**Limited Data**

Hsin-Fu Yeh

sons with field experimental observations, allowed to identify new challenges in northern

Moisture evaporation from porous media is studied by its importance in drying of foods and building materials and biological products such as biopesticides. An exciting foray into the current state of knowledge in the frame of physics of moisture evaporation process from porous media is the subject of the last chapter. Further, the authors aim to establish theoreti‐ cal support for designing biopesticides able to ensure efficient and effective fight against

**Prof. Dr. Daniel Bucur,**

Romania

University of Applied Life Sciences and Environment in Iasi,

agroecosystems.

VIII Preface

harmful insects in agricultural crops.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68495

#### Abstract

A limited number of parameters or a single meteorological parameter was used in this study to estimate evapotranspiration. The main objectives of this study are as follows. (1) The Penman-Monteith method was used to estimate ET. The empirical formula published by the Food and Agriculture Organization (FAO) was applied via substitution to compare situations that were missing certain meteorological parameters. (2) Radiation-based methods and temperature-based methods were compared with the Penman-Monteith method to estimate ET and discuss their applicability in the study area. With Tainan Weather Station of Taiwan as the study area, this study selected the Penman-Monteith method as well as six other radiation-based estimation formulas: Makkink, Turc, Jensen-Haise, Priestley-Taylor, Doorenbos-Pruit, and Abtew methods. The other four temperature-based estimation formulas, namely, Thornthwaite, Blaney-Criddle, Hamon, and Linacre methods, were used to estimate ET and compare the differences and the results were compared with the Penman-Monteith method. The results showed that there was little effect on estimating ET using the Penman-Monteith method when the wind speed data was missing or insufficient. The Turc method was the best among the six radiation-based estimation formulas, while the Linacre method was the best temperature-based estimation formula. Generally speaking, radiation-based estimation formulas were more accurate than temperature-based estimation formulas.

Keywords: evapotranspiration, Penman-Monteith, radiation method, temperature method

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 1. Introduction

Evapotranspiration (ET) is a basic element of the hydrologic cycle as well as a key factor in water balance [1]. According to statistics, global average annual rainfall is around 973 mm, and about 64% of surface water is lost through ET [2]. Therefore, ET is considered to be an indispensable parameter in hydrologic studies, such as irrigation scheduling and management, crop water demand, and environmental impact assessment [3]. Hence, effective evaluation of ET is important for the management and planning of water resources. In previous studies, many formulas of empirical or physical methods have been used to estimate ET in various climatic conditions; examples include the Makkink method [4], Priestley-Taylor method [5], lysimeter method [6], and micro-meteorological observation method [7]. The empirical formula of the Penman-Monteith method released by the Food and Agriculture Organization (FAO) is the method most internationally used [8]. This method requires consideration of a variety of meteorological parameters, such as temperature, radiation, relative humidity, and wind speed. These data, however, are frequently missing or hard to collect, resulting in difficulties in estimation [9]. In particular, reliable meteorological data, such as radiation, relative humidity, and wind speed, are rather difficult to collect in some areas. In addition, the maintenance of meteorological stations requires substantial funding and the installation is complex.

Therefore, in previous studies, many scholars have used a limited number of parameters or a single meteorological parameter to easily estimate ET and simplify the estimation methods, which are classified into five major categories based on the required meteorological parameters: (1) water balance method, (2) mass transfer method, (3) mixing method, (4) radiationbased method, and (5) temperature-based method [10]. Except for the last two methods, the other three methods require a variety of meteorological parameters to estimate ET, thus causing obstacles in data collection and obtaining complete meteorological information. Furthermore, studies have found that the results of empirical methods should be compared with the Penman-Monteith method and released by FAO so as to carry out accurate estimation in each region [11].

In this study, a single meteorological parameter was applied, as well as the Penman-Monteith method, six radiation-based methods, and four temperature-based methods, to effectively estimate ET. The main objectives of this study are as follows: (1) when radiation, wind speed, and relative humidity data were missing, empirical formulas were used for substitution in the Penman-Monteith method to compare the estimation results; (2) the regional applicability of the radiation- and temperature-based methods were compared so as to make these methods more suitable for the study area.

### 2. Material and methods

This study mainly discussed the effective evaluation of ET using limited meteorological parameters. With the Penman-Monteith method as the standard for estimation, ET was calculated using substitution formulas when radiation, wind speed, or relative humidity data were missing in the Penman-Monteith method. Six radiation-based methods and four temperature-based methods were selected to discuss their applicability in the study area. In this study, mean bias error, root mean square error, and the Pearson-type goodness-of-fit index were used to analyze and investigate the differences among ET estimations using the empirical formulas of temperature and radiation methods. Meanwhile, this study strived to determine the method with a simpler empirical formula to address the difficulties caused by a shortage of meteorological parameter data.

#### 2.1. Penman-Monteith method

1. Introduction

2 Current Perspective to Predict Actual Evapotranspiration

installation is complex.

each region [11].

more suitable for the study area.

2. Material and methods

Evapotranspiration (ET) is a basic element of the hydrologic cycle as well as a key factor in water balance [1]. According to statistics, global average annual rainfall is around 973 mm, and about 64% of surface water is lost through ET [2]. Therefore, ET is considered to be an indispensable parameter in hydrologic studies, such as irrigation scheduling and management, crop water demand, and environmental impact assessment [3]. Hence, effective evaluation of ET is important for the management and planning of water resources. In previous studies, many formulas of empirical or physical methods have been used to estimate ET in various climatic conditions; examples include the Makkink method [4], Priestley-Taylor method [5], lysimeter method [6], and micro-meteorological observation method [7]. The empirical formula of the Penman-Monteith method released by the Food and Agriculture Organization (FAO) is the method most internationally used [8]. This method requires consideration of a variety of meteorological parameters, such as temperature, radiation, relative humidity, and wind speed. These data, however, are frequently missing or hard to collect, resulting in difficulties in estimation [9]. In particular, reliable meteorological data, such as radiation, relative humidity, and wind speed, are rather difficult to collect in some areas. In addition, the maintenance of meteorological stations requires substantial funding and the

Therefore, in previous studies, many scholars have used a limited number of parameters or a single meteorological parameter to easily estimate ET and simplify the estimation methods, which are classified into five major categories based on the required meteorological parameters: (1) water balance method, (2) mass transfer method, (3) mixing method, (4) radiationbased method, and (5) temperature-based method [10]. Except for the last two methods, the other three methods require a variety of meteorological parameters to estimate ET, thus causing obstacles in data collection and obtaining complete meteorological information. Furthermore, studies have found that the results of empirical methods should be compared with the Penman-Monteith method and released by FAO so as to carry out accurate estimation in

In this study, a single meteorological parameter was applied, as well as the Penman-Monteith method, six radiation-based methods, and four temperature-based methods, to effectively estimate ET. The main objectives of this study are as follows: (1) when radiation, wind speed, and relative humidity data were missing, empirical formulas were used for substitution in the Penman-Monteith method to compare the estimation results; (2) the regional applicability of the radiation- and temperature-based methods were compared so as to make these methods

This study mainly discussed the effective evaluation of ET using limited meteorological parameters. With the Penman-Monteith method as the standard for estimation, ET was Penman-Monteith method was recommended by the FAO in the 1998 FAO-56 report for the assessment of ET, and it is currently used internationally [12]. After years of study by domestic scholars, it is believed that the Penman-Monteith method is quite suitable in Taiwan [13–15]. Its formula can be expressed as follows:

$$\text{ET} = \frac{0.408\Delta(\text{R}\_{\text{n}} - \text{G}) + \gamma \frac{900}{\text{T} + 2\text{T}3} \mathbf{u}\_{2}(\text{e}\_{\text{s}} - \text{e}\_{\text{a}})}{\Delta + \gamma (1 + 0.34 \mathbf{u}\_{2})} \tag{1}$$

In Eq. (1), ET represents evapotranspiration (mm d�<sup>1</sup> ); Δ represents the slope of air pressure curve (kPa �C�<sup>1</sup> ); T is the average temperature (�C); Rn is net radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ); G is the soil thermal flux (MJ m�<sup>2</sup> d�<sup>1</sup> ); γ is the humidity constant (kPa �C�<sup>1</sup> ); u2 is the wind speed measured at the height of 2 m (m s�<sup>1</sup> ); and (es�ea) is the difference between saturated and actual vapor pressure (kPa). For field applications, Eq. (1) was calculated with monthly air temperature, humidity, radiant energy, wind speed, and other parameters [12].

When data of some meteorological parameters could not be obtained or were incomplete, for instance, radiation, relative humidity, and wind speed, a calculation was conducted using the following empirical formula:

1. When data of relative humidity could not be obtained or was incomplete:

$$\mathbf{e\_a} = 0.611 \text{exp}\left(\frac{17.27 \text{T}\_{\text{min}}}{\text{T}\_{\text{min}} + 237.3}\right) \tag{2}$$

In Eq. (2), Tmin represents minimum temperature (�C).

2. When radiation data could not be obtained or was incomplete:

$$\mathbf{R\_s} = \mathbf{k\_{Rs}}\sqrt{(\mathbf{T\_{max}} - \mathbf{T\_{min}})} \mathbf{R\_a} \tag{3}$$

In Eq. (3), kRs is the empirical coefficient (kRs = 0.19); Ra is extraterrestrial solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ).

3. When data of wind speed could not be obtained or was incomplete:

When there is no record of wind speed in the evaluation area, the average Taiwan wind speed of 1.83 m s�<sup>1</sup> was used, which was estimated with the data collected by 20 central meteorological observatories in Taiwan during 1990–2008 [15]. In addition, wind speed at a height of 2 m above the ground was primarily used in the estimation of wind speed. Provided that the measurement height was not 2 m, the following formula was applied:

$$\mathbf{u}\_2 = \mathbf{u}\_z \frac{4.87}{\ln(67.8\mathbf{z} - 5.42)}\tag{4}$$

In Eq. (4), uz is the wind speed measured at a meteorological station (m s�<sup>1</sup> ); z is the height of the anemometer above the ground (m).

#### 2.2. Radiation-based methods

Priestley and Taylor [5] proposed that the estimation of ET could be explored from the perspective of energy conversion on the water surface. Evapotranspiration increased with an increase of radiation. Hence, radiation was taken as a vital meteorological parameter for ET assessment. Radiation-based methods were mainly based on the simplified principle of energy balance to estimate ET. Therefore, ET could be evaluated using a single meteorological parameter, and, in general, the form of radiation-based methods is as follows:

$$\text{ET} = \frac{\text{C}\_{\text{r}}}{\lambda} (\text{wR}\_{\text{s}}) \text{ or } \text{ET} = \frac{\text{C}\_{\text{r}}}{\lambda} (\text{wR}\_{\text{n}}) \tag{5}$$

λ represents the latent heat of evaporation (MJ kg�<sup>1</sup> ); Cr represents the generated empirical coefficient based on the relative humidity and wind speed; w is the generated empirical coefficient in accordance with temperature and latitude; Rs represents the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ); and Rn is the net radiation (W m�<sup>2</sup> d�<sup>1</sup> ).

Six radiation-based methods that are used internationally to assess evapotranspiration were selected in this study, including Makkink [4], Turc [16], Jensen-Haise [17], Priestley and Taylor [5], Doorenbos and Pruitt [18], and Abtew [19]. The methods are described as follows:

#### 2.2.1. Makkink method

$$\text{ET} = \alpha \times \left(\frac{\Delta}{\Delta + \gamma} \frac{\mathcal{R}\_s}{\lambda}\right) - \beta \tag{6}$$

Rs represents the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ); Δ is the slope of the saturated vapor pressure curve (kPa �C�<sup>1</sup> ); γ represents the humidity constant (kPa �C�<sup>1</sup> ); λ is the latent heat of evaporation (MJ kg�<sup>1</sup> ); and α = 0.61, β = 0.12.

#### 2.2.2. Turc method

1. Average relative humidity RH < 50%

Comparison of Evapotranspiration Methods Under Limited Data http://dx.doi.org/10.5772/intechopen.68495 5

$$\text{ET} = 0.013 \left( \frac{\text{T}}{\text{T} + 15} \right) \times (\text{R}\_s \times 23.8846 + 50) \times \left( 1 + \frac{50 - \text{RH}}{70} \right) \tag{7}$$

2. Average relative humidity RH > 50%

$$\rm{ET} = 0.013 \left( \frac{\rm{T}}{\rm{T} + 15} \right) (R\_s \times 23.8846 + 50) \tag{8}$$

In Eq. (8), T represents the average temperature (�C); Rs is the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ); and RH represents average relative humidity (%).

2.2.3. Jensen-Haise method

When there is no record of wind speed in the evaluation area, the average Taiwan wind speed of 1.83 m s�<sup>1</sup> was used, which was estimated with the data collected by 20 central meteorological observatories in Taiwan during 1990–2008 [15]. In addition, wind speed at a height of 2 m above the ground was primarily used in the estimation of wind speed. Provided that the measurement height was not 2 m, the following formula was applied:

4:87

lnð67:8z � <sup>5</sup>:42<sup>Þ</sup> <sup>ð</sup>4<sup>Þ</sup>

<sup>λ</sup> <sup>ð</sup>wRnÞ ð5<sup>Þ</sup>

); Cr represents the generated empirical

� β ð6Þ

); Δ is the slope of the saturated vapor

); λ is the latent heat of

).

); z is the height

u2 ¼ uz

eter, and, in general, the form of radiation-based methods is as follows:

ET <sup>¼</sup> Cr

λ represents the latent heat of evaporation (MJ kg�<sup>1</sup>

Rs represents the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup>

1. Average relative humidity RH < 50%

); and α = 0.61, β = 0.12.

of the anemometer above the ground (m).

4 Current Perspective to Predict Actual Evapotranspiration

2.2. Radiation-based methods

radiation (MJ m�<sup>2</sup> d�<sup>1</sup>

2.2.1. Makkink method

pressure curve (kPa �C�<sup>1</sup>

evaporation (MJ kg�<sup>1</sup>

2.2.2. Turc method

In Eq. (4), uz is the wind speed measured at a meteorological station (m s�<sup>1</sup>

Priestley and Taylor [5] proposed that the estimation of ET could be explored from the perspective of energy conversion on the water surface. Evapotranspiration increased with an increase of radiation. Hence, radiation was taken as a vital meteorological parameter for ET assessment. Radiation-based methods were mainly based on the simplified principle of energy balance to estimate ET. Therefore, ET could be evaluated using a single meteorological param-

<sup>λ</sup> <sup>ð</sup>wRs<sup>Þ</sup> or ET <sup>¼</sup> Cr

coefficient based on the relative humidity and wind speed; w is the generated empirical coefficient in accordance with temperature and latitude; Rs represents the amount of solar

Six radiation-based methods that are used internationally to assess evapotranspiration were selected in this study, including Makkink [4], Turc [16], Jensen-Haise [17], Priestley and Taylor [5], Doorenbos and Pruitt [18], and Abtew [19]. The methods are described as follows:

> Δ Δþγ

); γ represents the humidity constant (kPa �C�<sup>1</sup>

Rs λ 

); and Rn is the net radiation (W m�<sup>2</sup> d�<sup>1</sup>

ET ¼ α �

$$\rm ET = \rm C\_{\Gamma} \times (T - T\_{\infty}) \times R\_{\ast} \tag{9}$$

CT represents the temperature constant, and its calculation method is listed below:

$$\mathbf{C\_{T}} = \frac{1}{(\mathbf{C\_{1}} + \mathbf{C\_{2}} \times \mathbf{C\_{H}})} \tag{10}$$

$$\mathbf{C}\_{1} = 68 - 3.6 \times \frac{\mathbf{h}\_{\text{j}}}{1000} \tag{11}$$

$$\mathbb{C}\_2 = \text{13} \tag{12}$$

$$\mathbf{C}\_{\text{h}} = \frac{50}{\mathbf{e}\_{\text{s}}(\mathbf{T}\_{\text{max}}) - \mathbf{e}\_{\text{s}}(\mathbf{T}\_{\text{min}})} \tag{13}$$

hj is the sea surface height of the meteorological station; esðTmaxÞ � esðTminÞ represents the saturated vapor pressure at the highest temperature and the lowest temperature, respectively; T is the average temperature (�F); and Tx represents the temperature-axis intercept constant, and its formula is as follows:

$$\mathbf{T\_{x}} = 27.5 - 0.25 \times \left( \mathbf{e(T\_{max})} - \mathbf{e(T\_{min})} \right) - \frac{\mathbf{h}}{1000} \tag{14}$$

2.2.4. Priestley-Taylor method

$$\text{ET} = \alpha\_{\text{PT}} \frac{\Delta}{\Delta + \gamma} \frac{\mathbf{R\_n}}{\lambda} \tag{15}$$

Δ represents the slope of the saturated vapor pressure curve (kPa �C�<sup>1</sup> ); γ is the humidity constant (kPa �C�<sup>1</sup> ); Rn is the net radiation (W m�<sup>2</sup> d�<sup>1</sup> ); G represents soil thermal flux (MJ m�<sup>2</sup> d�<sup>1</sup> ); and αPT represents the empirical coefficient (αPT = 1.26).

#### 2.2.5. Doorenbos-Pruitt method

$$\mathbf{ET} = \mathbf{a} + \mathbf{b} \times \left(\frac{\Delta}{\Delta + \gamma} \frac{\mathbf{R\_s}}{\lambda}\right) \tag{16}$$

<sup>a</sup> <sup>¼</sup> <sup>1</sup>:<sup>066</sup> � <sup>0</sup>:<sup>13</sup> � <sup>10</sup>�<sup>2</sup> RH <sup>þ</sup> <sup>0</sup>:45 Uz � <sup>0</sup>:<sup>2</sup> � <sup>10</sup>�<sup>3</sup> RH � Uz � <sup>0</sup>:<sup>315</sup> � <sup>10</sup>�<sup>4</sup> RH2�0:<sup>11</sup> � <sup>10</sup>�<sup>2</sup> U2 z ð17Þ

$$\mathbf{b} = -0.3\tag{18}$$

Rs is the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ); Δ represents the slope of the saturated vapor pressure curve (kPa �C�<sup>1</sup> ); γ is the humidity constant (kPa �C�<sup>1</sup> ); λ represents the latent heat of evaporation (MJ kg�<sup>1</sup> ); Uz is the wind speed (m s�<sup>1</sup> ); and RH represents relative humidity (%).

#### 2.2.6. Abtew method

$$\text{ET} = \alpha \times \left(\frac{\text{R}\_{\text{s}}}{\lambda}\right) \tag{19}$$

In Eq. (19), Rs represents the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ); λ represents the latent heat of evaporation (MJ kg�<sup>1</sup> ); and α = 0.53.

#### 2.3. Temperature-based methods

Temperature was the easiest to obtain among the many meteorological parameters. Generally speaking, the form of temperature-based methods is as follows [10]:

$$\mathbf{ET} = \mathbf{c} \times \mathbf{T}^n \text{ or } \mathbf{ET} = \mathbf{c} \times \mathbf{d} \times \mathbf{T} (\mathbf{c}\_1 - \mathbf{c}\_2 \mathbf{h}) \tag{20}$$

In Eq. (20), T is the air temperature (�C); h represents humidity; c, c1, and c2 were constants; and d represents time.

Four temperature-based methods were chosen in this study to estimate ET, including the Thornthwaite [20], Blaney and Criddle [21], Hamon [22], and Linacre [23]. The methods are described below:

#### 2.3.1. Thornthwaite method

$$\text{ET} = \text{C} \times 16 \times \left(\frac{10 \text{T}}{\text{I}}\right)^{\text{a}} \tag{21}$$

In Eq. (21), T represents monthly average temperature of the air (�C); I is the thermal index, and its formula is as follows:

Comparison of Evapotranspiration Methods Under Limited Data http://dx.doi.org/10.5772/intechopen.68495 7

$$\mathbf{I} = \sum\_{\mathbf{j}=1}^{12} \mathbf{i}\_{\mathbf{j}} \tag{22}$$

$$\mathbf{i} = \left(\frac{\mathbf{T}}{\mathbf{5}}\right)^{1.51} \tag{23}$$

$$\mathbf{a} = 0.00000675\mathbf{I}^3 - 0.0000771\mathbf{I}^2 + 0.0179\mathbf{I} + 0.49239\tag{24}$$

C represents the correction coefficient.

$$\mathbf{C} = \frac{\mathbf{N}}{\mathbf{360}} \tag{25}$$

N represents monthly amount of daylight hours (h).

2.3.2. Blaney-Criddle method

2.2.5. Doorenbos-Pruitt method

6 Current Perspective to Predict Actual Evapotranspiration

<sup>a</sup> <sup>¼</sup> <sup>1</sup>:<sup>066</sup> � <sup>0</sup>:<sup>13</sup> � <sup>10</sup>�<sup>2</sup>

pressure curve (kPa �C�<sup>1</sup>

heat of evaporation (MJ kg�<sup>1</sup>

and d represents time.

2.3.1. Thornthwaite method

its formula is as follows:

described below:

2.3. Temperature-based methods

evaporation (MJ kg�<sup>1</sup>

2.2.6. Abtew method

Rs is the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup>

ET ¼ a þ b �

); γ is the humidity constant (kPa �C�<sup>1</sup>

ET ¼ α �

Temperature was the easiest to obtain among the many meteorological parameters. Generally

In Eq. (20), T is the air temperature (�C); h represents humidity; c, c1, and c2 were constants;

Four temperature-based methods were chosen in this study to estimate ET, including the Thornthwaite [20], Blaney and Criddle [21], Hamon [22], and Linacre [23]. The methods are

> 10T I <sup>a</sup>

ET ¼ C � 16 �

In Eq. (21), T represents monthly average temperature of the air (�C); I is the thermal index, and

ET <sup>¼</sup> <sup>c</sup> � Tn or ET <sup>¼</sup> <sup>c</sup> � <sup>d</sup> � <sup>T</sup>ðc1 � c2hÞ ð20<sup>Þ</sup>

Rs λ 

RH <sup>þ</sup> <sup>0</sup>:45 Uz � <sup>0</sup>:<sup>2</sup> � <sup>10</sup>�<sup>3</sup>

); Uz is the wind speed (m s�<sup>1</sup>

In Eq. (19), Rs represents the amount of solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup>

speaking, the form of temperature-based methods is as follows [10]:

); and α = 0.53.

Δ Δ þ γ

Rs λ 

RH � Uz � <sup>0</sup>:<sup>315</sup> � <sup>10</sup>�<sup>4</sup>

b ¼ �0:3 ð18Þ

); Δ represents the slope of the saturated vapor

); and RH represents relative humidity (%).

ð16Þ

U2 z ð17Þ

ð19Þ

ð21Þ

RH2�0:<sup>11</sup> � <sup>10</sup>�<sup>2</sup>

); λ represents the latent heat of

); λ represents the latent

$$\text{ET} = \mathbf{p} \times (0.46\mathbf{T} + 8.13) \tag{26}$$

P represents the annual daylight percentage of every month and T is the average temperature (�C).

2.3.3. Hamon method

$$\text{ET} = \mathbf{k} \times 0.1651 \times 216.7 \times \mathbf{N} \times (\frac{\mathbf{e}\_s}{\mathbf{T} + 273.3}) \tag{27}$$

In Eq. (27), k represents the empirical coefficient (k = 1.0); N represents daylight hours (h); es is the saturated vapor pressure (kPa); and T represents average temperature (�C).

2.3.4. Linacre method

$$\text{ET} = \frac{\frac{500\text{T}\_m}{100-\text{A}} + 15(\text{T} - \text{T}\_d)}{(80-\text{T})} \tag{28}$$

$$\mathbf{T\_m} = \mathbf{T} + \mathbf{0.006h} \tag{29}$$

T represents average temperature (�C); Td is the dew point temperature (�C); and A represents latitude (�).

#### 2.4. Statistical verification

In this study, the differences and correlations between the estimation results of the Penman-Monteith method and other formulas were compared and assessed using the following criteria:

#### 2.4.1. Mean bias error

The bias degree of the Penman-Monteith method and the other methods was determined from the mean bias error (MBE). A smaller value indicated a lower bias degree as well as a better result. The best fit was MBE = 0, and the formula is as follows:

$$\text{MBE} = \frac{\sum\_{i=1}^{n} (\text{E}\_i - \text{P}\_i)}{\text{n}} \tag{30}$$

Ei represents the estimated value of the empirical formula; Pi represents the estimated value of the Penman-Monteith method; and n is the total number of observations.

#### 2.4.2. Error percentage

$$\text{Error percentage} = \frac{\text{MBE}}{\overline{x}} \times 100\tag{31}$$

MBE represents the mean bias error of Eq. (30); and x represents the mean value.

#### 2.4.3. Root mean square error

$$\text{RMSE} = \sqrt{\frac{\sum\_{i=1}^{n} \left(\mathbf{E\_i} - \mathbf{P\_i}\right)^2}{\mathbf{n}}} \tag{32}$$

Root mean square error (RMSE) represents the variance degree of two estimated values. The best fit was RMSE = 0. In Eq. (32), Ei is the estimated value of empirical formula; Pi represents the estimated value of the Penman-Monteith method; and n is the total number of observations.

2.4.4. Pearson-type goodness-of-fit index (R2 )

$$\mathbf{R}^2 = \left[ \frac{\sum\_{i=1}^n (\mathbf{E}\_i - \overline{\mathbf{E}})(\mathbf{P}\_i - \overline{\mathbf{P}})}{\sqrt{\sum\_{i=1}^n (\mathbf{E}\_i - \overline{\mathbf{E}})^2} \sqrt{\sum\_{i=1}^n (\mathbf{P}\_i - \overline{\mathbf{P}})^2}} \right] \tag{33}$$

The Pearson-type goodness-of-fit index represents the degree of correlation between two estimation methods. The best fit was R<sup>2</sup> = 1.0. In Eq. (33), Ei represents the estimated value of the empirical formula; E is the average estimated value of the empirical formulas; Pi represents the estimated value of the Penman-Monteith method; P is the mean estimated value of the Penman-Monteith method; and n represents the total number of observations.

#### 2.5. Study area

2.4.1. Mean bias error

8 Current Perspective to Predict Actual Evapotranspiration

2.4.2. Error percentage

2.4.3. Root mean square error

of observations.

observations.

2.4.4. Pearson-type goodness-of-fit index (R2

R2 <sup>¼</sup>

2 6 4

The bias degree of the Penman-Monteith method and the other methods was determined from the mean bias error (MBE). A smaller value indicated a lower bias degree as well as a better

> X<sup>n</sup> i¼1

Ei represents the estimated value of the empirical formula; Pi represents the estimated value of

Error percentage <sup>¼</sup> MBE

s

Root mean square error (RMSE) represents the variance degree of two estimated values. The best fit was RMSE = 0. In Eq. (32), Ei is the estimated value of empirical formula; Pi represents the estimated value of the Penman-Monteith method; and n is the total number

MBE represents the mean bias error of Eq. (30); and x represents the mean value.

RMSE ¼

)

X<sup>n</sup> i¼1

X<sup>n</sup> i¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðEi � EÞ

The Pearson-type goodness-of-fit index represents the degree of correlation between two estimation methods. The best fit was R<sup>2</sup> = 1.0. In Eq. (33), Ei represents the estimated value of the empirical formula; E is the average estimated value of the empirical formulas; Pi represents the estimated value of the Penman-Monteith method; P is the mean estimated value of the Penman-Monteith method; and n represents the total number of

ðEi � PiÞ

x

Xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>n</sup> i¼1

n

ðEi � EÞðPi � PÞ

ðPi � PÞ <sup>2</sup> q

3 7

<sup>5</sup> <sup>ð</sup>33<sup>Þ</sup>

<sup>2</sup> q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X<sup>n</sup> i¼1

ðEi � PiÞ 2

<sup>n</sup> <sup>ð</sup>30<sup>Þ</sup>

� 100 ð31Þ

ð32Þ

result. The best fit was MBE = 0, and the formula is as follows:

MBE ¼

the Penman-Monteith method; and n is the total number of observations.

There is abundant precipitation in Taiwan. Its distribution, however, is uneven in both time and space. In addition to the significant precipitation difference between the wet season and dry season, the high mountains and steep slopes in Taiwan have insufficient reservoir storage as well as ET losses that collectively result in an extremely low amount of usable water. Water resource management could be achieved by accurately estimating ET to predict available water resources. In this study, the meteorological data recorded during the period of 1961–2013 by the Tainan weather station of Taiwan and provided by the Central Weather Bureau were considered (Figure 1). The collected meteorological parameters included temperature, wind speed, relative humidity, solar radiation, vapor pressure difference, daylight hours, and so on. Because the climatic factors that influenced ET might change with variation in the time scale, previous researches suggested that average monthly data would lead to a better result [24]. Therefore, this study used average monthly data for estimation.

Figure 1. Location of Tainan weather station.
