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

^{1}

^{2}

^{*}

## Abstract

**:**

_{tot}), which has a more significant impact in humid environments (70% to 90% of S

_{tot}), as observed in the areas of the Amazon biome in the state. Air relative humidity and wind speed have higher sensitivity indices during the dry season in the Cerrado biome (savanna) areas in Mato Grosso (20% and 30% of the S

_{tot}, respectively). Our findings show that changes in solar radiation, relative humidity, and wind speed are the main driving forces that impact the reference evapotranspiration.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}, 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 1200 to 2000 mm, with higher levels in the north and east-north regions and in areas with altitudes close to 800 m. The predominant climate is classified as Aw (tropical savanna climate) and Cwa (tropical climate) according to the Köppen classification [42].

^{−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 m (WS in m s

^{−1}). These data were collected from 33 automatic weather stations (AWSs) belonging to the National Institute of Meteorology (INMET), distributed throughout Mato Grosso, Brazil (Figure 1, Supplementary Table S1). Given that the wind speed variable in the INMET AWS is obtained at 10 m, the average wind speed at 2 m was calculated using Equation (1), as proposed by [1].

_{2}—is the wind speed at 2 m (m s

^{−1}), WS

_{10}—is the wind speed at 10 m (m s

^{−1}), and H—is the height at which the wind speed was obtained (m)—10 m.

#### 2.2. Data Quality Control and Homogeneity

^{−1}to 100 m s

^{−1}(0 ≤ WS ≤ 100).

#### 2.3. Reference Evapotranspiration (ETo) Calculation

^{−1}, and an albedo of 0.23 [1]. In this study, we employ the widely used FAO-56 Penman–Monteith method to calculate ETo [1], which is formulated as follows:

^{−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 ETo were obtained according to the procedure described in [1]. The G values were ignored for daily estimation (G = 0 MJ m

^{−2}d

^{−1}).

#### 2.4. Sobol’s Sensitivity Analysis Method

_{1}, X

_{2},…, X

_{d}) and their interaction has on the output.

_{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 the interaction of {X

_{1}, X

_{2},…, X

_{d}}. Subsequently, the method allows separating the contribution of each variable by estimating the Sobol’ first-order index (S

_{1}) (Equation (4)), as well as the variable interactions contribution by estimating the Sobol’ total-sensitivity index (S

_{tot}) (Equation (5)):

#### 2.5. Spatial Interpolation

^{®}10.8 (Environmental Systems Research Institute, Redlands, CA, USA), ordinary kriging model was used to create interpolation maps for ETo and its variables and sensitivity results across the entire study area. The ArcGIS Geostatistical Analyst tools were employed to generate a nested variogram with multiple model fits automatically. Additionally, a cross-validation method was executed to compare predicted and observed values, enabling the assessment of the model’s performance and fine-tuning of variogram parameters.

## 3. Results

#### 3.1. Climatic Variables and Penman–Monteith ETo Spatiotemporal Distribution

^{−1}), whereas their lowest mean happens at the end of the wet season (33.8 ± 1.3 °C and 0.9 ± 0.3 m s

^{−1}). Tmin and RH means range from 14.6 ± 1.0 °C to 20.0 ± 1.0 °C, 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 ETo (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}).

#### 3.2. Sobol’ Sensitivity Coefficients Spatiotemporal Distribution

_{1}) 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, ETo is highly sensitive to SRD, RH, and WS sensitive to Tmax; and insensitive to Tmin.

_{tot}, and 6% and 9% of the S

_{1}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 ETo 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%.

_{1}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 ETo 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 ETo to radiation ranges from 40% to 80%, indicating that RH, WS, and Tmax also influence ETo fluctuations in the Cerrado.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

^{−2}day

^{−1}) observed in the 33 AWSs of Mato Grosso State, Brazil. Table S3: Lower and upper sample limits (Inf—Sup), corresponding to the 2.5th and 97.5th percentiles, of the meteorological variable relative humidity (RH—%) observed in the 33 AWSs of Mato Grosso State, Brazil. Table S4: Lower and upper sample limits (Inf—Sup), corresponding to the 2.5th and 97.5th percentiles, of the meteorological variable maximum air temperature (Tmax—°C) observed in the 33 AWSs of Mato Grosso State, Brazil. Table S5: Lower and upper sample limits (Inf—Sup), corresponding to the 2.5th and 97.5th percentiles, of the meteorological variable minimum air temperature (Tmin—°C) observed in the 33 AWSs of Mato Grosso State, Brazil. Table S6: Lower and upper sample limits (Inf—Sup), corresponding to the 2.5th and 97.5th percentiles, of the meteorological variable wind speed at 2 m. (WS—m s

^{−1}) observed in the 33 AWSs of Mato Grosso State, Brazil.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Topographic and biomes maps, and location of INMET automatic weather stations (AWSs), in Mato Grosso, Brazil. Numerical identification according to Supplementary Table S1.

**Figure 2.**Boxplot showing the monthly variation of Penman–Monteith reference evapotranspiration (ETo) and its climatic variables obtained on 33 weather stations (AWSs) of Mato Grosso, Brazil. (Data series from 2008–2020). Climatic variables: SRD—downward solar radiation; RH—relative humidity; Tmax—maximum air temperature; Tmin—minimum air temperature; WS—wind speed at 2 m. The top and bottom edges of the box represent the 75th and 25th percentiles. The top and bottom whiskers represent the nonoutlier maximum and minimum. The line inside each box is the median. The gray-shaded background represents the rainy season. Empty and full boxes also indicate monthly values in the rainy and dry seasons, respectively. The different colors represent the meteorological variables evaluated (yellow—SRD; purple—RH; red—Tmax; blue—Tmin; green—WS; black—ETo).

**Figure 3.**Spatial distribution of Penman–Monteith reference evapotranspiration (ETo) and climatic variables in the 33 weather stations (AWSs) of Mato Grosso, Brazil. (Data series from 2008–2020). Climatic variables: SRD—downward solar radiation; RH—relative humidity; Tmax—maximum air temperature; Tmin—minimum air temperature; WS—wind speed at 2 m. Lines on the map represent the biome limits, and points represent the automatic weather stations (AWSs) of the state of Mato Grosso.

**Figure 4.**Annual and seasonal of Sobol’s first-order and total sensitivity indices of the climatic variables used to estimate Penman–Monteith reference evapotranspiration (ETo) in Mato Grosso, Brazil. Climatic variables: SRD—downward solar radiation; RH—relative humidity; Tmax—maximum air temperature; Tmin—minimum air temperature; WS—wind speed at 2 m.

**Figure 5.**Monthly variations of Sobol’s first-order and total sensitivity indices of the climatic variables used to estimate Penman–Monteith reference evapotranspiration (ETo) in Mato Grosso, Brazil. Climatic variables: SRD—downward solar radiation; RH—relative humidity; Tmax—maximum air temperature; Tmin—minimum air temperature; WS—wind speed at 2 m. The error bar represents the standard deviation of the indices across the 33 weather stations (AWSs) in the state of Mato Grosso, Brazil.

**Figure 6.**Spatial distribution of Sobol’s first-order and total sensitivity indices of the climatic variables used in the estimation of Penman–Monteith reference evapotranspiration (ETo) in the state of Mato Grosso, Brazil, on an annual and seasonal scale. Climatic variables: SRD—downward solar radiation; RH—relative humidity; Tmax—maximum air temperature; Tmin—minimum air temperature; WS—wind speed at 2 m. Lines on the map represent the biome limits, and points represent the automatic weather stations (AWSs) in Mato Grosso.

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**MDPI and ACS Style**

Sabino, M.; de Souza, A.P.
Global Sensitivity of Penman–Monteith Reference Evapotranspiration to Climatic Variables in Mato Grosso, Brazil. *Earth* **2023**, *4*, 714-727.
https://doi.org/10.3390/earth4030038

**AMA Style**

Sabino M, de Souza AP.
Global Sensitivity of Penman–Monteith Reference Evapotranspiration to Climatic Variables in Mato Grosso, Brazil. *Earth*. 2023; 4(3):714-727.
https://doi.org/10.3390/earth4030038

**Chicago/Turabian Style**

Sabino, Marlus, and Adilson Pacheco de Souza.
2023. "Global Sensitivity of Penman–Monteith Reference Evapotranspiration to Climatic Variables in Mato Grosso, Brazil" *Earth* 4, no. 3: 714-727.
https://doi.org/10.3390/earth4030038