HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-217-2017A Budyko framework for estimating how spatial heterogeneity and lateral moisture redistribution affect average evapotranspiration rates as seen from the atmosphereRouholahnejad FreundElhamelham.rouholahnejad@usys.ethz.chhttps://orcid.org/0000-0002-4316-2013KirchnerJames W.https://orcid.org/0000-0001-6577-3619Department of Environmental Systems Science, ETH Zurich, 8092 Zürich, SwitzerlandSwiss Federal Research Institute WSL, 8903 Birmensdorf, SwitzerlandDept. of Earth and Planetary Science, University of California, Berkeley, CA 94720, USAElham Rouholahnejad Freund (elham.rouholahnejad@usys.ethz.ch)11January201721121723318August201631August201616November201617November2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/217/2017/hess-21-217-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/217/2017/hess-21-217-2017.pdf
Most Earth system models are based on grid-averaged soil columns that do not
communicate with one another, and that average over considerable sub-grid
heterogeneity in land surface properties, precipitation (P), and potential
evapotranspiration (PET). These models also typically ignore topographically
driven lateral redistribution of water (either as groundwater or surface
flows), both within and between model grid cells. Here, we present a first
attempt to quantify the effects of spatial heterogeneity and lateral
redistribution on grid-cell-averaged evapotranspiration (ET) as seen from the
atmosphere over heterogeneous landscapes. Our approach uses Budyko curves, as
a simple model of ET as a function of atmospheric forcing by P and PET.
From these Budyko curves, we derive a simple sub-grid closure relation that
quantifies how spatial heterogeneity affects average ET as seen from the
atmosphere. We show that averaging over sub-grid heterogeneity in P and
PET, as typical Earth system models do, leads to overestimations of average ET.
For a sample high-relief grid cell in the Himalayas, this overestimation bias
is shown to be roughly 12 %; for adjacent lower-relief grid cells, it is
substantially smaller. We use a similar approach to derive sub-grid closure
relations that quantify how lateral redistribution of water could alter
average ET as seen from the atmosphere. We derive expressions for the maximum
possible effect of lateral redistribution on average ET, and the amount of
lateral redistribution required to achieve this effect, using only estimates
of P and PET in possible source and recipient locations as inputs. We show
that where the aridity index P/PET increases with altitude, gravitationally
driven lateral redistribution will increase average ET (and models that
overlook lateral redistribution will underestimate average ET). Conversely,
where the aridity index P/PET decreases with altitude, gravitationally
driven lateral redistribution will decrease average ET. The effects of both
sub-grid heterogeneity and lateral redistribution will be most pronounced
where P is inversely correlated with PET across the landscape. Our analysis
provides first-order estimates of the magnitudes of these sub-grid effects,
as a guide for more detailed modeling and analysis.
Introduction
The atmosphere integrates the fluxes of water, energy, and trace gases that
it receives from the spatially heterogeneous landscape beneath it. Earth
system models typically account for this spatial heterogeneity, and the
atmosphere's integration of it, only at scales larger than their relatively
coarse grid resolution. Accounting for the considerable heterogeneity of the
Earth's surface at smaller scales, and its consequences for fluxes from the
surface to the atmosphere, is a major challenge in Earth system modeling.
Sub-grid-scale surface heterogeneity and subsurface water redistribution
are unrepresented in Earth system models. At the 100 km by 100 km grid cell
scale, large mountain ranges (such as the Swiss Alps) become indistinct.
The grid resolution in Earth system models is typically translated directly
onto the Earth's surface, which is modeled as columns that are vertically
disaggregated into soil layers at scales of centimeters or meters, but are
horizontally averaged at the 1∘ by 1∘ (roughly 100 km by
100 km) scale of the overlying atmospheric model (Fig. 1). At this scale,
individual ridges and valleys disappear, and even major mountain ranges and
basins can become indistinct. Likewise, much of the variability in the
surface climatology of the landscape and its consequences for
land–atmosphere interactions are lost.
This loss of detail in land surface properties has important implications
for water fluxes in Earth system models. Given that ET may depend
nonlinearly on both water availability and atmospheric water demand, which
are both spatially variable at scales far below typical model grid scales,
the average ET over a heterogeneous landscape may differ substantially from
model ET estimates derived from spatially averaged land surface properties.
The potential importance of this issue has motivated research into methods
for capturing sub-grid-scale properties and processes within Earth system
models. For example, Beven (1995) discussed the scale dependence of
hydrological models and the pitfalls of using effective parameters to
reproduce the areal averages of sub-grid fluxes, especially where water
availability strongly influences the vapor flux. These aggregation problems
arise from the nonlinearity of the governing processes, together with
the spatial heterogeneity of the system.
Nesting higher-resolution regional models within global models represents an
obvious, but computationally demanding, approach to treating sub-grid-scale
heterogeneity. As described by Klink (1995), two broad classes of
aggregation schemes have been proposed to incorporate sub-grid heterogeneity
while keeping computational costs manageable. In “averaged” surface schemes,
the surface properties are averaged over each grid cell and the average is
applied directly in the model. In “mosaic” schemes, by contrast, individual
grid cells are partitioned into several surface types, the model is run for
each surface type separately, and the fluxes from each surface type are
area weighted to determine the average fluxes for the grid cell.
Numerous modeling studies over the past two decades have shown that, in
comparison to mosaic schemes and nested high-resolution models, averaged
surface schemes tend to overestimate evapotranspiration and sensible heat
flux (e.g., Klink, 1995; Giorgi and Avissar, 1997; Essery et al., 2003;
Teluguntala et al., 2011; Ershadi et al., 2013). Studies with nested
high-resolution models demonstrate that this overestimation bias is largest
where topographic effects play a major role (Giorgi and Avissar, 1997; Pope
and Stratton, 2002; Boyle and Klein, 2010; Bacmeister et al., 2014).
Another potential source of bias in Earth system models arises from their
neglect of surface and subsurface flows within and between grid cells.
Current Earth system models calculate infiltration and vertical transport of
water in each soil column, but the water that reaches the bottom of the
column is either stored as groundwater or simply disappears, reappearing
later in the ocean. In real-world landscapes, by contrast, significant
volumes of water are transported laterally, either via groundwater flow or
by rivers flowing from mountains into valleys and potentially redistributing
their water to valley ecosystems by infiltration into valley aquifers. These
lateral redistribution processes supply water for evapotranspiration in
groundwater-dependent ecosystems in the dry season (Fan and Miguez-Macho,
2010). Several case studies in the Amazon (Christoffersen et al., 2014),
central Argentina (Contreras et al., 2011; Jobbágy et al., 2011), and
other groundwater-dependent ecosystems (Eamus et al., 2015) demonstrate how
water supply can govern the seasonality and magnitude of evapotranspiration
in those regions. However, the potential effects of these lateral
redistribution processes on grid-scale ET, as viewed from the atmosphere,
are missing from current Earth system models, and the resulting biases in
modeled water fluxes are unknown.
The Earth system modeling community has recognized the need to determine how
sub-grid heterogeneity and lateral redistribution affect grid-scale
evapotranspiration rates as viewed from the atmosphere, and to develop
schemes that can efficiently account for these effects in land surface
models (Clark et al., 2015). A recent high-resolution modeling study for the
continental US (Maxwell and Condon, 2016) concluded that lateral
redistribution could substantially alter the partitioning of ET between
transpiration and bare-soil evaporation, but the net effect on the combined
ET flux remains unclear. The studies outlined above illustrate the potential
effects of spatial heterogeneity and lateral redistribution, but we
currently lack a general framework for estimating the resulting biases in
calculated evapotranspiration rates. Here, we present a first attempt to fill
this knowledge gap using an analysis based on Budyko curves as simple
semi-empirical estimators of ET. This analysis yields first-order estimates
of the potential effects of sub-grid heterogeneity and subsurface lateral
redistribution on ET fluxes from heterogeneous landscapes, as seen from the atmosphere.
A Budyko framework for estimating terrestrial water partitioning
The simplest widely used approach for estimating evapotranspiration rates
from the land surface is the Budyko framework (Turc, 1954; Mezentsev, 1955;
Pike, 1964; Budyko, 1974; Fu, 1981; Milly, 1993; Zhang et al., 2001; Yang et
al., 2007). Budyko showed empirically that under steady-state conditions in
catchments without significant groundwater inputs, losses, or storage
changes, the long-term annual average evapotranspiration (ET) rate is
functionally related to both the supply of moisture from the atmosphere
(precipitation, P) and net irradiance (available energy) as an estimator of
the evaporative demand for water by the atmosphere (potential
evapotranspiration, PET). Under arid conditions (that is, when P is much
smaller than PET), ET converges toward P, implying that ET is limited by the
available supply of water (Fig. 2, water limit line). Alternatively, under
humid conditions (that is, when P is much greater than PET), ET is limited
by atmospheric demand and E converges toward PET (Fig. 2, energy limit
line). Budyko's original work showed, and decades of studies have confirmed,
that, under the long-term steady-state assumptions outlined above,
hydrological systems typically operate close to either the energy or water constraints.
Several studies have explored how natural systems may violate the
assumptions of the Budyko approach. Net inputs or losses of groundwater, as
well as long-term changes in soil moisture and groundwater storage, have
been shown to alter the water balance sufficiently that measurements of P
and ET can produce points that fall far from the energy and water
constraints in Fig. 2. However, these apparent violations of the Budyko
approach can be corrected if the precipitation term P is replaced by an
effective precipitation that accounts for root zone water storage changes
and net inputs or losses of groundwater (Zhang et al., 2001,
2008; O'Grady et al., 2011; Istanbulluoglu et al., 2012; Wang, 2012; Chen et
al., 2013; Troch et al., 2013; Du et al., 2016).
The Budyko framework can be expressed in two different non-dimensional sets
of axes, depending on whether one wishes to focus on the effects of changing
water supply (P) or atmospheric water demand (PET). If one seeks to analyze
the effects of changing PET under a fixed P, it is most intuitive to
non-dimensionalize both axes by P, as shown in Fig. 2a. In this coordinate
space, translation along the horizontal axis represents a change in PET.
Our analysis, by contrast, focuses on how changes in water availability
affect ET under a fixed PET. For such questions, it is most intuitive to
non-dimensionalize the coordinate axes by PET, as shown in Fig. 2b. In this
coordinate space, translation left or right along the horizontal axis
represents changes in water availability. Thus, this coordinate space is
better suited to our analysis.
Budyko framework and energy and water limit lines. The blue cloud is
a smoothed scatterplot of the 30 arcsec resolution mean annual
precipitation (P), evapotranspiration (ET), and potential evapotranspiration (PET)
for continental Europe. ET and PET data are from MODIS (Mu et al., 2007), P
data set is from WorldClim (Hijmans et al., 2005).
Alternative empirical equations for mean annual evaporation rate in
Budyko framework: ET is mean annual evapotranspiration, P is mean annual
precipitation, PET is mean annual potential evapotranspiration (evaporative demand).
EquationParameterReferenceETPET=PPET1PPETn+11/n (1)n (dimensionless)Bagrov (1953), Turc (1954),modifies theMezentsev (1955), Pike (1964)partitioning of PChoudhury (1999), Zhang etbetween E and Qal. (2001), Milly and Dunne(2002), Yang et al. (2008)ETPET=PPET+ 1 -PPETω+11/ω (2)ω – similar to n,Fu (1981), Zhang et al. (2004),modifies theYang et al. (2007)partitioning of Pbetween E and QETPET=PPET+ω1+ωPPET-1+PPET (3)ω – coefficient ofZhang et al. (2001)vegetation andwater supplyETPET=PPETexpγ1-PPET-1expγ1-PPET-PPET (4)γ – the ratio ofMilly (1993), Porporato et al.soil water storage(2004)capacity toprecipitation
Table 1 presents several alternative empirical equations that have been
proposed for “Budyko curves” relating ET to P and PET. Our analysis will be
based on the Turc–Mezentsev equation (Eq. 1 in Table 1), because it is the
most widely used of the alternatives shown here. However, the differences
among these formulas are unimportant for the analysis presented below.
Here, we use Budyko curves as simple models for how ET is controlled by the
supply of available moisture (as represented by P) and evaporative demand
(as represented by PET). We could have used more complex ecohydrological
models to estimate ET instead, at the cost of increased complexity and
reduced transparency. However, any such models must obey the same energy and
water constraints that shape the behavior of catchments in the Budyko
framework, so we would not expect their behavior to deviate greatly from the
Budyko curves that are analyzed here. Thus, the Budyko curves that we analyze
here can be considered as approximations to the behavior of these more
complex models. They also have an important advantage for our purposes,
namely that they specify ET as an explicit function of its main drivers P
and PET, allowing us to derive general analytical results that might
otherwise be difficult to infer from sets of simulation results.
Effects of sub-grid heterogeneity on ET in a Budyko framework
The water and energy constraints that limit ET imply that ET is an
intrinsically nonlinear function of P and PET. Under arid conditions (with
P≪ PET), ET will increase almost linearly with P, but as
conditions become more humid and the supply of moisture exceeds the energy
available to evaporate it (P≫ PET), the energy
constraint will hold ET nearly constant as P increases. Conversely, under
humid conditions, ET will scale almost linearly with evaporative demand (as
expressed by PET), but as conditions become more arid and the supply of
moisture becomes limiting, ET will be constrained by P and will become
largely independent of PET.
As shown in Fig. 3, the nonlinear behavior of ET as a function of P and PET
is also reflected in Budyko curves, particularly near the transition between
humid and arid conditions (P/PET close to 1). This nonlinear behavior has
important implications for estimates of average ET in heterogeneous landscapes.
Illustration of heterogeneity bias in a Budyko curve (Eq. 5). The true
average (gray circle) of the ET values of locations 1 and 2 (black dots) is
less than the average ET that would be estimated from their average P/PET
(open circle). The size of the heterogeneity bias will be proportional to the
curvature in the ET function and proportional to the variability in P and PET
among the individual points (Eqs. 6–8).
As Fig. 3 illustrates, the average of a nonlinear function with
heterogeneous inputs will not, in general, be equal to the value of that
function evaluated at the average of the input values. That is, the average
of the function will not be the function of the average inputs (e.g.,
Rastetter et al., 1992; Giorgi and Avissar, 1997). One can visually see that
the resulting heterogeneity bias will depend on how strongly curved the
function is, and how widely its inputs are scattered. This intuitive concept
can be expressed mathematically by comparing the value of the function, and
the mean of its second-order Taylor expansion around the same point, to
estimate the heterogeneity bias (e.g., Kirchner et al., 1993).
We begin by restating Eq. (1) from Table 1 as an explicit function of P and
PET:
ET=f(P,PET)=PPPETn+11/n.
For a function of two variables, a second-order, second-moment expansion
leads directly to the following approximation for the mean of the function,
in terms of the function's value at the mean of its inputs:
ET‾≈fP‾,PET‾+12∂2f∂P2var(P)+12∂2f∂PET2var(PET)+∂2f∂P∂PETcov(P,PET),
where the derivatives are understood to be evaluated at P‾ and PET‾.
Evaluating the necessary derivatives using Eq. (5) directly
yields the following expression for the average ET:
ET‾≈fP‾,PET‾-(n+1)P‾n+1PET‾n+1P‾n+PET‾n2+1/n×12var(P)P‾2+12var(PET)PET‾2-cov(P,PET)P‾PET‾,
where the second term represents the heterogeneity bias (that is, the
difference between the average of the function and the function of the
average). The relative magnitude of this bias can be derived by combining
Eqs. (7) and (5), yielding
fP‾,PET‾-ET‾f(P‾,PET‾)≈(n+1)P‾PET‾n2+PET‾P‾n22×12var(P)P‾2+12var(PET)PET‾2-cov(P,PET)P‾PET‾.
From Eq. (8) one can directly see that the heterogeneity bias will depend on
the variances of P and PET, as well as their covariance (all
non-dimensionalized by their means). One can see that the heterogeneity bias
will generally be positive (that is, estimates based on P‾ and
PET‾ will overestimate ET‾), because the covariance term in
Eq. (8) will be less than the variance terms. One can also see that, all
else equal, a negative correlation between P and PET will amplify the
heterogeneity bias (because, in terms of the Budyko plot, this will lead to
greater scatter in P/PET). Furthermore, one can see that the relative
heterogeneity bias will be greatest when the term in curly brackets in
Eq. (8) will be as small as possible, which will occur at P‾/PET‾= 1 (the point of maximum curvature in
the Budyko curve). Finally, from Eq. (8) one can see that at higher values
of n, the peak heterogeneity bias will be greater (due to the
n+ 1 term), but will be more tightly focused around P‾/PET‾= 1
(due to the powers of n/2).
One-kilometer topography (a: SRTM; Jarvis et al., 2008) and
annual mean climatology for a 1∘ by 1∘ grid cell spanning the
Himalayan Front at 89–90∘ E, 27–28∘ N. Spatial patterns of
1 km resolution mean annual precipitation (b: WorldClim; Hijmans et al.,
2005), potential evapotranspiration (c: MODIS; Mu et al., 2007), and
(d) evapotranspiration (ET) calculated using the Budyko curve (Eq. 5).
Panel (e) shows a random sample of 50 points from (b), (c),
and (d), along with the average P, PET, and ET over the grid cell
(yellow circle), and the ET value estimated from Eq. (5) for the same average
P and PET (orange circle). This ET estimate is 921 mm yr-1, 11.8 %
more than the average of the 1 km resolution ET estimates.
Heterogeneity bias in average ET for the nine grid cells of the terrain
shown in Fig. 4a (88–91∘ E, 26–29∘ N) calculated from
high-resolution (1 km) spatial variation of mean annual P and PET in each
grid cell (Eq. 7). “True” heterogeneity bias is estimated by averaging
the ET predicted by the Budyko curve for each 1 km pixel, and comparing this
average with the ET predicted from the same curve using the average P and
PET in the corresponding grid cell. “Approximate” heterogeneity bias is
estimated from Eq. (8). The % bias is highest in cells with large standard
deviation in altitude and aridity index.
To estimate the heterogeneity bias that could result from grid-scale
averaging in Earth system models, we applied the analysis outlined above to
a 1∘ by 1∘ grid cell spanning the Himalayan Front in
west Bhutan (Fig. 4a). This grid cell spans a sharp north–south topographic
gradient, with altitudes ranging between ∼ 500 and
∼ 6500 m. Within this grid cell, we compiled 30 arcsec
values of P (WorldClim; Hijmans et al., 2005) and PET (MODIS; Mu et al.,
2007) to examine the finer-scale climatic drivers of variations in ET.
Because 30 arcsec is approximately 1 km, we will refer to these as 1 km
values for simplicity. P and PET values at 1 km resolution, as well as 1 km values of
ET estimated from these P and PET data using the Budyko curve (Eq. 5), vary
strongly in this 1∘ by 1∘ grid cell, as shown in
Fig. 4b–d. The averages of these P, PET, and ET values over the
1∘ by 1∘ grid cell will plot as the yellow circle in
Fig. 4e, lying well below the Budyko curve of the individual 1 km ET
estimates. If instead we estimated the average ET for the grid cell from its
average P and PET, we would obtain the orange circle on the Budyko curve,
corresponding to an 11.8 % overestimate of the true average of the 1 km ET values.
We repeated the same procedure to estimate the averaging bias in the eight grid
cells surrounding the one analyzed above (Fig. 5a). A comparison of these
nine grid cells shows that the averaging error is largest (around 13 %) when
the variability in the aridity index (AI =P/PET), driven in turn by
topographic variability, is largest (Fig. 5b–d).
(a) A conceptual two-column model. (b) Illustration of how the
two points representing the two columns shift towards each other in Budyko space
if water is transferred from the upper, wetter column to the lower, drier column.
Open circles represent columns without lateral transfer and solid circles
represent columns with lateral transfer.
Lateral redistribution by surface and subsurface flow, and its effects on average ET in a Budyko framework
Consider, as a thought experiment, an arid valley surrounded by high
mountains. Evapotranspiration in the valley may depend not only on local
precipitation in the valley but also on precipitation that falls in the
mountains and reaches the valley either by groundwater flow or by streamflow
that re-infiltrates into valley aquifers. The lateral transfer of water from
the mountains to the valley could clearly increase evapotranspiration rates
in the valley by making more water available for vegetation, but could
simultaneously make less water available for transpiration in the mountains.
Will the net effect of this lateral transfer be to increase, or decrease,
average ET as seen from the atmosphere?
The mountains, the valley, and the lateral transfer between them will all be
invisible at the grid scale of typical Earth system models. But the simple
scenario described above suggests that lateral transport could alter the
average ET over a model grid cell that incorporated both the mountains and
the valley. What properties of the landscape will control the sign and the
magnitude of the net effect on average ET? Here, we extend the Budyko
analysis presented above to estimate the potential effects of lateral
redistribution on average ET as seen from the atmosphere.
Our first step is to redefine the aridity index in the Budyko framework to
take account of water that becomes available for evapotranspiration either
through local precipitation or through net lateral transfer. In taking this
step, we are implicitly assuming that all water supplied to the ecosystem,
from any source, is equally available for evapotranspiration. We introduce
the term available water (AW), defined as
AW=P+nettransfer,
where net transfer represents the net influx of groundwater and
re-infiltrating streamflow. Substituting available water for precipitation
in the Turc–Mezentsev formula for the Budyko curve (Eq. 5), we obtain
ET=AWAWPETn+11/n=P+nettransferP+nettransferPETn+11/n,
where AW is available water and, as before, ET is actual evapotranspiration,
P is precipitation, PET is potential evaporation, and n
(dimensionless) is a catchment-specific parameter that modifies the
partitioning of P between E and Q. Our approach follows the lead of several
other investigators (Istanbulluoglu et al., 2012; Wang, 2012; Chen et al.,
2013; Du et al., 2016) who have expanded the “precipitation” term to account
for other sources of water in addition to precipitation per se. This
approach assumes that lateral transfer alters only the available water at
the two locations, and not their PETs.
Two-column model and lateral transfer in Budyko space: graphical interpretation of the concept
To continue the thought experiment outlined above, the mountain and valley
environments described above could be represented by two columns of a land
surface model, as shown in Fig. 6. Column 1 (the mountains, for instance) is a
“source” column for lateral transfer of available water to Column 2 (the
valley, for instance), which can be considered as a ”recipient” column for this
available water. In the example shown in Fig. 6, Column 1 has higher P
and/or lower PET than Column 2. Laterally transferring water from Column 1
to Column 2 will increase the water available for evapotranspiration (and
thus ET itself) in Column 2, and will reduce them in Column 1. But will the
increase in ET in Column 2 outweigh the decrease in ET in Column 1? That is,
will the average ET as seen from the atmosphere increase or decrease, and by how much?
We can graphically illustrate the effects of lateral redistribution between
the two columns in the Budyko framework as shown in Fig. 6b. The average ET
of Column 1 and Column 2 will always lie on the line connecting the
corresponding points on the Budyko plot (and thus below the Budyko curve
itself). As Fig. 6b shows, if we laterally transfer water from a more humid
column to a more arid column, the corresponding points on the Budyko plot
must move closer together, and the resulting average ET must move upward.
Conversely, if we laterally transfer water from a more arid column to a more
humid one, the corresponding points on the Budyko plot must move farther
apart, and the average ET must decrease.
Because lateral transfer will necessarily be driven by gravity (and thus
source locations will always lie above recipient locations), the
analysis shown in Fig. 6b leads directly to a simple general rule: wherever
higher locations are more humid, one should expect lateral redistribution to
result in a net increase in ET, and conversely, wherever higher locations
are more arid, lateral redistribution should result in a net decrease in ET.
As one can see from the graphical analysis shown in Figs. 6 and 7, the
magnitude of the net ET effect will depend primarily on the amount of
lateral redistribution (how far the points move along the Budyko curve) and
on the degree of curvature between them (and thus the
angle between the trajectories of the individual points). As
shown in Fig. 7, if both locations are humid
(and thus energy limited) or
both locations are arid (and thus water limited), lateral transfer from one
site to the other will have only a minimal effect on the average ET. If both
sites are energy limited (and remain energy limited), neither will respond
strongly to a change in the amount of water available for
evapotranspiration. If both sites are water limited (and remain
water limited), they will be almost equally sensitive to changes in
available water; thus, the increases in available water and ET at one site
will be nearly offset by the corresponding reductions at the other site. But
if one site is water limited and the other is energy limited, then the
responses of the two sites to changes in available water will be markedly
different, and lateral transfer from one to the other could substantially
affect the average ET over the two sites.
Four conceptual cases in a two-column model where Column 1 is topographically
always higher than Column 2 (water always moves from Column 1 to Column 2). Open
circles represent columns without lateral transfer and solid circles represent
columns with lateral transfer. Depending on the columns' wetness or dryness (P
and PET), lateral transfer can potentially (a) increase average ET
(the points representing Column 1 and Column 2 are pushed towards one another,
spanning significant curvature in the ET function), (b) decrease average
ET (points are pushed away from one another, spanning significant curvature in
the ET function), or (c, d) have little effect on average ET (the
columns shift almost collinearly along the energy-limit or water-limit limbs
of the curve).
Hypothetical numerical experiment with conceptual two-column model:
(a) no lateral transfer between columns, (b) 200 mm yr-1
lateral transfer from Column 1 (mountain) to Column 2 (valley) increases average
ET by 14 %. The magnitude of P (precipitation), PET (potential evapotranspiration),
ET (actual evapotranspiration), and R (recharge) are hypothetical and D is
drainage to deep groundwater or streamflow.
We emphasize that the analysis presented here is hypothetical. We are not
asserting that lateral transfer actually occurs between the two columns, or
even that it can occur between them, let alone what the magnitude of that
lateral transfer is. Instead, we are asking the hypothetical question: if
water flows from one column to the other, how much would we expect the
average ET to change, for each mm yr-1 of water that is lost from one
column and gained by the other?
Quantifying the effect of lateral transfer on average ET
We can make a first-order estimate of the net effect on ET using the Budyko
curve as a simple model of ET rates. An illustrative calculation, for an
extreme hypothetical case, is shown in Fig. 8. Column 1 is humid, with
2000 mm yr-1 of annual precipitation and a PET of 1000 mm yr-1 (AI
of 2.0), and Column 2 is arid, with 300 mm yr-1 of annual precipitation and
a PET of 2000 mm yr-1 (AI of 0.15). As Fig. 8b shows, laterally
transferring 200 mm yr-1from Column 1 to Column 2 would increase
average ET by about 85 mm yr-1, or 14 %.
We can generalize from this specific example by using Eq. (10) to calculate
the average ET as a function of the amount of available water that is
transferred from one column to the other:
ETavg=0.5P1-x×PET1P1-xn+PET1n1n+P2+x×PET2P2+xn+PET2n1n,
where x represents the net transfer from one column to the other.
Figure 9 depicts how the average ET and the AW / PET and ET / PET ratios of the
two sites change with lateral transfer. The average ET of the two columns
increases with increasing net transfer (x) up to a point, and then
decreases for higher values of x. One can see from Fig. 9 that
average ET reaches its maximum when x equalizes AW / PET (and thus
ET / PET) at the two sites (note that this does not imply that either AW or
PET are necessarily the same at the two sites).
We can verify this intuitive result by differentiating Eq. (11) by x:
dETavgdx=1P2+xPET2n+11+1/n-1P1-xPET1n+11+1/n.
At the maximum ETavg, dETavg/dx must equal
zero, which can only occur if xopt, the ET-maximizing rate of
lateral transfer, is such that the two terms in Eq. (12) are equal, implying that
P1-xoptPET1n+1=P2+xoptPET2n+1→P1-xoptPET1=P2+xoptPET2=P1+P2PET1+PET2,
which shows directly that AW/PET = (P±xopt)/PET in the two columns must be equal,
confirming the intuitive result from Fig. 9. Solving Eq. (13) for xopt leads to
xopt=P1PET2-P2PET1PET1+PET2=P1/PET1-P2/PET21/PET1+1/PET2.
The key result here (namely that ET is maximized when lateral transfer
equalizes the ratio AW / PET in the columns) is not restricted to two
columns, and is not specific to the particular curve that we have analyzed
here. Instead, it can be shown to be true for any downward-curving function
on a Budyko plot and for any number of interacting columns; for details, see
the Appendix.
Average ET is maximized for the rate of net transfer at which P/PET
and ET/PET of the two hypothetical columns of Fig. 8 cross one another.
The dimensionless quantity dETavg/dx (Eq. 12)
expresses the change in average ET per unit of lateral redistribution. One
quantity of particular interest could be the relative change in ET resulting
from the first unit of lateral transfer, which can be obtained directly from
Eq. (12) with x= 0:
dETavgdxatx=0=1P2PET2n+11+1/n-1P1PET1n+11+1/n.
This dimensionless number depends only on the aridity indices P/PET at the
two sites, and could be used as a screening tool to find regions where
lateral redistribution could potentially be most consequential.
Another benchmark for the potential importance of lateral transfer is the
maximum possible average ET rate, if lateral transfer took place at its
optimal value xopt. This quantity can be calculated by
substituting the optimal transfer rate xopt (Eq. 14) into our
modified Budyko formula (Eq. 11):
ETopt=P1+P22P1+P2PET1+PET2n+11/n.
Equation (16) shows that the optimal rate of ET (including lateral
redistribution) equals the Budyko curve estimate of ET at the average P and
average PET. As shown in the Appendix, this result is quite general, and
does not depend on the specific Budyko curve equation that we have used
here, nor on any specific number of columns. It requires only that all of
the columns are governed by the same downward-curving function in a
coordinate space defined by ET/PET and P/PET.
This result demonstrates an interesting connection with the analysis of
heterogeneity bias presented above. The maximum possible increase in ET from
lateral redistribution exactly equals the heterogeneity bias calculated in
the preceding section: both are equal to the ET function at the average P
and PET (e.g., Eq. 16 in the case of two columns), minus the average of the
ETs calculated for the individual columns using their individual P and
PET values. That is, both are equal to “the function of the averages”, minus “the
average of the functions”. Putting the same point differently, the ET that
an Earth system model calculates from average P and PET (the function of
the averages) is not just an overestimate of the true ET (as explained in
Sect. 3 above); it is the highest possible ET under
optimal redistribution of the available water.
This observation simplifies the problem of estimating the maximum possible
effect of lateral redistribution in heterogeneous terrain: one simply needs
to compare the average of the ETs calculated for every pixel within some
domain using those pixels' individual P and PET values, and the ET calculated
from the average P and average PET using the same Budyko curve.
Alternatively, one can approximate these quantities from the means and
variances of P and PET, using Eqs. (6)–(8).
Spatial patterns of altitude, precipitation (P), potential
evapotranspiration (PET), and aridity index (P/PET) in 1∘ by 1∘
grid cell in the Himalayas at 89–90∘ E, 26–27∘ N. There is a
sharp gradient in P, PET, and altitude in this grid cell. The labeled points 1,
2, and 3 correspond to the labeled points in Fig. 11.
Variation of precipitation (P), potential evapotranspiration (PET),
and aridity index (P/PET) with altitude in a 1∘ by 1∘ grid cell
of the Himalayas in the extent of Fig. 4 (89–90∘ E, 26–27∘ N).
P and PET for sites 1, 2, and 3 in Fig. 10 are marked in the graphs. Between
locations 3 and 2, P and aridity index increase and PET decreases with altitude.
Between points 2 and 1, P and aridity index sharply decrease and PET slightly
increases with altitude.
Of course, any of these estimates of the potential effects of lateral
redistribution ignore many real-world constraints, such as topographic or
lithologic barriers that could prevent lateral transfer between specific
locations (e.g., water will not flow uphill). Thus, this estimate should be
considered as only a theoretical upper bound.
Real-world example of redistribution effects on estimated ET
To illustrate the possible effects of lateral redistribution on average ET
in the real world, we will use the example of the 1∘ by
1∘ grid cell shown in the middle right of Figs. 4a and 5a, which
lies at the foot of the Himalayan Front at 89–90∘ E,
26–27∘ N. As before, we use 30 arcsec (∼ 1 km)
P, PET, and topographic data (from WorldClim, MODIS, and SRTM; Hijmans et
al., 2005; Mu et al., 2007; Jarvis et al., 2008) to represent the
finer-scale heterogeneity within this grid cell.
Figure 10 shows three locations that have been selected to illustrate the
possible effects of lateral redistribution on average ET. Location 3 is
close to sea level, whereas location 2 is at 300 m altitude and location 1
is at roughly 3000 m. We analyzed the effects of a hypothetical
redistribution of 500 mm yr-1 of water from location 1 to location 2,
and from location 2 to location 3.
As Fig. 11 shows, P and the aridity index increase dramatically from
location 1 (at 3000 m) to location 2 (at 300 m); that is, the landscape
becomes more humid as one moves downhill. Using the rule of thumb developed
above, one would expect that lateral transfer from location 1 to location 2
should result in a net decrease in average ET. Figure 12a confirms that, as
expected, lateral transfer would move the two points farther apart on the
Budyko curve, resulting in a net decrease of 9.3 % in the average ET of
the two locations.
Conversely, as Fig. 11 shows, as one moves downhill from location 2 to
location 3, the landscape becomes more arid (the aridity index decreases);
thus, the rule of thumb outlined above predicts that downhill lateral
transfer should result in a net increase in average ET. This expectation is
confirmed by Fig. 12b; the two locations move closer together on the
Budyko curve, resulting in a net 4 % increase in the average ET of the two locations.
Summary and discussion
The atmosphere mixes and integrates inputs from spatially heterogeneous
landscapes. Earth system models average over significant landscape
heterogeneity, which can lead to substantial biases in model results if the
underlying equations are nonlinear. Due to the mass and energy constraints
that limit evapotranspiration rates, ET will be a nonlinear concave-downward
function of P and PET, whether expressed by Budyko curves or by other ET
models. As a result, ET values calculated from averages of spatially varying
P and PET will overestimate the average of the spatially variable ET (the
function of the average will overestimate the average of the function).
Budyko curve and increase or decrease of average ET when transfer
of water from a higher location to a lower location is included.
(a) 500 mm yr-1 of transfer from site 1 (3000 m altitude, lower
aridity index) to site 2 (300 m altitude, higher aridity index) decreases the
average ET by 9.3 %. (b) 500 mm yr-1 of transfer of water from
location 2 (altitude 300 m, higher aridity index) to location 3 (altitude 10 m,
lower aridity index) increases the average ET by 4 %.
In Sect. 3, above we outlined an approach for estimating this heterogeneity
bias, using Budyko curves as a simple empirical ET model. One should keep in
mind that Budyko curves are empirically calibrated to catchment-averaged
precipitation and discharge (to calculate ET); thus, they already average
over the spatial heterogeneity within each calibration catchment. This
inherent spatial averaging should make Budyko curves smoother (less curved)
than the point-scale relationships that determine ET as a function of P and
PET. In other words, the true mechanistic equations that drive point-scale
ET may be much more sharply curved than Budyko curves (which already include
significant averaging, and thus must plot inside the curve of the raw
point-scale data, if such data were available). As a result, the effects of
sub-grid heterogeneity and lateral redistribution could potentially be
larger than what we have estimated here.
In Sect. 4, we explored the possibility that lateral transfers of water
from one location to another could change the average ET as seen from the
atmosphere. Exploring this question requires a modified Budyko framework, in
which one accounts for the water that is available for evapotranspiration
(P+ net transfer) rather than precipitation alone. This is consistent with
Budyko's original approach, which was based on mass balances in catchments
with no long-term groundwater gains or losses (i.e., with no net transfer,
and thus with the long-term supply of available water equal to
precipitation). Our analysis shows that in regions where the aridity index
increases with altitude, lateral redistribution would transfer water from
more humid uplands to more arid lowlands, resulting in a net increase in ET
(points would move closer together on the Budyko curve; Fig. 12b). Alternatively,
in regions where the aridity index decreases with altitude, lateral transfer
would redistribute water from more arid uplands to more humid lowlands,
resulting in a net decrease in average ET (Fig. 12a). We derived simple
analytical formulas for estimating the marginal ET effect of a unit of
lateral redistribution, as well as the maximum possible ET effect resulting
from an optimal (i.e., ET-maximizing) amount of lateral redistribution.
Water transfers will most strongly affect average ET if the source (or
recipient) location is energy limited and the recipient (or source) location
is water limited.
Our analysis of redistribution effects is based on the assumption that
lateral transfers will reduce the available water at the source location by
the same amount that they increase it at the receiving location. Thus, we are
assuming that water that is redistributed becomes unavailable for
evapotranspiration at the source location (for example, through rapid runoff
to channels or rapid infiltration to deep groundwater via preferential
flowpaths). Alternatively, if the redistributed water were assumed to come
only from surplus that is left over after evapotranspiration, the
available water (and thus ET) in the source location would not be reduced
while the available water (and thus ET) in the receiving location would be
increased. Under that assumption, any redistribution would increase average
ET, regardless of the climatic conditions in the source and receiving
locations. By assuming that available water is conserved (in the sense that
whatever is gained in one location is lost from another), our analysis may
underestimate the effect of redistribution on average ET.
It bears emphasis that our analysis of the effect of lateral redistribution
is inherently hypothetical. By estimating the ET effect of a (hypothetical)
transfer of water from one location to another, we are not implying that
such a transfer would actually take place at the assumed rate (or would even
occur at all) in the real world. Perhaps in reality there is no flowpath
connecting the two locations, for example, or perhaps its conductivity is
very low, or perhaps the putative source location lies downhill from the
putative recipient location. Likewise, although there may be an aquifer
connecting two locations, it may lie too deep below the rooting zone to have
any significant impact on evapotranspiration rates. Estimating the potential
effects of lateral redistribution on ET in real-world cases (rather than
hypothetical ones) will require careful attention to such matters, which are
beyond the scope of this paper.
The analysis that we have used to quantify the effects of spatial
heterogeneity and redistribution could also be used to study the effects of
temporal heterogeneity in water availability for evapotranspiration, and
temporal redistribution by storage of groundwater between wet and dry
seasons. Temporal heterogeneity (e.g., seasonality) in water availability
could substantially affect average ET, particularly in climates that shift
seasonally between water-limited and energy-limited conditions. In such
cases, ET estimates calculated from time-averaged P and PET will be higher
than the average of individual ET estimates derived from daily or monthly
values for available water and PET. Similarly, temporal redistribution of
available water between water-limited and energy-limited conditions
(through, e.g., interseasonal groundwater storage) could substantially
increase average ET. The formulas and approaches we have outlined above
could be straightforwardly applied to quantify these temporal heterogeneity
and redistribution effects (for a similar approach to temporal upscaling in
hydrological models, see Lim and Roderick, 2014). If, however, one bases
such an analysis on Budyko curves as an ET model, one should keep in mind
that these empirical curves are based on long-term catchment mass balances,
and thus they already average over seasonal and shorter-term variations in
water availability and PET. Thus, Budyko curves may already be substantially
smoother (less curved) than the short-term behavior that they average over.
As a result, any such analysis based on Budyko curves may underestimate the
impact of temporal heterogeneity and redistribution on average ET.
Our analysis does not explicitly account for how changes in ET may affect
atmospheric humidity and thus PET. This “complementarity” feedback between
ET and PET is potentially important for mechanistic models of the
evapotranspiration process, and could potentially change the magnitude
(though not the sign) of the ET effects that we have estimated in this
paper. Any such changes should be small, however, because Budyko curves are
empirical relationships derived from catchment mass balances, which already
subsume any feedbacks between ET and PET that arise in the calibration catchments.
The simplicity of the approach presented here is both a limitation and an
advantage. On the one hand, this simple approach necessarily overlooks, or
implicitly subsumes, many mechanistic relationships that would be explicitly
treated in more complex ecohydrological models. On the other hand, it avoids
the calibration issues and data constraints that may limit the applicability
of these more complex models. Our simple approach also has the advantage of
transparency; as Figs. 3, 4e, 6, and 12 show, one can directly visualize how
both spatial heterogeneity and lateral redistribution affect average ET,
using a simple graphical framework. This framework leads to relatively
simple analytical expressions and rules of thumb that can be used to gauge
where, and when, heterogeneity and lateral redistribution effects on ET are
likely to be most important.
An obvious next step is to use the framework developed here to make a
first-order estimate of the likely effects of spatial heterogeneity and
lateral redistribution on ET, as seen from the atmosphere at regional and
continental scales. The approach developed here is well suited to this task
because it is simple and relatively general, and its data requirements are
modest. Heterogeneity effects on ET can be estimated from the means,
variances, and covariance of P and PET, and, as we have shown, the maximum
hypothetical effect of lateral redistribution can be obtained directly from
the same analysis. Quantifying the likely real-world effects of lateral
redistribution will be much more challenging, since it necessarily requires
estimating the real-world magnitudes of these lateral redistribution fluxes.
Work on quantifying heterogeneity and redistribution effects on ET at
regional and continental scales is currently underway and will be the focus
of future papers.
Data availability
All the data in this study were retrieved from open-access data repositories.
The SRTM digital elevation database (Jarvis et al., 2008) can be downloaded from
http://www.cgiar-csi.org/data/srtm-90m-digital-elevation-database-v4-1.
The MODIS potential evapotranspiration data set (Mu et al., 2007) was downloaded
from http://www.ntsg.umt.edu/project/mod16. The WorldClim precipitation
data set (Hijmans et al., 2005) was downloaded from http://www.worldclim.org/current.
Generality of redistribution results
Here, we demonstrate that the optimal redistribution results presented in
Sect. 4.2 are also valid for any number of locations (not just two) and
for any downward-curving ET function that can be plotted on the Budyko axes
(not just Eq. 1, which was used to derive Eqs. 12–16 in Sect. 4.2).
We begin by assuming a set of N locations
i= 1 … N, each characterized by rates of precipitation Pi
and potential evapotranspiration PETi. In
keeping with the analysis of Sect. 4, we assume that the rate of
evapotranspiration at each location depends on its available water, AWi:
availablewateri=AWi=Pi±nettransfers,
and specifically on the ratio of available water to PET, which we denote for
future convenience as Ri,
Ri=AWiPETi.
We also assume that the evapotranspiration rates at all locations follow the
same functional dependence on AWi/PETi, and that
this functional relationship (denoted f) can be represented on
Budyko-type axes, that is,
ETiPETi=fAWiPETi=fRiorETi=PETifRi.
We impose no restrictions on the form of the function f, except that it must
be downward-curving; that is, its second derivative must be negative everywhere.
The first result to be demonstrated is if moisture is redistributed among
multiple locations, the highest possible average rate of ET will be achieved
when all locations have the same ratio
Ri= AWi / PETi (note that this does
not require that the AWi or the PETi are the
same). We begin by assigning all the locations the same R value, which
we denote Ropt (recognizing that its optimality
is not yet proven). We then show that any further redistribution of an
amount of water y from any location j to any other
location k (such that Rj<Ropt
and Rk>Ropt) will necessarily lead
to a decrease in overall ET. The transfer of y from location j to location k implies that
Rj=Ropt-yPETj,dRjdy=-1PETj
and
Rk=Ropt+yPETk,dRkdy=1PETk.
Taking the second-order Taylor expansion of Eq. (A3), one obtains for ETjETj=PETjfRopt+PETjdfdRdRdyy+PETj2d2fdR2dRdy2y2+…=PETjfRopt+PETjdfdR-1PETjy+PETj2d2fdR2-1PETj2y2+…=PETjfRopt-ydfdR+y22PETjd2fdR2+…,
and similarly for ETkETk=PETkfRopt+ydfdR+y22PETkd2fdR2+….
Thus, the net change in total ET for locations j and k together is
ETj+ETk-PETjfRopt+PETkfRopt=y212PETj+12PETkd2fdR2+….
Because the second derivative of f is always negative, the
right-hand side of Eq. (A8) will likewise be negative, implying a net
decrease in ET for locations j and k whenever y is
not zero. The stipulation that the second derivative of f is
negative everywhere guarantees that any higher-order terms that
have been omitted from the Taylor expansion must be too small to change the
sign of the right-hand side of Eq. (A8).
Thus, the general result is demonstrated for the individual pair of locations j
and k. Demonstrating that this result is true for this
pair of locations is sufficient to prove the general case, since any pattern
of water redistribution among any combination of locations is equivalent to
a linear combination of such pairwise water transfers.
The second result to be demonstrated is that, for any Budyko-type function f
and any combination of locations, the optimal rate of ET
(including lateral redistribution among the locations) will equal the Budyko
curve estimate at the average P and average PET. For a set of locations i,
Eq. (A3) implies an average ET of
ET‾=PETifRi‾,
where overbars indicate averages over all locations. As demonstrated above,
under optimal redistribution each location will have
Ri=Ropt, such that Eq. (A9) becomes
ET‾opt=PET‾fRopt.
What remains to be proven is that Ropt=P‾/PET‾. If we
denote the net transfer of water into each location as zi
(such that locations that have a net gain of available water have
zi> 0, and locations that have a net loss of
available water have zi< 0), for each location we can write
Ri=AWiPETi=Pi+ziPETiorRiETi=Pi+zi.
Summing Eq. (A11) over all locations, noting that under any mass-conserving
redistribution the zi values must sum to zero and under optimal
redistribution Ri=Ropt everywhere, we directly obtain
∑RiPETi=Ropt∑PETi=∑Pi+zi=∑Pi,
and therefore
Ropt=∑Pi∑PETi=P‾PET‾.
Combining Eqs. (A10) and (A13), we have
ET‾opt=PET‾fP‾PET‾,
thus proving the second general proposition.
Both authors have contributed equally to all aspects of this work.
The authors declare that they have no conflict of interest.
Acknowledgements
We thank Ying Fan Reinfelder for insightful discussions, and Ross Woods,
Michael Roderick, and two anonymous reviewers for their comments.
Edited by: R. Woods
Reviewed by: M. Roderick and two anonymous referees
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