---
title: "Optimum number of rain gauges"
subject: "Engineering Hydrology"
chapter: 2
key: "semester_5/engineering_hydrology:optimum-raingauge-number"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/optimum-raingauge-number
site: BCT Engineering
---

# Optimum number of rain gauges

Engineering Hydrology · Chapter 2: Precipitation

- Asked in: 2069 Chaitra, 6 marks
- Video: [Optimum number of rain gauges || Hydrology #civilengineering #hydrology](https://www.youtube.com/watch?v=POAAn-vg7hs) — CIVIL HUT
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/optimum-raingauge-number
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-2

## Question

How would you determine the optimum number of rain gauges to be installed in a given catchment?

## Short answer

Compute mean, standard deviation, and coefficient of variation Cv of annual rainfall from existing gauges. With permissible error ε%, optimum number N = (Cv/ε)². If N exceeds current count, add N−n gauges, considering spatial and elevation coverage. WMO density guidelines also apply.

## Answer

The optimum density of rain gauges balances accuracy and cost. The number needed to estimate the mean areal rainfall within a permissible error is found statistically.

#### Procedure
1. Use the existing gauges ($n$ stations) and their mean annual (or storm) rainfall values $P_1, \dots, P_n$.
2. Compute the mean $\bar P$ and the standard deviation

$$
\sigma_{n-1} = \sqrt{\frac{\sum (P_i - \bar P)^2}{n-1}}
$$

3. Compute the coefficient of variation $C_v = \dfrac{100\,\sigma_{n-1}}{\bar P}$.
4. Choose the permissible error $\varepsilon$ in the mean rainfall (usually 10%, or 5% for precise work).
5. The optimum number of gauges is

$$
N = \left(\frac{C_v}{\varepsilon}\right)^2
$$

6. If $N > n$, additional gauges $= N - n$ are installed (placed to cover under-represented areas and elevation zones).

#### Also considered
- WMO recommended minimum densities: flat regions of temperate, Mediterranean and tropical zones 1 gauge per 600-900 km²; mountainous regions of these zones 1 per 100-250 km²; small mountainous islands with irregular rainfall 1 per 25 km²; arid and polar zones 1 per 1500-10000 km² (with some relaxation allowed).
- Cost, accessibility of sites and purpose of data (design vs research).
- Because $N$ depends on $C_v$, the process may be repeated as new data come in.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the statistical method determine?
   A. Best gauge type
   B. Gauge height
   C. Storm duration
   D. Number of gauges for the mean rainfall within a permissible error
   Answer: D. N = (Cv/e)^2. (Key point: Purpose)

2. How is Cv defined?
   A. 100 sigma(n-1) / mean
   B. mean / sigma
   C. sigma^2 / mean
   D. 100 mean / sigma
   Answer: A. Cv = 100 sigma/ P-bar. (Key point: Cv definition)

3. When are additional gauges installed?
   A. If N < n
   B. If N > n, add N - n
   C. Always add N
   D. Never
   Answer: B. Additional = N - n, placed in under-represented areas. (Key point: Additional gauges)

4. What error is usually chosen?
   A. 50 %
   B. 0 %
   C. 10 % (5 % for precise work)
   D. 1 %
   Answer: C. Smaller error needs more gauges since N is proportional to 1/e^2. (Key point: Permissible error)

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