A technique for the determination of areal average rainfall - IAHS

Hydrological Sciences-Bulletin-des Sciences Hydrologiques, 23, 4, 12/1978
A technique for the determination
of areal average rainfall
PANDE, B. B. LAL and G. AL-MASHIDANI College of Engineering,
University of Mosul, Mosul, Iraq
Received 6 February 1978, revised 7 August 1978
Abstract. A method was developed for allocating an area likely to be shared by a raingauge station.
Average areal precipitation was found by weighting the rainfall at the gauging station in terms of
the product of distances enclosing it or radiating from it. The method was applied to two real and
two hypothetical basins and the results were compared with those obtained by existing techniques.
Une méthode pour la détermination de précipitation moyenne pour une zone donnée
Résumé. On a développé une méthode d'allouer une zone où on aménagera probablement une
station de jaugeage. On a trouvé la précipitation moyenne pour la zone en calculant la précipitation
à la station de jaugeage en fonction du produit des distances qui l'enferment ou qui y partent. On
a appliqué la méthode à deux bassins réels et à deux bassins hypothétiques et on a comparé les
résultats avec ceux qu'on a obtenus en utilisant des techniques actuelles.
INTRODUCTION
The usual techniques of determining average depth of rainfall over an area, viz. arithmetic
mean, Thiessen polygon and isohyetal method, are well known. Some new
methods have also been proposed in the literature. Akin (1971) developed the idea
of dividing an area into triangles and wrote computer programs for the necessary
calculations. Shaw & Lynn (1972) applied mathematical surface fitting techniques.
They used bi-cubic spline and multiquadric analysis techniques for computing average
rainfall. Hutchinson & Walley (1972) applied finite element techniques. Their method
also took into account an accurate representation of the shape and relief of the basin.
Bethlahmy (1976) developed a two-axis method. The method is simple and fast but
has not been applied to a real basin. Furthermore there is a considerable bias involved
in determining the two axes. Another serious limitation of the methodology is that
a particular raingauge, however far outside the boundary of the basin, will still have
some kind of effect in deciding average precipitation over the basin. Many such other
methods have been developed and the existing ones computerized. Somehow even
with all these developments, a practising hydrologist still depends upon Thiessen polygons.
The present paper suggests a method which is similar to the Thiessen method
but easier to apply.
0303-6936/78/1200-0445$02.00 © 1978 Blackwell Scientific Publications 445

446 Pande, B. B. Lai and G. Al-Mashidani
PROPOSED METHOD
The proposed method for determining areal average precipitation is based on three
assumptions which are similar to those of Thiessen. (1) A raingauge is significant only
up to the mid-distance towards another raingauge. (2) A raingauge near the centre of
the area is more significant than one far removed from the centre. Those outside the
basin are not equally significant with respect to average rainfall over the area. (3) The
significance of the raingauge is affected by the shape of the basin.
With reference to Figs 1 and 2 the proposed method is illustrated in a stepwise
manner as follows.
(1) Join the adjoining raingauge stations by straight lines and establish the mid
points of these lines.
(2) (a) If no central station is available, as in Fig. 1 establish the approximate
FIG. 1. No central station is available
geometric centre of the area by estimation, (b) Join the established centre point with
points established in Step 1 and extend the resulting lines to the boundary of the basin.
(3) (a) If a central station is available as in Fig. 2 join it to the various stations
lying around it by straight lines and mark the mid points of the resulting lines, (b) Join
these mid points so as to form a polygon around the central station, (c) Draw lines
from the mid points of the sides of the polygon of Step 3b to the points established
in Step 1 and extend these lines to the boundary of the basin.
°5
FIG. 2. Central station is available

Determination ofareal average rainfall 447
(4) Each of the raingauge stations is now enclosed by an area, the weight of which
can be calculated as follows, (a) The central station weight is calculated as half the sum
of the product of adjacent radial lines from the central station up to the enclosing
polygon or up to the basin boundary whichever is less, (b) Weights for outer stations
are obtained by the product of the length of adjacent lines, e.g. for station 2 of Fig. 1
the weight is computed as Oa2 X Oa3 and for station 2 of Fig. 2, the weight is computed
as 02a2 X 03a3.
The average precipitation of the basin is then calculated by summing the weighted
precipitation and dividing this by the sum of weights. Thus for Fig. 1 with no central
station:
n —1
S Pi w i + Pn Wn
' =1
where n — number of stations
Pi — value of precipitation at station i
wt = weight for station i = Oa{ X Oai+1
wn — weight for the nth station
= Oan X Oa1
and for Fig. 2 with a central station
V 1
Pavg
W, + W„
j=l L PiWi+Pn^n+Pc
£ Wi + wn+wc
i=i
where p = precipitation at station / or n or c according to the subscript
wt = weight at station i
= OpiX Oi+1ai+1
wn = weight at «th station
= Onan X Oxax
n — total number of outer stations (five in the present case)
wc = weight at the central station
= Vz "£" (pbj X cb]+1) + (cbn X cbx)
APPLICATION OF THE METHOD
The method has been applied to some real and hypothetical basins. Schulz (1973)
considers the basin of Kings River above Piedra, California (Fig. 3). Punmia et al.
(1969) consider the Canvery River basin in India (Fig. 4). The present technique was
applied to these basins and the results were compared with the arithmetic mean,
the Thiessen and the isohyetal methods as shown in Table 1. Details of computations
are shown in Tables 3 and 4.
Nemec (1973) considers a hypothetical basin with all raingauges lying beyond its
boundary as shown in Fig. 5. Bethlahmy (1976) also considers a hypothetical basin

448 Pande, B. B. Led and G. Al-Mashidani
Name of
basin
G (2495)
L (37.41 )

Name of
basin
Nemec's basin
Bethlahmy's basin
Storm 1
Storm 2
Storm 3
Storm 4
Storm 5
Determination of area! average rainfall
61 5mm
FIG. 5. Nemec's hypothetical basin.
FIG. 6. Bethlahmy's hypothetical basin.
TABLE 2. Comparison of results for two hypothetical basins
Arithmetic
mean
(cm)
66.8
2.00
2.00
2.00
2.00
2.00
Thiessen
method
(cm)
66.0
2.86
2.01
155
1,76
1.43
Isohyetal
method
(cm)
2.51
1.84
1.80
1.76
1.46
Two-axis
method
(cm)
2.38
2.14
1.81
1.87
1.80
Present
method
(cm)
66.9
2.50
1.90
1.65
1.62
2.04

450 Pande, B. B. Lai and G. Al-Mashidani
Name of
station
A
B
C
D
E
F
G
M
J
K
L
H
I
TABLE 3. Details of the calculations for the basin of Kings River above Piedra (Fig. 3)
Total
Rainfall
(in)
8.67
17.81
18.25
17.67
21.74
23.71
24.95
15.05
33.19
35.53
37.41
34.28
29.29
Details of w;
0.2 X 2.2
2.2 X 1.15
1.15 X 1.75
Vi (Sum of the product of
radial distances from D)
1.2 X 0.5 + 2.05 X 1.55
3.25 X 1.2 + 2.1 X 2.05
1.75 X 3.25
0.2 X 0.65
2.45 X 1.55
0.85 X 2.45
0.85 X 0.6
0.6 X 2.1
VÏ (Sum of the product of
radial distances from I)
Wi
0.44
2.53
2.0125
14.55
3.7775
8.205
5.6875
0.13
3.7975
2.0825
0.51
1.26
14.21
59.19375
Pi w i
3.814
45.059
36.7281
257.01
82.1228
194.54
141.9031
1.9565
126.039
73.99
19.0791
43.1928
416.3939
1441.8286
Sum of the product of radial distances from D
= 1.4 X 2.4 + 1.4 X 1.2 + 1.2 X 1.6 + 1.6 X 1.7 + 1.7 X 3.0 + 3.0 X 2.65 + 2.65 X 2.4
Sum of the product of radial distances from I
= 1.8 X 2.35 + 2.35 X 1.75 + 1.75 X 2.3 + 2.3 X 2.55 + 2.55 X 2.2 + 2.2 X 1.8
ZPiWj 1441.8286
Pave = = • = 24.36 inches
2 w< 59.19375
(Fig. 6) and applied his method to five hypothetical storms over it. The present technique
has also been applied to these problems and the results are compared in Table
2. Details of computations are given in Tables 5 and 6.
DISCUSSION
A study of Tables 1 and 2 shows that in general the results obtained by the present
method are close to those obtained using the isohyetal method. As such one can
presume that the method gives reasonable results. Besides, since the actual determination
of a particular area is in terms of the products of distances the present method is
faster than the Thiessen method and can be computerized more easily.
The method works when more than two stations are present in the locality. If,
however, only two stations are present then the method fails. Under such circumstances
it is suggested that a third hypothetical station may be located in the area so as to
form an equilateral triangle with the existing stations and having a rainfall value equal
to the arithmetic average of the existing two. With three stations in the area the
method can be applied in the usual manner.

Station
1
2
3
4
5
6
7
8
9
10
11
12
Determination ofareal average rainfall 451
TABLE 4. Details of the calculations for the basin of the Canveiy River (Fig. 4)
Total
Rainfall
(cm)
58
63
71
69
86
81
84
56
53
69
61
79
Details of w,-
2.3 X 1.3
1.3 X 0.2
0.2 X 2.2
2.2 X 2.6
2.6 X 1.7
1.7 X 1.2
1.2 X 1.8 + 3.25 X 1.75
3.25 X 1.9
1.9 X 1.4
1.8 X 2.7 + 1.4 X 1.75
2.3 X 2.7
Vi (Sum of product of radial
distance from Station 12)
Sum of the product of radial distance from Station 12
Station
Wf
2.99
0.26
0.44
5.72
4.42
2.04
7.848
6.175
2.66
7.31
6.21
14.106
60.1785
Pi w i
173.42
16.38
31.24
394.68
380.12
165.24
659.19
345.80
140.98
504.39
378.81
1114.4184
4304.668
= 2.0 X 1.225 + 1.225 X 2.075 + 2.075 X 1.575 + 1.575 X 1.6 + 1.6 X 1.775 + 1.775 X
X 1.7 + 1.7 X 2.075 + 2.075 X 1.975 + 1.975 X 2.0
XPiWf 4304.668
S Wl 60.1785
= 71.53 mm
TABLE 5. Details of the calculations for Nemec's hypothetical basin (Fig. 5)
Total
CONCLUSION
Rainfall
(mm) Details of w,- Pi w i
615
675
650
720
690
S Pi wt
1.1 X 2.05
2.05 X 1.1
1.1 X 1.1
1.1 X 1.75
1.75 X 1.1
6409.7
2 Wi 9.57
= 669.77 mm
2.255
2.255
1.21
1.925
1.925
9.57
1386.825
1522.125
786.50
1386.00
1328.25
6409.7
The present method of determining average rainfall is based on reasonable assumptions.
If the weight to be given to a particular station is estimated in terms of the product
of distances instead of actually measuring the area, the method becomes faster, and
gives reasonable results.

Pande, B. B. Lai and G. Al-Mashidani
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REFERENCES
Determination ofareal average rainfall 453
Akin, I.E. (1971) Calculation of the mean areal depth of precipitation./. Hydrol. 12,363-376.
Bethlahmy, N. (1976) The two axis method: a new method to calculate average precipitation
over a basin. Hydrol. Sci. Bull. 21, No. 3, 379-385.
Hutchinson, P. & Walley, W.J. (1972) Calculation of areal rainfall using finite element techniques
with altitudinal corrections. Hydrol. Sci. Butt. 17, No. 3,259-272.
Nemec, J. (1973) Engineering Hydrology: T.M.H. Edition. Tata McGraw-Hill Publishing Company,
New Delhi.
Punmia, B.C., Pande & Lai, B.B. (1969) Irrigation and Water Power Engineering. M/s Standard
Publishers & Distributors, 1705 B Nai Sarak, New Delhi, 6.
Schulz, E.F. (1973) Problems in Applied Hydrology. Water Resources Publications, P.O. Box 303,
Fort Collins, Colorado, U.S.A.
Shaw, E.M. & Lynn, P.P. (1972) Areal rainfall evaluation using two surface fitting techniques.
Hydrol. Sci. Bull. 17, No. 4,419^»33.