---
title: "Techniques for average depth of rainfall"
subject: "Hydro Power"
chapter: 2
key: "semester_6/hydro_power:average-rainfall-techniques"
source: https://www.bctengineer.com/q/semester_6/hydro_power/average-rainfall-techniques
site: BCT Engineering
---

# Techniques for average depth of rainfall

Hydro Power · Chapter 2: Introduction to Hydrology

- Asked in: 2081 Asoj, 6 marks
- Video: [Methods of Finding Average Rainfall Data - Hydrology - Water Resource Engineering 1](https://www.youtube.com/watch?v=U8AAU19CgB0) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/average-rainfall-techniques
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-2

## Question

Describe various techniques used to determine the average depth of rainfall over a river catchment area.

## Short answer

Average rainfall depth is determined by arithmetic mean (sum divided by number of gauges), Thiessen polygon method (area-weighted average using polygon areas), and isohyetal method (averaging between rainfall contours, most accurate, especially in hilly regions).

## Answer

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.

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

| Method | Accuracy | Effort |
|---|---|---|
| Arithmetic mean | Low | Least |
| Thiessen polygon | Medium | Medium |
| Isohyetal | High | Most |

**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.

## Check yourself

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

1. Why must the average depth of rainfall over a catchment be calculated instead of using one gauge reading?
   A. Gauges record only point values, but runoff depends on rainfall over the whole area
   B. Gauges are always inaccurate
   C. Rain falls only at gauge stations
   D. Runoff depends only on the rainfall at the catchment outlet
   Answer: A. A rain gauge gives point rainfall, whereas flow depends on rainfall over the entire catchment. (Key point: Why areal rainfall is needed)

2. When is the arithmetic mean method suitable?
   A. Hilly catchments with large rainfall variation
   B. Flat areas with uniformly spread gauges and small rainfall variation
   C. Catchments with very few gauges unevenly placed
   D. Only when isohyets are available
   Answer: B. The simple average ignores spacing and terrain, so it works only for flat, evenly gauged areas with little variation. (Key point: Arithmetic mean suitability)

3. For gauges with 80, 70 and 90 mm over 100, 150 and 80 km2, the Thiessen mean is 77.9 mm, below the arithmetic mean of 80 mm. Why?
   A. Thiessen always gives a lower value
   B. The 70 mm station covers the largest area and so gets the most weight
   C. The 90 mm gauge is ignored
   D. The areas are added instead of weighted
   Answer: B. The 70 mm station has the largest polygon (150 km2), pulling the area-weighted mean below the simple average. (Key point: Thiessen example)

4. In the isohyetal method, how is the mean rainfall computed?
   A. Average of all gauge readings
   B. Each gauge reading multiplied by its polygon area
   C. Mean of adjacent isohyet values weighted by the area between them
   D. Highest isohyet value only
   Answer: C. P = sum[ai (Pi + Pi+1)/2] / sum(ai), using the area between successive isohyets. (Key point: Isohyetal formula)

## Related in Hydro Power

- [Thiessen polygon method with rainfall data](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Athiessen-polygon-explain-numerical) — Hydro Power, chapter 2
- [Firm and secondary energy from average monthly flows](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Afirm-secondary-energy-2075) — Hydro Power, chapter 3
- [River discharge by mid-section method](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Amid-section-discharge) — Hydro Power, chapter 2
- [Discharge from depth and mean velocity data](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Adischarge-depth-mean-velocity) — Hydro Power, chapter 2
- [Arithmetic mean and Thiessen average precipitation](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aarithmetic-thiessen-numerical) — Hydro Power, chapter 2
- [Slope area method of discharge](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aslope-area-method) — Hydro Power, chapter 2

## Related in other subjects

- [Methods of average rainfall over catchment](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aavg-rainfall-methods) — Engineering Hydrology, chapter 2, asked 3×
- [Average rainfall by isohyetal method, Bagmati basin](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aisohyetal-bagmati-442) — Engineering Hydrology, chapter 2
- [Thiessen polygon mean rainfall, pentagon catchment](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Athiessen-pentagon-pqrs) — Engineering Hydrology, chapter 2
- [Sewer for 200 ha catchment at 0.6 depth](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Asewer-runoff-0p6-depth-200ha) — Sanitary Engineering, chapter 3, asked 2×