chapt4_average and areal rainfal

cmutsvangwa: Hydrology, Dept. of Civil and Water Eng., NUSTt, 16/02/2006 15:28:02

ANALYSIS OF PRECIPITATION

Distribution of rainfall Point rainfall It is rainfall at single station, measured at that particular rain gauge. If it is missing, point rainfall can be estimated from data from surrounding stations (Viessman et al, 1989),. The stations should be located near to each other, or are in the same hydrological zone Estimation of rainfall from surrounding stations Example Find point rainfall at station A from point rainfalls recorded from surrounding stations, B, C, D, E and F (10mm; 50mm; 80mm; 90mm; 58mm and 60mm respectively and as shown in Fig. 1) Solution From weighted averages:

• Plot y-axis and x-axis running through point A (Fig.1) • Find all coordinates of stations B; C; D; E and F ( yx ∆∆ ; ) • 222 yxD ∆+∆=

• 21

DW =

• Point rainfall at A, (PA) ( )∑

∑ ×=

WWP

PA

• P are point rainfalls recorded at surrounding stations

• ( )

mmW

WPPA 68

07.058.4

==×

=∑

∑

Table 1 Station Rainfall

(mm) x∆

y∆

2x∆

2y∆

22 yx ∆+∆

W PxW

A 50 0 0 0 0 0 0 0B 80 10 6 100 36 136 0.0074 0.37C 90 5 9 25 81 106 0.0094 0.75D 58 6 5 36 25 61 0.016 1.44E 60 4 6 16 36 52 0.019 1.10F 7 4 49 16 65 0.0153 0.92

∑ 338 ∑ 0.07 ∑ 4.58

Chapter 4: Average and areal rainfall

1


x

Cy

D B(x, y)

A

xE

F

Fig. 2: Coordinates of the stations Areal rainfall For large catchment areas, greater than 50km2, rainfall is not distributed uniformly. It is necessary to convert point values of rainfall to give an average rainfall or the areal precipitation over a certain catchment area. The three main methods for deriving areal precipitation are:

• arithmetic mean • Thiessen polygon • Isohyetal

Arithmetic mean It is the sum of all items divided by the number of the items. The mean of yearly rainfall observed for a period of consecutive years, usually 30 years and above.

• simple • good estimate in flat areas if gauges are uniformly distributed • applicable when rainfall at different stations does not vary

nP

P iavg

∑=

∑ iP =sum of the rainfall at different gauge stations n=number of stations

Median: The ordinate corresponding to 50% of the years. Items are arranged in ascending order: If the number of items is odd, then the middle item gives the mean and if the number of items is even, the average of the central items gives the median.


2


Thiessen polygon

• it allows for non uniform distribution of rainfall by producing a weighting factor for each gauge

• very accurate that the arithmetic mean • not suitable for mountainous areas because of the orographic influences

or for intense local storms Procedure

• stations are plotted on a map • stations are joined together by straight lines • perpendicular bisectors are drawn to the straight lines joining adjacent

stations to form polygons (Thiessen polygon). • gauges have to be properly located over the catchment area to get regular

shaped polygons • a new Thiessen diagram has to be constructed every time there is a change

in the rain gauge network • It is assumed that the area enclosed within the polygon has the same

amount of rainfall as the enclosed gauge. Each polygon is assumed to be influenced by the rain gauge station inside it.

• The areal average depth of rainfall of the entire catchment area is given by:

( )n

nn

i

iiavg AAA

PAPAPAAPA

P+++

+++== ∑

∑∑

..........

21

2211

The Thiessen coefficient is given as:

∑=

i

i

AAefficientThiessenCo

e.g. for rainfall at station 1, (A1) mentAreaTotalCatchAefficientThiessenCo 1=

Once these coefficients have been determined for a stable rain gauge network, the areal rainfall is very quickly computed for any set of rainfall measurements, and also to estimate missing data for one rain gauge station without redrawing the polygon. An example of the Thiessen method is shown in Fig 3(a). Isohyetal method The most accurate of all the three methods and is applicable when rainfall is controlled by relief. Procedure:

• Point rainfalls are plotted in a map


3


• lines of equal rainfall are drawn (isohyets) • measure area enclosed between successive isohyets with a planimeter • multiply each of these areas by the average rainfall between the isohyets

∑∑ ⎟

⎠⎞

⎜⎝⎛ +

×=

i

i

avg A

PPAP 2

21

Fig. 3: Examples of the Thiessen and Isohyetal methods

References 1. Viessman J.R., Lewis G. L., and Knapp J.W., (1989), Introduction to

hydrology, Harper Collins, USA


4

chapt4_average and areal rainfal

Documents