The numerator has two parts: a radiation (energy) term and an aerodynamic (drying power) term. In humid, calm climates the first dominates; in dry, windy climates the second can exceed it. The calculator below shows the split.
What is Reference Evapotranspiration?
Reference evapotranspiration (ET₀) is the rate of evaporation plus transpiration from a hypothetical, extensive, well-watered grass surface with a height of 0.12 m, a fixed surface resistance of 70 s/m and an albedo of 0.23 (Allen et al. 1998). Because the surface is fixed, ET₀ depends only on weather, so it is a climate variable that can be mapped and compared everywhere.
The Penman-Monteith form combines Penman's (1948) energy balance with Monteith's (1965) resistance concept. FAO adopted it in 1990 and published the standard procedure in Irrigation and Drainage Paper 56 (1998). The ASCE-EWRI (2005) standardized equation is identical for daily grass reference.
ET₀ drives irrigation scheduling, water balance and drought indices such as SPEI (see SPI and SPEI explained), and it is the evaporative demand term in most hydrological models.
Inputs You Need
| Input | Unit | If missing (FAO-56 Ch. 3) |
|---|---|---|
| Maximum and minimum air temperature | °C | Required |
| Humidity: RHmax and RHmin, or dew point, or RHmean | % or °C | ea ≈ e°(Tmin) (dew point ≈ Tmin) |
| Wind speed and anemometer height | m/s, m | Use 2 m/s as a global average |
| Solar radiation Rs or sunshine hours n | MJ m⁻² d⁻¹ or h | Rs = kRs√(Tmax − Tmin) Ra |
| Latitude, elevation, day of year | °, m, – | Required (from location and date) |
The 12 Calculation Steps
- Mean temperature: \(T=(T_{max}+T_{min})/2\).
- Atmospheric pressure and psychrometric constant: \(P=101.3\left(\dfrac{293-0.0065z}{293}\right)^{5.26}\), \(\gamma=0.000665\,P\).
- Saturation vapour pressure \(e^\circ(T)=0.6108\exp\!\left(\dfrac{17.27T}{T+237.3}\right)\); use \(e_s=\dfrac{e^\circ(T_{max})+e^\circ(T_{min})}{2}\), not \(e^\circ(T_{mean})\).
- Slope of the vapour pressure curve: \(\Delta=\dfrac{4098\,e^\circ(T)}{(T+237.3)^2}\).
- Actual vapour pressure: \(e_a=\dfrac{e^\circ(T_{min})\frac{RH_{max}}{100}+e^\circ(T_{max})\frac{RH_{min}}{100}}{2}\) (or \(e^\circ(T_{dew})\)).
- Extraterrestrial radiation: with \(\varphi\) latitude in radians, \(d_r=1+0.033\cos\frac{2\pi J}{365}\), \(\delta=0.409\sin\left(\frac{2\pi J}{365}-1.39\right)\), \(\omega_s=\arccos(-\tan\varphi\tan\delta)\): $$R_a=\frac{24(60)}{\pi}G_{sc}\,d_r\left[\omega_s\sin\varphi\sin\delta+\cos\varphi\cos\delta\sin\omega_s\right],\quad G_{sc}=0.0820$$
- Daylight hours: \(N=\dfrac{24}{\pi}\omega_s\).
- Solar radiation (Ångström): \(R_s=\left(a_s+b_s\dfrac{n}{N}\right)R_a\) with \(a_s=0.25,\ b_s=0.50\).
- Clear-sky radiation and net shortwave: \(R_{so}=(0.75+2\times10^{-5}z)R_a\), \(R_{ns}=(1-0.23)R_s\).
- Net longwave radiation: $$R_{nl}=\sigma\frac{T_{max,K}^4+T_{min,K}^4}{2}\left(0.34-0.14\sqrt{e_a}\right)\left(1.35\frac{R_s}{R_{so}}-0.35\right),\ \sigma=4.903\times10^{-9}$$
- Net radiation: \(R_n=R_{ns}-R_{nl}\); for daily steps \(G\approx0\).
- Wind at 2 m and ET₀: \(u_2=u_z\dfrac{4.87}{\ln(67.8z-5.42)}\), then apply the equation above.
Worked Example: FAO-56 Example 18 (Brussels, 6 July)
Given: latitude 50°48′N, elevation 100 m, 6 July (J = 187); Tmax = 21.5 °C, Tmin = 12.3 °C; RHmax = 84%, RHmin = 63%; wind 10 km/h (2.78 m/s) measured at 10 m; 9.25 hours of bright sunshine.
| Quantity | Value | Unit |
|---|---|---|
| Mean temperature T | 16.9 | °C |
| Pressure P / psychrometric constant γ | 100.1 / 0.0666 | kPa, kPa/°C |
| Saturation vapour pressure es | 1.997 | kPa |
| Actual vapour pressure ea | 1.409 | kPa |
| Vapour pressure deficit es − ea | 0.589 | kPa |
| Slope Δ | 0.122 | kPa/°C |
| Extraterrestrial radiation Ra | 41.09 | MJ m⁻² d⁻¹ |
| Daylight hours N | 16.1 | h |
| Solar radiation Rs | 22.07 | MJ m⁻² d⁻¹ |
| Clear-sky radiation Rso | 30.90 | MJ m⁻² d⁻¹ |
| Net longwave Rnl | 3.71 | MJ m⁻² d⁻¹ |
| Net radiation Rn | 13.28 | MJ m⁻² d⁻¹ |
| Wind at 2 m, u2 | 2.08 | m/s |
| ET₀ | 3.88 ≈ 3.9 | mm/day |
These numbers come straight from the calculator below, which opens with this example loaded, and agree with FAO-56's published result of 3.9 mm/day.
FAO-56 ET₀ Calculator
Location & date
Temperature & humidity
Wind & radiation
Change in ET₀ (mm/day) when each input is changed by the amount shown, others held constant. Large bars mark the inputs worth measuring carefully at your site.
Missing Data: What FAO-56 Recommends
- No humidity: assume the dew point equals Tmin, so \(e_a=e^\circ(T_{min})\). In arid climates subtract 2–3 °C from Tmin first, because nights rarely reach saturation.
- No radiation: Hargreaves' radiation formula \(R_s=k_{Rs}\sqrt{T_{max}-T_{min}}\,R_a\), with kRs ≈ 0.16 for interior and 0.19 for coastal locations.
- No wind: use 2 m/s, or a regional average. ET₀ is usually less sensitive to wind than to radiation, except in dry climates.
- Only temperature: the Hargreaves-Samani (1985) equation \(ET_0=0.0023\,(T+17.8)\sqrt{T_{max}-T_{min}}\;0.408R_a\) is a simpler alternative; the calculator reports it alongside Penman-Monteith for comparison.
From ET₀ to Crop Water Requirement
Crop evapotranspiration under standard conditions is \(ET_c=K_c\,ET_0\). The crop coefficient \(K_c\) changes through the season (initial, development, mid-season, late). For example, FAO-56 Table 12 gives Kc,mid ≈ 1.20 for field maize. With ET₀ = 3.9 mm/day, mid-season maize would use about 4.7 mm/day. The irrigation requirement then subtracts effective rainfall; see our duty, delta and base period guide for the irrigation-engineering side.
Common Mistakes
es from Tmean
Because e°(T) is non-linear, e°(Tmean) underestimates es and therefore ET₀.
Wind height not converted
Using 10 m wind as u2 overestimates the aerodynamic term by about a third.
Radiation units
W/m² × 0.0864 = MJ m⁻² day⁻¹. Hourly MJ values must be summed, not averaged.
Latitude sign and radians
Southern latitudes are negative; trigonometric functions need radians.
Rs/Rso above 1
Limit the ratio to 1.0 in the longwave term; values above 1 indicate radiation sensor or input errors.
Kelvin conversion
The longwave term uses T + 273.16 to the fourth power; forgetting it gives absurd Rnl.
Related Tools and Guides
Frequently Asked Questions
What is the FAO-56 Penman-Monteith equation?
ET₀ = [0.408 Δ (Rn − G) + γ (900/(T + 273)) u₂ (es − ea)] / [Δ + γ (1 + 0.34 u₂)]. It gives the daily evapotranspiration (mm/day) of a hypothetical well-watered grass reference surface 0.12 m tall with a surface resistance of 70 s/m and an albedo of 0.23, from temperature, humidity, wind speed and radiation.
What is the difference between ET₀ and ETc?
ET₀ is the reference evapotranspiration of a standard grass surface and depends only on weather. Crop evapotranspiration ETc = Kc × ET₀, where the crop coefficient Kc accounts for the crop type and growth stage. FAO-56 tabulates Kc values for many crops.
Why is 0.408 in the Penman-Monteith equation?
0.408 converts energy in MJ m⁻² day⁻¹ to an equivalent depth of evaporated water in mm/day. It is the inverse of the latent heat of vaporization (about 2.45 MJ/kg): 1/2.45 = 0.408.
How do I convert wind speed measured at 10 m to 2 m?
Use the FAO-56 logarithmic profile u₂ = u_z × 4.87 / ln(67.8 z − 5.42), where z is the measurement height in metres. For a 10 m anemometer the factor is 0.748.
Can I calculate ET₀ with only temperature data?
Yes. FAO-56 allows estimating solar radiation from the temperature range (Rs = kRs √(Tmax − Tmin) Ra, with kRs ≈ 0.16 inland and 0.19 coastal), humidity from Tmin (ea ≈ e°(Tmin)) and using 2 m/s for wind. Alternatively use the Hargreaves equation ET₀ = 0.0023 (Tmean + 17.8)(Tmax − Tmin)^0.5 × 0.408 Ra. Both are less accurate than the full equation.
How do I convert solar radiation from W/m² to MJ/m²/day?
Multiply the daily average in W/m² by 0.0864. For example, 250 W/m² averaged over 24 hours equals 21.6 MJ m⁻² day⁻¹.
References
Allen, R. G., Pereira, L. S., Raes, D., & Smith, M. (1998). Crop Evapotranspiration: Guidelines for Computing Crop Water Requirements. FAO Irrigation and Drainage Paper 56, Rome.
ASCE-EWRI (2005). The ASCE Standardized Reference Evapotranspiration Equation. American Society of Civil Engineers, Reston, VA.
Monteith, J. L. (1965). Evaporation and environment. Symposia of the Society for Experimental Biology, 19, 205–234.
Penman, H. L. (1948). Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society of London A, 193, 120–145.
Hargreaves, G. H., & Samani, Z. A. (1985). Reference crop evapotranspiration from temperature. Applied Engineering in Agriculture, 1(2), 96–99.
Zotarelli, L., Dukes, M. D., Romero, C. C., Migliaccio, K. W., & Morgan, K. T. (2010). Step by Step Calculation of the Penman-Monteith Evapotranspiration (FAO-56 Method). UF/IFAS Extension AE459.