Appendix A. The Penman-Monteith formula and the Kimberly-Penman Equation

In 1948, Penman described a method to calculate actual evaporation using meteorological parameters as inputs into an energy balance equation (Penman 1948). In 1965, Monteith described the effects of wind speed, the size and distribution of leaf stomata, and the aerodynamic properties of plant surfaces on the rates of transpiration from plants (Monteith 1965). Together, these studies formed the basis of the Penman-Monteith formula, which can be expressed in mm/day as (Monteith 1973):

ET = 0.408 * ΔR_n + (105.028*(e_s-e_a)/r_s)

(Δ + 0.067*(1+(r_s/r_a))

Where e_a is the vapor pressure, e_s is the saturation vapor pressure at the current temperature, Δ is the slope of the saturation vapor pressure (e_s) curve at the given temperature, R_n is net radiation, r_a is the aerodynamic resistance to evaporation from vegetation, and r_s is the physiological resistance to transpiration from vegetation. The aerodynamic resistance, which is reciprocal to the roughness of the earth’s surface and the wind speed, tends to be higher over open water (110-125 s/m) than over grasslands (50-70 s/m), and the r_a over grasslands tends to be higher than that over forests (5-10 s/m) (Hendriks 2010). Surface resistance tends to be higher in forests (80-150 s/m) than in grasslands (40-70 s/m) (Hendriks 2010). Given the wide range in aerodynamic and surface resistance across landscapes, site specific measurements need to be made to produce accurate ET values in a given location. In lieu of site-specific measurements, estimates for these values may be made based on the aforementioned ranges given for various land cover types.

In 1982, Wright developed coefficients to apply to the Penman-Monteith equation to measure ET in agricultural fields for various crops (Wright 1982). The field measurements for that study were made in agricultural research fields in Kimberly, Idaho and the equations that were developed from it are known as the Kimberly-Penman equations. The Kimberly-Penman equation using alfalfa as a reference crop is as follows (Wright 1982):

λETr = (Δ/(Δ + γ))(R_n – G) + (γ/(Δ + γ))6.43W_f(e_s – e_a),

Where G is soil heat flux, γ is the psychrometric constant, and W_f is the dimensionless wind function. This is the standard equation used throughout the US Agricultural Research Service to estimate ET at agricultural sites. This equation is also used to calculate ET at fire weather stations, where ET is incorporated into drought indices to measure fire danger ratings (https://raws.dri.edu/).

Daily ET totals are calculated from the weather station data using the Kimberly-Penman equation with alfalfa as a reference crop. Each weather station collects data on air temperature, relative humidity, wind speed, and solar radiation, averaged over 15-minute intervals, and summarized as daily averages (or daily totals in the case of solar radiation). At the USGS stations, net radiation is measured directly. At the RAWS stations, only incoming solar radiation is measured, and net radiation is calculated using albedo and cloud cover values specific to the location and time of year (Dokter 1994). At the USGS stations, the atmospheric pressure is measured directly. At the RAWS stations, the standard atmospheric pressure estimated for the station’s elevation is used. At Tijeras ET, the soil heat flux is measured directly. At the other three stations, the soil heat flux is calculated using an equation developed through soil research in Kimberly, ID (Wright 1982). The equation is as follows:

G = 0.377(T_mean - T_pr)

where: T_mean is the mean daily temperature in ^oC, T_pr is the average of the previous three days’ mean temperatures in ^oC, and G is expressed in MJ/m²/day.

The ET calculations for the RAWS stations, Oak Flats and Sandia Lakes, are completed in the Desert Research Institute internal database. The daily ET values were downloaded directly from the database, and then were summarized into 8-day totals, to match the 8-day periods represented in the MODIS data. The ET values at the USGS stations, Sandia Mountains Upper Precip Site and Tijeras ET, were calculated by the author using an Excel spreadsheet. The meteorological data was summarized into daily averages, or daily totals in the case of net radiation and soil heat flux, and the appropriate calculations were made to derive 8-day evapotranspiration totals.

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