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**Research article**
18 Feb 2019

**Research article** | 18 Feb 2019

Potential evaporation at eddy-covariance sites across the globe

^{1}Laboratory of Hydrology and Water Management, Ghent University, Coupure Links 653, 9000 Gent, Belgium^{2}Department of Earth and Environmental Engineering, Columbia University, New York City, New York 10027, USA

^{1}Laboratory of Hydrology and Water Management, Ghent University, Coupure Links 653, 9000 Gent, Belgium^{2}Department of Earth and Environmental Engineering, Columbia University, New York City, New York 10027, USA

**Correspondence**: Wouter H. Maes (wh.maes@ugent.be)

**Correspondence**: Wouter H. Maes (wh.maes@ugent.be)

Abstract

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Potential evaporation (*E*_{p}) is a crucial variable for
hydrological forecasting and drought monitoring. However, multiple
interpretations of *E*_{p} exist, which reflect a diverse range of methods to
calculate it. A comparison of the performance of these methods against field
observations in different global ecosystems is urgently needed. In this
study, potential evaporation was defined as the rate of terrestrial
evaporation (or *evapotranspiration*) that the actual ecosystem would attain if it were to evaporate at
maximal rate for the given atmospheric conditions. We use eddy-covariance
measurements from the FLUXNET2015 database, covering 11 different
biomes, to parameterise and inter-compare the most widely used
*E*_{p} methods and to uncover their relative performance. For each of the 107 sites, we isolate
days for which ecosystems can be considered unstressed, based on both an
energy balance and a soil water content approach. Evaporation measurements
during these days are used as reference to calibrate and validate the
different methods to estimate *E*_{p}. Our results indicate that a simple
radiation-driven method, calibrated per biome, consistently performs best
against in situ measurements (mean correlation of 0.93; unbiased RMSE of
0.56 mm day^{−1}; and bias of −0.02 mm day^{−1}). A Priestley and Taylor method,
calibrated per biome, performed just slightly worse, yet substantially and
consistently better than more complex Penman-based, Penman–Monteith-based or
temperature-driven approaches. We show that the poor performance of
Penman–Monteith-based approaches largely relates to the fact that the
unstressed stomatal conductance cannot be assumed to be constant in time at
the ecosystem scale. On the contrary, the biome-specific parameters required
by simpler radiation-driven methods are relatively constant in time and per
biome type. This makes these methods a robust way to estimate *E*_{p} and a
suitable tool to investigate the impact of water use and demand, drought
severity and biome productivity.

How to cite

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How to cite.

Maes, W. H., Gentine, P., Verhoest, N. E. C., and Miralles, D. G.: Potential evaporation at eddy-covariance sites across the globe, Hydrol. Earth Syst. Sci., 23, 925–948, https://doi.org/10.5194/hess-23-925-2019, 2019.

1 Introduction

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Since its introduction 70 years ago by Thornthwaite (1948), the
concept of potential evaporation (*E*_{p}), defined as the amount of water
which would evaporate from a surface unconstrained by water availability,
has been widely used in multiple fields. It has been incorporated in
hydrological models dedicated to estimate runoff (e.g. Schellekens et al.,
2017) or actual evaporation (Wang and Dickinson, 2012), as well as in
drought severity indices (Sheffield et al., 2012; Vicente-Serrano et al.,
2013). Long-term changes in *E*_{p} have been regarded as a driver of
ecosystem distribution and aridity (Scheff and Frierson, 2013) and used to
diagnose the influence of climate change on ecosystems based on climate
model projections (e.g. Milly and Dunne, 2016). However, many different
definitions of *E*_{p} exist, and consequently many different methods are
available to calculate it. In recent years, there has been an increasing
awareness of the impact of the underlying assumptions and caveats in
traditional *E*_{p} formulations (Weiß and Menzel, 2008; Kingston et al.,
2009; Sheffield et al., 2012; Seiller and Anctil, 2016; Bai et al., 2016; Milly
and Dunne, 2016; Guo et al., 2017). As such, a global appraisal of the most
appropriate method for assessing *E*_{p} is urgently needed. Yet, current
formulations reflect a disagreement on the mere meaning of this variable,
which requires the definition of some form of reference system (Lhomme,
1997). *E*_{p} has been typically defined as the evaporation which would
occur in given meteorological conditions if water was not limited, either
(i) over open water (Shuttleworth, 1993); (ii) over a reference crop, usually a
wet (Penman, 1963) or irrigated (Allen et al., 1998) short green grass
completely shading the ground; or (iii) over the actual ecosystem transpiring
under unstressed conditions (Brutsaert, 1982; Granger, 1989).

A second source of disagreement on the definition of *E*_{p} relates to the
spatial extent of the reference system and the consideration (or not) of
feedbacks from the reference system on the atmospheric conditions. Several
authors found it convenient to define *E*_{p} taking an extensive area as
a reference system, because this reduces the influence of advection and
entrainment flows (Penman, 1963; Priestley and Taylor, 1972; Brutsaert,
1982; Shuttleworth, 1993). Such an idealised extensive and well-watered
ecosystem evaporating at maximal rate for the given atmospheric
conditions can be expected to raise air humidity until the vapour pressure
deficit (VPD) tends to zero. In this case, evaporation is only driven by
radiative forcing and no longer by aerodynamic forcing. Meanwhile, others have defended the
use of reference systems that are infinitesimally small (Morton,
1983; Pettijohn and Salvucci, 2009; Gentine et al., 2011b), in order to avoid
the feedback of the reference system on aerodynamic forcing. The effect of
the choice of reference system is best exemplified by the complementary
relationship framework (Bouchet, 1964), which uses both approaches to link
potential and actual evaporation, through $(\mathrm{1}+b){E}_{{\mathrm{p}}_{\mathrm{0}}}={E}_{\mathrm{pa}}+b{E}_{\mathrm{a}}$,
with *b* an empirical constant (Kahler and Brutsaert, 2006; Aminzadeh et al., 2016), ${E}_{{\mathrm{p}}_{\mathrm{0}}}$ the
evaporation from an extensive well-watered surface (i.e. in which the
feedback from the ecosystem on the VPD and aerodynamic forcing is considered
and where evaporation is only driven by a radiative forcing), *E*_{pa} the
evaporation from a well-watered but infinitesimally small surface (i.e.
where evaporation is driven by both radiative and aerodynamic forcing) and
*E*_{a} the actual evaporation (Morton, 1983).

In light of all this controversy, the net radiation of the reference system remains
another point of discussion: some scientists argue that the (well-watered)
reference system should have the same net radiation as the actual
(water-limited) system (e.g. Granger, 1989; Rind et al., 1990; Crago and
Crowley, 2005). Yet, this is inherently inconsistent as the surface
temperature reflects the surface energy partitioning; thus a well-watered
system transpiring at a potential rate is expected to have a lower surface
temperature (Maes and Steppe, 2012) and correspondingly a higher net
radiation (e.g. Lhomme, 1997; Lhomme and Guilioni, 2006). Meanwhile, to some
extent, the albedo also depends on soil moisture (Eltahir, 1998; Roerink et
al., 2000; Teuling and Seneviratne, 2008) and it can be argued that it should
be adjusted to reflect well-watered conditions (Shuttleworth, 1993).
Finally, extensive reference surfaces can be expected to exert a feedback,
not only on the aerodynamic forcing, but also on the incoming radiation (via
impacts on air temperature, humidity and cloud formation). Yet, these
larger-scale feedbacks are not acknowledged when computing *E*_{p}, even when
considering extensive reference systems.

As it can be concluded from the above discussion, a unique and universally
accepted definition of *E*_{p} does not exist, and the most appropriate
definition remains tied to the specific interest and application.
Nonetheless, as different applications make use of different *E*_{p}
formulations, a good understanding of the implications of the choice for a
specific method is required (Fisher et al., 2011). For terrestrial
ecosystems, the use of an open-water reference system is uninformative about
the actual available energy and the aerodynamic properties of the actual
ecosystem (Shuttleworth, 1993; Lhomme, 1997). The approach of considering an
idealised well-watered crop system only takes climate forcing conditions
into account and not the actual land cover; as such, it has become the
standard to estimate aridity and trends in global drought (Dai, 2011). When
the actual ecosystem transpiring at unstressed rates is considered as
a reference system, both climate forcing conditions and ecosystem properties
are taken into account. This has been the preferred approach when
calculating *E*_{p} as an intermediate step to estimate actual evaporation,
often by applying a multiplicative stress factor (*S*) varying between 0 and 1,
such that *E*_{a}=*S**E*_{p} (e.g. Barton, 1979; Mu et al., 2007; Fisher et
al., 2008; Miralles et al., 2011; Martens et al., 2017). This *S* factor can be
considered analogous to the *β* factor used in some land surface models
to incorporate the effect of soil moisture in the estimation of gross
primary production and surface turbulent fluxes (Powell et al., 2013).

Several studies have attempted to compare and evaluate different *E*_{p} methods.
Some of these studies have compared the performance of different *E*_{p}
formulations in hydrological (Xu and Singh, 2002; Oudin et al., 2005a; Kay and
Davies, 2008; Seiller and Anctil, 2016) or climate models (Weiß and
Menzel, 2008; Lofgren et al., 2011; Milly and Dunne, 2016). Others considered
the Penman–Monteith method as the benchmark to test less input-demanding
formulations (e.g. Chen et al., 2005; Sentelhas et al., 2010). All these
studies have their own merits, yet an evaluation of *E*_{p} methods based on
empirical data of actual evaporation measurements is to be preferred
(Lhomme, 1997). To date, such approaches have been hampered by limited data
availability (Weiß and Menzel, 2008). Lysimeters provide arguably the
most precise evaporation measurements available (e.g. Abtew, 1996; Pereira
and Pruitt, 2004; Yoder et al., 2005; Katerji and Rana, 2011), but are
sparsely distributed and not always representative of larger ecosystems. Pan
evaporation measurements are more easy to perform and are broadly available
(Zhou et al., 2006; Donohue et al., 2010; McVicar et al., 2012) but provide a
proxy of open-water evaporation, rather than actual ecosystem potential
evaporation; they also exhibit biases related to the location, shape and
composition of the instrument (Pettijohn and Salvucci, 2009).
Eddy-covariance measurements are an attractive alternative, but, apart from
an unpublished study by Palmer et al. (2012), have so far only been used in
*E*_{p} studies focusing on local to regional scales (Jacobs et al.,
2004; Sumner and Jacobs, 2005; Douglas et al., 2009; Li et al., 2016).

The overall purpose of the present work is to identify the most suitable
method to estimate *E*_{p} at the ecosystem-scale across the globe. Because
we are using an empirical dataset of actual evaporation at FLUXNET sites,
the reference system considered in this study is the actual ecosystem, so
*E*_{p} is defined as the evaporation of the actual ecosystem when it is
completely unstressed. As mentioned above, this definition is the most
suitable for hydrological studies, studies of ecosystem drought and
derivations of actual evaporation through constraining *E*_{p} calculations. Following
this definition, *E*_{p} is similar to ${E}_{{\mathrm{p}}_{\mathrm{0}}}$ in the complementary
relationship. We used the most recent and complete eddy-covariance database
available, i.e. the FLUXNET2015 archive (http://fluxnet.fluxdata.org/, last access: 14 February 2019). The
most frequently adopted *E*_{p} methods are applied based on standard
parameterisations as well as calibrated parameters by biome and are
inter-compared in order to gain insights into the most adequate means to
estimate *E*_{p} from ecosystem to global scales.

2 Material and methods

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Methods to calculate *E*_{p} can be categorised based on the amount and type
of input data required. In this overview, we will only discuss the ones
evaluated in our study, which are arguably the most frequently used.

The well-known Penman–Monteith equation (Monteith, 1965) expresses latent
heat flux *λ**E*_{a} (W m^{−2}) as

$$\begin{array}{}\text{(1)}& {\displaystyle}{\displaystyle}\mathit{\lambda}{E}_{\mathrm{a}}={\displaystyle \frac{s\left({R}_{\mathrm{n}}-G\right)+\frac{{\mathit{\rho}}_{\mathrm{a}}{c}_{p}\mathrm{VPD}}{{r}_{\mathrm{aH}}}}{s+\mathit{\gamma}+\mathit{\gamma}\frac{{r}_{\mathrm{c}}}{{r}_{\mathrm{aH}}}}},\end{array}$$

with *λ* being the latent heat of vaporisation (J kg^{−1}),
*E*_{a} the actual evaporation (kg m^{−2} s^{−1}), *s* the slope of the
Clausius–Clapeyron curve relating air temperature with the saturation vapour
pressure (Pa K^{−1}), *R*_{n} the net radiation (W m^{−2}), *G* the ground
heat flux (W m^{−2}), *ρ*_{a} the air density (kg m^{−3}),
*γ* the psychrometric constant (Pa K^{−1}), *c*_{p} the specific heat
capacity of the air (J kg^{−1} K^{−1}), VPD the vapour pressure deficit
(Pa), *r*_{aH} the resistance of heat transfer to air (s m^{−1})
and *r*_{c} the surface resistance of water transfer (s m^{−1}). While *λ*,
*c*_{p}, *s* and *γ* are air-temperature-dependent, *r*_{aH} is a complex
function of wind speed, vegetation characteristics and the stability of the
lower atmosphere (see Sect. 2.3). In most methods to estimate *E*_{a}
or *E*_{p}, *r*_{aH} is estimated from a simple function of wind speed.

The Penman–Monteith equation is frequently used to calculate *E*_{p} by
adjusting *r*_{c} to its minimum value (the value under unstressed
conditions). If the reference system is the actual system or a reference
crop, *r*_{c} is usually considered a fixed, constant value larger than zero
(even if the soil is well-watered). In this study, both a universal, fixed
value of *r*_{c} for reference crops and a biome-specific constant value are
used (see Sect. 2.5). When instead of a well-watered canopy a wet canopy
(i.e. a canopy covered by water) is considered, *r*_{c}=0 and Eq. (1) collapses to

$$\begin{array}{}\text{(2)}& {\displaystyle}{\displaystyle}\mathit{\lambda}{E}_{\mathrm{p}}={\displaystyle \frac{s\left({R}_{\mathrm{n}}-G\right)+\frac{{\mathit{\rho}}_{\mathrm{a}}{c}_{p}\mathrm{VPD}}{{r}_{\mathrm{aH}}}}{s+\mathit{\gamma}}}.\end{array}$$

Equation (2) is often referred to as the Penman (1948) formulation and can be
conveniently rearranged as $\mathit{\lambda}{E}_{\mathrm{p}}=\frac{s({R}_{\mathrm{n}}-G)}{s+\mathit{\gamma}}+\frac{{\mathit{\rho}}_{\mathrm{a}}{c}_{p}\mathrm{VPD}}{(s+\mathit{\gamma}){r}_{\mathrm{aH}}}$
to illustrate that *E*_{p} is driven by a radiative (left term) and an
aerodynamic (right term) forcing (Brutsaert and Stricker, 1979).

When the reference system is considered an idealised extensive area, or when
radiative forcing is very dominant, the aerodynamic component of Eq. (2) may
become negligible; thus the whole equation collapses to $\mathit{\lambda}{E}_{\mathrm{p}}=\frac{s({R}_{\mathrm{n}}-G)}{s+\mathit{\gamma}}$, which is
commonly referred to as “equilibrium evaporation” (Slatyer and McIlroy,
1961). Priestley and Taylor (1972) analysed time series of open water and
water-saturated crops and grasslands and found that the evaporation over
these surfaces closely matched the equilibrium evaporation corrected by a
multiplicative factor, commonly denoted as *α*_{PT}:

$$\begin{array}{}\text{(3)}& {\displaystyle}{\displaystyle}\mathit{\lambda}{E}_{\mathrm{p}}={\mathit{\alpha}}_{\mathrm{PT}}{\displaystyle \frac{s\left({R}_{\mathrm{n}}-G\right)}{s+\mathit{\gamma}}}.\end{array}$$

This formulation is known as the Priestley and Taylor equation. Because
usually a constant value of *α*_{PT} is adopted, it assumes that the
aerodynamic term in the Penman equation (Eq. 2) is a constant fraction of
the radiative term. Typically, *α*_{PT}=1.26 is considered, as
estimated by Priestley and Taylor (1972) in their original experiments. In
this study, we also include a biome-specific value to extend its
applicability to all biomes (see Sect. 2.5). Since this method does not
require wind speed or VPD as input, it is widely applied in hydrological
models (Norman et al., 1995; Castellvi et al., 2001; Agam et al., 2010),
remote sensing evaporation models (Norman et al., 1995; Fisher et al.,
2008; Agam et al., 2010; Miralles et al., 2011) and drought monitoring methods
(Anderson et al., 1997; Vicente-Serrano et al., 2018).

Other studies such as Lofgren et al. (2011), or the more recent Milly and
Dunne (2016), further simplified Eq. (3) to make it a linear function of the
available energy by defining a constant multiplier here referred to as *α*_{MD}:

$$\begin{array}{}\text{(4)}& {\displaystyle}{\displaystyle}\mathit{\lambda}{E}_{\mathrm{p}}={\mathit{\alpha}}_{\mathrm{MD}}\left({R}_{\mathrm{n}}-G\right).\end{array}$$

In the case of Milly and Dunne (2016) this equation was applied to climate
model outputs based on a constant and universal value of *α*_{MD}=0.8.
On a daily scale, (*R*_{n}−*G*) expresses the total amount of energy available for evaporation, and
the fraction of this energy that is actually used for evaporation is
typically referred to “evaporative fraction”, or $\mathrm{EF}=\frac{\mathit{\lambda}{E}_{\mathrm{a}}}{(H+\mathit{\lambda}{E}_{\mathrm{a}})}=\frac{\mathit{\lambda}{E}_{\mathrm{a}}}{({R}_{\mathrm{n}}-G)}$.
From Eq. (4), it follows that
the parameter *α*_{MD} can be interpreted as the EF of the
unstressed ecosystem. In this study, we test both the general value of
*α*_{MD}=0.8 and a biome-specific constant (see Sect. 2.5).

Of the many empirical methods to estimate *E*_{p}, temperature-based methods
have arguably been the most commonly used because of the availability of
reliable air temperature data. For an overview of these methods, we refer to
Oudin et al. (2005a). In this study, three methods are included. First,
Pereira and Pruitt (2004) formulated a daily version of the well-known
Thornthwaite (1948) equation:

$$\begin{array}{}\text{(5a)}& {\displaystyle}& {\displaystyle}{T}_{\mathrm{eff}}<\mathrm{0}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\lambda}{E}_{\mathrm{p}}=\mathrm{0},\text{(5b)}& {\displaystyle}& {\displaystyle}\mathrm{0}<{T}_{\mathrm{eff}}<\mathrm{26}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\lambda}{E}_{\mathrm{p}}={\mathit{\alpha}}_{\mathrm{Th}}{\left({\displaystyle \frac{\mathrm{10}{T}_{\mathrm{eff}}}{I}}\right)}^{b}\left({\displaystyle \frac{N}{\mathrm{360}}}\right),\text{(5c)}& {\displaystyle}& {\displaystyle}\mathrm{26}<{T}_{\mathrm{eff}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\lambda}{E}_{\mathrm{p}}=-c+d{T}_{\mathrm{eff}}-e{T}_{\mathrm{eff}}^{\mathrm{2}},\end{array}$$

with *T*_{eff} the effective temperature, based on maximum and minimal
temperatures (see Sect. 2.5); *α*_{Th} an empirical
parameter (see below); *I* the yearly sum of (*T*_{a_month}∕5)^{1.514},
with *T*_{a_month} the mean air
temperature for each month; *N* the number of daylight hours; *b* a parameter
depending on *I* and *c*; and *d* and *e* empirical constants (see Sect. 2.5). The general
value of *α*_{Th}=16 is often adopted; in this study, we will also
calculate and apply a biome-specific value.

The second temperature-based formulation is that proposed by Oudin et al. (2005a), selected and developed after comparing 27 physically based and empirical methods with runoff data from 308 catchments:

$$\begin{array}{}\text{(6a)}& {\displaystyle}& {\displaystyle}{T}_{\mathrm{a}}<\mathrm{5}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\lambda}{E}_{\mathrm{p}}=\mathrm{0},\text{(6b)}& {\displaystyle}& {\displaystyle}{T}_{\mathrm{a}}>\mathrm{5}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\lambda}{E}_{\mathrm{p}}={\displaystyle \frac{{R}_{e}}{{\mathit{\rho}}_{\mathrm{a}}}}{\displaystyle \frac{\left({T}_{\mathrm{a}}-\mathrm{5}\right)}{{\mathit{\alpha}}_{\mathrm{Ou}}}},\end{array}$$

with *T*_{a} the air temperature (^{∘}C) and
*R*_{e} the top-of-atmosphere radiation (MJ m^{−2} day^{−1}), depending on
latitude and Julian day. Oudin et al. (2005a) suggested the use of *α*_{Ou}=100.
This value will be used, in addition to a biome-specific value.

Finally, the third temperature-based method is the Hargreaves and Samani (1985)
formulation, which includes minimum (*T*_{min}) and maximum (*T*_{max})
daily temperature, next to *T*_{a} and *R*_{e}:

$$\begin{array}{}\text{(7)}& {\displaystyle}{\displaystyle}\mathit{\lambda}{E}_{\mathrm{p}}={\mathit{\alpha}}_{\mathrm{HS}}{R}_{e}\left({T}_{\mathrm{a}}+\mathrm{17.8}\right)\sqrt{{T}_{\mathrm{max}}-{T}_{\mathrm{min}}},\end{array}$$

with *α*_{HS} a constant, normally assumed to equal 0.0023. As for
the other methods, we additionally apply a biome-specific value. A detailed
description of the calibration of all *E*_{p} methods is given in Sect. 2.5.

The Tier 2 FLUXNET2015 database based on half-hourly or hourly measurements
from eddy-covariance sensors is used to evaluate the different *E*_{p}
formulations (http://fluxnet.fluxdata.org/data/fluxnet2015-dataset/, last access: 14 February 2019). Sites lacking at
least one of the basic measurements required for our analysis
(i.e. *R*_{n}; *G*; *λ**E*_{a}; *H*; precipitation; wind speed, *u*; friction
velocity, *u*_{*}; *T*_{a}; and relative humidity, RH, or VPD) were not considered further.
For latent and sensible heat fluxes, we used the data corrected
by the Bowen ratio method. In this approach, the Bowen ratio is assumed to
be correct, and the measured *λ**E*_{a} and *H* are multiplied by a
correction factor derived from a moving window method; see http://fluxnet.fluxdata.org/data/fluxnet2015-dataset/data-processing/ (last access: 14 February 2019) for
a detailed description of this standard procedure. Nonetheless, taking the
uncorrected *λ**E*_{a} instead did not impact the main findings (not
shown). For the main heat fluxes (*G*, *H*, *λ**E*_{a}), medium and poor
gap-filled data were masked out according to the flags provided by FLUXNET.
As no quality flag was available for *R*_{n} measurements, the flag of the
shortwave incoming radiation was used instead. All negative values for *H*
or *λ**E*_{a} were masked out, as these relate to periods of interception
loss and condensation when accurate measurements are not guaranteed
(Mizutani et al., 1997). Similarly, all negative values of *R*_{n} were masked out.

Finally, sub-daily measurements were aggregated to daytime composites (i) by
applying a minimum threshold of 5 W m^{−2} of top-of-atmosphere incoming
shortwave radiation and (ii) after excluding the first and last (half-)hours
from these aggregates. Based on these daytime aggregates, the daytime means
of *s*, *γ* and *ρ*_{a} were calculated using the parameterisation
procedure described by Allen et al. (1998). We used *T*_{a} to calculate *s*.
Only days in which more than 70 % of the data were measured directly
were retained, and days with rainfall (between midnight and sunset) were
removed from the analyses to avoid the effects of rainfall interception.
Furthermore, only sites with at least 80 retained days were used for the
further analysis. The global distribution of the final selection of sites is
shown in Fig. 1 and detailed information about these sites is provided in
Table S1 of the Supplement. The IGBP classification was used to assign a biome to each site.

Estimates of *r*_{aH} are required for the Penman and Penman–Monteith
equations. The resistance of heat transfer to air, *r*_{aH}, was calculated as

$$\begin{array}{ll}{\displaystyle}{r}_{\mathrm{aH}}& {\displaystyle}={\displaystyle \frac{u}{{u}_{*}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{k{u}_{*}}}\left[\mathrm{ln}\left({\displaystyle \frac{{z}_{\mathrm{0}\mathrm{m}}}{{z}_{\mathrm{0}\mathrm{h}}}}\right)+{\mathrm{\Psi}}_{\mathrm{m}}\left({\displaystyle \frac{z-d}{L}}\right)-{\mathrm{\Psi}}_{\mathrm{m}}\left({\displaystyle \frac{{z}_{\mathrm{0}\mathrm{h}}}{L}}\right)\right.\\ \text{(8)}& {\displaystyle}& {\displaystyle}\left.-{\mathrm{\Psi}}_{\mathrm{h}}\left({\displaystyle \frac{z-d}{L}}\right)+{\mathrm{\Psi}}_{\mathrm{h}}\left({\displaystyle \frac{{z}_{\mathrm{0}\mathrm{h}}}{L}}\right)\right],\end{array}$$

in which *k*=0.41 is the von Kármán constant; *z* the (wind) sensor
height (m); *d* the zero displacement height (m); *z*_{0m} and *z*_{0h} the roughness lengths
for momentum and sensible heat transfer (m), respectively; *L* the Obukhov
length (m); and Ψ_{m}(*X*) and Ψ_{h}(*X*) the Businger–Dyer
stability functions for momentum and heat for the variable *X*,
respectively. These were calculated based on the equations given by
Garratt (1992) and Li et al. (2017) for stable, neutral and unstable
conditions. Note that, in neutral and stable conditions, Ψ_{m}(*X*)=Ψ_{h}(*X*)
and that ${\mathrm{\Psi}}_{\mathrm{m}}\left(\frac{z-d}{L}\right)-{\mathrm{\Psi}}_{\mathrm{m}}\left(\frac{{z}_{\mathrm{0}\mathrm{h}}}{L}\right)-{\mathrm{\Psi}}_{\mathrm{h}}\left(\frac{z-d}{L}\right)+{\mathrm{\Psi}}_{\mathrm{h}}\left(\frac{{z}_{\mathrm{0}\mathrm{h}}}{L}\right)=\mathrm{0}$.
This is not the case for the unstable
conditions that usually prevail during the daytime. Daytime averages of all
variables were used as input in Eq. (8). The sensor height *z* was collected
individually for each tower through online and literature research or
through personal communication with the towers' principal investigator.
The Obukhov length *L* was calculated as

$$\begin{array}{}\text{(9)}& {\displaystyle}{\displaystyle}L={\displaystyle \frac{-{u}_{*}^{\mathrm{3}}{\mathit{\rho}}_{\mathrm{a}}{T}_{\mathrm{a}}\left(\mathrm{1}+\mathrm{0.61}{q}_{a}\right){c}_{p}}{kgH}},\end{array}$$

with *q*_{a} being the specific humidity (kg kg^{−1}) and *g*=9.81 m s^{−2}
the gravitational acceleration.

The displacement height *d* and the roughness length for momentum flux *z*_{0m}
were estimated as a function of the canopy vegetation height (VH),
as *d*=0.66 VH and *z*_{0m}=0.1 VH (Brutsaert, 1982). VH was estimated from
the flux tower measurements using the approach of Pennypacker and Baldocchi (2016):

$$\begin{array}{}\text{(10)}& {\displaystyle}{\displaystyle}\mathrm{VH}={\displaystyle \frac{z}{\mathrm{0.66}+\mathrm{0.1}\mathrm{exp}\left(\frac{ku}{{u}_{*}}\right)}}.\end{array}$$

This equation was applied to the full (half-)hourly database and only when
conditions were near-neutral ($|z/L|<\mathrm{0.01}$) and friction velocities lower than 1 standard
deviation below the mean *u*_{*}
at each site. The daily VH was then aggregated by averaging out
the half-hourly estimates to daily values, excluding the 20 % outliers of
the data, and then calculating a 30-day-window moving average on the
dataset. When not enough (half-)hourly vegetation height observations (<160)
were available, the site was excluded from the analysis.
This gave robust results for all remaining sites; an example of VH temporal
development for a specific location is given in Fig. 2a. For homogeneous
canopies, the VH calculated this way represents the true vegetation height.
For savannah-like ecosystems, it corresponds to the vegetation height that
the vegetation would have if it was represented by a single big leaf model.

The Stanton number (defined as $k{B}^{-\mathrm{1}}=\mathrm{ln}\left({z}_{\mathrm{0}\mathrm{m}}{z}_{\mathrm{0}\mathrm{h}}^{-\mathrm{1}}\right)$) was
calculated by assuming that the surface aerodynamic temperature *T*_{0}
(defined by $H={\mathit{\rho}}_{\mathrm{a}}{c}_{p}\frac{({T}_{\mathrm{0}}-{T}_{\mathrm{a}})}{{r}_{\mathrm{aH}}}$) is
equal to the radiative surface temperature *T*_{s} derived from the longwave
fluxes. Then, through an iterative approach, an optimal value of *z*_{0h} was
obtained, using the following equations for *T*_{0} (Garratt, 1992)
and *T*_{s} (Maes and Steppe, 2012):

$$\begin{array}{ll}{\displaystyle}{T}_{\mathrm{0}}& {\displaystyle}={T}_{\mathrm{a}}+\left({\displaystyle \frac{H}{k{u}_{*}{\mathit{\rho}}_{\mathrm{a}}{c}_{p}}}\right)\left[\mathrm{ln}\left({\displaystyle \frac{z-d}{{z}_{\mathrm{0}\mathrm{h}}}}\right)-{\mathrm{\Psi}}_{\mathrm{h}}\left({\displaystyle \frac{z-d}{L}}\right)\right.\\ \text{(11)}& {\displaystyle}& {\displaystyle}\left.+{\mathrm{\Psi}}_{\mathrm{h}}\left({\displaystyle \frac{{z}_{\mathrm{0}\mathrm{h}}}{L}}\right)\right],\end{array}$$

$$\begin{array}{}\text{(12)}& {\displaystyle}{\displaystyle}{T}_{\mathrm{s}}=\sqrt[\mathrm{4}]{{\displaystyle \frac{{\mathrm{LW}}_{\mathrm{out}}-(\mathrm{1}-\mathit{\epsilon}){\mathrm{LW}}_{\mathrm{in}}}{\mathit{\sigma}\mathit{\epsilon}}}},\end{array}$$

with *σ* the Stefan–Boltzmann constant and *ε* the
emissivity. The (half-)hourly data were used for this
calculation. Following the approach of Li et al. (2017), only summertime
data were used, and only when *H* was larger than 20 W m^{−2} and
*u*_{*} larger than 0.01 m s^{−1}. Summertime was defined as those months in which
the maximal daily value of *H* is at least 85 % of the 98th percentile of *H*
based on the entire tower time series. In addition,
(half-)hourly observations with counter-gradient heat fluxes were excluded
from the analysis. For each observation, *z*_{0h} was optimised by minimising
the difference between *T*_{0} and *T*_{s}. Then, *k**B*^{−1} was calculated at
each site based on its relation with the observed Reynolds number (*R**e*) by
fitting the following function, based on the work by Li et al. (2017):

$$\begin{array}{}\text{(13)}& {\displaystyle}{\displaystyle}k{B}^{-\mathrm{1}}={a}_{\mathrm{0}}+{a}_{\mathrm{1}}R{e}^{a\mathrm{2}}.\end{array}$$

Note that Eq. (12) requires a value for *ε*, which is often
assumed to be equal to 0.98 for all sites (e.g. Li et al., 2017; Rigden and
Salvucci, 2015). Under the assumption that *T*_{0}=*T*_{s},
*ε* can also be calculated separately per site. If *H*=0, it follows that
*T*_{0}=*T*_{a} and, from Eq. (12),

$$\begin{array}{}\text{(14)}& {\displaystyle}{\displaystyle}\mathit{\epsilon}={\displaystyle \frac{{\mathrm{LW}}_{\mathrm{out}}-{\mathrm{LW}}_{\mathrm{in}}}{\mathit{\sigma}{T}_{\mathrm{a}}^{\mathrm{4}}-{\mathrm{LW}}_{\mathrm{in}}}}.\end{array}$$

Here, *ε* was calculated for each site using (half-)hourly data,
selecting those measurements where *H* was close to 0 ($-\mathrm{2}<H<\mathrm{2}$ W m^{−2})
and excluding rainy days as well as measurements in which the
albedo (calculated as *α* = SW_{out}SW${}_{\mathrm{in}}^{-\mathrm{1}}$ with
SW_{in} the incoming and SW_{out} the outgoing shortwave radiation) was above 0.4,
to avoid influences of snow or ice. Negative estimates of *ε*
were masked out, and the *ε* of the site was calculated as the
mean, after excluding the outlying 20 % of the data. Equation (3) was
applied both with a fixed *ε* of 0.98 and with the observed *ε*,
and the *ε* value yielding the lowest RMSE in Eq. (12) was retained
for each site. An example of such a function between *k**B*^{−1} and *R**e* is shown in Fig. 2b.

Finally, the surface resistance *r*_{c} (s m^{−1}) was calculated by
inverting the Penman–Monteith equation as

$$\begin{array}{}\text{(15)}& {\displaystyle}{\displaystyle}{r}_{\mathrm{c}}={\displaystyle \frac{s\left({R}_{\mathrm{n}}-G\right){r}_{\mathrm{aH}}+{\mathit{\rho}}_{\mathrm{a}}{c}_{p}\mathrm{VPD}}{\mathit{\gamma}\mathit{\lambda}{E}_{\mathrm{a}}}}-{\displaystyle \frac{(s+\mathit{\gamma}){r}_{\mathrm{aH}}}{\mathit{\gamma}}}.\end{array}$$

We converted the *r*_{c} estimates to surface conductance *g*_{c}
(mm s^{−1}) using ${g}_{\mathrm{c}}=\mathrm{1000}{r}_{\mathrm{c}}^{-\mathrm{1}}$; we will continue
using *g*_{c} (rather than *r*_{c}) for the remainder of this paper.
Note that the approach of calculating *k**B*^{−1} directly requires a separate measurement
of LW_{in} and LW_{out}, which was only available in 95 of the 107 selected
sites. For the remaining sites, an alternative approach was used to
calculate *k**B*^{−1} (see Supplement).

We include two different approaches to identify a subset of measurements per eddy-covariance site in which the ecosystem was unstressed and provide the results for both methods. A first approach is based on soil moisture levels. For those sites where soil moisture measurements were available, the maximal soil moisture level for each site was determined as the 98th percentile of all soil moisture measurements. We then split up the dataset of each site in five equal-size classes based on evaporation percentiles. For each class, days having soil moisture levels belonging to the highest 5 % of soil moisture levels within each class were selected as unstressed days, but only if the soil moisture level of these selected days was above 75 % of the maximum soil moisture. This guaranties the sampling of unstressed evaporation during all seasons.

As soil moisture data were not available for a large number of sites, and
because using soil moisture data does not exclude days in which evaporation
may be constrained by other kinds of biotic or abiotic stress, a second
approach for defining unstressed days was applied based on an energy balance
criterion. We calculated the EF from the daytime *λ**E*_{a} and
*H* values and considered it as a direct proxy for evaporative stress; i.e. we
assumed that, under unstressed conditions, a larger fraction of the available
energy is used to evaporate water (Gentine et al., 2007, 2011a; Maes et al.,
2011). This approach is similar to those of other *E*_{p} studies using
eddy-covariance or lysimeter data, in which the Bowen ratio (e.g. Douglas et
al., 2009) or the ratio of *λ**E*_{a} and (SW_{in} + LW_{in})
(Pereira and Pruitt, 2004) are used to define unstressed days. The
unstressed record was comprised of all days with EF exceeding the 95th percentile
EF threshold for each particular site or, if fewer than 15 days
fulfilled this criterion, the 15 days with the highest EF. Consequently, we
assume that at each site during at least 5 % of the days the conditions
are such that evaporation is unstressed and *E*_{a} reflects *E*_{p}. The
measured *E*_{a} from the identified unstressed days is further referred to
as *E*_{unstr} (mm day^{−1}) and used as reference data to evaluate the
different *E*_{p} methods.

To assess whether the atmospheric conditions of the unstressed datasets are
representative for the FLUXNET sites as such, a random bootstrap sample
having the same number of records as the unstressed dataset was taken from
the entire dataset of daily records. The mean, standard deviation, and 2nd and
98th percentile of SW_{in}, *T*_{a}, VPD and *u* were calculated.
This procedure was repeated 1000 times per site. A *t* test comparing the
values of the unstressed subsample with those of the 1000 random samples was
used to analyse whether the atmospheric conditions of the unstressed subsample
were representative for the overall site conditions. This analysis was
carried out for both methods to select unstressed days: the soil moisture
threshold and the energy balance criterion.

An overview of the different methods to calculate *E*_{p} is given in
Table 1. If possible, three versions of each method were calculated: (1) a
reference crop version, (2) a standard version and (3) a biome-specific
version. The reference crop version calculates *E*_{p} for the reference
short turf grass crop, with *R*_{n} and other properties of this
reference crop as well. The standard version uses the same
non-biome-specific parameters of the reference crop but considers *R*_{n}
and other properties of the actual ecosystem. The biome-specific version of
each method applies a calibration of the key parameters per biome (Table 1)
and considers *R*_{n} and other properties of the actual ecosystem. These
calibrated values per biome are based on the mean value of this key
parameter for the unstressed dataset for each site, averaged out per biome type.

*N* is the number of daylight hours;
*T*_{max} and *T*_{min} are the maximum and minimum daily air
temperature; RH_{max} and RH_{min} are the maximum and minimum RH;
SW^{*} and LW^{*} are the net shortwave and net longwave
radiation; SW_{TOA} is the shortwave incoming radiation at the top of the
atmosphere; FAO-56 refers to the methodology described by Allen et al. (1998).

To estimate the radiation and crop properties of the reference crop
versions, the equations described by Allen et al. (1998) in the FAO-56
method (Food and Agricultural Organization) were used and G was considered
to be 0. *R*_{n} was calculated as

$$\begin{array}{}\text{(16)}& {\displaystyle}{\displaystyle}{R}_{\mathrm{n}}={\mathrm{SW}}_{\mathrm{in}}\left(\mathrm{1}-{\mathit{\alpha}}_{\mathrm{ref}}\right)+{\mathrm{LW}}^{*},\end{array}$$

where *α*_{ref}=0.23 (Allen et al., 1998) and LW^{*} is the net
longwave radiation, calculated after Allen et al. (1998; Eq. 39, Sect. 3).

In the case of the reference crop version of the Penman–Monteith equation
(Eq. 1), the FAO-56 method was used as described by Allen et al. (1998),
with *g*_{c_ref} fixed as 14.49 mm s^{−1} (corresponding
with *r*_{c_ref}=69 s m^{−1}) and using Eq. (16) to
calculate *R*_{n}. The standard version of the Penman–Monteith equation used
observed (*R*_{n}, *G*, VPD) and calculated (*s*, *γ*, *ρ*_{a},
*r*_{aH}) daytime values as described in Sect. 2.2 in Eq. (1), and it also
assumed *g*_{c_ref}=14.49 mm s^{−1}. The biome-specific
version was calculated with the same data but used a biome-dependent value
of *g*_{c}. First, for each individual site, the unstressed *g*_{c} was
calculated as the mean of the *g*_{c} values of the unstressed record (see
Sect. 2.4). The mean value per biome was then calculated from these
unstressed *g*_{c} values. Regarding the Penman method (Eq. 2), the
reference crop and standard versions were calculated using the same input
data as for the Penman–Monteith methods; given Penman's consideration of no
surface resistance, no biome-specific version was calculated.

The reference crop version of the Priestley and Taylor method was calculated
from Eq. (3) with *R*_{n} from Eq. (16), *s* and *γ* from the FAO-56
calculations, and with *α*_{PT}=1.26. The standard version used
the same value for *α*_{PT} but the observed daytime values
for *R*_{n} and *G*. The biome-specific version followed a calibration
of *α*_{PT} similar to the *g*_{c_ref} calculation. For each site,
the unstressed *α*_{PT} was calculated as the average *α*_{PT},
obtained by solving Eq. (3) for *α*_{PT} using the
unstressed dataset. Finally, the mean per biome was calculated and used in
the *E*_{p} estimation. Regarding the method by Milly and Dunne (2016)
(Eq. 4), the reference crop, standard and biome-specific calculation were
calculated accordingly, with *R*_{n} from Eq. (16) for the reference crop
version, *α*_{MD}=0.8 for the reference crop and standard version,
and a calibrated *α*_{MD} by biome type for the biome-specific version.

For the Thornthwaite, Oudin, and Hargreaves–Samani formulations (Eqs. 5–7),
only the standard and biome-specific versions were calculated.
The standard version of Thornthwaite's formulation used *α*_{Th}=16.
In the biome-specific version, this parameter was again
calculated per site as the mean value of the unstressed records (e.g. Xu and
Singh, 2001; Bautista et al., 2009), and then averaged per biome type. The
effective temperature *T*_{eff} was calculated as
${T}_{\mathrm{eff}}=\mathrm{0.36}(\mathrm{3}{T}_{\mathrm{max}}-{T}_{\mathrm{min}})$
(Camargo et al., 1999). The parameter *b* was calculated as
$b=(\mathrm{6.75}\times {\mathrm{10}}^{-\mathrm{7}}{I}^{\mathrm{3}})(\mathrm{7.71}\times {\mathrm{10}}^{-\mathrm{7}}{I}^{\mathrm{2}})+\mathrm{0.0179}I+\mathrm{0.492}$
and the parameters *c*–*e* in Eq. (5c) were 415.85,
32.24 and 0.43, respectively (Pereira and Pruitt, 2004). For Oudin's
temperature-based formulation, *α*_{Ou}=100 was taken for the
standard version (Eq. 6). In the biome-specific version, this value was
recalculated by calculating *α*_{Ou} for the unstressed records
through Eq. (6), calculating the mean *α*_{Ou} per site and finally
the biome-dependent *α*_{Ou}. Similarly, for the Hargreaves–Samani
method, *α*_{HS}=0.0023 is used for the standard version (Eq. 7),
whereas, in the biome-specific version, this value was calculated using
the unstressed records. Altogether, this exercise yielded a total of
17 different methods to estimate *E*_{p}, whose specificities are
documented in Table 1.

The influence of climatic forcing data on *E*_{unstr} and
on *g*_{c_ref}, *α*_{PT} and *α*_{MD} was
investigated. This was done by calculating for each individual site the
correlations between the daily estimates of the atmospheric conditions and
the daily values of the unstressed datasets. Analyses were then performed on
these correlations of all sites.

3 Results

Back to toptop
Table S2 provides an overview of the analyses used to verify if the climatic forcing data of the unstressed subsets were representative for the atmospheric conditions of the sites as such. For the subsets of both unstressed criteria, atmospheric conditions were very representative for the site conditions, including for VPD. For the energy balance criterion, for instance, only at one site, the unstressed subset of the 98th percentile was significantly different from the random sampling-based simulations and in only two sites the 2nd percentile was significantly different from the simulations.

We focus here on the parameter estimates of the unstressed record based on
the energy balance criterion (Sect. 2.4). Of the full dataset, 107 flux
sites meet all the selection criteria (i.e. at least 80 days without
rainfall, good quality measurements of radiation and main fluxes, and at
least 160 vegetation height observations – see Sect. 2.2 and 2.3).
Despite considerable variation, *g*_{c_ref} does not differ
statistically across biomes, in contrast with *α*_{PT} and *α*_{MD}.
Overall, croplands (CRO) are characterised by a higher
measured *E*_{unstr}, which translates into the highest *g*_{c_ref},
*α*_{PT} and *α*_{MD} of all biomes. Grasslands (GRA),
deciduous broadleaf forest (DBF) and evergreen broadleaf forest (EBF) also
have a relatively high *g*_{c_ref}, *α*_{PT} and
*α*_{MD}, whereas mixed forest (MF) and savannah ecosystems (closed
shrubland, CSH; woody savannah, WSA; open shrublands, OSH; and
savannah, SAV, ecosystems) are characterised by lower *g*_{c_ref},
*α*_{PT} and *α*_{MD}. Only five sites (DE-Kli and IT-BCi, CRO;
CA-SF3, OSH; AU-Rig, GRA; and AU-Wac, EBF) have mean values of *α*_{PT} higher than the typically
assumed 1.26 (Table 2). In contrast, 27 sites, including 9 croplands, have a
mean value of *α*_{MD} above 0.80, and 42 sites have
a mean *g*_{c_ref} above 14.49 mm s^{−1}. Finally, wetlands (WET)
show a large standard deviation of *α*_{PT} and *α*_{MD}
(Table 2) due to their location in tropical, temperate and in arctic
regions. The parameters of the three temperature-based methods differed
significantly across biomes, but trends were different for each key
parameter and did not always match those for *g*_{c_ref},
*α*_{PT} and *α*_{MD} (Table 2).

CRO: cropland; DBF: deciduous broadleaf forest; EBF: evergreen broadleaf forest; ENF: evergreen needleleaf forest; MF: mixed forest; CSH: closed shrubland; WSA: woody savannah; SAV: savannah; OSH: open shrubland; GRA: grasslands; WET: wetlands.

Next, the effect of the atmospheric conditions on *E*_{unstr} and on the
key parameters *g*_{c_ref}, *α*_{PT} and *α*_{MD} of
the unstressed dataset is investigated. Figure 3 gives the
distribution of the correlations between the climatic variables (*R*_{n},
*T*_{a}, VPD, *u* and [CO_{2}]) and *E*_{unstr}, *g*_{c_ref},
*α*_{PT} and *α*_{MD} of the unstressed records at each site. We did
not include *α*_{Th}, *α*_{HS} or *α*_{Ou} because
the temperature-based methods did not perform well (see Sect. 3.3).
*E*_{unstr} was strongly positively correlated with *R*_{n}, *T*_{a} and
VPD in most sites, but less with *u* (Fig. 3a, Table 3). Considering all sites, the
correlation between *g*_{c_ref} and the climatic variables is
not significantly different from zero for any climate variable, yet
*g*_{c_ref} is significantly negatively correlated with *T*_{air}
and with VPD in 40 % and 45 % of the flux tower sites,
respectively (Table 3, Fig. 3b). The two other parameters,
*α*_{PT} and *α*_{MD}, appear less correlated to any of the climatic
variables across all sites (Table 3). In the case of *α*_{MD}, in
particular, the distributions of the correlations against all climate
forcing variables peak around zero (Fig. 3d): *α*_{MD} is hardly
influenced by *R*_{n} and is overall not dependent on *u*, *T*_{a},
[CO_{2}] or VPD in most sites (Table 3).

We first list the results of the analysis using the energy balance criterion
for selecting the unstressed records (Sect. 2.4). The scatterplots of
measured *E*_{unstr} versus estimated *E*_{p} based on the 17 different
methods are shown in Fig. 4 for three sites belonging to different biomes.
Despite the overall skill shown by the different *E*_{p} methods,
considerable differences can be appreciated. In general, the methods
designed for reference crops (PM_{r}, Pe_{r}, PT_{r}, MD_{r})
overestimate *E*_{unstr} and only two methods, MD_{B} and PT_{B}, match
the measured *E*_{unstr} closely.

Table 4 gives the mean correlation per biome for each method. The results
are very consistent and reveal that the highest correlations for nearly all
biomes are obtained with the standard and biome-specific radiation-based
method (MD_{s} and MD_{b}), closely followed by the standard and
biome-dependent Priestley and Taylor method (PT_{s} and PT_{b}).
Temperature-based methods have the lowest overall mean correlation as well
as lower mean correlations per biome, with the Hargreaves–Samani method
performing slightly better than the other two temperature-based methods.
Note that the correlations are the same for the standard and biome-specific
version in the case of the Priestley and Taylor (PT_{s} and PT_{b}),
radiation-based (MD_{s} and MD_{b}), Oudin (Ou_{s} and Ou_{b}), and
Hargreaves–Samani (HS_{s} and HS_{b}) methods (Table 4) – this is to be
expected, as the only difference between the standard and biome-specific
version of these methods is the value of their key parameters (*α*_{PT},
*α*_{MD}, *α*_{Ou}, *α*_{HS}), which are
multiplicative (see Eqs. 3, 4, 6 and 7). Differences are however
reflected in the unbiased root mean square error (unRMSE) and mean bias – see
Tables 5 and 6. The biome-specific versions of the radiation-based
method (MD_{b}) and of the Priestley and Taylor method (PT_{b}) consistently have the lowest unRMSE for all biomes. Though the difference between
these two methods is small, MD_{b} performs slightly better. The
standard Penman method (Pe_{s}) has the highest unRMSE. All reference crop
methods (PM_{r}, Pe_{r}, PT_{r}, MD_{r}) have a mean unRMSE above
1 mm day^{−1}, and the temperature-based methods (Th_{s}, Ou_{s},
HS_{s}, Th_{b}, Ou_{b} and HS_{b}) also have a relatively high unRMSE.
Finally, bias estimates are given in Table 6. Again, MD_{b} is overall the
best-performing method (mean bias closest to 0 mm day^{−1}), closely
followed by the PT_{b} method. Both methods consistently have the bias
closest to zero among all biomes, except for wetlands. Most reference crop
methods (PM_{r}, Pe_{r}, PT_{r}, MD_{r}), as
well as Pe_{s}, overestimate *E*_{p} in all biomes.

The use of soil water content as the criterion to select unstressed days (see
Sect. 2.4) is explored. In total, 62 sites have soil water content data and
meet the other selection criteria documented in Sect. 2.2. The results of
this analysis are given in Tables S3–S5. To allow
for a fair comparison, the same statistics have also been computed for just
the same 62 tower sites with the energy balance criterion (Tables S6–S8).
Using the soil moisture criterion, the correlations are overall lower and
the results of the mean correlation, unRMSE and biases are less consistent.
However, the overall performance ranking of the different models remains
similar: PT_{b} is the best-performing method with overall the highest
mean correlation (*R*=0.84) and the lowest unRMSE (0.78 mm day^{−1}) and a bias
close to zero (−0.04), closely followed by the MD_{b} method, with
*R*=0.81, unRMSE = 0.89 and a mean bias of −0.12. More complex Penman-based
models, and especially the empirical temperature-based formulations, show
again a lower performance.

So far, all flux sites were used to calibrate the key parameters (Table 2)
and those same sites were also used for the evaluation of the different
methods. This was done to maximise the sample size. However, to avoid
possible overfitting, we also repeated the analysis after separating
calibration and validation samples. For each biome, two-thirds of the sites
were randomly selected as calibration sites and one-third as validation
sites. The key parameters were then calculated from the calibration subset
and applied to estimate *E*_{p} of the biome-specific methods of the
validation subset. This procedure was repeated 100 times and the mean
correlation, unbiased RMSE and bias per biome were calculated. Results show
no substantial differences in overall correlation, unRMSE and bias of each
method, which are provided in Tables S9–S11 for completeness.

4 Discussion

Back to toptop
We prioritised the energy balance over the soil moisture criterion to select
unstressed days (see Sect. 2.4), because it can be applied to sites without
soil moisture measurements and because it implicitly allows the exclusion of
days in which the ecosystem is stressed for reasons other than soil moisture
availability (e.g. insect plagues, phenological leaf-out, fires, heat and
atmospheric dryness stress, nutrient limitations). In addition, soil
moisture at specific depths can be a poor indicator of water stress, as
rooting depth can vary and is not accurately measured (Powell et al.,
2006; Douglas et al., 2009; Martínez-Vilalta et al., 2014). This is
confirmed by our results: sampling unstressed days based on the energy-balance-based
criterion resulted in higher correlations between *E*_{p}
and *E*_{unstr} for all methods (Table S6 versus Table S3) and in lower unRMSE, with
the exception of the temperature-based methods (Table S7 versus Table S4).

Nonetheless, it could be argued that, because the MD method assumes a
constant evaporative fraction, the use of the evaporative fraction as a
criterion for selecting unstressed days may favour the MD and even the
closely related PT formulation. However, the soil moisture criterion adopted
here provides an independent check of the results and confirms the robust
and superior performance of the energy-driven PT_{b} and MD_{b} methods,
independently of the framework chosen to select unstressed days. In the
following discussion, the primary focus is on the results of selecting
unstressed days based on the energy balance criterion.

The resulting biome-specific values of the key parameters in Table 2 are
within the range of values used in reference crop and standard applications
of the models (Table 1), with the exception of *α*_{PT}, which is
typically lower than the frequently adopted value of 1.26. Other studies
also found *α*_{PT} values far below 1.26 but within the range of
our study, mainly for forests (e.g. Shuttleworth and Calder,
1979; Viswanadham et al., 1991; Eaton et al., 2001; Komatsu, 2005) but also
for tundra (Eaton et al., 2001) and grassland sites (Katerji and Rana, 2011) – see
McMahon et al. (2013) for an overview. Our results and these studies
demonstrate that the standard level of *α*_{PT}=1.26 is close to
the upper bound and will lead to an overestimation of *E*_{p} at most flux
sites (Table 5).

The poor performance of the PM_{r}, PM_{s} and PM_{b} methods was
relatively unexpected. Because the Penman–Monteith method incorporates the
effects of *T*_{a}, VPD, *R*_{n} and *u*, it is often considered superior
(e.g. Sheffield et al., 2012) and is even used as reference to evaluate other
*E*_{p} formulations (e.g. Xu and Singh, 2002; Oudin et al., 2005b; Sentelhas
et al., 2010). However, in studies dedicated to estimate *E*_{a} at
eddy-covariance sites, in which *g*_{c} is adjusted so it reflects the actual
stress conditions in the ecosystem, the Penman–Monteith method has already
been shown to perform worse than other simpler methods, such as the
Priestley and Taylor equation (e.g. Ershadi et al., 2014; Michel et al.,
2016). It is well known that its performance depends on the reliability of a
wide range of input data and on the methods used to derive *r*_{aH}
and *g*_{c} (Singh and Xu, 1997; Dolman et al., 2014; Seiller and Anctil, 2016). In
our case, the strict procedure followed to select the samples of 107 FLUXNET
sites (see Sect. 2.2) ensured that all relevant variables were available,
and the meteorological measurements were quality-checked. Hence, in our
analysis, input quality is unlikely to be the cause of low performance.

We believe that the underlying assumption of a constant *g*_{c}, typically
adopted by PM methods (PM_{r}, PM_{s}, PM_{b}) when
estimating *E*_{p}, is a more likely explanation of the poor performance. PM was the
only method in which the biome-specific calibration did not improve the
performance. This is partially because of the large variation in *g*_{c_ref}
among the different flux sites of the same biome
type (Table 2). In addition, of all the key parameters,
*g*_{c_ref} showed the largest mean relative standard deviation
of the unstressed datasets of individual sites (results not shown). Surface conductance of
the unstressed dataset was significantly negatively correlated with VPD in
45 % of the sites (Fig. 3b, Table 3). The relationship between *g*_{c}
and VPD for two of these sites is illustrated in Fig. 5. It is clear that
*g*_{c} of unstressed days (red dots) is always high for a given VPD,
confirming the validity of the energy balance method. However, for these
sites, it is also shown that *g*_{c} of the unstressed days becomes very high
when VPD becomes very low. As a consequence, the mean value of *g*_{c} of the
unstressed records, used to ultimately calculate *g*_{c_ref} per
biome type, is highly influenced by local VPD and is not necessarily a
representative ecosystem property.

The dependence of *g*_{c} on VPD, even when soil moisture is not limiting, has
been well studied (e.g. Jones, 1992; Granier et al., 2000; Sumner and Jacobs,
2005; Novick et al., 2016) and incorporated in most conventional stomatal or
surface conductance models (e.g. Jarvis, 1976; Ball et al., 1987; Leuning,
1995). Yet, out of practical reasons, *g*_{c_ref} is usually
considered constant in *E*_{p} methods using the Penman–Monteith approach,
with the PM_{r} as the best illustration. Our data confirm that in unstressed
conditions stomata open maximally only when VPD is very low. As such, the
VPD dependence of *g*_{c} smooths the impact of VPD in the Penman–Monteith
equation – drops in VPD are compensated for by increases in *g*_{c}, and vice
versa, lowering the impact of VPD on *E*_{a} (Eq. 1). As such, assuming a
constant *g*_{c_ref} value overestimates the influence of VPD
(and wind speed) on *E*_{p}.

Moreover, assuming a constant *g*_{c_ref} value in the
Penman–Monteith method also ignores the influence of CO_{2} levels
on *g*_{c}. As a result, Milly and Dunne (2016) found that the Penman–Monteith
methods with constant *g*_{c_ref} overpredicted *E*_{p} in
models estimating future water use. Calibrating the sensitivity
of *g*_{c_ref} to VPD and [CO_{2}] in the Penman–Monteith
equation is outside the scope of this study, but could certainly
improve *E*_{p} calculations – yet, it would further increase the complexity of the
model. Finally, we note that the above discussion also applies to Penman's
method: taking a wet canopy as reference in the Penman method
(*g*_{c}=∞ or *r*_{c}=0) may not only overestimate *E*_{p}
(Table 6) but also the influence of VPD and *u* on *E*_{p}.

The simpler Priestley and Taylor and radiation-based methods came forward as
the best methods for assessing *E*_{p} with both criteria to define
unstressed days. These observations are in agreement with studies
highlighting radiation as the dominant driver of evaporation of saturated or
unstressed ecosystems (e.g. Priestley and Taylor, 1972; Abtew, 1996; Wang et
al., 2007; Song et al., 2017; Chan et al., 2018). They also agree with Douglas
et al. (2009), who found that PT outperformed the PM method for estimating
unstressed evaporation in 18 FLUXNET sites.

Both PT_{b} and MD_{b} are attractive from a modelling perspective, as
they require minimum input data. However, this simplicity can also hold
risks. The Priestley and Taylor method has been criticised for its implicit
assumptions, which are also present in the MD_{b} method. For instance, by
not incorporating wind speed explicitly, it is assumed that the effect
of *u* on *E*_{p} is somehow embedded within *α*_{PT}
(or *α*_{MD}). Yet, several studies indicate that wind speeds are decreasing
(“stilling”) globally (McVicar et al., 2008, 2012; Vautard et al., 2010).
McVicar et al. (2012) also reported an associated decreasing
trend in observed pan evaporation worldwide as well as in *E*_{p} calculated
with the PM_{r} method. With PT (or MD) methods, this trend cannot be
captured. A similar criticism can be drawn with regards to the effect of
[CO_{2}] on stomatal conductance, water use efficiency and thus potential
evaporation (Field et al., 1995). A separate question is whether more
complex *E*_{p} methods that incorporate the effects of *u*, [CO_{2}] or
VPD explicitly do this correctly; the above-mentioned issues about the fixed
parameterisation of the Penman–Monteith methods for estimating *E*_{p}
indicate that this may typically not be the case.

Regarding the non-explicit consideration of *u* by simpler methods, our records
show a limited effect of *u* on *E*_{a} and *E*_{p}, even when considering larger
temporal scales. Of the 16 flux towers with at least 10 years of evaporation
data, we calculated the yearly average *E*_{a} as well as the annual mean
climatic forcing variables. Yearly averages were calculated from monthly
averages, which in turn were calculated if at least three daytime
measurements were available. Despite a relatively large mean standard
deviation in yearly average *u* of 7.0 %, yearly average *u* was not
significantly correlated with *E*_{a} in any of these sites. In contrast,
yearly average *R*_{n} was positively correlated with yearly
average *E*_{a} in 7 of the 16 sites, with comparable mean standard deviation in
annual *R*_{n} (8.5 %). Moreover, looking at all individual towers and
using the daily estimates, neither *α*_{MD} nor *α*_{PT}
were correlated with *u* (Fig. 3c and d, Table 3). In fact, since
*α*_{MD} appears hardly affected by any climatic variable, and given the relatively
small range in *α*_{MD} values within each biome (Table 2), it
appears that *α*_{MD} is a robust biome property that can be used in
the seamless application of these methods at a global scale. The robustness
of *α*_{MD} as a biome property is furthermore confirmed by the analysis
with independent calibration and validation sites, which hardly affected the
unRMSE and bias (Tables S10 and S11).

The best-performing methods rely heavily on measurements of available energy
(*R*_{n}−*G*) (Eqs. 2 and 3). In Sect. 3.2, all *E*_{p} calculations used
available energy obtained during unstressed conditions. The question is
whether *E*_{p} can also be calculated correctly using the actual *R*_{n}
and *G* when the ecosystems are not unstressed. As mentioned in Sect. 1, there
is discussion on whether SW_{out} and LW_{out} should be considered
forcing variables or ecosystem responses (e.g. Lhomme, 1997; Lhomme and
Guilioni, 2006; Shuttleworth, 1993). Among other considerations, it is clear
that *T*_{s} will be lower if vegetation is healthy and soils are well-watered (Maes and Steppe, 2012), which results in lower LW_{out}
and higher *R*_{n}. Therefore, while using the observations of SW_{in}
and LW_{in} as forcing variables in the computation of *E*_{p} is defendable
(despite the potential atmospheric feedbacks that may derive from the
consideration of unstressed conditions), we agree that SW_{out},
LW_{out} and *G* should ideally reflect unstressed rather than actual
conditions to estimate *E*_{p}. Note that, in that case, *E*_{p} deviates
from ${E}_{{\mathrm{p}}_{\mathrm{0}}}$ in the complementarity relationship, in which atmospheric feedbacks
affecting incoming radiation or VPD are implicitly considered (Kahler and Brutsaert, 2006).

A method to derive the unstressed estimates of SW_{in} (through the
albedo, *α*), LW_{in} (through *T*_{s}) and *G* under stressed
conditions is presented in the Sect. S2 and is based on the
MD_{b} method and on flux data of the unstressed datasets. We further
refer to this method as the “unstressed” *E*_{p}. It requires a large amount of
input data and is not practically applicable at a global scale. Comparing the
mean unstressed *E*_{p} with the “actual” *E*_{p} (i.e. *E*_{p} calculated
from the actual *R*_{n} and *G*) for all sites reveals that the
actual *E*_{p} is 8.2±10.1 % lower than the unstressed *E*_{p}. There are no
significant differences between biomes, but the distribution of the
underestimation is left-skewed and the actual *E*_{p} is more than 10 %
lower than the unstressed *E*_{p} in 22 % of the sites (Fig. 6). The main
reason explaining about 65 % of the difference between the actual and the
unstressed *E*_{p} is the difference in *T*_{s}. Assuming that the
unstressed *T*_{s} can be estimated as the mean of *T*_{a} and the
actual *T*_{s} results in a straightforward alternative to approximate the
unstressed *E*_{p} with only data of *T*_{a} and radiation:

$$\begin{array}{ll}{\displaystyle}{E}_{\mathrm{p}}& {\displaystyle}={\mathit{\alpha}}_{\mathrm{MD}}\left((\mathrm{1}-\mathit{\alpha}){\mathrm{SW}}_{\mathrm{in}}+\mathit{\epsilon}{\mathrm{LW}}_{\mathrm{in}}-\mathrm{0.5}\mathit{\epsilon}{\mathrm{LW}}_{\mathrm{out}}\right.\\ \text{(17)}& {\displaystyle}& {\displaystyle}\left.-\mathrm{0.5}\mathit{\epsilon}\mathit{\sigma}{T}_{\mathrm{a}}^{\mathrm{4}}-G\right).\end{array}$$

This approach was also tested and resulted in a mean underestimation
of *E*_{p} of 2.6±5.8 % compared to the unstressed *E*_{p}, with a
mean median value at −0.1 % (Fig. 6). Given the low error and the
straightforward calculation, we recommend this method to calculate *E*_{p} at
global scales.

Figure 7 gives an example of the seasonal evolution of *E*_{a} and *E*_{p} and
the *S* factor ($S={E}_{\mathrm{a}}{E}_{\mathrm{p}}^{-\mathrm{1}}$) in a grassland (Fig. 7a) and a
deciduous forest site (Fig. 7b). The short growing season in the grassland
site, when *E*_{a} is close to *E*_{p} and values of *S* are close to 1, stands
in clear contrast with the winter period, when grasses have died off
and *E*_{a} and consequently also *S* are very low. In the relatively wet broadleaf
forest, *E*_{a} and *E*_{p} follow a similar seasonal cycle. In winter, when
total evaporation is limited to soil evaporation, *S* is very low; in spring,
when leaves are still developing, *E*_{a} lags *E*_{p}. In summer,
*S* remains high and close to one.

5 Conclusion

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Based on a large sample of eddy-covariance sites from the FLUXNET2015
database, we demonstrated a higher potential of radiation-driven methods
calibrated by biome type to estimate *E*_{p} than of more complex
Penman–Monteith approaches or empirical temperature-based formulations. This
was consistent across all 11 biomes represented in the database and for two
different criteria to identify unstressed days, one based on soil moisture
and the other on evaporative fractions. Our analyses also showed that the
key parameters required to apply the higher-performance radiation-driven
methods are relatively insensitive to climate forcing. This makes these
methods robust for incorporation into global offline models, e.g. for
hydrological applications. Finally, we conclude that, at the ecosystem
scale, Penman–Monteith methods for estimating *E*_{p} should only be
prioritised if the unstressed stomatal conductance is calculated dynamically
and high accuracy observations from the wide palate of required forcing
variables are available.

Data availability

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Data availability.

The FLUXNET2015 dataset can be downloaded from http://fluxnet.fluxdata.org/data/fluxnet2015-dataset/ (FLUXNET, 2016). The main script for calculating potential evaporation with the different method as well as the daily flux data of one site (AU-How), for which permission of distribution was granted, is available as supplement. For further questions, we ask readers to contact the corresponding author at wh.maes@ugent.be.

Appendix A

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Supplement

Back to toptop
Supplement.

The supplement contains a description of the calculation method
of *k**B*^{−1} for sites where radiance fluxes are not separately measured (Sect. S1),
a description of the method to estimate unstressed *E*_{p} (see Sect. 4.5)
from flux data (Sect. S2) and several tables (Sect. S3), including an overview of the FLUXNET
sites used in the study (Table S1); a table on the representativeness of the
climate forcing conditions of the unstressed datasets for the full dataset
(Table S2); and correlation, unbiased RMSE and bias tables for different
selection criteria (Tables S3–S11). The supplement related to this article is available online at: https://doi.org/10.5194/hess-23-925-2019-supplement.

Author contributions

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Author contributions.

WHM and DGM designed the research. PG and NECV assisted in developing the optimal method for analysing all flux tower data. WHM performed the calculations and analyses. WHM and DGM wrote the manuscript, with contributions from PG and NECV.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This study was funded by the Belgian Science Policy Office (BELSPO) in the
frame of the STEREO III program project STR3S (SR/02/329). Diego G. Miralles
acknowledges support from the European Research Council (ERC) under grant
agreement no. 715254 (DRY-2-DRY). This work used eddy-covariance data
acquired and shared by the FLUXNET community, including these networks:
AmeriFlux, AfriFlux, AsiaFlux, CarboAfrica, CarboEuropeIP, CarboItaly,
CarboMont, ChinaFlux, FLUXNET-Canada, GreenGrass, ICOS, KoFlux, LBA, NECC,
OzFlux-TERN, TCOS-Siberia, and USCCC. The ERA-Interim reanalysis data are
provided by ECMWF and processed by LSCE. The FLUXNET eddy-covariance data
processing and harmonisation were carried out by the European Fluxes Database
Cluster, AmeriFlux Management Project and Fluxdata project of FLUXNET, with
the support of CDIAC and ICOS Ecosystem Thematic Center and the OzFlux,
ChinaFlux and AsiaFlux offices. The Atqasuk and Ivotuk towers in Alaska were
supported by the Office of Polar Programs of the National Science Foundation (NSF)
awarded to DZ, WCO and DAL (award number 1204263) with additional
logistical support funded by the NSF Office of Polar Programs; by the
Carbon in Arctic Reservoirs Vulnerability Experiment (CARVE), an Earth
Ventures (EV-1) investigation, under contract with the National Aeronautics
and Space Administration; and by the ABoVE (NNX15AT74A; NNX16AF94A) program.
The OzFlux network is supported by the Australian Terrestrial Ecosystem
Research Network (TERN, http://www.tern.org.au, last access: 14 February 2019). We would also
like to thank the anonymous reviewers as well as Giovanni Ravazzani for their
useful suggestions on an earlier version of the manuscript.

Edited by: Alberto Guadagnini

Reviewed by: Giovanni Ravazzani and one anonymous referee

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Short summary

Potential evaporation (*E*_{p}) is the amount of water an ecosystem would consume if it were not limited by water availability or other stress factors. In this study, we compared several methods to estimate *E*_{p} using a global dataset of 107 FLUXNET sites. A simple radiation-driven method calibrated per biome consistently outperformed more complex approaches and makes a suitable tool to investigate the impact of water use and demand, drought severity and biome productivity.

Potential evaporation (*E*_{p}) is the amount of water an ecosystem would consume if it were not...

Hydrology and Earth System Sciences

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