Global Sensitivity of Penman-Monteith Reference Evapotranspiration to Climatic Variables in Mato Grosso, Brazil

1. Introduction

Reference evapotranspiration (ET₀) is defined as the potential evapotranspiration of a hypothetical well-watered and actively growing grass surface of uniform height [1]. Understanding the mechanism of ET₀ is crucial in agricultural and hydrological studies and projects at local, regional, or global scales [2,3,4,5,6,7,8], as it plays an essential role in the hydrological cycle.

Given the importance of ET₀, different methods have been developed to model it, which, based on their fundamental factors, can be divided into three categories: mass transfer, radiation, and temperature equations [9,10,11,12,13,14]. The Penman-Monteith equation is a physical model that combines mass transfer and radiation methods and is the recommended methodology recommended by the FAO to estimate reference evapotranspiration [1].

The FAO-Penman-Monteith method is jointly determined by climatic variables such as downwards solar radiation (SRD), relative humidity (RH), maximum and minimum temperature (Tmax and Tmin), and wind speed (WS). For instance, affected by climate change, an increase in ET₀ can lead to an increase in the frequency, duration, and severity of drought [15,16], resulting in a lower water supply available in lakes, reservoirs, and groundwater [17,18]. However, in many hydrological and water resources climate change analyses only the changes in temperature and precipitation are investigated using temperature-based ET₀ estimation methods [19,20,21], which could lead to inaccuracies by not considering changes in solar radiation, wind speed, and relative humidity.

Sensitivity analysis are methodologies that allow for evaluating the relationships between the input and output variables of a system [22]. The methodology has been widely employed in environmental sciences, ecology, and hydrology, among others [23]. In studies related to evapotranspiration, sensitivity analysis enables the inspection of the impact that changes in climate variables have on the ET₀ fluctuation, as well as the identification of its dominant variables [24,25,26].

The sensitivity analysis methodologies are commonly divided into local sensitivity analysis (LSA) and global sensitivity analysis (GSA). Although local sensitivity analysis has been widely used to identify the influence of climatic variables on ET₀ [27,28,29,30,31], the LSA method has no self-verification and only reflects the local effects of individual variables [32]. On the other hand, global sensitivity analysis has the advantage of estimating the impact of all inputs and their combined effects on output changes [33,34]. The Sobol method is a GSA method based on variance decomposition analysis that provides a more robust performance than nonlinear models like the FAO-Penman-Monteith ET₀ [35,36]. Additionally, the Sobol method can assess the impact of uncertainties in the variables by encompassing their entire range of values, thus providing more information to quantify the influence of meteorological variables on ET₀.

Among the Brazilian territory, the state of Mato Grosso stands out for its vast territorial area, which encompasses three biomes: the Pantanal wetlands, savanna formations (Cerrado), and the Amazon rainforest, as well as two climate types: Aw climate (tropical savanna climate) and Cwa (tropical climate) [37]. The main economic activity in the state is agricultural production, with Mato Grosso being the largest grain producer in Brazil [38]. However, despite the high demand for water resources and the potential for expanding irrigated agriculture systems, there is still a lack of research focused on the impacts that changes in climate variables have on the fluctuation of evapotranspiration in the state of Mato Grosso [39,40].

Therefore, as it is critical to identify the dominant climatic variables that control ET₀ in the region to better plan the use of water resources, this study has three objectives: i) Analyzing the spatiotemporal variations in ET₀ and its related climatic variables (SRD, RH, Tmax, Tmin, and WS) in the state of Mato Grosso, Brazil; ii) Determining and discussing the sensitivity of ET₀ to the climatic variables; iii) Determining the dominant climatic variable attributed to ET₀ variability in the region.

2. Materials and Methods

2.1. Study Area and data

The state of Mato Grosso is located between the coordinates 06°00'S, 19°45'S and 50°06'W, 62°45'W and has a large territorial extension, with an area of 903,202,446 km², which represents 10.61% of the territory of Brazil. Climate-wise, the state has two well-defined seasons: the wet season, from October to April, and the dry season, from May to September. The total annual precipitation ranges from 1,200 to 2,000 mm, with higher levels in the north and east-north regions, as well as in areas with altitudes close to 800 m. The predominant climate is classified as Aw (Tropical savanna climate) and Cwa (tropical climate) according to Köppen classification [37].

The study used daily time-series data from 2008 to 2020 of the climatic variables: downwards solar radiation (SRD in MJ m^-2 day^-1), relative air humidity (RH in %), maximum air temperature (Tmax in °C), minimum air temperature (Tmin in °C), and mean wind speed at 2 meters (WS in m s^-1). These data were collected from 33 automatic weather stations (AWS) belonging to the National Institute of Meteorology (INMET), distributed throughout Mato Grosso, Brazil (Table 1, Figure 1).

2.2. Reference evapotranspiration (ET0) calculation

The FAO-Penman-Monteith (FAO-PM) is a standardized method recognized by the Food and Agriculture Organization for estimating reference evapotranspiration (ET₀). This method is based on physics and combines physiological and meteorological parameters (i.e., SRD, RH, Tmax, Tmin, and WS) to evaluate ET₀ at a hypothetical grass crop reference surface with an assumed crop height of 0.12 m, a fixed surface resistance of 70 s·m^-1, and an albedo of 0.23 [1]. In this study, we employ the widely used FAO-56 Penman-Monteith method to calculate ET₀ [1], which is formulated as follows:

$${\mathrm{ET}}_0=\frac{0.408\text{∆}\left({\mathrm{R}}_{\mathrm{n}}-\mathrm{G}\right)+\mathsf{\gamma }\left(\frac{900}{\mathrm{T}+273}\right)\mathrm{WS}\left(\mathrm{es}-\mathrm{ea}\right)}{\text{∆}+\mathsf{\gamma }\left(1+0.34\mathrm{WS}\right)},$$

(1)

where ET₀ = reference evapotranspiration (mm d^-1), Rn = net radiation at the crop surface (MJ·m^-2·day^-1), G = soil heat flux density at the soil surface (MJ·m^-2·day^-1), T = mean daily air temperature at 2 m height (mean value of Tmax and Tmin, ˚C), WS = wind speed at 2 m height (m·s^-1), es = saturation vapor pressure (kPa), ea = actual vapor pressure (kPa), Δ = slope of saturation vapor pressure versus air temperature curve (kPa·˚C^-1) and γ = psychometric constant (kPa·˚C^-1). The detailed calculations of Rn, Δ, γ and other parameters needed for computing ET₀ were obtained according to the procedure described in [1]. The G values were ignored for daily estimations purposes (G = 0 MJ m^-2 d^-1).

2.3. Sobol’s Sensitivity Analysis Method

Sobol's sensitivity analysis method [41] was utilized to evaluate the monthly sensitivity of PET to the five meteorological variables (SRD, RH, Tmax, Tmin, WS). The choice of Sobol's method was based on its ability to perform variance decomposition analysis and its superior performance in sensitivity analysis of nonlinear and nonmonotonic models [35,36,42]. Moreover, this method allows for examining the individual variables and their interactions and their impact on the model outputs [43]. The fundamental concept behind the Sobol's method is to decompose the model function f(X) (Equation 2) into increasing-dimensional summands, as illustrated in Equation 3:

$$\mathrm{Y}=\mathrm{f}\left(\mathrm{X}\right)=\mathrm{f}({\mathrm{X}}_1,{\mathrm{X}}_2,\text{⋯}{\mathrm{X}}_{\mathrm{d}}),$$

(2)

$$\mathrm{Y}={\mathrm{f}}_0+\sum _{\mathrm{i}=1}^{\mathrm{d}}{\mathrm{f}}_{\mathrm{i}}\left({\mathrm{X}}_{\mathrm{i}}\right)+\sum _{\mathrm{i}\text{<}\mathrm{j}}^{\mathrm{d}}{\mathrm{f}}_{\mathrm{ij}}\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{j}}\right)+\text{⋯}+{\mathrm{f}}_{1,2,\text{⋯},\mathrm{d}}\left({\mathrm{X}}_1,{\mathrm{X}}_1,\text{⋯},{\mathrm{X}}_{\mathrm{d}}\right),$$

(3)

where Y indicates the model function f₀ is the mean value and X are function variables. Considering that all the terms in the functional decomposition are orthogonal, the variance of model output Var(Y) can be decomposed as shown in Equations 4 - 6:

$$V\mathrm{ar}\left(\mathrm{Y}\right)=\sum _{\mathrm{i}=1}^{\mathrm{d}}{\mathrm{V}}_{\mathrm{i}}+\sum _{\mathrm{i}\text{<}\mathrm{j}}^{\mathrm{d}}{\mathrm{V}}_{\mathrm{ij}}+\text{⋯}+{\mathrm{V}}_{1,2,\text{⋯},\mathrm{d}},$$

(4)

$${\mathrm{V}}_{\mathrm{i}}={\mathrm{Var}}_{{\mathrm{X}}_{\mathrm{i}}}\left({\mathrm{E}}_{{\mathrm{X}}_{\stackrel{\mathrm{~}}{1}}}\left(\mathrm{Y}/{\mathrm{X}}_{\mathrm{i}}\right)\right),$$

(5)

$${\mathrm{V}}_{\mathrm{ij}}={\mathrm{Var}}_{{\mathrm{X}}_{\mathrm{ij}}}\left({\mathrm{E}}_{{\mathrm{X}}_{\stackrel{\mathrm{~}}{1\mathrm{j}}}}\left(\mathrm{Y}/{\mathrm{X}}_{\mathrm{ij}},{\mathrm{X}}_{\mathrm{j}}\right)\right)-{\mathrm{V}}_{\mathrm{i}}-{\mathrm{V}}_{\mathrm{j}},$$

(6)

where V_i is the contribution to Var(Y) due solely to the effect of X_i, and V_i,2,…,d is the contribution to Var(Y) due to interaction of {X₁, X₂,…, X_d}. Subsequently, the first-order index (S₁) (Equation 7), and total-sensitivity index (S_tot) (Equation 8) were calculated as:

$${\mathrm{S}}_1=\frac{{\mathrm{V}}_{\mathrm{i}}}{\mathrm{Var}\left(\mathrm{Y}\right)},$$

(7)

$${\mathrm{S}}_{\mathrm{tot}}=1-\frac{{\mathrm{Var}}_{{\mathrm{X}}_{\stackrel{\mathrm{~}}{\mathrm{i}}}}}{\mathrm{Var}\left(\mathrm{Y}\right)},$$

(8)

The variances in the above calculation process were estimated by approximate Monte Carlo numerical integration and the specific calculation details for quantifying the sensitivity can be found in [44]. This study uses quasi-random sequence sampling since it samples points more uniformly along with the Cartesian grids than uncorrelated random sampling [44]. The number of samples for the quasi-random Monte Carlo was set to 20000 monthly simulations in each AWS. The maximum and minimum range of the variables were set as the 2.5 and 97.5 percentiles of the daily observed data of each month per AWS (Supplementary: Tables S1–S5). We chose this approach to avoid setting the ranges based on outliers of the variable.

Since the minimum temperature in the same sample cannot be higher than the maximum temperature, this study adopts the approach suggested by [34,45]. Firstly, we calculate the mean and difference between Tmax and Tmin. Next, Sobol's sequence sampling is performed on the mean temperature and temperature difference. Finally, the minimum and maximum temperatures are restored from the generated samples.

3. Results

3.1. Climatic variables and Penman-Monteith ET0 spatiotemporal distribution

The monthly mean of Solar Radiation (SRD,) relative humidity (RH), maximum and minimum air temperature (Tmax, Tmin), wind speed (WS), and reference evapotranspiration (ET0) at the 33 AWS in Mato Grosso, Brazil are represented by the boxplots in Figure 2. All variables follow a seasonal pattern between wet and dry seasons. Tmax and WS reach their highest mean at the end of the dry season (38.4 ± 1.3 ℃ and 1.3 ± 0.5 m s^-1) while their lowest mean happens at the end of the wet season (33.8 ± 1.3 ℃ and 0.9 ± 0.3 m s^-1). Tmin and RH mean ranges from 14.6 ± 1.0 ℃ to 20.0 ± 1.0 ℃, and 52.6 ± 7.1% to 83.0 ± 2.5%, with a maximum in the wet season and minimum in the dry season. The lowest mean SRD is during the beginning of the dry season (16.2 ± 2.2 MJ m^-2 d^-1) and gradually increases until the end of the dry season when it reaches its highest mean (19.1 ± 1.4 MJ m^-2 d^-1). The beginning of the dry season also exhibits the lowest ET₀ (3.5 ± 0.3 mm d^-1). The evapotranspiration demand increases during the dry period reaching its maximum in September (5.0 ± 0.6 mm d^-1) and remaining relatively constant during the wet season (about 4.1 ± 0.3 mm d^-1).

The spatial distribution of evapotranspiration and its climatic variables on an annual and seasonal scale is displayed in Figure 3. The results show a zonal variation that follows the distribution of biomes and topography in Mato Grosso. The Amazon region of the state exhibits the lowest values of SRD, Tmax, WS, and ET₀, and the highest values of Tmin and RH. In the Cerrado region, the influence of topography can also be observed in higher altitude areas (southeast part of the state), where there is a decrease in SRD, Tmax, Tmin, and ET₀ compared to the rest of the biome in the state. The spatial distribution of the variables in the state is less evident during the wet season, as there is homogeneity.

in the means of SRD, RH, and Tmin between the two biomes of Mato Grosso.

3.2. Sobol sensitivity coefficients spatiotemporal distribution

Figure 4 shows the variation of sensitivity indices for the meteorological variables of ET₀ obtained from the 33 EMAs on an annual and seasonal time scale. Figure 5 presents the monthly variation of these sensitivity indices. Based on the method proposed by [44], the sensitivity indices of the first order (S₁) and total (S_tot) can be classified into three levels of sensitivity: insensitive (S < 0.01), sensitive (S ≥ 0.01), and highly sensitive (S ≥ 0.1). Thus, among the five meteorological variables, ET₀ is highly sensitive to SRD, RH, and WS; sensitive to Tmax; and insensitive to Tmin.

On the annual scale, the sensitivity indices of ET₀ for each variable generally followed the following order: SRD > WS > RH > Tmax > Tmin. Solar radiation (SRD) accounted for the largest variance, representing at least 59% of the total and first-order sensitivity indices during the dry period and reaching an average of 91% during the months of the rainy season. RH and WS explained approximately 8% and 11% of the S_tot, and 6% and 9% of the S₁ variation on an annual scale, respectively. RH and WS also alternated as the second most influential variable during the seasonal periods, with RH dominating during the rainy season and WS during the dry season. The temperature variables had a lesser influence on ET₀ in Mato Grosso. Throughout the year, the sensitivity indices of Tmax ranged from 3% to 10%, while Tmin reached a maximum value of approximately 1%.

The spatial distributions of the annual and seasonal averages of S₁ and S_tot are presented in Figure 6. The results indicate that, in addition to the seasonal behavior, the coefficients exhibit zonal characteristics that resemble the distribution of biomes in the state of Mato Grosso. Although SRD is the dominant variable in determining ET₀ in the state, the influence of radiation differs in the Amazon and Cerrado biomes. While in most of the Amazon region, the sensitivity indices of SRD are on average higher than 80%, in the Cerrado biome, the sensitivity of ET₀ to radiation ranges from 40% to 80%, indicating that RH, WS, and Tmax also influence ET₀ fluctuations in the Cerrado.

During the dry season of the Cerrado biome, when the sensitivity indices of SRD are around 50%, it is possible to observe that the second dominant variable varies according to the spatial distribution. In the southeast of the Cerrado, RH becomes the second dominant variable, accounting for 20% to 30% of the ET₀ variation. In the northeastern region of the Cerrado, Tmax becomes the second dominant variable, with approximately 20% to 30% influence. In the central region of the Cerrado, the second dominant variable is WS, with sensitivity indices ranging from 30% to 40%.

4. Discussion

Understanding the causes of ET₀ variations is essential for water resources management and agriculture in Mato Grosso. The results of this study have highlighted the high sensitivity of Penman-Monteith ET₀ to solar radiation in the study area. Overall, the sensitivity of ET₀ to the five meteorological variables in the state can be ranked as follows: SRD > WS ≥ RH > Tmax > Tmin. Although SRD is the dominant climatic variable, explaining 60% to 90% of the ET0 variability throughout the year, the sensitivity of ET₀ the climatic variables, similar to findings in the literature, showed strong indications of spatial variability (Figure 4 6; [25,47,48], with RH playing an important role in determining the spatial and seasonal variations of ET₀ sensitivity indices in Mato Grosso.

In the Mato Grosso Amazon biome portion, higher and relatively constant percentages of relative humidity are observed throughout the year, along with lower values and greater fluctuation of incident radiation measurements (Figure 4 3). The main cause for this pattern is that the Amazon region is characterized as a humid tropical area where convective activity forces the upward movement of moisture, forming a thicker cloud cover compared to the Cerrado region of Mato Grosso [49]. Consequently, in this biome, not only there is a greater attenuation of solar radiation, but there is also a greater reduction in temperature amplitude. This combination of factors could explain why solar radiation is the limiting factor in the evapotranspiration process in the Amazon biome.

These results are consistent with those found in other humid subtropical and tropical areas, such as in the forests of China [48,50], where it is reported that changes in incident solar radiation are the main driving force for ET₀ variability. Similar behavior can also be observed in the Cerrado region of Mato Grosso during the months of the wet season when relative humidity generally exceeds 80%. During this season, maximum temperature and temperature amplitude are lower, and cloud cover increases due to the predominant circulation of the continental equatorial mass (mEc) and the South Atlantic Convergence Zone (ZCAS), which transport moisture from the Amazon to the Cerrado regions (Figure 6) [49,51,52], impacting the solar radiation sensitivity indices.

During the dry season in the Cerrado biome region, the spatial distribution of sensitivity indices indicates three areas of influence analogous to the subdivision of the biome proposed by [51]: Cerrado Meridional (southeastern region of the Mato Grosso Cerrado), Cerrado Central, and Cerrado Setentrional (northeastern region of the Mato Grosso Cerrado). In the Cerrado Meridional region, the occasional incursion of air circulation masses from the polar mass (mP), combined with the topography, leads to increased cloudiness and lower average incident radiation (Figure 6) [52,53,54]. However, due to lower averages and greater fluctuations in relative humidity (RH) compared to the Amazon biome, an increase in RH sensitivity indices is observed. In the Cerrado Central and Setentrional regions, the reduction in atmospheric humidity caused by the tropical Atlantic mass and the flat topography creates clearer sky conditions [49,51,52,54], thereby reducing fluctuations in incident solar radiation. In this scenario, the increase in sensitivity indices for the variables is linked to their ability to increase water retention in the atmosphere. Therefore, in the Cerrado Setentrional region, where Tmax reaches the highest values in the state, Tmax has greater sensitivity than RH and WS, as it allows for an increase in the saturation vapor pressure deficit. On the other hand, in the Cerrado Central region, where Tmax is lower and WS is higher, wind speed has a greater influence due to disturbances and movement in the atmospheric boundary layer.

These results are consistent with those found by [36,48,55,56] which indicate that atmospheric air circulation directly influences the spatial patterns of global ET0 sensitivity. These authors also agree that high cloudiness and relative humidity conditions are the main factors that increase the sensitivity indices of radiation, while clear sky conditions and low relative humidity increase the sensitivity indices of wind speed. Thus, in arid and semi-arid regions, ET₀ is more sensitive to changes in SRD, WS, and RH variables.

The temperature variables (Tmax and Tmin) generally accounted for less than 10% of ET₀. Although higher temperatures, resulting in reduced RH and increased energy supply to the process, tend to increase ET₀, the effects of Tmax and Tmin on ET₀ sensitivity, as observed in studies by [36,48,55,57], were almost insignificant compared to other factors.

The Sobol method has provided valuable insights for understanding the impact of climatic variables on ET₀. Based on our results, caution should be exercised when making future ET₀ estimates in Mato Grosso state using formulations that do not account for changes in solar radiation, wind speed, and relative humidity. It is crucial to pay special attention to the estimation of Rs, as it is the dominant variable in Penman-Monteith ET₀ calculations. Furthermore, in the Mato Grosso Cerrado region, particularly during the dry period, the interactions between relative humidity and wind speed cannot be disregarded when determining evapotranspiration using the Penman-Monteith method.

5. Conclusions

The global sensitivity of ET₀ estimated by the Penman-Monteith method to meteorological variables (solar radiation downward- SRD, relative humidity - RH, maximum air temperature - Tmax, minimum air temperature - Tmin, and wind speed at 2 meters – WS) exhibits seasonal and spatial distribution in Mato Grosso state.

Regardless of the period, the dominant variable for evapotranspiration in Mato Grosso is solar radiation (SRD), which has the greatest influence during the rainy season and in humid regions such as the Amazon. The other variables that have a secondary impact on sensitivity follow the ranking of WS > RH > Tmax > Tmin.

ET₀ estimated by the Penman-Monteith method is most sensitive to changes in WS, RH, and Tmax in arid and semi-arid environments, such as the Cerrado during the dry season. However, the sensitivity to RH and WS is higher in the Meridional and Central regions of the Cerrado biome. In contrast, in the Setentrional region, the secondary variable with the highest sensitivity is Tmax.

The meteorological variable Tmin does not significantly influence ET₀ estimated by the Penman-Monteith method in the Mato Grosso state.

The estimation of ET₀ using the Penman-Monteith method for the Mato Grosso state needs to be more accurate and precise data on incident solar radiation (SRD), whether obtained from observed data series at meteorological stations or estimated through gap-filling methods.