Evaporation of grass under non-restricted soil moisture conditions

Hydrological Sciences-Journal-des Sciences Hydrologiques, 45(3) June 2000 391

Evaporation of grass under non-restricted soil moisture conditions

H. A. R. DE BRUIN Meteorology and Air Quality Group, Wageningen Agricultural University, Duivendaal 2, 6701 AP Wageningen, The Netherlands e-mail: [email protected]

J. N. M. STRICKER Department of Water Resources, Wageningen Agricultural University, de Nieuwlanden, Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands e-mail: [email protected]

Abstract The behaviour of various formulas for évapotranspiration of grass in non-restricted soil water conditions is considered. These are the expressions based on the Penman formula, i.e. "old" Penman, Penman-Monteith, Thorn-Oliver and the version recommended more recently by the FAO. Moreover, the Priestley-Taylor and the Makkink formulas are considered, which are radiation-based. Comparisons are made between daily mean values estimated with these formulas and direct measurements. The latter were collected over grass in the period 1979-1982 in the catchment area of the Hupselse Beek (The Netherlands). It was found that if all required input data were measured, the Priestley-Taylor and the "old" Penman formula yielded the best results. The assumption that soil heat flux can be neglected introduces a systematic and a random error of roughly 5%. The empirical estimates for net radiation from sunshine duration, temperature and humidity appear to perform rather poorly. These estimates improved significantly if solar radiation was measured directly. The empirical expression proposed by Slob (unpublished) that requires incoming solar radiation only as input, provided better results than the other more complicated expressions. Moreover, this study reveals that evaporation of unstressed grass is primarily determined by the available energy, i.e. good evaporation estimates can be obtained by using simply XE = 0.86(i?„ - G). The Makkink method appears to be attractive for practical applications. These findings support the use of Makkink's formula for routine calculations of crop-reference évapotranspiration as has been done by the Royal Netherlands Meteorological Institute since 1987.

Evaporation d'une pelouse dont le sol est bien alimenté en eau Résumé Dans cette étude, nous avons étudié le comportement de plusieurs formules estimant l'évapotranspiration d'une pelouse dans des conditions où le sol est bien alimenté en eau. Ces formules sont toutes basées sur la formule de Penman. Il s'agit de la formule de Penman classique, de la formule de Penman-Monteith, de la formule de Thom-Olivier et de la version récemment recommandée la FAO. Nous avons également considéré les formules de Priestley-Taylor et de Makkink qui sont basées sur la radiation. Nous avons comparé les valeurs des moyennes journalières estimées par ces formules à des mesures directes. Ces dernières sont été réalisées au cours de la période 1979-1982 sur des pelouses du bassin versant de Hupselse Beek (Pays-Bas). Nous avons constaté que, si toutes les données d'entrées nécessaires sont mesurées, la formule de Priestley-Taylor et la formule de Penman classique donnent les meilleurs résultats. L'hypothèse selon laquelle le flux de chaleur du sol pourrait être négligé introduit une erreur systématique et aléatoire de l'ordre de 5%. Les estimations empiriques de la radiation nette basées sur la durée d'ensoleillement, la température et l'humidité paraissent donner de mauvais résultats. Ces estimations sont significative-ment améliorées si la radiation solaire est mesurée directement. L'expression empirique proposée par Slob (non publiée) qui exige seulement la radiation solaire en entrée, fournit des résultats meilleurs que ceux d'autres expressions plus complexes. Cette étude révèle en outre que l'évaporation des pelouses hors conditions de stress hydrique dépend principalement de l'énergie disponible, c'est-à-dire que des bonnes

Open for discussion until I December 2000

mailto:[email protected]

mailto:[email protected]

392 H. A. R. de Bruin & J. N. M. Strieker

estimations de l'évaporation sont obtenues par la seule utilisation de l'expression XE = O.S6(R„ - G). La méthode de Makkink paraît être la plus attractive en vue des applications pratiques. Les résultats de nos recherches encouragent l'utilisation de la formule de Makkink pour les calculs de routine de l'évapotranspiration des cultures de référence entrepris depuis 1987 par l'Institut Météorologique Royal Néerlandais.

INTRODUCTION

For various practical problems in hydrology, meteorology and agriculture estimation of evaporation is required. Over the last fifty years a large number of equations, which require standard meteorological data, have been developed for this purpose. Without any doubt, the formula by Penman (1948) is best known and most widely used. Other examples are the related expressions proposed by Monteith (1965), resulting in the Penman-Monteith equation, and by Rijtema (1965), who derived a similar expression. Thom & Oliver (1977) presented a variant of the Penman-Monteith equation for hydrological applications using a more robust wind function. More recently, Allen et al. (1994) presented a revised version of the Penman-Monteith equation that is recommended by the FAO for calculation of the crop-reference évapotranspiration. This revised approach replaces the procedure recommended by Doorenbos & Pruitt (1977), that was based on the "old" Penman formula.

Experience has shown that, for short crops such as grass under non-limited soil moisture conditions, the Penman equation and related formulas (Thorn-Oliver and Penman-Monteith) yield reliable estimates of actual evaporation. On the other hand, it has also been found, that with a much simpler approach, based on observations of radiation and air temperature, estimates of actual evaporation of grassland can also be obtained. Examples are the formula by Makkink (1957) (as mentioned by Doorenbos & Pruitt, 1977) and by Priestley & Taylor (1972), published 15 years later. De Bruin (1981) found that a modified form of Makkink's equation yields good estimates of evaporation of a grass field at Cabauw, The Netherlands.

The usefulness for practical applications of the various evaporation formulas for short unstressed grass is investigated herein. Note that grass is often chosen as a reference crop in irrigation practice. Comparisons have been made with the actual evaporation of a grassland site in the Hupselse Beek catchment area (Province of Gelderland, The Netherlands) in the period 1979-1982, determined with the energy balance-Bowen ratio method and the energy balance-flux profile method described by Strieker & Brutsaert (1978).

Not all input parameters required for the full Penman-Monteith equation are always available. This applies in particular to net radiation, soil heat flux and, to a lesser extent, solar radiation. For that reason, the behaviour of the various evaporation formulas was investigated for the following cases: (a) all required input parameters are measured; (b) as (a), except that soil heat flux density, G is ignored; (c) net radiation is estimated from standard weather data, but with the incoming solar

radiation measured, and G = 0; and (d) net radiation and solar radiation estimated from weather data, and G - 0.

Furthermore, only daily mean values for days with no significant water stress were considered.

Evaporation of grass under non-restricted soil moisture conditions 393

EXPERIMENTAL SITE AND AVAILABLE DATA

Data collected during 1979-1982 in the Hupselse Beek Experimental Catchment were analysed. The catchment covers an area of 6.5 km" and is situated in the sandy region of the eastern Netherlands. It is an agricultural area consisting mainly of grassland. At the meteorological station, measurements were made of net radiation (/?„), solar radiation (S), wind speed (w), dry and wet bulb temperature (T and Tw), humidity, soil heat flux density (G), soil moisture and ground water depth. Wind speed and wet and dry bulb temperature data were collected at three levels using cup anemometers (KNMI-type) and modified Friedrich psychrometers installed on a ten-metre mast. The psychrometers were calibrated each month and the anemometers every six months. Annual calibration was applied to the other instruments.

During the growing season (mid April-mid October), actual evaporation was determined with the energy balance-Bowen ratio method, i.e.:

in which E is the evaporation, X is latent heat of vaporization and P is the Bowen ratio, which can be determined with:

P = y f (2) Ae

where AT and Ae are the differences between the temperature and water vapour pressure at two levels, and y the psychrometric constant. Averages were taken over 20 min. For periods where wet-bulb temperature data were missing, first, the sensible heat flux density H was determined using Monin-Obukhov's similarity theory and next XE was found from the energy balance equation for the Earth's surface:

XE = Rn-G-H (3)

Net radiation was measured with a CSERO CN1 net radiometer, while the incoming solar radiation S was observed with a Kipp solarimeter. Sunshine duration was measured with a Campbell-Stokes sunshine recorder during 1979 and with a Haenni solar meter (1980 onwards). In the first part of the observation period the soil heat flux was derived from soil temperature and soil moisture profiles using a calori-metric method (until 1 June 1980). Later, soil heat flux plates were used. More details of the experimental aspects are given by Wentholt (1989).

In this study 24 h mean data of mid April-mid October for the years 1979-1982, have been analysed.

METHODS

Makkink (1957) showed that the following formula yields reliable estimates of the evaporation of grass:

XE = cl-^— S + c, (4) s + y

394 H. A. R. de Bruin & J. N. M, Strieker

in which S the incoming solar radiation (also denoted as global radiation), 5 the slope of the saturation water vapour pressure vs temperature curve at air temperature T, and c\ andc'2 are empirical constants (0.63 and about 14 W va2, respectively),

De Bruin (1981) analysed a micrometeorological data set collected over grass at Cabauw (see De Bruin & Holtslag, 1982) and found that a one-constant version of Makkink's formula even yields good estimates for grass. In this study a modified Makkink formula has been used that also includes the soil heat flux, i.e.:

À£ = c ' — ( S - 2 G ) (5) s + y

with c' being another empirical constant. Here, c' = 0.63.This equation is very similar to the Priestley-Taylor formula published 15 years after Makkink:

XE = a-^—(Rn-G) (6) s + y

where a is the Priestley-Taylor parameter, which for "potential" conditions appears to have a value close to 1.3 (Priestley & Taylor, 1972). Here, a = 1.28 as found by Strieker (1981) for this data set. Note that 2c is about 1.26, which equals the Priestley-Taylor parameter a. This fact also justifies the use of 2G in equation (3) giving the soil heat flux equal weight in both expressions. Note also that, for the growing season, daily R„ ~ 0.5 S and G is often small compared to R„. From these features the modified formula of Makkink equation (5) follows directly from equation (4).

Historically, both Makkink (1957) and Priestley & Taylor (1972) considered their formula a simplification of Penman's equation (Penman, 1956):

s(R„-G)+^[ew(T)-e)

XE = 5 (7) s + y

in which p is the density of air, cp the specific heat at constant pressure of air, ew(T) the saturation water vapour pressure at air temperature (7), e the actual water vapour pressure and ra the aerodynamic resistance. In his original publication, Penman (1956) used a wind function f(u) instead of ra. These terms are related as follows:

'•-^h (8)

Yf(») Penman's equation has been used in this study with:

f(w) = 7.4(0.5 + 0.54M) W m-2 mK1 (9)

Penman derived his formula for a water surface. Monteith (1965) applied similar physics to vegetation which completely covers the soil and obtained:

s(Rn-G)+9^[ew{T)~e]

XE^ ^ r (10) r,

s + y 1 + ^L r

J


where rs is the surface resistance. In the literature, equation (10) is known as the Penman-Monteith equation.

The Penman-Monteith equation was applied here with rs = 60 s m"1 being a good estimate for the basic crop resistance, and with the empirical expression proposed by Thorn & Oliver (1977) for ra, i.e.:

4.72 In r2^

1 + 0.54M7

(ID

in which M2 is the wind speed at 2 m and zo the roughness length of the surface, which was taken as 1 cm. Note that rs and zo have been chosen in such a manner that the slope of the regression line between the actual and the calculated E using Penman-Monteith is one.

Thorn & Oliver (1977) pointed out that success of the original Penman equation for short vegetation is rather fortuitous, because the wind function given by equation (8) refers to a water surface (evaporation pan), which has a much smaller roughness length than grass. The effect of this is counterbalanced by ignoring a surface resistance in the denominator of equation (7). In order to account for these effects, they proposed the following version of the Penman-Monteith formula:

XE ,r s(Rn -G)+ my{l + 0.54«Xgw(T)-e] ( 1 2 )

s + y(l + n)

in which n is the mean ratio of ra and rs, taken here as 1.2, and TO is a factor accounting for the difference in roughness length of Penman's evaporation pan and of the actual surface. In this study, m = 1.2 (Strieker, 1981).

Recently, FAO recommended that, for the so-called grass reference évapotranspiration, a reduced form of the Penman-Monteith equation be used:

0.408 s{Rn-G) +y-^zzu2[ew(T)-e] E = , l + A I \ (13)

5 + y(l + 0.34«2) v

In addition, an empirical formula is recommended for R„ and the incoming solar radiation (see later). For further details, the reader is referred to Allen et al. (1989), Smith et al. (1991) and Allen et al. (1994). Here, equation (13) has been used, where E is converted to energy units (W m2).

In many practical applications, direct observations of soil heat flux and net radiation are not available. In most cases G is simply ignored, and R„ is estimated with an empirical expression. In order to investigate the impact of this practical feature E was evaluated with the different formulas for the cases (a)-(d) as described in the Introduction.

For the estimation of Rn two expressions were used. The first has been applied for many years by the Royal Netherlands Meteorological Institute (KNMI) in its routine calculations of evaporation according to the Penman formula (see De Bruin, 1981, 1987), containing empirical constants tested for sites in The Netherlands:

396 H. A. R, de Bruin & J. N. M. Strieker

Rn=(l-$S^(l-e.itm)oT.A;\0.2 + 0.8^ (14)

S0 'o .2 + 0 .6—| (15)

in which (3 is the albedo, 5 the incoming solar radiation, 5o the extra-terrestrial radiation, n the number of hours of bright sunshine, N the day length, rabs the absolute air temperature, eatm the effective emissivity of the cloudless sky and a the Stephan-Boltzman constant (= 5.67 10"8 W irf2 K"4).The constants a = 0.2 and b = 0.6 in the Angstrom formula (equation (15)) were obtained by Frantzen & Raaff (1975), who tested this formula for Dutch conditions. Here, the albedo was taken as (3 = 0.23 (Strieker, 1981) and Brunt's formula for eatm, was applied, i.e.:

eatm =0.53 + 0.067 -fe (16)

where the water vapour pressure e is expressed in mbar. The constants in (equation (16)) were as recommended and used by the Royal Netherlands Meteorological Institute in similar calculations (Kramer, 1956).

The FAO (Smith et al., 1991) recommended the following empirical expression for R„ and 5:

i ? „ = ( l ^ ) 5 - ( l - 8 , K , / 0.1 + 0.9—1 (17)

N

again using Brunt's formula for eatm, but with constants different from those used in (equation (16)), namely:

eatra =0.66 + 0.044377 (18)

and 5 is estimated with the Angstrom formula also, but with a = 0.25 and b = 0 .50:

5 = 5, Jo 0.25 + 0.50—1 (19)

Hereafter, the empirical expressions (14)—(16) are denoted as the KNMI method and the equations (17)—(19) as the FAO method. They differ only in the numerical values chosen for the various empirical constants.

Considering days without significant water stress, it is assumed that for days with XETO - kEaet < 25 W m"2 water stress can be ignored. This selection criterion was based on the experiences gained by Strieker (1981) with the Thom-Oliver (kET0) approach for the same data set. The period considered herein was mid-April to mid-October for 1979-1982 with 448 selected days.

RESULTS

In Table 1, a statistical summary is provided of the comparison between the measured daily mean evaporation and the estimates obtained with Makkink (equation (5)), Priestley-Taylor (equation (6)), Penman (equation (7)), Penman-Monteith


Table 1 Mean daily evaporation in energy units (W m"'1) evaluated with different methods for the four cases considered.

Method

Measured

Makkink Priestley-Taylor Penman Penman-Monteith Thom-Oliver FAO




Case

a a a a a a

b b b b b b

c=b c c c c c

d d d d d d

Mean (W rn2) 70.7

68.8 65.0 70.0 69.6 67.1 62.2

72.6 68.8 73.0 71.9 69.2 64.7

72.6 79.6 81.4 78.0 75.0 77.6

65.0 67.7 72.2 71.3 68.6 75.5

Ratio of means

1.028 1.088 1.010 1.016 1.054 1.137

0.974 1.028 0.968 0.983 1.022 1.093

0.974 0.888 0.869 0.906 0.943 0.911

1.088 1.044 0.979 0.992 1.031 0.936

Slope c

1.018 1.074 1.006 1.007 1.042 1.128

0.954 0.996 0.955 0.967 1.003 1.074

0.954 0.865 0.862 0.895 0.930 0.904

1.030 0.997 0.959 0.969 1.006 0.985

R2

0.77 0.89 0.88 0.82 0.79 0.86

0.71 0.76 0.82 0.77 0.76 0.80

0.71 0.71 0.79 0.75 0.75 0.79

0.41 0.52 0.67 0.67 0.66 0.68

e (W nf 2)

11.4 7.9 8.2

10.1 10.9 8.9

12.8 11.7 10.1 11.4 11.7 10.6

12.8 12.8 10.9 11.9 11.9 10.9

18.3 16.5 13.7 13.7 13.9 13.5

^-corrected

(W rn 2)

9.3 9.6

11.3 12.0 10.2

(equation (10)), Thom-Oliver (equation (12)) and FAO (equation (13)), respectively, for the four cases (a)-(d) considered. The factor c is the slope of the regression line forced through the origin: À£meas = ckE^, where XEmas and A.Ecaic are the measured and calculated daily XE values, respectively. Furthermore, the ratio of the mean daily values of XEmeiiS and A,£caic, the R2 values (R is the correlation coefficient), and the standard error of estimate e are shown. For case (a), a corrected e is also listed (explained later).

In Fig. 1, scatter plots are presented in which the measured daily evaporation (in energy units) is plotted against the modelled values for case (a). In Fig. 2, similar plots are shown for case (d).

For case (a), all methods have a slope close to 1, except the Priestley-Taylor equation and the FAO method, which have slopes of about 1.07 and 1.13, respectively. Apparently, the value of a = 1.28, as used earlier by Strieker (1981), seems somewhat too low. However, Strieker based his value on data of the mid-summer period and it is known that the Priestley-Taylor method starts to underestimate evaporation structurally outside this period for the mid-latitudes. The Priestley-Taylor and Penman approaches show the highest R2 values. The formulas based on Penman-Monteith (for which the slope is forced to be one) perform slightly worse. The Makkink method has the lowest R2, but its value differs only slightly from the more complicated Thom-Oliver method.


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For case (b), all slopes decrease about 5% compared to case (a). In addition, the R2

values decrease. This means that, for calculations of daily evaporation, ignoring G introduces a random error around a systematic one leading to an overestimation of the calculated evaporation.

In case (c), i.e. where G is ignored and net radiation is estimated from standard weather data and measured solar radiation, the results again change substantially, except for Makkink's equation which gives identical results for (b) and (c). The slopes become about 10% less than one, which indicates the introduction of a second systematic error arising from the radiation term. The values of R2 decrease further. Now the Priestley-Taylor formula has the lowest value. The Penman and FAO methods have the lowest scatter, but their slope is 0.9 or less.

Things become worse if solar radiation is estimated also, i.e. in case (d). Surprisingly, now all slopes are close to one (within 5%), but the R2 values are very low. Apparently, the empirical formulas for the estimation of net radiation do not perform well enough on a daily basis. This issue is discussed later. From the scatter plots shown in Fig. 2, it may be seen that these net radiation estimates lead to a nonlinear deviation.

It is evident from these results that the formulas that contain the most physics, i.e. the Penman-Monteith equation and the related Thom-Oliver and FAO methods, perform less well than those of Penman & Priestley-Taylor. The most simple formula, that of Makkink, has the most stable slope (it never deviates more than 5% from one). It should be noted that Makkink is the only formula that uses entirely independent input data, because it doesn't require the available energy A = (Rn - G). All other approaches need A as input, including the measured XE. Because of this, an artificial correlation is introduced between estimated and measured results. This is shown clearly in Fig. 3. The slope of the regression line forced through the origin is 0.86 and the corresponding R2 value is 0.87. This explains in part the relatively larger scatter obtained for Makkink. This feature is discussed later.

Because net radiation estimates appear not to perform too well, the estimated and measured net and solar radiation were compared separately. The results for net radiation are shown in Figs 4 and 5. In Fig. 4, the daily net radiation estimated with expressions (14)-(16), which is the KNMI method, are compared with the corresponding measured net radiation values, using measured solar radiation

0 20 40 60 80 100 120 140 160 Net Available Energy: A W m "2

Fig. 3 Measured evaporation in energy units vs measured available energy A.


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200

150

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The accuracy of net radiation estimates thus depends strongly on that of the solar radiation estimates. In Fig. 6 a comparison is made between 5 obtained with equation (15) and the corresponding measured values. It is seen that the scatter is quite high, whereas the relationship between measured and estimated S is nonlinear. This explains why the plots of measured vs estimated evaporation, in which net radiation and solar radiation are estimated, are also nonlinear (see Fig. 2). The linear regression line forced through the origin is Smeas = 1.075est, so there is also a systematic bias. Apparently the constants a and b in the Angstrom type of formula in the KNMI and FAO formulas cause systematic errors in S. For the data set used here, S = 50(0.27 + 0A6[n/N]) was found to be the best fit, as shown in Fig. 7, in which 5/50

is plotted against n/N. From this plot it is clear that the Angstrom type of formula behaves nonlinearly and that, at low values of n/N, the scatter is very large.

It is obvious that the empirical radiation formulas, which were obtained from the literature, do not perform very well. Considering the physics behind these empirical expressions, it is not surprising. It is clear also that the use of direct measurement of S will improve the net radiation estimates significantly. Slob (personal communications) showed several years ago that, if solar radiation is measured, daily mean net radiation can be estimated with the following simple expression:

R =(l-a)S-c,— 1 Sn

(20)


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0 50 100 150 200 Net Rad. Estimated Slob W m '2

Fig. 8 Measured net radiation vs net radiation estimated with the method of Slob (personal communication).

In Fig. 8 a scatter plot for this method is shown. The regression line through the origin is ^„.meas = 1.01/?„jest and R2 = 0.94, with a = 0.23 and cs = 135 W m"2. It should be noted that only c has been adjusted such that the slope of the regression line became close to 1. The R2 value of Slob's method is slightly better than that found for the KNMI (R2 = 0.93) or FAO (R2 = 0.92) estimates. But the latter methods require, besides S, temperature, water vapour and relative sunshine duration. This result is a strong indication that the empirical formulas for the longwave radiation are not very accurate. It is outside the scope of this paper to discuss the details of this item.

DISCUSSION

A micrometeorological data set, gathered over grassland in the Hupselse Beek Catchment over a period of four complete growing seasons (1979-1982) was analysed. Grass is often used as a reference crop in evaporation calculations based on standard weather data. Various methods for the crop reference évapotranspiration, as found in the literature, have been compared. Because, in practice, all required input data are often not available from direct measurements, four cases were considered.

It was found that, on a daily basis, the assumption that the soil heat flux can be ignored introduces—beside a random error (lower R2 compared to (a))—a systematic error of about 5%. The average value was 4.8 W nf2. For comparison, the mean net radiation was 86.6 W irf2 and the mean actual daily latent heat flux density was 70.7 W rn 2.

In the case that all input parameters are measured, the "old" Penman formula appears to perform better than all versions of the Penman-Monteith equation, here denoted FAO. At first sight this is surprising, because the Penman-Monteith formula is more physically based than Penman's equation. As pointed out by Thorn & Oliver (1977), the aerodynamic resistance term in the "old" Penman formula was derived empirically for matching the water loss of an evaporation pan, whose water surface is smoother than a grass surface but is considered to have no surface resistance. These two "errors" appear to cancel out each other and restrict the contribution of the


second term (rjra - 0). This is confirmed by the results presented here. The radiation based formula of Priestley-Taylor performs as well as the "old" Penman formula. Because the evaporation of short grass appears primarily to be determined by the available energy A, the second term, which contains the aerodynamic resistance (or wind function) and water vapour deficit, must play a minor role. This might explain the fact that the "old" Penman yields even better results than the different versions of Penman-Monteith. One may say that the Penman-Monteith-like equations give too much relative weight to the second term, thereby creating extra scatter.

Makkink's formula, which uses only incoming solar radiation and air temperature, has the lowest R2 value. However, it must be realized that for cases (a) and (b) all other methods contain measured A and Rn respectively, which, in turn, are highly correlated with the measured evaporation. This is shown clearly in Fig. 6, where the measured daily evaporation (in energy units) is plotted directly against A. The line is forced through the origin yielding a slope of 0.86 with R2 = 0.87. This last result is very close to, or even larger than, the R2 obtained for the Penman, Penman-Monteith, Thorn-Oliver and FAO methods. In this way, a very simple evaporation estimate, XE = 0.86A is obtained. The Priestley-Taylor method is slightly better, which means that the effect of the temperature dependence on s/(s + y) has some significance. An important conclusion is that evaporation of non-stressed grass in The Netherlands is primarily determined by the available energy A. Other factors, such as temperature, wind speed and water vapour deficit have second-order effects.

Another consequence of the strong impact of A is that all methods which are using measured A as input to estimate E will, inevitably, be highly correlated with the measured E in this study. Measurement errors in A hardly influence the correlation. Thus, one may consider the R2 values in Table 1 being the upper values for these methods. It is difficult to account for this extra correlation effect. A first estimate of it can be obtained by assuming for case (a) that A is affected by random errors only. The random measuring error in A for daily values was estimated to be at least 10% of the mean, which corresponds to about 8 W m"2. Since XE is about 0.86A, this corresponds to a random error of about 7 W nf2 in XE. In the estimations of XE, the corresponding error will be less, depending on the method. For the sake of simplicity, a single value of 5 W m"2 was used for the random error for all methods with A as input. Next, it was noted that, in general, the standard error of estimate is:

in which indices "est" and "meas" denote "estimated" and "measured" respectively and Gmeas is the standard deviation of the measured XE, which is 23.8 W m"2. In Table 1 the e values obtained for the various methods and cases are listed. The impact of random measuring errors in A, which are assumed to lead to a mean random error in

the evaporation estimates of 5 W m"2, can now be estimated withecomcted = ve 2 + 52

(W m"2), where eCOrrected is the corrected standard error of estimate. In Table 1 £COrrected is listed for case (a). It is seen that the skill of the Makkink method is very close to that of the Penman-Monteith type of estimates, which require more input data. The Priestley-Taylor and "old" Penman formulas perform still better, but strictly speaking


a higher value than 5 W m"2 in the eCOrrected formula should have been used for these methods, in which A has a greater weight.

Taking into account that the slope of the Makkink formula did not change too much for the four cases considered, it is concluded that a Makkink type of equation is very suitable for practical applications.

Because evaporation of non-stressed grass is strongly related to net radiation, the accuracy of the empirical formulas to estimate Rn largely determines the accuracy of evaporation estimates when net radiation is not measured directly. In this study, the Angstrom type equation, as used in the literature for estimation of incoming solar radiation, was found not to be very accurate. In order to avoid systematic deviations, it must be calibrated for local conditions.

CONCLUSIONS AND SOME RECOMMENDATIONS

The behaviour of different formulas for crop reference evaporation published in the literature has been investigated. It was found that if all required input data are measured, the Priestley-Taylor and the "old" Penman formulas yield the best results. The assumption that soil heat flux can be neglected introduces—apart from a random error—a systematic error of roughly +5% in the evaporation flux for the growing season and for the conditions used.

The empirical expressions used to estimate net radiation from sunshine duration, temperature and humidity appeared to perform rather poorly. These net radiation estimates improved significantly if solar radiation was measured directly. The empirical expression proposed by Slob (unpublished) that requires only incoming solar radiation as input, yielded similar or even better results than the other more complicated expressions. It can be concluded that the formulas to estimate net longwave radiation as used by Penman and Brunt are not very accurate.

This study reveals that evaporation of unstressed grass is primarily determined by the available energy A, i.e. net radiation minus soil heat flux density. Indeed, very good evaporation estimates were obtained by using \E = 0.86A. The fact that the \E is strongly dependent on A relatively flattered the results obtained with the methods by Priestley-Taylor, Penman & Penman-Monteith. When a correction for this effect was introduced, it was found that the Makkink approach yielded results that are only slightly less accurate than those approaches. This makes the Makkink method very attractive for practical applications. These findings support the use of Makkink's formula for routine calculations of crop-reference évapotranspiration as has been carried out by the Royal Netherlands Meteorological Institute since 1987 (De Bruin & Lablans, 1998). Makkink's formula becomes additionally attractive for routine calculations of crop-reference évapotranspiration, because its outcome is primarily determined by the incoming solar radiation, which can be retrieved from hourly images of geostationary satellites. In this way the daily crop-reference évapotranspiration can be mapped at one central location for large regions, or even countries, on a scale of about 5 x 5 km and can be made available to the user community by Internet.

Since incoming solar radiation is a quantity that is independent of surface properties, and, since solarimeters are very accurate nowadays, it is strongly recommended to include solar radiation sensors on standard meteorological stations.

4 0 6 H. A. R. de Bruin & J. N. M. Strieker

Net radiometers should be most useful, but their measurements are sensitive to the surface albedo and surface temperature. Therefore it requires much more intensive maintenance of the grass cover on standard climatologie al stations. In dry climates an additional problem is that the grass has to be irrigated to keep it green.

Acknowledgements The authors would like to thank the Ministry of Public Works (Rijkswaterstaat), Arnhem, The Netherlands for processing the data of the Hupselse Beek for so many years. They are also grateful to Jacques Kole who is in charge of managing the availability of the Hupselse Beek data records.

REFERENCES

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De Bruin, H. A. R. (1987) From Penman to Makkink. In: Evaporation and Weather (ed. by C. Hooghart), 5-30. Proceedings and Information no. 39, Commission for Hydrological Research TNO, The Hague, The Netherlands.

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Makkink, G. F. (1957) Testing the Penman formula by means of lysimeters. J. Instn Wat. Engrs 11, 277-288. Monteith, J. L. (1965) Evaporation and environment. Proc. Symp. Soc. Exper. Biol. 19,205-234. Penman, H. L. (1948) Natural evaporation from open water, bare soil, and grass. Proc. Roy. Soc. A. 193T, 120-145. Penman, H. L. (1956) Evaporation: an introductory survey. Neth. J. Agric. Sci. 4, 9-29. Priestley, C. H. B. & Taylor, R. J. (1972) On the assessment of surface heat flux and evaporation using large-scale

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Received 13 November 1998; accepted 29 December 1999

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Evaporation of grass under non-restricted soil moisture conditions