Techniques for average depth of rainfall

The average depth of rainfall over a catchment is needed because rain gauges record only point values, while runoff and hydropower flow depend on rainfall over the whole area. The three standard techniques are given below.

Methods of finding mean precipitation over an area

1. Arithmetic mean method

$$\bar{P} = \frac{P_1 + P_2 + \dots + P_N}{N}$$

Simple; good only for flat areas with uniformly spread gauges and small rainfall variation.

2. Thiessen polygon method

• Plot stations on the map and join adjacent stations by straight lines to form triangles.
• Draw perpendicular bisectors of these lines; they form a polygon around each station.
• Measure the polygon area $A_i$ inside the catchment for each station.

$$\bar{P} = \frac{\sum P_i A_i}{\sum A_i}$$

Accounts for uneven gauge spacing; can use stations just outside the catchment. Does not consider orography; polygons must be redrawn if a gauge is added or removed.

3. Isohyetal method

• Draw isohyets (lines of equal rainfall) by interpolating between station values.
• Measure the area $a_i$ between successive isohyets $P_i$ and $P_{i+1}$.

$$\bar{P} = \frac{\sum a_i \left(\frac{P_i + P_{i+1}}{2}\right)}{\sum a_i}$$

Most accurate, especially in hilly areas like Nepal, as the analyst can use knowledge of topography; but needs skill and a dense network.

  Isohyets (mm)          Thiessen polygons
   ___80___               +-----+-----+
  /  _60_  \              |  A  |  B  |
 |  / 40 \  |             |  *  |  *  |
  \ \____/ /              +--+--+--+--+
   \______/                  | C *  |
                             +------+

Example (Thiessen): stations with 80, 70, 90 mm over areas 100, 150, 80 km² give $\bar{P} = (8000 + 10500 + 7200)/330 = 77.9$ mm, while the arithmetic mean is 80 mm.