J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116 109
c
p
the specific heat of air and K
V
and K
H
the eddy diffusivities for water vapour and heat, respectively.
From Eqs 2, 5 and 6 and assuming that K
V
≈ K
H
, the Bowen ratio can be expressed as follows: β = γ
δT δz δeδz
7 In measurement practice, the gradients in Eq. 7 are
replaced by differential quotients Black and Mc- Naughton, 1971; Monteith, 1975
∂e ∂z
= 1e
1z =
e
2
− e
1
z
2
− z
1
8 ∂T
∂z =
1T 1z
= T
2
− T
1
z
2
− z
1
9 where e
1
and e
2
are water vapour pressure measured at two levels 1 and 2, T
1
and T
2
the air temperatures measured at two levels 1 and 2, and z
1
and z
2
the height above the ground of two measurement levels 1 and 2.
The Bowen ratio method of LE and S estimation described above, is commonly used by many inves-
tigators Spittlehouse and Black, 1981 and there are even commercial measurement units, based on this
method. Unfortunately, using only two measurement levels there is a possibility of errors when measure-
ments are carried out in a patchy landscape. Tem- perature and water vapour sensors must be located
within the internal boundary layer characteristic of the ecosystem under investigation fetch. The only way
to meet the fetch requirements, using only two layers of measurement, is to carry out the measurements of
latent and sensible heat fluxes above relatively large fields. In Middle Europe the landscape is often very
patchy and it is hard to meet the fetch requirements. Therefore, the modification of Bowen ratio method
was proposed by Olejnik, 1996 in which, air temper- ature and water vapour are measured at five heights.
The advantage of several measurement heights is that height of the adjusted surface layer or internal bound-
ary layer can be identified. Having the data on air tem- perature T
z
and vapour pressure e
z
at five levels it is possible to estimate both as a function of height as
follows: T
z
= f
1
z 10
e
z
= f
2
z 11
where z is height above the ground. In Eqs. 8 and 9, the differential quotients can be
replaced by the derivatives of functions described in Eqs. 10 and 11 i.e. df
1
dz and df
2
dz. The calculation of latent LE and sensible S heat
fluxes by the use of modified Bowen ratio method is shown on the basis of one selected measurement
cycle of air temperature and vapour pressure profiles Fig. 1. Using the psychrometric equation the water
vapour data are calculated on the basis of dry- and wet-bulb temperatures at five levels. The results of the
calculations for 40 profile series during one measure- ment cycle is shown in Fig. 1a. On the basis of these
profiles, mean values of air temperature T and wa- ter vapour pressure e at five levels were calculated
Fig. 1b. Using statistical and numerical methods, the air temperature and vapour pressure, as functions of
height, can be found Eqs. 10 and 11, solid line in Fig. 1b. The result of calculations of LE and S are
shown in Fig. 1c.
After determination of all four components of heat balance equation Eq. 1, evapotranspiration can be
calculated from latent heat flux density. These values obtained from measurements were then compared with
evapotranspiration simulated by THESEUS.
3.3. Measurement periods and types of surfaces The measurements were carried out during 8 pe-
riods in the fields near Turew and Müncheberg. In Turew, measurements were made in alfalfa and sugar
beet crops as well as a bare soil, while at Müncheberg, measurements were made in oat, wheat and sun-
flower crops. Measurements during period from 1–4 were carried out in Turew and periods from 5–8 in
Müncheberg. The lengths of measurement periods varied from 7 days periods 6–8 in Müncheberg
to 16 days periods 3 and 4 in Turew. The whole measurement set consisted of 81 days.
4. Results and discussion
For all measurement periods P1–P8, the follow- ing average data are shown in Fig. 2: air tempera-
ture, heat balance components, evapotranspiration es- timated by the Bowen ratio method ETRB and sim-
ulated by THESEUS ETRT as well as the average ratio ETRETP. In addition, the sums of precipitation
110 J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116
Fig. 1. The results of applications of measurement system for modified Bowen ratio method: a results of measurements of air temperature and water vapour pressure profiles; b statistical approximation of functions e
z
and T
z
on the basis of measurements and calculations of mean profiles; c latent LE and sensible S heat fluxes estimation as a function of height z.
for all measurement periods are shown in the upper part of Fig. 2. The daily rate of evapotranspiration
measured and modelled for all eight measurement periods is shown in Fig. 3. The measurement sites
and time of investigation were chosen to cover not only different plants see Section 3.3 but also differ-
ent weather conditions. Therefore, measurements were carried out in varied conditions from the point of view
of mean daily air temperature T, daily rainfall P and type plant cover: dry and warm with plants P1,
T =
14.2
◦
C, P = 1.7 mm per day, alfalfa, dry and warm without plants P2, T = 14.2
◦
C, P = 1.7 mm per day, bare soil, dry and cool with plants P4, T =
9.3
◦
C, P = 1.6 mm per day, sugar beet, dry and cool without plants P3, T = 9.4
◦
C, P = 1.7 mm per day, bare soil, wet and warm with fully developed plants
P7, T = 13.2
◦
C, P = 7.4 mm per day, wheat and wet and warm with not fully developed plants P6,
T = 13.3
◦
C, P = 7.4 mm per day, sunflower. Dur- ing the measurement periods different distributions of
precipitation were noted: during measurement periods P3 and P4 there were 8 days with no rain but during
J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116 111
Fig. 2. Comparison of averages or sums of meteorological parameter, heat balance components and evapotranspiration of eight measurement periods P1–P8.
periods P6 and P7 the significant precipitation was ob- served every second day including a heavy storm on
13 June 1995 when rainfall P = 23.9 mm. Because of such different conditions the results of measurements
of heat balance components varied significantly: net radiation R
n
from 9 Wm
− 2
on 24 September 1997 for bare soil to 232 Wm
− 2
on 12 June 1994 for oat, in these days latent heat flux varied from −10 Wm
− 2
to −
203 Wm
− 2
, respectively. Also, the ratio of ETRETP varied significantly, from 0.21 on 6 September 1995
— bare soil, to almost 3 on 9 June 1995 — wheat that day, the average temperature decreased by 2.8
◦
C in comparison with the day before, ETP was only 0.3 mm
while ETR 0.9 mm. The different canopy and weather conditions caused variation of daily evapotranspiration
rate: from 0.3 mm on 24 September 1997 P3, bare soil to 7.2 mm on 12 June 1994 P5 oat Fig. 3.
Average values calculated for measurement pe- riods from P1 to P8 of Bowen ratio varied from
− 0.04 for period P4 sugar beet to 1.0 for P2 bare
soil Fig. 2. The average value of net radiation R
n
varied from 50 Wm
− 2
P2, bare soil to 147 Wm
− 2
P5, oat and for the same periods average latent heat flux varied from −24 Wm
− 2
P2 to 133 Wm
− 2
P5, which gives ETR of 0.8 and 4.7 mm per day, respec-
tively. The minimum sum of evapotranspiration was equal to 8.6 mm P2, bare soil, 10 days while the
maximum was equal to 37.6 mm P5, oat, 8 days Fig. 2. The average ETRETP ratio depends on
canopy and weather conditions and varied from 0.4
112 J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116
Fig. 3. Comparison of evapotranspiration estimated by THESEUS model ETRT and calculated from latent heat flux measurements by the use of modified Bowen ratio method ETRB for 81 days 8 measurement periods P1–P8.
period P2 to 1 and more periods P5–P8. The high values of ETRETP ratio were noted during measure-
ment periods with favourable conditions for strong evapotranspiration well developed plant canopy, high
temperature and moist upper soil layer. The comparison of daily rates of evapotranspira-
tion in mm calculated from latent heat flux den- sity ETRB and simulated by THESEUS ETRT
shows a very good agreement. For all 81 measure- ment days the sum of ETRB and ETRT were 174.1
J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116 113
and 165.2 mm, respectively. The difference is 8.9 mm means that the relative error of estimation of evapo-
transpiration by the THESEUS model is about 5 for 81 days of comparison. The relative error of estima-
tion of the daily rate of evapotranspiration was calcu- lated using the following formula:
REE = ETRT
x
− ETRB
x
ETRB
x
12 where REE is relative error of estimation, x the day
number in the individual measurement period. For the whole set the average absolute daily error
is equal to 0.1 mm. In Fig. 4, the relationship between the daily values
of evapotranspiration estimated by both methods are shown. The linear regression was calculated on the
basis of the whole data set. It is easy to see that the THESEUS model underestimates daily rate of evap-
otranspiration particularly for evapotranspiration rate greater than 2 mm per day. The regression equation
for the whole set is as follows r = 0.958:
ETRB = 1.15 ETRT − 0.22 13
Additionally, in Fig. 4 the results of REE calcula- tions Eq. 12 are presented for all eight measure-
ment periods separately and for the whole data set.
Fig. 4. Comparison of evapotranspiration simulated by THESEUS ETRT and calculated from latent heat measured by the use of
modified Bowen ratio method ETRB.
In the case of the three periods, the REE was ≥10. The measurement of periods 1 and 2 were carried out
at two fields P1 alfalfa, P2 bare soil in the same time from date 2 September 1995 to 11 September 1995.
The summer 1995 was extremely dry only about 40 of long term average precipitation did occur. The
measured cumulative evapotranspiration of alfalfa was 18.0 mm, while the simulated value was 16.2 mm. In
this case, THESEUS underestimated the daily evapo- transpiration because the model is based on calculat-
ing the soil moisture of the upper layer 0–90 cm. The upper layer of soil in that summer was very dry but
alfalfa has a very deep root system which allowed the use of soil water from deeper layers. Under such ex-
treme precipitation conditions the relative error of the THESEUS estimation is still at an acceptable level. At
the same time measurements were carried out on bare soil period P2. This time the evapotranspiration esti-
mation by THESEUS was higher by about 30. The measurement site was called bare soil but in fact about
30–40 of the field was covered with straw lying in rows after harvesting. The straw layer decreases sig-
nificantly the rate of evaporation under the straw the soil was wet. The THESEUS model structure does
not allow the inclusion of such conditions straw on bare soil, and consequently, for that period period
P2 the estimation by the model was made for bare soil. So, the very high relative error of evapotranspira-
tion estimation for period P2 30 results not from an incorrect parameterisation for bare soil used in THE-
SEUS but rather from the absence of an appropriate part in the THESEUS model bare soil partly covered
by straw.
For period P4 there is also a relatively high error of evapotranspiration estimation by the model −14.9.
The field was covered with sugar beet in good condi- tion and was located close to the bare soil field. The
sugar beet still had a water demand and were able to evapotranspire more than 2 mm per day. The weather
during this period was not too warm and the net radia- tion was lower than normal. Consequently, the deficit
of energy for evapotranspiration could only be ac- counted for by advected sensible heat it causes the
highest ratio of ETRETP, Fig. 2. The average sen- sible heat flux density during the measuring period
P4 was the only one with positive values in average +
2 Wm
− 2
; Fig. 2, P4. The maximum value of sensi- ble heat coming from the atmosphere to the sugar beet
114 J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116
Fig. 5. Relative error of evapotranspiration estimation by THESEUS model for 81 measurement days.
field was 14 Wm
− 2
on 22 September 1997. For the whole period P4 the ratio ETRETP was equal to 1.26
Fig. 2. The comparison of ETRB and ETRT for period P4 suggests that if the THESEUS is used for
evapotranspiration estimation on the landscape scale special attention should be paid to areas with possi-
ble advection, otherwise the error of estimation can be significantly high.
The REE of the daily rate of evapotranspiration Eq. 12 decreased with increasing daily evapotran-
spiration rate Fig. 5. If the daily rate of evapotranspi- ration is low 0.5 mm or less then the REE can be very
high for 0.5 mm about 45. The measurement points with high relative error mainly come from period P2
bare soil with straw. The reason for such a poor es-
Fig. 6. Relative error of evapotranspiration estimation by THESEUS model in function of integration time separate for eight measurement periods.
timation for this period was described earlier. There are also some measurement points with high error of
estimation in other periods but the daily evapotran- spiration rate for these days is very low 0.4–0.8 mm
per day. Although these estimations have a very high error, because of very low ETR it does not influence
significantly the evapotranspiration estimation for a longer period week, month or season. For higher
rates of evapotranspiration right part of Fig. 5 the evapotranspiration estimated by the model shows an
excellent agreement with measurements. It means that for conditions which increase the evapotranspiration
high soil moisture, high temperature, strong wind, high R
n
and well developed plant canopy a very good parameterisation was achieved. For such conditions
J. Olejnik et al. Agricultural and Forest Meteorology 106 2001 105–116 115
the REE of daily rate of modelled evapotranspiration is about 7. On day 12 June 1994 period P5 — oat
the highest evapotranspiration was measured, 7.2 mm per day Fig. 3, P5, the average sensible heat flux
density was −28 Wm
− 2
while LE was −203 Wm
− 2
. The value of 7.2 mm per day is close to the maximum
value that is possible under Central Europe climato- logical condition. Even for such a day extreme high
net radiation, R
n
was equal to 232 Wm
− 2
the THE- SEUS simulation shows a very good agreement with
the measurements: ETRT is equal to 6.1 mm per day, which is only 15 less than measured.
Finally, the changes of average relative error of es- timation AREE of the evapotranspiration simulation
by the THESEUS model was calculated as a function of integration time. In Fig. 6, the results of such an
analysis are shown. The time of integration for calcu- lations of AREE was extended to the length of every
measurement period from 7 to 16 days — Fig. 6. It can be seen that AREE only in periods P5 and P8 oats
and wheat for 1 day of integration is smaller than 10. But if the AREE is calculated for time of integra-
tion longer than 5 days then it comes to that level for almost all measurement periods. Only in case of peri-
ods P2 and P4 is the AREE value still high even for the maximum time of integration 10 and 16 days, respec-
tively. For period P2 it is again the “straw problem” of bare soil evaporation simulation by the THESEUS
model. For period P4 it is the influence of advection on the heat balance structure of the sugar beet field,
which currently cannot be included in the model.
5. Summary and conclusions
