Daily Maximum (Tmax) and Minimum Temperature (Tmin)

Using data from CONAGUA (https://smn.conagua. gob.mx/es/climatologia/informacion-climatologica/ informacion-estadistica-climatologica) [29], daily series of Tmax and Tmin were obtained from 70 weather stations in Sinaloa for the period 1942–2019. These same series were previously obtained by [34]. Through a review of the availability of recent information (< 5% missing data), in this study, it was decided to work with seven weather stations (Culiacán, El Playón, Las Tortugas, Rosario, La Concha, Ixpalino, and Sanalona II) for the period 1979–2017 (Fig. 1).

Imputation of Missing Data, Homogenization of Series, and Determination of the Mean Daily Temperature (Tmn)

Using RStudio software, with the Climatol library [35] and the orthogonal regression method, missing daily data of Tmax and Tmin were estimated by imputation. Using the standard normal homogeneity test (SNHT) [36] method, with Climatol, the series was also homogenized. By means of the semi-sum of the complete and homogeneous series of Tmax and Tmin, the daily series of Tmn was determined.

In general, the greatest thermal extremes were registered in Ixpalino (Tmax = 46.50°C day⁻¹), El Playón and Las Tortugas (Tmin = −6.00°C day⁻¹), and El Playón (Tmn = 38.00°C day⁻¹, Table 1).

Wind Speed at 10 m Height (U_z)

Through the National Oceanic and Atmospheric Administration (NOAA) [37] (https://downloads.psl.noaa. gov/Datasets/ncep.reanalysis2/Monthlies/gaussian_grid/), the monthly series (Jan–Dec) of wind speed at a height of 10 m (U_z) were obtained for the period 1979–2017. Due to the availability of satellite information, U_z was obtained for only two coordinates in the state of Sinaloa: 1) 25°43′14″ N by 108°45′00″ W and 2) 23°55′42″ N by 106°46′48″ W.

Fig. 1.

Study area, Sinaloa state.

Wind Speed at 2 m Height (U₂)

Although [9, 38] states that wind speed is not very relevant for estimating PM_C in semi-arid regions, in this study, using Equation 1, the wind speed was obtained at a height of 2 m (U₂) [19, 22].

where U₂ = monthly average wind speed at a height of 2 m (m s^–1), U_z = average wind speed measured at a height of 10 m (m s^–1), and z = measurement height of U_z (m).

Since PM_C is the international standard because of its greater measurement accuracy [9], in this study, PM_C was estimated daily using Equations 2–10. These equations, which are recommended by [19, 22] when there are missing data, are given as follows:

where e_aC = calculated actual vapor pressure and Tmin = daily minimum air temperature (°C).

where e_s = saturation vapor pressure (kPa) and Tmn = daily mean air temperature (°C).

where ∆ = slope of the saturated vapor pressure curve (kPa °C^–1).

where SR = calculated incident solar radiation (MJ m^–2 day^–1), KRS = solar radiation adjustment coefficient (dimensionless), with a value of 0.16 for continental conditions and 0.19 for coastal conditions (in this study, KRS = 0.16 was used), Tmax = maximum daily air temperature (°C), and Ra = extraterrestrial solar radiation (obtained through tabulated values with respect to latitude, MJ m^–2 day^–1).

where SRo = incident solar radiation with clear sky (MJ m^–2 day^–1).

where Rnl = net longwave radiation (MJ m^–2 day^–1), σ = Stefan–Boltzmann constant (0.4903×10^–8 MJ K^–4 m^–2 day^–1) and TmnK = mean daily air temperature (°K⁴).

where Rns = net shortwave radiation (MJ m^–2 day^–1).

where Rn = net radiation (MJ m^–2 day^–1).

where PM_C = Penman–Monteith reference grass evapotranspiration (mm day^–1, calculated with missing data), PM_O = Penman–Monteith observed grass reference evapotranspiration (mm day^–1, at the Acaponeta station), G = soil heat flux density (MJ m^–2 day^–1, null for daily estimates), and γ = psychrometric constant (0.067 kPa°C^–1; tabulated value by [19, 22], for stations with altitudes ranging from 0 to 100 masl).

Normality Test and Correlation Coefficients

A Shapiro–Wilk normality test was applied to all PM_C and HA_C series [39]. To find out whether PM_C and HA_C were significantly correlated, a Pr was applied to the series that presented normality, and a Spearman correlation (Sr) was applied to the series that did not present normality.

Simple Linear Regressions (SLR) and Simple Nonlinear Regressions (SNR)

To generate sensitive models [7] to predict PM_R (dependent variable) based on HA_C̄ (independent variable), SLR were initially fitted (Equation 12). A Shapiro–Wilk normality test was applied to the SLR residuals. When the residuals were not normal, an SNR (10 different functions) was applied, fitting a curvilinear estimate. Of the 10 functions, the following were chosen: a) exponential function (monthly series, Equation 13) and potential function (seasonal series, Equation 14), which were selected due to the highest R² recorded.

where e = Euler number (2.7182) and a, b = regression coefficients that describe the relationship between PM_R and HA_C.

Hypothesis Test

For each SLR and SNR, the Pr and Sr were obtained by the square root of R². To find out if each Pr and Sr were significantly different from zero, hypothesis tests were applied [31, 40]. Each Pr and Sr were compared with a Pcr = |0.316| (Equation 15) and a critical Spearman correlation coefficient (Scr = 0.318, Equation 16).

where t_c = critical value of the student t statistic and df = degrees of freedom (n–2).

where z = 1.96, n = 39 (for the period 1979–2017).

The design of the hypotheses is shown in Equations 17–18:

Also, the root mean square error (RMSE) between PM_C and PM_R was calculated.

Monthly (Jan–Dec), Seasonal (Mar–Aug), and Annual (Jan–Dec) Series

For PM_C–Ixpalino in all months (Jan–Dec), p(normal), and W ranged from 0.090 to 0.623 and from 0.951 to 0.978, respectively (Fig. 3a)). In total, 37 monthly series did not present normality; PM_C = 15 series and HA_C = 22 series (Fig. 3a)). The seasonal series (Mar–Aug) that did not present normality [p(normal) < 0.05] were PM_C–El Playón [p(normal) = 7.8×10^–5], HA_C–El Playón [p(normal) = 1.7×10^–6], HA_C–Las Tortugas [p(normal) = 0.003], and HA_C–Rosario [p(normal) = 0.008, Fig. 3b)]. The annual series (Jan–Dec) without normality were PM_C–El Playón [p(normal) = 2.3×10^–4], HA_C–El Playón [p(normal) = 9.4×10^–7], HA_C–Las Tortugas [p(normal) = 0.006] and HA_C–Ixpalino [p(normal) = 0.018, Fig. 3b)]. According to [42], the results of the PM_C–Ixpalino series (Fig. 3a)) present normality. According to [43], in the results of Fig. 3b), the seasonal series (Mar–Aug) that did not present normality were: PM_C–El Playón, HA_C– El Playón, HA_C–Las Tortugas, and HA_C–Rosario, because they did not present the condition of p(normal) > 0.05. According to [43, 44], the annual series (Jan–Dec) that did not present normality were PM_C–El Playón, HA_C–El Playón, HA_C–Las Tortugas, and HA_C–Ixpalino, this is due to p(normal) < 0.05.

Fig. 3.

Normality of the monthly series (Jan–Dec) of calculated reference evapotranspiration: Penman–Monteith (PM_C) and Hargreaves (HA_C) methods, for the period 1979–2017 (dimensionless).

Monthly Correlations (Jan–Dec)

As shown in Table 2, the correlations ranged from rP = 0.443 (El Playón–Jul) to rP = 0.929 (Las Tortugas– Jan). All rP and rS were significantly different from zero (rP > rcP = |0.316| and rS > rcS = |0.318|). According to [31, 40], the results of Table 2 establish the significant monthly relationship (Jan–Dec) of PM_C vs HA_C, so monthly modeling of PM_R is appropriate (Equations 12–14), applying SLR and SNR, as recommended by [19, 22] and applied by [27, 45].

Seasonal (Mar–Aug) and Annual (Jan–Dec) Correlations

As shown in Table 3, all seasonal (Mar–Aug) and annual (Jan–Dec) rP and rS were significantly different from zero (rP > rcP = 0.316 and rS > rcS = 0.318). Seasonal correlations (Mar–Aug) ranged from rP = 0.693 (Sanalona II) to rP = 0.907 (La Concha). The annual correlations (Jan–Dec) ranged from rS = 0.831 (El Playón) to rP = 0.921 (La Concha). Because all the correlations in Table 3 were significant [31, 40], SLR and SNR can be applied to estimate PM_R with a seasonal (Mar–Aug) and annual (Jan– Dec) scale, with HA_C as the independent variable [27–45].

Fig. 4.

Normality of regression residuals between reference evapotranspiration calculated from Penman–Monteith (PM_R) and Hargreaves (HA_C): a) monthly (Jan–Dec) and b) seasonal (Mar–Aug) and annual (Jan–Dec) (dimensionless).

Normality Test of Monthly (Jan–Dec), Seasonal (Mar–Aug), and Annual (Jan–Dec) Residuals

The only series of monthly residuals (Jan–Dec) that did not present normality was Sanalona II–Oct [p(normal) = 0.046 and W = 0.942, Fig. 4a)]. In the normal monthly series, the p(normal) values ranged from 0.059 (El Playón–May) to 0.951 (Culiacán–Nov, Fig. 4a)). As seen in Fig. 4b), Las Tortugas for the seasonal period (Mar–Aug) [p(normal) = 0.028 and W = 0.936] was the only series that did not register normality. In the normal seasonal series, the p(normal) values ranged from 0.237 (Ixpalino) to 0.445 (Rosario). In the normal annual series, the p(normal) values ranged from 0.221 (Ixpalino) to 0.964 (La Concha). According to [43] in the results of Fig. 4b), and for the seasonal period (Mar–Aug), the only series that did not present normality was Las Tortugas, because p < 0.05. All series that did present the condition of p > 0.05 were considered normal series [42].

Monthly Coefficients and Goodness of Fit (Jan–Dec)

The fit ranged from R² = 0.196 (rP = 0.443, El Playón–Jul) with RMSE = 0.274 mm day^–1 to R² = 0.863 (rP = 0.929, Las Tortugas–Jan) with RMSE = 0.218 mm day^–1 (Table 4). For the SNR–exponential function (Sanalona II–Oct, Table= 4), R² = 0.706 (rS = 0.840 > rcS = |0.318|). All RLS (Table 4) exceeded rcP = |0.316| [31, 40] (significant correlation) and did not register a trend towards underestimation or overestimation (RMSE < 0.300 mm day^–1) [9]. In Table 4 and for Sanalona II–Oct (SNR–exponential function), the results were rS = 0.840 > rcS = |0.318| [31, 40] (significant correlation) [31, 40]. The methodology of Table 4 was applied to obtain more accurate estimates [19, 22] and was previously applied by [26, 27].

Coefficients and Seasonal (Mar–Aug) and Annual (Jan–Dec) Goodness of Fit

Seasonal fit (Mar–Aug, Table 5) ranged from R² = 0.480 (rP = 0.693, Sanalona II) with RMSE = 0.156 mm day^–1 to R² = 0.823 (rP = 0.907, La Concha) with RMSE = 0.117 mm day^–1. Annual fit (Jan– Dec) ranged from R² = 0.719 (rP = 0.848, Sanalona II) with RMSE = 0.112 mm day^–1 to R² = 0.848 (rP = 0.921, La Concha) with RMSE = 0.082 mm day^–1. For the SNR– potential function (Las Tortugas–seasonal, Table 5), the fit was R² = 0.699 (rS = 0.836 > rcS = |0.318|). All SLR (seasonal and annual, Table 5) exceeded rcP = |0.316| [31, 40] (significant correlation) and showed no trend towards underestimation or overestimation (RMSE < 0.300 mm day^–1) [9]. These results are in agreement with [19, 22], who state that PM_R models are more accurate than when only Equations 1–10 (PM_C) are used. [46] also state that HA_C estimation is the most recommended method when data is not available to estimate PM_C.

Simple Linear Regressions (SLR) between Calculated and Observed Values from: 1) Incident Radiation (SR vs SRg), 2) Penman–Monteith Reference

Evapotranspiration, Calculated with Equations (PM_C vs PM_O), and 3) Calculated with Regressions (PM_R vs PM_O)

All the monthly average SLR (Fig. 5a)–5c)) recorded rP significantly different from zero (rP > rcP = |0.576|, for n = 12). Specifically for SR vs SRg, the measures of fit were: R² = 0.905, rP = 0.951, and RMSE = 0.684 mm day^–1 (Fig. 5a)). In PM_C vs PM_O, the measures of fit were R² = 0.350, rP = 0.592, and RMSE = 0.590 mm day^–1 (Fig. 5b)). For PM_R vs PM_O, the measures of fit were R² = 0.391, rP = 0.625, and RMSE = 0.578 mm day^–1 (Fig. 5c)). The residuals of the three SLR presented normality: p(normal) = 0.193 and W = 0.907 (Fig. 5a)), p(normal) = 0.344 and W = 0.927 (Fig. 5b)), and p(normal) = 0.464 and W = 0.937 (Fig. 5c)). In validation, the three SLRs (Fig. 5a)–5c)) performed well (RMSE < 1.0 mm día^–1) [47]. In this study, SR was highly influential (approximately 90.5%) for estimating PM_C, which agrees with [8], who points out that SRg in Sinaloa is decisive for the estimation of PM_C. According to the results of PM_C vs PM_O (Fig. 5b)), Equations 1–10 were reliable and sensitive for estimating PM_C, even when the series presented missing data [9, 48, 49]. The results of Fig. 5c) and, according to [19, 22, 26, 27], the models of this study are also reliable and sensitive for predicting PM_R. Finally, because the residuals of the three SLRs (Fig. 5a)–5c)) presented normality, the SLR is an appropriate statistical tool to use for comparison of calculated and observed data [7].