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 (

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

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.

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,

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

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 (

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 (

Alternative empirical equations for mean annual evaporation rate in
Budyko framework: ET is mean annual evapotranspiration,

Table 1 presents several alternative empirical equations that have been
proposed for “Budyko curves” relating ET to

Here, we use Budyko curves as simple models for how ET is controlled by the
supply of available moisture (as represented by

The water and energy constraints that limit ET imply that ET is an
intrinsically nonlinear function of

As shown in Fig. 3, the nonlinear behavior of ET as a function of

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

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

One-kilometer topography (

Heterogeneity bias in average ET for the nine grid cells of the terrain
shown in Fig. 4a (88–91

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

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

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

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

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 (

Hypothetical numerical experiment with conceptual two-column model:

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

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

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:

We can verify this intuitive result by differentiating Eq. (11) by

Average ET is maximized for the rate of net transfer at which

The dimensionless quantity dET

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

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

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

Spatial patterns of altitude, precipitation (

Variation of precipitation (

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.

To illustrate the possible effects of lateral redistribution on average ET
in the real world, we will use the example of the 1

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

As Fig. 11 shows,

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.

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

Budyko curve and increase or decrease of average ET when transfer
of water from a higher location to a lower location is included.

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

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
(

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

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

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

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

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

Thus, the general result is demonstrated for the individual pair of locations

The second result to be demonstrated is that, for any Budyko-type function

Both authors have contributed equally to all aspects of this work.

The authors declare that they have no conflict of interest.

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