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# Penman-Monteith

Like the Penman equation, the Penman-Monteith equation requires daily mean temperature, wind speed, relative humidity, and solar radiation to predict net evapotranspiration. Other than radiation, these parameter are implicit in the derivation of $Delta$, $c_p$, and $delta_q$, if not conductances below.

The United Nations Food & Agriculture Organization (FAO) standard methods for modeling Evapotranspiration (ET) use a Penman-Monteith equation. The ASCE standard methods modify that Penman-Monteith equation for use with an hourly time step. The SWAT model is one of many GIS integrated hydrologic models estimating ET using Penman-Monteith equations.

Evapotranspiration contributions are very significant in a watershed's water balance, yet are often not emphasized in results because the precision of this component is often weak relative to more directly measured phenomena, eg. rain & stream flow. In addition to weather uncertainties, the Penman-Monteith equation is sensitive to vegetation specific parameters, eg. stomatal resistance or conductance. Gaps in knowledge of such are filled by educated assumptions, until more specific data accumulates.

Various forms of crop coefficients (Kc) account for differences between specific vegetation modeled and a Reference Evapotranspiration (RET or ET0) standard. Stress coefficients (Ks) account for reductions in ET due to environmental stress (eg. Soil saturation reduces root zone O2, low soil moisture induces wilt, air pollution effects, and salinity). Models of native vegetation cannot assume crop management to avoid recurring stress.

## Equation

$overset\left\{text\left\{Energy flux rate\right\}\right\}\left\{lambda_v E=frac\left\{Delta R_n + rho_a c_p left\left( delta q right\right) g_a \right\}$
{Delta + gamma left ( 1 + g_a / g_s right)}} ~ iff ~ overset{text{Volume flux rate}}{ET_o=frac{Delta R_n + rho_a c_p left( delta q right) g_a } { left( Delta + gamma left ( 1 + g_a / g_s right) right) lambda_v }}

λv = Latent heat of vaporization. Energy required per unit mass of water vaporized. (J/g)
Lv = Volumetric latent heat of vaporization. Energy required per water volume vaporized. (Lv = 2453 MJ m-3)

E = Mass water evapotranspiration rate (g s-1 m-2)
ETo = Water volume evapotranspired (m3 s-1 m-2)

Δ = Rate of change of saturation specific humidity with air temperature. (Pa K-1)
Rn = Net irradiance (W m-2), the external source of energy flux
cp = Specific heat capacity of air (J kg-1 K-1)
ρa = dry air density (kg m-3)
δe = vapor pressure deficit, or specific humidity (Pa)
ga = Conductivity of air, atmospheric conductance (m s-1)
gs = Conductivity of stoma, surface conductance (m s-1)
γ = Psychrometric constant (γ ≈ 66 Pa K-1)

(Monteith, 1965):

Note: Often resistances are used rather than conductivities.

$g_a = tfrac\left\{1\right\}\left\{ r_a\right\} ~ ~ And ~ ~ g_s = tfrac\left\{1\right\}\left\{ r_s\right\} = tfrac\left\{1\right\}\left\{ r_c\right\}$
where rc refers to the resistance to flux from a vegetation canopy to the extent of some defined boundary layer.

Also note that $g_s$ varies over each day, and in response to conditions as plants adjust such traits as stoma openings. Being sensitive to this parameter value, the Penman-Monteith equation obviates the need for more more rigorous treatment of $g_s$ perhaps varying within each day. Penman's equation was derived to estimate daily ET from daily averages.

A derivation of this equation may be found at http://biomet.ucdavis.edu/evapotranspiration/PMDerivation/PMD.htm
This also explains relations used to obtain $delta q$ & $Delta$ in addition to assumptions key to reaching this simplified equation.