Contents preface (VII) introduction 1—37

2.3.3.1. Arithmetic Mean Method
This is the simplest method in which average depth of rainfall is obtained by obtaining the

sum of the depths of rainfall (say P1, P2, P3, P4	.... Pn) measured at stations 1, 2, 3, ..... n and
dividing the sum by the total number of stations i.e. n. Thus,
			P + P + P + ...... P	1	n
		∑
			n		n		n
P =	123	=					P	(2.2)
									i

i = 1
This method is suitable if the rain gauge stations are uniformly distributed over the entire area and the rainfall variation in the area is not large.
2.3.3.2. Theissen Polygon Method
The Theissen polygon method takes into account the non-uniform distribution of the gauges by assigning a weightage factor for each rain gauge. In this method, the enitre area is divided into number of triangular areas by joining adjacent rain gauge stations with straight lines, as shown in Fig. 2.7 (a and b). If a bisector is drawn on each of the lines joining adjacent rain gauge stations, there will be number of polygons and each polygon, within itself, will have only one rain gauge station. Assuming that rainfall P_i recorded at any station i is representative rainfall of the area A_i of the polygon i within which rain gauge station is located, the weighted
average depth of rainfall P for the given area is given as

			=	1 ^n	P A	(2.3)
	P
			A ∑
						i i
							i = 1
					n
where,			A = ∑ Ai	= A1 + A2 + A3 + ...... An

i = 1

HYDROLOGY

47

1

20.3
Here, ^A_Aⁱ is termed the weightage factor for ith rain gauge.

² 3 2 ₃

1. 
   _60.9
   

	10	11				10
7	140.6	4	7		11
60.0	8	154.0	54.7		4
84.0	9				9
				8
6		93.2				1	1
		6
45.6	5

	48.1	5
(a)		(b)

2

		11	150
7	8	120	10
		90	9
	60
6
30			5

(c)
3

4

Fig. 2.7 Areal averaging of precipitation (a) rain gauge network, (b) Theissen polygons (c) isohyets (Example 2.1)
This method is, obviously, better than the arithmetic mean method since it assigns some weightage to all rain gauge stations on area basis. Also, the rain gauge stations outside the catchment can also be used effectively. Once the weightage factors for all the rain gauge
stations are computed, the calculation of the average rainfall depth P is relatively easy for a given network of stations.

While drawing Theissen polygons, one should first join all the outermost raingauge stations. Thereafter, the remaining stations should be connected suitably to form quadrilaterals. The shorter diagonals of all these quadrilaterals are, then, drawn. The sides of all these triangles are, then bisected and, thus, Theissen polygons for all raingauge stations are obtained.

2.3.3.3. Isohyetal Method
An isohyet is a contour of equal rainfall. Knowing the depths of rainfall at each rain gauge station of an area and assuming linear variation of rainfall between any two adjacent stations, one can draw a smooth curve passing through all points indicating the same value of rainfall, Fig. 2.7 (c). The area between two adjacent isohyets is measured with the help of a planimeter.
The average depth of rainfall P for the entire area A is given as

			1		[Area between two adjacent isohyets]
P =	A ^Σ
× [mean of the two adjacent isohyet values]		(2.4)

Since this method considers actual spatial variation of rainfall, it is considered as the best method for computing average depth of rainfall.

Example 2.1 The average depth of annual precipitation as obtained at the rain gauge stations for a specified area are as shown in Fig. 2.7 (a). The values are in cms. Determine the average depth of annual precipitation using (i) the arithmetic mean method, (ii) Theissen polygon method, and (iii) isohyetal method.
Solution: (i) Arithmetic mean method :
Using Eq. (2.2), the average depth of annual precipitation,
1

P = ₁₁ [20.3 + 88.1 + 60.9 + 54.7 + 48.1 + 45.6 + 60.0 + 84.0 + 93.2 + 140.6 + 154.0]

1

= ₁₁ (849.5) = 77.23 cm.

48 IRRIGATION AND WATER RESOURCES ENGINEERING
(ii) Theissen polygons for the given problem have been shown in Fig. 2.7 (b). The compu-tations for the average depth of annual precipitation are shown in the following Table :

	Rainfall	Area of	Weightage
Rain Gauge	^Pi	Polygon, Ai		factor (%)		Ai
Station	(cm)	(km^2)		Ai	× 100	P
		ΣA
				ΣA	i
							i
					i
1	20.3	22	1.13		0.23
2	88.1	0	0		0
3	60.9	0	0		0
4	54.7	0	0		0
5	48.1	62	3.19		1.53
6	45.6	373	19.19		8.75
7	60.0	338	17.39	10.43
8	84.0	373	19.19	16.12
9	93.2	286	14.71	13.71
10	140.6	236	12.14	17.07
11	154.0	254	13.07	20.13
Total		1944	100.01	87.97

Average annual precipitation = ΣP_{i Σ}^A_Aⁱ_i = 87.97 cm

≈ 88 cm
(iii) Isohyetal method : Isohyets are shown in Fig. 2.7(c). The computations for the aver-age depth of annual precipitation are shown in the following Table :

Isohyets	Net area	Average Precipitation
(cm)	^Ai		^Pi		^Pi^Ai
	(km)^2		(cm)
< 30	96		25		2400
30–60	600		45		27000
60–90	610		75		45750
90–120	360		105		37800
120–150	238		135		32130
> 150	40		160		6400
Total	1944				151480
∴ Average annual precipitation for the basin =	151480	= 77.92 cm.
1944