sums up findings regarding rainfall variability, observed in a small catchment in n o r t h e r n Tunisia during a period of 2 yr. Point and areal intensities are presented for the ten most high-intensive storms observed. Storm-centered areal reduction factors are calculated for different durations and areas.
INTRODUCTION
In all practical rainfall-runoff calculations, hydrologists have to estimate the areal rainfall from point measurements. This can be done with different methods ranging from simple arithmetic averaging to sophisticated computerized interpolation and extrapolation techniques. However, as regards localized rainfall of the convective type, the number of rain-gauges is usually insufficient for making reliable areal estimates. In this context it is important to know the magnitude of the spatial variation in rainfall during different time periods. In connection with urban design problems, the spatial variability during short-time periods becomes interesting. Several studies of point-to-area rainfall relationships have been made during the last few years (see RodriguezIturbe and Mejia, 1974; Rao and Chenchayya, 1975; Bell, 1976; Schumann, 1983; Nguyen, 1984; Niemczynowicz, 1984). All these studies deal with humid climate regions. However, the largest temporal and spatial variations in rainfall are to be found in arid and semi-arid regions. The short-term variability of high-intensive rains in these climates has not been paid much attention so far. The present paper addresses this aspect. It sums up findings regarding rainfall variability observed in a small catchment in northern Tunisia during a period of 2 yr. Point and areal intensities are presented for the ten most high-intensive
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© 1986 Elsevier Science Publishers B.V.
286
MAIN WATER COURSE MAIN CONOUITS . . . . ..........
SUBCATCHMENT 01V~C)ER DIVIDINGLINE FOR URBAN
Fig. 1. Catchment topography and instrumentation.
288 CATCHMENT BOROER
400
/
ISOHYE TS (ram)
,~50 ~ / /
i ~
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350
,
300 \x
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ISOMYETS OF: STORM 1g~..12.05 02.30- 0330
(mini
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.
.
9
.
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Fig. 2. Rainfall distribution for the rainy period in 1982/83 compared to a separate high-intensive rainstorm.
289
the altitude of the stations is decisive for the rainfall depth. Short highintensive rainstorms often have an areal distribution different from the common daily or monthly pattern, as can be seen in Fig. 2. During the rainy period in 1982-83, about 230 to 460 mm were recorded in the seven rain-gauges. A marked gradient in rainfall from east to west was observed, as shown in Fig. 2. For the high-intensive rainstorm recorded during 1 h on December 5, 1982, shown in the figure for purposes of comparison, the recorded rainfall varied between about 1 and 21 mm. The local character of high-intensive storms can be shown by use of correlation techniques. A common method for studying rainfall variability and for transforming point rainfall into areal rainfall is to use spatial correlation structures (see for example Rodriguez-Iturbe and Mejia, 1974; Niemczynowicz and JSnsson, 1981; Simeonidis, 1984). The dependence structure among time series is determined by the lag-k cross-correlation between two series (Salas et al., 1980): N-k -
r~ =
N
g-k -
t=l
(xt+k-
t=, "
11/2
(1)
(x,+k
t=l
where r~ = lag-k cross-correlation coefficient; x~i), x~)= time series; ~i) = mean of the first N - k values of series i; xt+k -0) = mean of the last N - k values of series j. Equation (1) was used in order to compare the variability of rainfall within the catchment during different time periods. Figure 3 gives an example of this method for the rainy season in 1982-83 on a daily and monthly basis. As a comparison, hourly values of the ten most intensive rainfalls observed during the 2 yr are also shown. In the figure, the lag-zero cross-correlation for monthly and daily rainfall displays great similarities. The correlation between stations gradually decreases with distance. A high correlation is maintained up to about 2000-3000 m. The 1-h storms, however, display a very low correlation between stations. Only gauges very close to each other show any significant dependence. The correlation structure is usually not isotropic, as shown by Sharon (1974) and Tase (1976). By taking the orientation of the line connecting two stations into consideration, the cellular nature of high-intensive rainfall can be shown. In Fig. 4 a comparison of spatial correlation, considering the interstation distance and the orientation for monthly rainfall and the ten 1-h rainstorms, respectively, is shown. It can be seen t h a t the ten rainstorms occur as cellular correlation isolines. The typical cell size, as defined by Niemczynowicz and JSnsson (1981) as being the area within the 0.7 correlation isoline, seems to be about 6 to 7 km ~ for a 1-h duration. The monthly correlations, on the other hand, display no cellular patterns. Instead the isolines seem to run parallel to a line in NE-SW direction, t h a t is normal to the isohyets. This is, however, not a very clear observation. Because of the orientation of the rain-gauges, which
290 correlation coeff monthly v a l u e l
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0 -0.2
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Fig. 3. Lag-zero cross-correlation coefficient for different periods versus the interstation distance.
is mainly in a NW-SE direction, the diagramme gives information mainly for this direction. VARIABILITY OF SHORT-TERM RAINSTORMS
The storm intensity profiles are interesting from an urban hydrological point of view when deciding the general shape of design storms. The relationship between the proportional rainfall depth and the rainfall duration for the ten rainstorms is shown in Fig. 5. The proportional rainfall depth is defined as the rainfall depth at a given duration divided by the total rainfall depth. The figure gives a comparison between point and areal values. The areal values were calculated by use of Thiessen polygons, expanding outwards from the gauge with maximum intensity. The point intensity profiles show a con-
291 distance
O06?~~j~0
north
~/ (kin) ~-5.0
distance
I/
corr coeft
monthly values
w.,Oo: t i
north cart
hourlycoeffvalues
o.
• o,s,o'......
,
0.6
.... t~ o.g Fig. 4. Correlation fields for monthly rainfall and ten 1-h rainstorms. siderable scatter. Both rainfalls with more and less rain in the first and second h alf of the dur a t i on period are represented. However, a general trend, which is amplified in the areal intensity profiles, can be seen. Typical mass curves seem to be characteristically concave-upwards in the first half of the duration and concave-downwards in the second ha l f of the duration. This supports findings of East African storm characteristics which were shown to have this general p atter n by Sumner (1984). It is interesting to see t h a t the largest scatter in the intensity profiles for the point values seem to disappear when an areal value of 10 km 2 is considered. The scatter from 10 to 20 km 2is not changed considerably. The rainstorms were f ur t her analyzed as regards i n t e n s i t y - d u r a t i o n relationships. Figure 6 shows point and areal i n t e n s i t y ~ l u r a t i o n curves for the ten rainstorms. It can be seen t h a t in the case of the recorded storms the 5-min period of maximum intensity varied between about 0.5 and 2.0 mm min 1 (point values). Areal intensity (10km ~ and 20km 2) was calculated by use of Thiessen polygons, expanding outwards from the gauge with maximum point intensity. As a comparison, the theoretical point i n t e n s i t y - d u r a t i o n curves with a r e t u r n period of 1, 2 and 5 yr for Tunis according to T hi rri ot et al. (1981) are shown in the figure. It can be seen t h a t for short durations (5 rain) one of the recorded storms (No. 7) seems to represent a rainfall with a 5-yr r e t u r n period, anot her storm (No. 8) seems to represent a rainfall with a 2-yr r e t u r n period and several others (Nos. 3, 5, 6, 9) represent rainfalls with a yearly r e t u r n period. For longer durations (60min) several recorded storms (Nos. 5, 7, 8) seem to represent rainfalls with a 5-yr r e t u r n period. There may be two explanations of the fact t h a t high-intensive rainstorms were over-represented during the r a t h e r short observation period (2-yr). It may have been an unusual year from a hydrological point of view or the theoretical i n t e n s i t y - d u r a t i o n curves may be based on too low values. Irrespective of which explanation is adopted, it must be borne in mind t h a t the annual variations in this region are extreme and t hat observa-
292 proportional rainfall depth
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: I0 km2
0 (rnin}
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Fig. 5. Point and areal storm intensity profiles for ten rainstorms (the figures 1, 2..... 10 refer to rainstorm number).
tions from short periods may not be directly comparable to statistical r e t u r n periods based on longer records. From Fig. 6 and Table 1 it can be seen t h a t rainfall intensity is quickly reduced when a larger area and a longer durat i on are considered. For 5-min durations the mean areal rainfall intensity for 10 km 2 is only about 60~/o compared to the point value. For 20 km 2, the mean areal intensity is about 40% compared to the point value. For longer durations (60 min) the same values are about 80 and 60% for 10 and 20 km 2 respectively. From Fig. 6 and Table 1 it can also be noted t hat high variabilities in rainfall occur during the shortest time periods. Increasing duration and increasing area reduce the standard deviation rapidly. As can be seen from Fig. 6, rainfall intensity does not invariably decrease with increasing dur a t i on (for example rain No. 5). This is due to the occurrence of more t h a n one raincell or rainfall intensity maximum for the given duration. These local intensity maxima become more important the larger the area or the longer the dur a t i on considered, causing a local rise in the i n t e n s i t y ~ l u r a t i o n curves. A measure of the rainfall variability in time over an area can be provided by correlation analysis as described above. Hershfield (1984) showed t h a t the co r r elati on coefficient between simultaneous incremental rainfall depths for a given storm, as computed from all pairs of gauges in the network, decays rapidly with decreasing incremental value. Consequently, the rates of the incoming storm rainfall can var y greatly over a small area, but the cumulations during the storm period tend to equalize the storm totals. In order to show this for the ten rainstorms analyzed, the correlation coefficients for different rainfall increments were calculated and plotted against distance between the gauges. In order to avoid a false rise in correlation, increment events with no rainfall recorded at any of the stations were excluded. The result displayed a
293 rainfall intensitY(minim in) k 7~
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45
50
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60
time
(mln)
Fig. 6. Point and areal intensity~luration curves for ten rainstorms (the figures 1, 2..... 10 refer to rainstorm number). very wide scatter irrespective of increment period. However, a closer examination by use of correlation fields s h o w e d t h a t typical correlation isolines are elliptically drawn out. The elliptical character of the correlation isolines seems to be more p r o n o u n c e d for longer increments. The 0.7 correlation isoline for 5, 10 and 20-min increments represented an area of about 2, 7 and 10 km 2 respectively. This s h o w s t h a t correlation increases w i t h increasing increment period also for short h i g h - i n t e n s i v e storms, t h o u g h but w e a k l y and depending on direction.
294 TABLE 1 Statistical p r o p e r t i e s of rainfall i n t e n s i t y depending on d u r a t i o n and a r e a for ten r a i n s t o r m s (£ = mean, m = median, s = s t a n d a r d d e v i a t i o n ) ( m m h -1) A r e a (kin 2)
D u r a t i o n (min) 5
10
20
30
60
Point £ m s
0.99 0.90 0.46
0.71 0.63 0.37
0.48 0.41 0.26
0.38 0.32 0.19
0.24 0.21 0.12
10 ~ m s
0.62 0.52 0.33
0.48 0.41 0.28
0.35 0.29 0.17
0.30 0.25 0.14
0.19 0.18 0.08
20 £ m s
0.37 0.33 0.18
0.30 0.30 0.13
0.24 0.26 0.09
0.21 0.20 0.08
0.14 0.15 0.05
AREAL REDUCTION FACTORS
The spatial properties of rainfall can also be described by areal reduction factors. Storm-centered areal reduction factors for the ten rainstorms were calculated as described by Bell (1976) and Niemczynowicz and JSnsson (1981). Storm-centered areal reduction factors are mainly used for converting point estimates of probable maximum precipitation to areal estimates when no frequency estimates are involved (Hershfield, 1962). Storm-centered areal reduction factors are defined as the ratio between the maximum areal rainfall within the storm zone for the given area and duration, and the maximum point rainfall within the same storm for the same duration. Figure 7 shows the stormcentered areal reduction factors for the ten rainstorms depending on area and duration. For each rainstorm, calculations started from the area represented by the gauge with the highest rainfall intensity. The areas represented by gauges were determined by use of the Thiessen polygon method. It can be seen t h a t in most cases the areal reduction factor decreases rather rapidly with increasing area. However, for some rainstorms the reduction factor exceeds the value 1.0. As mentioned before this depends on the occurrence of different raincells within the area covered by the gauging network. Occurrence of different raincells is also indicated by temporarily increasing reduction factors, especially for longer durations when they are influenced by incoming new raincells. From Table 2 and Fig. 7 it can be seen t h a t on an average and for a given area, the reduction factor increases with increasing duration. This means t h a t when the reduction factor increases, the reduction in intensity becomes smaller (by definition of reduction factor). The areal reduction is the most significant for short durations. For durations of 5 and 10 min and a considered area of 20 km 2, the reduction in intensity is over 50%. For a duration of 60 min,
295 areal reduction L factor
.
.
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8
,2
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~ - - r 1'6
-~--~
20
....
{ kin2}
duration30 rain
08 05 04 1 o.2J 0
- -
r
r
7-
k 8 areal reduction f a c t o r ~ 10 t ~
/2
16 duration
0.8
I
20 area (kin 2) 60 rain I
06 ~
~
5
04
?
0.2 0
0
I*
8
I'2
l'B
2'0
area
(kin2)
Fig. 7. Storm-centered areal reduction factors for ten rainstorms depending on duration (the figures 1, 2 ..... 10 refer to rainstorm number).
296 TABLE 2 Statistical p r o p e r t i e s of areal r e d u c t i o n factors depending on d u r a t i o n and a r e a for ten r a i n s t o r m s (~ = mean, m = median, s = s t a n d a r d deviation) A r e a (km 2)
D u r a t i o n (min) 5
10
20
30
60
5 ~ m s
0.83 0.84 0.10
0.85 0.84 0.08
0.88 0.85 0.07
0.90 0.90 0.05
0.94 0.95 0.08
10 ~ m s
0.64 0.67 0.14
0.69 0.69 0.17
0.76 0.72 0.15
0.80 0.80 0.11
0.87 0.86 0.15
20 2 m s
0.39 0.37 0.14
0.48 0.41 0.19
0.57 0.45 0.23
0.62 0.54 0.20
0.67 0.70 0.19
considering the same area, the reduction is over 30%. The standard deviation does not change significantly with increasing duration but instead with increasing area. The larger the area considered, the larger is the standard deviation. DISCUSSION AND CONCLUSION
Because of the larger variability of rainfall in arid and semi-arid areas a basic understanding of the storm development and intensity characteristics is important. Especially in an urban hydrological context it is essential to have some knowledge of short high-intensive rainstorms and their extension. In this respect, case studies of selected storms, as in this paper, may provide a greater understanding of the rainfall process. The ten most high-intensive rainstorms observed in a small catchment in northern Tunisia during the period 1981/82-1982/83 have been examined. In spite of the rather short observation period, high-intensive rainstorms seem to have been well represented. For short durations (5 min) one of the recorded storms represented a rainfall with a 5-yr return period. For longer durations several other recorded storms also represented rainfalls with a 5-yr return period. Using spatial correlation between rain-gauges combined by twos, it was shown t h a t compared to monthly and daily rainfall figures, the ten analyzed rainfall s, each with a set duration of 1 h, showed a very weak distance-correlation dependence. Only the closest stations with a distance of about 1000 m, displayed a significant correlation around 0.8 to 0.9. The storm-intensity profiles showed a considerable scatter for point values. However, a general trend, displaying concave-upwards patterns in the first half of the duration and concave-downwards patterns in the second half of the
297 duration, was found. This trend was amplified when areal values were considered. From the intensity-duration curves it was found t h a t for short durations (5 min) the mean areal rainfall intensity for 10km 2 was only about 60% as compared to the point value. For 20 km 2, the mean areal intensity was only about 40% as compared to the point value. For longer durations (60 min) the same values were about 80% and 60% for 10 km 2 and 20 km 2, respectively. Storm-centered areal reduction factors showed to be the most significant for short durations. For durations of 5 and 10 min and a considered area of 20 km 2, the reduction in intensity was over 50%. For longer durations (60 min), considering the same area the reduction was over 30%. The results presented in this paper can be compared to similar studies in a humid climate. Compared to studies made in Sweden (Niemczynowicz and JSnsson, 1981) regarding the spatial and temporal variability of high-intensive rainstorms, great similarities can be observed. The rain-producing cell size, defined as the 0.7 correlation isoline, has about the same areal extension both in Sweden and in Tunisia. Also the areal reduction of point rainfall observations has, on an average, approximately the same values. However, the occurrence of multiple rain-producing cells seems to be more frequent in the Tunisian catchment, occasionally causing larger areal variability. This also leads to temporarily increasing reduction factors, especially for longer durations when they are influenced by incoming new raincells. For practical applications, the most common procedure is to process the rainfall series (usually one-point data) so that some kind of synthesis is obtained, such as for instance intensity-duration-frequency relationships. Then the different types of design storms with known return periods can be derived and used as input for runoff modelling. Design storms are usually assumed to be uniformly distributed over a catchment, regardless of the size of the catchment or its geographical orientation. When it comes to storm-centered areal reduction factors, the practical applications are not obvious. Storm-centered areal reduction factors only give a description of the areal properties of individual storms. They cannot tell a n y t h i n g about the rainfall frequency distribution for a given area. Such a description is not easily generalized for other storms, as the area with maximum precipitation varies from event to event and the location of this area cannot be predicted. However, comparison of storm-centered areal reduction factors from different locations is of great interest. Observed similarities in spatial and temporal variability of high-intensive rainstorms in Sweden and Tunisia have very interesting implications as to similarity of rainfall producing mechanisms in different climatic zones. The fixed area areal reduction factors, however, have been recognized as more useful for practical applications, being more statistical in character, as they represent the ratio between areal and point rainfall with the same return period. Storm-centered areal reduction factors can, however, be used for converting point estimates of
298 p r o b a b l e m a x i m u m r a i n f a l l to a r e a l e s t i m a t e s w h e n no f r e q u e n c y e s t i m a t e s are involved. T h e i n v e s t i g a t i o n , s u m m a r i z e d i n t h i s p a p e r , is o n l y r e p r e s e n t a t i v e i n t h e case of small c a t c h m e n t s in the p r e v a i l i n g climate type. B e c a u s e of the arbit r a r y c r i t e r i o n for s e l e c t i o n o f t h e r a i n s t o r m e v e n t s , t h e r e s u l t s c a n o n l y be g e n e r a l i z e d for s m a l l , l o c a l i z e d s t o r m s . I t is, h o w e v e r , t h e o p i n i o n of t h e a u t h o r s t h a t this type of r a i n s t o r m s has the most s i g n i f i c a n t i n f l u e n c e on r u n o f f in urban areas. REFERENCES Bell, F., 1976. The areal reduction factor in rainfall frequency estimation. Institute of Wallingford, Wallingford, Rep. No. 35. Hershfield, D., 1962. Extreme rainfall relationships. J. Hydraul. Div. Proc. Am. Soc. Civ. Eng., HY6: 73~79. Hershfield, D., 1984. Some statistical properties of short-duration rainfall. In: P. Harremo~s (Editor), Rainfall as the Basis for Urban Runoff Design and Analysis. Proceedings of a Specialised Seminar held in Copenhagen, Denmark, 24~26 August, 1983. Pergamon Press, Oxford. Nguyen, V., 1984. Frequency analysis of point-to-area rainfall ratios. In: Proc. Third Int. Conference on Urban Storm Drainage, GSteborg, Sweden, June 4-8, 1984, Vol. 1, Analysis and Design of Stormwater Systems. GSteborg. Niemczynowicz, J., 1984. An investigation of the areal and dynamic properties of rainfall and its influence on runoff generating processes. Dissertation, Department of Water Resources Engineering, Lund Institute of Technology, University of Lund, Lund, Rep. 1005. Niemczynowicz, J. and JSnsson, O., 1981. Extreme rainfall events in Lund 1979-80. Nord. Hydrol., 12: 129-142. Rao, R. and Chechayya, B., 1975. Comparative analysis of short time increment urban precipitation characteristics. Proc. National Symposium on Precipitation Analysis for Hydrologic Modeling, June 2~28, 1975. AGU, Davis, Calif. Rodriguez-Iturbe, I. and Mejia, J., 1974. On the transformation of point rainfall to areal rainfall. Water Resour. Res., 10(4): 729-735. Salas, J., Delleur, J., Yevj evich, V. and Lane, W., 1980. Applied modeling of hydrologic time series. Water Resources Publications, Fort Collins, Colo. Schumann, D., 1983. Zur Fl~iche-HShe-Beziehung einzelner Regenf~ille. Z. Meteorol., 33(3): 179~183. Sharon, D., 1974. The spatial pattern of convective rainfall in Sukumaland, Tanzania: A statistical analysis. Arch. Meteorol. Geophys. Bioklimatol. Ser. B, 22: 201-218. Simeonidis, A., 1984. Precision in estimating mean areal precipitation for small watersheds. Dissertation, Department of Physical Geography, Uppsala University, Uppsala, UNGI Report No. 60. Sumner, G., 1984. Characteristics of storms over urban areas in East Africa, In: P. Harremo~s (Editor), Rainfall as the Basis for Urban Runoff Design and Analysis. Proceedings of a specialised seminar held in Copenhagen, Denmark, 24-26 August 1983. Pergamon Press, Oxford. Tase, N., 1976. Area-deficit-intensitycharacteristics of droughts. Colorado State University, Fort Collins, Color., Hydrol. Pap. 87. Thirriot, C., Mallel, K. and Triki, M., 1981. Fonction de r~partition des averses en Tunisie. Houille Blanche, 7/8: 541-548.
285
[1] S P A T I A L A N D T E M P O R A L CHARACTERISTICS OF H I G H - I N T E N S I V E R A I N F A L L IN N O R T H E R N T U N I S I A
RONNY BERNDTSSON and JANUSZ NIEMCZYNOWICZ
Department of Water Resources Engineering, Lund Institute of Science and Technology, University of Lund, Lund (Sweden) (Received February 25, 1986; accepted for publication May 20, 1986)
ABSTRACT Berndtsson, R. and Niemczynowicz, J., 1986. Spatial and temporal characteristics of high-intensive rainfall in n o r t h e r n Tunisia. J. Hydrol., 87: 285-298. When dealing with u r b a n design, it is essential to have knowledge of the magnitude of the spatial variability of short high-intensive rainstorms. Most of the studies of high-intensive rainfall, conducted during recent years, concern regions with a humid climate. Still, the largest temporal and spatial variations in rainfall are to be found in arid and semi-arid regions. This paper sums up findings regarding rainfall variability, observed in a small catchment in n o r t h e r n Tunisia during a period of 2 yr. Point and areal intensities are presented for the ten most high-intensive storms observed. Storm-centered areal reduction factors are calculated for different durations and areas.
INTRODUCTION
In all practical rainfall-runoff calculations, hydrologists have to estimate the areal rainfall from point measurements. This can be done with different methods ranging from simple arithmetic averaging to sophisticated computerized interpolation and extrapolation techniques. However, as regards localized rainfall of the convective type, the number of rain-gauges is usually insufficient for making reliable areal estimates. In this context it is important to know the magnitude of the spatial variation in rainfall during different time periods. In connection with urban design problems, the spatial variability during short-time periods becomes interesting. Several studies of point-to-area rainfall relationships have been made during the last few years (see RodriguezIturbe and Mejia, 1974; Rao and Chenchayya, 1975; Bell, 1976; Schumann, 1983; Nguyen, 1984; Niemczynowicz, 1984). All these studies deal with humid climate regions. However, the largest temporal and spatial variations in rainfall are to be found in arid and semi-arid regions. The short-term variability of high-intensive rains in these climates has not been paid much attention so far. The present paper addresses this aspect. It sums up findings regarding rainfall variability observed in a small catchment in northern Tunisia during a period of 2 yr. Point and areal intensities are presented for the ten most high-intensive
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286
MAIN WATER COURSE MAIN CONOUITS . . . . ..........
SUBCATCHMENT 01V~C)ER DIVIDINGLINE FOR URBAN
Fig. 1. Catchment topography and instrumentation.
288 CATCHMENT BOROER
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Fig. 2. Rainfall distribution for the rainy period in 1982/83 compared to a separate high-intensive rainstorm.
289
the altitude of the stations is decisive for the rainfall depth. Short highintensive rainstorms often have an areal distribution different from the common daily or monthly pattern, as can be seen in Fig. 2. During the rainy period in 1982-83, about 230 to 460 mm were recorded in the seven rain-gauges. A marked gradient in rainfall from east to west was observed, as shown in Fig. 2. For the high-intensive rainstorm recorded during 1 h on December 5, 1982, shown in the figure for purposes of comparison, the recorded rainfall varied between about 1 and 21 mm. The local character of high-intensive storms can be shown by use of correlation techniques. A common method for studying rainfall variability and for transforming point rainfall into areal rainfall is to use spatial correlation structures (see for example Rodriguez-Iturbe and Mejia, 1974; Niemczynowicz and JSnsson, 1981; Simeonidis, 1984). The dependence structure among time series is determined by the lag-k cross-correlation between two series (Salas et al., 1980): N-k -
r~ =
N
g-k -
t=l
(xt+k-
t=, "
11/2
(1)
(x,+k
t=l
where r~ = lag-k cross-correlation coefficient; x~i), x~)= time series; ~i) = mean of the first N - k values of series i; xt+k -0) = mean of the last N - k values of series j. Equation (1) was used in order to compare the variability of rainfall within the catchment during different time periods. Figure 3 gives an example of this method for the rainy season in 1982-83 on a daily and monthly basis. As a comparison, hourly values of the ten most intensive rainfalls observed during the 2 yr are also shown. In the figure, the lag-zero cross-correlation for monthly and daily rainfall displays great similarities. The correlation between stations gradually decreases with distance. A high correlation is maintained up to about 2000-3000 m. The 1-h storms, however, display a very low correlation between stations. Only gauges very close to each other show any significant dependence. The correlation structure is usually not isotropic, as shown by Sharon (1974) and Tase (1976). By taking the orientation of the line connecting two stations into consideration, the cellular nature of high-intensive rainfall can be shown. In Fig. 4 a comparison of spatial correlation, considering the interstation distance and the orientation for monthly rainfall and the ten 1-h rainstorms, respectively, is shown. It can be seen t h a t the ten rainstorms occur as cellular correlation isolines. The typical cell size, as defined by Niemczynowicz and JSnsson (1981) as being the area within the 0.7 correlation isoline, seems to be about 6 to 7 km ~ for a 1-h duration. The monthly correlations, on the other hand, display no cellular patterns. Instead the isolines seem to run parallel to a line in NE-SW direction, t h a t is normal to the isohyets. This is, however, not a very clear observation. Because of the orientation of the rain-gauges, which
290 correlation coeff monthly v a l u e l
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Fig. 3. Lag-zero cross-correlation coefficient for different periods versus the interstation distance.
is mainly in a NW-SE direction, the diagramme gives information mainly for this direction. VARIABILITY OF SHORT-TERM RAINSTORMS
The storm intensity profiles are interesting from an urban hydrological point of view when deciding the general shape of design storms. The relationship between the proportional rainfall depth and the rainfall duration for the ten rainstorms is shown in Fig. 5. The proportional rainfall depth is defined as the rainfall depth at a given duration divided by the total rainfall depth. The figure gives a comparison between point and areal values. The areal values were calculated by use of Thiessen polygons, expanding outwards from the gauge with maximum intensity. The point intensity profiles show a con-
291 distance
O06?~~j~0
north
~/ (kin) ~-5.0
distance
I/
corr coeft
monthly values
w.,Oo: t i
north cart
hourlycoeffvalues
o.
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.... t~ o.g Fig. 4. Correlation fields for monthly rainfall and ten 1-h rainstorms. siderable scatter. Both rainfalls with more and less rain in the first and second h alf of the dur a t i on period are represented. However, a general trend, which is amplified in the areal intensity profiles, can be seen. Typical mass curves seem to be characteristically concave-upwards in the first half of the duration and concave-downwards in the second ha l f of the duration. This supports findings of East African storm characteristics which were shown to have this general p atter n by Sumner (1984). It is interesting to see t h a t the largest scatter in the intensity profiles for the point values seem to disappear when an areal value of 10 km 2 is considered. The scatter from 10 to 20 km 2is not changed considerably. The rainstorms were f ur t her analyzed as regards i n t e n s i t y - d u r a t i o n relationships. Figure 6 shows point and areal i n t e n s i t y ~ l u r a t i o n curves for the ten rainstorms. It can be seen t h a t in the case of the recorded storms the 5-min period of maximum intensity varied between about 0.5 and 2.0 mm min 1 (point values). Areal intensity (10km ~ and 20km 2) was calculated by use of Thiessen polygons, expanding outwards from the gauge with maximum point intensity. As a comparison, the theoretical point i n t e n s i t y - d u r a t i o n curves with a r e t u r n period of 1, 2 and 5 yr for Tunis according to T hi rri ot et al. (1981) are shown in the figure. It can be seen t h a t for short durations (5 rain) one of the recorded storms (No. 7) seems to represent a rainfall with a 5-yr r e t u r n period, anot her storm (No. 8) seems to represent a rainfall with a 2-yr r e t u r n period and several others (Nos. 3, 5, 6, 9) represent rainfalls with a yearly r e t u r n period. For longer durations (60min) several recorded storms (Nos. 5, 7, 8) seem to represent rainfalls with a 5-yr r e t u r n period. There may be two explanations of the fact t h a t high-intensive rainstorms were over-represented during the r a t h e r short observation period (2-yr). It may have been an unusual year from a hydrological point of view or the theoretical i n t e n s i t y - d u r a t i o n curves may be based on too low values. Irrespective of which explanation is adopted, it must be borne in mind t h a t the annual variations in this region are extreme and t hat observa-
292 proportional rainfall depth
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Fig. 5. Point and areal storm intensity profiles for ten rainstorms (the figures 1, 2..... 10 refer to rainstorm number).
tions from short periods may not be directly comparable to statistical r e t u r n periods based on longer records. From Fig. 6 and Table 1 it can be seen t h a t rainfall intensity is quickly reduced when a larger area and a longer durat i on are considered. For 5-min durations the mean areal rainfall intensity for 10 km 2 is only about 60~/o compared to the point value. For 20 km 2, the mean areal intensity is about 40% compared to the point value. For longer durations (60 min) the same values are about 80 and 60% for 10 and 20 km 2 respectively. From Fig. 6 and Table 1 it can also be noted t hat high variabilities in rainfall occur during the shortest time periods. Increasing duration and increasing area reduce the standard deviation rapidly. As can be seen from Fig. 6, rainfall intensity does not invariably decrease with increasing dur a t i on (for example rain No. 5). This is due to the occurrence of more t h a n one raincell or rainfall intensity maximum for the given duration. These local intensity maxima become more important the larger the area or the longer the dur a t i on considered, causing a local rise in the i n t e n s i t y ~ l u r a t i o n curves. A measure of the rainfall variability in time over an area can be provided by correlation analysis as described above. Hershfield (1984) showed t h a t the co r r elati on coefficient between simultaneous incremental rainfall depths for a given storm, as computed from all pairs of gauges in the network, decays rapidly with decreasing incremental value. Consequently, the rates of the incoming storm rainfall can var y greatly over a small area, but the cumulations during the storm period tend to equalize the storm totals. In order to show this for the ten rainstorms analyzed, the correlation coefficients for different rainfall increments were calculated and plotted against distance between the gauges. In order to avoid a false rise in correlation, increment events with no rainfall recorded at any of the stations were excluded. The result displayed a
293 rainfall intensitY(minim in) k 7~
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Fig. 6. Point and areal intensity~luration curves for ten rainstorms (the figures 1, 2..... 10 refer to rainstorm number). very wide scatter irrespective of increment period. However, a closer examination by use of correlation fields s h o w e d t h a t typical correlation isolines are elliptically drawn out. The elliptical character of the correlation isolines seems to be more p r o n o u n c e d for longer increments. The 0.7 correlation isoline for 5, 10 and 20-min increments represented an area of about 2, 7 and 10 km 2 respectively. This s h o w s t h a t correlation increases w i t h increasing increment period also for short h i g h - i n t e n s i v e storms, t h o u g h but w e a k l y and depending on direction.
294 TABLE 1 Statistical p r o p e r t i e s of rainfall i n t e n s i t y depending on d u r a t i o n and a r e a for ten r a i n s t o r m s (£ = mean, m = median, s = s t a n d a r d d e v i a t i o n ) ( m m h -1) A r e a (kin 2)
D u r a t i o n (min) 5
10
20
30
60
Point £ m s
0.99 0.90 0.46
0.71 0.63 0.37
0.48 0.41 0.26
0.38 0.32 0.19
0.24 0.21 0.12
10 ~ m s
0.62 0.52 0.33
0.48 0.41 0.28
0.35 0.29 0.17
0.30 0.25 0.14
0.19 0.18 0.08
20 £ m s
0.37 0.33 0.18
0.30 0.30 0.13
0.24 0.26 0.09
0.21 0.20 0.08
0.14 0.15 0.05
AREAL REDUCTION FACTORS
The spatial properties of rainfall can also be described by areal reduction factors. Storm-centered areal reduction factors for the ten rainstorms were calculated as described by Bell (1976) and Niemczynowicz and JSnsson (1981). Storm-centered areal reduction factors are mainly used for converting point estimates of probable maximum precipitation to areal estimates when no frequency estimates are involved (Hershfield, 1962). Storm-centered areal reduction factors are defined as the ratio between the maximum areal rainfall within the storm zone for the given area and duration, and the maximum point rainfall within the same storm for the same duration. Figure 7 shows the stormcentered areal reduction factors for the ten rainstorms depending on area and duration. For each rainstorm, calculations started from the area represented by the gauge with the highest rainfall intensity. The areas represented by gauges were determined by use of the Thiessen polygon method. It can be seen t h a t in most cases the areal reduction factor decreases rather rapidly with increasing area. However, for some rainstorms the reduction factor exceeds the value 1.0. As mentioned before this depends on the occurrence of different raincells within the area covered by the gauging network. Occurrence of different raincells is also indicated by temporarily increasing reduction factors, especially for longer durations when they are influenced by incoming new raincells. From Table 2 and Fig. 7 it can be seen t h a t on an average and for a given area, the reduction factor increases with increasing duration. This means t h a t when the reduction factor increases, the reduction in intensity becomes smaller (by definition of reduction factor). The areal reduction is the most significant for short durations. For durations of 5 and 10 min and a considered area of 20 km 2, the reduction in intensity is over 50%. For a duration of 60 min,
295 areal reduction L factor
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/2
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06 ~
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Fig. 7. Storm-centered areal reduction factors for ten rainstorms depending on duration (the figures 1, 2 ..... 10 refer to rainstorm number).
296 TABLE 2 Statistical p r o p e r t i e s of areal r e d u c t i o n factors depending on d u r a t i o n and a r e a for ten r a i n s t o r m s (~ = mean, m = median, s = s t a n d a r d deviation) A r e a (km 2)
D u r a t i o n (min) 5
10
20
30
60
5 ~ m s
0.83 0.84 0.10
0.85 0.84 0.08
0.88 0.85 0.07
0.90 0.90 0.05
0.94 0.95 0.08
10 ~ m s
0.64 0.67 0.14
0.69 0.69 0.17
0.76 0.72 0.15
0.80 0.80 0.11
0.87 0.86 0.15
20 2 m s
0.39 0.37 0.14
0.48 0.41 0.19
0.57 0.45 0.23
0.62 0.54 0.20
0.67 0.70 0.19
considering the same area, the reduction is over 30%. The standard deviation does not change significantly with increasing duration but instead with increasing area. The larger the area considered, the larger is the standard deviation. DISCUSSION AND CONCLUSION
Because of the larger variability of rainfall in arid and semi-arid areas a basic understanding of the storm development and intensity characteristics is important. Especially in an urban hydrological context it is essential to have some knowledge of short high-intensive rainstorms and their extension. In this respect, case studies of selected storms, as in this paper, may provide a greater understanding of the rainfall process. The ten most high-intensive rainstorms observed in a small catchment in northern Tunisia during the period 1981/82-1982/83 have been examined. In spite of the rather short observation period, high-intensive rainstorms seem to have been well represented. For short durations (5 min) one of the recorded storms represented a rainfall with a 5-yr return period. For longer durations several other recorded storms also represented rainfalls with a 5-yr return period. Using spatial correlation between rain-gauges combined by twos, it was shown t h a t compared to monthly and daily rainfall figures, the ten analyzed rainfall s, each with a set duration of 1 h, showed a very weak distance-correlation dependence. Only the closest stations with a distance of about 1000 m, displayed a significant correlation around 0.8 to 0.9. The storm-intensity profiles showed a considerable scatter for point values. However, a general trend, displaying concave-upwards patterns in the first half of the duration and concave-downwards patterns in the second half of the
297 duration, was found. This trend was amplified when areal values were considered. From the intensity-duration curves it was found t h a t for short durations (5 min) the mean areal rainfall intensity for 10km 2 was only about 60% as compared to the point value. For 20 km 2, the mean areal intensity was only about 40% as compared to the point value. For longer durations (60 min) the same values were about 80% and 60% for 10 km 2 and 20 km 2, respectively. Storm-centered areal reduction factors showed to be the most significant for short durations. For durations of 5 and 10 min and a considered area of 20 km 2, the reduction in intensity was over 50%. For longer durations (60 min), considering the same area the reduction was over 30%. The results presented in this paper can be compared to similar studies in a humid climate. Compared to studies made in Sweden (Niemczynowicz and JSnsson, 1981) regarding the spatial and temporal variability of high-intensive rainstorms, great similarities can be observed. The rain-producing cell size, defined as the 0.7 correlation isoline, has about the same areal extension both in Sweden and in Tunisia. Also the areal reduction of point rainfall observations has, on an average, approximately the same values. However, the occurrence of multiple rain-producing cells seems to be more frequent in the Tunisian catchment, occasionally causing larger areal variability. This also leads to temporarily increasing reduction factors, especially for longer durations when they are influenced by incoming new raincells. For practical applications, the most common procedure is to process the rainfall series (usually one-point data) so that some kind of synthesis is obtained, such as for instance intensity-duration-frequency relationships. Then the different types of design storms with known return periods can be derived and used as input for runoff modelling. Design storms are usually assumed to be uniformly distributed over a catchment, regardless of the size of the catchment or its geographical orientation. When it comes to storm-centered areal reduction factors, the practical applications are not obvious. Storm-centered areal reduction factors only give a description of the areal properties of individual storms. They cannot tell a n y t h i n g about the rainfall frequency distribution for a given area. Such a description is not easily generalized for other storms, as the area with maximum precipitation varies from event to event and the location of this area cannot be predicted. However, comparison of storm-centered areal reduction factors from different locations is of great interest. Observed similarities in spatial and temporal variability of high-intensive rainstorms in Sweden and Tunisia have very interesting implications as to similarity of rainfall producing mechanisms in different climatic zones. The fixed area areal reduction factors, however, have been recognized as more useful for practical applications, being more statistical in character, as they represent the ratio between areal and point rainfall with the same return period. Storm-centered areal reduction factors can, however, be used for converting point estimates of
298 p r o b a b l e m a x i m u m r a i n f a l l to a r e a l e s t i m a t e s w h e n no f r e q u e n c y e s t i m a t e s are involved. T h e i n v e s t i g a t i o n , s u m m a r i z e d i n t h i s p a p e r , is o n l y r e p r e s e n t a t i v e i n t h e case of small c a t c h m e n t s in the p r e v a i l i n g climate type. B e c a u s e of the arbit r a r y c r i t e r i o n for s e l e c t i o n o f t h e r a i n s t o r m e v e n t s , t h e r e s u l t s c a n o n l y be g e n e r a l i z e d for s m a l l , l o c a l i z e d s t o r m s . I t is, h o w e v e r , t h e o p i n i o n of t h e a u t h o r s t h a t this type of r a i n s t o r m s has the most s i g n i f i c a n t i n f l u e n c e on r u n o f f in urban areas. REFERENCES Bell, F., 1976. The areal reduction factor in rainfall frequency estimation. Institute of Wallingford, Wallingford, Rep. No. 35. Hershfield, D., 1962. Extreme rainfall relationships. J. Hydraul. Div. Proc. Am. Soc. Civ. Eng., HY6: 73~79. Hershfield, D., 1984. Some statistical properties of short-duration rainfall. In: P. Harremo~s (Editor), Rainfall as the Basis for Urban Runoff Design and Analysis. Proceedings of a Specialised Seminar held in Copenhagen, Denmark, 24~26 August, 1983. Pergamon Press, Oxford. Nguyen, V., 1984. Frequency analysis of point-to-area rainfall ratios. In: Proc. Third Int. Conference on Urban Storm Drainage, GSteborg, Sweden, June 4-8, 1984, Vol. 1, Analysis and Design of Stormwater Systems. GSteborg. Niemczynowicz, J., 1984. An investigation of the areal and dynamic properties of rainfall and its influence on runoff generating processes. Dissertation, Department of Water Resources Engineering, Lund Institute of Technology, University of Lund, Lund, Rep. 1005. Niemczynowicz, J. and JSnsson, O., 1981. Extreme rainfall events in Lund 1979-80. Nord. Hydrol., 12: 129-142. Rao, R. and Chechayya, B., 1975. Comparative analysis of short time increment urban precipitation characteristics. Proc. National Symposium on Precipitation Analysis for Hydrologic Modeling, June 2~28, 1975. AGU, Davis, Calif. Rodriguez-Iturbe, I. and Mejia, J., 1974. On the transformation of point rainfall to areal rainfall. Water Resour. Res., 10(4): 729-735. Salas, J., Delleur, J., Yevj evich, V. and Lane, W., 1980. Applied modeling of hydrologic time series. Water Resources Publications, Fort Collins, Colo. Schumann, D., 1983. Zur Fl~iche-HShe-Beziehung einzelner Regenf~ille. Z. Meteorol., 33(3): 179~183. Sharon, D., 1974. The spatial pattern of convective rainfall in Sukumaland, Tanzania: A statistical analysis. Arch. Meteorol. Geophys. Bioklimatol. Ser. B, 22: 201-218. Simeonidis, A., 1984. Precision in estimating mean areal precipitation for small watersheds. Dissertation, Department of Physical Geography, Uppsala University, Uppsala, UNGI Report No. 60. Sumner, G., 1984. Characteristics of storms over urban areas in East Africa, In: P. Harremo~s (Editor), Rainfall as the Basis for Urban Runoff Design and Analysis. Proceedings of a specialised seminar held in Copenhagen, Denmark, 24-26 August 1983. Pergamon Press, Oxford. Tase, N., 1976. Area-deficit-intensitycharacteristics of droughts. Colorado State University, Fort Collins, Color., Hydrol. Pap. 87. Thirriot, C., Mallel, K. and Triki, M., 1981. Fonction de r~partition des averses en Tunisie. Houille Blanche, 7/8: 541-548.