Keywords

6.1 Introduction

Most ET estimation models are empirical. Usually, the models are statistical correlations of evaporation with minimum, maximum, or mean meteorological parameters. Performances differ from location to location, and sometimes application to a specific location requires recalibration. The Penman–Monteith method is a complex method that is closest to a physical model, accounting for mass, momentum, and energy transfer with external and internal resistance and conductance terms incorporated. ET estimation method selection depends on the availability and quality of meteorological data and site features. The subject of meteorological data quality is discussed in detail in Chap. 2. Simple methods require fewer input parameters and could satisfy needs in many regions where intensive data collection networks are not available and are costly.

6.2 Simple Methods

6.2.1 Pan Method

Estimating lake evaporation from pan evaporation is the simplest method but has many challenges. Evaporation from a small metallic pan usually placed on dry site is higher than evaporation from a lake. Advective energy, heat storage difference, and higher vapor pressure deficit due to the site environment result in higher evaporation. A coefficient, K p, is used to reduce pan evaporation to estimate lake evaporation (Eq. 6.1). Reference crop evapotranspiration (Eq. 6.2) is also estimated from pan evaporation using a coefficient (C et). Coefficients K p and C et are dependent on the type of pan, environment at the site, and pan operation. Wide ranges of these coefficients are available indicating that pan evaporation measurements are affected by site-specific factors. These factors include site location, type of pan, quality of measurements, and operations and maintenance. Comparison of pan evaporation data from seven sites around Lake Okeechobee in south Florida resulted in pan coefficients ranging from 0.64 to 0.95, demonstrating that each pan is influenced by local environment and operations (Abtew 2001; Abtew et al. 2011). Spatial variation of the pan evaporation to surface water evaporation ratio over the United States for May through October (warm months) is mapped with a range of 0.64–0.88 (Farnsworth et al. 1982). On this map, pan coefficients for south Florida range from 0.72 to 0.74. Reference crop coefficient, C et, is dependent on meteorological conditions such as wind speed and humidity, and the values range from 0.35 to 0.85 (Jensen 1974):

$$ {E_{\text{o}}} = { }{K_{\text{p}}}{E_{\text{pan}}} $$
(6.1)
$$ {E_{\text{tp}}} = {C_{\text{et}}}{E_{\text{pan}}} $$
(6.2)

Historical pan evaporation data are usually plagued with outliers, gaps, and data of questionable quality. Variations in pan evaporation data within relatively small distances indicate the challenges of acquiring consistent observations from pans. The wide range of pan coefficients tends to overemphasize the shortcoming of pan data (Shuttleworth 1993; Abtew et al. 2011).

Pan evaporation data from south Florida were analyzed to see if the data quality is sufficient to determine evaporation trends (Abtew et al. 2011). A total of nine pan evaporation sites with varying lengths of record from 1916 to 2009 were used for this analysis. Missing data less than a week were estimated mainly by interpolation. Months and years with too many missing days were excluded. In many cases where several daily data were available as a cumulative value on the last day of record, the values were redistributed equally for each day of accumulation. The maximum annual record at a site was 210 cm, and the minimum record was 119 cm. The mean pan annual evaporation for all sites was 156 cm with a standard deviation of 18.5 cm. The distance between gauges was a maximum of 109 km and a minimum of 0.2 km. These ranges of records reflect the challenges of acquiring good quality pan evaporation observations rather than actual variation in the parameter in a subregion. Provided water is available for evaporation and the site environment and operation are similar, the annual spatial variation in evaporation should not be as large as recorded at these sites. Probably, these challenges are common at pan evaporation sites at other parts of the world. Even if there was no error in observations, the role of the microclimate and environment differences between sites and differences in site operations could produce varying results within a subregion (Abtew et al. 2011). Figure 6.1 depicts monthly pan evaporation from eight sites in south Florida showing variation in measurement and recording. Screening of data for quality and assembling of data from many sites could provide a set of pan evaporation data for applications. Lake evaporation and crop evapotranspiration estimation from pan evaporation are discussed in Chaps. 3 and 8. Well-installed, maintained, and operated evaporation pans could consistently provide good-quality data, and evaporation can be estimated with locally calibrated pan coefficients.

Fig.6.1
figure 1

Annual pan evaporation data from eight sites in south Florida showing variation in measurements and recording

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6.2.2 Temperature-Based Methods

6.2.2.1 Blaney–Criddle Method

Temperature-based ET estimation methods are the simplest methods. The most commonly applied temperature-based evapotranspiration estimation method is the Blaney–Criddle method. The measured climatic variable input is air temperature, and the equation is as follows (Eq. 6.3):

$$ {\text{E}}{{\text{T}}_{\text{o}}} = p(0.46\,{T_{\text{avg}}} + { }8) $$
(6.3)

where ETo is reference crop evapotranspiration (mm/day) average for the month, T avg is mean daily temperature (°C) for the month, and p is mean daily percentage of annual daytime hours for the month (Table 6.1).

Table 6.1 Mean daily percentage (p) of annual sunshine hours for different latitudes
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The FAO (Food and Agricultural Organization, UN) Blaney–Criddle method is a modified Blaney–Criddle that accounts for the effect of other weather parameters on crop water requirements (Doorenbos and Pruitt 1977). The method accounts for temperature, relative humidity, sunshine hours, site elevation, and wind speed to estimate reference crop evapotranspiration. The FAO Blaney–Criddle method has been in use in the western states of Nevada, Washington, Idaho, Oregon, and California to estimate irrigation requirements on a state-wide basis (Allen and Pruitt 1986). The equation is given as follows (Eq. 6.4):

$$ {\text{E}}{{\text{T}}_{\text{o}}} = c(p(0.46{T_{\text{avg}}} + 8.13))\left\{ {1 + 0.1\frac{\text{elev}}{{1{,}000}}} \right\} $$
(6.4)

where ETo is estimated evapotranspiration in mm day−1 from grass reference crop (8–15 cm tall and well watered) for the month of consideration, T avg is mean daily temperature in °C for the month, p is mean daily percentage of total annual daytime hours for the month and latitude, and c is adjustment factor which depends on minimum relative humidity, wind speed, and sunshine hours. To avoid the use of graphs and tables to interpolate the correlation factor, c, two coefficients, a and b (Eqs. 6.5 and 6.6), were formulated to replace c (Frevert et al. 1983).

$$ a = 0.0043{\text{R}}{{\text{H}}_{{\min }}} - \frac{n}{N} - 1.41 $$
(6.5)

where RHmin is average minimum relative humidity for the month and n/N is mean actual to possible sunshine ratio.

$$ \begin{array}{lllll} b & = 0.819 - 0.00409{\text{R}}{{\text{H}}_{{\min }}} + 1.07\frac{n}{N} + 0.0656{u_{\text{day}}}\frac{{0.00597{\text{R}}{{\text{H}}_{{\min }}}}}{N}n \\ & \quad - 0.000597{\text{R}}{{\text{H}}_{{\min }}}{u_{\text{day}}} \end{array} $$
(6.6)

where u day is daytime wind speed in m s−1 for the month. In cases where sunshine data is not available but solar radiation data is available, Eq. 6.7 can be used to deduct n/N (Jensen 1974).

$$ \frac{n}{N} = 2.08\frac{{{R_{\text{s}}}}}{{{R_{\text{A}}}}} - 0.48 $$
(6.7)

where R s is solar radiation and R A is extraterrestrial radiation. A demonstration of the application of the modified Blaney–Criddle method (Eq. 6.4) to estimate reference evapotranspiration in the Everglades Agricultural Area in south Florida is shown in Table 6.2.

Table 6.2 Application of the modified Blaney–Criddle method to estimate reference evapotranspiration in south Florida
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The annual reference evapotranspiration estimate is 1,459 mm. The seasonal pattern follows the seasonal evapotranspiration pattern of the region, but this reference evapotranspiration is 5.6% higher than the potential evapotranspiration, as computed by the simple Abtew method. The Blaney–Criddle reference evapotranspiration estimates and the simple Abtew method potential evapotranspiration compare closely from December to April. The Blaney–Criddle estimates for the remaining months are higher (Fig. 6.2).

Fig. 6.2
figure 2

Modified Blaney–Criddle method reference ET and simple Abtew method potential evapotranspiration for south Florida

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Other temperature-based ET estimation methods include the Thornthwaite method where monthly potential ET is estimated from mean monthly air temperature, daytime hours, and 12-month sum of heat index (Jensen 1974).

6.2.2.2 Hargreaves–Samani Method

The Hargreaves–Samani method is not truly a temperature-based method because it has a radiation term in it. Since measurement is not needed for the extraterrestrial radiation (R A), this method may be classified as a temperature-based method. The Hargreaves equation is given by Eq. 6.8 (Hargreaves and Samani 1985):

$$ {\text{E}}{{\text{T}}_{\text{r}}} = a{R_{\text{A}}}{({T_{{\max }}} - { }{T_{{\min }}})^{{0.5}}}({T_{\text{avg}}} + 17.8) $$
(6.8)

where ETr is grass reference ET (mm day−1), T max and T min (°C) are maximum and minimum daily air temperature, T avg is mean daily air temperature (average of daily maximum and minimum), and R A is extraterrestrial radiation (mm day−1). This method has been applied or tested in many places and widely published.

6.2.3 Radiation-Based Methods

In parts of the world where solar radiation explains most of the variation in evaporation and evapotranspiration, a simple equation may be calibrated to estimate ET from one variable, solar radiation.

6.2.3.1 The Simple Abtew Method

The simple Abtew method (Eq. 6.9) has been applied to estimate lake evaporation, wetland evapotranspiration, and potential evapotranspiration. This equation was developed from open water evaporation and wetland evapotranspiration lysimeter studies in south Florida:

$$ {\text{ET}} = {K_1}\frac{{{R_{\text{s}}}}}{\lambda } $$
(6.9)

where ET is daily wetland evapotranspiration or shallow open water evaporation or potential evapotranspiration (mm day−1), Rs is solar radiation (MJ m−2 day−1), λ is latent heat of vaporization of water (MJ kg−1), and K 1 is a dimensionless coefficient (0.53). The mm day−1 unit is derived from the fact that a kilogram of water is 1,000 cc (106 mm3) and a square meter is 106 mm2.

Application of this method is shown in Chaps. 7 and 8. The simple Abtew method has been successfully applied in many parts of the world. Evaporation from Lake Ziway in the Ethiopian Rift Valley was estimated with this method, and results were comparable to the energy balance and Penman equation (Melesse et al. 2009). Satisfactory results of reference ET estimation for the Fogera flood plain in Ethiopia, with the simple method, are reported with adjustment of the K 1 to 0.48 (Enku et al. 2011). The simple Abtew method was applied to estimate ET in Gansu province, northwest China, with recalibrated coefficients (Zhai et al. 2009). The simple method was applied to estimate evaporation from Lake Titicaca, South America. It was found to be the best method compared to eight evaporation models (Delclaux and Coudrain 2005). Comparative application of the simple method further demonstrates its usefulness. In an effort to identify the most relevant approach to calculate potential evapotranspiration for use in daily rainfall–runoff models, 27 potential ET models were compared for stream flow simulation from 308 catchments in France, the United States, and Australia. Each potential ET model estimate was applied to four continuous daily lumped rainfall–runoff models, and the simple Abtew method had a comparable goodness-of-fit measure (Oudin et al. 2005).

Shoemaker and Sumner (2006) applied the simple Abtew method to estimate potential evapotranspiration from open water, saw grass, and bullrush marsh and compared it to the Priestley–Taylor and Penman methods. Out of the eight sites of measurement, the simple method had the smallest standard error for two sites. The low cost of monitoring needed for this method was pointed out as a positive attribute compared to other methods that require more parameters. Xu and Singh (2000) evaluated various radiation-based methods for calculating evaporation and concluded that the simple Abtew method, referenced as the simple Abtew equation, can be used when available data is limited to radiation data. The simple method is applicable to remote sensing where the input, solar radiation, is acquired through satellite observations (Jacobs et al. 2002).

In a U.S. Geological Survey (USGS) study, nine sites in the Everglades of south Florida were instrumented with sensors to determine evapotranspiration from different features using the Bowen ratio–energy balance method (German 2000). Figure 6.3 shows the nine USGS sites and site characteristics where evapotranspiration was measured with the Bowen ratio–energy balance method. Field data is available with varying lengths of record, from 1996 to 2000. The field instrumentation had net radiometer, pyranometer, wind speed and direction sensors, air temperature and humidity sensors, rain gauge, storage battery, solar panel, data logger, and cellular phone. Pictures of a site with instrumentation are shown in Chap. 7. The Bowen ratio–energy balance method is a micrometeorological method for measurement of evaporation (latent heat) with an approximate accuracy of 10% (Dugas et al. 1991). Details of the Bowen ratio ET measurement are given in Chaps. 3 and 7. Mean Bowen ratio ET measurement and estimates by the simple Abtew method are shown in Fig. 6.4 for seven of the sites. The mean square error for all sites is 0.06 mm, showing a very good estimation.

Fig. 6.3
figure 3

Wetland evapotranspiration study sites in south Florida (German 2000; U.S. Geological Survey; Abtew 2005)

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Fig. 6.4
figure 4

Comparison of Bowen ratio wetland ET measurements and Simple Abtew method estimates at seven sites in the Everglades

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Lake Ziway is located in the Ethiopian Rift Valley with an average surface area of 490 km2 at an elevation of 1,636 m msl. Monthly and annual average Lake Ziway evaporation estimates have been published. The estimates vary from method to method of evaporation estimation. Annual lake evaporation estimates by Coulomb et al. (2001) estimated with the energy balance, Penman, and Complementary Relationship Lake Evaporation (CRLE) methods were 1,777, 1,875, and 1,728 mm, respectively. The coefficient for the simple method can be adjusted to the results of the three methods or to the one that is believed to be closer to the true estimates. As an illustration, the energy balance and the simple Abtew method were compared with the K value in Eq. 6.9 revised to 0.57 from 0.53. The results are shown in Fig. 6.5. Detail of the energy balance method application for lake evaporation is presented in Chap. 8.

Fig. 6.5
figure 5

Comparison of energy balance and simple Abtew method evaporation estimates for Lake Ziway, Ethiopia

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6.2.3.2 Makkink Method

The Makkink method (1957) is classified as radiation-based method, although it requires mean air temperature (°C), relative humidity, and air pressure (mb) to calculate the slope of saturation vapor pressure curve (Δ) and the psychrometric constant (γ). The original Makkink equation is given as follows (Eq. 6.10):

$$ {\text{ET}} = 0.61\frac{{\Delta {R_{\text{s}}}}}{{(\Delta + \gamma )\lambda }} $$
(6.10)

where ET is potential evapotranspiration from grass (cm day−1), R S is solar radiation in cal cm−2 day−1, Δ is the slope of saturation vapor pressure curve (mb °C), γ is the psychrometric constant (mb °C), and λ is latent heat of vaporization (cal gm−1). Δ, γ, and λ are computed by Eqs. 6.11, 6.12, and 6.13 (Maidment 1993).

$$ \Delta = \frac{{4098{e_{\text{s}}}}}{{{{(237.3 + T)}^2}}} $$
(6.11)

where e s (kPa) is saturation vapor pressure and T is given air temperature (°C).

$$ \gamma = 0.0016286\frac{P}{\lambda } $$
(6.12)

where P (kPa) is atmospheric pressure.

$$ \lambda = 2.501 - (0.00236*{T_{\text{s}}}) $$
(6.13)

where T s (°C) is surface temperature of water.

For south Florida, the Makkink method to estimate potential evapotranspiration was calibrated to the simple Abtew method and is shown in Eq. 6.14. ET for 2007 was 1,330 and 1,322 mm for the Makkink and simple Abtew methods, respectively. Comparison of daily estimates by the two methods is depicted in Fig. 6.6:

$$ {\text{ET}} = 0.743\frac{{\Delta {R_{\text{s}}}}}{{(\Delta + \gamma )\lambda }} $$
(6.14)

where ET is in mm day−1, Δ and γ are in kPa °C−1, R s is in MJ m−2 day−1, and λ is MJ kg−1. Equation 6.14 is close to what was proposed by Hansen (1984) in the Netherlands with a coefficient of 0.7.

Fig. 6.6
figure 6

Modified Makkink and simple Abtew methods potential evapotranspiration estimation for south Florida

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6.2.3.3 Priestley–Taylor Method

The Priestley–Taylor method is similar to the Makkink method (1957), but net solar radiation (R n) is used instead of solar radiation (R s). Since R n is smaller than R s, the coefficient in Priestley–Taylor is higher (α = 1.26) for compensation. The Priestley–Taylor method is expressed by Eq. 6.15. Measuring R n is more problematic than measuring R s, as discussed in Chap. 2:

$$ {\text{ET}} = 1.26\frac{{\Delta {R_{\text{n}}}}}{{(\Delta + \gamma )\lambda }} $$
(6.15)

where other terms are similar to Eq. 6.14.

ET estimates with the Priestley–Taylor method reflect R n input data quality, and the application is limited by this data availability. Figure 6.7 depicts Makkink, simple Abtew, and Priestley–Taylor methods application for 1 year for south Florida. The coefficient alpha (α) was 1.18 based on previous work where the method was calibrated to lysimeter measurements (Abtew and Obeysekera 1995). From Fig. 6.7, the seasonal pattern of the Priestley–Taylor estimates is different than expected. ET rate increased in late summer and fall while it is expected to decline. The cause could be R n data quality. Application of this method is further provided in Chaps. 7 and 8.

Fig. 6.7
figure 7

Modified Makkink, simple Abtew, and Priestley–Taylor methods potential evapotranspiration estimation for south Florida

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6.2.3.4 Turc Method

Methods that use both solar radiation and air temperature attempt to explain more ET variation by adding a second input. One of these methods is the Turc method adjusted for different units than the original equation (Eq. 6.16):

$$ {\text{E}}{{\text{T}}_{\text{p}}} = {K_2}\frac{{(23.89{R_{\text{s}}} + 50)T_{{_{\text{avg}}}}}}{{({T_{\text{avg}}} + 15)}} $$
(6.16)

where ETp is potential evapotranspiration in mm day−1, K 2 is coefficient (0.013), R s is solar radiation in MJ m−2 day−1, and T avg is average air temperature (°C). The original Turc method estimates are lower in the first half of the year in south Florida (Fig. 6.8). In a previous study (Abtew 1996), using daily maximum air temperature (T max) instead of average temperature provided a better fit to measured data in south Florida (Fig. 6.8).

Fig. 6.8
figure 8

Simple Abtew, Turc, and modified Turc methods for potential evapotranspiration estimation for south Florida

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Estimates for 2007 were 1,322, 1,291, and 1,390 mm for simple Abtew, Turc, and modified Turc methods. Application of the modified Turc method is presented in Chaps. 7 and 8.

6.2.4 Solar Radiation–Maximum Temperature Method

The solar radiation–maximum temperature method was developed by the author based on lysimeter studies in south Florida reflecting radiation and maximum air temperature to explain a large portion of variation in ET in south Florida and similar environments (Abtew 1996). The method is presented by Eq. 6.17 where K 3 is a coefficient with a dimension (°C):

$$ {\text{ET}} = \frac{1}{{{K_3}}}\frac{{{R_{\text{s}}}{T_{{\max }}}}}{\lambda } $$
(6.17)

Figure 6.9 depicts a comparison of daily potential evapotranspiration (evaporation) estimates by the simple Abtew and solar radiation–maximum temperature methods. With a K 3 value of 53.5°C, ET for 2007 for the two methods was 1,322 and 1,323 mm for the year, respectively. Further application of this method is given in Chaps. 7 and 8.

Fig. 6.9
figure 9

Simple Abtew and solar radiation–maximum temperature methods potential evapotranspiration estimation for south Florida

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6.2.5 Mass Transfer Method

The mass transfer method is based on the vapor pressure gradient from the water surface to the air above and vapor transport by wind. The common form of the formulation is given in Eq. 6.18:

$$ E = {k_{\text{m}}} \ u{e_{\text{s}}} - {e_{\text{a}}} $$
(6.18)

where E is evaporation from water surface, k m is a mass transfer limiting term, u is wind speed, e s is saturation vapor pressure at water surface, and e a is actual vapor pressure of the air.

Details of application of the mass transfer method are presented in Chap. 8. In Chap. 8, application of the mass transfer method to a lake with lake surface water temperature and temperature of the air above the water is presented. The estimates are compared to energy balance evaporation estimation for a lake. Application was for 1 month. It is shown that when the vapor pressure deficit and wind speed are high, the mass transfer method gives high estimates beyond the energy available to sustain such an evaporation rate. At lower vapor pressure deficit, the method provides evaporation estimates that are too low. The method does not account for available energy. That is why the Penman method better described the evaporation process by combining mass transfer and energy balance components of the evaporation process.

6.3 Complex Methods

6.3.1 Energy Balance Methods

Details and application of the energy balance method are presented in Chaps. 4, 7, 8, and 10. The energy balance method accounts for energy inflow (Energyin), energy outflow (Energyout), and change in energy storage (ΔEnergys) but does not include available moisture and mechanism of mass transfer. e is errors. A simplified form of the energy balance method is shown by Eq. 6.19:

$$ {\text{Energ}}{{\text{y}}_{\text{in}}} - {\text{Energ}}{{\text{y}}_{\text{out}}} = \Delta {\text{Energ}}{{\text{y}}_{\text{s}}}\pm e $$
(6.19)

The vertical energy balance at the water surface of a lake is expressed by Eq. 6.20 dropping the advective energy term:

$$ \lambda E = {R_{\text{n}}} - H - G $$
(6.20)

where λE is latent heat flux, H is sensible heat flux, and G is heat gained or lost. λ is latent heat of vaporization of water and E is evaporation. G is computed from temperature change and heat storage terms, R n is measured, and H is estimated by equations that involve temperature and wind speed gradients. Details of dew evaporation, wetland evapotranspiration, and lake evaporation with the energy balance method are presented in Chaps. 4, 7, and 8.

6.3.2 The Penman Method

The Penman method is the basis for most preferred methods of evapotranspiration estimation at this time. Howard Penman in 1948 developed an equation to describe evaporation from an open water surface. The Penman equation was complete in describing the evaporation process because it has a moisture availability component, mass transfer component, and required energy for evaporation component. It requires daily mean temperature, wind speed, relative humidity, and solar radiation. Penman’s equation incorporates concepts from other equations. Dalton’s equation of mass moisture flux is a function of vapor pressure deficit, wind speed, and surface resistance. The resistance offered for water molecules to leave the water surface and move into the air is a function of air density, specific heat of air, psychrometric constant, latent heat of vaporization, surface resistance, and wind speed. With respect to the energy needed for evaporation, net solar radiation is divided between sensible heat and latent heat (evaporation), assuming no energy loss or gain to the ground. Sensible heat loss or gain results in change of temperature.

6.3.2.1 Mass Transfer (Sink Strength)

As shown by Dalton’s equation, earlier attempts to formulate evaporation focused on mass transfer and aerodynamic components, as shown in Eq. 6.21 (Penman 1948):

$$ E = ({e_{\text{ss}}} - { }{e_{\text{dd}}})f{(}u{)} $$
(6.21)

where E is evaporation per unit time, e ss is vapor pressure at the evaporating surface, e dd is vapor pressure in the atmosphere above, and f(u) is a function of the horizontal wind. Depending on the units used for vapor pressure and wind speed, various equations had been calibrated with coefficients to estimate evaporation from vapor pressure gradient and wind speed. Application of the mass transfer equation to lake evaporation is given in Chap. 8. This approach lacks accounting for energy required for evaporation and subject to influence by wind speed and vapor pressure gradient only.

Although there could be a vapor pressure gradient, the presence of resistance at the water and air interface was realized early on. Momentum, mass, and energy transfer from a surface to the air above are a complex phenomenon. Air flowing over a surface develops a logarithmic profile as a result of a drag created by the surface (Monteith 1973; Abtew et al. 1989). Although vapor pressure deficit could exist between the surface and the air above, there are forces that resist vapor molecules from leaving the surface. On a smooth surface, the logarithmic wind velocity profile breaks close to the surface as a result of interaction with surface roughness, and a small layer of laminar flow develops transitioning to turbulent flow above. The reaction to the surface resistance force is shear stress force over the surface created from wind speed gradient. Appreciation of the complex nature of the tiny layer and forces involved is presented in detail by Monteith (1973).

Figure 6.10 illustrates the unmodified and modified turbulent layers over a rough surface with a distinct fraction of cover and boundary layer over the roughness objects. The density or fraction of cover (F c) and height of the roughness objects (h) determines the thickness of the boundary layer and changes in the wind profile (Abtew et al. 1989). The momentum flux is highest in the unmodified flow layer, followed by the modified flow layer, and least close to the roughness surface. Eddies or still air may exist in between the roughness objects below height d + z o. On a smooth flat surface, the laminar layer should be small on top of the surface. Roughness objects can be rigid or nonrigid (crop). Undulations as waves on open water act as roughness and affect the wind profile. Roughness height for water waves can be estimated (Abtew 2001). Details on wind profile are presented in Chap. 2.

Fig. 6.10
figure 10

Boundary layers when wind flows over a rough surface (roughness height (h c), displacement height (d), and aerodynamic roughness (z o))

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Attempts to expand the aerodynamic influence in evaporation resulted in equations with more coefficients. A simplified form is shown in Eq. 6.22 referred by Penman as sink strength (Penman 1948):

$$ E = 0.376({e_{\text{s}}} - {e_{\text{d}}})u_2^{{0.76}} $$
(6.22)

where E is evaporation in mm day−1, e s and e d are in mm mercury, and u 2 is wind speed at 2-m height measured in mph. Equation 6.22 was applied for a month period for estimating evaporation from Lake Okeechobee in south Florida. Water surface temperature was used to compute e s (saturation vapor pressure), and air temperature was used to compute e d (actual vapor pressure). This method is sensitive to changes in vapor pressure deficit.

Figure 6.11a depicts a comparison of evaporation computed by Eq. 22 (Penman sink strength) and the simple Abtew method. Figure 6.11b depicts daily average wind speed and vapor pressure deficit used in the calculation. Equation 6.22 was applied with the same coefficients, and the total evaporation for the month was 133 mm while the simple Abtew method gave 145 mm.

Fig. 6.11
figure 11

(a) Penman (sink strength) and simple Abtew method evaporation estimation in south Florida; (b) wind speed and vapor pressure deficit

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6.3.2.2 Combination of Sink Strength and Energy Balance

Penman combined the sink strength and energy balance methods to develop the Penman evaporation equation (Eq. 6.23):

$$ E = \frac{{\Delta {R_{\text{n}}} + \gamma \delta ef(u)}}{{(\Delta + \gamma )}} $$
(6.23)

where E is latent heat of flux of evaporation (kW m−2), Δ is slope of the vapor pressure curve (kPa °C−1), R n is net radiation (kW m−2), δe is vapor pressure deficit (kPa), f(u) is wind function (m s−1), and γ is psychrometric constant (kPa °C−1). The wind function, f(u), is expressed by Eq. 6.24 (Allen et al. 1989):

$$ f{(}u{)} = 6.43({a_{\text{w}}} + {b_{\text{w}}}{u_2}) $$
(6.24)

where u 2 is wind speed at 2-m height (m s−1) and a w and b w are coefficients computed on daily basis for south Florida by Eqs. 6.25 and 6.26 (Abtew and Obeysekera 1995; Abtew 1996).

$$ {a_{\text{w}}} = 0.10 + 0.2\exp \left\{ { - {{\left[ {\frac{{J - 173}}{{58}}} \right]}^2}} \right\} $$
(6.25)
$$ {b_{\text{w}}} = 0.04 + 0.2\exp \left\{ { - {{\left[ {\frac{{J - 243}}{{80}}} \right]}^2}} \right\} $$
(6.26)

where J is day of the year. Application of the Penman combination method in south Florida is shown in Fig. 6.12a where estimates are compared to the simple Abtew method estimates. The simple Abtew method does not have mass transfer or sink strength component. The Penman combination method annual ET estimate (1,374 mm) is greater by 3.8% compared to the simple Abtew method (1,322 mm). Generally, during the dry season (May through November), the simple Abtew method estimates are higher. During the wet, humid months, the Penman method has higher estimates. Figure 6.12b depicts solar radiation (R s) used in the simple method and net solar radiation (R n) used in the Penman method.

Fig. 6.12
figure 12

(a) Penman combination method and simple Abtew method s for evaporation estimation in south Florida, (b) solar and net radiation, (c) vapor pressure deficit and wind speed

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Figure 6.12c depicts daily vapor pressure deficit and wind speed at 2-m height. It is clearly shown that the Penman method is sensitive to vapor pressure deficit.

6.3.2.3 The Penman–Monteith Method

The Penman–Monteith (P–M) equation is the closest to a physical evapotranspiration estimation model and applicable at shorter time periods than a day. Energy balance, momentum transfer, and mass transfer are accounted, and internal and external resistance or conductance to the evapotranspiration process is accounted. In this section, the P–M equation is presented with details of each parameter or coefficient used. The P–M equation has been accepted as the standard to compute reference evapotranspiration (Eq. 6.27):

$$ {\text{ET}} = \frac{1}{\lambda }\frac{{\Delta ({R_{\text{n}}} - G) + \rho {c_{\text{p}}}({e_{\text{a}}} - {e_{\text{d}}})\frac{1}{{{r_{\text{a}}}}}}}{{\Delta + \gamma \left( {1 + \frac{{{r_{\text{c}}}}}{{{r_{\text{a}}}}}} \right)}} $$
(6.27)

where ET is evapotranspiration in mm day−1, Δ is the slope of the vapor pressure curve (kPa °C−1), γ is psychrometric constant (kPa °C−1), R n is net radiation (MJ m−2 day−1), G is heat flux (MJ m−2 day−1), ρ is atmospheric density (kg m−3), c p is specific heat of moist air (kJ kg−1 °C−1), (e a − e d) is vapor pressure deficit (kPa), r c is canopy resistance, and r a is aerodynamic resistance. This method has the most measured, derived, and estimated inputs, as shown in Table 6.3.

Table 6.3 Input required for the Penman–Monteith method
Full size table

Change in heat storage (G) in soil or water is computed by Eq. 6.28:

$$ G = {c_{\text{s}}}{d_{\text{s}}}({T_n} - {T_{{n - 1}}}) $$
(6.28)

where c s is soil or water heat capacity (2.100 MJ m–3 °C–1 or 4.18 MJ m–3 °C–1, respectively), d s is effective depth (m), and T n and T n – 1 are average air temperature on day n and previous day.

6.3.2.4 Canopy Conductance (g c) and Canopy Resistance (r c)

According to Monteith (1973), water vapor loss from a leaf by diffusion is equivalent to an electrical circuit with cuticular resistances being analogous to resistance to current flow. Canopy resistance is the inverse of canopy conductance. Various authors have presented methods to estimate canopy conductance and resistance (Weert and Kamerling 1974; Slabbers 1977; Katerji and Perrier 1983; Lafleur and Rouse 1988; Allen et al. 1989; Kim and Verma 1991; Saugier and Katerji 1991; Steiner et al. 1991; Lafleur and Roulet 1992; Todorovic 1999; Katerji and Rana 2008). In FAO-P-M daily reference ET estimation method from a reference crop of known height, fixed canopy resistance (70 s m−1) is recommended (Allen et al. 1998). Theoretical and empirical approaches have been used to show canopy resistance is related to soil moisture, available energy, vapor pressure deficit, and aerodynamic resistance (Gharsallah et al. 2009). Choudhury and Idso (1985) proposed that wheat stomatal conductance is a function of net solar radiation and canopy resistance is a function of leaf area index by canopy strata and canopy conductance (Eq. 6.29):

$$ {g_{\text{c}}} = \sum\limits_{{j = 1}}^n {{L_j}{S_{\text{cj}}}} $$
(6.29)

where g c is canopy conductance, L j is leaf area index for canopy strata j, and S cj is stomatal conductance of leaf strata L j . Currently, Eq. 6.30 is widely used for estimating canopy resistance (Allen et al. 1989):

$$ {r_{\text{c}}} = \frac{{{r_{\text{s}}}}}{{0.5{\text{LAI}}}} $$
(6.30)

where r c is average daily bulk canopy resistance (s m−1), r s is average minimum daytime value of stomatal resistance (s m−1) for a single leaf, and LAI is leaf area index. Equations 6.29 and 6.30 are similar in form when resistance is substituted for conductance in Eq. 6.29. Canopy resistance was reported to be related to water stress ranging from 30 to 100 s−1 m−1 for the equatorial forest of Kenya (Szeicz and Long 1969).

Abtew et al. (1995) conducted experimental work by measuring stomatal conductance with a porometer on cattail plants in a lysimeter to develop a canopy resistance parameter. The design and operations of the cattail lysimeter are presented in Chaps. 3 and 7. The objective was to develop a canopy resistance parameter (r c), measure all weather parameters needed to compute ET, measure ET with the lysimeter, and apply the P–M method with the developed r c value. Then, compare the ET estimates to lysimeter measurements. The weather station measured solar radiation, net solar radiation, photosynthetic photon density flux (PPFD), air temperature, humidity, atmospheric pressure, and water temperature. Wind speed was measured at 1, 2.6, and 10 m for the purpose of developing the wind profile and estimating aerodynamic resistance (r a) so that the only variable left is r c. Wind speed was measured every 10 s and averaged every 15 min. All other parameters were measured every 5 min and averaged every 15 min. Leaf conductance and transpiration were measured with a LI-1600M steady state porometer with an aperture area of 1 cm2.

Cattails have symmetrically arranged leaves ranging from two to four leaves on each side. Adaxial (back) leaf surfaces are concave, and abaxial (front) leaf surfaces are convex. A sampling method was used to select representative locations for measurement of conductance and transpiration from a plant. Measurements were made on the sunlit side of each plant, in the middle of the upper half (apical) and in the middle of the lower half (basal) sections of inner, middle, and outer leaves. Leaf conductance (gm), leaf temperature, photosynthetic photon flux density (PPFD), and leaf transpiration were measured between 9:45 am and 5:00 pm on April 8 and 29, 1993. Measurements were made during clear skies. A total of 208 leaf conductance measurements were made from 30 plants (Abtew et al. 1995). Figure 6.13 depicts each observation (one side of leaf) of stomatal conductance with a porometer. The following equation (Eq. 6.31) was used for computing stomatal conductance, g s (LI-COR, Inc., 1989):

$$ {g_{\text{s}}} = \frac{{{g_{\text{b}}}{g_{\text{m}}}}}{{{g_{\text{b}}} - {g_{\text{m}}}}} $$
(6.31)

where g b is the boundary layer conductance inside the porometer cubicle and g m is the measured conductance of the leaf (sum of abaxial and adaxial sides) in mol m−2 s−1. The boundary conductance, g m, measured in the laboratory with a wet filter was 2.26 mol m−2 s−1. Molar conductance units were converted to velocity units based on Eq. 6.32 (LI-COR, Inc., 1989).

$$ {g_{\text{sv}}} = \frac{{8.314{g_{\text{s}}}({T_{\text{avg}}} + 273)}}{P} $$
(6.32)

where g sv is stomatal conductance (mm s−1), g s is stomatal conductance (mol m−2 s−1), 8.314 is the gas constant (Pa m3 mol−1 K−1), T avg is average leaf temperature (°C), and P is average atmospheric pressure (Pa). Observed and computed parameters from the experiment are shown in Table 6.4 adopted from Abtew et al. (1995). The objective of the experiment was to derive canopy resistance (r c) for cattails, a parameter needed for application of the Penman–Monteith equation from measured stomatal conductance. Canopy resistance (r c) is the inverse of canopy conductance. Canopy conductance was derived from mean stomatal conductance as a summation of leaf area-weighted stomatal conductance of apical and basal sections of the upper, middle, and inner leaves of 30 plants (Eq. 6.33). Comparative approaches are reported in the literature (Roberts et al. 1980; Saugier and Katerji 1991).

$$ {g_{\text{c}}} = \sum\limits_j^P {\sum\limits_i^L {\left( {{{({g_{{{\text{sv}}ji}}})}^l} + {{({g_{{{\text{sv}}ji}}})}^u}} \right)} } {\text{LAI}} $$
(6.33)

where g c is canopy conductance (mm s−1); g svji is mean stomatal conductance (mm s−1) for adaxial leaf side (l), abaxial leaf side (u), profile (section) j, and leaf (layer) i; P is leaf profile; L is leaf layer; and LAI is leaf area index.

Fig. 6.13
figure 13

Stomatal conductance observations over 2 days (single side of leaf)

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Table 6.4 Cattail leaf conductance (g m), stomatal conductance (g s, g sv), and other parameters
Full size table

The leaf area-weighted and boundary layer-corrected canopy conductance (g c) for cattails was 19.9 mm s−1. Canopy resistance (r c) is the inverse of canopy conductance (g c) and is reported in m s−1 unit. Canopy resistance is derived as follows and is reported in s m−1 (Eq. 6.34):

$$ {r_{\text{c}}} = \frac{{1{,}000}}{{{g_{\text{c}}}}} $$
(6.34)

where r c is m s−1 and g c is in mm s−1. The seasonal canopy resistance for cattails in south Florida, derived from Eq. 6.30, is 50.3 m s−1. Comparison of the sum of squares of error between lysimeter measurements and Penman–Monteith model ET computation (r c = 50.3 m s−1) shows that r c values of 40–70 m s−1 produce very close results.

6.3.2.5 Aerodynamic Resistance (r a)

The aerodynamics resistance has been commonly presented as mainly a function of surface characteristics and wind speed. Equation 6.35 (Allen et al. 1989) has been in use for a while:

$$ {r_{\text{a}}} = \frac{{\ln \frac{{(z - d)}}{{{z_{\text{om}}}}}}}{{{k^2}}} \times \frac{{\ln \frac{{({z_{\text{h}}} - d)}}{{{z_{\text{oh}}}}}}}{{{u_{\text{z}}}}} $$
(6.35)

where r a is aerodynamic resistance (s m−1), z is the height of wind measurement (m), z h is the height of air temperature and humidity measurement (m), d is displacement height (m), z om is roughness length for momentum transfer, z oh is roughness length for vapor and heat transfer, k is the von Karman constant for turbulent diffusion (0.41), and u z is wind speed measurement at height z. Application of Eq. 6.35 on a daily basis over a year in south Florida resulted in a mean r a of 83.3 s m−1. Variation of the daily mean r a is shown in Fig. 6.14.

Fig. 6.14
figure 14

Variation of daily mean aerodynamic resistance (r a)

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There are several equations to estimate displacement height (d) and aerodynamic roughness (z o or z om), as shown in Abtew et al. (1989). Applying the author’s methods (Eqs. 6.36 and 6.37), wind speed at 2-m height is estimated (Abtew et al. 1989). Z oh is estimated by Eq. 6.38 (Allen et al. 1989):

$$ d = {F_{\text{c}}}{h_{\text{c}}} $$
(6.36)

where F c is fraction of surface cover and h c is average height of cover.

$$ {z_{\text{om}}} = 0.13({h_{\text{c}}} - d) $$
(6.37)
$$ {z_{\text{oh}}} = 0.1{z_{\text{om}}} $$
(6.38)

Application of the Penman–Monteith method is shown in Chaps. 7 and 9. Measured input of known data quality from weather stations and derived and estimated parameters from wetland surfaces were used.

6.4 Remote Sensing Methods

The latest technology of satellite-based environmental monitoring holds promise for providing meteorological variable observations for large areas. Satellite-based evapotranspiration estimation methods and applications are presented in detail in Chaps. 10, 11, and 12.

6.5 Summary

In this chapter, several ET estimation methods are presented along with input requirements. While complex methods approach physical representation of the ET process, the number of input parameters required increases. The cost of acquiring input parameters and maintaining acceptable data quality increases with the complexity of method. The virtue of application of simpler methods is well demonstrated.