Regional Evapotranspiration and Bouchet's Hypothesis

Regional Evapotranspiration and Bouchet's Hypothesis
The complementary relationship in estimation of regional
evapotranspiration: The Complementary Relationship
Areal Evapotranspiration and Advection-Aridity models
Michael T. Hobbins and Jorge A. Ramı́rez
Department of Civil Engineering, Colorado State University, Fort Collins, Colorado
Thomas C. Brown
Rocky Mountain Research Station, U.S. Forest Service, Fort Collins, Colorado
Lodevicus H. J. M. Claessens
The Ecosystems Center, Marine Biological Laboratory, Woods Hole, Massachusetts
Abstract. Two implementations of the complementary relationship hypothesis for
regional evapotranspiration, the Complementary Relationship Areal Evapotranspiration
(CRAE) model and the Advection-Aridity (AA) model, are evaluated against independent
estimates of regional evapotranspiration derived from long-term, large-scale water
balances (1962–1988) for 120 minimally impacted basins in the conterminous United
States. The CRAE model overestimates annual evapotranspiration by 2.5% of mean
annual precipitation, and the AA model underestimates annual evapotranspiration by
10.6% of precipitation. Generally, increasing humidity leads to decreasing absolute errors
for both models, and increasing aridity leads to increasing overestimation by the CRAE
model and underestimation by the AA model, with the exception of high, arid basins,
where the AA model overestimates evapotranspiration. Overall, the results indicate that
the advective portion of the AA model must be recalibrated before it may be used
successfully on a regional basis and that the CRAE model accurately predicts monthly
regional evapotranspiration.
In problems of regional hydrology, and therefore of global
climatology, understanding the large-scale behavior of processes defining fluxes of sensible and latent heat is of paramount importance. In estimating evapotranspiration, the physics of energy and mass transfer at the land surface–atmosphere
interface can be successfully modeled at small temporal and
spatial scales [Parlange and Katul, 1992a, 1992b], yet our understanding of the processes at the monthly, or seasonal, and
regional scales necessary for use by hydrologists, water managers, and climate modelers is limited.
Consequently, our ability to estimate actual regional evapotranspiration (ETa ) is often constrained by models that treat
potential evapotranspiration (ETp ) as an independent climatic
forcing process, often an empirical function of pan evaporation
(ETpan) observed at nearby weather stations, or by models that
tend to rely on gross assumptions as to the nature of moisture
dynamics in each of the components of the land surface–
atmosphere interface and of the interactions between them.
However, at regional scales, ETa and ETp are not independent
of each other. Complex feedback interactions between processes governing their rates are established based on the degree to which the soil can satisfy the atmospheric demand for
water vapor and on the resultant effect on energy distribution
Copyright 2001 by the American Geophysical Union.
Paper number 2000WR900358.
at the land-atmosphere interface. There is therefore a need for
models of regional evapotranspiration that incorporate these
feedback mechanisms, while avoiding the difficulties inherent
in explicitly coupling the microscale (surface) and macroscale
(free atmosphere) phenomena in the soil-plant-atmosphere
system. Complementary relationship models are such models.
The hypothesis of the complementary relationship between
ETa and ETp in regional evapotranspiration was first proposed
by Bouchet [1963]. Using such models, ETa is estimated using
data that describe the conditions of the overpassing air, obviating the need for locally optimized coefficients and surface
The most widely known of these models are the Complementary Relationship Areal Evapotranspiration (CRAE)
model [Morton, 1983] and the Advection-Aridity (AA) model
[Brutsaert and Stricker, 1979]. In earlier versions of the CRAE
model [Morton, 1976], good agreement was shown between
model estimates and water budget estimates of regional evapotranspiration (ETa ) for basins in Canada, Ireland, Kenya, and
the United States. However, these versions predicted potential
evapotranspiration (ETp ) with the Penman equation and did
not invoke the assumption of an “equilibrium temperature”
T p . The version used in the work reported herein was calibrated by Morton [1983] on a monthly basis and applied on an
annual basis to 143 basins in Canada, the United States, Ireland, Africa, Australia, and New Zealand. The mean absolute
error of 19.6 mm yr⫺1 convinced Morton [1983, p. 34] that
“[T]he environmental diversity, the calibration technique, and
the good fit of average annual data 䡠 䡠 䡠 combine to provide
Figure 1. Schematic representation of the complementary relationship in regional evapotranspiration (assuming constant energy availability).
assurance that the monthly estimates of areal evaporation are
The Advection-Aridity model was used by Brutsaert and
Stricker [1979] to calculate regional evapotranspiration over
time steps of 3 days for a small, rural catchment in the Netherlands during the drought of 1976, when ETp far exceeded
ETa . The model was found to yield a good match with energy
budget estimates. However, the AA model has not been tested
on a continental-scale basis, over a wide variety of climates, nor
on a monthly basis, all features which are essential in hydrological and climatological modeling.
Aside from Morton’s [1965, 1983] and Morton et al.’s [1985]
numerous publications, several studies have addressed the validity of complementary relationship models, whether through
comparison with other evapotranspiration estimators [BenAsher, 1981; Sharma, 1988; Doyle, 1990; Lemeur and Zhang,
1990; Chiew and McMahon, 1991; Parlange and Katul, 1992a],
through analysis based on meteorological observations and/or
modeling results [LeDrew, 1979; McNaughton and Spriggs,
1989; Granger and Gray, 1990; Lhomme, 1997; Kim and Entekhabi, 1997], or by defining improvements to existing models
[Kovacs, 1987; Granger, 1989; Parlange and Katul, 1992b]. Although these studies demonstrated varied success for complementary relationship models, the hypothesis is considered an
important concept for hydrologic modeling [Nash, 1989;
Dooge, 1992]. However, with the exception of Morton [1983]
none of these studies evaluated the complementary relationship over a wide range of climatologically varying conditions
(e.g., continental size areas); thus its value as an operational
tool has not been well established.
This study compares the CRAE model to the AA model
regarding formulation and performance, and, most important,
it evaluates the complementary relationship hypothesis in the
context of large-scale, long-term water balances. Specifically,
the authors sought to (1) construct monthly surfaces for the
conterminous United States of the components of the complementary relationship using the CRAE [Morton, 1983; Morton et
al., 1985] and AA [Brutsaert and Stricker, 1979] models; (2)
examine the spatial distribution of these surfaces across the
conterminous United States; (3) compare the models’ estimates of actual evapotranspiration to independent estimates of
regional evapotranspiration provided by long-term, large-scale
water balances for undisturbed basins in the conterminous
United States; and (4) examine the errors invoked in closing
the water balances and the relationships of these closure errors
to climatological and physical basin characteristics.
Complementary Relationship
Complementary Relationship Hypothesis
Bouchet’s [1963] hypothesis of a complementary relationship
states that over areas of a regional size and away from sharp
environmental discontinuities, there exists a complementary
feedback mechanism between actual (ETa ) and potential
(ETp ) evapotranspiration. In this context, ETp is defined as the
evapotranspiration that would take place from a moist surface
under the prevailing atmospheric conditions, limited only by
the amount of available energy. Under conditions where ETa
equals ETp , this rate is referred to as the wet environment
evapotranspiration (ETw ). The general complementary relationship is then expressed as
ETa ⫹ ETp ⫽ 2ETw.
The complementary relationship hypothesis is essentially
based on empirical observations, supported by a conceptual
description of the underlying interactions between evapotranspiring surfaces and the atmospheric boundary layer. Bouchet
[1963] hypothesized that when, under conditions of constant
energy input to a given land surface–atmosphere system, water
availability becomes limited, ETa falls below its potential, and
a certain amount of energy becomes available. This energy
excess, in the form of sensible heat and/or long wave back
radiation, increases the temperature and humidity gradients of
the overpassing air and leads to an increase in ETp equal in
magnitude to the decrease in ETa . If water availability is increased, the reverse process occurs, and ETa increases as ETp
decreases. Thus ETp ceases to be an independent causal factor,
or climatologically constant forcing function, and instead is
predicated upon the prevailing conditions of moisture availability. Figure 1 illustrates the complementary relationship.
For a more detailed analysis of this hypothesis, see Bouchet’s
[1963] seminal paper.
Advection-Aridity Model
In examining the CRAE and AA models, it will be useful to
bear in mind Penman’s [1948] list of the two requirements for
evaporation: first, the mechanism for removing the water vapor, or sink strength, and, second, supply of energy to provide
the latent heat of vaporization, or energy balance.
The AA model combines, first, bulk mass transfer and, second, energy budget considerations in a convex linear combination of terms representing these two phenomena in the familiar Penman expression for evaporation from a wet surface,
or potential evapotranspiration ETpAA:
p ⫽
Q ⫹␭
E a.
⌬⫹␥ n
The first term of (2) represents a lower limit on the evaporation from moist surfaces, as the second term tends toward zero
over large, homogeneous surfaces under steady state conditions. The second term represents the effects of large-scale
advection. Here ␭ represents the latent heat of vaporization, ⌬
is the slope of the saturated vapor pressure curve at air temperature, ␥ is the psychrometric constant, and Q n is the net
available energy at the surface, usually approximated by the
net absorbed radiation at the surface minus the diffusive
ground heat flux, R n ⫺ G. E a is known as the “drying power
of the air” and is a product of a function of the wind speed at
height Z r above the evaporating surface, or “wind function”
f(U r ), and the difference between the saturated vapor pressure (e *a ) and vapor pressure of the overpassing air (e a ), or
“vapor pressure deficit.” E a takes the following general form:
E a ⫽ f共U r兲共e *a ⫺ e a兲.
better estimated using an approximation to the surface temperature by expanding the back radiation (long wave radiation)
term about the air temperature T a , lead to inaccurate estimates of the energy available for evaporation.
To calculate ETp using the so-called “climatological approach,” Morton [1983] decomposes the Penman equation into
two separate equations describing the energy balance and vapor transfer process. The refinement proposed by Kohler and
Parmele [1967] is developed further by the use of an “equilibrium temperature” T p . T p is defined as the temperature at
which Morton’s [1983] energy budget method and mass transfer
method for a moist surface yield the same result for ETp , and
it is used to adjust the surface energy budget for differences in
back radiation ⌬LW and sensible heat ⌬H as follows:
ETp兩 T⫽Tp ⫽ ETp兩 T⫽Ta ⫺
⌬LW ⫹ ⌬H
In (7), ⌬LW is approximated by a first-order Taylor expansion
of the black body radiation about T p :
⌬LW ⫽ 4 E␴ T 3p共T p ⫺ T a兲,
where E represents the emissivity of the evaporating surface
and ␴ represents the Stefan-Boltzmann constant. The psychrometric constant ␥ can be expressed as
ep ⫺ ea
pC p
H ep ⫺ ea
0.622 ␭
T p ⫺ T a ␭ ET T p ⫺ T a
The AA model uses a simple, empirically based, linear approximation for the wind function f(U r ) proposed by Penman
where ␤ is the Bowen ratio, the ratio of sensible heat flux H to
latent heat flux ␭ET; p is pressure, and C p is the specific heat
of air at constant pressure. ⌬H can be then expressed as
f共U r兲 ⬇ f共U 2兲 ⫽ 0.26共1 ⫹ 0.54U 2兲 ␩ .
⌬H ⫽ f T共T p ⫺ T a兲.
The original expression, f(U 2 ) ⫽ 0.26 (1 ⫹ 0.54 U 2 ),
required wind speeds at 2-m elevation in m s⫺1 and vapor
pressures in mbars to yield E a in mm d⫺1. The factor ␩ in (4)
is required to produce dimensional homogeneity in the SI
system. Substituting this approximation and (3) into the Penman equation (2) yields the expression for ETpAA in (5) used by
Brutsaert and Stricker [1979] in the original AA model:
p ⫽
Qn ⫹ ␭
f共U 2兲共e *a ⫺ e a兲.
In formulating the AA model for use in 3-day time steps,
Brutsaert and Stricker [1979] ignore any effect of atmospheric
instability in the wind function term.
In their original model, Brutsaert and Stricker [1979] calculated ETAA
over 3-day periods using the Priestley and Taylor
[1972] equation for partial equilibrium evaporation:
␭ ETwAA ⫽ ␣
Q ,
⌬⫹␥ n
where the value of the constant ␣ is 1.28. The value of this
constant and its influence on the performance of the AA
model are examined by Hobbins et al. [this issue].
2.3. Complementary Relationship Areal
Evapotranspiration Model
Morton [1983] states that for nonhumid environments both
the Penman approach and the improvements to it suggested by
Kohler and Parmele [1967], that the net radiation term can be
In his formulation of the CRAE model, Morton [1983] replaced the wind function f(U r ) with a calibrated vapor transfer
coefficient f T defined in (11) below, which is constant for a
given atmospheric pressure and independent of wind speed:
␭ f共U r兲 ⬇ f T ⫽ 共 p 0/p兲 0.5f Z␨ ⫺1.
Morton [1983] assumes f T to be independent of wind speed for
the following reasons. First, vapor transfer increases with both
surface roughness and wind speed, but these two are negatively
correlated; vapor transfer increases with atmospheric instability, which is more pronounced at lower wind speeds. Morton
[1983] assumes that as a result of these two complementary
mechanisms, no net change in vapor transfer occurs because of
variations in wind speed. Second, Morton [1983] questions the
reliability of climatological observations of wind speed because
of instrumental and station variability. In (11), ␨ represents a
dimensionless stability factor with values greater than or equal
to 1, p is the atmospheric pressure, and p 0 is the atmospheric
pressure at sea level. Here f Z is a coefficient whose value is
28.0 W m⫺2 mbar⫺1 for above-freezing temperatures. For below-freezing temperatures the value of f Z is increased by a
factor of 1.15, the ratio of the latent heat of sublimation to the
latent heat of vaporization. The exponent 0.5 represents the
effect of atmospheric pressure on the evapotranspiration process and the vapor transfer coefficient. Combining the expressions for ⌬LW and ⌬H and substituting f T yields Morton’s
[1983] energy budget (12) and mass transfer (13) expressions
for ETpCRAE at the equilibrium temperature T p :
⫽ Q n ⫺ 关 ␥ f T ⫹ 4 E␴ T 3p兴共T p ⫺ T a兲,
⫽ f T共e *p ⫺ e a兲.
In (13), e *p is the saturated vapor pressure at T p , and e a is the
actual vapor pressure at T a . ETpCRAE is then defined as the
evapotranspiration that would take place at T p .
Morton [1983] modifies the Priestley-Taylor partial equilibrium evaporation equation (6) to account for the temperature
dependence of both the net radiation term and the slope of the
saturated vapor pressure curve ⌬. The Priestley-Taylor factor ␣
is replaced by a smaller factor b 2 ⫽ 1.20, while the addition
of b 1 ⫽ 14 W m⫺2 accounts for large-scale advection during
seasons of low or negative net radiation and represents the
minimum energy available for ETw but becomes insignificant
during periods of high net radiation. As in the ETpCRAE expression (12), disparities between the surface temperature and
the air temperature at potential conditions are considered by
subtracting ⌬LW (8) from the radiation budget at the surface.
is calculated as
␭ ETwCRAE ⫽ b 1 ⫹ b 2
⫽ b1 ⫹ b2
关Q n ⫺ 4 E␴ T 3p共T p ⫺ T a兲兴
⌬p ⫹ ␥
Q *,
⌬p ⫹ ␥ n
where ⌬ p and Q *n are the slope of the saturated vapor pressure
curve and the net available energy adjusted to the equilibrium
temperature T p , respectively.
Although some writers have claimed that one of the advantages of the CRAE model is that it does not require any
calibration of parameters, this is only true in a local sense. The
CRAE model incorporates global calibration of parameters b 1 ,
b 2 , and f T , using data collected in arid regions for 154 station
months with precipitation totals sufficiently small that they
could be substituted for ETa [Morton, 1983]. In accordance
with the complementary relationship the sum of the computed
ETp and the precipitation was taken to be twice ETw .
In implementing the AA and CRAE models on a monthly
basis, the ground heat flux G is neglected, and ETa is calculated as a residual of (1).
Monthly evapotranspiration was estimated for the conterminous United States using the CRAE and AA models. Model
estimates were then compared with evapotranspiration estimates for selected basins computed from water balances. On
the basis of the record lengths of the available data sets the
study was confined to the water years 1962–1988.
Model Data Sets and Spatial Interpolation
The main advantage of complementary relationship models
is that they rely solely on routine climatological observations.
Local temperature and humidity gradients in the atmospheric
boundary layer respond to, and obviate the necessity for information regarding, the conditions of moisture availability at the
surface. The models bypass the complex and poorly understood soil-plant processes and thus do not require data on soil
moisture, stomatal resistance properties of the vegetation, or
any other aridity measures. Neither do they require local calibration of parameters beyond those built into the models.
The models require data on average temperature, wind
speed, solar radiation, humidity, albedo, and elevation. The
meteorological input data sets are in discrete format, i.e., point
values at station locations. In order to generate estimates of
areal evapotranspiration a spatial interpolation technique was
applied to the point observations of these variables, and evapotranspiration was calculated at each resulting grid cell.
An analysis of the estimation error invoked by various grid
cell sizes conducted by Claessens [1996] indicated that for cell
sizes larger than 10 km the resultant increase in the variance of
the distribution of the estimation error is unacceptable, while
smaller cell sizes result in excessive additional computational
burden with only a relatively minor decrease of the estimation
error variance. Thus all spatial interpolation and analysis was
conducted at a 10-km cell size.
Kriging was the a priori preference for spatial interpolation
of those climatological inputs (i.e., minimum and maximum
temperature) whose station networks would support the inherent semivariogram estimation procedure [Tabios and Salas,
1985]. Otherwise, an inverse distance weighted (IDW) scheme
was used (i.e., for solar radiation, humidity, and wind speed).
For each spatial variable, refinements were made to the chosen
scheme in order better to describe the spatial estimates of the
variables. These refinements are more fully covered in other
sources [Tabios and Salas, 1985; Kitanidis, 1992; Bras and Rodrı́guez-Iturbe, 1993].
The validity of calculating ETa with interpolated observations of the meteorological variables was tested by comparing
the observed relationship between average annual values of
precipitation and ETa derived using a subset of the basins in
this analysis with that derived from data obtained directly (i.e.,
without spatial interpolation) at the meteorological stations.
Significance tests on the regression parameters of the resulting
relationships indicated that those obtained for the basin subset
results were not significantly different from those of the station
results, thus validating this approach.
Temperature data were obtained from the National Climate
Data Center (NCDC) data set (EarthInfo, NCDC Summary of
the Day (TD-3200 computer file), Boulder, Colorado 1998).
Average temperature was estimated as the mean of the average monthly maximum and average monthly minimum temperatures. Claessens [1996] presented results from crossvalidation analysis testing spatial interpolation schemes for
temperature and humidity. It was shown that spatial interpolation of average temperature could be improved by incorporating a simple adiabatic adjustment into the interpolation
scheme. The adiabatic adjustment consists of three steps: (1)
transforming the temperature values to residuals of potential
temperature by subtracting the effects due to elevationdependent adiabatic expansion (9.8⬚C 1000 m⫺1), (2) carrying
out ordinary kriging across the entire area of study on the
transformed data set, and (3) reversing the potential temperature transformation.
Wind speed data were taken from the Solar and Meteorological Surface Observation Network (SAMSON) [National
Oceanic and Atmospheric Administration (NOAA), 1993] and
U.S. Environmental Protection Agency Support Center for
Regulatory Air Models. The former source contains hourly
data on wind speed collected at 217 stations within the conterminous United States. The latter contains data for 29 National Weather Service (NWS) stations in the later years in the
record (i.e., 1984 –1988). The raw data were interpolated using
a simple IDW scheme without a trend surface.
Solar radiation was estimated from the SAMSON [NOAA,
1993]. This data set contains both observed solar radiation
from first-order weather stations and modeled solar radiation
for selected second-order weather stations. There are a total of
215 stations in the conterminous United States. The data
record covers the period 1961 through 1990. In order to improve the interpolation of solar radiation the climatological
station monthly means were regressed on station latitude, longitude, and elevation taken individually and in all combinations. For all months, trend surfaces were generated, and the
solar radiation data were then interpolated using IDW with
Humidity was estimated from the NCDC data set (EarthInfo, NCDC Surface Airways (TD-3280 computer file), Boulder, Colorado, 1998). This data set contains long-term records
of dew point temperature for first- and second-order NWS
stations, with 323 stations in the conterminous United States.
The data record for most stations starts in 1948 and is updated
continuously. The adiabatic adjustment described for the temperature data set was also applied to dew point temperature,
with the IDW scheme used for interpolation of the adiabatically adjusted residuals.
Albedo was estimated using an update from the Gutman
[1988] average monthly albedo surfaces (G. Gutman, personal
communication, 1995). This data set contains albedo estimates
derived from the advanced very high resolution radiometer,
with an original spatial resolution of about 15 km. Elevation
was taken from a 30-arc-sec digital elevation model (National
Geophysical Data Center).
Long-Term, Large-Scale Water Balances
Water balances can be used to estimate evapotranspiration
only for timescales over which the surface and subsurface storage changes and diversions are zero or known with some degree of certainty. Assuming stationarity conditions for the climatic forcing, the long-term (i.e., climatological), large-scale
water balance for an undisturbed basin should lead to negligible net changes in overall basin moisture storage. For a control
volume including the ground surface and transpiring canopy
and extending to the groundwater aquifer, the long-term,
steady state water balance can be expressed as
ET*a ⫽ P ⫺ Y,
where P represents basin-wide precipitation and Y represents
basin yield, both expressed as depth equivalents. Basin yield Y
includes contributions from both surface and groundwater flow
and is estimated by the observed streamflow in the manner of
Eagleson [1978]. Thus a water balance estimate of the longterm average annual evapotranspiration ET*a can be obtained
from independent data on precipitation and streamflow. ET*a
can then be compared with the long-term average annual value
as obtained from monthly evapotranspiration estimates using
the complementary relationship models (ETMODEL
) and proa
vides a means to verify the models.
To estimate ET*a , streamflow data were taken from Wallis et
al. [1991] and Slack and Landwehr [1992]. These two data sets
contain only those basins with little or no regulation and include corrections for missing values and station relocations.
Between them they contain data for about 1475 gauging stations. Both data sets cover the water years 1948 through 1988.
Precipitation estimates were taken from the Parameterelevation Regressions on Independent Slopes Model (PRISM)
[Daly et al., 1994]. This data set combines climatological and
Table 1. Classification of Selected Basins by Size
Eastern Basin Set
Complete Basin Set
Basin Area, km
HUC, hydrologic unit codes.
statistical concepts in an objective precipitation interpolation
model and currently contains data at a grid cell size of 4 km
and a monthly time step for the period 1940 –1999. PRISM was
selected as it yielded better results (i.e., lower cross-validation
bias and absolute error) than kriging techniques [Daly et al.,
1994] and was assumed to be the best available estimate of
precipitation fields, particularly over the complex terrain that
dominates large portions of the western United States.
Basin Selection
Because the long-term analysis of the water balance components required that basins meet a criterion of minimal anthropogenic impact, only and all basins included in the two streamflow data sets listed above were considered. It was assumed
that interbasin and intrabasin diversions and groundwater
pumping were insignificant for the selected basins. In addition
to having a relatively low level of intrabasin diversion, Ramı́rez
and Claessens [1994] concluded, based on two U.S. Geological
Survey (USGS) interbasin transfer inventories [Petsch, 1985;
Mooty and Jeffcoat, 1986], that the basins used in this study
were only minimally affected by interbasin diversions.
At the time of this study a comprehensive digital data set of
USGS-gauged watershed boundaries did not exist; thus the
digital delineation of the USGS eight-digit hydrologic unit
codes (HUCs) were used, combined with published sizes of
gauge drainage areas. Only and all gauges for which the associated HUCs constituted from 85% to 115% of the gauge
drainage area were considered, which together with the requirement of minimum impact resulted in the selection of 139
basins containing a total of 351 HUCs and covering approximately 17.4% of the conterminous United States. Table 1 classifies the selected basins by size. The 120 basins to the east of
the Continental Divide, containing a total of 309 HUCs, are
the primary focus of this study.
Water Balance Closure Errors
The average annual water balance closure error ␧ represents
the error invoked in closing a large-scale, long-term water
balance using ETMODEL
and is henceforth referred to as the
“closure error.” Here ␧ is calculated for each basin, as a percentage of average annual precipitation, from
⫺ ET*a共i, j兲兲
共i, j兲
i⫽1 j⫽1
P 共i, j兲
i⫽1 j⫽1
where i and j are the water year and month, respectively.
Nonzero water balance closure errors must first be considered to be either an overestimation (positive closure error) or
underestimation (negative closure error) of evapotranspiration
by the models. Other possible explanations, however, which
were not quantified in this study, are (1) violations of the
assumption of undisturbed conditions, through the effects of
groundwater pumping and/or surface water diversions, which
were minimized for the gauged basins by the selection criteria;
(2) violations of the assumption of negligible net groundwater
flow out of the basin; (3) violations of the assumption of stationarity in climatological forcing; (4) errors in the hydroclimatological record; and (5) errors induced by spatial interpolation of the climatic variables.
4.1. Spatial Patterns of the Complementary
Relationship Components
For the 27 years of record (1962–1988), monthly surfaces
were constructed for wind speed, solar radiation, dew point
temperature, and average temperature. These surfaces were
used as inputs to either the CRAE model or the AA model, or
both, resulting in monthly surfaces of ETp , ETw , and ETa .
Average annual surfaces of ETw , ETp , and ETa and associated
comparative surfaces are presented in Figures 2, 3, and 4,
Both models predict ETw with a strong negative latitudinal
trend (Figures 2a and 2b), which is a direct result of a similar
gradient in the solar radiation-forcing field. However, ETCRAE
is consistently higher than ETAA
w : The mean excess (Figure 2c),
averaged across the entire conterminous United States, is 170
mm yr⫺1. In calculating ETCRAE
, the increase in ETw due to
the addition of the b 1 term, which is equivalent to adding 179
mm yr⫺1, and the upward adjustment of ⌬p at T p far outweigh
the opposite effect of the correction for back radiation at T p
(i.e., the ⌬LW term in (7)). The difference in the radiative
terms of the ETCRAE
and ETAA
parameterizations resulting
from the use of T p and T a , respectively, can be expressed as
follows (using (6) and (14)):
␭ ETwCRAE ⫺ b 1 ␭ ETwAA
Q *n ⫺
Qn ⫽
⌬p ⫹ ␥
The spatial mean value (averaged across the entire conterminous United States) for this difference (Figure 2d) is found to
be only 39 mm yr⫺1 (equivalent to 3 W m⫺2). The effect
of using T p in the ETCRAE
parameterization is to increase
over most of the study area. Generally, the radiative
term of the CRAE model adjusted for T p exceeds the radiative
term of the AA model across the area to the east of the
Continental Divide, the excess increasing with aridity and decreasing with increasing latitude; the radiative term of the
CRAE model adjusted for T p exceeds that of the AA model in
northern Maine and in the northern Great Plains and Great
Lakes regions. To the west of the Continental Divide the difference pattern is more heterogeneous, reflecting a strong topographical influence: The Central Valley of California and
the Sonoran and Mojave Deserts are all positive (i.e., CRAE
term exceeds AA term), while the higher elevations of the
western United States (i.e., all significant mountain ranges) are
negative (i.e., on the right-hand side of (17) the AA term
exceeds CRAE term).
ETp (Figures 3a and 3b) displays a negative latitudinal gradient similar to ETw in both models. In the western half of the
study area the pattern is complicated by the heterogeneous
topography and the decrease in precipitation, resulting in a
limitation of moisture supply and subsequent decrease in ETa
and increase in ETp . Figures 3a and 3b compare the ETp
estimates to observations of class A pan evaporation estimates
(ETpPAN) from 1931 to 1960 interpolated across the conterminous United States as independent estimates of ETp [U.S.
Geological Survey, 1970]. While the general spatial patterns of
both models are broadly similar to the ETpPAN map, the pan
values are more closely mimicked by ETpCRAE. The maximum
ETp values occur in the desert Southwest, with a maximum
ETpCRAE of around 3000 mm yr⫺1 in the Imperial and Death
Valleys of southwest California and the Sonoran Desert of
southern Arizona (peaking at 3046 mm yr⫺1 in Death Valley)
and maximum ETpPAN (⬎1440 mm yr⫺1) across southwest California. ETpPAN slightly exceeds ETpCRAE (by about 100 mm
yr⫺1) in the High Plains region of the Texas Panhandle, western Oklahoma, Kansas, and southern Nebraska. ETpCRAE exceeds ETpPAN (by about 100 mm yr⫺1) in the Southeast (northern Florida, Georgia, South Carolina, Arkansas, and the
southern portions of Mississippi, Alabama, and Louisiana), in
Illinois, western Indiana, northern Kentucky, southern Pennsylvania, and in northern New England.
The ETpAA estimates are very similar to the ETpPAN estimates
across the eastern half and the northern tier of the United
States. In the Southwest, ETpPAN exceeds ETpAA by about 500
mm yr⫺1, although the spatial patterns are similar: Both ETpAA
and ETpPAN attain maximum values in western Texas, southern
Arizona, and southwestern California. Both predict lower ETp
with increasing elevation and have lobes of highest ETp across
the Southwest from the Central Valley of California through
southern portions of Nevada and Arizona, New Mexico, and
western portions of Texas, Oklahoma, and Kansas.
A surface representing the ETpCRAE ⫺ ETpAA difference
(Figure 3c) indicates that ETpCRAE is consistently and significantly higher than ETpAA; the mean excess is 199 mm yr⫺1. The
region of greatest excess is, similar to the ETw results, the
desert Southwest, where the excess is of the order of 400 –725
mm yr⫺1. It is only in isolated pockets, northern Texas, southwestern Kansas, southeastern Minnesota, Long Island, and
Cape Cod, that ETpAA exceeds ETpCRAE; here the excess is
0 –120 mm yr⫺1.
Comparison of the ETp surfaces suggests a potential difficulty with the wind field input to ETpAA. Because of the simplicity of the IDW interpolation scheme as applied to U 2 , the
wind surfaces are very station-oriented, inasmuch as stations
with extreme values affect their surrounding regions disproportionately. This leads to higher ETpAA values around stations
with extremely high U 2 estimates, and vice versa. This effect is
most pronounced in the eastern half of the study area; in the
western half the effect is generally confused by the multifarious
effects of the heterogeneous topography on the other variables.
Both models predict the highest values of ETa (Figure 4a
and 4b) around the coastline of the Gulf of Mexico, particularly in southern Florida. ETa decreases north and west away
from the gulf. In the western United States the patterns reflect
the complex topography, with local maximum ETa values observed over higher ground, particularly the Sierra Nevada and
Rocky Mountains. The lowest values are in the desert south-
Figure 2. (a) Mean annual Complementary Relationship Areal Evapotranspiration (CRAE) model wet
environment evapotranspiration ETCRAE
. (b) Mean annual Advection-Aridity (AA) model wet environment
evapotranspiration ETAA
w . (c) Mean annual wet environment evapotranspiration difference (CRAE ⫺ AA).
(d) Mean annual difference between T p and T a parameterizations of ETw (i.e., left-hand side of equation (17),
west: the Sonoran Desert in southern parts of California, Arizona, and Nevada for the CRAE model and southern parts of
California and Nevada for the AA model.
Figure 4c, which shows the ETCRAE
indicates that ETCRAE
exceeds ETAA
over the entire eastern
United States, with the exception of isolated areas in the southeast. The areas of greatest excess of ETCRAE
over ETAA
in the central Great Plains, south Texas, and south Florida.
Figure 2. (continued)
Throughout the most arid areas of the west, ETAA
Figures 5a and 5b present average annual surfaces of the
difference between precipitation and ETMODEL
expressed as
an average annual depth. This difference represents the average annual yield and includes the effects of contributions from
surface runoff, interflow, and groundwater discharge/recharge.
Estimates of basin yield are essential products of any evapo-
transpiration model. For this reason, and because such surfaces facilitate comparison with reports from other evapotranspiration models, they are included here. Generally, yields
appear to decrease in a westerly direction as a result of the
effects of elevation and aridity on ETMODEL
and precipitation.
For both models, predicted ETa exceeds precipitation in many
areas, indicating overestimation by the evapotranspiration
models, underestimation of precipitation by the PRISM
Figure 3. (a) Mean annual CRAE potential evapotranspiration ETpCRAE (matched to ETpan contours). (b)
Mean annual AA potential evapotranspiration ETpAA (matched to ETpan contours). (c) Mean annual potential
evapotranspiration difference (CRAE ⫺ AA).
model, or violations of the water balance assumptions (e.g.,
groundwater depletion, surface water diversions into or from
the area, or nonstationarity in climatological forcing). These
areas are predominantly to the west of the Great Plains. For
the CRAE model, negative yields occur over southern Texas,
the High Plains, North and South Dakota, the Basin and
Range country of Nevada and Oregon, the southern and central Rocky Mountains, the Snake River Valley, and the Columbia Plateau. The AA model predicts negative yields over a
smaller area including southern Texas, the Basin and Range
Figure 3. (continued)
country of Nevada and Oregon, the southern and central
Rocky Mountains, the Snake River Valley, and the Columbia
and Colorado Plateaus.
Figure 6 presents the relationship between average annual
values of standardized precipitation and the standardized values of ET*a , ETpMODEL, and ETMODEL
generated in this study.
The evapotranspiration rates have been standardized by expressing them as a fraction of ETMODEL
. Standardized annual
precipitation has been used as a surrogate for moisture availability. Figure 6 indicates that on an average annual basis,
there is a clear relationship between ET*a and moisture availability and that this is complementary to the one between ETp
and moisture availability, as conceptualized in Figure 1. The
resemblance of Figure 1, the theoretical complementary relationship, to Figure 6 is limited to the area to the left (i.e., arid,
semiarid, and subhumid) side of the convergence of ETa and
ETp on ETw shown in Figure 1. This results from the fact that
on an annual and regional basis, natural land surfaces in even
the wettest regions will not approach saturation, and hence
ETa will always be significantly below its limiting value of ETw .
Water Balance Closure Errors
Figure 7 shows the relationship between the ETMODEL
timates and the ET*a observations for all 139 basins. Ideally, all
points would lie on the 1:1 relation indicated on the graph or,
more realistically, would be normally distributed around it with
a low variance. For the 120 basins to the east of the Continental Divide, which are the primary focus of this paper,
is significantly related to ET*a (R 2 ⫽ 0.89 and p ⬍
0.05 for CRAE slope; R 2 ⫽ 0.87 and p ⬍ 0.05 for AA
slope). For the east the greatest divergence between ETCRAE
and ET*a occurs for basins where ET*a is under 500 mm yr⫺1.
In these cases, ETCRAE
overestimates ET*a . Substantial divera
gence between ETAA
ET*a occurs below about 800 mm
yr⫺1 of ET*a , where ETaAA predominantly underestimates
ET*a .
Figure 8 presents the empirical distribution of the 139 water
balance closure errors for both models. Summary statistics are
listed in Table 2. The ranges are approximately ⫺25% to
⫹20% for the CRAE model, with one high outlier, and are
⫺30% to ⫹15% for the AA model, with five high outliers.
Neither model yields closure errors that are normally distributed. For the CRAE model the closure errors are positively
skewed (␥ ⫽ 0.9908), with mean ⫹2.35% and standard deviation 7.69%. The distribution of the closure errors for the AA
model appears bimodal, with modes at ⫺5% and ⫺20%, and is
positively skewed (␥ ⫽ 1.7501), with mean ⫺7.92% and standard deviation 12.67%.
Figures 9a and 9b show the spatial distribution of the water
balance closure errors for the CRAE and AA models, respectively. The distribution of the closure errors is similar to that of
the average annual surfaces of yield (Figures 5a and 5b). Areas
where precipitation ⫺ ETMODEL
is negative force positive
closure errors, whereas negative closure errors are only found
in areas where precipitation ⫺ ETMODEL
is positive. Figure 9a
shows that small ␧CRAE (⫹/⫺ 5%) predominate in basins in
the central parts of the study area (the Midwest and the central
and southern Great Plains) and the Southeast. The more extreme closure errors occur in the western half of the study area,
with negative closure errors in the desert Southwest and positive closure errors in New England, the northern Great Plains,
southern Texas, and particularly the Basin and Range country
of Nevada, Oregon, and Utah.
The ␧AA (Figure 9b) are small (⫹/⫺ 5%) in the Southeast,
New England, southern Texas, and isolated basins in North
Dakota and the northern Great Lakes region, which are all
areas of low elevation or proximity to the ocean. Broadly
Figure 4. (a) Mean annual CRAE actual evapotranspiration ETCRAE
. (b) Mean annual AA actual evapoa
transpiration ETAA
a . (c) Mean annual actual evapotranspiration difference (CRAE ⫺ AA).
speaking, negative closure errors are limited to the area to the
east of the Continental Divide, and positive closure errors are
limited to the mountainous areas surrounding, and to the west
of, the divide. Exceptions are few: The basins in the Rio
Grande valley and isolated basins in southern Texas, the lower
Mississippi valley, the central Appalachians, and the Upper
Peninsula of Michigan display positive closure errors, and isolated basins in the Columbia River valley in Washington state,
the Basin and Range country in southern Nevada, and in
southern Arizona and New Mexico display negative closure
With both models, areas to the west of the Continental
Figure 4. (continued)
Divide display the worst results. The one CRAE and five AA
high outliers mentioned previously represent high, arid basins
in the desert Southwest and Rocky Mountains (the CRAE
outlier and one of the AA outliers represent a basin high in the
Rocky Mountains of Colorado, three AA outlier basins are in
eastern Arizona and western New Mexico, and one AA outlier
basin is in northern Nevada). The apparent failure of the
models in these basins may be due to two basic factors. First,
the complementary relationship assumes a boundary layer that
reflects thorough mixing of the effects of surface environmental discontinuities [Bouchet, 1963]. Such mixing occurs over
length scales of the order of 100 –1000 m [Bouchet, 1963; Davenport and Hudson, 1967] and is ultimately dependent on the
scale of discontinuity and the prevailing atmospheric conditions. The assumption of a well-mixed boundary layer is suspect in many areas to the west of the Continental Divide,
where surface heterogeneities abound. The CRAE model, at
least, has been shown [Claessens, 1996] to perform poorly in
areas of high elevation and complex relief. Second, the interpolation of the input variables for estimating the ETMODEL
and precipitation are increasingly suspect in areas of heterogeneous relief and may lead to significant basin-wide closure
Although the basins in the Pacific Northwest are also in high,
rugged terrain, they show better results than do the outlier
basins. Large errors in application of the complementary relation are unlikely to be found in more humid basins such as
these, where, as shown in Figure 1, the estimates of ETa are
closer to those of ETp. Hence not only do these basins potentially defy the assumption of a well-mixed boundary layer, but
the effects of any misapplication of the complementary relationship here will be obscured by the mitigating effects of
greater humidity. It can therefore be assumed that these basins
do not clarify the utility of the complementary relationship
models in estimating ETa .
Because of the potential data and model problems in western basins all basins to the west of the Continental Divide are
left for future research and excluded from further analysis
herein: All results reported henceforth, unless otherwise
stated, are for the basins to the east of the Continental Divide,
and these basins are referred to as the “eastern basins.” These
basins cover approximately 21% of the area of the conterminous United States to the east of the Continental Divide.
However, for comparison purposes the western basins are still
included on the graphs.
Figure 10 shows the distribution of closure errors for the 120
eastern basins. Excluding the western basins from the distribution eliminates the most extreme closure errors (particularly
the positive errors). The mean ␧CRAE is slightly increased (i.e.,
from ⫹2.35% to ⫹2.51%), the minimum ␧CRAE is increased
from ⫺24.87% to ⫺10.67%, and the maximum ␧CRAE is decreased from ⫹43.13% to ⫹22.85%. As shown in the histograms (Figures 4 and 5) and reflected in the skewness values
(Table 2), the distribution of CRAE closure errors becomes
less biased toward positive values (␥ ⫽ 0.8153). The effect on
the AA closure error set is to decrease the mean ␧AA from
⫺7.92% to ⫺10.59% and to decrease the maximum ␧AA from
⫹48.71% to ⫹9.70%, leaving the AA distribution more symmetrically distributed around its mean than before. Although
the AA distribution appears to be bimodal, with modes at
⫺5% and ⫺15%, the skewness (or g statistic in the skewness
test) value of ␥ ⫽ 0.1179 falls within the bounds of normality
at the 5% significance level. Thus the hypothesis of normality
cannot be rejected for this distribution.
Figures 11a through 11c depict the effects of basin climatology upon the performance of the models, as measured by the
Figure 5. (a) Mean annual CRAE yield. (b) Mean annual AA yield.
water balance closure errors ␧CRAE and ␧AA. Each of the
dependent variables, average annual precipitation (Figure
11a), average annual precipitation minus streamflow (Figure
11b), and average annual relative evapotranspiration
/ETpMODEL) (Figure 11c), is a measure of moisture
availability (i.e., aridity or humidity) of the basin, with aridity
increasing toward the left in each graph. To some degree these
aridity measures may be considered surrogates of the degree of
continentality of the basins.
Figure 11a relates closure error to mean annual basin-wide
precipitation. For the sake of clarity the independent variable
is shown up to 1600 mm only; an outlier at 3000 mm is not
shown, but it does not affect the regression results. The ␧CRAE
decrease with precipitation (R 2 ⫽ 0.15 and p ⬍ 0.05),
Figure 6. Complementary relationship diagrams of the eastern subset: (a) CRAE model and (b) AA model.
Rates have been standardized by expressing them as a fraction of ETMODEL
Figure 7. ETMODEL
versus ET*a .
Figure 8. Histogram of closure errors ␧CRAE and ␧AA of the complete basin set.
tending toward zero at approximately 1200 mm yr⫺1. The ␧AA
increases with precipitation (R 2 ⫽ 0.31 and p ⬍ 0.05),
tending toward zero at about 1400 mm yr⫺1. Thus the predictive powers of both models increase with humidity.
These trends are also evident in the relationship between
␧MODEL and the independent estimates of evapotranspiration
ET*a , provided by (15) (Figure 11b). The ␧CRAE decreases with
ET*a (R 2 ⫽ 0.18 and p ⬍ 0.05) and converges to zero at
approximately 800 mm yr⫺1. The ␧AA increases with ET*a
(R 2 ⫽ 0.18 and p ⬍ 0.05) and converges to zero at about 900
mm yr⫺1. The scatter of ␧AA increases with aridity. Again, for
both models the ETMODEL
estimate improves with humidity.
The effect of the relative degree of soil control or climate
control of evapotranspiration (i.e., effect of moisture availability at the land surface) on water balance closure can be assessed by relating the closure error to relative evapotranspiration ETa /ETp (Figure 11c). Increasing this ratio is equivalent
to moving to the right in Figure 1: ETa and ETp converge
toward ETw . The ␧CRAE are slightly negatively correlated with
relative evapotranspiration (R 2 ⫽ 0.01 and p ⬍ 0.05), although very scattered. The ␧AA display a consistently positive
relationship with relative evapotranspiration (R 2 ⫽ 0.43 and
p ⬍ 0.05) and converge toward zero; for relative evapotranspiration values below approximately 0.36, there is increased
scatter in ␧AA. These results indicate that the performance of
the CRAE model is nearly independent of the degree of sat-
uration of the basin land surface, whereas ETAA
improve with increasing basin saturation.
The relationship between the closure errors and the mean
annual basin-wide evaporation temperature difference (T p ⫺
T a ) is shown in Figure 11d. Note that the range for this temperature difference is approximately ⫹/⫺2.5⬚C. The relationship between evaporation temperature difference and ␧CRAE is
slightly negative (R 2 ⫽ 0.06 and p ⬍ 0.05). The ␧CRAE
appear to be centered around zero for basins where T p exceeds
T a . The relationship between evaporation temperature difference and ␧AA is significantly positive (R 2 ⫽ 0.34 and p ⬍
0.05). The ␧AA are widely scattered for basins where T a exceeds T p but converge toward zero for basins where T p exceeds T a .
Figure 11d indicates that the two ETMODEL
estimates are
most similar for basins where T p exceeds T a . This result is
somewhat more subtle than that stated by Morton [1983, p. 21],
who claimed that “the analytical solution of Penman
[1948] 䡠 䡠 䡠 [is] accurate only under relatively humid conditions
where the equilibrium temperature [T p ] is near the air temperature [T a ].”
Morton’s [1983] statement above and the fact that the CRAE
model was calibrated using data from arid basins imply that the
CRAE model should significantly outperform the AA model in
arid regions. This implication is borne out by examining the
closure errors at the arid end of the climatic spectrum. For the
Table 2. Summary Statistics for Water Balance Closure Errors
Eastern Basin Set
Complete Basin Set
Mean (percent precipitation)
Median (percent precipitation)
Minimum (percent precipitation)
Maximum (percent precipitation)
Standard deviation (percent precipitation)
Figure 9. (a) Geographic distribution of CRAE closure errors of the complete basin set. (b) Geographic
distribution of AA closure errors of the complete basin set.
cluster of one CRAE and five AA positive outliers the temperatures at which the evaporative processes are evaluated are
significantly different (T a exceeds T p by 1.5⬚–2⬚C), yet for four
of the five outliers the CRAE model performs well (␧CRAE
ranges from ⫹7.58% to ⫺1.93%).
In the conterminous United States, continental-scale precipitation displays an overall, climatological gradient from a hu-
mid climate in the east to a semiarid climate in the west. As
continental-scale elevation generally increases similarly in an
east-west direction, there is a strong negative correlation between precipitation and elevation, particularly across the study
area to the east of the Continental Divide. Thus, when comparing ␧MODEL with mean basin elevation, one would expect a
similar relationship to that observed between closure error and
Figure 10. Histogram of closure errors ␧CRAE and ␧AA of the eastern subset.
average annual basin precipitation (Figure 11a). Figure 12
demonstrates that for the CRAE model, closure errors do
indeed increase with elevation (R 2 ⫽ 0.05 and p ⬍ 0.05),
with ␧CRAE tending toward zero for the lowest basins. The ␧AA
exhibit a strong negative correlation with basin elevations
(R 2 ⫽ 0.14 and p ⬍ 0.05).
As expected from a misapplication of the complementary
relationship in the rugged terrain of the western basins, the set
of basins with the highest elevations, and therefore the most
rugged terrain, exhibits the most scatter. The cluster of
␧MODEL above ⫹30% is for high, arid basins in the desert
Southwest and the Rocky Mountains. Of the western basins
the highest closure errors (␧CRAE ⫽ ⫹43.13% and ␧AA ⫽
⫹48.71%) occur at the greatest elevation (2966 m) (Figure 12).
Figure 13 shows the relationship between ␧MODEL and mean
annual basin wind speed and demonstrates the effects of the
different treatments of advection in the two models. The
␧CRAE are weakly positively correlated with wind speed (R 2 ⫽
0.07 and p ⬍ 0.05). They are clustered around zero for the
lowest wind speeds and increase in variability with increasing
wind speed. This near independence of the CRAE model’s
performance with wind speed appears to support Morton’s
[1983] treatment of advection. The ␧AA are strongly negatively
correlated with, and hence the AA model is very sensitive to,
wind speed (R 2 ⫽ 0.50 and p ⬍ 0.05). In fact, mean annual
wind speed exhibits the strongest relationship with ␧AA of any
climatic variable. This suggests that the first step in improving
the AA model should be to reparameterize the wind function
f(U 2 ). This reparameterization is the subject of Hobbins et al.
[this issue].
The fact that the positive outliers in particular and the western basins in general have wind speeds toward the low end of
the range (i.e., ⬍4 m s⫺1) points to a deficiency in the spatial
interpolation of the wind speed data in the mountainous areas,
as one would expect to find higher mean annual basin-wide
wind speed for the highest, most rugged basins. An attempt
was made to improve the interpolation of the wind speed fields
by using trend surfaces. For each month, climatological (i.e.,
across the length of the record) mean wind speed values were
generated for each station, and these values were regressed on
the station latitude, longitude, and elevation taken individually
and in all combinations. However, no distinct trends could be
found, and the attempt was abandoned.
Summary and Conclusions
Average annual surfaces of ETw and ETp indicate gradients
that are a result of gradients in radiative forcing and a combination of radiative and precipitation forcing, respectively. The
effects of elevation, through precipitation forcing and/or the
vapor transfer function parameter f T , are most evident in the
modeled surfaces of ETp . For both components of the complementary relationship, ETp and ETw , the estimates generated by the CRAE model exceed those of the AA model. The
resulting modeled ETa surfaces show a negative latitudinal
gradient in the eastern half of the conterminous United States
and a positive elevational gradient for western areas. In the
western part of the study area, regions were identified where
modeled ETa exceeds precipitation on an average annual basis. For the AA model these areas correspond to the areas of
lowest precipitation, whereas for the CRAE model such a
correspondence is less evident. These regions of negative yield
are generally associated with irrigated agriculture, groundwater depletion, and/or surface water diversions. However, the
spatial extent and magnitude of this precipitation deficit could
be an indication that the ETa estimates produced by the models are too high. This can be attributed to either the models
themselves or to the inappropriateness of the meteorological
forcing fields, either through low data quality or inadequacies
involved in their spatial interpolation.
The b 1 term in ETCRAE
, included to allow for any periods of
negative radiation, increases ETCRAE
by approximately 179
mm yr⫺1, which almost exactly accounts for the excess of longCRAE
term average ETw
over ETw
. If b 1 were neglected,
would be very close to the standard Priestley-Taylor
parameterization used for ETAA
w .
All of the positive AA and CRAE closure errors in the Basin
and Range country, the northern Great Plains, and southern
Texas are a direct result of the negative yields predicted for
Figure 11. (a) Closure errors versus mean annual basin-wide precipitation. (b) Closure errors versus mean
annual basin-wide ET*a . (c) Closure errors versus mean annual basin-wide relative ETMODEL
. (d) Closure
errors versus mean annual basin-wide evaporation temperature difference.
these areas. These could result from the effect of irrigated
agriculture, through groundwater depletion and surface water
diversions, which would be a violation of the assumption of
minimum impact. Most other CRAE closure errors are of the
order of ⫹/⫺5% and do not appear spatially biased. Almost all
other AA closure errors are negative and may result from a
poorly calibrated wind function.
Both models’ performances are affected by basin climatology. The CRAE model underestimates ETa slightly in humid
climates and overestimates slightly in arid climates, which supports the conclusions in other work by Hobbins et al. [1999]
where it was found that the CRAE model performs less well
under arid and advective conditions (i.e., that it tends to overestimate ETa ). The AA model generally underestimates ETa
in all but the most extreme arid climates at high elevations.
Scatterplots of closure error versus aridity measures (Figures
11a, 11b, and 11c) show that both models yield closure errors
that increase in variability with increasing aridity but converge
toward zero with increasing humidity. These trends are also
reflected in the relationship of the closure errors with mean
basin elevation (Figure 12), which in the case of the eastern
United States may be considered a surrogate for continentality
or aridity.
Moving toward the other end of the climatic spectrum (i.e.,
increasing moisture availability), the predictive powers of both
models increase in moving toward regions of increased climate
control of evapotranspiration rates and decrease in moving
toward regions of increasing soil control. Increased climate or
soil control in this context refers to increased and decreased
moisture availability. Because irrigated agriculture is often associated with areas of low moisture availability, these trends
could be a direct reflection of anthropogenic influences (i.e.,
through net groundwater withdrawals and net diversion of
surface waters) not sufficiently mitigated against in the basin
Figure 11. (continued)
selection procedure. However, it should be noted that groundwater pumping in a study basin affects only the independent
evapotranspiration estimate ET*a not the ETMODEL
Indeed, one of the primary advantages of complementary relationship models is that the assumption of the integration of
atmospheric moisture accounts for all surface hydrology, and
therefore their utility is unaffected in basins where groundwater pumping is present.
The models differ significantly in their treatments of the
temperature at which the evaporative processes are evaluated.
The AA model, in using the Penman equation, uses the air
temperature T a . The CRAE model hypothesizes an equilibrium temperature T p at which the radiative and vapor transfer
components of the evaporative process are equal. The other
significant difference between the two models is their treatments of advection. The AA model uses actual wind speed data
to calculate the drying power of the air E a in the expression for
ETpAA, whereas the CRAE model calculates ETpCRAE by use of
a vapor transfer coefficient f T , independent of wind speed.
The near-zero mean annual CRAE closure error appears to
support Morton’s [1983] reasoning for using a wind transfer
coefficient that is independent of observed wind speeds. However, the positive correlation observed between the CRAE
closure errors and climatological basin-wide wind speed, although weak, suggests that there is some opportunity to improve this model’s treatment of advection. The elevational
effects on ETCRAE
could be better modeled by reparametera
izing the vapor transfer function f T in (11) and reestimating the
values of the b 1 and b 2 parameters in the expression for
(14) by another regional calibration in representative
arid, mountainous areas.
The strong negative correlation of the AA closure errors
with wind speed clearly demonstrates the sensitivity of this
model to the wind function f(U 2 ) described in (4), which was
first proposed [Brutsaert and Stricker, 1979] for use in the AA
model operating at a temporal scale of the order of days. These
results highlight the need for a reparameterization of this component of ETpAA to yield both accurate ETp estimates and
Figure 12. Closure errors versus mean basin elevation.
unbiased water balance estimates on a monthly basis (i.e., a
parameterization specific to regional spatial scales and
monthly temporal scales). These potential improvements are
the subject of Hobbins et al. [this issue].
On a mean annual basis the relationships observed between
the ETpMODEL components and the independent estimates of
regional evapotranspiration ET*a are very similar to the theoretical complementary relationship. Although the performances of both models improve markedly with increasing humidity, it must be noted that in humid basins the ETMODEL
estimate more closely approaches the ETpMODEL estimate.
Hence the complementary behavior underpinning observation
in such basins may not be as apparent.
Although an independent, theoretical proof of the complementary relationship hypothesis has not yet been established,
this study provides indirect evidence supporting its plausibility.
However, it is important to mention that the observed performance of the models should not only be attributed to their
basic, driving hypothesis but also to Morton’s [1983] and Brutsaert and Stricker’s [1979] choice of governing equations and
their applicability at the regional spatial scale and monthly
temporal scale. Within this context, LeDrew’s [1979, p. 500]
statement regarding the CRAE model that “For long-term
means its success must depend on calibration
procedures 䡠 䡠 䡠 and errors which are self-compensating in the
long run” should be borne in mind. Overall, the excellent
performance of the CRAE model, at least in the eastern
United States, demonstrates its utility in providing independent estimates of ETa in regions across the climatic spectrum,
while further enhancing the importance of the concept of the
complementary relationship for hydrometeorological applications.
Figure 13. Closure errors versus mean annual basin-wide wind speed.
Acknowledgments. This work was partially supported by the U.S.
Forest Service and the National Institute for Global Environmental
Change through the U.S. Department of Energy (Cooperative Agreement DE-FC03-90ER61010). In addition, one of the authors (Jorge A.
Ramı́rez) received partial support from the Colorado Water Resources Research Institute. This paper benefited greatly from comments by Guido Salvucci and two other anonymous reviewers.
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T. C. Brown, Rocky Mountain Research Station, U.S. Forest Service, Fort Collins, CO 80526-2098. ([email protected])
L. H. J. M. Claessens, The Ecosystems Center, Marine Biological
Laboratory, Woods Hole, MA 02543. ([email protected])
M. T. Hobbins and J. A. Ramı́rez, Department of Civil Engineering,
Colorado State University, Fort Collins, CO 80523-1372.
([email protected]; [email protected])
(Received December 20, 1999; revised November 2, 2000;
accepted November 2, 2000.)
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