Reference Evapotranspiration Formula (Penman-Monteith)
Calculate reference evapotranspiration (ET0) using the Penman-Monteith equation.
Essential for irrigation scheduling and crop water demand estimation.
The Formula (Penman-Monteith)
Reference Evapotranspiration (ET₀) is the rate at which water evaporates from a well-irrigated grass reference surface under given weather conditions. It represents the atmospheric demand for water — essentially, how thirsty the environment is on a given day. The Penman-Monteith equation, adopted by the FAO in 1998, is the world standard for calculating ET₀.
The formula combines energy balance (solar radiation) with aerodynamic effects (wind and humidity). It requires several meteorological inputs but gives highly accurate results for irrigation planning.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| ET₀ | Reference evapotranspiration | mm/day |
| Rₙ | Net radiation at the crop surface | MJ/m²/day |
| G | Soil heat flux density (often ≈ 0 for daily calculations) | MJ/m²/day |
| T | Mean daily air temperature at 2 m height | °C |
| u₂ | Wind speed at 2 m height | m/s |
| eₛ | Saturation vapor pressure | kPa |
| eₘ | Actual vapor pressure | kPa |
| eₛ−eₘ | Saturation vapor pressure deficit | kPa |
| Δ | Slope of the vapor pressure-temperature curve | kPa/°C |
| γ | Psychrometric constant (≈ 0.0665 kPa/°C at sea level) | kPa/°C |
Simplified Approximation (Hargreaves Method)
When full meteorological data is unavailable, the Hargreaves equation gives a good approximation using only temperature and extraterrestrial radiation (Rₘ, available from solar tables by latitude and day of year):
Example 1 — Hot Summer Day (California)
Central Valley, California in July: T = 25°C, high solar radiation (Rₙ = 22 MJ/m²/day), moderate wind (u₂ = 2.5 m/s), low humidity (vapor deficit ≈ 2.5 kPa).
High radiation + hot temperature + low humidity + moderate wind all increase ET₀
ET₀ ≈ 6–8 mm/day — a hot irrigation day requiring significant water application
Example 2 — Mild Spring Day
Temperate spring conditions: T = 15°C, moderate radiation (Rₙ = 12 MJ/m²/day), calm wind (u₂ = 1.0 m/s), moderate humidity (vapor deficit ≈ 0.8 kPa).
Lower temperature + less radiation + calm conditions reduce ET₀
ET₀ ≈ 2.5–3.5 mm/day — moderate irrigation demand
Reference ET₀ by Climate Zone
| Climate Zone | Season | ET₀ (mm/day) |
|---|---|---|
| Humid tropical | Year-round | 3.5–5.0 |
| Semi-arid (Mediterranean) | Summer | 6.0–9.0 |
| Semi-arid (Mediterranean) | Winter | 1.0–2.5 |
| Arid (desert) | Summer | 8.0–12.0 |
| Temperate continental | Summer | 3.5–5.5 |
| Temperate continental | Winter | 0.5–1.5 |
| Cool/sub-polar | Summer | 1.5–3.0 |
Crop Water Demand
To find the actual water demand of a specific crop, multiply ET₀ by the crop coefficient (Kc):
Kc varies by crop and growth stage. Grass Kc = 1.0 (by definition). Maize at peak growth Kc ≈ 1.20. Wheat Kc ≈ 1.10. Tomatoes Kc ≈ 1.15.
When to Use It
Use the reference evapotranspiration formula when:
- Designing drip or sprinkler irrigation systems for farms and orchards
- Scheduling automated irrigation controllers based on weather data
- Estimating crop water demand for agricultural water budgeting
- Analyzing climate change impact on regional water availability
- Calculating water footprints for food products
- Planning municipal parks and golf course irrigation in water-scarce regions