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The hourly Penman-Monteith equation for estimating reference evapotranspiration used on the CIMIS computer is a version described in the Food and Agricultural Organization's Irrigation and Drainage Paper No. 56 (Allen, R.K., L.S. Pereira, D. Raes, and M. Smith. 1998). However, bulk surface resistance is adapted from the ASCE Task Committee on Standardization of Reference Evapotranspiraton (2000). Two reference evapotranspiration equations are employed, one for a grass reference and the other for alfalfa.
Reference Evapotranspiration for Grass
(EQ1)
Where
  • ETo = grass reference evapotranspiration (mm h-1)
  • Δ = slope of saturation vapor pressure curve (kPa °C-1) at mean air temperature (T)
  • Rn = net radiation (MJ m-2 h-1)
  • G = soil heat flux density (MJ m-2 h-1)
  • γ = psychrometric constant ( kPa°C-1)
  • Ta = mean hourly air temperature ( °C)
  • U2 = wind speed at 2 meters (m s-1)
  • es = saturation vapor pressure (kPa) at the mean hourly air temperature (T) in °C
  • ea = actual vapor pressure (kPa) at the mean hourly air temperature (T) in °C
  • lamda = latent heat of vaporization in (MJ kg-1)
  • Cd = bulk surface resistance and aerodynamic resistance coefficient

  • When Rn>0 (daytime)
    Soil heat flux (MJ m-2 h-1)
    (EQ2)
    Bulk surface resistance and aerodynamic resistance coefficient
    (EQ3)
  • When Rn <= 0 (nighttime)
    Soil heat flux (MJ m-2 h-1)
    (EQ4)
    Bulk surface resistance and aerodynamic resistance coefficient
    (EQ5)
Reference Evapotranspiration for Alfalfa
(EQ6)
Where
  • ETr = alfalfa reference evapotranspiration (mm h-1)
  • Δ = slope of saturation vapor pressure curve (kPa °C-1) at mean air temperature (T)
  • Rn = net radiation (MJ m-2 h-1)
  • G = soil heat flux density (MJ m-2 h-1)
  • γ = psychrometric constant ( kPa °C-1)
  • Ta = mean hourly air temperature ( °C)
  • U2 = wind speed at 2 meters (m s-1)
  • es = saturation vapor pressure (kPa) at the mean hourly air temperature (T) in °C
  • ea = actual vapor pressure (kPa) at the mean hourly air temperature (T) in °C
  • lamda = latent heat of vaporization in (MJ kg-1)
  • Cd = bulk surface resistance and aerodynamic resistance coefficient

  • When Rn >0 (daytime)
    Soil heat flux (MJ m-2 h-1)
    (EQ7)
    Bulk surface resistance and aerodynamic resistance coefficient
    (EQ8)
  • When Rn <= 0 (nighttime)
    Soil heat flux (MJ m-2 h-1)
    (EQ9)
    Bulk surface resistance and aerodynamic resistance coefficient
    (EQ10)
Other components in Eq.1 and 6 are calculated as follows:
  1. Delta
    (EQ11)
    Ta = mean hourly air temperature ( °C)
  2. Net radiation (MJ m-2 h-1)
    (EQ12)
    Rs = solar radiation (MJ m-2 h-1)
    Rnl = net longwave radiation (MJ m-2 h-1)
  3. Psychrometric constant
    (EQ13)
    P = barometric pressure (kPa)
    (EQ14)
    A = station elevation above mean sea level (meters)
  4. Saturation vapor pressure (kPa)
    (EQ15)
    Ta = mean hourly air temperature (°C)
  5. Actual vapor pressure (kPa)
    (EQ16)
    es = saturation vapor pressure (kPa)
    RH = relative humidity (%)
  6. Latent heat of vaporization (MJ kg-1)
    (EQ17)
    Ta = mean hourly air temperature (°C)
Extraterrestrial Radiation Equation
To calculate net longwave radiation in Eq. 10, extraterrestrial radiation, Ra, is required. The calculation of Ra is based on Duffie and Beckman (1991).
*
(EQ18)
  • GSC = solar constant (0.082 MJ m -2 min-1)
  • dr = inverse relative distance Earth-Sun (correction for eccentricity of Earth's orbit around the sun)
  • ω1 = Solar time angle 1/2 hour before ω, that is, at the beginning of period (radians)
  • ω2 = Solar time angle 1/2 hour after ω, at end of period (radians)
  • fe = Station latitude (radians)
  • δ = Declination of the sun above the celestial equator (radians)
Where
  • (EQ19)

    J = day of year (1-366)
  • (EQ20)
    (EQ21)
  • Solar time angle ω at midpoint of the hourly period is given by:
    (EQ22)

    t = standard clock time (1-24)

    Lm = longitude of measurement location (weather station) in degrees

    Lz = longitude of the local time meridian (degrees West), 120° for Pacific time zone

    (EQ23)
    (EQ24)

    J = day of year (1-366)

    (EQ25)

    Y = latitude (decimal degrees)
    (EQ26)
  • Long wave Radiation Equation (MJ m-2 h-1)
    (EQ27)

    f = cloudiness factor

    epsilon = net emissivity of the surface

    σ = Steffan-Boltzman constant (2.04*10-10) MJ m -2 h-1 K-4

    Tak = mean hourly temperature (Kelvin)

    For daytime hours defined as θ >10°
    (EQ28)

    Rs = solar radiation (MJ m -2 h-1)

    Rso = clear sky total global radiation at the earth surface (MJ m-2 h-1)
    For nighttime hours, θ <= 10 °
    (EQ29)

    Rspm = solar radiation (MJm-2 h-1) during the last hour for which θ>10 °

    Rsopm = clear sky solar radiation (MJ m-2h-1) during the last hour for which θ>10 °

    When f<0 set f = 0.595, and when f>1.0 set f=1.0

    (EQ30)

    A = station elevation above mean sea level (meters)

    (EQ31)

    Solar altitude is calculated as:
    (EQ32)
All terms have been defined above.
If you have questions, please contact the following person(s):
Bekele Temesgen
Department of Water Resources
Office of Water Use Efficiency
P.O. Box 942836
Sacramento, CA 94236-0001
temesgen@water.ca.gov
Simon Eching
Department of Water Resources
Office of Water Use Efficiency
P.O. Box 942836
Sacramento, CA 94236-0001
seching@water.ca.gov
References
  • Allen, R.G., M. Smith, L.S. Pereira, A. Perrier. 1994. An update for the calculation of reference evapotranspiration. ICID Bulletin 1994 Vol 43 No 2.
  • Allen,R.K., L.S. Pereira, D. Raes, and M. Smith. 1998. Crop evapotranspiration. Guideline for computing crop water requirements. FAO Irrigation and Drainage Paper No. 56. United Nations Food and Agricultural Organization, Rome.
  • Duffie, J.A., and Beckman, W.A. (1981). Solar engineering of thermal processes, 2nd ed., John Wiley & Sons, Inc., New York, N.Y.
  • Eching, S.O, and D. Moellenberndt. 1998. Technical elements of CIMIS, the California Irrigation Management Information System : State of California, Resources Agency, Dept. of Water Resources, Division of Planning and Local Assistance, 63 p.
  • Walter, I.A., R.G. Allen, R. Elliott, M.E. Jensen, D. Itenfisu, B. Mecham, T.A. Howell, R. Snyder, P. Brown, S. Eching, T. Spofford, M. Hattendorf, R.H. Cuenca, J.L. Wright, D. Martin. 2000. ASCE's Standardized Reference Evapotranspiration Equation. Proc. of the Watershed Management 2000 Conference, June 2000, Ft. Collins, CO, American Society of Civil Engineers, St. Joseph, MI.
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