118 D. Amarakoon et al. Agricultural and Forest Meteorology 102 2000 113–124
5. Data analysis
The data analysis involved the estimation of day- time 07:00–18:00 hours values of i 20 min and
hourly latent heat fluxes, ii average evaporation for the three periods of data collection and iii evapora-
tion on individual days using Eq. 6, and followed by comparisons with the corresponding measured values.
The 20 min and hourly values of latent heat flux were estimated using a data set formed by combining data
from all the three periods, January to February of 1994, January to February of 1990 and August to September
of 1989. The individual days used in the calculations are listed in Table 3 including daytime average air
temperature θ , vapor pressure deficit δe which is equal to e−e and wind speed u. The days listed in
Table 3 had data for at least 8 h during daytime. Two values for the parameter α
′
in Eq. 6 were used in the calculations and β was kept at 20 W m
− 2
. One value of α
′
was 0.95, which was the average value deduced by De Bruin and Holtslag 1982 using daytime hourly
values of latent heat flux for their normal period, and the other was 1.05. The motivation for testing 1.05
for α
′
was the following. As stated in the study of De Bruin and Holtslag 1982, Priestley–Taylor pa-
rameter α and the parameter α
′
have shown similar behavior and daily mean value of α has appeared to be
about 10 greater than its hourly value during day- time. Therefore we could expect α
′
to be about 10 greater when daily values are considered, for a given
β . Thus, it is justified to test the value of 1.05 for α
′
. For further evaluation of the results, we estimated i
daytime evaporation on days listed in Table 3 using
Table 3 Mean values of daytime temperature θ , vapor pressure deficit δe and wind speed u for selected days
Day
a
θ
◦
C δ
e mb
u m s
− 1
Day θ
◦
C δ
e mb
u m s
− 1
1989227 31.4
14.1 4.7
199035 25.4
10.8 1.5
1989228 31.0
14.5 2.4
199038 27.4
14.5 2.9
1989243 31.1
11.8 3.4
199040 27.6
13.6 3.0
1989247 32.2
13.2 2.3
199041 27.3
13.7 2.9
1989248 31.6
13.0 2.2
199042 28.1
14.5 4.0
199020 30.2
21.2 2.9
199432 24.4
9.8 1.2
199021 30.0
18.4 4.8
199435 26.7
11.3 1.3
199027 29.2
17.1 2.2
199436 26.9
9.9 2.4
199033 28.2
14.8 4.4
199439 28.0
15.3 2.0
199034 29.0
17.7 5.0
199440 27.6
19.2 3.9
a
Day is given as: year-day number.
Penman 1948 scheme given by 1 with fu from Eq. 2 and Priestley and Taylor 1972 scheme given by
Eq. 4 with 1.26 and ii daytime 20 min and hourly values of latent heat flux using Eq. 5 with α=1.26
for the combined data set. The values estimated were compared with the corresponding measured values.
A value of 0.674 mb K
− 1
was used for γ in the analysis. This value corresponds to p=1014 mb av-
erage measured pressure, c
p
= 1005 J kg
− 1
K
− 1
and λ=
2.43×10
6
J kg
− 1
Brutsaert, 1982. The values of 1
and e were estimated employing Lowe’s polyno- mials Lowe, 1977 for 1 and e, and our air temper-
ature measurements.
6. Results and discussion
Table 4 presents the linear regression results from the comparison of the measured 20 min and hourly la-
tent heat flux with the corresponding values estimated using Eq. 6 and Priestley and Taylor 1972 scheme
with α=1.26. The first value inside the parentheses in every column corresponds to α
′
= 1.05 and the sec-
ond value is from Priestley and Taylor 1972 scheme with α=1.26. The value outside the parentheses cor-
responds to α
′
= 0.95. Figs. 1 and 2 illustrate the com-
parison of hourly values for α
′
= 0.95 and α
′
= 1.05,
respectively. The results in Table 4 show that for the 20 min and hourly time steps, the correlation coeffi-
cients are better than 0.95 for all the cases considered, indicating good correlation between the measured
values and the values estimated using Eq. 6 with α
′
= 0.95 and 1.05, and Eq. 5 with α=1.26. The
D. Amarakoon et al. Agricultural and Forest Meteorology 102 2000 113–124 119
Table 4 Linear regression results of the comparison between the measured
latent heat flux and the estimated flux using i De Bruin and Holtslag scheme given by Eq. 6 and ii Priestley–Taylor scheme
given by Eq. 5 with α=1.26
Quantity
a
20 min values Hourly values
b
M 1639
423 Y
av
W m
− 2
256.2 284.2
S 1.11 1.0, 0.83
1.12 1.02, 0.84 I
W m
− 2
− 5.7 −3.6, 16.4
− 8.4 −6.2, 14.6
η W m
− 2
56.5 51, 68 50.8 42, 60.4
r 0.954
0.961 η
Y
av
22 20, 26 18 15, 21
a
M= number
of data
values, S=
slope, I=
intercept, Y
av
= average measured latent heat flux, r=correlation coeffi-
cient and η=root mean square error given by: {6measured
value−estimated value
2
M }
12
. The first value inside parentheses is from Eq. 6 with α
′
= 1.05 and the second value is from Eq.
5 with α=1.26. The value outside the parentheses is from Eq. 6 with α
′
= 0.95. The data set used is the combination from the
three periods of data collection.
b
Hourly values refer to average of three successive 20 min values within 1 h.
slopes and the root mean square errors, however, show differences. For both time steps, the application of Eq.
6 has produced slopes greater than 1.0. Based on the closeness to 1.0, values for the slope with α
′
= 1.05 are
Fig. 1. Comparison of the measured daytime hourly latent heat flux with the flux estimated using Eq. 6 and α
′
= 0.95, for the
combined data set. Fig. 2. Comparison of the measured daytime hourly latent heal
flux with the flux estimated using Eq. 6 and α
′
= 1.05, for the
combined data set.
better than those with 0.95. The improvement is about 11 and it is visible in Figs. 1 and 2, especially at high
l values. The root mean square errors as a percentage
of the measured average values are smaller than 22 with 2–3 improvement by the use of 1.05 for α
′
. It is seen from the results in Table 4 that the use of
Eq. 5 with α=1.26 has produced slopes 0.84 and 0.83 which are much smaller about 17 than 1.0
and root mean square errors higher 3–6 than those resulted by the use of Eq. 6. The slope values much
smaller than 1.0 is an indication of the fact that the val- ues estimated by Eq. 5 with α=1.26 are higher than
the measured values, on the average. This implies that the blending of data from different periods and hours
has resulted in a data set that corresponds to a situa- tion where the average surface moisture level is below
saturation. This type of situation is useful in testing Eq. 6 because the period of study in De Bruin and
Holtslag 1982 for which α
′
= 0.95 and β=20 W m
− 2
were suitable has been referred to as the normal pe- riod, and not wet or saturated. The mean value of 1.12
reported in the study of De Bruin and Holtslag 1982, for hourly values of the latent heat flux, supports the
classification of the period as a normal period. The mean value of α, based on our hourly values of latent
heat flux is 1.17. A comparison of our correlation
120 D. Amarakoon et al. Agricultural and Forest Meteorology 102 2000 113–124
Fig. 3. Daytime variation of the global short-wave radiation for the periods: August to September of 1989, January to February
of 1990 and January to February of 1994.
coefficient and the root mean square error obtained by applying Eq. 6 and α
′
= 0.95 with the corresponding
De Bruin and Holtslag 1982 values indicates that our results are closer within 1 to theirs. The correla-
tion coefficient obtained by them was 0.97 and the root mean square error was 17 of the average measured
hourly latent heat flux. A comparison of our measured average latent heat flux with the average latent heat
flux of De Bruin and Holtslag 1982, for hourly val- ues, shows that our value 284 W m
− 2
is 2.27 times greater than theirs 125 W m
− 2
. The latent heat flux levels in the tropics are higher because of higher ra-
diation levels, shown in Fig. 3. The above results in- dicate that the skill of De Bruin and Holtslag 1982
scheme is consistent even at higher flux levels and slightly wetter conditions our average value for α is
greater than theirs by 5.
Table 5 Daytime 07:00–18:00 hours measured average evaporation and estimated values using Eq. 6 with α
′
= 0.95 and 1.05, for the three
periods of data collection
a
Quantity August–September 1989
January–February 1990 January–February 1994
E mm, measured
4.21 3.65
2.89 E
mm, using Eq. 6 with α
′
= 0.95
3.73 11.4 3.35 8.2
2.87 1 E
mm, using Eq. 6 with α
′
= 1.05
4.09 2.8 3.68 1
3.14 8.7
a
The percentages inside the parentheses are the deviations from the measured values.
Table 5 presents daytime average evaporation mea- sured and calculated using Eq. 6 with α
′
= 0.95 and
1.05 for the three data collection periods. The percent- age deviations given inside the parentheses are less
than 11.4 with α
′
= 0.95 and 8.7 with α
′
= 1.05.
These results indicate that Eq. 6 satisfactorily esti- mates daytime average evaporation over larger time
steps of the order of several days, 28–32 days, and α
′
= 1.05 provides better results. The improvement in
August to September of 1989 and January to Febru- ary of 1990 by the use of α
′
= 1.05 in Eq. 6 is more
than the offset in January to February of 1994. The percentage deviation d as used in our work is given
by Eq. 14.
d = 100 ×
measured value − estimated value measured value
14 Table 6 presents the results from the comparison of
estimated daytime evaporation with the correspond- ing measured values on individual days listed in Table
3. The evaporation was estimated using i De Bruin and Holtslag 1982 scheme with α
′
= 0.95 and 1.05,
ii Priestley and Taylor 1972 scheme with α=1.26 and iii Penman 1948 scheme given by Eq. 1 with
f u from Eq. 2. Figs. 4 and 5 illustrate these com-
parisons. A total of 20 days were used and these days had data covering at least eight daytime hours. From
the results for the correlation coefficients, root mean square errors and average percentage deviations in
Table 6, following observations can be made. i Priest- ley and Taylor 1972 scheme, given by Eq. 4 in our
work, with α=1.26 estimates evaporation well. This agrees with the observation of De Bruin and Holtslag
1982 on daily values. An inspection of Fig. 5 re- veals that the estimates are mainly overestimates 14
from 20 values with a few about four values hav- ing higher percentage deviations, greater than about
11. ii The results, using Eq. 6 with α
′
= 0.95,
D. Amarakoon et al. Agricultural and Forest Meteorology 102 2000 113–124 121
Table 6 Results from the comparison of daytime measured evaporation and evaporation estimated using i De Bruin and Holtslag scheme given
by Eq. 6 with α
′
= 0.95 and 1.05, ii Priestley–Taylor scheme, given by Eq. 4 with α=1.26 and iii Penman scheme, given by Eq. 1
with fu from Eq. 2, for the days in Table 3 Quantity
a
Eq. 6 with α
′
= 0.95
Eq. 6 with α
′
= 1.05
Eq. 4 with α=1.26 Eqs. 1 and 2
S 1.27
1.15 0.95
0.91 I
mm −
0.34 −
0.32 0.02
0.08 r
0.95 0.95
0.95 0.88
η mm
0.566 0.365
0.347 0.584
E
av
mm 3.34
3.34 3.34
3.34 η
E
av
17 11
10 17
d
av
12.2 3.8
5.5 9.3
a
The definitions of the quantities S, I, r, and η are the same as those in Table 4, except that the analysis is for the evaporation E. E
av
= average of measured evaporations and d
av
= average of percentage deviations.
are satisfactory but there is good 6–8 improve- ment in the root mean square error value and average
percentage deviation with α
′
= 1.05. The improvement
is visible in Fig. 4. Regarding the improvement and the use of 1.05 for α
′
, it is necessary to mention that in a few cases 4 out of the 20 the use of 1.05 for
α
′
worsened the percentage deviation by about 4. But this offset was tolerable compared to the overall
improvements produced. iii The level of agreement between values estimated by Eq. 1 with fu from
Fig. 4. Comparison of the daytime measured evaporation on days given in Table 3 with values estimated using Eq. 6.
Eq. 2 and measured values is acceptable, but lower than the level of agreement from the use of Eqs. 4
or 6. This indicates that Eq. 6 with α
′
= 1.05 or
Eq. 4 with α=1.26 are better choices for estimat- ing daytime evaporation on a daily basis for the type
of study we conducted. However, from Fig. 5 it can be seen that on some days about 6 percentage devi-
ations are small, less than about 6, indicating that the conditions had been favorable for potential evapo-
transpiration.
Fig. 5. Comparison of the daytime measured evaporation on days given in Table 3 with values estimated using Eqs. 1 and 4.
Wind function fu is from Eq. 2.
122 D. Amarakoon et al. Agricultural and Forest Meteorology 102 2000 113–124
The results from Eq. 4 with α=1.26 and those from Eq. 6 with α
′
= 1.05 are very close as seen
in Table 6. Therefore, it appears that Eq. 4 with α=
1.26 is as equally good as Eq. 6 with α
′
= 1.05
for estimating daytime evaporation on a daily basis in practice. But a careful analysis will indicate that Eq.
6 with α
′
= 1.05 or 0.95 may be better in practical ap-
plications. A closer look at Figs. 4 and 5 reveals that, the days on which Eq. 4 with α=1.26 produces large
negative percentage deviations are well modeled by Eq. 6 with α
′
= 1.05 or 0.95. Higher negative percent-
age deviations imply larger overestimates for evapo- ration by Eq. 4 and indicate below saturation surface
moisture conditions. These conditions normally exist in practice, especially on agricultural lands. Such sit-
uations can be better described, including the daytime variation of evaporation, by Eq. 6 as the example in
Fig. 6 illustrates. Fig. 6 shows the daytime variation of evaporation in 20 min time steps for the 40th day
of January 1994. Evaporation measured, evaporation calculated using Eq. 4 with α=1.26, evaporation
calculated using Eq. 6 with α
′
= 0.95 and evapora-
Fig. 6. Daytime variation of global short-wave radiation, evapora- tion measured, evaporation estimated using Eq. 6 with α
′
= 0.95
and 1.05, and Eq. 4 with α=1.26, for the 40th day of January 1994. The total evaporation values are given inside parentheses.
tion calculated using Eq. 6 with α
′
= 1.05 are shown
as a function of time. The graphs show that Eq. 6 models the variation well, but Eq. 4 fails d=18.6
especially around mid-day. The daytime evaporation estimated using Eq. 6 agree well with the measured
value. The percentage deviation with α
′
= 0.95 was
2.8 and the percentage deviation with α
′
= 1.05 was
6.8, which are good. Apparently, this day was one of the days for which the value 1.05 for α
′
produced an offset of 4. This type of departure of the evaporation
estimated using Eq. 4 with α=1.26, around mid-day, was also observed with other days having higher
percentage deviations. A reason can be the surface moisture depletion due to higher evaporation around
mid-day influenced mainly by radiation. For the 40th day of January 1994, the variation of the global
short-wave radiation is also shown in Fig. 6. The global short-wave radiation around noon on this day
is about 850 W m
− 2
. Evaporation around mid-day is about 0.225 mm for 20 min. There is good correlation
between evaporation and radiation. The correlation coefficient between evaporation and global short-wave
radiation for this day is 0.91. Apart from radiation, the other factors that can influence evaporation are vapor
pressure deficit and wind speed. The average values of daytime vapor pressure deficit and wind speed for the
40th day of January 1994 were 19.2 mb and 3.9 m s
− 1
, respectively. The correlation coefficients for the com-
parison of evaporation with vapor pressure deficit and wind speed on this day were 0.89 and 0.73, respec-
tively, which indicate that radiation has a stronger influence on evaporation followed by vapor pressure
deficit. Further studies, however, are necessary to substantiate our observations on moisture depletion.
7. Concluding remarks
