Agricultural Hydrology, Precision Irrigation & Micrometeorology
FAO-56 Penman-Monteith Reference Evapotranspiration (ET0) Formula
Calculate FAO-56 Radiation-Driven ET Term (mm/d), Wind-VPD Aerodynamic ET Term (mm/d), Total Reference Grass Evapotranspiration ET0 (mm/d), and Weekly Water Volume per Hectare (m³/ha).
Why FAO-56 embeds stomatal canopy resistance (1 + 0.34 * u2) in the denominator—separating crop ET0 from open-pan evaporation!
The global United Nations FAO Irrigation and Drainage Paper 56 (FAO-56 Penman-Monteith) standardizes daily Reference Evapotranspiration ET0 (mm/d) for a well-watered 0.12 m clipped grass canopy with fixed bulk surface resistance r_s = 70 s/m and aerodynamic resistance r_a = 208 / u_2: ET0 = (0.408 * Delta * (R_n - G) + gamma * (900 / (T_mean + 273)) * u_2 * VPD) / (Delta + gamma * (1 + 0.34 * u_2))! Notice how wind speed u_2 (m/s at 2 m height) appears in BOTH the aerodynamic numerator AND the stomatal-resistance denominator gamma * (1 + 0.34 * u_2)—meaning stronger hot dry winds boost aerodynamic Vapor Pressure Deficit (VPD = e_s - e_a) demand while slightly attenuating the radiation term!
Verify on a Sunny Weather-Station Day—Slope_Vapor_Curve Delta = 0.145 kPa/°C (at 20°C, B2), Net_Rad_Minus_Soil_Flux (R_n - G) = 14.0 MJ/(m²·d) (B3), Psychrometric_Constant gamma = 0.067 kPa/°C (B4), Mean_Air_Temp T = 20.0°C (B5), Wind_Speed_2m u_2 = 2.0 m/s (B6), and Vapor_Pressure_Deficit VPD = 1.10 kPa (B7): Radiation Component contributes 3.22 mm/d (E2), Aerodynamic Component adds 1.76 mm/d (E3), Total Reference ET0 equals 4.97 mm/d (E4), and 7-Day Reference Water Demand per Hectare (1 mm = 10 m³/ha) equals 4.97 * 70 = 347.90 m³/ha (E5)!
| FAO-56 Weather-Station KPI (Delta=0.145, Rn-G=14, T=20°C, u2=2 m/s, VPD=1.1 kPa) | Formula | Verified Output |
|---|---|---|
| Radiation ET0 Term (mm/d) | (0.408 * Delta * RnG) / (Delta + gamma*(1+0.34*u2)) | 3.22 mm/d (2.54 @ u2 = 5 m/s) |
| Aerodynamic ET0 Term (mm/d) | (gamma*(900/(T+273))*u2*VPD) / Denom | 1.76 mm/d (3.47 @ u2 = 5 m/s) |
| Total FAO-56 Reference ET0 (mm/d) | Radiation_Term + Aerodynamic_Term | 4.97 mm/d (6.01 @ u2 = 5 m/s) |
| 7-Day Water Demand (m³/ha/wk) | ROUND(ET0_mm_d * 7 * 10, 2) | 347.90 m³/ha (420.70 @ u2 = 5 m/s) |
Complete FAO-56 Penman-Monteith Daily ET0 Formula
Validate Delta > 0, (R_n - G) >= 0, gamma > 0, Mean_Temp > -40°C, Wind_Speed u_2 >= 0, and VPD >= 0!
=LET(dl_t, B2, rn_g, B3, gm_p, B4, tm_c, B5, u2_w, B6, vp_d, B7, IF(OR(dl_t <= 0, rn_g < 0, gm_p <= 0, tm_c <= -40, u2_w < 0, vp_d < 0), "Invalid FAO56 Input", LET(dn_m, dl_t + gm_p * (1 + 0.34 * u2_w), rd_t, (0.408 * dl_t * rn_g) / dn_m, ar_t, (gm_p * (900 / (tm_c + 273)) * u2_w * vp_d) / dn_m, ROUND(rd_t + ar_t, 2))))Advective Dry-Wind Surge Check (Set 2m Wind Speed u_2 = 5.0 m/s)
Raise Wind_Speed_2m B6 from 2.0 m/s to 5.0 m/s: Radiation Term E2 attenuates from 3.22 to 2.54 mm/d due to the (1 + 0.34*u2) denominator, yet Aerodynamic Term E3 surges from 1.76 to 3.47 mm/d—lifting Total ET0 E4 to 6.01 mm/d (420.70 m³/ha/week in E5)!
=IF(ISTEXT(E4), E4, ROUND(E4 * 70, 2))Try the example
Use the original values or change them to test your understanding.