Published online 12 March 2007
Published in Agron J 99:585-590 (2007)
DOI: 10.2134/agronj2006.0159
© 2007 American Society of Agronomy
677 S. Segoe Rd., Madison, WI 53711 USA
Notes & Unique Phenomena
Explicit and Recursive Calculation of Potential and Actual Evapotranspiration
Robert J. Lascanoa,* and
Cornelius H. M. van Bavelb
a Texas A&M Univ. Res. and Ext. Center, 3810 4th Street, Lubbock, TX 79415
b Soil & Crop Sciences Dep., Texas A&M Univ., 245 Pecan Valley Rd., Center Point, TX 78010
* Corresponding author (r-lascano{at}tamu.edu)
Received for publication May 22, 2006.
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ABSTRACT
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The explicit combination method (ECM; Penman, 1948) to calculate potential evapotranspiration (ETp) is a physically based model using standard climatological data. It is based on an assumption regarding 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 ETp and of actual evapotranspiration (ETa) using hourly weather data collected on 45 d during the warm season in Lubbock, TX. Results show that on hot summer days ECM underestimated the daily value of ETp and of ETa 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.
Abbreviations: ECM, explicit combination method • ET, evapotranspiration (kg m–2 d–1 or mm d–1) • ETa, actual crop evapotranspiration (kg m–2 s–1 or mm d–1) • ETp, potential evapotranspiration (kg m–2 s–1 or mm d–1) • rc, canopy resistance (s m–1) • RCM, recursive combination method • Ts, surface temperature (°C)
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INTRODUCTION
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THE 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 ETa 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 ETa to be less than ETp.
Historically, the methods of relating ETp to weather parameters were empirical and lacked general validity. However, Penman (1948) and Budyko (1958) independently proposed methods to calculate ETp 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 ETp, see Sellers (1965) and Brutsaert (1982).
Penman (1948) derived an explicit equation for ETp 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 temperature. The object of this assumption was the elimination of the surface temperature from the set of equations used in the calculation of ETp (Milly, 1991). However, Sellers (1965, p. 169–170) pointed out that this procedure 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 authors and proposed an algebraic method based on higher derivatives of the relation between air temperature and saturation humidity. The resulting explicit formula (see his Eq. [23]) is complex, though a simplification (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 interest is his statement that only by iteration can complete 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, ETp and the surface temperature Ts, and the Goff-Gratch equation that relates the saturation humidity at the surface to that temperature. Starting with an initial value for Ts, the value of both unknowns is found by iteration, resulting in a value for ETp 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 ETp and ETa.
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 ETa is a two-step procedure (ASCE, 2005). First, one calculates a reference evapotranspiration (ETref) value using a single standardized reference ET equation for either a short (e.g., grass) or a tall (e.g., alfalfa) crop. Second, this ETref is multiplied by crop specific coefficients to estimate ETa. 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 ETref. 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 temperature, common in irrigated crops in hot and dry climates.
In contrast, the RCM finds the value of ETp or ETa with the required precision, but several repeated calculations are needed. First suggested by Budyko (1958), iteration has received attention (e.g., Tracy et al., 1984; McArthur, 1990), but without impact on the standard methods used to calculate ETa (e.g., Allen et al., 1998; ASCE, 2005). Thus, iterative methods are not new and they have been used to calculate water evaporation from the soil and plant using energy and water balance simulation models (e.g., Lascano and Van Bavel, 1983, 1986; Lascano et al., 1987). Bristow (1987) also used a Newton iterative technique to find the surface temperature in solving the energy balance equation. In view of the general availability of computing devices and commercially available software that can give solutions almost immediately, recursive techniques can easily be incorporated in algorithms to calculate ETp and ETa, including in automated weather stations.
Our objective is to demonstrate the difference between the values of ETp and ETa, as calculated respectively by the ECM and RCM methods, using the same meteorological input data for both.
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MATERIALS AND METHODS
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In this section we describe the procedures followed to calculate ETp and ETa by the explicit and the recursive method. The ECM is based on Penman (1948), and the RCM is based on Budyko (1958). This description is not meant to be exhaustive in terms of theories or to review all the different methods derived from or similar to the Penman (1948) formula, but rather to document how the four different values of ET and ensuing results were calculated. All calculations were done using the mathematical software Mathcad v. 13 (Mathsoft Engineering & Education, Cambridge, MA) and Microsoft Excel 2002 (Microsoft Corporation, Redmond, WA), both on a laptop computer.
Input Weather Data to Calculate Evapotranspiration Values
In all calculations we used as input hourly weather collected in Lubbock, TX (33°41' N, 101°49' W, and 998 m above sea level) at a screen height of 2.0 m as described in Lascano (2000). To establish a wide range of environmental conditions we selected 45 consecutive days in 1994, for which the average daily values of air temperature (Ta, °C), dewpoint temperature (Td, °C), windspeed (u, m s–1), and the daily total incoming short-wave irradiance (Rg, MJ m–2) are given in Table 1. Calculated from the hourly data, these daily averages were also used in an alternative computation of ETp and ETa using ECM and RCM.
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Table 1. Average daily values of Ta (air temperature), Td (dewpoint temperature), u (windspeed), and total Rg (incoming short-wave irradiance) for 45 d in Lubbock, TX, 1994.
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Explicit Combination Calculation of ETp and ETa
Potential Evapotranspiration
The ECM calculation of ETp was done using the well-known combination equation given by Penman (1948) and as used by others with a modified wind function (e.g., Van Bavel, 1966; Brutsaert, 1982), from
 | [1] |
where 
is the dimensionless ratio of the slope of the saturation vapor pressure vs. air temperature (kPa °C–1) to the psychrometric constant (kPa °C–1), Rn is the net irradiance (W m–2), G is the soil heat flux (W m–2, assumed to be 0.0 for a full crop canopy), 
is the latent heat of vaporization (J kg–1), 
s is the saturated air absolute humidity (kg m–3), a is the actual air absolute humidity (kg m–3), ra is the aerodynamic resistance (s m–1), and the ETp is in kg m–2 s–1.
The calculations of , , s, and a in Eq. [1] were done with procedures similar to those given by Van Bavel (1966) and Ham (2005). The value of ra was calculated as a function of the measured u (Sellers, 1965). The value of Rn, if not measured, can be calculated by the method of Kimball et al. (1982), although other methods are available (e.g., Ham, 2005). These procedures vary in the literature and we imply no preference regarding their accuracy, only that they do not affect the ECM or RCM procedures as such.
Actual Evapotranspiration
The ECM calculation of ETa was obtained by using a canopy resistance term (rc, s m–1) in the denominator of Eq. [1] as follows:
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Equation [2] can also be written by combining it with Eq. [1] and assuming G = 0.0 as
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Penman (1953), recognizing that the transpiration from well-watered vegetation involves a diffusion resistance of the leaf stomata, proposed that the expression for ETp, formulated in 1948, be modified as shown by his Eq. [9]. This equation is entirely analogous to Eq. [3] above, but the Penman procedure to find the proposed value of SD, a stomatal (S) and day-length (D) factor, together equivalent to rc, is neither practical, nor experimentally tested. A method to measure the leaf diffusion resistance, that would preempt its calculation, was proposed (Van Bavel et al., 1965) and improved by commercially available and portable instruments known as porometers. Nevertheless, a proven and general method to calculate the canopy resistance from the leaf resistance does not exist.
An experimental method to find the value of rc is to measure ETa with precision weighing lysimeters and to calculate rc by rewriting Eq. [3] (Van Bavel and Ehrler, 1968):
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in which ETp is calculated from Eq. [1]. Actual values for rc were obtained by this method in two field experiments in central Arizona with well-watered sorghum (Ehrler and Van Bavel, 1967; Van Bavel and Ehrler, 1968). Hourly ETp was calculated by the ECM method, ETa was measured hourly with three precision weighing lysimeters, and using Eq. [4], the hourly value of rc was determined. On 14 July 1965, the midday hourly value of rc was 39 s m–1. The experiment was repeated in 1966 and, 15 d after flood irrigation on 1 June, the average of three midday hourly values for rc was 28 s m–1 (Van Bavel and Ehrler, 1968). We emphasize that the value of 35 s m–1 is used only as an example to show the difference between ECM and RCM when used to compute the value of ETa. An improved method to find the hourly value of rc from a field experiment, using RCM, is discussed in the next section.
Recursive Combination Calculation of ETp and ETa
The RCM used to calculate ETp and ETa is based on the iterative procedure suggested by Budyko (1958), an outline of which is given by Sellers (1965, p. 168–170). The RCM does not assume the values for Ts and s at the surface, but rather it derives both values by solving the energy balance equation and Murray's equation (Murray, 1967) simultaneously. The unknowns are ETp or ETa, and the surface temperature Ts. The two equations are combined in a single implicit expression that must equal zero and is solved by iteration, starting with an initial value for Ts. The computation gives the value of Ts and of ETp if the resistance to water vapor transport is ra, or the value of ETa, if the resistance is ra + rc, the sum of the aerodynamic and the canopy resistance.
In these calculations of ETp and ETa with the RCM we used the software Mathcad v. 13; however, other similar and available commercial software could also be used. Specifically, in Mathcad v. 13, root, a built-in function, was used to solve for Ts by the Secant or Mueller method (e.g., Fröberg, 1965, p. 31–36). For example,
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which returns the value of var1 to make the function f equal to zero, and when a and b are specified, root finds var1 on the interval [a,b], while Ts = 10.0 is the initial guess for Ts. Equation [6] shows, in Matchcad syntax, the iterative calculation of Ts and ETa from the energy balance equation, in which the first term is the radiative component, the second term is the sensible heat flux, and the third term is the latent heat flux, where humidity terms are calculated using the Murray (1967) equation. By setting rc = 0.0, one will obtain the value of ETp and the corresponding value of Ts. In Eq. [6], d is the air density (kg m–3), cp is the specific air heat capacity (J kg–1 °C–1), and Rl is the sky long-wave irradiance (W m–2). One should note that Eq. [6] also accounts for the effect of Ts on the outgoing long-wave radiation (Kimball et al., 1982). However, if net irradiance is measured directly, its value supplants the first term in Eq. [6]. Evapotranspiration and Ts can also be calculated using the Solver (add-in feature) in Microsoft Excel 2002. We obtained identical results when using either Mathcad or Excel. A copy of the ECM and RCM algorithms used to calculate ETp and ETa using Mathcad v. 13 and Excel 2002 is available from the corresponding author on request.
Calculation of Canopy Resistance
In the calculation of ETa with the RCM method, we used an assumed value of 35 s m–1 for rc. However, the following experimental procedure is proposed and can be used with the RCM method to find rc. Given a measurement of hourly ETa with precision lysimeters, ETa is calculated from the pertinent weather data by Eq. [6], but rc is defined as a range variable in Mathcad, rather than as a constant, for example from 10 to 100 s m–1 in steps of 10 s m–1. This automatically produces a table and a graph of ETa as a function of rc, which gives a computed value of rc for the measured value of ETa, obtained during daytime hours, when stomata are fully open.
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RESULTS AND DISCUSSION
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Table 1 shows the daily averages of the weather parameters recorded in Lubbock during a 45-d period in 1994, from Day 175 (24 June) through Day 219 (7 August), a typical range of summer weather conditions.
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A second-degree polynomial regression of the ECM hourly values of ETp (y) obtained from the sum of the calculated hourly values over the corresponding RCM ETp values (x) was calculated as y = –0.0249 x2 + 1.1643x – 0.2637 with R2 = 0.98. This regression equation was used to plot a graph of the value of (RCM – ECM) vs. RCM, shown in Fig. 1a
. The regression equation using daily weather values for ETp was y = –0.0269x2 + 1.1899x – 0.1312 with R2 = 0.97, and the corresponding plot of (RCM – ECM) vs. RCM is also shown in Fig. 1a. These results demonstrate two facts: the first being that when ETp is found iteratively and exceeds 9 mm d–1, ETp found explicitly is an underestimate from 1 to 4 mm d–1, or from 11 up to 25%. The second conclusion, based on the totals during the 45-d period, shown in Table 2, is that the use of the daily averages of the input weather data gives overall somewhat lower values for ETp, amounting to about 8% using RCM and 4% using ECM.
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Fig. 1. Difference of (RCM – ECM) vs. RCM for (a) ETp (potential evapotranspiration) and (b) ETa (actual transpiration) using hourly and daily weather data. RCM = recursive combination method and ECM = explicit combination method.
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Table 2. Cumulative calculated values of ETp (potential evapotranspiration) and ETa (actual transpiration) for the 45-d period using hourly and daily weather input values using ECM (explicit combination method) and RCM (recursive combination method) procedures.
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A similar analysis of the calculation of ETa is also based on the weather data in Table 1, with an assumed value for rc of 35 s m–1. As previously noted, this value serves only as an example and should be experimentally determined for each specific crop cover and growth stage of development. The polynomial regression equation for ETa, as found by ECM (y) over those found by RCM (x) for hourly weather data, yielded y = –0.0251x2 + 1.1682x – 0.2337, R2 = 0.98; for daily weather data, y = –0.0306x2 + 1.2389x – 0.1994, R2 = 0.98. These two regression equations were used to plot a graph of (RCM – ECM) vs. RCM, and are given in Fig. 1b for both the hourly and daily input values. It shows that, when ETa > 8 mm d–1, the difference between the RCM and the ECM value is >0.5 mm d–1 and becomes as large as 3.8 mm d–1, or 24% of the RCM value. The difference between the ETa values derived from the hourly data and daily ones, shown in Table 2, is somewhat less than the corresponding value for ETp, but in both instances the difference is small enough to justify the use of daily data if no hourly weather values are available.
A remaining question is how ETa values compare with the corresponding ones for ETp. The 45-d totals in Table 2 yielded a ratio of 0.89 for ETa/ETp. Obviously, this ratio depends on the value of rc, for which we assumed a value of 35 s m–1.
A detailed illustration of the difference between the results obtained by the ECM and the RCM methods is a plot of the hourly ETp, as found by both procedures on Day 178, 1994, shown in Fig. 2
, and of the corresponding calculated surface temperature Ts, shown in Fig. 3
. The hourly value of ET was always greater when computed by the RCM than by the ECM. The difference peaked at 0.39 mm h–1 during the afternoon, a difference of 48% of the RCM result for the same time of day. The corresponding difference in the surface temperature, an automatic product of the iterative procedure, shows it to be up to 6°C lower in the early evening when the RCM is used, compared with the ECM.
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Fig. 2. Hourly values of calculated potential evapotranspiration (ETp) on Day 178, 1994 at Lubbock, TX, as found from the explicit (ECM) and the recursive (RCM) combination methods. The total daily evapotranspiration is also shown for ECM and RCM.
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Fig. 3. Hourly values of calculated surface temperature (Ts) obtained with the explicit (ECM) and the recursive (RCM) combination methods on Day 178, 1994 at Lubbock, TX.
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CONCLUSIONS
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In sum, the Budyko (1958) or RCM procedure for calculating either the ETp or the ETa makes no assumptions regarding the temperature and the saturation humidity at the evaporating surface. Any procedure that is based on the assumptions introduced by Penman (1948) will likely underestimate ETp and ETa. However, the error depends on the weather evaporative demand, increasing as the latter increases.
The recursive method for calculating ETp and ETa, based on the Budyko (1958) model and well-established theory, needs experimental verification, particularly in arid and semiarid climates where normally water demand for irrigation far exceeds availability. Such field tests require precision weighing lysimeters (e.g., Schneider et al., 1998) to obtain hourly values of ETa and a separate determination of rc, as suggested in the description of the RCM procedure. The calculations needed for the iterative method are fast and accurate using available mathematical software. The calculation of daily ETp or ETa from either hourly or daily input values can be performed by automated weather stations in real time that, moreover, can control the irrigation water application (Lascano, 2000). Our results suggest that the RCM procedure does not require any more calculating effort than the traditional ECM procedure.
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REFERENCES
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ABSTRACT
INTRODUCTION
