The methodological steps of the next sections are used to develop a global map of local coefficients for correcting the original potential evapotranspiration formula of Thornthwaite following a calibration and a validation procedure.

The derivation–calibration procedure was performed at a global scale using global gridded data from two databases. The first database of Hijmans et al. (2005) provides gridded data of mean monthly precipitation P and mean monthly temperature T of the period 1950–2000 (WorldClim version 1.2) at 30 arcsec spatial resolution (∼ 1 × 1 km at the Equator) (Fig. 1a, b). The second database is of Aschonitis et al. (2017) and provides gridded data of mean monthly reference evapotranspiration Er of the period 1950–2000 at five different resolutions (30 arcsec, 2.5 arcmin, 5 arcmin, 10 arcmin, and 0.5∘) (Fig. 1c). The method used for estimating Er is the ASCE-standardized method (formerly FAO-56 Penman–Monteith), which estimates reference evapotranspiration for short, clipped grass (Allen et al., 2005). The database of Er (Aschonitis et al., 2017) was built using the temperature from the first database of Hijmans et al. (2005) at 30 arcsec resolution, and for this reason the two gridded databases are compatible.

The validation procedure was performed using raw data of stations from three different databases. The first database is the CIMIS database (California Irrigation Management System – CIMIS, https://cimis.water.ca.gov/stations.aspx, last access: 1 October 2020), which includes stations from California, USA, and it was selected because it provides a dense and descriptive network of stations for a specific region that combines semiarid/temperate coastal, plain, and mountain environments. In total 60 stations (Fig. 2a) were used from the CIMIS database that have at least 15 years of observations, with a significant part of their observations after 2000. The second database is the AGBM database (Australian Government – Bureau of Meteorology, http://www.bom.gov.au, last access: 1 June 2020). This database includes many stations from Australia and was selected because the station's network covers a large territory with a large variety of climate classes from desert to tropical climate. The selection of stations was performed in order to cover all the possible existing Köppen–Geiger climatic types (Peel et al., 2007) and altitude ranges that exist in the Australian territory. In total 80 stations were used (Fig. 2b) that have at least 15 years of observations, with a significant part of their observations after 2000. The third database is the ECAD database (European Climate Assessment & Database, https://www.ecad.eu, last access: 1 February 2020). This database is a network that contains more than 20 000 stations throughout Europe and provides daily observations of climatological parameters. In this study, a final number of 385 stations (Fig. 2c) was selected from this database because they contained complete data of precipitation, temperature, solar radiation, relative humidity, and wind speed for a period of at least 20 years with a significant part of their observations after 2000. Some additional stations from the three databases (CIMIS, AGBM, ECAD), which do not have at least 15 years of observations, were selected due to their special Köppen–Geiger climate class or the high altitude of their location. The total final number of stations used in the study from the three databases is 525, and their full description is given in Table S1 of the Supplement.

The monthly potential evapotranspiration Ep using the Thornthwaite (1948) method after its adjustment for variable daylight and month lengths (Willmott et al., 1985) is estimated as follows. 1Ep=16⋅10⋅TmeanJa⋅N⋅n365 2aJ=∑i=112ji2bji=Tmean,i51.5142cα=6.75⋅10-7⋅J3-7.71⋅10-5⋅J2+1.79⋅10-2⋅J+0.492 3aN=24π⋅ωs3bωs=π2-arctan⁡-tan⁡φ⋅tan⁡δX0.5 where X=1-[tan⁡(φ)]2⋅[tan⁡(δ)]2, if X≤0 then X=0.00001 3cδ=0.409⋅sin⁡2⋅π⋅dj365-1.39 Here Ep is the mean monthly potential evapotranspiration or potential evapotranspiration of month i (millimeters per month), Tmean,i is the mean monthly temperature (∘C), n is the number of days in the month, N is the mean length of daylight of the days of the month (hours), J is the annual heat index, ji is the monthly heat index, α is the function of the annual heat index, and dj is the Julian day.

The benchmark method that was used for developing correction coefficients for the temperature-based method of Thornthwaite Ep is the ASCE-standardized method (formerly FAO-56 Penman–Monteith), which estimates reference evapotranspiration from short, clipped grass as follows (Allen et al., 2005): 4Er=0.408⋅Δ⋅Rn-G+γ⋅u2⋅es-ea⋅CnTmean+273.16Δ+γ⋅1+Cd⋅u2, where Er is the reference evapotranspiration (mm d-1), Δ is the slope of the saturation vapor pressure–temperature curve (kPa ∘C-1), Rn is the net radiation at the crop surface (MJ m-2 d-1), G is the soil heat flux density at the soil surface (MJ m-2 d-1), γ is the psychrometric constant (kPa ∘C-1), u2 is the wind speed at 2 m above the soil surface (m s-1), es is the saturation vapor pressure (kPa), ea is the actual vapor pressure (kPa),Tmean is the mean daily air temperature (∘C), and Cn and Cd are constants, which vary according to the time step and the reference crop type and describe the bulk surface resistance and aerodynamic roughness. Equation (3) can be applied for two types of reference crop (i.e., short and tall). The short reference crop (ASCE-short) corresponds to clipped grass of 12 cm height and surface resistance of 70 s m-1, where the constants Cn and Cd have the values 900 and 0.34, respectively (Allen et al., 2005). The use of Eq. (3) in daily or monthly steps for short reference crop is equivalent to the FAO-56 method (Allen et al., 1998), and this is how it is used in this study.

The derivation of a correction coefficient for Eq. (1) using as a benchmark the values of Eq. (3) is performed based on the same procedure proposed by Aschonitis et al. (2017) that has been used before for developing partially weighted annual correction coefficients for Priestley–Taylor and Hargreaves–Samani evapotranspiration methods. The procedure starts with the derivation of the monthly coefficient cth,i for each month i based on Eq. (5). Applying this procedure, 12 values of monthly cth,i are produced. The 12 monthly cth coefficients are then used to build mean annual coefficients. As was mentioned in Aschonitis et al. (2017), the efficiency of mean annual correction coefficients is mainly associated with their ability to better describe the larger values of the dependent variable (i.e., the values of Er during summer/hot months) and not the smaller values during cold periods when the absolute errors (ei=Er,i-Ep,i) are smaller. For this reason, weighted annual averages based on the monthly cth,i coefficients are estimated considering the participation weight of each month in the annual Er. Moreover, under cold conditions, the monthly coefficients cth,i may present unrealistic values that significantly affect the weighted averages. To solve this problem, threshold values for the monthly Ep,i and Er,i were used before the inclusion of their cth,i in the weighted average estimations. Preliminary analysis showed that when the mean monthly Ep,i and/or Er,i values are below ∼ 45 mm per month (∼ 1.5 mm d-1), then unrealistic mean monthly cth,i values occur (as unrealistic values are considered those that are at least 1 order of magnitude larger or smaller than 1). Taking into account the above, the following procedure was performed in order to obtain a partially weighted average based on monthly cth,i values after excluding those months with Er and/or Ep≤45 mm per month as follows. 5cth,i=Er,i/Ep,i6IfEr,i>45mm per month, then Fr,i=1 otherwise=07IfEp,i>45mm per month, then Fm,i=1 otherwise=08Er,iadj=Er,i⋅Fr,i⋅Fm,i9AEradj=∑i=112Er,iadj10Cth=∑i=112Er,iadjAEradj⋅cth,i Here cth,i is the monthly correction coefficient, Fr,i is the filter function for the reference method (ASCE) with values of 0 or 1, Fm,i is the filter function for the understudy model (Thornthwaite formula) with values of 0 or 1, Er,iadj is the adjusted monthly value of Er,i from the ASCE-short method that becomes 0 when Fr,i or Fm,i is 0, AEradj is the annual sum of the monthly Er,iadj adjusted values, Cth is the annual partially weighted average (p.w.a.) of the monthly cth,i coefficients for short reference crop, and i is the index of each month. Considering the above, the final corrected Thornthwaite formula for monthly calculations is given by the following equation: 11Eps,i=Cth⋅Ep,i, where Eps,i is the corrected temperature-based short reference crop evapotranspiration (millimeters per month) of month i.

The above procedure was followed to calibrate the annual partially weighted average Cth (Eq. 10) for every location on the globe based on mean monthly Er and Ep of 1950–2000 using

the gridded mean monthly temperature data of Hijmans et al. (2005) that were further used to estimate the original mean monthly gridded Thornthwaite Ep (Eq. 1) for the period 1950–2000 (in the form of 12 raster datasets of Ep for each month) and

The validation procedure with the data of the 525 stations was performed by comparing the mean monthly and the mean annual benchmark values of Er (Eq. 4) versus the original Ep (Eq. 1) and versus the corrected Eps Thornthwaite formula (Eq. 11) considering the annual partially weighted average coefficients Cth at the location of each station. The validation was performed separately for each database of stations (ECAD, AGBM, CIMIS) but also all together using the following five statistical criteria. 12MAE=1N∑i=1NSi-Oi13ME=1N∑i=1NSi-Oi14RMSE=1N∑i=1NSi-Oi215RSqr=∑i=1NOi-O‾iSi-S‾i∑i=1NOi-O‾i2∑i=1NSi-S‾i2216d=1-∑i=1NSi-Oi2∑i=1NSi-O‾i+Oi-O‾i2 Here MAE is the mean absolute error, ME the mean error, RMSE the root-mean-square error, RSqr the coefficient of determination, d the index of agreement, O the observed or benchmark value (i.e., Er), S the value simulated by the model (i.e., Ep or Eps), N the number of observations, and i the subscript referring to each observation. The value of perfect fit is 0 for the criteria MAE, ME, and RMSE while 1 is a perfect fit for the criteria RSqr and d. The values of the MAE, ME, and RMSE criteria have the same units as the observed and simulated data while RSqr and d are unitless.

The role of the new corrected formula of Thornthwaite (Eq. 11) as an internal parameter of aridity indices was also evaluated against the original method (Eq. 1). For this purpose, the AIUNEP (UNEP, 1997) and AITH (Thornthwaite, 1948) aridity indices were used. The difference between the two indices is that AIUNEP does not consider seasonality. The two indices estimated based on Er (Eq. 4) were used as the benchmark in order to compare the respective indices calculated with the original Thornthwaite Ep (Eq. 1) and the corrected Eps (Eq. 11) using the 525 stations' data. The evaluation was performed

by comparing the estimated aridity classes of 525 stations produced by the benchmark AIUNEP and AITH values using Er versus the classes of the two indices using Ep and Eps, respectively, and

The AIUNEP aridity index is the simpler method for hydroclimatic analysis, and it is given by the following equation: 17AIUNEP=PyEy, where Py is mean annual precipitation (mm yr-1) and Ey is mean annual potential evapotranspiration (mm yr-1). The values of Eq. (16) are classified according to the following (UNEP, 1997; Cherlet et al., 2018):

AIUNEP < 0.05 → hyper-arid

The classes for AIUNEP > 0.65 are usually given as one humid class. The UNEP index does not consider the effect of seasonal variation in precipitation and potential evapotranspiration.

The AITH aridity index is calculated as follows: 18aS=∑i=112Pi-Ei and 18bD=∑i=112Ei-Pi, 19AITH=100S-0.6DEy, where Pi and Ei are the monthly precipitation and potential evapotranspiration of month i, respectively. S (mm yr-1) considers only the positive values of (Pi-Ei) > 0, while (Pi-Ei) < 0 values are set to 0. In the case of D (mm yr-1), only the positive values of (Ei-Pi) > 0 are considered while for (Ei-Pi) < 0 they are set to 0. The various climatic types according to AITH values are the following.

-60 > AITH → hyper-arid (HE)

The global map of the Cth correction coefficient was developed following the procedure described in Sect. 2.2, and it is given in Fig. 3. The validation of the derived Cth coefficients was performed for each one of the three datasets of stations (California, CIMIS; Australia, AGBM; Europe, ECAD), separately, by comparing the performance of mean monthly values (Fig. S1a–f, Supplement) and the performance of mean annual values (Fig. S2a–f, Supplement) of Ep (Eq. 1) and Eps (Eq. 11) versus the benchmark values of Er (Eq. 4). The statistical criteria (Eqs. 12–16) for both monthly and annual comparisons for each one of the three datasets of stations are given in Table 1. The respective monthly and annual comparisons after merging all the stations from the three datasets are also presented in Fig. 4a–d. From the results shown in Figs. S1, S2, and 4 and Table 1, a much better performance of Eps compared to the original Thornthwaite formula Ep is observed in all cases, providing not only better monthly but also better annual reference evapotranspiration estimations that approximate the values of ASCE for short reference grass.

The use of Cth coefficients in AIUNEP and AITH aridity indices was also evaluated based on the raw data of all 525 stations (California, CIMIS; Australia, AGBM; Europe, ECAD).

The aridity classes of 525 stations given by the benchmark AIUNEP using Er were 76 % identical with the classes of the AIUNEP using Ep and 93 % identical with the classes of the AIUNEP using Eps. Similarly, the aridity classes of 525 stations given by the benchmark AITH using Er were 52 % identical with the classes of the AITH using Ep and 58 % identical with the classes of the AITH using Eps. Eps showed better performance compared to Ep at correctly identifying the aridity classes in both indices. The lower percentages of success in the case of AITH for both Ep and Eps are due to the double number of classes of AITH in comparison to AIUNEP. Merging the B and A classes of AITH to one humid class, as in the case of AIUNEP, the successful identical codes with Er are raised to 63 % for Ep and 76 % for Eps.

The 1 : 1 log-log plots of AIUNEP using Er versus the AIUNEP using Ep and Eps are given in Fig. 5a, b, respectively, while the same comparisons using AITH are given in Fig. 6a, b. The visual inspection of Figs. 5 and 6 clearly shows that Eps outperforms Ep in the range of non-humid classes of both AIUNEP and AITH. To highlight this result, the statistical metrics (Eqs. 12–16) were estimated after splitting the stations into two groups (non-humid and humid) based on the respective thresholds of humid classes of each index calculated using Er (Table 2). Table 2 verifies the better performance of Eps compared to Ep in both AIUNEP and AITH aridity indices for the non-humid classes.

On the other hand, the statistics showed that Ep showed better performance in both AIUNEP and AITH aridity indices for their respective humid classes. This result is of less importance since Eps showed better performance compared to Ep at correctly identifying the aridity classes in both indices based on all stations despite the fact that the stations belonging to humid classes were more in both indices (Table 2). Moreover, in the case of AIUNEP, there is only one humid class (AIUNEP > 0.65), and thus there is no point in comparing the performance of Ep and Eps from a statistical point of view since their values will always lead to the same classification code/characterization (i.e., humid). In the case of AITH > 20, the same justification of AIUNEP could be used since the detailed division of five humid classes (B1, B2, B3, B4, A) provided by AITH was proposed for the alternative use of the index as a “humidity index” (Thornthwaite, 1948).

The derivation of local Cth coefficients at a global scale was performed using the mean monthly grid datasets of 1950–2000 assuming stationary climate conditions, while the validation was performed using stations' raw data from California and Australia that are expanded up to 2016 and stations' raw data from Europe that are expanded up to 2020 (Table S1). The reasons for choosing the specific grid datasets for the derivation of Cth coefficients are the following.

They are in the form of high-resolution grids (30 arcsec, ∼ 1 km at Equator), which have been developed using interpolation techniques that include the effects of latitude, longitude, and elevation. These grids allow us to derive more representative Cth values for every position even when weather stations do not locally exist.

On the other hand, several works have shown climate differences after 2000 (Hansen et al., 2010; McVicar et al., 2012a, b; Wild et al., 2013; Willet et al., 2014; Sun et al., 2017). Such changes could possibly affect the validity of Cth coefficients and the final estimated values of Er for periods beyond 2000. For this reason, the Cth values and the mean monthly Er values of the grids of Aschonitis et al. (2017) of the period 1950–2000 were extracted from the positions of all 525 stations and compared with the respective values of computed Er and Cth using stations' raw data, which go beyond 2000. The results of this comparison are given in Fig. S3a, b (Supplement) and clearly show that the gridded Er data and Cth of 1950–2000 do not show serious deviations from their respective values for periods beyond 2000, allowing their safe use. Moreover, the fact that the original Thornthwaite (1948) formula was built before 1950 using data from the eastern and central USA and that the Cth values of the specific territories range between 0.9–1.1 for 1950–2000 (Fig. 3), it is not only a verification of the Cth derivation methodology but also an additional indication of a generalized temporal stability of Cth.

In the case of Fig. S3b, there is a distinctly deviated Cth pair of values from the 1 : 1 line (point indicated by a red arrow), which is associated with a specific station belonging to the Centro de Investigación Atmosférica de Izaña. This station is an exceptional case since it is at the top of a mountain at 2371 m a.s.l. on Tenerife (Canary Islands). The derived Cth of this station from the grid of the period 1950–2000 is almost half (Cth value equal to 1.37) of the one estimated using stations' raw data (Cth value equal to 2.44). This large difference is not the result of climate difference before and after 2000, but it is fully justified by the fact that the Cth value of the grid corresponds to an area of ∼ 1 km2 while the specific position of the station is unique, which can be described as the most extreme position within this pixel. There are also three stations on Tenerife in lowland areas where the derived Cth values of 1950–2000 are in agreement with those estimated by the stations' raw data.

The case of Izana station on Tenerife was the perfect example for triggering further investigation for the possible effects of scale in similar environments with extremely variable topography. Investigating the individual stations with the larger percentage of deviation of Eps from Er, a relative systematic deviation was observed at some stations of the CIMIS (California) database, which are concentrated at the coastline between Los Angeles and San Diego. The specific region is a narrow (∼ 20–30 km), highly urbanized coastal zone of ∼ 200 km, which is enclosed between the coastline and a hilly/mountainous zone. At the specific stations, the average of Cth values of the period 1950–2000 from the position of these stations was 1.85, while the average of Cth values using their raw data was estimated at 1.46. Apart from the large topographic variation, another reason for the Cth differences at these stations could be the bias that has been removed by clearing extreme flagged wind values in the data of the CIMIS database, which are probably associated with frequent extreme events in this region (extreme winds, droughts including wildfires, and heavy precipitation). This could justify the fact that the gridded Cth values of 1950–2000 at the positions of the stations are greater than the Cth values estimated by their raw data from CIMIS after removing flagged extreme values.

An additional analysis based only on the stations of California was made to show that a wider regional mean value of Cth coefficient could also be an additional option, especially when the whole territory is described by local Cth coefficients that are only > 1 or only < 1 (in California all local Cth coefficients are > 1). For this analysis, the average value of Cth=1.66 was estimated based on the values of local Cth coefficients of 1950–2000 from the locations of all stations of CIMIS (California). The mean monthly and mean annual Eps values of these stations were computed using Cth=1.66 for all of them and compared with the respective Er values estimated with stations' raw data (Fig. S4a, b, Supplement). The results of Fig. S4a, b showed that even a regional average of Cth values for California can lead to better results of Eps compared to Ep as it was given for monthly and annual estimations in Figs. S1a and S2a, respectively.

The initial trials to derive annual correction coefficients Cth of this study were made using the average value of the 12 monthly cth,i values of each i month. This procedure led to unreasonably high values due to the extreme high values during winter. An example of this problem based on the gridded data used in the calibration–derivation procedure is given in Fig. S5a (Supplement), which corresponds to a position close to Lake Garda in Italy (45.45∘ N, 10.124∘ E). According to Fig. S5a, the annual average of monthly cth,i values for this location is equal to 2.4 due to the extremely high values during winter and especially during January. Using the 2.4 value as the annual correction coefficient, the Eps,i value of July becomes equal to 338 mm, which is 203 mm larger than the respective Er,i value of July (Fig. S5a). The specific procedure for deriving annual Cth coefficients was rejected due to this problem. A second approach was to use the 12 pairs of monthly Er and Ep for each position on the grid in order to perform regression analysis based on the form y=a⋅x without intercept based on the form of Er=Cth⋅Ep. An example of the specific procedure is given in Fig. S5b using the data of Fig. S5a, where the annual Cth value was found equal to 0.98. The specific procedure provides annual Cth values, which are always closer to the monthly coefficients of the warmer months since optimization algorithms try to minimize the total error, which mainly originates in the months that show larger evapotranspiration values. Despite the fact that the specific procedure pays less attention to the monthly cth,i values of colder months, it was considered acceptable since most of the hydroclimatic applications require higher accuracy for the larger evapotranspiration values rather than the lower ones.

A similar approach with the one of Fig. S5b was performed by Cristea et al. (2013) for deriving annual correction coefficients for the Priestley–Taylor method for 106 stations across the contiguous USA. The correction coefficients were estimated for each station by minimizing the sum of the squared residuals between Priestley–Taylor and the benchmark FAO-56, considering data only for the period April–September (warmer semester). The obtained optimized values of the correction coefficients for each station were then interpolated to produce a map of the Priestley–Taylor correction coefficients. For our study, the specific procedure was found to be extremely demanding in computing requirements since it was impossible to be performed pixel by pixel (777.6 million pixels) with a conventional computer unit for the whole globe using as input 24 rasters of extremely high resolution (∼ 1 km) with a total size of ∼ 70 GB. In order to solve this problem, the method using partially weighted averages (Eqs. 5–10) developed by Aschonitis et al. (2017) was used, which provides similar results to the regression analysis of y=a⋅x but allows us to perform calculations step by step with a conventional computer unit in the GIS environment using large gridded databases. For the data of Fig. S5a, the partially weighted average method provided a Cth value equal to 0.99, which is almost equal to 0.98 of Fig. S5b. The method of partially weighted averages is also extremely efficient since it is not restricted only to the warmer semester or to any other predefined period like the case of Cristea et al. (2013) since it controls all months one by one using the threshold of 45 mm per month, which is more appropriate for global applications and especially for applications of high-resolution data, giving the appropriate weight to the months with significant values of evapotranspiration.

The threshold of 45 mm per month was derived empirically after analyzing many datasets using monthly and mean monthly data. In the case of monthly data, a representative example is given in Fig. S6a, b (Supplement) using the monthly data of Embrun station in France (44.57∘ N, 6.50∘ E) 1980–2020. Figure S6a shows the box–whisker plots of monthly Er,i values of the station, while Fig. S6b shows the respective box–whisker plots of monthly cth,i values. The maximum cth,i values of December, January, and February are outside the plot of Fig. S6b, with values of 30.1, 129.4, and 210.1, respectively. Figure S6a, b show that the monthly cth,i values of months with Er,i < 45 mm per month are extremely unstable, and their mean monthly value, even if it seems normal, cannot guarantee its safe use. In the case of mean monthly data, a representative example is given in Fig. 7, where the 6300 mean monthly cth,i values derived by the raw data of the 525 stations were plotted against their respective mean monthly Er values using a 2D density scatter plot. Figure 7 shows that the mean monthly cth,i values of the stations start to exhibit extremely high dispersion below the threshold of 45 mm per month, with values reaching 1 order of magnitude larger than unity. In the case where there is a location where all months show Er or Ep values below 45 mm per month, it is suggested to either use the non-zero Cth value of the closer location in the map of Fig. 3 or directly use the original Thornthwaite formula without correction.