What is reference evapotranspiration (ETo)?
Reference evapotranspiration is the water use of a hypothetical, well-watered grass reference surface under a given day's weather. FAO Irrigation and Drainage Paper No. 56 maintains the Penman-Monteith equation as the sole standard method for computing it from meteorological data, and it's the foundation nearly every other water-use calculator on this site builds on: multiply ETo by a crop coefficient, plant factor or landscape coefficient, and you get the actual water use of a specific crop or planting.
This calculator runs the full daily Penman-Monteith procedure, exactly as FAO-56 lays it out in its calculation sheet, and shows every intermediate value along the way so you can check the working, not just the final number.
If you only have maximum and minimum temperature, without humidity, wind or radiation data, the full Penman-Monteith method isn't available to you; see the Hargreaves ET calculator instead, which FAO-56 presents as a simpler alternative for that situation. Also referred to as an FAO-56 Penman-Monteith calculator, this tool covers the daily time step; the irrigation water requirement calculator takes ETo the rest of the way to a net or gross irrigation depth.
How this reference ET calculator works
FAO-56's calculation sheet works through several intermediate quantities before combining them into the final equation. This calculator follows the same sequence.
1. Mean temperature and the slope of the vapor pressure curve
Tmean and Δ
Tmean = (Tmax + Tmin) ÷ 2
Δ = 4098 × [0.6108 × exp(17.27 Tmean ÷ (Tmean + 237.3))] ÷ (Tmean + 237.3)²
2. Atmospheric pressure and the psychrometric constant
P and γ
P = 101.3 × [(293 − 0.0065z) ÷ 293]5.26, where z = elevation (m)
γ = 0.665 × 10-3 × P
3. Vapor pressure
Saturation and actual vapor pressure
e°(T) = 0.6108 × exp(17.27T ÷ (T + 237.3)) · es = [e°(Tmax) + e°(Tmin)] ÷ 2
ea depends on which humidity data you provide: [e°(Tmin)×RHmax/100 + e°(Tmax)×RHmin/100] ÷ 2 from max/min RH; es × RHmean/100 from mean RH; or e°(Tdew) from dewpoint temperature.
4. Extraterrestrial radiation and daylight hours
These depend only on latitude and day of year (J), using standard astronomical relationships for solar declination (δ), the sunset hour angle (ωs), and the inverse relative Earth-Sun distance (dr).
Ra and N
Ra = (24×60/π) × Gsc × dr × [ωs sinφ sinδ + cosφ cosδ sinωs]
N = 24/π × ωs
5. Solar radiation, clear-sky radiation, and net radiation
Rs (from sunshine hours)
Rs = (0.25 + 0.50 n/N) × Ra
Enter Rs directly instead if you already have measured solar radiation.
Rso, Rns, Rnl, Rn
Rso = (0.75 + 2z/100,000) × Ra · Rns = 0.77 × Rs
Rnl = σ × [(Tmax,K⁴ + Tmin,K⁴)/2] × (0.34 − 0.14√ea) × (1.35 Rs/Rso − 0.35)
Rn = Rns − Rnl
6. Wind speed at 2 m
u2
u2 = uz × 4.87 ÷ ln(67.8z − 5.42), where z = measurement height (m)
7. The Penman-Monteith equation
ETo
ETo = [0.408Δ(Rn−G) + γ(900/(Tmean+273))u2(es−ea)] ÷ [Δ + γ(1+0.34u2)]
G (soil heat flux) is taken as zero for this daily calculation, per FAO-56.
When this calculator isn't the right tool
Use a different approach for:
- Temperature-only data, without humidity, wind or radiation. See the Hargreaves ET calculator.
- Estimating ETo from a pan evaporation reading instead of weather data. See the pan evaporation to ET calculator.
- Turning ETo into a net or gross irrigation depth. See the irrigation water requirement calculator.
Reference ET calculation example
This reproduces FAO-56's own published daily example: 6 July in Uccle (Brussels, Belgium), 50°48'N, 100 m elevation. Tmax 21.5°C, Tmin 12.3°C, RHmax 84%, RHmin 63%, wind measured at 10 m = 2.78 m/s, sunshine 9.25 hours.
- Tmean: (21.5+12.3)/2 = 16.9°C. Δ: 0.122 kPa/°C.
- P: 100.1 kPa. γ: 0.0666 kPa/°C.
- es: 1.997 kPa. ea (from RHmax/RHmin): 1.409 kPa. es−ea: 0.589 kPa.
- u2 (converted from 10 m): 2.078 m/s.
- Ra: 41.09 MJ/m²/day. N: 16.1 hours. Rs: 22.07 MJ/m²/day.
- Rso: 30.90. Rns: 17.00. Rnl: 3.71. Rn: 13.28 MJ/m²/day.
- ETo: 2.81 (radiation term) + 1.07 (aerodynamic term) ≈ 3.9 mm/day, matching FAO-56's own published result.
The same procedure in US units
15 July at a mid-latitude US site (45.72°N), 656 ft elevation, Tmax 80°F, Tmin 59°F, RHmax 80%, RHmin 40%, wind 5 mph measured at 2 m, sunshine 9 hours:
- Tmean: 20.8°C (69.5°F). Δ: 0.151 kPa/°C. γ: 0.0658 kPa/°C.
- es: 2.602 kPa. ea: 1.382 kPa. es−ea: 1.220 kPa.
- Ra: 40.55 MJ/m²/day. N: 15.2 hours. Rs: 22.16 MJ/m²/day. Rn: 13.02 MJ/m²/day.
- ETo: 3.01 + 2.06 ≈ 5.07 mm/day ≈ about 0.20 in/day.
Both examples follow the exact sequence the calculator runs; enter your own day's data above for your result.
Practical guidance
Use the humidity method that matches the data you actually have; FAO-56's calculation sheet treats maximum and minimum relative humidity as the primary method, with mean relative humidity or dewpoint as alternatives. Where wind speed is measured somewhere other than 2 m, such as a rooftop anemometer or a 10 m weather-station mast, always enter the real measurement height so the conversion is applied rather than assuming the reading is already at 2 m.
For solar radiation, prefer an actual measured value or sunshine-hour reading for your location and date over a seasonal average; ETo can vary considerably day to day with cloud cover. If you're estimating ETo for a period rather than a single day, FAO-56 notes that mean ten-day or monthly data can be used the same way, though the soil heat flux term this calculator sets to zero would need including for monthly calculations when the ground is warming or cooling.
Local calibration matters. FAO-56 is the internationally standard method, but real ETo can be affected by local conditions not fully captured by any single equation. Compare against local weather-network ETo values where available.
What changes ETo
| Input | Effect | What to check |
|---|---|---|
| Temperature | Higher temperatures raise the saturation vapor pressure and Δ, generally raising ETo. | Use the actual day's max/min, not a monthly average, for a daily estimate. |
| Humidity | Lower humidity (higher vapor pressure deficit) raises the aerodynamic term. | Use humidity data from the same day as temperature and wind. |
| Wind speed | Higher wind raises the aerodynamic term, especially when the air is dry. | Confirm the measurement height for an accurate 2 m conversion. |
| Radiation / sunshine | More solar radiation raises net radiation and the radiation term, usually the larger of the two terms. | Use actual sunshine or radiation data rather than a clear-sky assumption. |
| Latitude and date | Both set day length and sun angle, which set extraterrestrial radiation (Ra). | Confirm latitude and the correct day of year. |
| Elevation | Higher elevation lowers atmospheric pressure and the psychrometric constant, and raises clear-sky radiation slightly. | Use the site's actual elevation, not sea level. |
Frequently asked questions
What is reference evapotranspiration (ETo)?
Reference evapotranspiration is the water use of a hypothetical, well-watered grass reference surface under the local weather conditions, per FAO Irrigation and Drainage Paper No. 56. It's a standard against which actual crop or landscape water use can be compared using a crop or plant factor, and FAO-56 maintains the Penman-Monteith equation as the sole standard method for calculating it from weather data.
What is the FAO-56 Penman-Monteith equation?
It's a physically based equation combining a radiation term and an aerodynamic (wind and humidity) term to estimate ETo from temperature, humidity, wind speed and net radiation, using the crop height, albedo and resistance values of the reference grass surface. FAO-56 presents it as the sole standard method for computing ETo from meteorological data.
What if I only have temperature data, not humidity, wind and radiation?
This calculator needs the full Penman-Monteith data set. If you only have maximum and minimum temperature, use the Hargreaves ET calculator instead, which FAO-56 presents as a simpler alternative equation for that situation.
See the Hargreaves ET calculator.
How do I enter humidity if I only know the mean relative humidity, not the max and min?
Choose the mean relative humidity option. FAO-56's calculation sheet lists this as a valid method, alongside using maximum and minimum relative humidity or a dewpoint temperature, though it notes deriving actual vapor pressure from mean relative humidity alone is less recommended than the other methods because of non-linearities in the underlying relationship.
Where do I get sunshine hours or solar radiation data?
Many weather stations and agricultural weather networks report daily sunshine duration or solar radiation directly. If you only have sunshine hours, this calculator converts them to solar radiation using the Angstrom formula with FAO-56's default coefficients; if you already have measured solar radiation, you can enter it directly instead.
Why does wind speed need to be measured at 2 meters?
The FAO-56 Penman-Monteith equation is standardized to wind speed at 2 m height. FAO-56 provides a logarithmic wind speed profile equation to convert measurements taken at other heights down to the 2 m standard; this calculator applies that conversion automatically from the measurement height you enter.
Why does this calculator show so many intermediate values?
The Penman-Monteith equation is built from several physical quantities, such as the slope of the saturation vapor pressure curve, net radiation and the psychrometric constant, calculated in sequence. Showing each one lets you check the calculation step by step against FAO-56's own calculation sheet, rather than only seeing the final number.
Can I use this for a period longer than one day?
This calculator is built for a daily calculation, per the FAO-56 daily time-step procedure. FAO-56 notes that mean ten-day or monthly weather data can also be used with the same equation to get a ten-day or monthly average ETo, but the soil heat flux term, assumed to be zero here, would need to be included for monthly periods when the ground is warming or cooling significantly.
Who built and checked this calculator
Licensed Professional Engineer and Irrigation Association Certified Irrigation Designer with more than 20 years in irrigation design, system auditing and landscape water management. Full profile
How this page was checked
- Every equation was implemented exactly as FAO-56's calculation sheet (Box 11) presents it, in the same sequence.
- The metric worked example reproduces FAO-56's own published Example 18 (Uccle, Belgium) to the decimal; the US example was hand-calculated through the full sequence and traced against the code.
- All divisions (pressure, wind profile, Angstrom formula) are guarded so no input combination returns a blank or invalid result.
Sources
- Food and Agriculture Organization of the United Nations. Allen, R.G., Pereira, L.S., Raes, D. & Smith, M. FAO Irrigation and Drainage Paper No. 56, Chapter 4: Determination of ETo — the Penman-Monteith equation, calculation sheet (Box 11), and Example 18.
- Food and Agriculture Organization of the United Nations. FAO Irrigation and Drainage Paper No. 56, Chapter 3: Meteorological Data — extraterrestrial radiation, daylight hours, and wind speed profile equations.