GATE 2023 CE (CE2) – Question 62
A catchment may be idealized as a circle of radius 30 km. There are five rain gauges, one at the center of the catchment and four on the boundary (equi-spaced), as shown in the figure (not to scale). The annual rainfall recorded at these gauges in a particular year are given below.
| Gauge | G1 | G2 | G3 | G4 | G5 |
|---|---|---|---|---|---|
| Rainfall (mm) | 910 | 930 | 925 | 895 | 905 |
Using the Thiessen polygon method, what is the average rainfall (in mm, rounded off to two decimal places) over the catchment in that year?
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Correct answer: 908.00 to 917.12
Explanation
**Step 1: Thiessen areas.** The catchment is a circle of radius 30 km with gauge G1 at the centre and G2 to G5 on the boundary (north, east, south and west).
- The perpendicular bisector between the centre and each boundary gauge lies 15 km from the centre. These four lines bound the central polygon, a **square of side 30 km**: area $A_1=30\times30=900$ km².
- The total catchment area is $\pi(30)^2=2827.4$ km². The remaining area is shared equally by the four boundary gauges:
$$A_2=A_3=A_4=A_5=\frac{2827.4-900}{4}=481.9\text{ km}^2 .$$
**Step 2: weighted average.**
$$\bar P=\frac{910(900)+(930+925+895+905)(481.9)}{2827.4}$$
$$\bar P=\frac{819\,000+3655\times481.9}{2827.4}=\frac{819\,000+1\,761\,300}{2827.4}\approx\mathbf{912.56\ mm}.$$