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Simplified forms for the standardized FAO-56 Penman–Monteith reference evapotranspiration using limited weather data
Accepted Manuscript Simplified forms for the standardized FAO-56 Penman-Moneith reference evap‐ otranspiration using limited weather data John D. Valiantzas PII: DOI: Reference:
S0022-1694(13)00650-1 http://dx.doi.org/10.1016/j.jhydrol.2013.09.005 HYDROL 19085
To appear in:
Journal of Hydrology
Received Date: Revised Date: Accepted Date:
28 May 2013 5 September 2013 7 September 2013
Please cite this article as: Valiantzas, J.D., Simplified forms for the standardized FAO-56 Penman-Moneith reference evapotranspiration using limited weather data, Journal of Hydrology (2013), doi: http://dx.doi.org/10.1016/j.jhydrol. 2013.09.005
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1
1
Simplified forms for the standardized FAO-56 Penman-Moneith
2
reference evapotranspiration using limited weather data
3
John D. Valiantzas*
4 5
Abstract
6
New simple algebraic expressions equivalent in accuracy to the “standardized” FAO-
7
56 Penman-Monteith daily reference crop evapotranspiration (ET0) computation
8
procedure are derived. The suggested formulas are based on extensions made to a
9
previously developed simple algebraic formula for the Penman evaporation equation.
10
The derivation of the new formulas is based on simplifications made and the
11
systematic analysis on the correspondence between the FAO-56 Penman-Monteith
12
equation and the standardized Penman’s equation. The ET0 calculated by the new
13
formulas is easy to use for routine hydrologic applications requiring routine weather
14
records usually available at standard weather stations: air temperature, T (oC), solar
15
radiation RS (MJ/m2/d), relative humidity, RH (%), and wind velocity, u (m/s). For
16
places where not all these data are available (or reliable), new expressions which does
17
not require wind speed and/ or solar radiation data are proposed. A simplified formula
18
for estimating reference crop evapotranspiration
19
maximum and minimum air temperatures), and Tdew (the dew point temperature) or
20
RH data alone is derived. The performance of the new derived formulas was tested
21
under various climatic conditions using a global climatic data set including monthly
22
data as well as daily data obtained from weather stations.
requiring Tmax and Tmin ( the
23 24
Key-words: Penman-Monteith, Penman equation,
25
hydrologic models, water resources management
26 27 28
*
reference evapotranspiration,
Division of Water Resources Management, Department of Natural Resources and Agricultural Engineering, Agricultural
University of Athens, 75 Iera Odos, 11855, Athens, Greece Email:[email protected]
1
2
1
1. Introduction
2
The FAO-56 methodology (Allen et al., 1998) for estimating reference crop
3
evapotranspiration, ET0, (daily or monthly data), recommends the sole use of the
4
Penman-Monteith equation. The standardized scheme is considered as the “standard”
5
method in hydrological and irrigation applications at well-watered meteorological
6
stations under varying locations and climatic conditions. According to Allen et al.
7
(1998) the recommended form of the FAO-56 Penman-Monteith equation is ( P−M ) ( P−M ) ET0( P − M ) = Erad + Eaero
8
⎧ 0.408Δ( Rn − G ) ⎫ ⎧ 900γ uD ⎫ =⎨ ⋅ ⎬+ ⎨ ⎬ ⎩ Δ + γ (1 + 0.34u ) ⎭ ⎩[Δ + γ (1 + 0.34u )] (T + 273) ⎭
(1)
9
( P−M ) where the term Erad , is the radiation term of the FAO-56 Penman-Monteith
10
( P−M ) equation, and Eaero is the aerodynamic component, Rn is the net radiation at the
11
surface (MJ/m2/d); Δ is the slope of the saturation vapor pressure curve (kPa/ oC); γ is
12
psychrometric coefficient (kPa/ oC); G is soil heat flux density (MJ/m2/d) and u is
13
wind speed at 2 m height (m/s). The application of the FAO-56 Penman Montheith,
14
Eq. (1), requires the commonly weather station measured meteorological observations
15
e. g. maximum and minimum air temperatures, Tmax, and Tmin, , solar radiation, RS,
16
maximum and minimum relative humidity RHmax , RHmin respectively, wind speed, u,
17
as well as site details of latitude and altitude.
18
Penman (1948) published the radiation-aerodynamic combination equation to predict
19
evaporation from open water, bare soil, and grass.
20
6.43 ( fU ) D ⎫ ⎧ Δ ( Rn ) ⎫ ⎧ γ (P) ( P) + Eaero =⎨ ⋅ ⋅ EPEN = Erad ⎬+⎨ ⎬ λ ⎩Δ + γ λ ⎭ ⎩Δ + γ ⎭
21
(2)
2
3
1
(P) where the term Erad , is the radiation term of the Penman combination equation
2
corresponding to the incoming net short wave radiation component, and the outgoing
3
(P) net long wave radiation component, and Eaero is the aerodynamic component λ is
4
latent heat of vaporization (MJ/kg); fU is the Penman’s wind function.
5
The original Pemman (1948, 1963) equation as well as its various modifications have
6
also been widely used to estimate reference crop evapotranspiration. In a recent paper,
7
Valiantzas (2006) developed a simple algebraic explicit formula, equivalent in
8
accuracy to the 1963-Penman equation for estimating ET0. The derivation of the
9
Valiantzas (2006) formula depending explicitly on the commonly available weather
10
station measured variables T, RS , RH, and u is based on the systematic analysis and
11
mathematical simplifications made to the “standardized” computation procedure
12
recommended by Shuttleworth (1993) and Allen et al. (1998).
13
simplified formulas suggested by Valiantzas (2006) were recommended, validated
14
and successfully applied by various researches as Lewis and Lamoureux (2010),
15
MacDonald et al. (2009), Rimmer et al. (2009), McMahon et al. (2013), and
16
D'Agostino (2013) and others. In recent papers, Valiantzas (2012) and Valiantzas
17
(2013 a,b,c) suggested formulas that are based on simplifications made to the original
18
Valiantzas (2006) ET-formula and they are all equivalent in accuracy to the 1963-
19
Penman model suggested for estimating ET0,
20
However, according to the results of Jensen et al. (1990), Itenfisu et al. (2000), and
21
others the 1963-Pennman method for estimating ET0 is not as accurate as the Penman-
22
Monteith method. Jensen et al. (1990) evaluated 20 ET0 methods and compared
23
against lysimeter measurements at 11 locations located in different climatic zones
3
The previous
4
1
around the world. The Penman-Monteith method ranked as the best method for all
2
climatic conditions. The 1963-Penman method ranked fourth of all methods. Standard
3
error of estimate for the 1963-Penman method was 0.57 mm/d compared against
4
monthly lysimeter data. This compared to 0.36 mm/d for the Penman-Monteith
5
method.
6
Itenfisu et al. (2000) reported that the 1963-Penman method yielded higher ET0
7
values when compared to the ASCE-Penman Monteith equation (following similar
8
ET0 trends with FAO-56 Penman-Monteith) in Jensen et al. (1990) with the ratio of
9
1.0 to 1.10. Jensen et al. (1990) reported that all the Penman models (except the
10
Penman-Monteith) overestimated lysimeter measurements in humid locations.
11
Valiantzas (2013b) using data from a global dataset has shown that the formula
12
equivalent to the original 1963-Penman model (Valiantzas 2006 ET-formula),
13
although resulting in relatively good estimates of ET0 compared to the FAO-56
14
Penman-Monteith “standard” method tends to overestimate ET0.
15
On the other hand, it is universally accepted that the FAO-56 Penman-Monteith is the
16
“standard” method for estimating daily or monthly ET0.
17
A disadvantage in the application of the FAO-56 Penman Montheith using Eq. (1) is
18
that the main weather variables appearing directly in the equation are T, Rn, D, Δ, γ,
19
and u. Although there is specific instruments to measure Rn, and D, the usually
20
available weather records in standard meteorological station are T, RH, RS, and u.
21
(Shuttleworth, 1993). The structure of Eq. (1) suggests that the commonly measured
22
inputs appear explicitly in the computation of Rn, D, Δ, and γ in Eq. (1). The FAO-56
23
methodology represented by Eq. (1), is actually an abbreviation form of a complex
24
algorithm comprising a plethora of specific supporting equations adopted to convert
4
5
1
the input measured variables into a number of other intermediate estimated
2
parameters. A plethora of intermediate parameters appear in the application of the
3
FAO-56 Penman-Monteith procedure, such as the latent heat of vaporization, the
4
saturation vapor pressure, the actual vapor pressure, the psychrometric coefficient, the
5
slope vapor pressure curve, the atmospheric pressure, the effective emissivity of the
6
surface, the clear-sky solar radiation, the Stephan-Boltzman constant, the cloudiness
7
factor, and many others. The complexity of calculations increases as each of these
8
parameters could be expressed by a variety of units. The use of all these parameters
9
could create confusion in the calculation steps during the application of the FAO-56
10
Penman-Monteith procedure, thus resulting in significant errors should the appearing
11
parameters not be expressed in the appropriate units.
12
In this paper, an algebraic formula equivalent in accuracy to the FAO-56 Penman
13
Monteith algorithm is developed for calculating directly ET0 using the input routinely
14
measured variables Tmax, and Tmin , RS , RH, u, and Z only, where Z (m) is the
15
elevation of the site. The only additional parameter to be estimated appearing in the
16
suggested formula is the extraterrestrial radiation, RA.
17
The formula is obtained by analysing the correspondence of radiation and
18
aerodynamic terms between the two models (Penman and FAO-56 Penman-Monteith
19
model) Subsequently, by varying the meteorological variables over their typical range
20
of variation using numerical simulations and regression procedures, a series of new
21
empirical simplified expressions of simple mathematical form approximating the
22
standardized components of the FAO-56 Penman-Monteith procedure were
23
developed.
5
6
1
Since the computation of variable RA in daily basis requires a rather complicated
2
numerical procedure another more simplified formula, easy to use for routine
3
hydrologic applications, is also developed to compute directly ET0 from the input
4
measured variables Tmax, and Tmin , RS , RH, u, Z, φ, and J only, where φ is latitude
5
of the site (radians) and J is the Julian day number.
6
The proposed ET0 algebraic formulas (full or limited set of data) explicit to routinely
7
measured data could easily incorporated to regression procedures applied to routinely
8
measured meteorological data or lysimeter data to build up ET0 empirical accurate
9
formulas at local scale. The suggested formulas could also easily adopted to apply
10
regionalization procedures for estimating ET0 at large scale. Such algebraic formulas might
11
facilitate the investigation of ET0 trends or sensitivity analysis (Linacre 2004).
12
A major disadvantage to application of the standardized FAO-56 Penman-Monteith
13
procedure is the relatively high data demand requiring measurements of T, RH, RS,
14
and u. The number of weather stations where all these parameters are available is
15
limited especially in developing countries. Another problem is linked to data quality.
16
Wind speed data are rarely available or of questionable precision (Jensen et al. 1997;
17
Allen 1996). Therefore, for such cases, a new expression which does not require wind
18
speed data is also proposed, in this paper.
19
Further, solar radiation data are not always reliable (Last and Snyder, 1998) whereas
20
some older electronic sensors for relative humidity measurement produce commonly
21
errors (Allen 1996).
22
Lastly, another serious problem is related to the cost of instrumentation for collecting
23
the required meteorological data in automated weather stations. Using methods that
24
require limited set of data might reduce the cost drastically. Recently, Exner-
6
7
1
Kittridge and Rains (2010) evaluated various alternative combinations of limited set
2
of data ET0 methods for their accuracy in conjunction with their corresponding cost.
3
They introduced the cost effectiveness index. Valiantzas (2012) and Exner-Kittridge
4
(2012) concluded that if the addition of RH measurements to the air temperature data,
5
T, improve the accuracy of the ET0 estimation, then the cost effectiveness of a RH-T
6
method could increase dramatically compared to other alternatives of limited set of
7
data methods
8
extremely low additional cost of the RH sensor..
9
Therefore, further simplifications on the FAO-56 Penman-Monteith formula was
10
made leading to a temperature-humidity based formula not requiring u and RS data
11
measurements. The performance of the derived formulas for estimating reference crop
12
evapotransiration is also tested.
13
2. Theory-The development of the new simplified versions of standardized FAO-
14
56 Penman-Monteith
15
For the development of the simplified versions it is initially assumed that Z=0, where
16
Z is elevation of the site (m).
17
For Z=0 the coefficient γ takes the single constant value of γ0 =0.0671 kPa.
18
Furthermore, as the value of λ varies only slightly over a normal temperatures range, a
19
single constant value (for T=20 oC) is considered λ0=2.45 MJ/kg.
20
( P−M ) ( P−M ) (P) 2.1 Approximation of the Erad . On the correspondence between Erad and Erad
21
The variable Δrad0, expressing the difference of the radiation terms of the two models
22
reduced by dividing by the variable Rn is given as
23
⎡ ⎤ ( P−M ) ( P) Δ rad 0 = ( Erad − Erad ) / Rn = 0.408 ⎢ Δ + (1 + Δ0.34u)γ − Δ +Δγ ⎥ 0 0⎦ ⎣
(requiring additional u and/or Rs instrument sensors), due to the
7
(3)
8
1
( P−M ) (P) Since Erad / Rn depends on T and u, and Erad / Rn depends on T, the variable Δrad0,
2
expressing the reduced difference of the two radiation terms is generally affected by
3
the values of wind speed, u, and average temperature, T.
4
In this section it will be shown that there is a considerably high dependence of the
5
Δrad0 on the u variable, whereas the effect of the variable T on this term is rather
6
insignificant.
7
To investigate the behavior of this term a series of numerical simulations were carried
8
out to generate the “accurate” values of Δrad0. Combinations of the required input
9
mean temperatures values, T, and wind speed values u were generated by varying T
10
and u over typical range of variations. The values of T varied between 0 and 35 0C
11
whereas u varied between 0.5 to 8.5 m/s. Series of data sets were generated by
12
combining the previous “typical” input values of T and u in all possible ways. For
13
each given set of data the values of the term Δrad0 were computed from Eq. (3).
14
Afterwards the “accurate” values of Δrad0 obtained from simulations were plotted
15
against the values of u. The results from 18,000 numerical simulations (Fig.1) show
16
that there is a high dependence of the computed Δrad0 on the u variable whereas the
17
effect of the values of T is rather insignificant. Using the least square procedure the
18
variation of Δrad0 with u can be approximated by the following relationship;
19
Δ rad 0 = − 0.03u 0.7
20
Results indicated a good agreement between approximate and exact values. Statistical
21
regression results for Eq. 4 are Y=0.973X and R2=0.959, where Y is approximate
22
estimation, X is exact value, and R2 is coefficient of determination.
(4)
8
9
1
Furthermore, the variable Rn may be rounded by an empirical relationship to Rs
2
(Linacre, 1993), e. g. Rn ≈ 0.55RS
3 4
5
( P−M ) Therefore E rad may be approximated by combining the previous formula
(approximating Rn) and Eq. (4) with Eq. (3) as ( P−M ) ( P) ( P) Erad ≈ Erad − 0.03Rnu 0.7 ≈ Erad − 0.03 ⋅ 0.55RS u 0.7
(5)
6
On the other hand, according to Valiantzas (2006) the two component of the radiation
7
( P) ( P) term of the Penman combination equation, E radS , and E radL corresponding to the
8
incoming net short wave radiation component and the outgoing net long wave radiation
9
component respectively can be accurately approximated as
10
( P) E radS ≈ 0.051(1 − α ) RS T + 9.5
11
⎛R ⎞⎛ RH ⎞ ( P) ≈ 0.188(T + 13) ⎜ S − 0.194 ⎟ ⎜⎜1 − 0.00015(T + 45)2 EradL ⎟ 100 ⎟⎠ ⎝ RA ⎠⎝
12
Finally, since
13
(6)
(P) (P) (P) Erad = EradS − EradL
(7)
(8)
14
( P−M ) then the radiation term of the FAO-56 Penman-Monteith procedure, Erad , can be
15
calculated by substituting Eqs (6) and (7) into Eq.(8) and the resultant equation in to
16
Eq. (5)
17
Further Simplifications
18
( P) The component of the radiation term of Penman’s equation, E radL , can be further
19
simplified according to Valiantzas (2006) as
20
E
(P) radL
⎛R ⎞ ≈ 2.4⎜⎜ S ⎟⎟ ⎝ RA ⎠
2
⎛ RH ⎞ + 0.024(T + 20 )⎜1 − ⎟ ⎝ 100 ⎠
9
(9)
10
1
In Eq. (9) the computation of variable RA in daily basis requires a rather complicated
2
numerical procedure. Therefore its term [(RS/RA)] from Eq. (9) is replaced intuitively
3
by a term of the following simple mathematical form, not requiring calculation of RA.
4
⎛ RS ⎞ ⎛ RS C2 ⎞ ⎜ ⎟ ≈ C1 ⎜ 2 ϕ ⎟ + C3 ⎝N ⎠ ⎝ RA ⎠
5
where C1, C2, and C3 are empirical coefficients that should be identified. The
6
maximum possible duration of daylight (hrs), N, can be approximately estimated as
7
N ≈ 9.8δϕ + 12
8
where δ is solar declination (radians) given by
9
δ = 0.409 sin(
(10)
(10a)
2π J − 1.39) 365
(10b)
10
φ is the latitude of the station expressed in radians and J is Julian day number. For
11
monthly estimations the Julian day corresponding to ith month is calculated as
12
J = INT (30.5i − 14.6)
13
A calibration procedure was applied to identify the three regression coefficients, C1,
14
C2, and C3 using a global climatic data set that includes monthly data (the FAO-
15
CLIMWAT, Smith 1993). Climatic data from thirteen countries with relatively high
16
quality records (Temesgen et al., 1999), that essentially cover all the typical range of
17
variation of the input weather variables, were selected: Spain (58 meteorological
18
stations), France (42), Italy (60), Greece (20), and Cyprus (27) in Europe; Pakistan
19
(23), Lebanon (16) and India (18) in Asia; Egypt (28), Tunisia (19), Algeria (22),
20
Ethiopia (142), and Sudan (63) in Africa. The total numbers of the selected stations is
21
535. These data were selected for the calibration of the derived formula because they
22
essentially cover all the typical range of variation of the input weather variables T, RS,
(10c)
10
11
1
RH, and u. The latitude for the selected stations varies from 3 to 51o, and the elevation
2
varies from 0 to 3000 m. Some of the countries (France, Italy and Spain) are selected
3
to represent humid and semi-humid temperate climates and others (Ethiopia, Sudan,
4
Egypt and Pakistan) to represent dry arid and semi-arid tropical climates (Temesgen
5
et al., 1999). All the stations are located in the northern hemisphere.
6
Calibration procedure of the no=6,420 data of the database leads to the values of
7
coefficients of C1=3.9, C2= 0.15, and C3=0.16.
8
The statistical results for Eq. (10) (shown in Fig. 2) using as input the meteorological
9
data from the FAO-CLMIWAT database indicated a good agreement between
10
approximate and exact values.
11
( P−M ) ( P−M ) (P) 2.2 Approximation of the Eaero . On the correspondence between Eaero and Eaero
12
The aerodynamic term of the Penman’s procedure is given as
13
14 15
16
(P) Eaero =
γ0 6.43 fU (u ) D( s tan d ) λ0 Δ + γ 0 1
(11)
whereas the aerodynamic term of the Penman-Monteith according to the standardized FAO-56 computational procedure is calculated as ( P−M ) Eaero =
uD( s tan d ) 900γ 0 ⋅ [Δ + γ 0 (1 + 0.34u )] (T + 273)
(12)
17
where D(stand) is the mean vapor pressure deficit computed according to
18
recommendation of the standardized FAO-56 procedure.
19
( P−M ) (P) It is assumed that the Eaero can be approximated from Eaero by an empirical
20
(P) relationship according to which Eaero is multiplied by an unknown empirical function
21
of temperature
11
12
1
( P−M ) (P) Eaero ≈ Eaero ⋅ fT (T ) ≈ D( s tan d )
γ0 6.43[ fU (u ) ⋅ fT (T )] λ0 Δ + γ 0 1
(13)
2
where fU (u ) and fT (T ) are considered as purely empirical functions of wind and
3
temperature respectively assumed to have the following simple mathematical forms
4
fU (u ) = c1u c2
5
fT (T ) =
(13a)
1 c3 − T
(13b)
6
where c1, c2 and c3 are empirical regression coefficients that should be identified.
7
To demonstrate the validity of empirical Eq. (13), and identify the three regression
8
parameters, a series of numerical simulations was conducted to generate synthetic
9
meteorological data used for the calibration. The numerical simulations were carried
10
( P−M ) out to generate “exact” [ Eaero /D(stand)] values (depending on T and u only) according
11
to Eq. (12). Combination of typical ranges of T and u, with 0< T< 35 oC and 0.5< u
12
( P−M ) <8.5 m/s, were used to calculate the “exact” values of [ Eaero /D(stand)]. Then the
13
regression
14
1 γ0 ( P−M ) ⎡⎣ Eaero / D( s tan d ) ⎤⎦ ≈ 6.43 ⎡⎣c1u c2 / (c3 − T ) ⎤⎦ λ0 Δ + γ 0
15
as given by Eq. (13).
of
the
empirical
equation
has
the
following
form
(14)
16
Further simplifications could be made on the initial regression Eq. (14). Valiantzas
17
(2006) has demonstrated that the following relationship is an accurate approximation:
18
e0 (T ) ⋅ [
γ0 6.43] ≈ 0.048(T + 20) λ0 Δ + γ 0 1
(15)
12
13
1
where e0(T) is the value of the saturation vapor pressure curve, corresponding to the
2
temperature value of
3
expression for the Clausius-Calpeyron equation ; Stull, 2000)
4
⎛ 17.27 ⋅ Τ ⎞ e0 (Τ) = 0.6108 ⋅ exp ⎜ ⎟ ⎝ Τ + 237.3 ⎠
T and calculated by the Teten's equation, (the empirical
(16)
5 6
Here T is temperature in oC. Regression statistics of Eq. (15) yields Y=0.994X and
7
R2=0.9996.
8
Substituting Eq. (15) into Eq. (14), together with the results of the regression Eq. (14)
9
is transformed into the following simplified form
10
( P−M ) ⎡⎣ Eaero / D( s tan d ) ⎤⎦ e0 (T ) ≈ c1u c2 [ 0.048(T + 20) / (c3 − T ) ]
(17)
11
Using the least squares regression procedure on Eq. (17), and using the synthetic
12
meteorological data, the three regression parameters were determined (c1=426,
13
c2=0.75 and c3=400). Approximate values obtained from the right hand side of Eq.
14
( P−M ) (17) are compared with the “exact” values of [ Eaero /D(stand)]e0(T) (Figure not shown)
15
. Results indicated that empirical formula, Eq. (17) yields a very good approximation
16
of
17
Furthermore, the temperature term [ (T + 20) / (400 − T ) ] appearing in the right hand
18
side of Eq. (17) (with c3=400) can be approximated by the following simple
19
expression: 349(T+17) (this is demonstrated performing numerical simulations with
20
0< T <35 oC, with regression results Y=1.001X, and R2 =0.9995). Finally, substituting
21
the previously obtained simple expression in to Eq. (17) the following formula is
22
( P−M ) obtained to approximate [ Eaero /D(stand)]
( P−M ) [ Eaero /D(stand)] e0(T). Regression yields Y=1.001X, and R2 =0.975 .
13
14
1
( P−M ) ⎡⎣ Eaero / D( s tan d ) ⎤⎦ e0 (T ) ≈ 0.0585(T + 17)u 0.75
(18)
2
( P−M ) The generated synthetic “exact” values of e0 (T ) [ Eaero /D(stand)] obtained from the
3
previous series of numerical simulations according to Eq. (12) were plotted against
4
the approximate values of 0.0585(T + 17)u 0.75 (Eq. (18)). The results from 18,000
5
numerical simulations are reported in Fig. 3. Statistical regression results for Eq. 18
6
(shown in Fig. 3) are Y=1.003X and R2=0.974.
7
Aprroximation of D(stand)
8
The computation of the mean “standardized” vapor pressure deficit, D(stand), according
9
to the recommendation of the standardized FAO-56 Penman-Monteith procedure is
10
as follows:
11
D( s tan d ) = eS ( s tan d ) − ea ( s tan d )
12
where eS ( s tan d ) and ea ( s tan d ) are the mean “standardized” saturation and actual vapor
13
pressures respectively computed according to the FAO-56 Penman-Monteith
14
procedure as:
15
eS ( s tan d ) = 0.5 ⎡⎣ e 0 (Tmax ) + e 0 (Tmin ) ⎤⎦
16
where e0(Tmax) and e0(Tmin) are saturated vapor pressures corresponding to the
(19)
(20)
17
temperature values of t=Tmax and t=Tmin respectively, computed by Eq. (16) and
18
ea ( s tan d ) =
19
Eq. (21) is applied when only mean relative humidity data, RH, are available.
20
Alternatively, when RHmax and RHmin data are available, then the following
21
relationship is recommended for computing eS ( s tan d )
RH 2 ⋅ 0 100 ⎡⎣1 / e (Tmax ) + 1 / e 0 (Tmin ) ⎤⎦
(21)
14
15
RH max 0 RH min 0 ⋅ e (Tmax ) + ⋅ e (Tmin ) 100 100
1
ea ( s tan d ) =
2
Systematic analysis of the daily meteorological data obtained from various weather
3
stations of the CIMIS database in California have shown that both equations (Eqs. 21
4
and 22) provided values for ea ( s tan d ) that can be considered almost identical for
5
practical applications.
(22)
6
The ( eS ( s tan d ) ) computed from Eq. (20) is affected only by the values of temperature
7
Tmax and Tmin. To investigate the behavior of the previous term, a series of numerical
8
simulations were carried out to generate the “exact” values of eS ( s tan d ) from Eq. (20).
9
Combination of typical ranges of Tmax, and Tmin were implicitly generated as
10
Tmax=T+TR/2 and Tmin=T-TR/2 by varying the mean temperature T and the difference
11
of temperatures TR=(Tmax-Tmin) over a typical range of variations, 3< TR< 24 oC and
12
2< T< 37 oC. Data sets for which Tmax>46 oC or Tmin<-5 oC were excluded from
13
simulations. Afterwards, the “exact” values of the term [eS ( s tan d ) / e0 (T )] obtained
14
from simulations were plotted against the values of the term TR (Figure not shown). It
15
is shown that there is a significantly high dependence of the computed [eS ( s tan d ) / e0 ]
16
term on the TR variable (with a coefficient of determination R2=0.944). Using the
17
least square procedure to the results from no=3,557 simulations the following
18
approximate relationship for ( eS ( s tan d ) ) presented in Fig. 4 was obtained
19
eS ( s tan d ) ≈ e 0 (T ) ⋅ (1 + 0.00043 ⋅ TR 2 )
20
Results presented in Fig. 4 indicated a very good agreement between approximate and
21
exact values. Regression yields Y=0.991X, and R2 =0.9995
(23)
15
16
1
A similar procedure using the same synthetic data is applied to investigate the
2
behavior of the term [e0(Tmax) · e0(Tmin)]. It is shown that there is a significantly high
3
dependence of the computed [e0(Tmax) · e0(Tmin)] term on e0(T). Using the least square
4
procedure the following approximate relationship was obtained
5
e 0 (Tmax ) ⋅ e0 (Tmin ) ≈ e0 (T ) 2
6
Statistical regression results for Eq. (24) are Y=0.983X and R2=0.9996.
7
Substituting Eqs. (23) and (24) into Eq.(21), an approximation of ea ( s tan d ) is obtained.
8
Afterwards, substituting the resultant approximation as well as Eq. (23) in to Eq.(19)
9
after manipulation, D(stand) is finally approximated as
10
D( s tan d )
(24)
⎡⎣(1 + 0.00043 ⋅ TR 2 ) 2 − RH /100 ⎤⎦ ≈ e (T ) ⋅ (1 + 0.00043 ⋅ TR 2 ) 0
(25)
11 12
A similar procedure using synthetic meteorological input data TR and RH leads to
13
further simplification of Eq. (25)
14
D( s tan d ) ≈ e0 (T ) ⎡⎣(1.03 + 0.00055 ⋅ TR 2 ) − RH /100⎤⎦
15
3. Results
16
3.1 The new expressions
17
Expressions requiring full set of data
18
( P−M ) An accurate approximate of Erad is obtained by substituting Eqs. (6) and (7) into Eq.(8)
19
( P−M ) and the resultant equation in to Eq. (5). Similarly, an accurate approximate of Eaero is
20
obtained by substituting the approximate Eq. (25), into approximate Eq. (18), after
21
( P−M ) manipulation. Finally substituting the previously obtained approximates of Eaero and
16
(26)
17
1
( P−M ) Erad in to Eq. (1) an accurate approximate version for the FAO-56 Penman-Monteith
2
procedure for estimating ET0 in daily basis is obtained
3
( referred to in this paper as “Fo1-PM (Rs,T,RH,u)”):
4
ET0( P − M ) ≈ 0.051(1 − α ) RS T + 9.5 5
⎛R ⎞⎛ RH ⎞ 0.7 −0.188(T + 13) ⎜ S − 0.194 ⎟ ⎜⎜1 − 0.00015(T + 45) 2 ⎟⎟ − 0.0165RS u 100 ⎠ ⎝ RA ⎠⎝ ⎡(1 + 0.00043 ⋅ TR 2 ) 2 − RH /100⎤⎦ 0.75 ⎣ +0.0585(T + 17)u + 0.0001Z (1 + 0.00043 ⋅ TR 2 )
(27)
6
RS and RA should be expressed in MJ/m2/d (1 MJ/m2/d =23.88 cal/cm2/d = 0.408
7
mm/d (equivalent evaporation) =11.57 W/m2, T, Tmax, Tmin, and TR= (Tmax- Tmin) in
8
o
C, u in m/s, RH (%), Z in m, and α=0.23. Note that the above formula depends on T
9
but also on Tmax and Tmin values. The last term of the right-hand side equation
10
(indicating the effect of Z) is obtained by following a similar procedure with this
11
applied for the derivation of simplified Penman’s version in Valiantzas (2006).
12
Subsequently, another simplified version is obtained when the approximate
13
simplifications, Eqs. (9) and (26) were used
ET0( P − M ) ≈ 0.051(1 − α ) RS T + 9.5 2
14
⎛R ⎞ ⎛ RH ⎞ 0.7 −2.4 ⎜ S ⎟ − 0.024 (T + 20 ) ⎜1 − ⎟ − 0.0165RS u R 100 ⎝ ⎠ ⎝ A⎠
(28)
+0.0585(T + 17)u 0.75 ⎡⎣(1.03 + 0.00055 ⋅ TR 2 ) − RH /100⎤⎦ + 0.0001Z 15
In Eqs. (27) and (28) the computation of variable RA in daily basis requires a rather
16
complicated numerical procedure. A further simplified version not requiring
17
calculation of RA is proposed using the simplified approximate Eq. (10)
18
(referred to in this paper as “Fo2-PM (Rs,T,RH,u)”):
17
18
1
ET0( P − M ) ≈ 0.051(1 − α ) RS T + 9.5 2
2
⎛ ⎞ ⎛ ⎞ RSϕ 0.15 ⎛ RH ⎞ 0.7 0.92 − ⎜ 22.46 ⎜ + ⎟ − 0.024 (T + 20 ) ⎜1 − ⎟ ⎟ − 0.0165RS u 2 ⎝ 100 ⎠ ⎝ [4sin(2π J / 365 − 1.39) ⋅ ϕ + 12] ⎠ ⎝ ⎠ +0.0585(T + 17)u 0.75 ⎡⎣(1.03 + 0.00055 ⋅ TR 2 ) − RH /100 ⎤⎦ + 0.0001Z
(29)
3 4
the latitude of the station φ is expressed in radians and J is Julian day number. For
5
monthly estimations the Julian day corresponding to ith month is calculated as
6
J = INT (30.5i − 14.6)
7
Expression not requiring wind speed data
8
( referred to in this paper as “Fo-PM(Rs,T,RH)”)
9
In places where no wind data are available the average value of u=2 m/s of 2,000
10
stations over the globe (Allen et al., 1998) can be used in the FAO-56 Penman-
11
Monteith equation. Substituting the value of u=2 m/s in the previously developed
12
formula, Eq. (29), the following approximate formula is proposed for ET0 estimation,
13
when wind speed data are missing
14
ET0( P − M ) ≈ 0.0393RS T + 9.5 2
15
⎛ ⎞ ⎛ ⎞ RSϕ 0.15 ⎛ RH ⎞ − ⎜ 22.46 ⎜ + 0.92 ⎟ − 0.024 (T + 20 ) ⎜1 − ⎟ − 0.0268RS 2 ⎟ π ϕ [4sin(2 / 365 1.39) 12] 100 J − ⋅ + ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ +0.0984(T + 17) (1.03 + 0.00055 ⋅ TR 2 − RH /100 )
16 17 18
Expression requiring temperature-humidity data alone (referred to in this paper as “Fo-PM(T,RH)”)
18
(30)
19
1
Hargreaves and Samani (1982) suggested the following radiation empirical formula
2
for the estimation of solar radiation
3
RS ≈ kRS ⋅ RA (Tmax − Tmin )0.5
4
where kRs is empirical radiation adjustment coefficient which generally differs from
5
location to location from 0.12 to 0.25 in average (Samani, 2004). There are several
6
values adopted for kRs depending on various topographical and climatological factors.
7
The conventionally default “average” adopted value was kRs≈0.17. Thus inherent to its
8
empirical nature, there is some uncertainty relatively to this coefficient.
9
Allen et al. (1998) suggested that the value of minimum temperature Tmin is a
10
substitute of the dew point temperature, Tdew, (at well-watered meteorological
11
stations) e.g. Tdew≈Tmin. The dew point temperature, Tdew (oC), can be computed by the
12
following formula (Allen et al, 1998):
13
Tdew =
14
where ea is the actual vapour pressure that can be estimated as
15
⎡ 17.27T ⎤ RH ⋅ ea = 0.6108 ⋅ exp ⎢ ⎣ T + 237.3 ⎥⎦ 100
16
The value of Tmin in the radiation formula Eq. (31) is intuitively substituted by the
17
value of Tdew as calculated by Eq. (32), depending on T and RH alone. Then, the
18
following formula for estimating Rs is obtained
19
RS ≈ kRS ⋅ RA (Tmax − Tdew )0.5
20
The values of kRs estimated by the empirical radiation formula, Eq. (31) and the
21
proposed modified radiation formula Eq. (33) were computed from data obtained
22
from FAO-CLIMWAT database from the 535 stations over the globe. From the full
(31)
116.91 + 237.3ln(ea ) 16.78 − ln(ea )
(32)
(32 a)
(33)
19
20
1
set of data, the no=4,461 monthly estimates corresponding to well watered conditions
2
(Temesgen et al., 1999) were retained. The variation of the values of kRs calculated by
3
the two radiation formulas was plotted (Figures not depicted). The results indicated
4
that when the proposed Eq. (33) is applied the spread of kRs coefficient from its default
5
value of 0.17 is less enough than using Eq. (30). Therefore Eq. (33) yields more
6
accurate values than Eq. (31). This conclusion is clearly demonstrated in Figs. 5a and
7
5b where the estimations provided by the radiation formula (31) and the modified
8
suggested one (33) were compared with the measured values of RS from the FAO-
9
CLIMWAT.
10
In this paper, the values of estimation methods were compared with the values of the
11
“standard” method by using simple error analysis and linear regression, i.e. Y=SX ,
12
where S=regression coefficient (slope of the linear curve), Y= reference values
13
obtained by the “standard” method; X= correspondents estimates by the comparison
14
method. Additionally statistical parameters were calculated: the standard error of the
15
estimate, SEE =
16
well as the long term average ratio, rt = X av / Yav where Xav and Yav = long term
17
average value of approximate and “standard” estimates respectively. The traditional
18
coefficient of determination, R2, was also used.
19
The statistical results of the correlation between the original radiation formula, Eq.
20
(31), with measured RS are S=1.044, R2=0.765 and SEE= 3.21 MJ/m2/d, whereas for
21
the modified formula, Eq. (33), the results are S=1.012, R2=0.849 and SEE=2.39
22
MJ/m2/d. The original formula Eq. (31) produce poorer estimates than the suggested
23
Eq. (33), it produced higher scatter of the results than Eq. (33), and a higher SEE.
{∑ (Y − X ) n
1
i
i
2
}
/ (nO − 1)
0.5
where nO=total number of observations as
20
21
1
Substituting the previously modified radiation suggested formula, Eq. (33), in Eq.
2
(28) the following approximate formula is proposed for ET0 estimation, when wind
3
speed and solar radiation data are missing
⎛ RH ⎞ ET0( P − M ) ≈ 0.00668RA (T + 9.5)(Tmax − Tdew ) − 0.0696(Tmax − Tdew ) − 0.024 (T + 20 ) ⎜1 − ⎟ ⎝ 100 ⎠
4
−0.00455RA (Tmax − Tdew )0.5 + 0.0984(T + 17) (1.03 + 0.00055 ⋅ TR 2 − RH /100 )
5
Simplified 1963-Penman ET-formula
6
(referred to in this paper as “Fo-PENM (Rs,T,RH,u)”)
7
Valiantzas (2006) suggested the following simplified ET-formula that approximates
8
the 1963-Penman -for grass- scheme:
9
ET0 ≈ 0.051(1 − α ) R S
⎛R T + 9.5 − 2.4⎜⎜ S ⎝ RA
(34)
2
⎞ RH ⎞ ⎛ ⎟⎟ + 0.048(T + 20 )⎜1 − ⎟(0.5 + 0.536u ) + 0.00012Z ⎝ 100 ⎠ ⎠
(35)
10 11
Valiantzas (2006) suggested to use α=0.25.
12
Reduced set FAO-56 Penman-Monteith method requiring temperature-humidity data
13
alone
14 15
(referred to in this paper as “FAO-PM (T, RH) ” The reduced to wind and solar radiation data FAO-56 Penman-Monteith method
16
requires measured T , and RH data alone and uses estimations for wind speed and
17
solar radiation data. According to Allen et al. (1998) a single constant value of u=
18
2m/s was assumed in Eq. (1) Furthermore, the RS values are estimated by the
19
Hargreaves and Samani (1982) radiation empirical formula, Eq. (31).
20
Hargreaves-Samani temperature method
21
(referred to in this paper as “HARG (T) ”
21
22
1
The solar radiation- based equation of Hargreaves (1975) requiring only the data RS,
2
Tmax, and Tmin, is:
3
ET0 ≈ 0.0135 ⋅ 0.408R S (T + 17.8)
4
Substituting approximate Eq. (31) in to Eq. (36) the well-known temperature-based
5
equation is obtained.
6
Although the temperature- based equation of Hargreaves requires T data only, it is
7
also included for the comparisons with the other methods. Note that Hargreaves
8
formula is developed from calibration of meteorological data at the sites of California
9
(Davis).
(36)
10
3.2 Testing the formulas - Comparisons
11
The ET0 estimated by the new (full or limited set of data) suggested formulas as well
12
as the simplified 1963-Penman ET-formula developed by Valiantzas(2006), Fo-
13
PENM(Rs,T,RH,u), the reduced FAO-PM (T, RH), and the Hargreaves formula were
14
compared with the conventionally considered as a reference method for comparisons
15
standardized FAO56-PM scheme using monthly or daily time scale data.
16
Firstly, monthly data extracted from the 535 stations of the FAO-CLIMWAT database
17
were used.
18
Subsequently, because the mean monthly data from CLIMWAT refer to the long-
19
term average year, a detailed dataset including particularly high quality daily data
20
from real years, the CIMIS data base (http://wwwcimis.water.ca.gov) has been also
21
used for validation and comparison purposes of the suggested formulas. Most CIMIS
22
stations conform to the basic definition of a reference weather station corresponding
23
to well-watered conditions. Recorded daily data selected from 7 sites of California has
24
been used for comparisons. An effort was made to select stations representing distinct
22
23
1
climates as much as possible covering a wide range of weather parameters. Two
2
locations were characterized by a humid climate where other by semi-arid or arid
3
climate. The long term average wind ranged approximately from 1.0 to 4.0 m/s. Table
4
1 lists the characteristics of the selected stations.
5
Comparisons were made using graphics and simple linear regression. The statistical
6
results of the correlation, rt, R2, and SEE were used in comparing ET0 values
7
estimated by the different methods.
8
Monthly data
9
Figure 6 presents the comparisons of methods for the 535 stations over the globe
10
using the monthly data from the FAO-CLIMWAT database. For all the sites, the
11
suggested full set of data expressions, Fo1-PM (Rs,T,RH,u) and Fo2-PM (Rs,T,RH,u)
12
(not requiring the detailed computation of RA) produced estimates in perfect
13
agreement with the standardized FAO56-PM. The bias error of the suggested new full
14
set of data formulas is practically negligible, the scatter of the results for both
15
methods is insignificant, R2=0.998 and 0.996 respectively, and the value of SEE is
16
rather
17
Valiantzas(2006), Fo-PENM (Rs,T,RH,u), produced poorer estimates than the new
18
suggested ones, it produced higher scatter of the results, R2=.978, and a significantly
19
higher SEE by significant percentage of about 220% (Fig. 6 a, b, and c).
20
Comparing the limited data formulas, the formula requiring T and RH data alone,
21
Fo-PM (T,RH),
22
additional Rs measurements e. g. Fo-PM (Rs,T,RH). It produced higher scatter of the
23
results, and a higher SEE by a percentage of about 31% (Fig. 6 d, and e). However,
negligible.
The
simplified
1963-Penman
ET-formula
developed
by
produced less accurate estimations than the formula requiring
23
24
1
the Fo-PM (T,RH) proposed formula produced more accurate estimations than the
2
reduced FAO-PM (T,RH) method requiring the same set of data (Fig. 6 e and f).
3
The Hargreaves equation, HARGR (T), produced the worst results among all the
4
methods. Compared with the Fo-PM(T,RH), the HARGR(T) produced higher scatter
5
of the results, and a higher SEE by a percentage of about 33% (Fig. 6 e, and g).
6
Daily data
7
Daily ET0 estimations obtained by the previously described approximate formulas
8
were compared with FAO56-PM procedure at the 7 sites of California. Statistical
9
results of comparisons were presented at Tables 2, and 3. Fig. 7 also presents the
10
correlations graphs of comparisons for the Davis site. A similar behaviour concerning
11
the performance of the previously described formulas was observed when daily data
12
were applied.
13
Both full set of data newly developed expressions Fo1-PM (Rs,T,RH,u) and Fo2-PM
14
(Rs,T,RH,u) are in perfect agreement with FAO-56 Penman-Monteith methods for all
15
7 sites ( the Fo2-PM is slightly inferior than the Fo1-PM). The rt value does not vary
16
significantly from 1 (maximum bias error is about 2.5% for both formulas), whereas
17
the spread of data is insignificant for all the 7 stations (minimal value of R2 is 0.993).
18
On the contrary, the simplified 1963-Penman ET-formula (full set of data) developed
19
by Valiantzas(2006), Fo-PENM (Rs,T,RH,u), produced poorer estimates than the new
20
suggested ones for all the 7 sites. The spread of data increases for all sites (minimal
21
value of R2 decreased to 0.951), and it produced a significantly higher average SEE by
22
the significant percentage of about 100% (Table 2, Figs. 7 a, b, and c).
23
In a recent paper, Valiantzas (2013c) has performed comparisons for 17 stations in
24
California indicating that the reduced set FAO-56 Penman-Monteith procedure not
24
25
1
requiring wind data has provided weaker results at all stations when compared to the
2
suggested in the same paper modification of simplified 1963-Penman ET-formula (not
3
requiring wind data also). Respectively, the Fo-PM (Rs,T,RH)
4
results than the simplified 1963-Penman ET-formula (not requiring wind data) for all
5
the 7 stations studied in the present paper (results not shown).
6
Again, the new suggested formula using a limited set of data, T and RH alone, Fo-
7
PM(T,RH), performed better than the reduced FAO-PM (T,RH) method requiring the
8
same set of data for all the 7 stations (Table 3, Figs 7 e, and f). The Fo-PM(T,RH)
9
performed better than the temperature-based formula of Hargreaves, HARGR(T) also,
10
for all the 7 stations (Table 3, Figs 7 e, and g). The HARGR(T) produced higher
11
scatter of the results for all the 7 stations, and a higher average SEE by a percentage
12
of about 22%.
13
4. Summary and Conclusion
14
A new series of simplified formulas, easy to use for routine hydrologic applications,
15
was derived to approximate the FAO-56 Penman-Monteith computational procedure.
16
The first one can be used to estimate the reference crop evapotranspiration directly
17
from the commonly measured weather data, solar radiation, maximum and minimum
18
temperature, relative humidity, and wind velocity. The second formula is a simplified
19
version of the first one and does not require the computation of the extraterrestrial
20
radiation.
21
For places where wind speed data are not readily available another formula which is a
22
simplified version not requiring wind speed data, is also proposed.
23
Further simplifications were applied resulting to an even more simplified version
24
which requires relative humidity and maximum - minimum temperature data alone.
25
provided weaker
26
1
The suggested new formulas were tested using the measured long-term monthly data
2
of a global climatic data set (FAO CLIMWAT) as well as the measured daily data
3
from 7 weather stations. Both full set of data newly developed expressions performed
4
better than the previously developed simplified 1963-Penman approximate ET0 full
5
set of data formula (Valiantzas, 2006 ET-formula). The new suggested formula using
6
a limited set of data, T and RH alone, performed better than the reduced FAO-56
7
Penman-Moneith method requiring the same set of data, and the temperature-based
8
formula of Hargreaves.
9 10
References
11
Allen, R.G., Pereira, L.S., Raes D., Smith, M., 1998. Crop evapotranspiration:
12
guidelines for computing crop water requirements. FAO Irrigation and Drainage
13
Paper, 56, Rome, 300pp.
14 15 16
D'Agostino, D., 2013. Moisture dynamics in an historical masonry structure: The Cathedral of Lecce (South Italy). Building and Environment, 63, 122-133 Exner-Kittridge, M. G. , 2012, Closure of “Case Study on the Accuracy and
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Cost/Efectiveness in Simulating Reference Evapotranspiration in West-Central
18
Florida” by Exner-Kittridge, M. G, Rains. M. C., Journal of Hydrologic Engineering,
19
17(1), 225-226.
20
Exner-Kittridge, M. G.. Rains, M. C.,2010. Case Study on the Accuracy and
21
Cost/Effectiveness in Simulating Reference Evapotranspiration in West-Central
22
Florida. J. Hydrol. Eng., 15(9), 696-703
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Hargreaves G.H., 1975. Moisture availability and crop production. Trans. Am. Soc. Agric. Eng., 18(5), 980-984.
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27
1 2
Hargreaves, G.H., Samani, Z.A., 1982. Estimating potential evapotranspiration. J. Irrig. Drain. Div., 108(3), 225–230.
3
Itenfisu D., Elliot, R. L., Allen, R.G., Walter I.A., 2000. Comparison of
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reference evapotranspiration calculations across a range of climates. Proc.,4th Nat.
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Irrig. Symp., ASAE, St. Joseph, Mich.,216-227.
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Jensen, M.E.,Burman R.D., Allen, R.G.,1990. Evapotranspiration and
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irrigation water requirements. ASCE Manuals and Reports in Engineering Practice, no
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70, 332pp.
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Lewis T., and Lamoureux S. F., 2010. Twenty-first century discharge and
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sediment yield predictions in a small high Arctic watershed. Global and Planetary
11
Change, 71(1-2), 27-41
12 13
Linacre E.T.,1993. Data sparse estimation of potential evaporation usind a simplified Penman equation., Agric. Forest Meteorol., 64, 225-237
14
Linacre E.T., 2004. Evaporation trends., Theor. Appl. Climatol., 79, 11-21
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MacDonald, R. J., Byrne, J. M., Kienzle, S. W., 2009. A physically based
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daily hydrometeorological model for complex mountain terrain. Journal of
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Hydrometeorology, 10(6), 1430-1446
18
McMahon, T.A. , Peel, M.C., Lowe, L., Srikanthan, R., McVicar, T.R., 2013,
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Estimating actual, potential, reference crop and pan evaporation using standard
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meteorological data: A pragmatic synthesis. Hydrology and Earth System Sciences,
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17( 4), 1331-1363
22 23
Penman H.L., 1948. Natural evaporation from open water, bare and grass. Proc. R. Soc. Lond., Ser. A 193, 120-145.
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28
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Penman H.L., 1963. Vegetation and hydrology. Technical Communication no 53, Commonwealth Bureau of Soils, Harpenden, England.
3
Rimmer, A., Samuels, R., Lechinsky, Y., 2009. A comprehensive study
4
across methods and time scales to estimate surface fluxes from Lake Kinneret, Israel.
5
J. Hydrol., 379(1-2), 181-192
6
Samani, Z., 2004. Discussion of ‘‘History and evaluation of Hargreaves
7
evapotranspiration equation’’ by George H. Hargreaves and Richard G. Allen. J. Irrig.
8
Drain. Eng. 130(5), 447–448.
9 10 11 12 13 14 15 16
Shuttleworth, W.J., 1993. Evaporation, Ch. 4. ed. D.R. Maidment, McGrawHill, New York, 4.1-4.53. Smith M., 1993. CLIMWAT for CROPWAT: a climatic database for irrigation planning and management. FAO Irrigation and Drainage Paper, 49, Rome. Stull R.B., 2000: Meteorology for Scientist and Engineers 2nd Edition. Brooks/Cole Thomson Learning, Page 98 Temesgen, B., Allen R.G., Jensen D.T., 1999, Adjusting temperature parameters to reflect well-watered conditions. J. Irrig. Drain. Eng., 125(1), 26-33
17
Valiantzas J.D., 2006. Simplified versions for the Penman evaporation
18
equation using routine weather data. Journal of Hydrology, 331(3-4), 690-702.
19
Valiantzas, J. D., 2012, Discussion of “Case Study on the Accuracy and
20
Cost/Efectiveness in Simulating Reference Evapotranspiration in West-Central
21
Florida” by Exner-Kittridge, M. G, Rains. M. C., Journal of Hydrologic Engineering,
22
17(1), 224-225.
28
29
Valiantzas, J. D., 2013 a. Simplified Reference Evapotranspiration Formula
1 2
Using an Empirical Impact Factor for Penman's Aerodynamic Term. Journal of
3
Hydrologic Engineering, 18 (1), 108-114. Valiantzas, J.D. 2013 b. Simple ET0 forms of Penman’s equation without wind
4 5
and/or humidity data I Theoretical development ” Journal of Irrigation and Drainage
6
Engineering, ASCE, 139 (1), 2013, 1-8.
7
Valiantzas, J. D., 2013 c. Simple ET0 Forms of Penman’s Equation without
8
Wind and/or Humidity Data II: Comparisons with Reduced Set-FAO and other
9
Methodologies. J. Irrig. Drain. Eng., 139(1), 9-19.
10 11
Figure Captions
12
Fig. 1 Variation of accurate values of the reduced difference of the radiation
13
terms of Penman and Penman-Monteith methods, Δrad0, with wind speed , u, for series
14
of computations over typical range of input meteorological variables.
15
Fig. 2 Statistical regression graph of the left-hand term ( RS / RA ) versus the
16
right-hand term 3.9 RSϕ 0.15 / N 2 + 0.16 of the approximate Eq. (10), for long-term
17
mean monthly meteorological data from 535 stations of the global CLIMWAT data
18
base .
19
Fig. 3
Variation of approximate and accurate values of the function
20
( P−M ) [ Eaero /D(stand)] e0(T) with approximate 0.0585(T + 17)u 0.75 values, for series of
21
computations over typical range of input meteorological variables,
29
30
1
Fig. 4
Statistical regression graph of the left-hand term eS ( s tan d ) versus the
2
right-hand term e 0 (T ) ⋅ (1 + 0.00043 ⋅ TR 2 ) of the approximate Eq. (23), for series of
3
computations over typical range of input meteorological variables.
4
Fig. 5 (a) Comparison of approximate empirical radiation formula, Eq. (31),
5
with exact values of the solar radiation variable, and (b) comparison of the proposed
6
modified radiation formula, Eq. (33), with exact values of the solar radiation variable,
7
for long-term mean monthly meteorological data from 535 stations of the global
8
CLIMWAT data base.
9
Fig.
6
a-g
Long-term
mean
monthly
values
of
reference
crop
10
evapotranspiration estimated by the tested formulas versus the standardized FAO-56
11
Penman-Moteith scheme, for 535 stations of the global CLIMWAT data set.
12
Fig. 7 a-g Daily values of reference crop evapotranspiration estimated by the
13
various methods versus the standardized FAO-56 Penman-Moteith scheme, for Davis.
14
Table Legends
15
Table 1 Station name/number, Elevation (m), Latitude (deg), Longitude (deg),
16
Climate Conditions, Average Wind Speed (m/s) and Period of Data Used of the 7
17
stations of CIMIS-California selected for comparisons.
18
Table 2 Statistical summary of comparison between the daily values of ET0
19
estimated by different full-set of data formulas and the reference method standardized
20
FAO-56 Penman-Monteith scheme, for the 7 stations.
21
Table 3 Statistical summary of comparison between the daily values of ET0
22
estimated by different limited data formulas and the reference method standardized
23
FAO-56 Penman-Monteith scheme, for the 7 stations.
30
31
1
Table 1.
Arid
Average Wind (m/s) 2.0
1995-2005
121.91
Humid
2.29
1995-2001
38.28
121.79
Arid
3.90
1998-2004
7.6
38.13
121.38
Humid
1.09
2000-2004
Davis/6
18.3
38.54
121.78
Arid
2.68
1994-1999
Parlier/39
102.7
36.6
119.5
Arid
1.73
1995-2005
Victorville
880.9
34.48
117.26
Arid
2.76
1995-1997
Station name/number
Elevation (m)
Latitude (deg)
Longitude (deg)
Climate
McArthur/43
1008.9
41.07
121.45
Zamora/27
15.2
38.81
HastingsTract/122
3
Lodi West/166
2
31
Data Period
32
1 2 3 4 5 6
TABLE 2
Station name 1. McArthur 2. Zamora 3.Hastings Tract 4.Lodi West 5.Davis 6. Parlier 7. Victorville
Average
Fo1-PM (RS,T,RH,u) vs FAO56PM SEE rt R2 (mm/d)
Fo2-PM (RS,T,RH,u) vs FAO56-PM SEE rt R2 (mm/d)
Fo-PENM (RS,T,RH,u) vs FAO56-PM SEE rt R2 (mm/d)
1.014
0.997
0.116
1.020
0.996
0.190
0.976
0.983
0.290
1.010
0.995
0.151
1.022
0.997
0.159
0.986
0.986
0.255
1.002
0.993
0.217
1.020
0.994
0.248
1.009
0.951
0.588
1.022
0.998
0.114
1.027
0.995
0.187
1.039
0.995
0.195
0.994
0.996
0.157
1.014
0.997
0.159
0.978
0.984
0.324
1.003
0.998
0.095
1.000
0.998
0.114
0.986
0.993
0.201
0.985
0.997
0.161
1.012
0.994
0.225
0.991
0.969
0.438
1.004
0.996
0.144
1.016
0.996
0.183
0.996
0.980
0.327
7 8 9 10
32
33
1 2 3 4 5 6
TABLE 3
Fo-PM(T,RH) vs FAO56-PM
Fo-PM(RS,T,RH) vs FAO56-PM Station name 1. McArthur 2. Zamora 3.Hastings Tract 4.Lodi West 5.Davis 6. Parlier 7. Victorville
FAO-PM(T, RH) vs FAO56-PM
HARG(T) vs FAO56PM
rt
R2
SEE (mm/d)
rt
R2
SEE (mm/d)
rt
R2
SEE (mm/d)
rt
R2
SEE (mm/d)
1.036
0.970
0.413
1.002
0.954
0.474
1.071
0.951
0.526
1.044
0.942
0.509
1.014
0.925
0.588
1.016
0.907
0.641
1.036
0.895
0.705
1.060
0.829
0.886
0.870
0.937
0.926
0.830
0.928
1.105
0.830
0.917
1.143
0.842
0.852
1.250
1.173
0.961
0.700
1.190
0.928
0.801
1.210
0.917
0.887
1.234
0.910
0.924
0.949
0.931
0.670
0.945
0.915
0.734
0.950
0.903
0.771
0.949
0.838
0.957
1.037
0.976
0.374
1.051
0.965
0.450
1.077
0.956
0.544
1.085
0.941
0.572
0.933
0.890
0.872
0.948
0.886
0.881
0.931
0.849
0.982
0.848
0.845
1.228
1.001
0.941
0.649
0.997
0.926
0.727
1.015
0.912
0.794
1.009
0.880
0.925
Average
7
33
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
(a)
(b)
Figure 6
g
Figure 7
g
34
1 2 3 4
• • •
New simple expressions for the “standardized” Penman-Monteith ET0 New temperature – humidity ET0 formula New solar radiation empirical formula
5 6
34
S0022-1694(13)00650-1 http://dx.doi.org/10.1016/j.jhydrol.2013.09.005 HYDROL 19085
To appear in:
Journal of Hydrology
Received Date: Revised Date: Accepted Date:
28 May 2013 5 September 2013 7 September 2013
Please cite this article as: Valiantzas, J.D., Simplified forms for the standardized FAO-56 Penman-Moneith reference evapotranspiration using limited weather data, Journal of Hydrology (2013), doi: http://dx.doi.org/10.1016/j.jhydrol. 2013.09.005
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1
1
Simplified forms for the standardized FAO-56 Penman-Moneith
2
reference evapotranspiration using limited weather data
3
John D. Valiantzas*
4 5
Abstract
6
New simple algebraic expressions equivalent in accuracy to the “standardized” FAO-
7
56 Penman-Monteith daily reference crop evapotranspiration (ET0) computation
8
procedure are derived. The suggested formulas are based on extensions made to a
9
previously developed simple algebraic formula for the Penman evaporation equation.
10
The derivation of the new formulas is based on simplifications made and the
11
systematic analysis on the correspondence between the FAO-56 Penman-Monteith
12
equation and the standardized Penman’s equation. The ET0 calculated by the new
13
formulas is easy to use for routine hydrologic applications requiring routine weather
14
records usually available at standard weather stations: air temperature, T (oC), solar
15
radiation RS (MJ/m2/d), relative humidity, RH (%), and wind velocity, u (m/s). For
16
places where not all these data are available (or reliable), new expressions which does
17
not require wind speed and/ or solar radiation data are proposed. A simplified formula
18
for estimating reference crop evapotranspiration
19
maximum and minimum air temperatures), and Tdew (the dew point temperature) or
20
RH data alone is derived. The performance of the new derived formulas was tested
21
under various climatic conditions using a global climatic data set including monthly
22
data as well as daily data obtained from weather stations.
requiring Tmax and Tmin ( the
23 24
Key-words: Penman-Monteith, Penman equation,
25
hydrologic models, water resources management
26 27 28
*
reference evapotranspiration,
Division of Water Resources Management, Department of Natural Resources and Agricultural Engineering, Agricultural
University of Athens, 75 Iera Odos, 11855, Athens, Greece Email:[email protected]
1
2
1
1. Introduction
2
The FAO-56 methodology (Allen et al., 1998) for estimating reference crop
3
evapotranspiration, ET0, (daily or monthly data), recommends the sole use of the
4
Penman-Monteith equation. The standardized scheme is considered as the “standard”
5
method in hydrological and irrigation applications at well-watered meteorological
6
stations under varying locations and climatic conditions. According to Allen et al.
7
(1998) the recommended form of the FAO-56 Penman-Monteith equation is ( P−M ) ( P−M ) ET0( P − M ) = Erad + Eaero
8
⎧ 0.408Δ( Rn − G ) ⎫ ⎧ 900γ uD ⎫ =⎨ ⋅ ⎬+ ⎨ ⎬ ⎩ Δ + γ (1 + 0.34u ) ⎭ ⎩[Δ + γ (1 + 0.34u )] (T + 273) ⎭
(1)
9
( P−M ) where the term Erad , is the radiation term of the FAO-56 Penman-Monteith
10
( P−M ) equation, and Eaero is the aerodynamic component, Rn is the net radiation at the
11
surface (MJ/m2/d); Δ is the slope of the saturation vapor pressure curve (kPa/ oC); γ is
12
psychrometric coefficient (kPa/ oC); G is soil heat flux density (MJ/m2/d) and u is
13
wind speed at 2 m height (m/s). The application of the FAO-56 Penman Montheith,
14
Eq. (1), requires the commonly weather station measured meteorological observations
15
e. g. maximum and minimum air temperatures, Tmax, and Tmin, , solar radiation, RS,
16
maximum and minimum relative humidity RHmax , RHmin respectively, wind speed, u,
17
as well as site details of latitude and altitude.
18
Penman (1948) published the radiation-aerodynamic combination equation to predict
19
evaporation from open water, bare soil, and grass.
20
6.43 ( fU ) D ⎫ ⎧ Δ ( Rn ) ⎫ ⎧ γ (P) ( P) + Eaero =⎨ ⋅ ⋅ EPEN = Erad ⎬+⎨ ⎬ λ ⎩Δ + γ λ ⎭ ⎩Δ + γ ⎭
21
(2)
2
3
1
(P) where the term Erad , is the radiation term of the Penman combination equation
2
corresponding to the incoming net short wave radiation component, and the outgoing
3
(P) net long wave radiation component, and Eaero is the aerodynamic component λ is
4
latent heat of vaporization (MJ/kg); fU is the Penman’s wind function.
5
The original Pemman (1948, 1963) equation as well as its various modifications have
6
also been widely used to estimate reference crop evapotranspiration. In a recent paper,
7
Valiantzas (2006) developed a simple algebraic explicit formula, equivalent in
8
accuracy to the 1963-Penman equation for estimating ET0. The derivation of the
9
Valiantzas (2006) formula depending explicitly on the commonly available weather
10
station measured variables T, RS , RH, and u is based on the systematic analysis and
11
mathematical simplifications made to the “standardized” computation procedure
12
recommended by Shuttleworth (1993) and Allen et al. (1998).
13
simplified formulas suggested by Valiantzas (2006) were recommended, validated
14
and successfully applied by various researches as Lewis and Lamoureux (2010),
15
MacDonald et al. (2009), Rimmer et al. (2009), McMahon et al. (2013), and
16
D'Agostino (2013) and others. In recent papers, Valiantzas (2012) and Valiantzas
17
(2013 a,b,c) suggested formulas that are based on simplifications made to the original
18
Valiantzas (2006) ET-formula and they are all equivalent in accuracy to the 1963-
19
Penman model suggested for estimating ET0,
20
However, according to the results of Jensen et al. (1990), Itenfisu et al. (2000), and
21
others the 1963-Pennman method for estimating ET0 is not as accurate as the Penman-
22
Monteith method. Jensen et al. (1990) evaluated 20 ET0 methods and compared
23
against lysimeter measurements at 11 locations located in different climatic zones
3
The previous
4
1
around the world. The Penman-Monteith method ranked as the best method for all
2
climatic conditions. The 1963-Penman method ranked fourth of all methods. Standard
3
error of estimate for the 1963-Penman method was 0.57 mm/d compared against
4
monthly lysimeter data. This compared to 0.36 mm/d for the Penman-Monteith
5
method.
6
Itenfisu et al. (2000) reported that the 1963-Penman method yielded higher ET0
7
values when compared to the ASCE-Penman Monteith equation (following similar
8
ET0 trends with FAO-56 Penman-Monteith) in Jensen et al. (1990) with the ratio of
9
1.0 to 1.10. Jensen et al. (1990) reported that all the Penman models (except the
10
Penman-Monteith) overestimated lysimeter measurements in humid locations.
11
Valiantzas (2013b) using data from a global dataset has shown that the formula
12
equivalent to the original 1963-Penman model (Valiantzas 2006 ET-formula),
13
although resulting in relatively good estimates of ET0 compared to the FAO-56
14
Penman-Monteith “standard” method tends to overestimate ET0.
15
On the other hand, it is universally accepted that the FAO-56 Penman-Monteith is the
16
“standard” method for estimating daily or monthly ET0.
17
A disadvantage in the application of the FAO-56 Penman Montheith using Eq. (1) is
18
that the main weather variables appearing directly in the equation are T, Rn, D, Δ, γ,
19
and u. Although there is specific instruments to measure Rn, and D, the usually
20
available weather records in standard meteorological station are T, RH, RS, and u.
21
(Shuttleworth, 1993). The structure of Eq. (1) suggests that the commonly measured
22
inputs appear explicitly in the computation of Rn, D, Δ, and γ in Eq. (1). The FAO-56
23
methodology represented by Eq. (1), is actually an abbreviation form of a complex
24
algorithm comprising a plethora of specific supporting equations adopted to convert
4
5
1
the input measured variables into a number of other intermediate estimated
2
parameters. A plethora of intermediate parameters appear in the application of the
3
FAO-56 Penman-Monteith procedure, such as the latent heat of vaporization, the
4
saturation vapor pressure, the actual vapor pressure, the psychrometric coefficient, the
5
slope vapor pressure curve, the atmospheric pressure, the effective emissivity of the
6
surface, the clear-sky solar radiation, the Stephan-Boltzman constant, the cloudiness
7
factor, and many others. The complexity of calculations increases as each of these
8
parameters could be expressed by a variety of units. The use of all these parameters
9
could create confusion in the calculation steps during the application of the FAO-56
10
Penman-Monteith procedure, thus resulting in significant errors should the appearing
11
parameters not be expressed in the appropriate units.
12
In this paper, an algebraic formula equivalent in accuracy to the FAO-56 Penman
13
Monteith algorithm is developed for calculating directly ET0 using the input routinely
14
measured variables Tmax, and Tmin , RS , RH, u, and Z only, where Z (m) is the
15
elevation of the site. The only additional parameter to be estimated appearing in the
16
suggested formula is the extraterrestrial radiation, RA.
17
The formula is obtained by analysing the correspondence of radiation and
18
aerodynamic terms between the two models (Penman and FAO-56 Penman-Monteith
19
model) Subsequently, by varying the meteorological variables over their typical range
20
of variation using numerical simulations and regression procedures, a series of new
21
empirical simplified expressions of simple mathematical form approximating the
22
standardized components of the FAO-56 Penman-Monteith procedure were
23
developed.
5
6
1
Since the computation of variable RA in daily basis requires a rather complicated
2
numerical procedure another more simplified formula, easy to use for routine
3
hydrologic applications, is also developed to compute directly ET0 from the input
4
measured variables Tmax, and Tmin , RS , RH, u, Z, φ, and J only, where φ is latitude
5
of the site (radians) and J is the Julian day number.
6
The proposed ET0 algebraic formulas (full or limited set of data) explicit to routinely
7
measured data could easily incorporated to regression procedures applied to routinely
8
measured meteorological data or lysimeter data to build up ET0 empirical accurate
9
formulas at local scale. The suggested formulas could also easily adopted to apply
10
regionalization procedures for estimating ET0 at large scale. Such algebraic formulas might
11
facilitate the investigation of ET0 trends or sensitivity analysis (Linacre 2004).
12
A major disadvantage to application of the standardized FAO-56 Penman-Monteith
13
procedure is the relatively high data demand requiring measurements of T, RH, RS,
14
and u. The number of weather stations where all these parameters are available is
15
limited especially in developing countries. Another problem is linked to data quality.
16
Wind speed data are rarely available or of questionable precision (Jensen et al. 1997;
17
Allen 1996). Therefore, for such cases, a new expression which does not require wind
18
speed data is also proposed, in this paper.
19
Further, solar radiation data are not always reliable (Last and Snyder, 1998) whereas
20
some older electronic sensors for relative humidity measurement produce commonly
21
errors (Allen 1996).
22
Lastly, another serious problem is related to the cost of instrumentation for collecting
23
the required meteorological data in automated weather stations. Using methods that
24
require limited set of data might reduce the cost drastically. Recently, Exner-
6
7
1
Kittridge and Rains (2010) evaluated various alternative combinations of limited set
2
of data ET0 methods for their accuracy in conjunction with their corresponding cost.
3
They introduced the cost effectiveness index. Valiantzas (2012) and Exner-Kittridge
4
(2012) concluded that if the addition of RH measurements to the air temperature data,
5
T, improve the accuracy of the ET0 estimation, then the cost effectiveness of a RH-T
6
method could increase dramatically compared to other alternatives of limited set of
7
data methods
8
extremely low additional cost of the RH sensor..
9
Therefore, further simplifications on the FAO-56 Penman-Monteith formula was
10
made leading to a temperature-humidity based formula not requiring u and RS data
11
measurements. The performance of the derived formulas for estimating reference crop
12
evapotransiration is also tested.
13
2. Theory-The development of the new simplified versions of standardized FAO-
14
56 Penman-Monteith
15
For the development of the simplified versions it is initially assumed that Z=0, where
16
Z is elevation of the site (m).
17
For Z=0 the coefficient γ takes the single constant value of γ0 =0.0671 kPa.
18
Furthermore, as the value of λ varies only slightly over a normal temperatures range, a
19
single constant value (for T=20 oC) is considered λ0=2.45 MJ/kg.
20
( P−M ) ( P−M ) (P) 2.1 Approximation of the Erad . On the correspondence between Erad and Erad
21
The variable Δrad0, expressing the difference of the radiation terms of the two models
22
reduced by dividing by the variable Rn is given as
23
⎡ ⎤ ( P−M ) ( P) Δ rad 0 = ( Erad − Erad ) / Rn = 0.408 ⎢ Δ + (1 + Δ0.34u)γ − Δ +Δγ ⎥ 0 0⎦ ⎣
(requiring additional u and/or Rs instrument sensors), due to the
7
(3)
8
1
( P−M ) (P) Since Erad / Rn depends on T and u, and Erad / Rn depends on T, the variable Δrad0,
2
expressing the reduced difference of the two radiation terms is generally affected by
3
the values of wind speed, u, and average temperature, T.
4
In this section it will be shown that there is a considerably high dependence of the
5
Δrad0 on the u variable, whereas the effect of the variable T on this term is rather
6
insignificant.
7
To investigate the behavior of this term a series of numerical simulations were carried
8
out to generate the “accurate” values of Δrad0. Combinations of the required input
9
mean temperatures values, T, and wind speed values u were generated by varying T
10
and u over typical range of variations. The values of T varied between 0 and 35 0C
11
whereas u varied between 0.5 to 8.5 m/s. Series of data sets were generated by
12
combining the previous “typical” input values of T and u in all possible ways. For
13
each given set of data the values of the term Δrad0 were computed from Eq. (3).
14
Afterwards the “accurate” values of Δrad0 obtained from simulations were plotted
15
against the values of u. The results from 18,000 numerical simulations (Fig.1) show
16
that there is a high dependence of the computed Δrad0 on the u variable whereas the
17
effect of the values of T is rather insignificant. Using the least square procedure the
18
variation of Δrad0 with u can be approximated by the following relationship;
19
Δ rad 0 = − 0.03u 0.7
20
Results indicated a good agreement between approximate and exact values. Statistical
21
regression results for Eq. 4 are Y=0.973X and R2=0.959, where Y is approximate
22
estimation, X is exact value, and R2 is coefficient of determination.
(4)
8
9
1
Furthermore, the variable Rn may be rounded by an empirical relationship to Rs
2
(Linacre, 1993), e. g. Rn ≈ 0.55RS
3 4
5
( P−M ) Therefore E rad may be approximated by combining the previous formula
(approximating Rn) and Eq. (4) with Eq. (3) as ( P−M ) ( P) ( P) Erad ≈ Erad − 0.03Rnu 0.7 ≈ Erad − 0.03 ⋅ 0.55RS u 0.7
(5)
6
On the other hand, according to Valiantzas (2006) the two component of the radiation
7
( P) ( P) term of the Penman combination equation, E radS , and E radL corresponding to the
8
incoming net short wave radiation component and the outgoing net long wave radiation
9
component respectively can be accurately approximated as
10
( P) E radS ≈ 0.051(1 − α ) RS T + 9.5
11
⎛R ⎞⎛ RH ⎞ ( P) ≈ 0.188(T + 13) ⎜ S − 0.194 ⎟ ⎜⎜1 − 0.00015(T + 45)2 EradL ⎟ 100 ⎟⎠ ⎝ RA ⎠⎝
12
Finally, since
13
(6)
(P) (P) (P) Erad = EradS − EradL
(7)
(8)
14
( P−M ) then the radiation term of the FAO-56 Penman-Monteith procedure, Erad , can be
15
calculated by substituting Eqs (6) and (7) into Eq.(8) and the resultant equation in to
16
Eq. (5)
17
Further Simplifications
18
( P) The component of the radiation term of Penman’s equation, E radL , can be further
19
simplified according to Valiantzas (2006) as
20
E
(P) radL
⎛R ⎞ ≈ 2.4⎜⎜ S ⎟⎟ ⎝ RA ⎠
2
⎛ RH ⎞ + 0.024(T + 20 )⎜1 − ⎟ ⎝ 100 ⎠
9
(9)
10
1
In Eq. (9) the computation of variable RA in daily basis requires a rather complicated
2
numerical procedure. Therefore its term [(RS/RA)] from Eq. (9) is replaced intuitively
3
by a term of the following simple mathematical form, not requiring calculation of RA.
4
⎛ RS ⎞ ⎛ RS C2 ⎞ ⎜ ⎟ ≈ C1 ⎜ 2 ϕ ⎟ + C3 ⎝N ⎠ ⎝ RA ⎠
5
where C1, C2, and C3 are empirical coefficients that should be identified. The
6
maximum possible duration of daylight (hrs), N, can be approximately estimated as
7
N ≈ 9.8δϕ + 12
8
where δ is solar declination (radians) given by
9
δ = 0.409 sin(
(10)
(10a)
2π J − 1.39) 365
(10b)
10
φ is the latitude of the station expressed in radians and J is Julian day number. For
11
monthly estimations the Julian day corresponding to ith month is calculated as
12
J = INT (30.5i − 14.6)
13
A calibration procedure was applied to identify the three regression coefficients, C1,
14
C2, and C3 using a global climatic data set that includes monthly data (the FAO-
15
CLIMWAT, Smith 1993). Climatic data from thirteen countries with relatively high
16
quality records (Temesgen et al., 1999), that essentially cover all the typical range of
17
variation of the input weather variables, were selected: Spain (58 meteorological
18
stations), France (42), Italy (60), Greece (20), and Cyprus (27) in Europe; Pakistan
19
(23), Lebanon (16) and India (18) in Asia; Egypt (28), Tunisia (19), Algeria (22),
20
Ethiopia (142), and Sudan (63) in Africa. The total numbers of the selected stations is
21
535. These data were selected for the calibration of the derived formula because they
22
essentially cover all the typical range of variation of the input weather variables T, RS,
(10c)
10
11
1
RH, and u. The latitude for the selected stations varies from 3 to 51o, and the elevation
2
varies from 0 to 3000 m. Some of the countries (France, Italy and Spain) are selected
3
to represent humid and semi-humid temperate climates and others (Ethiopia, Sudan,
4
Egypt and Pakistan) to represent dry arid and semi-arid tropical climates (Temesgen
5
et al., 1999). All the stations are located in the northern hemisphere.
6
Calibration procedure of the no=6,420 data of the database leads to the values of
7
coefficients of C1=3.9, C2= 0.15, and C3=0.16.
8
The statistical results for Eq. (10) (shown in Fig. 2) using as input the meteorological
9
data from the FAO-CLMIWAT database indicated a good agreement between
10
approximate and exact values.
11
( P−M ) ( P−M ) (P) 2.2 Approximation of the Eaero . On the correspondence between Eaero and Eaero
12
The aerodynamic term of the Penman’s procedure is given as
13
14 15
16
(P) Eaero =
γ0 6.43 fU (u ) D( s tan d ) λ0 Δ + γ 0 1
(11)
whereas the aerodynamic term of the Penman-Monteith according to the standardized FAO-56 computational procedure is calculated as ( P−M ) Eaero =
uD( s tan d ) 900γ 0 ⋅ [Δ + γ 0 (1 + 0.34u )] (T + 273)
(12)
17
where D(stand) is the mean vapor pressure deficit computed according to
18
recommendation of the standardized FAO-56 procedure.
19
( P−M ) (P) It is assumed that the Eaero can be approximated from Eaero by an empirical
20
(P) relationship according to which Eaero is multiplied by an unknown empirical function
21
of temperature
11
12
1
( P−M ) (P) Eaero ≈ Eaero ⋅ fT (T ) ≈ D( s tan d )
γ0 6.43[ fU (u ) ⋅ fT (T )] λ0 Δ + γ 0 1
(13)
2
where fU (u ) and fT (T ) are considered as purely empirical functions of wind and
3
temperature respectively assumed to have the following simple mathematical forms
4
fU (u ) = c1u c2
5
fT (T ) =
(13a)
1 c3 − T
(13b)
6
where c1, c2 and c3 are empirical regression coefficients that should be identified.
7
To demonstrate the validity of empirical Eq. (13), and identify the three regression
8
parameters, a series of numerical simulations was conducted to generate synthetic
9
meteorological data used for the calibration. The numerical simulations were carried
10
( P−M ) out to generate “exact” [ Eaero /D(stand)] values (depending on T and u only) according
11
to Eq. (12). Combination of typical ranges of T and u, with 0< T< 35 oC and 0.5< u
12
( P−M ) <8.5 m/s, were used to calculate the “exact” values of [ Eaero /D(stand)]. Then the
13
regression
14
1 γ0 ( P−M ) ⎡⎣ Eaero / D( s tan d ) ⎤⎦ ≈ 6.43 ⎡⎣c1u c2 / (c3 − T ) ⎤⎦ λ0 Δ + γ 0
15
as given by Eq. (13).
of
the
empirical
equation
has
the
following
form
(14)
16
Further simplifications could be made on the initial regression Eq. (14). Valiantzas
17
(2006) has demonstrated that the following relationship is an accurate approximation:
18
e0 (T ) ⋅ [
γ0 6.43] ≈ 0.048(T + 20) λ0 Δ + γ 0 1
(15)
12
13
1
where e0(T) is the value of the saturation vapor pressure curve, corresponding to the
2
temperature value of
3
expression for the Clausius-Calpeyron equation ; Stull, 2000)
4
⎛ 17.27 ⋅ Τ ⎞ e0 (Τ) = 0.6108 ⋅ exp ⎜ ⎟ ⎝ Τ + 237.3 ⎠
T and calculated by the Teten's equation, (the empirical
(16)
5 6
Here T is temperature in oC. Regression statistics of Eq. (15) yields Y=0.994X and
7
R2=0.9996.
8
Substituting Eq. (15) into Eq. (14), together with the results of the regression Eq. (14)
9
is transformed into the following simplified form
10
( P−M ) ⎡⎣ Eaero / D( s tan d ) ⎤⎦ e0 (T ) ≈ c1u c2 [ 0.048(T + 20) / (c3 − T ) ]
(17)
11
Using the least squares regression procedure on Eq. (17), and using the synthetic
12
meteorological data, the three regression parameters were determined (c1=426,
13
c2=0.75 and c3=400). Approximate values obtained from the right hand side of Eq.
14
( P−M ) (17) are compared with the “exact” values of [ Eaero /D(stand)]e0(T) (Figure not shown)
15
. Results indicated that empirical formula, Eq. (17) yields a very good approximation
16
of
17
Furthermore, the temperature term [ (T + 20) / (400 − T ) ] appearing in the right hand
18
side of Eq. (17) (with c3=400) can be approximated by the following simple
19
expression: 349(T+17) (this is demonstrated performing numerical simulations with
20
0< T <35 oC, with regression results Y=1.001X, and R2 =0.9995). Finally, substituting
21
the previously obtained simple expression in to Eq. (17) the following formula is
22
( P−M ) obtained to approximate [ Eaero /D(stand)]
( P−M ) [ Eaero /D(stand)] e0(T). Regression yields Y=1.001X, and R2 =0.975 .
13
14
1
( P−M ) ⎡⎣ Eaero / D( s tan d ) ⎤⎦ e0 (T ) ≈ 0.0585(T + 17)u 0.75
(18)
2
( P−M ) The generated synthetic “exact” values of e0 (T ) [ Eaero /D(stand)] obtained from the
3
previous series of numerical simulations according to Eq. (12) were plotted against
4
the approximate values of 0.0585(T + 17)u 0.75 (Eq. (18)). The results from 18,000
5
numerical simulations are reported in Fig. 3. Statistical regression results for Eq. 18
6
(shown in Fig. 3) are Y=1.003X and R2=0.974.
7
Aprroximation of D(stand)
8
The computation of the mean “standardized” vapor pressure deficit, D(stand), according
9
to the recommendation of the standardized FAO-56 Penman-Monteith procedure is
10
as follows:
11
D( s tan d ) = eS ( s tan d ) − ea ( s tan d )
12
where eS ( s tan d ) and ea ( s tan d ) are the mean “standardized” saturation and actual vapor
13
pressures respectively computed according to the FAO-56 Penman-Monteith
14
procedure as:
15
eS ( s tan d ) = 0.5 ⎡⎣ e 0 (Tmax ) + e 0 (Tmin ) ⎤⎦
16
where e0(Tmax) and e0(Tmin) are saturated vapor pressures corresponding to the
(19)
(20)
17
temperature values of t=Tmax and t=Tmin respectively, computed by Eq. (16) and
18
ea ( s tan d ) =
19
Eq. (21) is applied when only mean relative humidity data, RH, are available.
20
Alternatively, when RHmax and RHmin data are available, then the following
21
relationship is recommended for computing eS ( s tan d )
RH 2 ⋅ 0 100 ⎡⎣1 / e (Tmax ) + 1 / e 0 (Tmin ) ⎤⎦
(21)
14
15
RH max 0 RH min 0 ⋅ e (Tmax ) + ⋅ e (Tmin ) 100 100
1
ea ( s tan d ) =
2
Systematic analysis of the daily meteorological data obtained from various weather
3
stations of the CIMIS database in California have shown that both equations (Eqs. 21
4
and 22) provided values for ea ( s tan d ) that can be considered almost identical for
5
practical applications.
(22)
6
The ( eS ( s tan d ) ) computed from Eq. (20) is affected only by the values of temperature
7
Tmax and Tmin. To investigate the behavior of the previous term, a series of numerical
8
simulations were carried out to generate the “exact” values of eS ( s tan d ) from Eq. (20).
9
Combination of typical ranges of Tmax, and Tmin were implicitly generated as
10
Tmax=T+TR/2 and Tmin=T-TR/2 by varying the mean temperature T and the difference
11
of temperatures TR=(Tmax-Tmin) over a typical range of variations, 3< TR< 24 oC and
12
2< T< 37 oC. Data sets for which Tmax>46 oC or Tmin<-5 oC were excluded from
13
simulations. Afterwards, the “exact” values of the term [eS ( s tan d ) / e0 (T )] obtained
14
from simulations were plotted against the values of the term TR (Figure not shown). It
15
is shown that there is a significantly high dependence of the computed [eS ( s tan d ) / e0 ]
16
term on the TR variable (with a coefficient of determination R2=0.944). Using the
17
least square procedure to the results from no=3,557 simulations the following
18
approximate relationship for ( eS ( s tan d ) ) presented in Fig. 4 was obtained
19
eS ( s tan d ) ≈ e 0 (T ) ⋅ (1 + 0.00043 ⋅ TR 2 )
20
Results presented in Fig. 4 indicated a very good agreement between approximate and
21
exact values. Regression yields Y=0.991X, and R2 =0.9995
(23)
15
16
1
A similar procedure using the same synthetic data is applied to investigate the
2
behavior of the term [e0(Tmax) · e0(Tmin)]. It is shown that there is a significantly high
3
dependence of the computed [e0(Tmax) · e0(Tmin)] term on e0(T). Using the least square
4
procedure the following approximate relationship was obtained
5
e 0 (Tmax ) ⋅ e0 (Tmin ) ≈ e0 (T ) 2
6
Statistical regression results for Eq. (24) are Y=0.983X and R2=0.9996.
7
Substituting Eqs. (23) and (24) into Eq.(21), an approximation of ea ( s tan d ) is obtained.
8
Afterwards, substituting the resultant approximation as well as Eq. (23) in to Eq.(19)
9
after manipulation, D(stand) is finally approximated as
10
D( s tan d )
(24)
⎡⎣(1 + 0.00043 ⋅ TR 2 ) 2 − RH /100 ⎤⎦ ≈ e (T ) ⋅ (1 + 0.00043 ⋅ TR 2 ) 0
(25)
11 12
A similar procedure using synthetic meteorological input data TR and RH leads to
13
further simplification of Eq. (25)
14
D( s tan d ) ≈ e0 (T ) ⎡⎣(1.03 + 0.00055 ⋅ TR 2 ) − RH /100⎤⎦
15
3. Results
16
3.1 The new expressions
17
Expressions requiring full set of data
18
( P−M ) An accurate approximate of Erad is obtained by substituting Eqs. (6) and (7) into Eq.(8)
19
( P−M ) and the resultant equation in to Eq. (5). Similarly, an accurate approximate of Eaero is
20
obtained by substituting the approximate Eq. (25), into approximate Eq. (18), after
21
( P−M ) manipulation. Finally substituting the previously obtained approximates of Eaero and
16
(26)
17
1
( P−M ) Erad in to Eq. (1) an accurate approximate version for the FAO-56 Penman-Monteith
2
procedure for estimating ET0 in daily basis is obtained
3
( referred to in this paper as “Fo1-PM (Rs,T,RH,u)”):
4
ET0( P − M ) ≈ 0.051(1 − α ) RS T + 9.5 5
⎛R ⎞⎛ RH ⎞ 0.7 −0.188(T + 13) ⎜ S − 0.194 ⎟ ⎜⎜1 − 0.00015(T + 45) 2 ⎟⎟ − 0.0165RS u 100 ⎠ ⎝ RA ⎠⎝ ⎡(1 + 0.00043 ⋅ TR 2 ) 2 − RH /100⎤⎦ 0.75 ⎣ +0.0585(T + 17)u + 0.0001Z (1 + 0.00043 ⋅ TR 2 )
(27)
6
RS and RA should be expressed in MJ/m2/d (1 MJ/m2/d =23.88 cal/cm2/d = 0.408
7
mm/d (equivalent evaporation) =11.57 W/m2, T, Tmax, Tmin, and TR= (Tmax- Tmin) in
8
o
C, u in m/s, RH (%), Z in m, and α=0.23. Note that the above formula depends on T
9
but also on Tmax and Tmin values. The last term of the right-hand side equation
10
(indicating the effect of Z) is obtained by following a similar procedure with this
11
applied for the derivation of simplified Penman’s version in Valiantzas (2006).
12
Subsequently, another simplified version is obtained when the approximate
13
simplifications, Eqs. (9) and (26) were used
ET0( P − M ) ≈ 0.051(1 − α ) RS T + 9.5 2
14
⎛R ⎞ ⎛ RH ⎞ 0.7 −2.4 ⎜ S ⎟ − 0.024 (T + 20 ) ⎜1 − ⎟ − 0.0165RS u R 100 ⎝ ⎠ ⎝ A⎠
(28)
+0.0585(T + 17)u 0.75 ⎡⎣(1.03 + 0.00055 ⋅ TR 2 ) − RH /100⎤⎦ + 0.0001Z 15
In Eqs. (27) and (28) the computation of variable RA in daily basis requires a rather
16
complicated numerical procedure. A further simplified version not requiring
17
calculation of RA is proposed using the simplified approximate Eq. (10)
18
(referred to in this paper as “Fo2-PM (Rs,T,RH,u)”):
17
18
1
ET0( P − M ) ≈ 0.051(1 − α ) RS T + 9.5 2
2
⎛ ⎞ ⎛ ⎞ RSϕ 0.15 ⎛ RH ⎞ 0.7 0.92 − ⎜ 22.46 ⎜ + ⎟ − 0.024 (T + 20 ) ⎜1 − ⎟ ⎟ − 0.0165RS u 2 ⎝ 100 ⎠ ⎝ [4sin(2π J / 365 − 1.39) ⋅ ϕ + 12] ⎠ ⎝ ⎠ +0.0585(T + 17)u 0.75 ⎡⎣(1.03 + 0.00055 ⋅ TR 2 ) − RH /100 ⎤⎦ + 0.0001Z
(29)
3 4
the latitude of the station φ is expressed in radians and J is Julian day number. For
5
monthly estimations the Julian day corresponding to ith month is calculated as
6
J = INT (30.5i − 14.6)
7
Expression not requiring wind speed data
8
( referred to in this paper as “Fo-PM(Rs,T,RH)”)
9
In places where no wind data are available the average value of u=2 m/s of 2,000
10
stations over the globe (Allen et al., 1998) can be used in the FAO-56 Penman-
11
Monteith equation. Substituting the value of u=2 m/s in the previously developed
12
formula, Eq. (29), the following approximate formula is proposed for ET0 estimation,
13
when wind speed data are missing
14
ET0( P − M ) ≈ 0.0393RS T + 9.5 2
15
⎛ ⎞ ⎛ ⎞ RSϕ 0.15 ⎛ RH ⎞ − ⎜ 22.46 ⎜ + 0.92 ⎟ − 0.024 (T + 20 ) ⎜1 − ⎟ − 0.0268RS 2 ⎟ π ϕ [4sin(2 / 365 1.39) 12] 100 J − ⋅ + ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ +0.0984(T + 17) (1.03 + 0.00055 ⋅ TR 2 − RH /100 )
16 17 18
Expression requiring temperature-humidity data alone (referred to in this paper as “Fo-PM(T,RH)”)
18
(30)
19
1
Hargreaves and Samani (1982) suggested the following radiation empirical formula
2
for the estimation of solar radiation
3
RS ≈ kRS ⋅ RA (Tmax − Tmin )0.5
4
where kRs is empirical radiation adjustment coefficient which generally differs from
5
location to location from 0.12 to 0.25 in average (Samani, 2004). There are several
6
values adopted for kRs depending on various topographical and climatological factors.
7
The conventionally default “average” adopted value was kRs≈0.17. Thus inherent to its
8
empirical nature, there is some uncertainty relatively to this coefficient.
9
Allen et al. (1998) suggested that the value of minimum temperature Tmin is a
10
substitute of the dew point temperature, Tdew, (at well-watered meteorological
11
stations) e.g. Tdew≈Tmin. The dew point temperature, Tdew (oC), can be computed by the
12
following formula (Allen et al, 1998):
13
Tdew =
14
where ea is the actual vapour pressure that can be estimated as
15
⎡ 17.27T ⎤ RH ⋅ ea = 0.6108 ⋅ exp ⎢ ⎣ T + 237.3 ⎥⎦ 100
16
The value of Tmin in the radiation formula Eq. (31) is intuitively substituted by the
17
value of Tdew as calculated by Eq. (32), depending on T and RH alone. Then, the
18
following formula for estimating Rs is obtained
19
RS ≈ kRS ⋅ RA (Tmax − Tdew )0.5
20
The values of kRs estimated by the empirical radiation formula, Eq. (31) and the
21
proposed modified radiation formula Eq. (33) were computed from data obtained
22
from FAO-CLIMWAT database from the 535 stations over the globe. From the full
(31)
116.91 + 237.3ln(ea ) 16.78 − ln(ea )
(32)
(32 a)
(33)
19
20
1
set of data, the no=4,461 monthly estimates corresponding to well watered conditions
2
(Temesgen et al., 1999) were retained. The variation of the values of kRs calculated by
3
the two radiation formulas was plotted (Figures not depicted). The results indicated
4
that when the proposed Eq. (33) is applied the spread of kRs coefficient from its default
5
value of 0.17 is less enough than using Eq. (30). Therefore Eq. (33) yields more
6
accurate values than Eq. (31). This conclusion is clearly demonstrated in Figs. 5a and
7
5b where the estimations provided by the radiation formula (31) and the modified
8
suggested one (33) were compared with the measured values of RS from the FAO-
9
CLIMWAT.
10
In this paper, the values of estimation methods were compared with the values of the
11
“standard” method by using simple error analysis and linear regression, i.e. Y=SX ,
12
where S=regression coefficient (slope of the linear curve), Y= reference values
13
obtained by the “standard” method; X= correspondents estimates by the comparison
14
method. Additionally statistical parameters were calculated: the standard error of the
15
estimate, SEE =
16
well as the long term average ratio, rt = X av / Yav where Xav and Yav = long term
17
average value of approximate and “standard” estimates respectively. The traditional
18
coefficient of determination, R2, was also used.
19
The statistical results of the correlation between the original radiation formula, Eq.
20
(31), with measured RS are S=1.044, R2=0.765 and SEE= 3.21 MJ/m2/d, whereas for
21
the modified formula, Eq. (33), the results are S=1.012, R2=0.849 and SEE=2.39
22
MJ/m2/d. The original formula Eq. (31) produce poorer estimates than the suggested
23
Eq. (33), it produced higher scatter of the results than Eq. (33), and a higher SEE.
{∑ (Y − X ) n
1
i
i
2
}
/ (nO − 1)
0.5
where nO=total number of observations as
20
21
1
Substituting the previously modified radiation suggested formula, Eq. (33), in Eq.
2
(28) the following approximate formula is proposed for ET0 estimation, when wind
3
speed and solar radiation data are missing
⎛ RH ⎞ ET0( P − M ) ≈ 0.00668RA (T + 9.5)(Tmax − Tdew ) − 0.0696(Tmax − Tdew ) − 0.024 (T + 20 ) ⎜1 − ⎟ ⎝ 100 ⎠
4
−0.00455RA (Tmax − Tdew )0.5 + 0.0984(T + 17) (1.03 + 0.00055 ⋅ TR 2 − RH /100 )
5
Simplified 1963-Penman ET-formula
6
(referred to in this paper as “Fo-PENM (Rs,T,RH,u)”)
7
Valiantzas (2006) suggested the following simplified ET-formula that approximates
8
the 1963-Penman -for grass- scheme:
9
ET0 ≈ 0.051(1 − α ) R S
⎛R T + 9.5 − 2.4⎜⎜ S ⎝ RA
(34)
2
⎞ RH ⎞ ⎛ ⎟⎟ + 0.048(T + 20 )⎜1 − ⎟(0.5 + 0.536u ) + 0.00012Z ⎝ 100 ⎠ ⎠
(35)
10 11
Valiantzas (2006) suggested to use α=0.25.
12
Reduced set FAO-56 Penman-Monteith method requiring temperature-humidity data
13
alone
14 15
(referred to in this paper as “FAO-PM (T, RH) ” The reduced to wind and solar radiation data FAO-56 Penman-Monteith method
16
requires measured T , and RH data alone and uses estimations for wind speed and
17
solar radiation data. According to Allen et al. (1998) a single constant value of u=
18
2m/s was assumed in Eq. (1) Furthermore, the RS values are estimated by the
19
Hargreaves and Samani (1982) radiation empirical formula, Eq. (31).
20
Hargreaves-Samani temperature method
21
(referred to in this paper as “HARG (T) ”
21
22
1
The solar radiation- based equation of Hargreaves (1975) requiring only the data RS,
2
Tmax, and Tmin, is:
3
ET0 ≈ 0.0135 ⋅ 0.408R S (T + 17.8)
4
Substituting approximate Eq. (31) in to Eq. (36) the well-known temperature-based
5
equation is obtained.
6
Although the temperature- based equation of Hargreaves requires T data only, it is
7
also included for the comparisons with the other methods. Note that Hargreaves
8
formula is developed from calibration of meteorological data at the sites of California
9
(Davis).
(36)
10
3.2 Testing the formulas - Comparisons
11
The ET0 estimated by the new (full or limited set of data) suggested formulas as well
12
as the simplified 1963-Penman ET-formula developed by Valiantzas(2006), Fo-
13
PENM(Rs,T,RH,u), the reduced FAO-PM (T, RH), and the Hargreaves formula were
14
compared with the conventionally considered as a reference method for comparisons
15
standardized FAO56-PM scheme using monthly or daily time scale data.
16
Firstly, monthly data extracted from the 535 stations of the FAO-CLIMWAT database
17
were used.
18
Subsequently, because the mean monthly data from CLIMWAT refer to the long-
19
term average year, a detailed dataset including particularly high quality daily data
20
from real years, the CIMIS data base (http://wwwcimis.water.ca.gov) has been also
21
used for validation and comparison purposes of the suggested formulas. Most CIMIS
22
stations conform to the basic definition of a reference weather station corresponding
23
to well-watered conditions. Recorded daily data selected from 7 sites of California has
24
been used for comparisons. An effort was made to select stations representing distinct
22
23
1
climates as much as possible covering a wide range of weather parameters. Two
2
locations were characterized by a humid climate where other by semi-arid or arid
3
climate. The long term average wind ranged approximately from 1.0 to 4.0 m/s. Table
4
1 lists the characteristics of the selected stations.
5
Comparisons were made using graphics and simple linear regression. The statistical
6
results of the correlation, rt, R2, and SEE were used in comparing ET0 values
7
estimated by the different methods.
8
Monthly data
9
Figure 6 presents the comparisons of methods for the 535 stations over the globe
10
using the monthly data from the FAO-CLIMWAT database. For all the sites, the
11
suggested full set of data expressions, Fo1-PM (Rs,T,RH,u) and Fo2-PM (Rs,T,RH,u)
12
(not requiring the detailed computation of RA) produced estimates in perfect
13
agreement with the standardized FAO56-PM. The bias error of the suggested new full
14
set of data formulas is practically negligible, the scatter of the results for both
15
methods is insignificant, R2=0.998 and 0.996 respectively, and the value of SEE is
16
rather
17
Valiantzas(2006), Fo-PENM (Rs,T,RH,u), produced poorer estimates than the new
18
suggested ones, it produced higher scatter of the results, R2=.978, and a significantly
19
higher SEE by significant percentage of about 220% (Fig. 6 a, b, and c).
20
Comparing the limited data formulas, the formula requiring T and RH data alone,
21
Fo-PM (T,RH),
22
additional Rs measurements e. g. Fo-PM (Rs,T,RH). It produced higher scatter of the
23
results, and a higher SEE by a percentage of about 31% (Fig. 6 d, and e). However,
negligible.
The
simplified
1963-Penman
ET-formula
developed
by
produced less accurate estimations than the formula requiring
23
24
1
the Fo-PM (T,RH) proposed formula produced more accurate estimations than the
2
reduced FAO-PM (T,RH) method requiring the same set of data (Fig. 6 e and f).
3
The Hargreaves equation, HARGR (T), produced the worst results among all the
4
methods. Compared with the Fo-PM(T,RH), the HARGR(T) produced higher scatter
5
of the results, and a higher SEE by a percentage of about 33% (Fig. 6 e, and g).
6
Daily data
7
Daily ET0 estimations obtained by the previously described approximate formulas
8
were compared with FAO56-PM procedure at the 7 sites of California. Statistical
9
results of comparisons were presented at Tables 2, and 3. Fig. 7 also presents the
10
correlations graphs of comparisons for the Davis site. A similar behaviour concerning
11
the performance of the previously described formulas was observed when daily data
12
were applied.
13
Both full set of data newly developed expressions Fo1-PM (Rs,T,RH,u) and Fo2-PM
14
(Rs,T,RH,u) are in perfect agreement with FAO-56 Penman-Monteith methods for all
15
7 sites ( the Fo2-PM is slightly inferior than the Fo1-PM). The rt value does not vary
16
significantly from 1 (maximum bias error is about 2.5% for both formulas), whereas
17
the spread of data is insignificant for all the 7 stations (minimal value of R2 is 0.993).
18
On the contrary, the simplified 1963-Penman ET-formula (full set of data) developed
19
by Valiantzas(2006), Fo-PENM (Rs,T,RH,u), produced poorer estimates than the new
20
suggested ones for all the 7 sites. The spread of data increases for all sites (minimal
21
value of R2 decreased to 0.951), and it produced a significantly higher average SEE by
22
the significant percentage of about 100% (Table 2, Figs. 7 a, b, and c).
23
In a recent paper, Valiantzas (2013c) has performed comparisons for 17 stations in
24
California indicating that the reduced set FAO-56 Penman-Monteith procedure not
24
25
1
requiring wind data has provided weaker results at all stations when compared to the
2
suggested in the same paper modification of simplified 1963-Penman ET-formula (not
3
requiring wind data also). Respectively, the Fo-PM (Rs,T,RH)
4
results than the simplified 1963-Penman ET-formula (not requiring wind data) for all
5
the 7 stations studied in the present paper (results not shown).
6
Again, the new suggested formula using a limited set of data, T and RH alone, Fo-
7
PM(T,RH), performed better than the reduced FAO-PM (T,RH) method requiring the
8
same set of data for all the 7 stations (Table 3, Figs 7 e, and f). The Fo-PM(T,RH)
9
performed better than the temperature-based formula of Hargreaves, HARGR(T) also,
10
for all the 7 stations (Table 3, Figs 7 e, and g). The HARGR(T) produced higher
11
scatter of the results for all the 7 stations, and a higher average SEE by a percentage
12
of about 22%.
13
4. Summary and Conclusion
14
A new series of simplified formulas, easy to use for routine hydrologic applications,
15
was derived to approximate the FAO-56 Penman-Monteith computational procedure.
16
The first one can be used to estimate the reference crop evapotranspiration directly
17
from the commonly measured weather data, solar radiation, maximum and minimum
18
temperature, relative humidity, and wind velocity. The second formula is a simplified
19
version of the first one and does not require the computation of the extraterrestrial
20
radiation.
21
For places where wind speed data are not readily available another formula which is a
22
simplified version not requiring wind speed data, is also proposed.
23
Further simplifications were applied resulting to an even more simplified version
24
which requires relative humidity and maximum - minimum temperature data alone.
25
provided weaker
26
1
The suggested new formulas were tested using the measured long-term monthly data
2
of a global climatic data set (FAO CLIMWAT) as well as the measured daily data
3
from 7 weather stations. Both full set of data newly developed expressions performed
4
better than the previously developed simplified 1963-Penman approximate ET0 full
5
set of data formula (Valiantzas, 2006 ET-formula). The new suggested formula using
6
a limited set of data, T and RH alone, performed better than the reduced FAO-56
7
Penman-Moneith method requiring the same set of data, and the temperature-based
8
formula of Hargreaves.
9 10
References
11
Allen, R.G., Pereira, L.S., Raes D., Smith, M., 1998. Crop evapotranspiration:
12
guidelines for computing crop water requirements. FAO Irrigation and Drainage
13
Paper, 56, Rome, 300pp.
14 15 16
D'Agostino, D., 2013. Moisture dynamics in an historical masonry structure: The Cathedral of Lecce (South Italy). Building and Environment, 63, 122-133 Exner-Kittridge, M. G. , 2012, Closure of “Case Study on the Accuracy and
17
Cost/Efectiveness in Simulating Reference Evapotranspiration in West-Central
18
Florida” by Exner-Kittridge, M. G, Rains. M. C., Journal of Hydrologic Engineering,
19
17(1), 225-226.
20
Exner-Kittridge, M. G.. Rains, M. C.,2010. Case Study on the Accuracy and
21
Cost/Effectiveness in Simulating Reference Evapotranspiration in West-Central
22
Florida. J. Hydrol. Eng., 15(9), 696-703
23 24
Hargreaves G.H., 1975. Moisture availability and crop production. Trans. Am. Soc. Agric. Eng., 18(5), 980-984.
26
27
1 2
Hargreaves, G.H., Samani, Z.A., 1982. Estimating potential evapotranspiration. J. Irrig. Drain. Div., 108(3), 225–230.
3
Itenfisu D., Elliot, R. L., Allen, R.G., Walter I.A., 2000. Comparison of
4
reference evapotranspiration calculations across a range of climates. Proc.,4th Nat.
5
Irrig. Symp., ASAE, St. Joseph, Mich.,216-227.
6
Jensen, M.E.,Burman R.D., Allen, R.G.,1990. Evapotranspiration and
7
irrigation water requirements. ASCE Manuals and Reports in Engineering Practice, no
8
70, 332pp.
9
Lewis T., and Lamoureux S. F., 2010. Twenty-first century discharge and
10
sediment yield predictions in a small high Arctic watershed. Global and Planetary
11
Change, 71(1-2), 27-41
12 13
Linacre E.T.,1993. Data sparse estimation of potential evaporation usind a simplified Penman equation., Agric. Forest Meteorol., 64, 225-237
14
Linacre E.T., 2004. Evaporation trends., Theor. Appl. Climatol., 79, 11-21
15
MacDonald, R. J., Byrne, J. M., Kienzle, S. W., 2009. A physically based
16
daily hydrometeorological model for complex mountain terrain. Journal of
17
Hydrometeorology, 10(6), 1430-1446
18
McMahon, T.A. , Peel, M.C., Lowe, L., Srikanthan, R., McVicar, T.R., 2013,
19
Estimating actual, potential, reference crop and pan evaporation using standard
20
meteorological data: A pragmatic synthesis. Hydrology and Earth System Sciences,
21
17( 4), 1331-1363
22 23
Penman H.L., 1948. Natural evaporation from open water, bare and grass. Proc. R. Soc. Lond., Ser. A 193, 120-145.
27
28
1 2
Penman H.L., 1963. Vegetation and hydrology. Technical Communication no 53, Commonwealth Bureau of Soils, Harpenden, England.
3
Rimmer, A., Samuels, R., Lechinsky, Y., 2009. A comprehensive study
4
across methods and time scales to estimate surface fluxes from Lake Kinneret, Israel.
5
J. Hydrol., 379(1-2), 181-192
6
Samani, Z., 2004. Discussion of ‘‘History and evaluation of Hargreaves
7
evapotranspiration equation’’ by George H. Hargreaves and Richard G. Allen. J. Irrig.
8
Drain. Eng. 130(5), 447–448.
9 10 11 12 13 14 15 16
Shuttleworth, W.J., 1993. Evaporation, Ch. 4. ed. D.R. Maidment, McGrawHill, New York, 4.1-4.53. Smith M., 1993. CLIMWAT for CROPWAT: a climatic database for irrigation planning and management. FAO Irrigation and Drainage Paper, 49, Rome. Stull R.B., 2000: Meteorology for Scientist and Engineers 2nd Edition. Brooks/Cole Thomson Learning, Page 98 Temesgen, B., Allen R.G., Jensen D.T., 1999, Adjusting temperature parameters to reflect well-watered conditions. J. Irrig. Drain. Eng., 125(1), 26-33
17
Valiantzas J.D., 2006. Simplified versions for the Penman evaporation
18
equation using routine weather data. Journal of Hydrology, 331(3-4), 690-702.
19
Valiantzas, J. D., 2012, Discussion of “Case Study on the Accuracy and
20
Cost/Efectiveness in Simulating Reference Evapotranspiration in West-Central
21
Florida” by Exner-Kittridge, M. G, Rains. M. C., Journal of Hydrologic Engineering,
22
17(1), 224-225.
28
29
Valiantzas, J. D., 2013 a. Simplified Reference Evapotranspiration Formula
1 2
Using an Empirical Impact Factor for Penman's Aerodynamic Term. Journal of
3
Hydrologic Engineering, 18 (1), 108-114. Valiantzas, J.D. 2013 b. Simple ET0 forms of Penman’s equation without wind
4 5
and/or humidity data I Theoretical development ” Journal of Irrigation and Drainage
6
Engineering, ASCE, 139 (1), 2013, 1-8.
7
Valiantzas, J. D., 2013 c. Simple ET0 Forms of Penman’s Equation without
8
Wind and/or Humidity Data II: Comparisons with Reduced Set-FAO and other
9
Methodologies. J. Irrig. Drain. Eng., 139(1), 9-19.
10 11
Figure Captions
12
Fig. 1 Variation of accurate values of the reduced difference of the radiation
13
terms of Penman and Penman-Monteith methods, Δrad0, with wind speed , u, for series
14
of computations over typical range of input meteorological variables.
15
Fig. 2 Statistical regression graph of the left-hand term ( RS / RA ) versus the
16
right-hand term 3.9 RSϕ 0.15 / N 2 + 0.16 of the approximate Eq. (10), for long-term
17
mean monthly meteorological data from 535 stations of the global CLIMWAT data
18
base .
19
Fig. 3
Variation of approximate and accurate values of the function
20
( P−M ) [ Eaero /D(stand)] e0(T) with approximate 0.0585(T + 17)u 0.75 values, for series of
21
computations over typical range of input meteorological variables,
29
30
1
Fig. 4
Statistical regression graph of the left-hand term eS ( s tan d ) versus the
2
right-hand term e 0 (T ) ⋅ (1 + 0.00043 ⋅ TR 2 ) of the approximate Eq. (23), for series of
3
computations over typical range of input meteorological variables.
4
Fig. 5 (a) Comparison of approximate empirical radiation formula, Eq. (31),
5
with exact values of the solar radiation variable, and (b) comparison of the proposed
6
modified radiation formula, Eq. (33), with exact values of the solar radiation variable,
7
for long-term mean monthly meteorological data from 535 stations of the global
8
CLIMWAT data base.
9
Fig.
6
a-g
Long-term
mean
monthly
values
of
reference
crop
10
evapotranspiration estimated by the tested formulas versus the standardized FAO-56
11
Penman-Moteith scheme, for 535 stations of the global CLIMWAT data set.
12
Fig. 7 a-g Daily values of reference crop evapotranspiration estimated by the
13
various methods versus the standardized FAO-56 Penman-Moteith scheme, for Davis.
14
Table Legends
15
Table 1 Station name/number, Elevation (m), Latitude (deg), Longitude (deg),
16
Climate Conditions, Average Wind Speed (m/s) and Period of Data Used of the 7
17
stations of CIMIS-California selected for comparisons.
18
Table 2 Statistical summary of comparison between the daily values of ET0
19
estimated by different full-set of data formulas and the reference method standardized
20
FAO-56 Penman-Monteith scheme, for the 7 stations.
21
Table 3 Statistical summary of comparison between the daily values of ET0
22
estimated by different limited data formulas and the reference method standardized
23
FAO-56 Penman-Monteith scheme, for the 7 stations.
30
31
1
Table 1.
Arid
Average Wind (m/s) 2.0
1995-2005
121.91
Humid
2.29
1995-2001
38.28
121.79
Arid
3.90
1998-2004
7.6
38.13
121.38
Humid
1.09
2000-2004
Davis/6
18.3
38.54
121.78
Arid
2.68
1994-1999
Parlier/39
102.7
36.6
119.5
Arid
1.73
1995-2005
Victorville
880.9
34.48
117.26
Arid
2.76
1995-1997
Station name/number
Elevation (m)
Latitude (deg)
Longitude (deg)
Climate
McArthur/43
1008.9
41.07
121.45
Zamora/27
15.2
38.81
HastingsTract/122
3
Lodi West/166
2
31
Data Period
32
1 2 3 4 5 6
TABLE 2
Station name 1. McArthur 2. Zamora 3.Hastings Tract 4.Lodi West 5.Davis 6. Parlier 7. Victorville
Average
Fo1-PM (RS,T,RH,u) vs FAO56PM SEE rt R2 (mm/d)
Fo2-PM (RS,T,RH,u) vs FAO56-PM SEE rt R2 (mm/d)
Fo-PENM (RS,T,RH,u) vs FAO56-PM SEE rt R2 (mm/d)
1.014
0.997
0.116
1.020
0.996
0.190
0.976
0.983
0.290
1.010
0.995
0.151
1.022
0.997
0.159
0.986
0.986
0.255
1.002
0.993
0.217
1.020
0.994
0.248
1.009
0.951
0.588
1.022
0.998
0.114
1.027
0.995
0.187
1.039
0.995
0.195
0.994
0.996
0.157
1.014
0.997
0.159
0.978
0.984
0.324
1.003
0.998
0.095
1.000
0.998
0.114
0.986
0.993
0.201
0.985
0.997
0.161
1.012
0.994
0.225
0.991
0.969
0.438
1.004
0.996
0.144
1.016
0.996
0.183
0.996
0.980
0.327
7 8 9 10
32
33
1 2 3 4 5 6
TABLE 3
Fo-PM(T,RH) vs FAO56-PM
Fo-PM(RS,T,RH) vs FAO56-PM Station name 1. McArthur 2. Zamora 3.Hastings Tract 4.Lodi West 5.Davis 6. Parlier 7. Victorville
FAO-PM(T, RH) vs FAO56-PM
HARG(T) vs FAO56PM
rt
R2
SEE (mm/d)
rt
R2
SEE (mm/d)
rt
R2
SEE (mm/d)
rt
R2
SEE (mm/d)
1.036
0.970
0.413
1.002
0.954
0.474
1.071
0.951
0.526
1.044
0.942
0.509
1.014
0.925
0.588
1.016
0.907
0.641
1.036
0.895
0.705
1.060
0.829
0.886
0.870
0.937
0.926
0.830
0.928
1.105
0.830
0.917
1.143
0.842
0.852
1.250
1.173
0.961
0.700
1.190
0.928
0.801
1.210
0.917
0.887
1.234
0.910
0.924
0.949
0.931
0.670
0.945
0.915
0.734
0.950
0.903
0.771
0.949
0.838
0.957
1.037
0.976
0.374
1.051
0.965
0.450
1.077
0.956
0.544
1.085
0.941
0.572
0.933
0.890
0.872
0.948
0.886
0.881
0.931
0.849
0.982
0.848
0.845
1.228
1.001
0.941
0.649
0.997
0.926
0.727
1.015
0.912
0.794
1.009
0.880
0.925
Average
7
33
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
(a)
(b)
Figure 6
g
Figure 7
g
34
1 2 3 4
• • •
New simple expressions for the “standardized” Penman-Monteith ET0 New temperature – humidity ET0 formula New solar radiation empirical formula
5 6
34