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**Hydrology and Earth System Sciences**
An interactive open-access journal of the European Geosciences Union

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**Research article**
28 Aug 2018

**Research article** | 28 Aug 2018

Exploring the relationships between warm-season precipitation, potential evaporation, and “apparent” potential evaporation at site scale

^{1}College of Arts and Sciences, Department of Geography and Geographic Information Science, University of Cincinnati, Cincinnati, USA^{2}College of Engineering and Applied Science, Department of Civil Engineering and Architectural Engineering and Construction Management, University of Cincinnati, Cincinnati, USA

^{1}College of Arts and Sciences, Department of Geography and Geographic Information Science, University of Cincinnati, Cincinnati, USA^{2}College of Engineering and Applied Science, Department of Civil Engineering and Architectural Engineering and Construction Management, University of Cincinnati, Cincinnati, USA

**Correspondence**: Xi Chen (xi.chen@uc.edu)

**Correspondence**: Xi Chen (xi.chen@uc.edu)

Abstract

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Bouchet's complementary relationship and the Budyko hypothesis are two classic frameworks that are inter-connected. To systematically investigate the connections between the two frameworks, we analyze precipitation, pan evaporation, and potential evaporation data at 259 weather stations across the United States. The precipitation and pan evaporation data are from field measurement and the potential evaporation data are collected from a remote-sensing dataset. We use pan evaporation to represent “apparent” potential evaporation, which is different from potential evaporation. With these data, we study the correlations between precipitation and potential evaporation, and between precipitation and “apparent” potential evaporation. The results show that 93 % of the study's weather stations exhibit a negative correlation between precipitation and “apparent” potential evaporation. Also, the aggregated data cloud of precipitation vs. “apparent” potential evaporation with 5312 warm-season data points from 259 weather stations shows a negative trend in which “apparent” potential evaporation decreases with increasing precipitation. On the other hand, no significant correlation is found in the data cloud of precipitation vs. potential evaporation, indicating that precipitation and potential evaporation are independent. We combine a Budyko-type expression, the Turc–Pike equation, with Bouchet's complementary relationship to derive upper and lower Bouchet–Budyko curves, which display a complementary relationship between “apparent” potential evaporation and actual evaporation. The observed warm-season data follow the trend of the Bouchet–Budyko curves. Our study shows the consistency between Budyko's framework and Bouchet's complementary relationship, with the distinction between potential evaporation and “apparent” potential evaporation. The formulated complementary relationship can be used in quantitative modeling practices.

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How to cite.

Chen, X. and Buchberger, S. G.: Exploring the relationships between warm-season precipitation, potential evaporation, and “apparent” potential evaporation at site scale, Hydrol. Earth Syst. Sci., 22, 4535–4545, https://doi.org/10.5194/hess-22-4535-2018, 2018.

1 Introduction

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Potential evaporation (*E*_{p}) is a widely used physical variable in
hydrologic frameworks. It is the evaporation rate under unlimited land
surface water supply (Thornthwaite, 1948). Pan evaporation (*E*_{pan})
measurement is often used as a surrogate of potential evaporation. However,
these two variables are not the same (Brutsaert and Parlange, 1998; Roderick
et al., 2009). A stipulation is added in the potential evaporation definition
in Van Bavel (1966) and further clarified in Brutsaert (2015) that “the
surface vapor pressure be saturated, so that it can be found from the surface
temperature.” Therefore, the main difference between potential evaporation
and pan evaporation is that pan evaporation is not measured under saturated
surface vapor pressure. As a result, potential evaporation can be considered
to depend only on the energy supply of climate, while pan evaporation is
driven by both energy supply and humidity deficit in the atmosphere (Rotstayn
et al., 2006). In Brutsaert and Parlange (1998), the term “apparent”
potential evaporation (*E*_{pa}) is introduced to distinguish pan
evaporation from potential evaporation. “Apparent” potential evaporation
can be measured by an evaporation pan, while potential evaporation cannot. We
acknowledge that there are different definitions of potential evaporation in
the literature (Aminzadeh et al., 2016). Our study follows the definition of
potential evaporation in Brutsaert and Parlange (1998) and Brutsaert (2015).

Because potential evaporation is energy-driven, it can be used as a physical
variable to describe the energy supply in a hydrologic system. For instance,
the well-established Budyko framework (Budyko, 1958, 1974) uses precipitation
(*P*) and potential evaporation to represent the relationship between water
supply and energy supply, and therefore to describe the impact of long-term
climate on the hydrologic cycle. The Budyko framework has been extensively
used to analyze interactions between hydrology, climate,
vegetation, and other
elements in watersheds (Milly, 1994; Zhang et al., 2001; Yang et al., 2007,
2011; Donohue et al., 2007; Xu et al., 2014; Zhou et al., 2015, 2016).
Furthermore, the Budyko framework, which was originally applicable at the
long-term mean annual scale, has been extended to shorter timescales, such as
annual (Wang and Alimohammadi, 2012; Zhang et al., 2008) and intra-annual
periods (Chen et al., 2013).

Several studies have made connections between the Budyko framework and Bouchet's complementary relationship (CR) (Bouchet, 1963). Yang et al. (2006) used the Fu equation (Fu, 1981), which is one of the commonly used equations to represent the Budyko curve, to describe the relationship between actual evaporation and potential evaporation in the CR. Roderick et al. (2009) presented a complementary relationship normalized by net irradiance and compared it with the Budyko framework. Lhomme and Moussa (2016) combined the Turc–Pike equation (Turc, 1954; Pike, 1964), which is another commonly used Budyko-type equation, with the CR to show the dependence of the Budyko curve on the drying power of the air.

When linking the Budyko framework with the CR, it is crucial to have a clear
definition of different types of evaporation used in these two frameworks.
Brutsaert and Parlange (1998) and Brutsaert (2015) generalized the CR and
provided definitions of the evaporation terms in the CR, namely actual
evaporation (*E*), potential evaporation (*E*_{p}), and “apparent”
potential evaporation (*E*_{pa}; see Fig. 1a). Brutsaert and Parlange
(1998) point out that the complementary relationship is between actual
evaporation and “apparent” potential evaporation, not between actual
evaporation and potential evaporation. In the Budyko framework (Fig. 1b), the
definition of potential evaporation follows Van Bavel (1966)'s potential
evaporation definition that it is under unlimited land surface water supply
without the effect of humidity deficit (Budyko, 1974), which is the same as
the *E*_{p} definition in the generalized CR. The definitions of
evaporation, potential evaporation, and “apparent” potential evaporation in
these different frameworks are summarized in Table 1.

Process-based speaking, the CR suggests a connection between evaporation and
“apparent” potential evaporation (Fig. 1a), which is driven by the energy
feedbacks between atmosphere and land surface. During the drying process at
the land surface, the excessive energy that is not used for evaporation will
be available for the increase in sensible heat. The rise in air temperature
will lead to an increase in the rate of “apparent” potential evaporation
(Brutsaert and Parlange, 1998; Brutsaert, 2005; Aminzadeh et al., 2016). This
connection between *E*_{pa} and *E* also suggests a connection between
*E*_{pa} and *P*, since the water supply from precipitation will affect
the rate of evaporation. In terms of the Budyko framework, *E*_{p} and
*P* are used as the representations of energy supply and water supply,
respectively. The ratio between *E*_{p} and *P* is the primary
controlling factor of the ratio of *E* over *P* in watersheds at the
long-term mean annual timescale (Fig. 1b). The ratio of *E*_{p} over *P*
is also called the aridity index, which represents the dryness of the climate
in a watershed. The ratio of *E* over *P* increases with the increase in the
aridity index, indicating that more water from precipitation will become
evaporation rather than runoff under a drier climate (Arora, 2002). No
connection between *E*_{p} and *P* is suggested in the Budyko framework.

In order to explore the connections between the Budyko framework and the CR,
our study investigates the relationships between precipitation and potential
evaporation as well as between precipitation and “apparent” potential
evaporation. We collect warm-season precipitation, potential evaporation, and
pan evaporation data from 259 weather stations across the contiguous US.
Studying the relationships between *P*, *E*_{p}, and *E*_{pa}
advances our understanding of the well-established classic Budyko framework
and the CR. Furthermore, based on insights provided by previous studies (Yang
et al., 2006; Roderick et al., 2009; Lhomme and Moussa, 2016), we use a
Budyko-type expression to develop a new formulation for the CR.

2 Methodology

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The Budyko curve (Fig. 1b) describes the relationship between long-term water partitioning, represented by the ratio of actual evaporation over precipitation, and long-term climate, represented by the ratio of potential evaporation over precipitation, namely the aridity index (Budyko, 1958, 1974). In recent decades, the Budyko framework has been examined with annual data (e.g., Yang et al., 2007; Potter and Zhang, 2009; Cheng et al., 2011). A number of Budyko-type functions have been developed to mathematically describe the Budyko curve (Turc, 1954; Fu, 1981; Zhang et al., 2001; Yang et al., 2008; Wang and Tang, 2014). Within these functions, the Turc–Pike equation is a parsimonious single-parameter equation (Turc, 1954; Pike, 1964):

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{E}{P}}={\left[\mathrm{1}+{\left({\displaystyle \frac{{E}_{\text{p}}}{P}}\right)}^{-v}\right]}^{-{\scriptscriptstyle \frac{\mathrm{1}}{v}}},\end{array}$$

where *E* is actual evaporation, *E*_{p} is potential evaporation, *P* is
precipitation, and *v* is a parameter to represent landscape properties such
as vegetation coverage and soil properties (Zhang et al., 2001; Yang et al.,
2008). The parameter *v* needs to be a positive number, and its typical
value is 2.0.

Bouchet's complementary relationship (Bouchet, 1963) describes the
relationship between actual evaporation *E* and potential evaporation *E*_{p}.
Brutsaert and Parlange (1998) introduced the term “apparent” potential
evaporation *E*_{pa} and clarified that the CR is between *E* and *E*_{pa}, not
*E* and *E*_{p} (Fig. 1a). They also proposed a generalized complementary
relationship:

$$\begin{array}{}\text{(2)}& bE+{E}_{\text{pa}}=(\mathrm{1}+b){E}_{\text{p}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{0}\le E\le {E}_{\text{p}}\le {E}_{\text{pa}},\end{array}$$

where *b* is a proportionality parameter not less than one. When *b* is equal
to one, Eq. (2) represents the original complementary relationship (Kahler
and Brutsaert, 2006). “Apparent” potential evaporation will be higher than
potential evaporation, especially under dry conditions, while it gradually
approaches potential evaporation as the ratio of *E* over *E*_{pa}
increases (Fig. 1a). As suggested by Morton (1976) and Brutsaert and Stricker
(1979), potential evaporation can be estimated using the Priestley–Taylor
equation (Priestley and Taylor, 1972), which is also called the equilibrium
evaporation (Brutsaert and Chen, 1995; Jiang and Islam, 2001). “Apparent”
potential evaporation can be estimated using the Penman equation (Penman,
1948; Linacre, 1994; Rotstayn et al., 2006) or using data measured at
evaporation pans (Brutsaert, 1982; Brutsaert and Parlange, 1998):

$$\begin{array}{}\text{(3)}& {E}_{\text{pa}}=a{E}_{\text{pan}},\end{array}$$

where *E*_{pan} is the pan evaporation and *a* is the pan coefficient.
The pan coefficient varies from location to location (Stanhill, 1976;
Linacre, 1994). In Kahler and Brutsaert (2006), a pan coefficient of *a*=1.0
is recommended for mixed natural vegetation, which will be used in this
study. It should be noted that the linear relationship between *E*_{pa}
and *E*_{pan} given in Eq. (3) and the choice of “*a*” value will not
affect the correlations between *P*, *E*_{p}, and *E*_{pa}.

The *x* axis of the complementary relationship is a ratio between *E* and
*E*_{pa} (Bouchet, 1963). Ramírez et al. (2005) used the
water-energy framework to link the CR with the Budyko approach and changed
the *x* axis in the CR to moisture availability. Following this idea, several
studies have used precipitation or wetness index (*P*∕*E*_{p}) to
represent moisture availability in the CR (Yang et al., 2006; Roderick et
al., 2009). In this study, we also use *P* to represent moisture availability
in the CR. *E*_{p} is a horizontal line in the CR that is parallel to
the *x* axis (Fig. 1a). Therefore, the modified CR indicates that *P* and
*E*_{p} are independent. On the other hand, the upper curve of the CR,
representing “apparent” potential evaporation *E*_{pa}, declines along
the *x* axis, indicating that *E*_{pa} and *P* are not independent. For
a dimensionless CR, we normalize the *x* and *y* axes. The normalized CR
describes the relationship between $\frac{{E}_{\text{pa}}}{{E}_{\text{p}}}$,
$\frac{E}{{E}_{\text{p}}}$, and $\frac{P}{{E}_{\text{p}}}$ (Fig. 2).

To connect the Budyko framework with the normalized CR toward formulating the Bouchet–Budyko curves, we first transform Eq. (1) into a relationship between $\frac{E}{{E}_{\text{p}}}$ and $\frac{P}{{E}_{\text{p}}}$:

$$\begin{array}{}\text{(4)}& {\displaystyle \frac{E}{{E}_{\text{p}}}}={\left[{\left({\displaystyle \frac{P}{{E}_{\text{p}}}}\right)}^{-v}+\mathrm{1}\right]}^{-{\scriptscriptstyle \frac{\mathrm{1}}{v}}}.\end{array}$$

Yang et al. (2006) did a similar transformation using the Fu equation (Fu,
1981). Dividing both sides of Eq. (2) by *E*_{p} yields

$$\begin{array}{}\text{(5)}& b{\displaystyle \frac{E}{{E}_{\text{p}}}}+{\displaystyle \frac{{E}_{\text{pa}}}{{E}_{\text{p}}}}=\mathrm{1}+b.\end{array}$$

Combining Eqs. (4) and (5) gives a relation between $\frac{P}{{E}_{\text{p}}}$ and $\frac{{E}_{\text{pa}}}{{E}_{\text{p}}}$:

$$\begin{array}{}\text{(6)}& {\displaystyle \frac{{E}_{\text{pa}}}{{E}_{\text{p}}}}=b+\mathrm{1}-{\left[{\left({\displaystyle \frac{P}{{E}_{\text{p}}}}\right)}^{-v}+\mathrm{1}\right]}^{-\mathrm{1}/v}{E}_{\text{pa}}\ge {E}_{\text{p}}.\end{array}$$

Equations (4) and (6) represent the lower and upper curves of the normalized
CR, respectively (Fig. 2). Roderick et al. (2009) presented a similar
framework, without the formulation of the curves. To verify the relationships
between *P*, *E*_{p}, and *E*_{pa}, and to examine the
Bouchet–Budyko curves in Eqs. (4) and (6), we analyze climate data from 259
weather stations across the contiguous US.

Monthly precipitation and pan evaporation are collected from the National Oceanic and Atmospheric Administration (NOAA) at the National Climatic Data Center (NCDC). The data can be downloaded at https://www.ncdc.noaa.gov/IPS/cd/cd.html (last access: 17 August 2018). The precipitation data are measured using a standard rain gauge and the pan evaporation data using Class A evaporation pans. We collect data for the period 1984–2015 from a total of 259 weather stations (Fig. 3a). Since pan evaporation is collected only during warm months (when temperatures remain above freezing), the weather stations in cold regions have less than 12 months of pan readings in a year. We call the period of warm months in a year a “warm season”. We calculate the monthly average pan evaporation and precipitation using only the warm months for each year at each weather station. For short, they are called warm-season data (i.e., warm-season pan evaporation, warm-season precipitation). We also calculate the annually averaged warm-season data to represent the long-term average level of pan evaporation and precipitation at each station. For short, they are called long-term average data. Over the 259 selected stations, there is an average of 7 months per year with available pan evaporation data. As Fig. 3 shows, the number of available months decreases from the southern regions to the northern regions. For stations in the southern states with all 12 months of available data in a year, the full year will be considered a warm season. The northern state stations have fewer warm months, and, accordingly, the warm season is much shorter. On the other hand, not all 259 weather stations have the full record from 1984 to 2015; the average number of years with available data for each location is 18. A complete summary of the information available at all 259 weather stations is provided in Table S1 in the Supplement. In order to minimize the uncertainty from various warm periods in a year from station to station, we repeat the analysis using an alternative source of pan evaporation in the NCDC dataset containing homogenized warm month data from May to October (Hobbins, et al., 2017). A total of 93 weather stations overlap both sets of pan evaporation data for the period 1984 to 2001 (Fig. 3b). We convert pan evaporation in the NCDC dataset to “apparent” potential evaporation using Eq. (3).

The *E*_{p} data are collected from a remote-sensing dataset (Zhang et
al., 2010), which is generated using the Priestley–Taylor equation with
remotely sensed net radiation:

$$\begin{array}{}\text{(7)}& \mathit{\lambda}{E}_{\text{p}}=\mathit{\alpha}{\displaystyle \frac{\mathrm{\Delta}}{\mathrm{\Delta}+\mathit{\gamma}}}({R}_{\text{n}}-G),\end{array}$$

where *λ* (J kg^{−1}) is the latent
heat of vaporization; *λ**E*_{p} (W m^{−2}) is the latent heat
flux; *α* is a coefficient to account for the effect of surface
characteristics and vegetation, and is set to 1.26; Δ
(Pa ^{∘}C^{−1}) is the slope of the saturated vapor pressure curve;
*γ* (Pa ^{∘}C^{−1}) is the psychometric constant; *R*_{n}
(W m^{−2}) is the net radiation; and *G* (W m^{−2}) is the heat flux
into the ground. The *E*_{p} data cover the period 1983–2006. Similarly
to *P* and *E*_{pa}, we calculate the warm-season *E*_{p} and
long-term annually averaged *E*_{p} based on the monthly *E*_{p}
data.

Using the collected weather station data of precipitation and pan evaporation
for the period 1984 to 2015, we first calculate the Pearson correlation
coefficient between warm-season *P* and warm-season *E*_{pa} for each
location (Fig. 3a). We then perform the same correlation analysis of *P* and
*E*_{pa} using the homogenized pan evaporation dataset (Hobbins et al.,
2017) (Fig. 3b). Secondly, we use data of warm-season *P* and warm-season
*E*_{p} for the period of 1984 to 2006, which is the period when both
*P* and *E*_{p} data are available, to investigate the correlation
between *P* and *E*_{p}. Finally, to validate the newly derived
Bouchet–Budyko curves, the relationship between $\frac{P}{{E}_{\text{p}}}$ and
$\frac{{E}_{\text{pa}}}{{E}_{\text{p}}}$ is plotted using the collected data at both
seasonal and long-term average timescales.

3 Results

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In the 259 weather stations, 93 % of the stations have a negative
correlation between *P* and *E*_{pa} (Fig. 4a), but only 43 % of the
stations are statistically significant (*p*<0.05; Fig. 4b). All significant
*P*–*E*_{pa} correlations are negative. The weather stations located in
the western region (regions with longitude higher than the weather station
average longitude of 94.81^{∘} W) are more likely to have a significant
*P*–*E*_{pa} negative correlation than those located in the east
(regions with longitude lower than 94.81^{∘} W). This spatial
difference may be related to climate characteristics: the eastern region has
higher precipitation (average 105.5 mm month^{−1}) and lower “apparent”
potential evaporation (average 145.3 mm month^{−1}), while the western
region has lower precipitation (average 44.6 mm month^{−1}) and higher
“apparent” potential evaporation (average 203.5 mm month^{−1}).
Bouchet's complementary relationship is more significant in arid regions
(Ramírez et al., 2005), corresponding to the left side of the CR curves,
while it is less significant in humid regions, corresponding to the right
side of the CR curves (Fig. 1a). As a result, the negative correlation
between precipitation and “apparent” potential evaporation is more
significant in the west than in the east.

All the warm-season *P* vs. *E*_{pa} relations (i.e., all years, all
seasons, for a total of 5312 data points) are shown in Fig. 5a. The data
cloud shows a negative trend in general. We also plot the long-term annually
averaged values of warm-season *P* and *E*_{pa} of the 259 weather
stations (Fig. 5b), which shows a similar negative trend. Hobbins et
al. (2004) showed a similar negative trend between precipitation and pan
evaporation with watershed-scale data. To represent the spatial distribution
of the weather stations, we color code the data points based on their spatial
coordinates of latitude and longitude. The climate in the eastern US is much
wetter than the western US, and therefore the data cloud of *E*_{pa}
vs. *P* is separated into two parts horizontally. The right side of the cloud
represents the northeastern and southeastern US (green and brown,
respectively), while the left side of the cloud generally represents the
northwestern and southwestern US (yellow and red, respectively).

As explained before, we also use an alternative pan evaporation dataset
(Hobbins et al., 2017) to further validate our analysis result. This dataset
is homogenized to have the same period of a pan evaporation data record in
each year from May to October. In order to minimize the data heterogeneity
caused by station move and human errors, this dataset compiled pan
evaporation data from 247 stations across the US with thorough quality
control. It is derived from the same dataset as our data, namely the NCDC
dataset. Based on the homogenized pan evaporation data, 85 stations out of 93
(91 %) have a negative correlation between *P* and *E*_{pa}. Of
these, 41 % of the stations have a statistically significant relationship
(*p*<0.05), all negative. This result is consistent with the analysis result
based on our collected data from 259 weather stations. We also use the data
cloud to show the relationship between *P* and *E*_{pa} in the warm
period of May to October in each year at each of the 93 stations (Fig. 5c),
as well as the relationship of long-term annually averaged warm-period *P*
and *E*_{pa} (Fig. 5d). The trend of the data cloud is similar to the
data cloud trend using our collected data at both seasonal and long-term
average timescales. In other words, both datasets show a negative
relationship between *P* and *E*_{pa}.

The *P* and *E*_{p} data are shown in Fig. 5e, f. At both seasonal and
long-term average timescales, there is no clear relationship shown between
*P* and *E*_{p}, confirming the independence between *P* and
*E*_{p} discussed in Sect. 2.1.3. This result shows the difference
between *E*_{p} and *E*_{pa}, that *E*_{p} is independent of
*P* but *E*_{pa} is not. Therefore, it is important to distinguish
*E*_{pa} from *E*_{p} and to understand the different physical
mechanisms of the two processes (Brutsaert, 2015).

To present the *P*, *E*_{p}, and *E*_{pa} relationships at
individual locations and therefore to further investigate the dependence
between the three variables, we select four weather stations from the four
quadrants of the contiguous US (Fig. 3a), to show the warm-season *P*,
*E*_{p}, and *E*_{pa} in time series (Fig. 6). The two stations in
the southern regions have data in all 12 months of a year, while the two
stations in the northern regions only have *E*_{pa} data for 6 months of
each year. All four stations show negative correlations between *P* and
*E*_{pa}. This negative correlation at the weather station in Florida is
not statistically significant (Fig. 6g, h). As mentioned before, the *P* and
*E*_{pa} correlation is less significant in the eastern region than in
the west, because of the wetter climate in the east. On the other hand, at
the other three locations, the warm-season *P* and *E*_{pa} are
relatively symmetric to each other (Fig. 6a–f). During years when one series
is above average, the other tends to be below average and vice versa. In
terms of the relationship between *P* and *E*_{p}, all four locations
show no significant correlations between the two variables (*p*>0.05). This
is consistent with the independence of *P* and *E*_{p} shown in
Fig. 5e, f.

There are two Bouchet–Budyko curves (Fig. 2). The upper curve describes the
relationship between *E*_{pa}, *E*_{p}, and *P* (Eq. 6) and the
lower curve describes the relationship between *E*, *E*_{p}, and *P*
(Eq. 4). The lower curve is derived from the Budyko curve based on the
Turc–Pike equation. This relationship between *E*, *E*_{p}, and *P* has
been studied extensively following the Budyko framework and, therefore, it is
not the focus of this study. This study investigates the relationship between
*E*_{pa}, *E*_{p}, and *P*, which is represented by the upper
Bouchet–Budyko curve. Since the collected weather station data of *P* and
*E*_{pa} are available from 1984 to 2015 and the *E*_{p} data
collected from the remote-sensing dataset are available from 1983 to 2006, we
examine the relationship between *P*∕*E*_{p} and *E*_{pa}∕*E*_{p}
in the overlapping period of 1984 to 2006 (Fig. 7). Using Eq. (6) three
curves with different *b* values (1, 2, and 3) are shown in Fig. 7. The *v*
value is set at 2, which is commonly used in the Budyko framework. When *b*
equals one, the two CR curves are symmetric. When *b* exceeds one, the two CR
curves are asymmetric. This asymmetry is discussed in previous studies
(Kahler and Brutsaert, 2006; Brutsaert, 2015). One explanation of this
asymmetry between *E* and *E*_{pa} is that the evaporation pan will
receive more heat than the surrounding area (Kahler and Brutsaert, 2006).
Brutsaert (2015) reports an even higher *b* value of 4.5. The horizontal
solid black line in Fig. 7 is the boundary of the upper Bouchet–Budyko
curve, above which *E*_{pa} exceeds *E*_{p}.

4 Discussion

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With the weather station data, a negative correlation between warm-season *P*
and *E*_{pa} is shown in 242 out of the 259 weather stations (93 %).
The negative correlation between *P* and *E*_{pa} is linked by the
humidity deficit. The formation of precipitation is positively related to the
local level of humidity (Pal et al., 2000; Sheffield et al., 2006; An et al.,
2017), while “apparent” potential evaporation is inversely related to
humidity or positively related to the humidity deficit (Penman, 1948; Allen
et al., 1998). As a result, precipitation and “apparent” potential
evaporation will tend to exhibit a negative correlation. According to
Bouchet's complementary relationship, this negative correlation between *P*
and *E*_{pa} is more pronounced in arid regions than in humid regions.

On the other hand, *P* and *E*_{p} show no significant correlation at
both the seasonal and long-term average timescales. As a result, our study
indicates that potential evaporation and precipitation, the representations
of energy supply and water supply, are likely to be independent. This
independence is currently under investigation with field data. It should be
noted that the relationships between *P* and *E*_{p} and between *P* and
*E*_{pa} found in this study are not direct causal relationships, but
rather the result of interactions between a number of physical variables,
such as net radiation, wind speed, humidity, and so forth. Further
investigation into the physical mechanisms connecting these variables is
underway.

Combining Bouchet's complementary relationship and the Budyko framework leads
to two dimensionless CR curves, normalized by *E*_{p} (Fig. 2). The
upper Bouchet–Budyko curve is derived from the connection between the Budyko
framework and the CR, and the lower Bouchet–Budyko curve is derived directly
from the Budyko framework, based on the Turc–Pike equation. The companion CR
curves show that as the wetness index *P*∕*E*_{p} decreases, the
difference between *E* and *E*_{pa} grows. This indicates the
complementary relationship between *E* and *E*_{pa} is most pronounced
in arid environments, that is, the CR is more significant under water-limited
conditions. As discussed in Ramírez et al. (2005), the CR can be
considered an extension of the Budyko framework.

The *P*, *E*_{p}, and *E*_{pa} collected in this study follow the
general trend of the upper Bouchet–Budyko curve (Fig. 7). The remote-sensing
data of *E*_{p} may not have the same level of accuracy as the field
measured *P* and *E*_{pa}. The value of *α* in Eq. (7) may vary
from location to location (Chen and Brutsaert, 1995; Brutsaert and Chen,
1995). Such factors may explain the deviation of some data points from the CR
curve in Fig. 7.

This upper Bouchet–Budyko curve can be used to estimate the *E*_{pa}
based on the data of *P* and *E*_{p}. The “apparent” potential
evaporation can be measured by evaporation pan, but this measurement has its
limitations. For example, it is only available for warm periods. The
collected data with time-averaged pan evaporation levels over weeks, months,
and years may lead to systematic error in surface flux calculations
(Brutsaert, 1982; Kahler and Brutsaert, 2006). The Bouchet–Budyko curve can
help us to estimate *E*_{pa} without the limitations of evaporation
pans. Compared with more physically based *E*_{pa} quantification
approaches, such as the Penman equation (Penman, 1948) and the “PenPan”
model (Rotstayn et al., 2006), our equations are derived from conceptual
frameworks and therefore may provide top–down insights into the
*E*_{pa} level in hydrologic systems.

Similarly to the Budyko framework, the Bouchet–Budyko curves can be used in hydrologic models and climate models. These Bouchet–Budyko curves can be used to examine the fidelity of simulated precipitation and evaporation sequences routinely produced by general circulation models to drive climate change investigations.

5 Conclusions

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We collected warm-season precipitation, potential evaporation, and
“apparent” potential evaporation data at 259 weather stations in the US to
investigate the correlation among these three physical variables. The results
showed a negative correlation between *P* and *E*_{pa} at 93 % of
the stations. The physical reason for the *P*–*E*_{pa} negative
correlation could be related to the humidity variability. When humidity
increases, the likelihood of precipitation increases while the rate of
“apparent” potential evaporation decreases. On the other hand, our study
results supported the assumption that *P* and *E*_{p} are independent.
Combining the CR with a Budyko-type equation, we formulated the companion CR
curves, showing the connection between the Bouchet and Budyko frameworks.
These insights may encourage hydrologists to further explore the strong link
between the Budyko framework and the CR, promoting new ways of hydrologic
modeling. Future work will investigate the physical mechanisms behind the
newly derived Bouchet–Budyko curves and explore the application of these
companion curves.

Data availability

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Data availability.

The data of precipitation and pan evaporation measurements can be downloaded from the National Climatic Data Center website: https://www.ncdc.noaa.gov/IPS/cd/cd.html. The homogenized pan evaporation data can be downloaded from the USGS ScienceBase: https://www.sciencebase.gov/catalog/ (Hobbins, 2017). The data of remote-sensing-based potential evaporation are provided by the Numerical Terradynamic Simulation Group at the University of Montana, based on the study of Zhang et al. (2010). The data can be downloaded from their website: http://www.ntsg.umt.edu/about/default.php (Zhang, 2010).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/hess-22-4535-2018-supplement.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank the editor and the three anonymous reviewers for their insightful
and critical comments and valuable suggestions.

Edited by: Bob Su

Reviewed by: three
anonymous referees

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Short summary

Based on warm season data from 259 weather stations across the US, we analyze the correlation between precipitation, potential evaporation, and “apparent” potential evaporation (measured by pan evaporation). Over 93 % of the stations show negative correlation between precipitation and

apparentpotential evaporation, but no clear relationship is shown between precipitation and potential evaporation. The collected data points follow the trend of the newly derived Bouchet–Budyko curve.

Based on warm season data from 259 weather stations across the US, we analyze the correlation...

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