the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Correcting Thornthwaite potential evapotranspiration using a global grid of local coefficients to support temperaturebased estimations of reference evapotranspiration and aridity indices
Dimos Touloumidis
MarieClaire ten Veldhuis
Miriam CoendersGerrits
Thornthwaite's formula is globally an optimum candidate for largescale applications of potential evapotranspiration and aridity assessment at different climates and landscapes since it has lower data requirements compared to other methods and especially from the ASCEstandardized reference evapotranspiration (formerly FAO56), which is the most datademanding method and is commonly used as the benchmark method. The aim of the study is to develop a global database of local coefficients for correcting the formula of monthly Thornthwaite potential evapotranspiration (E_{p}) using as benchmark the ASCEstandardized reference evapotranspiration method (E_{r}). The validity of the database will be verified by testing the hypothesis that a local correction coefficient, which integrates the local mean effect of wind speed, humidity, and solar radiation, can improve the performance of the original Thornthwaite formula. The database of local correction coefficients was developed using global gridded temperature, rainfall, and E_{r} data of the period 1950–2000 at 30 arcsec resolution (∼ 1 km at Equator) from freely available climate geodatabases. The correction coefficients were produced as partial weighted averages of monthly ${E}_{\text{r}}/{E}_{\text{p}}$ ratios by setting the ratios' weight according to the monthly E_{r} magnitude and by excluding colder months with monthly values of E_{r} or E_{p} < 45 mm per month because their ratio becomes highly unstable for low temperatures. The validation of the correction coefficients was made using raw data from 525 stations of Europe; California, USA; and Australia including data up to 2020. The validation procedure showed that the corrected Thornthwaite formula E_{ps} using local coefficients led to a reduction of RMSE from 37.2 to 30.0 mm m^{−1} for monthly step estimations and from 388.8 to 174.8 mm yr^{−1} for annual step estimations compared to E_{p} using as a benchmark the values of the E_{r} method. The corrected E_{ps} and the original E_{p} Thornthwaite formulas were also evaluated by their use in Thornthwaite and UNEP (United Nations Environment Program) aridity indices using as a benchmark the respective indices estimated by E_{r}. The analysis was made using the validation data of the stations, and the results showed that the correction of the Thornthwaite formula using local coefficients increased the accuracy of detecting identical aridity classes with E_{r} from 63 % to 76 % for the case of Thornthwaite classification and from 76 % to 93 % for the case of UNEP classification. The performance of both aridity indices using the corrected formula was extremely improved in the case of nonhumid classes. The global database of local correction factors can support applications of reference evapotranspiration and aridity index assessment with the minimum data requirements (i.e., temperature) for locations where climatic data are limited. The global grids of local correction coefficients for the Thornthwaite formula produced in this study are archived in the PANGAEA database and can be assessed using the following link: https://doi.org/10.1594/PANGAEA.932638 (Aschonitis et al., 2021).
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The assessment of potential or reference evapotranspiration is among the most important components for many hydroclimatic applications such as irrigation design and management, water balance assessment studies, and assessment of aridity classification and drought indices (Weiß and Menzel, 2008; Wang and Dickinson, 2012; McMahon et al., 2013; Aschonitis et al., 2017).
Such applications, and especially applications of aridity classification and drought indices (UNEP, 1997; Thornthwaite, 1948; Palmer, 1965; Holdridge, 1967; Beguería et al., 2014) that are usually employed at large scales, require estimations of potential or reference evapotranspiration of respective scale. The major problem in such applications is not only the limited availability of stations per se but also the limitation of many stations to provide data for a complete set of parameters (i.e., precipitation, temperature, solar radiation, wind speed, humidity). A complete set of climate parameters is prerequisite for accurate estimations of potential or reference evapotranspiration using integrated methods such as these of Penman (1948), Shuttleworth (1993), Allen et al. (1998, 2005), and others which are expressions of energy balance. Unfortunately, largescale applications suffer from these limitations, and the common solution is to use temperaturebased formulas (Thornthwaite, 1948; McCloud, 1955; Hamon, 1961, 1963; Baier and Robertson, 1965; Malmström, 1969; Hargreaves and Samani, 1982; Camargo et al., 1999; Droogers and Allen, 2002; Pereira and Pruitt, 2004; Oudin et al., 2005; Trajkovic, 2005, 2007; Trajkovic and Kolakovic, 2009a, b; Almorox et al., 2015; Aschonitis et al., 2017; Sanikhani et al., 2019; Quej et al., 2019; Trajkovic et al., 2020). However, extensive literature shows that temperaturebased formulas are inherently of low performance because temperature cannot properly describe the evaporative flux, while various studies have shown differences among the Penman–Monteithbased and temperaturebased potential evapotranspiration assessments such as the one of Thornthwaite (1948), which is the most popular in aridity and drought index applications (Sheffield et al., 2012; Dai, 2013; van der Schrier et al., 2013; Trenberth et al., 2014; Yuan and Quiring, 2014; Zhang et al., 2015; Asadi Zarch et al., 2015).
The formula of Thornthwaite (1948) was firstly proposed as the internal part of the respective Thornthwaite aridity–humidity index, and it was calibrated based on measured monthly evapotranspiration from some wellwatered grasscovered lysimeters in the eastern and central USA (Willmott et al., 1985; Van Der Schrier et al., 2011). The specific formula overestimates the potential evapotranspiration in humid climates, and underestimates it in arid climates (Pereira and Pruitt, 2004; Castañeda and Rao, 2005; Trajkovic and Kolakovic, 2009a, b). Thus, a number of efforts have been made to amend the parameters or constants of the empirical formula to adapt it to various geographical zones (Jain and Sinai, 1985; Pereira and Pruitt, 2004; Castañeda and Rao, 2005; Zhang et al., 2008; Bakundukize et al., 2011; Yang et al., 2017). Indicative modifications were proposed by Willmott et al. (1985) using an additional parametrization presented for mean monthly temperature above 26.5 ^{∘}C and an adjustment for variable daylight and month lengths. Camargo et al. (1999) substituted the mean monthly temperature by another factor called effective temperature considering the amplitude between maximum and minimum temperature. Jain and Sinai (1985) modified the constant in the general formula based on the min–max range of the annual mean air temperature to calculate the evapotranspiration for semiarid conditions. Pereira and Pruitt (2004) proposed an adaptation of the Thornthwaite scheme to estimate the daily reference evapotranspiration on two contrasting environments in the USA and Brazil. Castañeda and Rao (2005) recalibrated the coefficient of the general formula based on estimations of potential evapotranspiration using the FAO Penman–Monteith method in southern California while Bautista et al. (2009) performed a similar procedure for stations located in a coastal semiarid climate and inland tropical subhumid climate in regions of Mexico. Zhang et al. (2008) used a modified formula to estimate the actual evapotranspiration in cropland, shrubland, and forest located in the subalpine region of southwestern China. Bakundukize et al. (2011) used two modifications of the original Thornthwaite method for groundwater recharge estimations in the interlacustrine zone of east Africa. Yang et al. (2017) presented a method to quantitatively identify the differences in the spatiotemporal variabilities of global drylands between the Thornthwaite and Penman–Monteith parameterizations. Trajkovic et al. (2019, 2020) provided successful corrections of the original formula based on the FAO Penman–Monteith method for stations located in Hungary, Serbia, Romania, Croatia, and Slovakia.
In recent years, advanced interpolation techniques, climatic models, and other methods have achieved gridded datasets of various climatic parameters (Hijmans et al., 2005; Sheffield et al., 2006; Osborn and Jones, 2014; Harris et al., 2014; Brinckmann et al., 2016; Liu et al., 2020), facilitating attempts to develop global maps of potential/reference evapotranspiration and to investigate the accuracy of formulas of reduced parameters versus benchmark methods at global scale (Droogers and Allen, 2002; Weiß and Menzel, 2008; Zomer et al., 2008; Aschonitis et al., 2017). A similar attempt is performed in this study, aiming to develop a global database of local correction coefficients for the original Thornthwaite formula. This attempt aims to support all hydroclimatic applications and specifically to support largescale applications of aridity indices, which are highly affected by the use of different potential evapotranspiration methods (Proutsos et al., 2021). The hypothesis that is tested in this work is that a global grid of local correction coefficients that integrates the local mean effects of wind speed, humidity, and solar radiation can improve the performance of the original potential evapotranspiration formula of Thornthwaite by converting it into a formula of reference evapotranspiration for short reference crop based on the FAO Penman–Monteith concept.
2.1 Data
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 E_{r} 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 E_{r} is the ASCEstandardized method (formerly FAO56 Penman–Monteith), which estimates reference evapotranspiration for short, clipped grass (Allen et al., 2005). The database of E_{r} (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.
2.2 Derivation and validation of Thornthwaite correction coefficients for short reference crop based on ASCEstandardized method
The monthly potential evapotranspiration E_{p} using the Thornthwaite (1948) method after its adjustment for variable daylight and month lengths (Willmott et al., 1985) is estimated as follows.
where $X=\mathrm{1}\left[\mathrm{tan}\right(\mathit{\phi}){]}^{\mathrm{2}}\cdot [\mathrm{tan}\left(\mathit{\delta}\right){]}^{\mathrm{2}}$, if X≤0 then X=0.00001
Here E_{p} is the mean monthly potential evapotranspiration or potential evapotranspiration of month i (millimeters per month), T_{mean,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, j_{i} is the monthly heat index, α is the function of the annual heat index, and d_{j} is the Julian day.
The benchmark method that was used for developing correction coefficients for the temperaturebased method of Thornthwaite E_{p} is the ASCEstandardized method (formerly FAO56 Penman–Monteith), which estimates reference evapotranspiration from short, clipped grass as follows (Allen et al., 2005):
where E_{r} is the reference evapotranspiration (mm d^{−1}), Δ is the slope of the saturation vapor pressure–temperature curve (kPa ^{∘}C^{−1}), R_{n} 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}), u_{2} is the wind speed at 2 m above the soil surface (m s^{−1}), e_{s} is the saturation vapor pressure (kPa), e_{a} is the actual vapor pressure (kPa),T_{mean} is the mean daily air temperature (^{∘}C), and C_{n} and C_{d} 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 (ASCEshort) corresponds to clipped grass of 12 cm height and surface resistance of 70 s m^{−1}, where the constants C_{n} and C_{d} 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 FAO56 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 c_{th,i} for each month i based on Eq. (5). Applying this procedure, 12 values of monthly c_{th,i} are produced. The 12 monthly c_{th} 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 E_{r} during summer/hot months) and not the smaller values during cold periods when the absolute errors (${e}_{i}={E}_{\text{r},i}{E}_{\text{p},i}$) are smaller. For this reason, weighted annual averages based on the monthly c_{th,i} coefficients are estimated considering the participation weight of each month in the annual E_{r}. Moreover, under cold conditions, the monthly coefficients c_{th,i} may present unrealistic values that significantly affect the weighted averages. To solve this problem, threshold values for the monthly E_{p,i} and E_{r,i} were used before the inclusion of their c_{th,i} in the weighted average estimations. Preliminary analysis showed that when the mean monthly E_{p,i} and/or E_{r,i} values are below ∼ 45 mm per month (∼ 1.5 mm d^{−1}), then unrealistic mean monthly c_{th,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 c_{th,i} values after excluding those months with E_{r} and/or E_{p}≤45 mm per month as follows.
Here c_{th,i} is the monthly correction coefficient, F_{r,i} is the filter function for the reference method (ASCE) with values of 0 or 1, F_{m,i} is the filter function for the understudy model (Thornthwaite formula) with values of 0 or 1, ${E}_{\text{r},i}^{\text{adj}}$ is the adjusted monthly value of E_{r,i} from the ASCEshort method that becomes 0 when F_{r,i} or F_{m,i} is 0, ${\mathrm{AE}}_{\text{r}}^{\text{adj}}$ is the annual sum of the monthly ${E}_{\text{r},i}^{\text{adj}}$ adjusted values, C_{th} is the annual partially weighted average (p.w.a.) of the monthly c_{th,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:
where E_{ps,i} is the corrected temperaturebased short reference crop evapotranspiration (millimeters per month) of month i.
The above procedure was followed to calibrate the annual partially weighted average C_{th} (Eq. 10) for every location on the globe based on mean monthly E_{r} and E_{p} 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 E_{p} (Eq. 1) for the period 1950–2000 (in the form of 12 raster datasets of E_{p} for each month) and

the respective mean monthly grids of E_{r} based on ASCEstandardized for short reference crop (Eq. 1) from Aschonitis et al. (2017) (in the form of 12 raster datasets of E_{r} for each month).
The validation procedure with the data of the 525 stations was performed by comparing the mean monthly and the mean annual benchmark values of E_{r} (Eq. 4) versus the original E_{p} (Eq. 1) and versus the corrected E_{ps} Thornthwaite formula (Eq. 11) considering the annual partially weighted average coefficients C_{th} 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.
Here MAE is the mean absolute error, ME the mean error, RMSE the rootmeansquare error, R_{Sqr} the coefficient of determination, d the index of agreement, O the observed or benchmark value (i.e., E_{r}), S the value simulated by the model (i.e., E_{p} or E_{ps}), 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 R_{Sqr} and d. The values of the MAE, ME, and RMSE criteria have the same units as the observed and simulated data while R_{Sqr} and d are unitless.
2.3 Evaluating the use of correction coefficients in aridity indices based on station data
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 AI_{UNEP} (UNEP, 1997) and AI_{TH} (Thornthwaite, 1948) aridity indices were used. The difference between the two indices is that AI_{UNEP} does not consider seasonality. The two indices estimated based on E_{r} (Eq. 4) were used as the benchmark in order to compare the respective indices calculated with the original Thornthwaite E_{p} (Eq. 1) and the corrected E_{ps} (Eq. 11) using the 525 stations' data. The evaluation was performed

by comparing the estimated aridity classes of 525 stations produced by the benchmark AI_{UNEP} and AI_{TH} values using E_{r} versus the classes of the two indices using E_{p} and E_{ps}, respectively, and

by comparing the respective values of the indices using 1 : 1 plots and the statistical metrics of Eqs. (12)–(16).
The AI_{UNEP} aridity index is the simpler method for hydroclimatic analysis, and it is given by the following equation:
where P_{y} is mean annual precipitation (mm yr^{−1}) and E_{y} 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):

AI_{UNEP} < 0.05 → hyperarid

0.03 ≤ AI_{UNEP} < 0.2 → arid

0.2 ≤ AI_{UNEP} < 0.5 → semiarid

0.5 ≤ AI_{UNEP} < 0.65 → dry subhumid

0.65 < AI_{UNEP} → humid
The classes for AI_{UNEP} > 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 AI_{TH} aridity index is calculated as follows:
and
where P_{i} and E_{i} are the monthly precipitation and potential evapotranspiration of month i, respectively. S (mm yr^{−1}) considers only the positive values of (P_{i}−E_{i}) > 0, while (P_{i}−E_{i}) < 0 values are set to 0. In the case of D (mm yr^{−1}), only the positive values of (E_{i}−P_{i}) > 0 are considered while for (E_{i}−P_{i}) < 0 they are set to 0. The various climatic types according to AI_{TH} values are the following.

−60 > AI_{TH} → hyperarid (HE)

−60 ≤ AI_{TH} < −40 → arid (E)

−40 ≤ AI_{TH} < −20 → semiarid (D)

−20 ≤ AI_{TH} < 0 → dry subhumid (C1)

0 ≤ AI_{TH} < 20 → moist subhumid (C2)

20 ≤ AI_{TH} < 40 → low humid (B1)

40 ≤ AI_{TH} < 60 → moderate humid (B2)

60 ≤ AI_{TH} < 80 → highly humid (B3)

80 ≤ AI_{TH} < 100 → very humid (B4)

100 ≤ AI_{TH} → hyperhumid (A)
3.1 Derivation and validation of the C_{th} correction coefficients
The global map of the C_{th} correction coefficient was developed following the procedure described in Sect. 2.2, and it is given in Fig. 3. The validation of the derived C_{th} 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 E_{p} (Eq. 1) and E_{ps} (Eq. 11) versus the benchmark values of E_{r} (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 E_{ps} compared to the original Thornthwaite formula E_{p} 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.
3.2 Evaluating the use of C_{th} coefficient in AI_{UNEP} and AI_{TH} aridity indices
The use of C_{th} coefficients in AI_{UNEP} and AI_{TH} 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 AI_{UNEP} using E_{r} were 76 % identical with the classes of the AI_{UNEP} using E_{p} and 93 % identical with the classes of the AI_{UNEP} using E_{ps}. Similarly, the aridity classes of 525 stations given by the benchmark AI_{TH} using E_{r} were 52 % identical with the classes of the AI_{TH} using E_{p} and 58 % identical with the classes of the AI_{TH} using E_{ps}. E_{ps} showed better performance compared to E_{p} at correctly identifying the aridity classes in both indices. The lower percentages of success in the case of AI_{TH} for both E_{p} and E_{ps} are due to the double number of classes of AI_{TH} in comparison to AI_{UNEP}. Merging the B and A classes of AI_{TH} to one humid class, as in the case of AI_{UNEP}, the successful identical codes with E_{r} are raised to 63 % for E_{p} and 76 % for E_{ps}.
The 1 : 1 loglog plots of AI_{UNEP} using E_{r} versus the AI_{UNEP} using E_{p} and E_{ps} are given in Fig. 5a, b, respectively, while the same comparisons using AI_{TH} are given in Fig. 6a, b. The visual inspection of Figs. 5 and 6 clearly shows that E_{ps} outperforms E_{p} in the range of nonhumid classes of both AI_{UNEP} and AI_{TH}. To highlight this result, the statistical metrics (Eqs. 12–16) were estimated after splitting the stations into two groups (nonhumid and humid) based on the respective thresholds of humid classes of each index calculated using E_{r} (Table 2). Table 2 verifies the better performance of E_{ps} compared to E_{p} in both AI_{UNEP} and AI_{TH} aridity indices for the nonhumid classes.
On the other hand, the statistics showed that E_{p} showed better performance in both AI_{UNEP} and AI_{TH} aridity indices for their respective humid classes. This result is of less importance since E_{ps} showed better performance compared to E_{p} 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 AI_{UNEP}, there is only one humid class (AI_{UNEP} > 0.65), and thus there is no point in comparing the performance of E_{p} and E_{ps} 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 AI_{TH} > 20, the same justification of AI_{UNEP} could be used since the detailed division of five humid classes (B1, B2, B3, B4, A) provided by AI_{TH} was proposed for the alternative use of the index as a “humidity index” (Thornthwaite, 1948).
4.1 Validity of the derived C_{th} for periods beyond the calibration period
The derivation of local C_{th} 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 C_{th} coefficients are the following.

They are in the form of highresolution 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 C_{th} values for every position even when weather stations do not locally exist.

They cover a large period of time (i.e., 1950–2000), so they can provide more representative mean annual p.w.a. C_{th} values. The upper threshold of the year 2000 of these grids also allows the validation dataset of stations to be more valid since the larger part of their data is after 2000, and this reduces the possibility of having been used in grids' development.
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 C_{th} coefficients and the final estimated values of E_{r} for periods beyond 2000. For this reason, the C_{th} values and the mean monthly E_{r} 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 E_{r} and C_{th} 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 E_{r} data and C_{th} 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 C_{th} values of the specific territories range between 0.9–1.1 for 1950–2000 (Fig. 3), it is not only a verification of the C_{th} derivation methodology but also an additional indication of a generalized temporal stability of C_{th}.
In the case of Fig. S3b, there is a distinctly deviated C_{th} 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 C_{th} of this station from the grid of the period 1950–2000 is almost half (C_{th} value equal to 1.37) of the one estimated using stations' raw data (C_{th} 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 C_{th} value of the grid corresponds to an area of ∼ 1 km^{2} 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 C_{th} values of 1950–2000 are in agreement with those estimated by the stations' raw data.
4.2 Scale and other effects on the accuracy of the derived C_{th}
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 E_{ps} from E_{r}, 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 C_{th} values of the period 1950–2000 from the position of these stations was 1.85, while the average of C_{th} values using their raw data was estimated at 1.46. Apart from the large topographic variation, another reason for the C_{th} 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 C_{th} values of 1950–2000 at the positions of the stations are greater than the C_{th} 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 C_{th} coefficient could also be an additional option, especially when the whole territory is described by local C_{th} coefficients that are only > 1 or only < 1 (in California all local C_{th} coefficients are > 1). For this analysis, the average value of C_{th}=1.66 was estimated based on the values of local C_{th} coefficients of 1950–2000 from the locations of all stations of CIMIS (California). The mean monthly and mean annual E_{ps} values of these stations were computed using C_{th}=1.66 for all of them and compared with the respective E_{r} values estimated with stations' raw data (Fig. S4a, b, Supplement). The results of Fig. S4a, b showed that even a regional average of C_{th} values for California can lead to better results of E_{ps} compared to E_{p} as it was given for monthly and annual estimations in Figs. S1a and S2a, respectively.
4.3 Justifications about the methodology for deriving annual C_{th} correction coefficients based on partially weighted averages
The initial trials to derive annual correction coefficients C_{th} of this study were made using the average value of the 12 monthly c_{th,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 c_{th,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 E_{ps,i} value of July becomes equal to 338 mm, which is 203 mm larger than the respective E_{r,i} value of July (Fig. S5a). The specific procedure for deriving annual C_{th} coefficients was rejected due to this problem. A second approach was to use the 12 pairs of monthly E_{r} and E_{p} for each position on the grid in order to perform regression analysis based on the form $y=a\cdot x$ without intercept based on the form of ${E}_{\text{r}}={C}_{\text{th}}\cdot {E}_{\text{p}}$. An example of the specific procedure is given in Fig. S5b using the data of Fig. S5a, where the annual C_{th} value was found equal to 0.98. The specific procedure provides annual C_{th} 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 c_{th,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 FAO56, 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 $y=a\cdot x$ 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 C_{th} 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 highresolution 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 E_{r,i} values of the station, while Fig. S6b shows the respective box–whisker plots of monthly c_{th,i} values. The maximum c_{th,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 c_{th,i} values of months with E_{r,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 c_{th,i} values derived by the raw data of the 525 stations were plotted against their respective mean monthly E_{r} values using a 2D density scatter plot. Figure 7 shows that the mean monthly c_{th,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 E_{r} or E_{p} values below 45 mm per month, it is suggested to either use the nonzero C_{th} value of the closer location in the map of Fig. 3 or directly use the original Thornthwaite formula without correction.
The produced global database of local C_{th} coefficients of this study has been archived in PANGAEA and can be assessed using the following link: https://doi.org/10.1594/PANGAEA.932638 (Aschonitis et al., 2021). The database is provided at five different resolutions (30 arcsec, 2.5 arcmin, 5 arcmin, 10 arcmin, 0.5^{∘}). The coarser resolutions are provided in order to cover the observed resolution range in the initial climatic data used for developing the published E_{r} gridded data by Aschonitis et al. (2017) (e.g., the temperature data of Hijmans et al., 2005, were provided at 30 arcsec resolution, while the solar radiation, humidity, and wind speed data of Sheffield et al., 2006, were provided at 0.5^{∘} resolution and rescaled to 30 arcsec using bilinear interpolation). The data of different resolutions can be used as a tool to assess uncertainties associated with temperature variation effects within different resolution pixels or to estimate average values of the coefficients for larger territories, which have problems at coarse resolutions (e.g., coastlines or islands that do not exist in 0.5^{∘} resolution), taking into account the concept and concerns of Daly (2006).
A global database of local correction coefficients for improving the performance of the monthly temperaturebased Thornthwaite potential evapotranspiration method was built using gridded data covering the period 1950–2000. The method for developing the correction coefficients was based on partially weighted averages of their respective mean monthly values estimated as the monthly ratios between the benchmark ASCEstandardized E_{r} method (formerly FAO56) versus the original Thornthwaite E_{p}. The correction coefficients were produced as partially weighted averages of monthly ${E}_{\text{r}}/{E}_{\text{p}}$ ratios by setting the ratios' weight according to the monthly E_{r} magnitude and by excluding colder months because the ${E}_{\text{r}}/{E}_{\text{p}}$ ratio becomes highly unstable for low temperatures. The correction coefficients were validated using raw data from 525 stations of California, Australia, and Europe that include independent data beyond 2000 up to 2020. The results showed that the correction coefficients significantly improved the monthly and annual results of the original Thornthwaite method E_{p}. The use of E_{p} with or without correction coefficients was also evaluated through their use in the aridity indices of Thornthwaite and UNEP versus the respective indices estimated based on the benchmark ASCEstandardized E_{r}. The results showed again that the correction coefficients significantly improved the performance of the indices compared to the original Thornthwaite method, especially in nonhumid environments. The global database of local correction coefficients supports applications of reference evapotranspiration and aridity index assessment with minimum data requirements (i.e., mean temperature) for locations where climate data are limited. Uncertainties in the values of correction coefficients were observed in regions of high topographic variability, and a recommendation for such cases is the use of a regional average of correction coefficients or the use of local C_{th} values based on the available coarser resolutions provided in the database. The methods and results presented in this study and the observed uncertainties can be used as a base for future works focusing on (a) the validation of the correction coefficients for other places in the world, (b) comparison with other models of low data requirements, and (c) use of the p.w.a. method for recalibrating correction coefficients using station or climate models' data of recent periods.
The supplement related to this article is available online at: https://doi.org/10.5194/essd141632022supplement.
The idea behind the work was conceived by VA, the data processing was carried out by VA and DT, and quality control, visualization, and writing were completed by VA, DT, MCG, and MCtV.
The contact author has declared that neither they nor their coauthors have any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This paper was edited by David Carlson and reviewed by one anonymous referee.
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