Optimum number of rain gauges

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.