The objective of the study is to provide global grids
(0.5

The reference crop evapotranspiration ET

The Priestley–Taylor method requires net solar radiation and
temperature data. The P-T formula includes an empirical factor known as
advection coefficient

The Hargreaves–Samani (H-S) method requires only temperature data, including
four empirical factors (or three depending on the formula). A part of the
equation empirically describes the incident solar radiation

The analysis of ET

The objectives of the study are

to develop mean monthly maps of
ET

to develop global maps that provide the possible annual
error in ET

to develop global maps of readjusted annual coefficients for the H-S and P-T evapotranspiration methods for both short and tall reference crop based on a new method that estimates partial weighted averages of the monthly coefficients (the same procedure was also followed for the coefficients of the H-S radiation formula)

to validate the results of the readjusted P-T and H-S coefficients using data from meteorological stations from different locations with different climatic conditions and

to compare the predictive ability of the readjusted P-T and H-S coefficients for the short reference crop evapotranspiration with the respective predictions obtained from other models that have low data requirements.

The analysis presented in this study was based on global climatic data
obtained from the following databases:

The database of Hijmans et al. (2005) provides mean monthly values for the
parameters of precipitation and maximum, minimum and mean temperature at 30 arcsec spatial resolution. The data are provided as grids of mean monthly
values of the period 1950–2000 (

The database of Sheffield et al. (2006) provides monthly values of
parameters such as wind speed at the height of 10 m above the ground
surface, solar radiation, specific humidity, precipitation and temperature
for the period 1948–2006 at 0.5

The database of Peel et al. (2007) provides the revised global Köppen–Geiger climate map. The data are provided in raster form with 0.1 degree spatial resolution. The climate map was developed using the GHCN (Global Historical Climatology Network) version 2.0 dataset (Peterson and Vose, 1997), which includes precipitation data from 12 396 stations and temperature data from 4844 station data for the periods 1909–1991 and 1923–1993, respectively. The Köppen–Geiger map was used to obtain the climatic type of the meteorological stations used in the validation dataset.

All the calculations presented in the next sections were performed in an ArcGIS 9.3 ESRI environment in a WGS84 ellipsoid coordinate system. For area coverage calculations or for estimations of mean global values of various parameters, coordinate system conversions were performed from WGS84 to projected Cylindrical Equal Area system (any percentage global coverage calculations and the derivation of mean global values of various parameters are performed without considering Antarctica).

The estimation of ET

The calculation of Priestley–Taylor (P-T) method is performed by the
following equation (Priestley and Taylor, 1972):

The Hargreaves–Samani (H-S) method (Hargreaves and Samani, 1982, 1985) for
ET

In order to reduce the errors of the aforementioned methods in the high
latitudes and altitudes (polar and alpine environments) where negative
temperatures exist, a filter was used in all methods to set mean monthly
ET

The first step of the analysis includes the estimation of mean monthly and
mean annual ET

In the case of mean annual ET

In the case of mean monthly ET

The procedures of MAD %,

For the case of P-T, the readjustment of the mean monthly

For the case of the H-S method, the readjustment of the coefficients was
performed in two stages. In the first stage, the readjustment was performed
in the radiation formula (Eq. 3) only for the

The new mean monthly

For estimating the PWA of mean monthly

A similar procedure (using the set of Eqs. 7) was also followed to obtain the
PWA of mean monthly

Stations from two databases (the California Irrigation Management System CIMIS
database,

Meteorological stations from the USA (California; USA-CA) (CIMIS database) and Australia (AGBM database).

Continued.

Continued.

Position of stations

In the case of CIMIS stations, the monthly data for all climatic parameters
were obtained, including ET

The validation procedure was performed using the data of the stations in
Table 1 by comparing the mean monthly values of ET

The selected models have been calibrated by either using global data or a representative amount of data from California or Australia. Models that have been tested for California and Australia and showed good performance were also included.

The selected models showed better performance in comparison to other models when tested using the validation datasets of Californian and Australian stations, but they also showed good performance with regard to other regions based on studies from the literature. It has to be mentioned that an extremely large amount of models were examined taking into account the modified H-S and P-T models obtained from literature that has already been cited in the introduction and the large lists of models presented in the work of Valipour (2015a, b, 2017) and Valipour et al. (2017). Strict modifications of the P-T and H-S models with fixed coefficients calibrated for local conditions were not used because they cannot adapt their coefficients to the large climatic variability in the validation dataset.

The majority of the selected models require additional parameters in comparison to the P-T and H-S models. This criterion was used in order to compare the strength of the readjusted P-T and H-S coefficients versus such models.

Two modified models of H-S by Droogers and Allen (2002), where the second one uses precipitation as an additional parameter. The models were calibrated using global data.

Three models of reduced parameters given by Valiantzas (2013a, b, 2014), which were calibrated using 535 stations from Europe, Asia and Africa. The first model uses temperature and radiation data, while the other two use temperature, radiation and humidity data. The models have been tested for Californian (Valiantzas, 2013c) and Australian conditions (Ahooghalandari et al., 2017).

Two models of reduced parameters by Ahooghalandari et al. (2016) calibrated/validated using stations from various locations in Australia. The models use temperature and relative humidity data.

The Copais model of Alexandris et al. (2006) that uses temperature, radiation and humidity data. The model was calibrated/validated using data from Greece, California and Oregon (USA), while it has shown a very good response to many other regions of the world including Australia (Ahooghalandari et al., 2017).

Additional models of reduced parameters obtained from the
international literature, which provide equivalent results to ET

*

The following statistical criteria were used in the validation procedure:
the coefficient of determination (

Mean annual values (mm year

Mean annual values of

The global maps of mean monthly ET

The MAD % (Eq. 5) maps of the ASCE-tall, standard P-T and standard H-S methods
versus ASCE-short are given in Fig. 4a, b and c, while the MAD % of the standard solar radiation formula of H-S versus
the

Mean annual difference percentage (MAD %) of ET

The percent coverage

The case of MAD % between the ET

Even though MAD %,

Spatial extent of the major climatic groups CGs of the Köppen–Geiger climate map (Peel et al., 2007); spatial extent of DMAD classes (P-T versus H-S) within each CG.

The spatial variation in MAD % for the case of

The PWAs of mean monthly

Partial weighted averages of mean monthly

As regards the spatial variation in

As regards the spatial variation in

Partial weighted averages of mean monthly

In the case of

The validation of the readjusted

Comparative

Statistical criteria from the comparisons

Table 5a and Fig. 8 show the ET

The P-T(PWA-s) and H-S(PWA-s) models (Fig. 8b, d) outperformed the respective standard P-T(1.26) and H-S(0.0023) models (Fig. 8a, c) with regard to all the statistical criteria (Table 5a), reducing the RMSE values at 40 and 25 %, respectively.

The comparison of statistical criteria between H-S(0.0023), H-S(PWA-s),
DRAL1 and DRAL2, which follow the general formula of the H-S method and are
based on calibrations with global data, showed the following order of
accuracy H-S(PWA-s)

The standard P-T(1.26) showed the worst results for all criteria (Table 4a), while the use of P-T(PWA-s) succeeded in improving the predictions, giving better results for the H-S(0.0023), DRAL2, VAL1 and AKJ2 models.

The H-S(PWA-s) provided better results for DRAL1, DRAL2, VAL1, AKJ1 and AKJ2 where the latter four require data for more climatic parameters.

The order of accuracy of the models was the following: VAL3

Table 5b and Fig. 9a and b show the ET

Table 5c and Fig. 10a and b show the comparisons between the

The recalibrated coefficients of the H-S and P-T methods were estimated
using raster datasets that cover the period 1950–2000 assuming stationary
climate conditions, while the validation datasets of Californian (USA) and
Australian stations are expanded up to 2016. Recent studies have shown
changes/anomalies after 2000 in temperature (Hansen et al., 2010; Sun et
al., 2017), solar radiation (Wild et al., 2013), wind speed (McVicar et al.,
2012a, b) and atmospheric humidity (Willet et al., 2014) and such changes
could affect the validity of the revised coefficients. The comparisons of

Possible changes in wind speeds after 2000, since the majority of wind speed data in the station datasets corresponds to periods after 2000.

Uncertainties in the Sheffield et al. (2006) wind data due to the scarce existing wind data for calibrating their model on a global scale during the period of 1950–2000 and especially during the years belonging to the first half of the simulation period.

The effect of the equation

The bias that may have been introduced after cleaning extreme wind values in the data of USA-CA stations, which may be associated with hurricane events. The region of California is strongly affected by hurricanes and the higher wind speeds in the rasters of the Sheffield et al. (2006) data may partly have occurred because they included such events in their climatic simulations.

The bias that may have been introduced by the wind data of Australian stations. The AGBM database provides 12 values of mean monthly wind speeds of the total observation periods for 09:00 and another 12 values for 15:00 local time. In order to get the mean monthly wind speeds of the stations, the average value of 09:00 and 15:00 conditions was used for each month.

Combinations of all the aforementioned cases.

Comparative

Comparative

Uncertainties may also exist in the case of DE

Thus, uncertainties exist in both rasters and station data. In future studies, further improvements in the revised coefficients can be made by using global raster data, which incorporate the conditions after 2000, and by solving many of the aforementioned problems related to both station data and raster data produced by climatic models.

The analysis presented in this study passed through various stages before the selection of the annual PWA form of the coefficients (Eq. 7). Some steps in the preliminary analysis were to analyse (a) the different forms of averages (e.g. mean, mode, median, geometric mean, harmonic mean) for deriving annual coefficients and (b) the strength of the derived mean monthly and seasonal coefficients versus the annual PWA coefficients and versus the coefficients of the standard methods,.

As regards the use of the weighted annual average (w.a.) of the mean monthly
coefficients instead of other forms of averages (e.g. mean, mode, median,
geometric mean – g.m., harmonic mean – h.m.), preliminary analysis was
performed using data extracted by the climatic rasters from many positions
in the world. During this analysis, trials to derive annual coefficients
were made using an optimization algorithm separately for each position. The
results showed that the optimized annual values were always closer to the
monthly coefficients of the warmer months since the optimization algorithms
try to reduce the total error, which is mainly dominated by the months that
show larger ET

The case of mean monthly coefficients was also examined (results not shown).
The results showed that the assessment of annual ET

It is also important to note that the derivation of annual coefficients is a
pure optimization problem when station data are used. For example, Cristea
et al. (2013) derived coefficients of the P-T method for 106 stations that
represent a range of climates across the contiguous USA. The coefficients
were estimated for each station by minimizing the sum of the squared
residuals between the benchmark FAO-56 and P-T using data only for the
period April–September. The obtained optimized values of the coefficients
were interpolated in order to make a map of the

Special attention was also given in the case of

The uncertainties, which may be introduced by climate disturbances/changes or other uncertainties related to the data used for calibrating the coefficients, can be reduced by taking into account some of the following observations and recommendations.

A separate analysis using only the stations of California showed that a
regional mean value of the coefficients derived by PWA values may present
even better performance because it probably counterbalances other
uncertainties associated with the spatial climatic variability within a
specific region. A factor for such uncertainties may be rainfall, which may
not show significant seasonal deviations or deviations from the expected
annual values for a large region but may show different spatial patterns
every year within the region affecting the accuracy of the coefficients. The
aforementioned observation was verified by the application of the H-S method for
ET

The comparison between P-T and H-S evapotranspiration methods with
readjusted coefficients but also their comparison with the other models in Table 2 also provided significant information. From the comparison between
P-T and H-S with readjusted coefficients, it was observed that H-S provided
better results in both the short and the tall reference crop. The better performance of H-S
can be attributed to the fact that more than

A very interesting observation was also made about the tall reference crop
based on the results of the MAD % map (Fig. 4a). In the MAD % class of

The datasets produced in this study have been archived in the PANGAEA database
(

The study provided global grids of revised annual coefficients for the P-T and H-S methods for estimating
ET

The methods used in this study, their respective results and the observed
uncertainties can be used as a base for future work focusing on

the validation of the coefficients for other parts of the world, especially using climatic data obtained after 2000, and the comparison with other models of low data requirements

the recalibration of the coefficients using data from climatic models that include observations from more recent years and an analysis of climate change effects on the coefficients

the use of the available climatic datasets obtained from climatic models in order to calibrate models of the coefficients for various locations providing values that are not fixed, such as the ones given in this study,

analysis of alternative methods for deriving annual coefficients that approximate optimized values or the incorporation of optimization algorithms in a GIS environment to capture the optimum solution per pixel

the confrontation of uncertainties related to the data used for calibration and validation (e.g. low representativity of interpolated climatic parameters due to the lack of data in many parts of the world; errors associated with commonly used equations, such as the one used for adjusting wind data at 2 m height; uncertainties associated with the observed data).

The authors declare that they have no conflict of interest.

This study was performed in the context of two post-doctoral research studies by Vassilis Aschonitis financed by Ferrara University (Italy) and Aristotle University of Thessaloniki (Greece).Edited by: David Carlson Reviewed by: two anonymous referees