1. Introduction

The scarcity and the deficient quality of the discharge data are common problems in hydrological modeling. In fact, most of the river basins across the world are ungauged or poorly gauged, without in situ monitoring for the most relevant hydroclimatic variables [1], with emphasis on river discharges. Such whole spectra of cases are embraced now under the term "ungauged basins" meaning catchments where meteorological data or river flow, or both, is not measured [2]. The prediction in ungauged basins (PUB) is so relevant that in 2003, the International

**92**

*Current Practice in Fluvial Geomorphology - Dynamics and Diversity*

[1] Yilmaz L. Maximum entropy theory by using the meandering morphological investigation-II. Journal of RMZ-Materials and Geoenvironment. 2007

[2] Moisy F. 2008. Available from: https://www.math.dartmouth.edu// archive/m53f09/public\_html/proj/

[3] Normant F, Tricot G. Fractals in Engineering. 1995. Available from: https://www.google.com/search?q=Nor mant+and+Triart&tbm=isch&source= univ&sa=X&ved=2ahUKEwjgurqA5O\_ hAhVSyqQKHeK7ChAQsAR6BAgJEAE

Alexis\_writeup.pdf

&biw=1366&bih=6

2015;**80**(1):1-13

[4] Jiang B, Yin J. Ht-index for quantifying the fractal or scaling structure of geographic features. Annals of the Association of American Geographers. 2014;**104**(3):530-541

[6] Fractal Dimension and Self-

[7] Sierpinski S. 2002. Available from: https://www.mathsisfun.com/ sierpinski-triangle.html; https:// www mathworld.wolfram.com/

[8] Longley PA, Batty M. Spatial Analysis: Modelling in a GIS

https://books.google.com.tr/ books?isbn=0470236159

Environment. 1996. Available from:

SierpinskiSieve.html

[5] Jiang B. Geospatial analysis requires a different way of thinking: The problem of spatial heterogeneity. GeoJournal.

Similarity. Available from: https://www. math.dartmouth.edu//archive/m53f09/ public\_html/proj/Alexis\_writeup.pdf

Association of Hydrological Sciences (IAHS) launched an initiative for 10 years aiming at contributing to shift the scientific culture of hydrology toward improved scientific understanding of hydrological processes so that data scarcity or unavailability could be overcome [3].

moisture storage capacity. These models proved to be able to estimate monthly runoff [12] and provided the basis of many other two-parameter hydrological models [13, 14]. Several studies have shown that many models produce similar results to those simpler previous ones, e.g., [15–18]. The more recent water balance model of Temez model [19] also stands out among the available simplest models. Although the Thornthwaite-Mather model is quite old, its recurrent use over time for different water management issues in different hydrological environments and the fact that many recent studies continue to adopt such approaches demon-

Effect of the Evapotranspiration of Thornthwaite and of Penman-Monteith in the Estimation…

As for the second one, it has been widely used in Spanish catchments [24]. However, both methods make use of potential evapotranspiration evaluation, which requires records of climatologic variables that are not usually readily avail-

In this context, the present study aims at understanding the role of different

streamflows obtained via the Thornthwaite-Mather monthly water balance model. The two models for the evapotranspiration were the Thornthwaite and the Penman-Monteith models. The former is recognizably simple since it only makes use of average monthly temperatures. Conversely, the latter requires records of several climatologic variables, which, in practical terms, makes it much more restrictive. The validity of the results obtained is restricted to Mainland Portugal, which means that there is an opportunity for additional research aiming at expanding the analysis

procedures to compute the evapotranspiration in the estimates of monthly

2. The potential evapotranspiration: the water balance model

EVP <sup>¼</sup> <sup>16</sup> � <sup>10</sup> Tmed

tion of EV0 (mm/day) for a given place can use Eq. (2):

Potential evapotranspiration (EVP) is the process of water transfer from the soil to the atmosphere, either directly or through the plants when the water required for such process is fully available. Potential and actual evapotranspiration are very rarely measured due to the complex, expensive, and hard methodology required (e.g., percolation gauge, weighing lysimeter). Several methods and models are available for indirect evaluation, such as temperature-based methods [10, 25, 26] and radiation-based methods [27] or combined methods, as the well-known

According to Thornthwaite [10], the EVP (mm/month) for 1 month with Nd

I

where Tmed is the average air temperature (°C) in that month; I is an annual heat index which depends on the monthly heat indexes which, in turn, are function of the average air temperatures along the several months of the year, each with Nd number of days; α is an exponent which also depends on I; and N/12 is the astronomical day expressed in 12 h units of a 30-day month at the latitude where EVP is

The Penman-Monteith method yields to the potential evapotranspiration for a soil wholly covered by a reference culture (grass in active growth, with uniform height, and free of water supply limitations) [29], and, for this reason, this evapotranspiration is frequently called reference evapotranspiration, EV0. The calcula-

�

N=12 � Nd

<sup>30</sup> (1)

<sup>α</sup>

strate its current effectiveness, e.g., [20–23].

DOI: http://dx.doi.org/10.5772/intechopen.88441

able, except for rainfall and temperature.

Penman-Monteith method [28].

days is given by Eq. (1):

to be calculated.

95

and its conclusions to other hydrological environments.

As the majority of worldwide basins [3], the Portuguese ones are also ungauged or poorly gauged. In fact, in Portugal, the systematic measurement of its river discharges started much later than that of other hydrometeorological variables resulting in much sparser monitoring networks with much more recent records. Additionally, most of the discharge data series thus acquired have recurrent gaps, either sporadic or for long periods, thereby limiting their use for both scientific hydrological studies and design purposes. These circumstances bring forward the need to apply hydrological models aiming not only at filling in the gaps in the records or at increasing their time spans but also at estimating discharges at ungauged catchments. Therefore, hydrological modeling can be looked as a tool to solve the problem of the lack of river discharge data or of the poor quality of the existing data.

However, almost regardless of the purpose of a hydrological model, its applicability depends on its simplicity and on the compatibility between its data requirements and the available information.

The hydrological models may have different degrees of complexity [4]. One of the disadvantages of using more complex hydrological models is that they require more data and more considerable effort in parameterization.

Greater complexity, however, does not mean that a model is better. According to Jakeman et al. [5], some simple models have performed more complex (or better than) alternatives in some cases. Over-parameterization and a lack of appropriate data for parameterization are a concern with complex models [6]. Fewer parameters that only treat the fundamental processes—a parsimonious approach to modeling—could sometimes be a better conceptualization of reality [7].

As a guiding principle, a relatively simple model is likely to be required if there are limited data. The simplest model that can be usefully applied is one that captures only those factors that are critical to the processes under study [8]. This was also stressed by Hillel [9], who stated the following principles of a model development:


The previous constraints make the water balance approach most suitable for many hydrological purposes, including those related to the improvement of the quality/length of the discharge series.

The first water balance model based exclusively on rainfall and temperature was developed in the 1940s by Thornthwaite [10] and later revised by Thornthwaite and Mather [11]. They proposed two different conceptual models based on two parameters: soil moisture capacity and water excess above the maximum soil

#### Effect of the Evapotranspiration of Thornthwaite and of Penman-Monteith in the Estimation… DOI: http://dx.doi.org/10.5772/intechopen.88441

moisture storage capacity. These models proved to be able to estimate monthly runoff [12] and provided the basis of many other two-parameter hydrological models [13, 14]. Several studies have shown that many models produce similar results to those simpler previous ones, e.g., [15–18]. The more recent water balance model of Temez model [19] also stands out among the available simplest models.

Although the Thornthwaite-Mather model is quite old, its recurrent use over time for different water management issues in different hydrological environments and the fact that many recent studies continue to adopt such approaches demonstrate its current effectiveness, e.g., [20–23].

As for the second one, it has been widely used in Spanish catchments [24]. However, both methods make use of potential evapotranspiration evaluation, which requires records of climatologic variables that are not usually readily available, except for rainfall and temperature.

In this context, the present study aims at understanding the role of different procedures to compute the evapotranspiration in the estimates of monthly streamflows obtained via the Thornthwaite-Mather monthly water balance model. The two models for the evapotranspiration were the Thornthwaite and the Penman-Monteith models. The former is recognizably simple since it only makes use of average monthly temperatures. Conversely, the latter requires records of several climatologic variables, which, in practical terms, makes it much more restrictive. The validity of the results obtained is restricted to Mainland Portugal, which means that there is an opportunity for additional research aiming at expanding the analysis and its conclusions to other hydrological environments.

## 2. The potential evapotranspiration: the water balance model

Potential evapotranspiration (EVP) is the process of water transfer from the soil to the atmosphere, either directly or through the plants when the water required for such process is fully available. Potential and actual evapotranspiration are very rarely measured due to the complex, expensive, and hard methodology required (e.g., percolation gauge, weighing lysimeter). Several methods and models are available for indirect evaluation, such as temperature-based methods [10, 25, 26] and radiation-based methods [27] or combined methods, as the well-known Penman-Monteith method [28].

According to Thornthwaite [10], the EVP (mm/month) for 1 month with Nd days is given by Eq. (1):

$$\text{EVP} = \left[ \mathbf{16} \times \left( \mathbf{10} \frac{\text{Tmed}}{\text{I}} \right)^{a} \right] \times \left[ \frac{\text{N} / \text{12} \times \text{N}\_{\text{d}}}{\text{30}} \right] \tag{1}$$

where Tmed is the average air temperature (°C) in that month; I is an annual heat index which depends on the monthly heat indexes which, in turn, are function of the average air temperatures along the several months of the year, each with Nd number of days; α is an exponent which also depends on I; and N/12 is the astronomical day expressed in 12 h units of a 30-day month at the latitude where EVP is to be calculated.

The Penman-Monteith method yields to the potential evapotranspiration for a soil wholly covered by a reference culture (grass in active growth, with uniform height, and free of water supply limitations) [29], and, for this reason, this evapotranspiration is frequently called reference evapotranspiration, EV0. The calculation of EV0 (mm/day) for a given place can use Eq. (2):

Association of Hydrological Sciences (IAHS) launched an initiative for 10 years aiming at contributing to shift the scientific culture of hydrology toward improved

or poorly gauged. In fact, in Portugal, the systematic measurement of its river discharges started much later than that of other hydrometeorological variables resulting in much sparser monitoring networks with much more recent records. Additionally, most of the discharge data series thus acquired have recurrent gaps, either sporadic or for long periods, thereby limiting their use for both scientific hydrological studies and design purposes. These circumstances bring forward the need to apply hydrological models aiming not only at filling in the gaps in the records or at increasing their time spans but also at estimating discharges at ungauged catchments. Therefore, hydrological modeling can be looked as a tool to solve the problem of the lack of river discharge data or of the poor quality of the

As the majority of worldwide basins [3], the Portuguese ones are also ungauged

However, almost regardless of the purpose of a hydrological model, its applicability depends on its simplicity and on the compatibility between its data require-

The hydrological models may have different degrees of complexity [4]. One of the disadvantages of using more complex hydrological models is that they require

Greater complexity, however, does not mean that a model is better. According to Jakeman et al. [5], some simple models have performed more complex (or better than) alternatives in some cases. Over-parameterization and a lack of appropriate data for parameterization are a concern with complex models [6]. Fewer parameters that only treat the fundamental processes—a parsimonious approach to modeling—could sometimes be a better conceptualization of reality [7].

As a guiding principle, a relatively simple model is likely to be required if there are limited data. The simplest model that can be usefully applied is one that captures only those factors that are critical to the processes under study [8]. This was also stressed by Hillel [9], who stated the following principles of a model development:

• Parsimony—the model should not be more complex than the required data and should include the smallest possible number of parameter with values to be

• Modesty—a model should not intend to do "too much"; there is no such thing

• Precision—a model should not intend to describe a phenomenon with precision

• Verifiability—a model must be verifiable, and it is always necessary to know its

The previous constraints make the water balance approach most suitable for many hydrological purposes, including those related to the improvement of the

The first water balance model based exclusively on rainfall and temperature was developed in the 1940s by Thornthwaite [10] and later revised by Thornthwaite and Mather [11]. They proposed two different conceptual models based on two parameters: soil moisture capacity and water excess above the maximum soil

more data and more considerable effort in parameterization.

scientific understanding of hydrological processes so that data scarcity or

Current Practice in Fluvial Geomorphology - Dynamics and Diversity

unavailability could be overcome [3].

ments and the available information.

computed from the data.

higher than the capacity to measure it.

as "the model".

limits of validity.

94

quality/length of the discharge series.

existing data.

Current Practice in Fluvial Geomorphology - Dynamics and Diversity

$$\text{EVO} = \frac{\mathbf{0.408\Delta(R\_n - g)} + \gamma \frac{900}{\text{Tmed} + 273} \mathbf{v\_2(e\_a - e\_d)}}{\Delta + \gamma (1 + \mathbf{0.34U\_2})} \tag{2}$$

around 70 (in the north) to 85% (in the south) of the annual precipitation occurs in

Effect of the Evapotranspiration of Thornthwaite and of Penman-Monteith in the Estimation…

The seasonal variability of the precipitation is due to the characteristics of the general circulation of the atmosphere and to regional climate factors, related to Portugal's geographic location, in the south-westerly extreme of the Iberian Peninsula (between 37° and 42°N and 6.5° and 9.5°W). The North Atlantic Oscillation (NAO) and other teleconnection indexes at the synoptic and smaller scales explain the interannual variability [38]. In terms of spatial variability, the mean annual precipitation varies from more than 2800 mm, in the northwestern region, to less than 400 mm, in the southern region, following a complex spatial pattern (N–S/E–W), in close connection with the relief, far beyond the most

Figure 1 shows the schematic location of the 16 climatological stations used in the study over a mean annual flow depth map (H in mm/year). The figure shows that the southern and the more north-eastern regions are characterized by water scarcity (rarely exceeding in average 150–200 mm/year) and that only in the center/north western region there is some surface water availability. The mean annual

The climatological stations were selected aiming at representing the different prevailing hydrological regimes in Portugal and especially at ensuring a common period with all the records required by the application of the Penman-Monteith, which in Portugal is not easy to get. The records at the previous stations were obtained from the Portuguese Institute for the Ocean and Atmosphere (IPMA), which has high data quality standards and is one of the main sources of Portuguese hydrological and hydrometeorological and also from the database AGRIBASE from the Instituto Superior de Agronomia (ISA), the School of Agriculture of the Lisbon University. Although the periods for which it was possible to obtain the required data are not very recent, this has no influence on the purpose of the study.

Some general characteristics of the previous stations are presented in Table 1 along with the mean monthly values of the following variables, computed based on

values of the precipitation and of the surface runoff over the country are

the wet semester—from October to March.

DOI: http://dx.doi.org/10.5772/intechopen.88441

determinant factor of the precipitation spatial pattern.

approximately 960 and 385 mm, respectively.

Figure 1.

97

Location of the climatological stations of Table 1.

where Tmed is the average air temperature (°C), Δ is the slope of the saturation vapor pressure temperature relationship (k Pa°C�<sup>1</sup> ), Rn is the net solar radiation (MJ m�<sup>2</sup> d�<sup>1</sup> ), g is the soil heat flux (MJ m�<sup>2</sup> d�<sup>1</sup> ), γ is the psychrometric constant (k Pa°C�<sup>1</sup> ), v2 is the mean wind velocity 2 m above the ground (ms�<sup>1</sup> ), ea is the vapor saturation tension at temperature T (kPa), and ed is the actual vapor tension (kPa). The calculation of some of the previous variables, besides its complexity, may also require the average maximum and average minimum air temperatures, the average air relative humidity, and the global solar radiation.

Thornthwaite method seems to underestimate the potential evapotranspiration in Mainland Portugal [30, 31], while the Penman-Monteith method tends to overestimate it [29]. Its results are, however, more satisfactory in a large number of different climatic, timescale, and location constraints [29].

The Thornthwaite-Mather water balance model applies the mass equation along time to an element of the terrestrial phase of the hydrologic cycle by calculating the water fluxes "entering" that element, those "leaving" it, and the variations in the water storage within that same element [10, 32–34] according to

$$\mathbf{P} = \mathbf{S} + \mathbf{EVA} + \Delta \mathbf{S} \tag{3}$$

where, for a given time interval, P is the rainfall, S is the water excess or superavit, EVA is the actual evapotranspiration, and ΔS is the water storage variation (all variables expressed in the same units).

The previous water balance model does not consider the heterogeneity of the watershed, the deep infiltration, and the complexity of the water movements (either on the surface or in the ground). In addition, it does not consider that surface runoff occurs whenever the rainfall intensity exceeds the infiltration rate. Despite these simplifications, it may be considered that the water excess or superavit, S, represents the upper limit of the surface runoff.

Within these conditions, the water balance model can be used to estimate the river discharges. In order to do so and after assigning to the soil a maximum useable water storage capacity (Smax), the model assumes that, as long as there is water availability (either storage in the ground or from the rainfall), the actual evapotranspiration rate is equal to the potential evapotranspiration; otherwise, it will occur at a lower rate. Furthermore, it also assumes that there is no onset of surface runoff if the capacity to store water in the soil is not filled up, even if the rainfall intensity exceeds the infiltration rate. The amount of water in the soil in the months where rainfall is lower than evapotranspiration can be calculated by Mendonça [35]:

$$\mathbf{AS}\_{\mathbf{i}} = \mathbf{AS}\_{\mathbf{i}-1} e^{\mathbf{L}\_{\mathbf{i}}/\text{Smax}} \tag{4}$$

where AS (mm) represents the water in the soil in month identified by the index, Smax is maximum useable water storage capacity, and Li (mm) is the water potential loss (i.e., the difference between the rainfall and the potential evapotranspiration) accumulated since the onset of the dry period up to month i.

#### 3. Case studies and data

The precipitation regime in Portugal is highly irregular both in space and in time and, in this last case, either within the year or among the years [36, 37]. On average,

### Effect of the Evapotranspiration of Thornthwaite and of Penman-Monteith in the Estimation… DOI: http://dx.doi.org/10.5772/intechopen.88441

around 70 (in the north) to 85% (in the south) of the annual precipitation occurs in the wet semester—from October to March.

The seasonal variability of the precipitation is due to the characteristics of the general circulation of the atmosphere and to regional climate factors, related to Portugal's geographic location, in the south-westerly extreme of the Iberian Peninsula (between 37° and 42°N and 6.5° and 9.5°W). The North Atlantic Oscillation (NAO) and other teleconnection indexes at the synoptic and smaller scales explain the interannual variability [38]. In terms of spatial variability, the mean annual precipitation varies from more than 2800 mm, in the northwestern region, to less than 400 mm, in the southern region, following a complex spatial pattern (N–S/E–W), in close connection with the relief, far beyond the most determinant factor of the precipitation spatial pattern.

Figure 1 shows the schematic location of the 16 climatological stations used in the study over a mean annual flow depth map (H in mm/year). The figure shows that the southern and the more north-eastern regions are characterized by water scarcity (rarely exceeding in average 150–200 mm/year) and that only in the center/north western region there is some surface water availability. The mean annual values of the precipitation and of the surface runoff over the country are approximately 960 and 385 mm, respectively.

The climatological stations were selected aiming at representing the different prevailing hydrological regimes in Portugal and especially at ensuring a common period with all the records required by the application of the Penman-Monteith, which in Portugal is not easy to get. The records at the previous stations were obtained from the Portuguese Institute for the Ocean and Atmosphere (IPMA), which has high data quality standards and is one of the main sources of Portuguese hydrological and hydrometeorological and also from the database AGRIBASE from the Instituto Superior de Agronomia (ISA), the School of Agriculture of the Lisbon University. Although the periods for which it was possible to obtain the required data are not very recent, this has no influence on the purpose of the study.

Some general characteristics of the previous stations are presented in Table 1 along with the mean monthly values of the following variables, computed based on


Figure 1. Location of the climatological stations of Table 1.

EV0 <sup>¼</sup> <sup>0</sup>:408Δð Þþ Rn � <sup>g</sup> <sup>γ</sup> <sup>900</sup>

vapor pressure temperature relationship (k Pa°C�<sup>1</sup>

), g is the soil heat flux (MJ m�<sup>2</sup> d�<sup>1</sup>

Current Practice in Fluvial Geomorphology - Dynamics and Diversity

average air relative humidity, and the global solar radiation.

different climatic, timescale, and location constraints [29].

variation (all variables expressed in the same units).

superavit, S, represents the upper limit of the surface runoff.

than evapotranspiration can be calculated by Mendonça [35]:

3. Case studies and data

96

ASi ¼ ASi�<sup>1</sup>e

where AS (mm) represents the water in the soil in month identified by the index, Smax is maximum useable water storage capacity, and Li (mm) is the water potential loss (i.e., the difference between the rainfall and the potential evapotranspiration) accumulated since the onset of the dry period up to month i.

The precipitation regime in Portugal is highly irregular both in space and in time and, in this last case, either within the year or among the years [36, 37]. On average,

water storage within that same element [10, 32–34] according to

(MJ m�<sup>2</sup> d�<sup>1</sup>

(k Pa°C�<sup>1</sup>

Tmedþ<sup>273</sup> v2ð Þ ea � ed

P ¼ S þ EVA þ ΔS (3)

Li=Smax (4)

), Rn is the net solar radiation

), γ is the psychrometric constant

(2)

), ea is the

Δ þ γð Þ 1 þ 0:34U2

where Tmed is the average air temperature (°C), Δ is the slope of the saturation

), v2 is the mean wind velocity 2 m above the ground (ms�<sup>1</sup>

in Mainland Portugal [30, 31], while the Penman-Monteith method tends to overestimate it [29]. Its results are, however, more satisfactory in a large number of

where, for a given time interval, P is the rainfall, S is the water excess or superavit, EVA is the actual evapotranspiration, and ΔS is the water storage

The previous water balance model does not consider the heterogeneity of the watershed, the deep infiltration, and the complexity of the water movements (either on the surface or in the ground). In addition, it does not consider that surface runoff occurs whenever the rainfall intensity exceeds the infiltration rate. Despite these simplifications, it may be considered that the water excess or

Within these conditions, the water balance model can be used to estimate the river discharges. In order to do so and after assigning to the soil a maximum useable water storage capacity (Smax), the model assumes that, as long as there is water availability (either storage in the ground or from the rainfall), the actual evapotranspiration rate is equal to the potential evapotranspiration; otherwise, it will occur at a lower rate. Furthermore, it also assumes that there is no onset of surface runoff if the capacity to store water in the soil is not filled up, even if the rainfall intensity exceeds the infiltration rate. The amount of water in the soil in the months where rainfall is lower

vapor saturation tension at temperature T (kPa), and ed is the actual vapor tension (kPa). The calculation of some of the previous variables, besides its complexity, may also require the average maximum and average minimum air temperatures, the

Thornthwaite method seems to underestimate the potential evapotranspiration

The Thornthwaite-Mather water balance model applies the mass equation along time to an element of the terrestrial phase of the hydrologic cycle by calculating the water fluxes "entering" that element, those "leaving" it, and the variations in the


#### Table 1.

Climatological stations. General features and mean monthly values of precipitation, P; mean, average maximum and average minimum air temperature,Tmed,Tmax and Tmin, respectively; air relative humidity, HR; number of sunny hours, I; and wind velocity 2 m above the ground, v.

Figure 2. Mean monthly values of P,Tmed,Tmax,Tmin, HR, I, and v, according to Table 1.

the data provided: precipitation, P; mean, average maximum and average minimum air temperatures, Tmed, Tmax and Tmin, respectively; air relative humidity, HR; number of sunshine hours, I; and wind velocity, v. In Figure 2, the corresponding mean monthly values are represented. The recording periods of Table 1 refer to hydrological years, which in Portugal start on October 1.

Figure 2 shows that from the north to the south of Mainland Portugal, the precipitation decreases and the temperature and the sunshine hours increase.

#### 4. Results

Based on the records of Table 1, the potential evapotranspirations of Thornthwaite (EVP) and Penman-Monteith (EV0) were computed, as well as the surface flows that they predict according to the Thornthwaite-Mather water balance model—Eqs. (3) and (4).

i. The values of the potential evapotranspiration of Thornthwaite (EVP) are systematically lower than those of the evapotranspiration of Penman-Monteith

Monthly potential evapotranspiration of Thornthwaite, EVP, and of Penman-Monteith, EV0. Linear regression line (continuous line), equation, and correlation coefficient, r. Line representing the equality between

Effect of the Evapotranspiration of Thornthwaite and of Penman-Monteith in the Estimation…

DOI: http://dx.doi.org/10.5772/intechopen.88441

(EV0), thus confirming the previous knowledge for Portugal [29]; the

ii. The correlation between EV0 and EVP is always very high (0.9 or higher), which, under scarcity of data, suggests the possibility to estimate EV0 based

The first conclusion anticipates that the application of Thornthwaite-Mather water balance model would yield rather distinct estimates of the surface runoff

on EVP.

monthly EVP and EV0 (dashed line).

Figure 3.

99

when based upon on EVP or on EV0.

differences between those values increase as the evapotranspirations increase.

In Figure 3, the Thornthwaite (EVP) and Penman-Monteith (EV0) monthly potential evapotranspirations are compared for each of the 16 climatologic stations. Each diagram contains the representation of the straight line from the linear regression analysis between EVP and EV0, its equation, and the respective correlation coefficient. There is also a second dashed straight line representing the equality between the two evapotranspirations under consideration. The scale of the axes is always the same in order to allow the comparison between those evapotranspirations and among the results from the different climatological stations. Figure 3 clearly highlights two relevant conclusions:
