Introduction
Estimating either natural evaporation rates from open water surfaces (E_{o})
or from wellwatered plant canopies (potential evapotranspiration or PET) require
quantifying the diabatic and adiabatic components of the latent heat transfer
from the atmosphere to the evaporating surface at a given temperature and pressure.
The former component depends on the net solar energy available for evaporation
at the surface and the latter on the state and movement of moistureunsaturated
air above the surface. Over 50 years ago, Howard Penman (Penman, 1948, 1956)
analyzed and combined these two thermodynamic processes into a single equation.
This equation (commonly called the Penman combination equation) integrated the
effects of net solar radiation, air temperature, air humidity and wind speed
conditions at a given location to obtain a semiempirical estimate of the evaporative
demand of the atmosphere above open water surfaces. However, in contrast to
E_{o}, PET also depends on soil and plant canopy characteristics. To
circumvent this difficulty, the PET estimated by Penmanbased equations was
defined as applicable to an idealized reference wellwatered plant canopy actively
growing under optimal conditions. The PET of any given canopy is then estimated
as the product of this reference PET and an empirically determined crop coefficient.
Although any reference canopy could be chosen, the most widely used have been
grass or alfalfa maintained at a specified height.
Since its introduction, continuous effort has been made to refine Penmanbased estimation of reference PET (Doorenbos and Pruitt, 1977; Monteith, 1965; Monteith and Unsworth, 1990; Allen et al., 1998). The primary scientific motivation underpinning this effort was (and still is) the need to more fully understand and realistically quantify the thermodynamics of evapotranspiration. As a result, there is a current consensus that Penmanbased approaches provide reference PET estimates that best match experimental observations. More recently the many variants of the Penmanbased approach have converged into the PenmanMonteith equation to estimate reference PET (Monteith, 1965; Monteith and Unsworth, 1990; Allen et al., 1998; Itenfisu et al., 2000). Currently, many consider the PenmanMonteith equation to be a more standardized and universally applicable PET estimator. Its adoption has been recommended worldwide (Walter et al., 2001; ASCEEWRI, 2004). The United Nations Food and Agricultural Organization (FAO) has recommended the PenmanMonteith equation as their sole standard method (Allen et al., 1998). This report (Allen et al., 1998) states that "… it (the PenmanMonteith equation) is a method with the strong likelihood of correctly predicting reference PET in a wide range of locations…" and that "… it overcomes shortcomings of the previous FAO Penman method and provides values more consistent with actual crop water use data worldwide…". In some areas where extensive lysimeter observations of reference PET were available, the tendency has been to adopt the PenmanMonteith PET estimator. Yet relatively more areas continue to use variants of the traditional Penman method. Applicability of the PenmanMonteith equation still needs to be demonstrated in these areas.
Monteith and Unsworth (1990) took Penman's original concept to a higher level of abstraction and formulated an equation (the PenmanMonteith equation) that can be utilized for the direct calculation of evapotranspiration from any canopy. He did this by incorporating air resistances in series within and above the canopy that controlled the evaporative fluxes the under water vapor and energy gradients that drive evapotranspiration rates. The first of these resistances termed as the bulk surface resistance (r_{s}) lumped the effect of specific soil and canopy characteristics on the fluxes within the canopy into a single parameter. The second, termed as the aerodynamic resistance (r_{a}), was used to characterize the effect of air movements assuming neutral vertical stability conditions of the atmospheric boundary layer above the canopy. Although conceptually more general and realistic, these resistance parameters change with canopy type and development and soil wetness over time. Quantifying them over time require sophisticated instrumentation. Consequently, the PenmanMonteith equation may offer little practical advantage over the traditional use of crop coefficients to estimate PET for different canopy types. However, there is evidence (Walter et al., 2001; ASCEEWRI, 2004) showing that it is more effective for estimating reference PET where the canopy type and soil wetness conditions are fixed. The resistances r_{s} and r_{a} have been reasonably well defined for a well watered grass reference crop with an assumed height of 0.12 m having an albedo (r) of 0.23 as r_{s} = 70 s m^{1} and r_{a} = (208/u) s m^{1} where, u is the measured windspeed in m s^{1} at 2 m above the ground.
Botswana is a landlocked country in Southern Africa between 22° and 26°
south latitude and between 21° and 28° east longitude. It falls in the
semiarid climatic category (Bsh climate according to the KoppenGeigerPohl
climatic classification system). The Kalahari semidesert covers about twothirds
of Botswana. Most of the population lives along the desert's fringe where the
monomodal summer rainfall is sufficiently reliable to sustain crop and livestock
production (Campbell, 1980). Evapotranspiration is a large component of the
soil water balance and therefore its accurate estimation is more important in
such low rainfall semiarid climates than in humid regions. Reasonably accurate
estimates of E_{o} and PET are essential for quantifying water balances
needed for effective, sciencebased hydrological engineering and water management
in Botswana. However, applicability of the PenmanMontieth equation has not
been evaluated in Botswana although a small set of lysimeter observations were
recorded in 1989 for the agrometeorological station located at the DMS headquarters
in Gaborone (24° 40' S, 25° 55' E, elevation 983 m). The question addressed
here was whether the Penman Monteith equation would better match these existing
lysimeter observations of reference PET in Botswana than their currentlyused
Penmanbased estimator and, if not why not.
Materials and Methods
Penmanbased equation for Botswana: The form of the Penmanbased equation currentlyused by the Department of Meteorological Services (DMS) in Gaborone, Botswana is tailored to the type of instrumentation and data collected at the few fully equipped agrometeorological stations in the country. The Penmanbased DMS equation is:
where PET = mean daily potential evapotranspiration in mm water; Δ = slope of the saturation vapor pressure versus temperature curve in mb°K^{1 }at the value for the mean daily air temperature T°K; γ = 0.66 mb°K^{1} is the wet and dry bulb psychrometer constant (Rose, 1966). It represents the value of aP = C_{p}P/ (0.622 L) in the wetanddrybulb equation. Under wellventilated conditions at sea level (P = 1013 mb) with C_{p} (specific heat at constant pressure of air) = 1005 J kg^{1}°K^{1} and L (latent heat of vaporization of water at 20°C) = 2.453x10^{6} J kg^{1}, the value of aP is 0.667 mb°K^{1}; R_{a} = mean daily Angot's value for a given location in mm water; r = reflection coefficient or albedo for a particular surface such as open water or fresh green vegetation; n = mean daily hours bright sunshine recorded on the StokesCampbell bright sunshine recorder; N = mean day length in hours for a given location and is calculated from astronomical formulae also given by Iqbal (1983) and Gommes (1983); σ = a pseudoconstant = StefanBoltzmann constant divided by L the latent heat of vaporization of water in cal g^{1}; T = mean daily temperature of the particular surface in°K; e_{s} (T) = saturation vapor pressure in mb at temperature T (°K).
The first two terms estimate the potential evaporative demand due to the net incident solar radiation (incident short wave minus the net outgoing longwave radiation) on the reference canopy. The third term represents the evaporative loss caused by advection of air at a given level of dryness and is usually termed as the aerodynamic component. The relative importance of these two components depends on climatic conditions. The temperature dependent ratio Δ/(Δ+γ) in effect weights the relative importance of the energy and aerodynamic components since Δ/(Δ+γ)] + γ/(Δ+ γ) = 1.
Equation 1 indicates that the psychrometric constant (γ), is independent of barometric pressure change due to station elevation. However, DMS indicated that in their algorithm the value of γ was corrected for pressure (P) decrease with station elevation (z) as recommended by McCollough (1965). The DMS algorithm used the hypsometric equation for a neutral standard atmosphere (i.e., with a temperature profile corresponding to the dry adiabatic lapse rate of 0.0065°K m^{1}) i.e., P(z) = P_{atm} [1 (0.0065 z / T_{o})] ^{5.264} where z is in meters, P_{atm }= atmospheric pressure of 1013 mb at mean sea level and T_{o} is the surface temperature taken as equal to the air temperature.
The coefficients a_{1} and a_{2} in the DMS equation represent the semiempirical regression coefficients in the Angstrom equation R_{s}/R_{a} = a_{1 }+ a_{2} n/N relating observed solar radiation and hours of bright sunshine. They were selected by DMS based on the study results of Andringa (1987) and Persaud et al. (1997) which supported earlier studies for similar climates by MartinezLozana et al. (1984). The DMS algorithm uses the coefficients in the second term (a_{3}, a_{4}, a_{5} and a_{6}) and in the third term (a_{7}, a_{8} and a_{9}) of Penman's equation as recommended by Doorenbos and Pruitt (1977). As reported by Vossen (1988). The value of r = 0.25 is generally accepted for short green grass in the middle and high latitudes.
Values of the saturation vapor pressure values for e_{s} (T) needed to obtain vapor pressure from relative humidity data were calculated using the following standard GoffGratch equation (Goff and Gratch,1946; Goff, 1957) as modified by Gommes (1983).
The values of day length N on a given date for the DMS headquarters station in Gaborone was calculated as N = (D/2) (24/π) with D/2 (radians) = arc cos {– [ (0.0148 + sin φ sin δ)/(cos φ cos δ)]}, where φ (radians) is the latitude (+ve for North, ve for South) and δ (radians) is the declination of the sun on a given date. Sin δ = sin ε sin l where ε = 23.45° = 0.4093 radians is the angle between the earth's equator and ecliptic and l (radians) is the celestial longitude measured anticlockwise from the earth's position at the vernal equinox. The Angot value (R_{a} in mm H_{2}O) received per unit area of the earth’s surface at a given location at this latitude was calculated as (Iqbal, 1983; Gommes, 1983):
Where J is the day counted from January 1.
The PenmanMonteith equation: The PenmanMontieth equation (Allen et al., 1998) is:
where PET = the reference evapotranspiration in mm day^{1}; L = latent heat of vaporization in MJ kg^{1}; R_{n} and G = the net radiation and soil heat flux, respectively in MJ m^{2} d^{1}; (e_{s}e_{a}) = the vapor pressure deficit of the air in mb; Δ = de_{s}/dT and γ = 0.67 = the psychrometric constant in mb°K^{1} (or more appropriate in this context in J m^{3}°K^{1}); K_{t }= is the number of seconds in the period over which PET values are calculated = 86,400 seconds for daily values; ρ_{a} = mean natural air density in kg m^{3} at atmospheric pressure; C_{p} is the specific heat of the air in MJ kg^{1}°K^{1}; r_{a} and rs = the aerodynamic and bulk surface resistance, respectively in s m^{1}.
L (the latent heat of evaporation in J kg^{1}) was assumed fixed and equal to 2.453 MJ kg^{1} as for the DMS equation. For daily periods G is effectively zero (Allen et al., 1998). Values of e_{s}(T), e and Δ in mb were calculated as described above for the DMS equation. Since direct measurements of R_{n} were unavailable anywhere in Botswana, its value (already in mm day^{1}) was taken as the same as that estimated for the DMS equation using Angot's value and hours bright sunshine as:
The density of moist air was approximated as that of dry air since the latter is only slightly greater. Using the ideal gas law ρ_{a} in kg m^{3} ≈ P_{atm}/(R_{d}T) where R_{d} is the specific gas constant for dry air = 287 J kg^{1}°K^{1}. As explained in the introduction, γ in mb°K^{1} at P_{atm} = C_{p}P_{atm}/(0.622 L) and therefore (K_{t} ρ_{a} C_{p})/(r_{a }L)= 86,400 [(0.622 γ)/(R_{d }T r_{a})] in mm H_{2}O°K^{1} day^{1}x100 (since γ is taken in mb°K^{1}). Substituting the appropriate numerical values, (K_{t} ρ_{a} C_{p})/(r_{a }L)=100x{86,400 (0.622 γ) [1/(287T)] (u/208) = 90 γu/T mm H_{2}O°K^{1} day^{1}. Also, substituting the numerical values for r_{s} and r_{a} gives 1 + r_{a}/r_{s} = 1 + 0.34 u. Consequently, the PenmanMonteith equation used for computational purposes in this study was:
Comparison of estimated and lysimeter PET values: Information on the set of PET values for grass was contained in a report by Sakamoto et al. (1990). They were measured in two free drainage lysimeters constructed in 1986 at the meteorological station at the located at the DMS headquarters in Gaborone. Records of thirtysix fiveday mean lysimeter values of PET under grass cover were available during the period March 21, 1989 through 20 March, 1990. As stated in the report, the 5day average was considered more realistic since it was not practical to maintain ideal steadystate, unsaturated water flow conditions in these lysimeters and there were periods when they were not maintained (Sakamoto et al., 1990). The 36 values were distributed as follows: fall 9, winter 8, spring 10, summer 9. These observed values were compared to corresponding estimates using the DMS Penmanbased equation and the PenmanMontieth equation.
Results
Figure 1 presents the measured 5day lysimeter means and
corresponding values using the DMS and Penman equations over the year starting
March 21, 1989. Both equations show the same pattern of estimated PET, but it
appears that the correspondence between measured values and estimates depended
on the season. The PenmanMontieth estimates were closer to measured values
during the fall and winter of 19891990 while the DMS equation better estimated
the observations during the spring and summer.

Fig. 1: 
Thirtysix measured 5day lysimeter averages means and corresponding
values using DMS and PenmanMonteith estimators over the year starting March
21, 1989 in Botswana 
Table 1: 
Mean sum of squares of deviations between measured lysimeter
values (5day averages) and corresponding values using DMS and PenmanMonteith
estimators over 4 seasons in Botswana starting March 21, 1989 

^{1}DMS = Department Meteorological Services, Gaborone,
Botswana 
Table 2: 
Longterm maximum and minimum monthly averages of variables
used for estimating sensitivity of reference PET estimates in Botswana to
coefficients in the Penmanbased DMS^{1} formula 

^{1}DMS = Department Meteorological Services, Gaborone,
Botswana 
In order to better quantify this finding, the mean sum of squares of deviations
between measured lysimeter 5day averages and corresponding values using the
DMS and PenmanMontieth equations were calculated for each of the seasons in
Botswana starting March 21, 1989. These results (Table 1)
confirmed the inference from Fig. 1. Table 1
also shows that for the entire year the DMS equation gave a lower mean sum of
squares of deviations.
The same meteorological variables were used went into both equations for calculation of R_{n}. Therefore the differences in the estimates (Fig. 1) were likely due to the coefficients a_{7}, a_{8}, a_{9} for the DMS equation and the parameters r_{s} and r_{a }of the PenmanMonteith equation. The question that needed answering was how sensitive were the PET estimates to these parameters.
In general the change ΔF in the value of a function F in response to small changes in n parameters a_{1}, a_{2}...a_{i}...a_{n} of the function is:
The coefficients in the DMS equation appear as linear relations and therefore the estimates of PET will respond linearly to step changes in these coefficients. Thus using sets of maximum and minimum values of the climatic variables associated with these coefficients, it was possible to estimate a maximum or minimum possible PET response to an upward or downward step change in each of the coefficients.
The longterm maximum and minimum values for the appropriate meteorological
variables were estimated from DMS mean monthly records for Gaborone and are
given in Table 2. As already discussed, the values of γ
in this table were corrected for the elevation (983 m) of the DMS headquarters
station in Gaborone. Substitution of these values into the appropriate derivatives
of the DMS equation showed that the response of the PET estimates were most
sensitive to changes a_{9} (Table 3). This suggested
that the DMS value of 0.01 for a_{9} may be too high for the winter
months. Lowering its value would tend to give better correspondence with the
observed lysimeter values. The value of the parameter a_{9} in Penmanbased
PET estimators has been contentious. In the original Penman equation for open
water a_{9} was set = 1/100 for u in miles day^{1} or 1/161
for u in km day^{1}. Doorenbos and Pruitt (1977) used a_{9}
= 1/100 for u in km day^{1} stating that the similarity to the value
for a_{9} in the original Penman equation was purely coincidental. Frère
and Popov (1979) suggested that in arid and semiarid conditions, whenever the
mean monthly temperature exceeds 5°C, a_{9} should be taken as a
function of ΔT the difference between the maximum and minimum air temperature
otherwise the value of a_{9} should be set equal to 1/161.
Table 3: 
Response of PET estimates to step increases in the coefficients
r, a_{7}, a_{8}, a_{9} for the DMS equation using
maximum and minimum values of the input variables 

^{1}Maximum and minimum values of variables given
in Table 2 

Fig. 2: 
Response of PET estimates to unit step increases (1 s m^{1})
over the range of possible values in Botswana for the parameter r_{a
}in the PenmanMonteith equation using maximum and minimum values of
the input variables with r_{s} = 70 s m^{1} 
Indeed, the long term mean monthly temperature records for Gaborone showed
that the mean temperature is above 5°C throughout the year. Clearly, there
is a need for experiments to better quantify the behavior of the wind coefficient
(a_{9}) in the aerodynamic term of the Penmanbased DMS equation.
Unlike the partial derivatives for the DMS equation, those for the PenmanMonteith
equation with respect to r_{s} and r_{a} are nonlinear. This
means ∂(PET)/∂r_{s} = f(r_{s}; r_{a}) and
∂(PET)/∂r_{a} = f(r_{a}; r_{s}). Since r_{a}
= 208/u, using values for u in Table 2, would give a possible
values for r_{a} = 82.4 s m^{1} for u_{max }= 218 km
day^{1} (2.523 m s^{1}) and 156.3 s m^{1} for u_{min
}= 115 km day^{1} (1.331 m s^{1}). In terms of r_{a}
and r_{s} Eq. 5 above can be rewritten as PET (mm
day^{1}) = u(r_{a}, r_{s})/v(r_{a}, r_{s})
with u = ΔRn + [18725 γ (e_{s}e)]/(T r_{a}) and v
= Δ + γ[1 + (r_{s}/r_{a})]. Based on the values for
n/N, e_{s} and e in Table 2, the maximum and minimum
values of R_{n }using Eq. 4 were, respectively 7.30
and 2.00 mm H_{2}O day^{1}. Values of Δ(PET) for unit
step increases over the possible range of r_{a} can be obtained using
Eq. 6 for the maximum and minimum variable values given in
Table 2. Similarly Δ(PET) can be obtained for unit step
increases of r_{a} = 70±20 s m^{1} (i.e., in a reasonable
neighborhood about the recommended value for r_{a} = 70 s m^{1}).
These results (Fig. 2 and 3) show that the
PET estimates using the PenmanMonteith equation were comparatively much less
sensitive to variations in r_{a} and r_{s }than the DMS equation
was to variation in a_{9}.

Fig. 3: 
Response of PET estimates to unit step increases (1 s m^{1})
over a range of possible values for the parameter r_{s }in the PenmanMonteith
equation using maximum and minimum values for the input variables and r_{a
}in Botswana 
Therefore it would mean unrealistic lowering of r_{a} and r_{s
}to adjust for the mismatch of the PenmanMonteith estimates of summer
PET in Botswana. On the other hand, adjusting a_{9} in the existing
DMS Penmanbased equation is justifiable.
Conclusions
The findings in their entirety show that the ability of both the DMS and PenmanMonteith equations to estimate the observed reference PET depended on the season. Estimated and observed values can be better matched by small adjustments in the windspeed coefficient in the aerodynamic term of the DMS equation. On the other hand, the appreciable lowering of the two resistance input parameters (r_{a} and r_{s}) needed to improve the correspondence between the PenmanMontieth estimates and the observed summer PET in Botswana would be unjustifiable. It is clear that further longterm, multilocational lysimeter measurements of reference crop PET in Botswana are needed to more fully evaluate the applicability of the PenmanMonteith equation.