Explicit and Recursive Calculation of Potential and Actual Evapotranspiration Robert J. Lascano* and Cornelius H. M. van Bavel ABSTRACT The explicit combination method (ECM; Penman, 1948) to cal- culate potential evapotranspiration (ET p ) is a physically based model using standard climatological data. It is based on an assumption re- garding the temperature and humidity at the evaporating surface that is not made in a recursive combination method (RCM; Budyko, 1958). Our objective was to compare the two methods by calculating values of ET p and of actual evapotranspiration (ET a ) using hourly weather data collected on 45 d during the warm season in Lubbock, TX. Re- sults show that on hot summer days ECM underestimated the daily value of ET p and of ET a by as much as 25% compared with RCM. The proposed RCM procedure is based on the same physical principles as ECM, but uses iteration to find an accurate answer. It can easily be used with commercially available mathematical software that has proven to be stable. The RCM needs experimental verification before implementation for crop irrigation. T HE TERM POTENTIAL EVAPOTRANSPIRATION, due to Thornthwaite (1948), stands for the maximum rate of water loss by evaporation from the land surface under given atmospheric conditions. The ET a represents values of evapotranspiration (ET) that, applied to well-watered agricultural crops, facilitate the planning and efficient use of water in crop production. It takes account of the role of leaf stomata in causing ET a to be less than ET p . Historically, the methods of relating ET p to weather parameters were empirical and lacked general validity. However, Penman (1948) and Budyko (1958) indepen- dently proposed methods to calculate ET p based on known physical principles and standard climatological data, commonly referred to as the combination method. The solution was obtained by combining the equations for the transport of water vapor and sensible heat from or to the land surface with an expression for the radiative energy balance of that surface. For reviews of methods to calculate ET p , see Sellers (1965) and Brutsaert (1982). Penman (1948) derived an explicit equation for ET p by making the assumption that the ratio between the temperature gradient between the surface and the air above and the corresponding humidity gradient, given saturation at the surface, would equal the value of the Clausius-Clapeyron equation at the ambient air tem- perature. The object of this assumption was the elimina- tion of the surface temperature from the set of equations used in the calculation of ET p (Milly, 1991). However, Sellers (1965, p. 169–170) pointed out that this pro- cedure is questionable under hot and arid conditions. Furthermore, the validity of the assumption of using a linear expansion of the saturation vapor pressure curve versus the air temperature has been questioned (e.g., Tracy et al., 1984; Paw U and Gao, 1988; McArthur, 1990, 1992; Milly, 1991; Paw U, 1992). Milly (1991) reviewed attempts to approximate the correct value of the surface temperature by several au- thors and proposed an algebraic method based on higher derivatives of the relation between air temper- ature and saturation humidity. The resulting explicit formula (see his Eq. [23]) is complex, though a simpli- fication (see his Eq. [25]) is more practical. However, its convergence is not assured, and Milly (1991) states that this simplification will probably have an error. Of in- terest is his statement that only by iteration can com- plete numerical accuracy be obtained, a view also shared by Tracy et al. (1984) and McArthur (1992). The earliest proposal to have it supplant the explicit solution seems to be a 1951 report by Budyko, cited by Milly (1991). In addition, Budyko (1958, p. 162–163) suggested, without making any assumptions, an energy balance equation that contains two unknowns, ET p and the surface tem- perature T s , and the Goff-Gratch equation that relates the saturation humidity at the surface to that tempera- ture. Starting with an initial value for T s , the value of both unknowns is found by iteration, resulting in a value for ET p that satisfies the energy balance. Outlines of both methods are given in Sellers (1965, p. 168–170). Hereafter, we will refer to the Penman (1948) formula as the ECM and to the Budyko (1958) procedure as the RCM to calculate both ET p and ET a . The ECM requires a single computation and has been widely used, tested, and adapted (e.g., Allen et al., 1998; ASCE, 2005). The recommended method to calculate ET a is a two-step procedure (ASCE, 2005). First, one calculates a reference evapotranspiration (ET ref ) value using a single standardized reference ET equation for either a short (e.g., grass) or a tall (e.g., alfalfa) crop. Second, this ET ref is multiplied by crop specific coef- ficients to estimate ET a . An essentially identical method used to calculate the crop water requirements is given by Allen et al. (1998). Both methods use the so-called Penman–Monteith equation to calculate ET ref . This equation involves an approximation of the saturation vapor pressure–temperature relation (e.g., McArthur, 1990; Milly, 1991), the quality of which deteriorates with increasing differences between surface and air temper- ature, common in irrigated crops in hot and dry climates. R.J. Lascano, Texas A&M Univ. Res. and Ext. Center, 3810 4th Street, Lubbock, TX 79415; and C.H.M. van Bavel, Soil & Crop Sciences Dep., Texas A&M Univ., 245 Pecan Valley Rd., Center Point, TX 78010. Received 22 May 2006. *Corresponding author (r-lascano@ tamu.edu). Published in Agron. J. 99:585–590 (2007). Notes & Unique Phenomena doi:10.2134/agronj2006.0159 ª American Society of Agronomy 677 S. Segoe Rd., Madison, WI 53711 USA Abbreviations: ECM, explicit combination method; ET, evapotrans- piration (kg m 22 d 21 or mm d 21 ); ET a , actual crop evapotranspira- tion (kg m 22 s 21 or mm d 21 ); ET p , potential evapotranspiration (kg m 22 s 21 or mm d 21 ); r c , canopy resistance (s m 21 ); RCM, recursive combination method; T s , surface temperature (jC). Reproduced from Agronomy Journal. Published by American Society of Agronomy. All copyrights reserved. 585 Published online March 12, 2007