## 1. Introduction

Subgrid inhomogeneity and aggregation may at first sight seem like different subjects. Using the scaling approach (Ren 2001), both issues can be addressed in the same general framework. Land surface heterogeneity is a persistent problem in land surface modeling as well as measurement bias quantification and has been addressed by many researchers (see, e.g., Mahrt 1987; Shuttleworth 1988; Henderson-Sellers and Pitman 1992; Koster and Suarez 1992; Seth et al. 1994; Brutsaert 1995; Heuvelink 1998; Panin et al. 1998; Rodriguez-Camino and Avissar 1999; Davey and Pielke 2005).

To suit a myriad of different uses, land surface modeling strategies taken for different climate problems are quite different in sophistication. Any such model, however, takes a set of land surface characteristics as input. This model is then calibrated and verified against micrometeorological measurements collected over field experiment sites with uniform conditions (and should be so doing). In reality, for a finite size area (under natural conditions and of infinite degree of heterogeneity) to be modeled by a land surface scheme, one basically has two choices: subdivide this area into numerous “uniform” plots and run a land surface model repeatedly for each and then weight average the output [e.g., mosaic approach of Koster and Suarez (Koster and Suarez 1992)]; or one can “properly” preprocess the input land surface characters and use these corresponding “effective” land surface characteristics to drive the land surface model [e.g., Wetzel and Chang 1988; statistical–dynamical approaches of Avissar (Avissar 1992)]. The first approach, especially used for a long time simulation, should put model effectiveness into consideration. What compounded the second approach is that we generally lack knowledge of the distribution characteristics, and even so we still are not sure which way is a better way to “average” the input parameters considering that latent heat flux is acutely nonlinearly dependent on surface characteristics such as soil moisture and vegetation.

Shuttleworth (Shuttleworth 1988), based on the analysis of three international land surface experiments, pointed out the necessity for aggregating land surface parameters as input to hydrological models serving the needs of AGCMs. This work, to a certain measure, echoes his call for improved definition of the effective average surface quantities. Unlike most of the previous works, which were based on sophisticated numerical models involving many detailed land surface processes, which, in turn, makes it hard to understand the causality mechanism for problems encountered during the aggregation process, in this study the discussion will concentrate on the roles played by soil and vegetation in transpiration processes, based on an analytical model with simplified soil hydrological processes. This approach obtained non-case-sensitive conclusions that are nearly impossible to draw for most of the previous studies. We chose the evapotranspiration rate as a measure also because previous studies (e.g., Li and Avissar 1994) indicate that latent heat flux is the most sensitive quantity to spatial variability of land surface characteristics.

For many study purposes, it is desirable to estimate if possible the range of over-/underestimation caused by taking an average of the continuous soil and vegetation properties for an upscaling process that aims to derive a mean flux over a large, heterogeneous region. This composes another aim of this study.

The work is organized as follows. The general philosophy in modeling soil hydrology for atmospheric numerical models is briefly discussed in section 2. A simplified analytical model is introduced to describe evapotranspiration and soil water diffusion processes for a vegetated surface and its formal solution is discussed in section 3, with a focus on discussing scaling issues. In section 4, two sets of numerical experiments are conducted based on normal soil and vegetation properties. Results from the numerical experiments are discussed in section 5.

## 2. Soil hydrology

The importance of land surface hydrological processes has been recognized for a long time by the climate modeling community, in order to describe in a consistent way all the components of the water and energy cycle over long periods of time. To this end, a variety of schemes has been devised. Because soil hydrological processes within the rhizosphere are influenced by the vegetation root system, any reasonably accurate mathematical description of water uptake by vegetation with heterogeneous soil and root properties is complicated. The reader is referred to Smith et al. (Smith et al. 2002) for a concise presentation of the governing equations in soil water transport.

For a smooth introduction of the analytical model, we present a modified Richards equation (Richards 1931) that includes the vegetation transpiration. The Richards equation, which is the key relationship for addressing basic soil hydrological processes within the rhizosphere and constitutes the basis for multiple-layer soil hydrology models, is based on the following two facts. (i) The water flux rate inside soil is proportional to the water potential gradient, the constant of proportionality being called the hydraulic conductivity (Darcy’s law; Hillel 1982; Jury et al. 1991); and (ii) the change in water content of a specific layer is due to the convergence/divergence of water fluxes (mass conservation).

*θ*is the volumetric water content (volume of water divided by total volume of soil);

*h*is the soil water pressure head in units of length;

*K*is the unsaturated hydraulic conductivity;

*t*is time (s);

*z*specifies the vertical coordinate, positive upward (m);

*ω*is a dimensionless function of soil water pressure head;

*R*is the root shape factor (m

_{s}^{−1}) usually expressed as a function of root length density distribution (Y. Luo 2002, personal communication); and

*T*is the transpiration (m s

_{r}^{−1}) rate by vegetation roots. Note that in Equation (1), gravitational acceleration is incorporated into

*K*, and the liquid water density is assumed to be a constant.

The last term of Equation (1) may be called the transpiration term. The effects from the soil water content, the ability of the soil to conduct water to the roots, and even the water logging and soil water salinity can be incorporated into *ω.* The contribution from the vegetation type as well as its developmental stage can be incorporated into *R _{s}*. The atmospheric demand (e.g., energy supply, vapor pressure deficit, and wind speed), as pioneered by Penman (Penman 1948), can be incorporated into

*T*.

_{r}*K*is usually parameterized using the saturated hydraulic conductivity

*K*by the relation

_{s}*K*=

*K*Θ

_{s}^{2}

^{b+}^{3}(Clapp and Hornberger 1978), where Θ is the volumetric water content relative to saturation (water volume per pore space), and

*b*is the Clapp–Hornberger parameter (the slope of the retention curve on a logarithmic graph). Here,

*h*is similarly parameterized as

*h*=

*h*Θ

_{s}*, and*

^{−b}*θ*is related to Θ by total porosity

*P*

_{total},

*θ*=

*P*

_{total}Θ. Transpiration extracted by plants in the root layer is usually weighted by a function that peaks at the top of the root layer and decreases linearly downward and becomes zero at a prespecified depth (e.g., Capehart and Carlson 1994). Following Feddes et al. (Feddes et al. 1978), a linear dependency of

*ω*upon

*h*is a viable assumption for a wide range of soil water contents (above wilting point and below the field capacity). We finally assume that the atmospheric demand can only be satisfied to such a degree that the overall effect on the last term in Equation (1) is a linear dependency on

*θ*(which is equivalent to an assumption of a Θ

^{b+1}dependency of

*T*), in the form of

_{r}*Wθ*, with

*W*being a gross coefficient in inverse seconds. Thus, Equation (1) is transformed toorwhere

*K*(Θ) = −

_{D}*Kbh*Θ

^{−1}

*P*

_{total}

^{−1}is the hydraulic diffusion coefficient (m

^{2}s

^{−1}), and

*G*(Θ) = (2

_{V}*b*+ 3)

*K*Θ

^{−1}

*P*

_{total}

^{−1}is the gravitational coefficient (m s

^{−1}).

The first term on the right-hand side of Equation (3) is a vertical diffusion term, the second term is considered as gravitational drainage, and the third term results from vegetation transpiration. Since *K _{D}* is a function of Θ, analytically, it means that Equation (3) is a complex nonlinear equation. However, it has some nice qualities under very weak requirements of

*K*; for example, the solution exists and is unique and the solution is confined (by boundary conditions and source/sink terms not shown here).

_{D}## 3. An analytical model describing the evapotranspiration process

### 3.1. Model description

*L*(Figure 1) is considered. Several simplifications and assumptions are necessary for obtaining an analytic solution. The hydraulic diffusivity

*K*is assumed to be constant (using the notation

_{D}*K*henceforth for simplicity). To cover a wide range of water distribution regimes, instead of putting an extra forcing term in the governing equation, three different initial soil moisture profiles were used to mimic different phases of drying down after rainfall. To simplify the solution process, the following further simplifications are made: soil moisture is constant at the bottom of this soil slab (i.e., the lower boundary is at or near the water table); that is, Θ =

*θ** at z =

*L*. At the surface, the soil moisture merges gradually with the atmosphere. This also implies that there is no direct evaporation from the bare ground surface. The moisture extraction weight

*W*is further assumed to be constant, although a variable

*W*as a function of

*z*only can be similarly solved. Under the above assumptions, the soil water redistribution process under constant root extraction can be described by this highly simplified equation:where Θ is volumetric soil water content relative to saturation (water volume per pore volume, m

^{3}m

^{−3});

*K*is the soil diffusion coefficient (m

^{2}s

^{−1});

*z*specifies the vertical coordinate, positive downward (m); and

*W*is the soil extraction weight. To solve Equation (4), the initial soil moisture profile must be specified since a well-posed parabolic partial differential equation like Equation (4) often arises in connection with an evolutionary system in which the flux is downgradient with respect to the field variable. Three different general forms (which can represent up to seven frequently observed soil moisture distributions of initial conditions) are discussed here.

- (a) The initial soil moisture profile is fixed as a quadratic form that reaches its minimum at
*z*= 0 and maximum at z =*L*; that is, Θ|_{t=0}= (*θ**/*L*^{2})*z*^{2}, 0 ≤*z*≤*L*. This case, denoted as initial condition 1 (IC1), may simulate the water contents of a very dry period. The soil moisture decreases all the way up to the surface. Taking a more general form of Θ|_{t=0}= (*θ** −*θ*_{wilt}/*L*^{2})*z*^{2}+*θ*_{wilt}, 0 ≤*z*≤*L*, where*θ*_{wilt}is the wilting point, does not change the following discussions substantially. - (b) The initial soil moisture profile is set as a cubic polynomial form that possesses the same values at both the upper and lower boundaries and may bulge in or bulge out in the middle, depending on the sign of the parameter
*λ*_{1}; that is,This case, IC2 henceforth, describes a picture of soil moisture briefly after a rainfall event. The surface is still wet but not as wet as the layer immediately below it, where the soil moisture content is even higher than that of the deep layer. - (c) There is a general lapse rate in the initial soil moisture profile that makes the soil water content at the upper and lower boundaries different. A possible bulge in or bulge out is also allowed (sign of integer
*m*); that is,This case, IC3 hereafter, may be used to describe the soil water condition immediately after a large rainfall event. The soil moisture is highest at the surface and decreases gradually downward.

Understanding that the bulge in and/or bulge out feature does not affect our conclusions qualitatively, the following discussions are confined to the cases of *λ*_{1} = 1 and *λ*_{2} = 0.05 with *m* = −1.

*W*∫

^{L}

_{0}

*θ*(

*z*)

*dz*. From Equation (5), it is straightforward to derive the water extraction rate (i.e., transpiration rate) as

The left-hand side of Equation (6) is directly related to the transpiration rate. Equation (6) shows that the transpiration rate for this highly idealized model will gradually approach a rate that is proportional to the depth of the hydraulically active soil layer, and the larger the soil diffusivity *K*, the larger the steady-state transpiration rate. This agrees with our intuition since the only mechanism providing water (for transpiration) in this analytical model is soil water diffusion. Also, analysis of the two transient terms shows that the shallower (smaller *L*) the soil slab, the faster it approaches the steady state; the larger the value of *W*, the faster it approaches the steady state. Equation (6) also possesses the potential to explain whether directly averaging the parameter *W* and/or *K* will systematically underestimate/overestimate latent heat flux over a heterogeneous region.

For two different locations, the moisture extraction coefficients are *W*_{1} and *W*_{2}, respectively (representing different vegetation covers). To calculate the average transpiration rate from an inhomogeneous region composed of these two vegetation types, direct averaging of *W*_{1} and *W*_{2} and substituting into Equation (6) will yield a different result from using *W*_{1} and *W*_{2} separately and then taking the average. Similarly, for a region composed of soil type 1 (of soil hydraulic diffusivity *K*_{1}) and soil type 2 (of soil hydraulic diffusivity *K*_{2}), it is worthwhile to study the error caused by direct averaging of *K*_{1} and *K*_{2} in calculating the area-averaged evapotranspiration rate.

_{t=0}=

*θ**[1.0 + (

*z*

^{2}/

*L*

^{2}) − (

*z*

^{3}/

*L*

^{3})], 0 ≤

*z*≤

*L*, is used as initial condition (IC2) in Equation (4), parallel to Equation (5), we will obtainThen the soil moisture extraction rate is

In Equation (4), parameters *K* and *W* are used to represent the soil and vegetation properties, respectively. Even without taking the natural range of their values into consideration, the behaviors of the solutions under different initial and boundary conditions share the commonality that as long as the summation of coefficients in the formal solutions converges, the formal solution is also the classic solution (with clear physical interpretations). For the boundary conditions specified in Equation (4), as long as the initial soil moisture profile is continuous to the second order with respect to *z*, Equation (4) with the auxiliary conditions is well posed to describe our problem, and the formal solutions as in Equations (5), (7), and (9) are all classic solutions.

Three initial soil moisture conditions are used in this study to simulate the soil moisture regime undergoing a drying-down process. The corresponding solutions are composed of a transient part that dwindles with time and a steady part that determines the final structure of the soil water content profiles. The integral forms of the solutions share yet another interesting feature: at the initial time, they are all independent of the parameter *K*. The transpiration rates are proportional to *θ** and increase as *W* and/or *L* increase. During the transient period, both *W* and *K* affect the transpiration rates nonlinearly. On reaching the steady state, all the differences in the initial conditions are forgotten [indicated by the exponential terms in the three expressions for transpiration rate, i.e., Equations (6), (8), and (10), respectively]. In reality, the soil profiles are continually being rewetted by rainfall, and an extended (the time scale will be evaluated later) drying period does not exist for most regions. That is also one reason for us to choose three initial soil profiles. Through studying the following drying-down process, we hoped that the results from the discussion of the scaling implications of this analytical model would be reasonably general.

### 3.2. Soil and vegetation parameters

From Dickinson et al. (Dickinson et al. 1993) and Saxton et al. (Saxton et al. 1986), the values of *K _{D}* for 11 U.S. Department of Agriculture (USDA) soil types can be obtained (columns 2 and 3 in Table 1), using the expression in Equation (3). The evapotranspiration rate for normal plants varies from 2.7 × 10

^{−8}to 2.7 × 10

^{−7}m s

^{−1}. Inasmuch as the depth of hydrological active layer

*L*equals 1 m, and the average volumetric water content is 50%, then the weights of transpiration must be about 5.4 × 10

^{−8}to 5.4 × 10

^{−7}s

^{−1}for normal vegetations. Table 2 (adapted from Allen et al. 1998) provides the general range of the potential evaporation rate.

## 4. Numerical design

In the following numerical experiments, the vertical wavenumber [the summation indices in Equations (5) through (10)] is truncated at *n* = 500, which is large enough to avoid the Gibbs phenomenon (Gibbs 1899; Walker 1988). The soil moisture content at the bottom of the soil layer is specified as the field capacity of the corresponding soil type (last column in Table 1). For convenience in the following discussion, the right-hand sides of Equations (6), (8), and (10) are denoted as *F*(*W*, *K*). If two patches of land surface possess different vegetation types, they are labeled as *W*_{1} and *W*_{2}; on the other hand, if they have different soil types, they are labeled as *K*_{1} and *K*_{2}, respectively. In other words, in Equation (4), moisture extraction weight and diffusivity are related to vegetation type and soil type, respectively.

A prerequisite for models intended to address the scaling issue of surface latent heat flux is the ability to accurately describe the evolution of water profiles. To determine soil moisture evolutions for different initial conditions, experiments are performed for sandy (curves based on *θ** = 0.424 and *K* = 5.7 × 10^{−7} m^{2} s^{−1}) and clay soil (curves based on *θ** = 0.667 and *K* = 2.69 × 10^{−8} m^{2} s^{−1}) as in Figure 2. Figure 2a suggests that all three initial profiles converge to the same final steady-state profile. For the parabolic soil moisture initialization (IC1), the soil moisture curve experiences a wettening process for either value of *θ** (Figure 2b). However, the shift to a wetter profile for sandy soil is much quicker than the counterpart of the clay soil, resulting from the fact that sandy soil conducts water better than does clay soil. If there is sufficient soil moisture for transpiration (as in IC2 and IC3), the general trend is to straighten the profiles (with a general left-shifting in the plotted time period) before shifting to the steady-state profiles. However, the drying-down process (controlled, on the one hand, by water extraction by vegetation and, on the other hand, by water redistribution through diffusion) develops much more quickly for sandy soil than for clay soil. In comparison to IC3, where the most prompt response to the drying down occurs at a deeper layer (around 1 m in Figure 2c), the drying-down process for IC2 is quicker at the shallower layer than the underlying layer.

Further analysis of Figures 2b–d shows that the soil water profiles resulting from IC2 and IC3 are all significantly changed during the drying, but that the general shape of the profile resulting from IC1 never changes fundamentally (it always possesses a pattern of “top layer dryer than bottom layer”). To further clarify whether these conclusions are dependent on the parameter values chosen, the same experiment was repeated with *L* = 0.15 m (figures not shown here). It is found that for all three stages of drying down (IC1–IC3), the evolutions of the soil moisture profiles to the final shape are faster than those with *L* = 1.5 m. It means that there is a shorter time scale for the transient period. Otherwise, the basic features are qualitatively the same as for the *L* = 1.5 m case.

As mentioned in section 3.1., one of the most important features of this analytical solution is that it is composed of a transient part and a steady part, which is inherent in the boundary structure of Equation (4) and is independent of initial conditions. To understand the scaling significance of the initial moisture condition, comparisons should be performed during the transient stages of drying down. On the other hand, if one desires to learn the effects of soil and vegetation heterogeneity on the area-averaged surface latent heat flux calculation for the steady state, an experiment with any one of these three ICs suffices. Thus, determination of the decaying time scale [the inverse to the coefficient of time *t* in the expression for the soil water profile in Equation (5), i.e., *τ* = (4*L*^{2}/4*L*^{2}*W* + *K*(2*n* + 1)^{2}*π*^{2})] is a necessary step in order to depict the numerical experiments’ suitably. The value of *τ* is plotted for *n* = 0, 1, 2, 3, 4, that is, the first five components in Equation (5). Figure 3 is plotted to illustrate the general relationship between time scale *τ* and *W* and *K*. To cover the natural range of values of *W* and *K*, Figure 3 is split into two panels. The first component has a decay time scale ranging from 28 days (at the high *K* end) to about half a year (at the very low *K* end), while the fifth component possesses a scale ranging from half a day (sandy end) to around 1 month (clay soil end). Also notice that, especially at the high *K* end, the value of the root layer depth influences the time scales. Generally speaking, the larger the value of *L*, the longer the decaying time scale. However, as will be seen later, different values of *L* only quantitatively affect the conclusions on the scaling issue. The following experiments are all performed for the *L* = 1.5 m case.

To further reveal the importance of determining the decay time scale, the time evolution of the transpiration rate as defined in Equations (6), (8), and (10) is plotted against the *W* and *K* values (Figure 4). At the beginning, as expected, the transpiration rate is independent of the value of *K*, resulting in the contour lines of transpiration rate being parallel to the horizontal axis (Figure 4a). In Figure 4a, IC1 gives a much weaker transpiration rate than IC2 and IC3, reflecting the initially very dry (especially for the upper layer) condition for IC1. The differences in the contour lines for each IC generally disappear after 240 days. The comparison between the three ICs in Figure 4 shows that, during the transient period, the contour lines from different ICs intersect each other (Figures 4a–c). At day 365, they overlap closely (Figure 4d). Figure 4d also suggests that the transpiration rate is not sensitive to the value of *K* for very small values of *W* and is not sensitive to the value of *W* as *K* becomes very small. Thus, in numerical experiments, the use of more than one value of *W* is almost indispensable for evaluating the effects of mixing different soil types on the calculated area-averaged transpiration rate. Similarly, to evaluate the effects of mixing two different vegetation types on the calculated area-averaged transpiration rate, experiments must be performed for more than one soil type (specified by the value of *K* in this model). From Figures 4b,c, it is also apparent that different initializations of soil moisture profiles matter for the interim stages of drying down. For the steady state or the very beginning stage, to show the relative error caused by taking the average, it suffices to analyze the results from one initialization scheme.

Suppose there are two different soil types (*K*_{1} and *K*_{2}) evenly distributed over a certain region. Either of the two methods are used here to calculate the area-averaged evapotranspiration rate.

- Taking the average of
*K*_{1}and*K*_{2}and using it in*F*(*W*,*K*) resulting in*F*[W,(*K*_{1}+*K*_{2}/2)]. - Taking the average of
*F*(*W*,*K*_{1}) and F(*W*,*K*_{2}), that is [*F*(*W*,*K*_{1}) +*F*(*W*,*K*_{2})/2].

Results from these two methods are denoted as *F*(*W*, *K** F _{K}*(

*W*,

*K*), respectively. The result obtained from the second method may act as a reference provided the lateral interactions within the soil are negligible. The difference between these two approaches will be analyzed for a normal range of

*K*values (

*K*in Table 1). The vegetation type is set as young forest (

_{D}*W*≈ 5 × 10

^{−8}s

^{−1},

*L*≈ 1.5 m). To facilitate the following discussion, for a (

*K*

_{1},

*K*

_{2}) pair, a percentage error is defined as

*P*(

_{K}*K*

_{1},

*K*

_{2})=(

*F*(

*W*,

*K*

*F*

_{K}*W*,

*K*)/

*F*

_{K}*W*,

*K*)) × 100%. A negative value of

*P*(

_{K}*K*

_{1},

*K*

_{2}) indicates an underestimation of the area-averaged latent heat flux and a positive value means an overestimation.

Percentage error defined in this way reduces the parameter space by one degree of freedom; that is, soil water content at the bottom of the hydraulically active layer *θ** has nothing to do with this percentage error. Instead, the percentage error is only a function of the initial soil profile shape, the values of *W* and *K*, and the evolution stage of the drying down. Although the value of *θ** by itself does not matter for the percentage error defined above, larger *θ** is usually associated with a heavier clay soil type. Note that *P _{K}*(

*K*

_{1},

*K*

_{2}) is also permutable with respect to

*K*

_{1}and

*K*

_{2}. For all the statements involving

*K*

_{1}and

*K*

_{2}together, the position of

*K*

_{1}and

*K*

_{2}are exchangeable. However, in all the plots with the same value ranges for both

*K*

_{1}and

*K*

_{2},

*K*

_{2}is always designated to represent the smaller one among a (

*K*

_{1},

*K*

_{2}) pair. In reading these figures, one should interpret

*K*

_{1}as “the larger

*K*value in the (

*K*

_{1},

*K*

_{2}) pair” and

*K*

_{2}as “the smaller

*K*value in the (

*K*

_{1},

*K*

_{2}) pair.”

Finally, in this analytical model, *L* affects the time scale (*e*-folding time before reaching the steady state) of the interim stage of drying down and also the magnitude of errors caused by averaging two distinct *K*s. However, *L* does not affect qualitatively the conclusions (over- or underestimation) made about the scaling issue using this simple model. For the numerical experiments performed, *L* is set to 1.5 m, as summarized in Table 3. Because the natural variation of *K* easily covers four orders of magnitude, if only one linear graph is used to describe the effects of mixing two *K*s, many minute but important features will be buried by the widely distinct scales. The full range of *K* is thus bisected: 1.5 × 10^{−8} to 1.5 × 10^{−6} m^{2} s^{−1} (e.g., sands) and 1.5 × 10^{−10} to 2.5 × 10^{−8} m^{2} s^{−1} (clays). Therefore, three experiments are required to fully cover the variation range of *K*. For each initial condition, the first experiment estimates the error caused by mixing two sandy soils; the second experiment estimates the error caused by mixing two clay types of soil; and the last experiment resembles the mixing of a clay type of soil and a sandy type of soil, where the largest errors are expected to occur.

In the same spirit, for two different vegetation types (*W*_{1} and *W*_{2}) evenly distributed over a certain area, two methods to calculate the area-averaged evapotranspiration rate are used here: *F*(*W**K*) = *F*[(*W*_{1} + *W*_{2}/2), *K*) or * F _{W}*(

*W*,

*K*) = (

*F*(

*W*

_{1},

*K*) +

*F*(

*W*

_{2},

*K*)/2]. Exactly analogous to the discussion of averaging two

*K*s, for a (

*W*

_{1}, W

_{2}) pair, a percentage error is defined as

*P*(

_{W}*W*

_{1},

*W*

_{2}) = [

*F*(

*W*

*K*) −

*F*

_{W}*W*,

*K*)/

*F*

_{W}*W*,

*K*)] × 100%. Moreover, it possesses all the properties of

*P*(

_{K}*K*

_{1},

*K*

_{2}). Experiments were performed for a wide range of values for (

*W*

_{1},

*W*

_{2}) pairs. The same experiment was repeated for two extreme soil types, that is, sandy and clay, representing the highest and lowest possible

*K*values. The numerical experiments are summarized in Table 4.

## 5. Analysis

This study has extracted the essence of transpiration and soil water diffusion processes and constructed a dynamic framework using a diffusion equation with hybrid boundary conditions. In this framework, differences in soil properties are represented as different hydraulic diffusivities and differences in vegetation properties are represented solely as different water extraction rates. Because *L* in this analytical model only affects the time scale of the transient stage of drying down and the magnitude of errors caused by averaging two distinct *K*s/*W*s, it does not affect qualitatively the overestimation/underestimation of directly averaging two soil or vegetation quantities. It is therefore sufficient to discuss the results from only one value of *L*. For the steady state, the errors caused by averaging two different soil types or vegetation categories can be estimated unambiguously, such that the analysis can be focused on the transition stage of drying down described by the analytical model.

### 5.1. Effects of averaging two different types of soil

Because the roles played by *K*_{1} and *K*_{2} are identical, for figures with the same range in the values of *K*_{1} and *K*_{2} (i.e., the first two experiments for each IC in Table 3), only half of the total plot is needed. Based on the information contained in Figure 3, where the scaling times for normal vegetations growing on sandy (Figure 3b) and clay (Figure 3a) soils are shown, 1 year is considered long enough to yield a complete picture of the soil profile evolution.

Figure 5 shows that for IC1 and relatively larger hydraulic diffusivity (2 × 10^{−8} to 1.5 × 10^{−6} m^{2} s^{−1}), averaging two different *K* values always causes an overestimation in the area-averaged transpiration rate for the period under consideration. The percentage error increases monotonically with time (cf. Figures 5a–d). After day 67, changes in the images become insignificant. The four panels share commonality in that the percentage error grows as the difference between the two *K* values becomes larger. It is noticeable that the lines are curved and not exactly parallel to the diagonal zero value line, showing the nonlinear nature of the averaging processes. The way the contour lines slant in Figure 5c implies that the percentage error is more sensitive to changes of the relatively smaller *K* (*K*_{2} in the plots) than to changes in the relatively larger *K* (*K*_{1} in the plots). In Figure 5d, the contour lines of percentage error are convex toward the 0 line. This convex nature is not so evident, so that the percentage error caused by one (*K*_{1}, *K*_{2}) pair never overlaps with another different (*K*_{1}, *K*_{2}) pair, and two different *K*_{2}s cannot result in the same percentage error as mixed with the same *K*_{1}. Thus, the effect of taking the average value of the two values for *K* is an overestimation for the area-averaged latent heat flux within the range of 1.5 × 10^{−8} to 1.5 × 10^{−6} m^{2} s^{−1} (sandy soil), and the overestimation gets worse at the later stage of the soil moisture evolution. The situations for mixing two clay soils are qualitatively the same as in Figure 5 but with significantly smaller percentage errors. This is mainly due to the value range of *K*, which in this case is two orders of magnitude smaller than that of sandy soil. In accordance with our expectation, the last experiment (KIC13) yields the largest error because it represents the mixing of two extreme types of soil. The magnitude of the percentage error is dominated by the variation of *K*_{2}. For example, for *K*_{2} = 7.5 × 10 ^{−7} m^{2} s^{−1}, a full range of variation of *K*_{1} changes the percentage error from 28% to 40% at most; however, for *K*_{1} = 7 × 10 ^{−9} m^{2} s^{−1}, a full range of variation of *K _{2}* can cause the percentage error to vary from 5% to 35%.

For IC2 and relatively larger values of *K*, the situation is very different in that there can be either an overestimation or underestimation for the same (*K*_{1}, *K*_{2}) pair, depending on the evolving stages. Over the first 10 days, there are constant underestimations to the evaporation rate, no matter where the (*K _{1}*,

*K*) pair is located. Then the overestimation region appears and drives the underestimation area to the lower right part of the (

_{2}*K*

_{1}–

*K*

_{2}) plots (

*K*

_{2}less than 2.5 × 10

^{−7}m

^{2}s

^{−1}). Forty-five days later, the underestimation region disappears completely and this pattern gives way to the one described by IC1. In addition, the percentage errors are of rather small magnitude (generally less than 0.5%, for both overestimation and underestimation). For the relatively smaller range in values of

*K*(2 × 10

^{−10}to 2 × 10

^{−8}m

^{2}s

^{−1}), that is, clay soils (experiment KIC22), the underestimation region occurs at the upper-left corner of the plots and never becomes significant in comparison to the adjacent overestimation region. Even for the overestimation region, the magnitudes of the errors are insignificant before reaching the steady state. For the mixing of sandy and clay soil types under IC2 (experiment KIC23), during the early stage of evolution, there is no overestimation region in the

*K*

_{1}–

*K*

_{2}graph. A slight underestimation (0.5%–1%) region dominates. Gradually, from the high value end of

*K*

_{2}, there develops an underestimation region (generally less than 1% in magnitude). This region moves downward and reaches its climax around day 30 and shrinks after day 35. The overestimation dominates after day 90, and this pattern continues until the steady state is attained.

For experiment KIC31 (IC3 initial profile and relatively larger values of *K*), there are alternating patterns of overestimation and underestimation during the transient state (figures not shown). The underestimation region tends to have the same order of magnitude as the overestimation region. Thus, a (*K*_{1}, *K*_{2}) pair, properly located, can experience first an overestimation, then an underestimation, and finally an overestimation again. Interpolated physically, there exists a large uncertainty in estimating the evapotranspiration rate from a heterogeneous region during this wet period (within 2 months for *L* = 1.5 m and *W* = 5.4 × 10^{−8} s^{−1}, i.e., forest) for the sandy and loamy soil types. If the whole area is extremely clayey, for example, as is the case in experiment KIC32, systematic overestimation dominates the whole period. However, the magnitude of the error is one order smaller than that for a sandy type of soil.

A comparison among these three ICs shows that IC1 has the largest magnitude of error. That is likely because it describes a stage of drying down when the water constraint exists and the diffusion rate controls the evapotranspiration rate. Then the soil property *K* plays an increasing role in the evapotranspiration process. During the transient period and for clay soil, IC2 yields the smallest error among the three ICs. By a further comparison of IC2 and IC3 (for clay soil type), one finds that the stage described by IC3, if it appears at all, lasts a very short period of time before evolving into the stages described by IC2. Interpreted physically, the moisture in the upper soil layer is extracted quickly by vegetation, and the diffusion process in clay soil is not strong enough to compensate for the water losses there. Thus the top wet pattern cannot be sustained for clay soil. For sandy soils, comparison of experiments KIC11, KIC21, and KIC31 further indicates that during the transient period, *K* plays a more important role for the drier period (IC1) since the magnitude of error there is apparently larger than the remaining two. Thus, it can be concluded that the error caused by the averaging of two different soil types matters less during the wetter period.

The pattern changes shown in KIC21 and KIC31 are rooted in the shape change of soil moisture profiles. Examining the evolution scenario of these two experiments (figures not shown here), IC3 may have experienced more complex structure change during the drying-down period than IC2 did, since KIC31 shows more complex pattern (overestimation/underestimation) transition than KIC21. This suggests that IC3 starts earlier than IC2 in the drying-down process. However, constraints from the diffusion equation prevent the evolving path of IC3 from merging exactly into that of IC2. Hence, there can only be a rough analogy between them.

The likely reason for the lack of underestimation periods for IC1 is that the soil is too dry and the shape of the vertical profile never changes sufficiently to be substantially different from the original one. The situations for IC2 and IC3 are different. From Figures 2c,d, the shape of the moisture profiles for IC2 and IC3 can be substantially different from the original ones. For example, IC3 changes from top-wetter to bottom-wetter after a month of drying down.

### 5.2. Effects of averaging two different types of vegetations

The discussion of the percentage error caused by averaging two different values of *W* shares many commonalities with the corresponding discussion (same IC) of averaging two values of *K*. However, comparisons of percentage errors caused by averaging different soils and by averaging different vegetations are not encouraged in this context because first, *W* and *K* play different roles in the simple analytical model; second, the entire right-hand side of Equations (6), (8), and (10) is discussed, which includes a base state (the first term) that is related only to *W*. The magnitude of this base state is quite large and only acts as a denominator in the definition of *P _{K}*. The third reason is that

*L*is set as 1.5 m. If 0.15 m is chosen, the role played by

*K*will be significantly enhanced.

For the experiments with *P _{W}*, the value range for both

*W*s is set to be 2.4 × 10

^{−9}to 2.4 × 10

^{−7}s

^{−1}, which is wide enough to cover the natural range for most kinds of vegetation. For each initial soil moisture condition, two parallel sets of experiments are performed for clay soil (

*K*= 2.07 × 10

^{−10}m

^{2}s

^{−1}) and sandy soil (

*K*= 1.26 × 10

^{−8}m

^{2}s

^{−1}). Comparison between all six experiments (WIC11–WIC32) shows that for the first 3 months, for either vegetation growing on sandy soil or clay soil, taking the average of the two

*W*values always gives an overestimation of the evapotranspiration rate. During this period, the percentage error of overestimation increases with time, but the magnitude is generally less than 6%.

Experiment WIC11 reveals a situation of systematic overestimation, and the error gets larger with time. The pattern becomes steady after day 170. For the first 99 days, WIC12 (clay soil starting with IC1) is similar to WIC11 (sandy soil starting with IC1). Afterward, an underestimation region appears from the left side of the overestimation region. These patterns continue until the steady state is reached. This suggests that, for clay soil, even for the steady state, there can be an overestimation for certain (*W*_{1}, *W*_{2}) pairs but an underestimation for other (*W*_{1}, *W*_{2}) pairs.

For experiment WIC21, the error increase with time is not monotonic since a temporary maximum value is reached on day 95. In WIC22, the onset for the underestimation region starts at nearly the same time as for the corresponding IC1 case (experiment WIC12), the magnitude of error being slightly different at the beginning and potentially quite different at some later stage. This implies that soil type plays a definitive role in the overestimation or underestimation (recall that for sandy soil, for any IC, there is no underestimation of significant magnitude). Different initial conditions do matter for the magnitudes of the error during the transient stages. The two experiments with IC3 (WIC31 and WIC32) further confirm this conclusion. A cross reference among the results from experiments WIC11, WIC21, and WIC31 shows that for sandy soil, under all ICs, it is unlikely for severe underestimation regions to appear on the *W*_{1}–*W*_{2} graph. However, for clay soil (cf. WIC12, WIC22, and WIC32), it is likely that severe underestimation and overestimation regions will coexist at later stages of the evolution. These three experiments share a similarity regarding the general pattern of overestimation and underestimation and are slightly different in timing and magnitude of error (the magnitude of error during day 240 through day 360 can be as large as 70% for IC3 and 50% for IC1).

Thus, for vegetation growing on clay soil, unlike the situation for parameter *K*, even after reaching the steady state, averaging of two different vegetation types can cause either overestimation or underestimation to the area-averaged transpiration rate, depending on the location of the (*W*_{1}, *W*_{2}) pair in the *W*_{1}–*W*_{2} parameter space. For different ICs, the time needed for reaching the steady state as well as the locations and degrees of underestimation can be quite different.

The patterns of underestimation and overestimation from the experiments with clay soils seem to suggest that the most severe underestimation occurs as the largest *W*_{1} (active forest) mixed with a relatively small *W*_{2} ≈ 5.5 × 10^{−8} s^{−1}. For a (*W*_{1}, *W*_{2}) pair composed of a value for *W*_{1} at the high value end and a *W*_{2} at the low value end, the underestimation/overestimation is very sensitive to slight variations in *W*_{2}. For example, in Figure 6, for *W*_{1} = 2 × 10^{−7} s^{−1}, a slight variation in *W*_{2}, say, from 2.5 × 10^{−8} to 1 × 10^{−8} s^{−1}, can change the percentage error from a 25% underestimation to a 25% overestimation. The least underestimation occurs as both *W*_{1} and *W*_{2} are greater than 1 × 10^{−7} s^{−1}, that is, both of them are active forests. Interestingly, the centers for severe underestimation do not appear exactly where two vegetation types have the largest difference in water extraction weight.

## 6. Conclusions

In latent heat flux estimation for a heterogeneous region, the error introduced by taking directly an average of soil/vegetation properties is directly related with the shape of the relation between latent heat flux and the parameter considered (Rodriguez-Camino and Avissar 1999). Heuristic examples of soil water content profiles are used in this paper to show that directly averaging soil and vegetation properties may cause systematic and significant overestimation/underestimations to the area-averaged mean latent heat flux. Simplicity of the analytical model and our heuristic initial soil profiles make the generalization easier than using sophisticated numerical models and make it clearer for physical interpretations.

Through the discussion of the solutions of a well-posed parabolic equation with hybrid boundary conditions [Dirichlet lower boundary condition and Neumann upper boundary condition (Riesz and Nagy 1955; Tychonov and Samarski 1964)], this study showed that the effects of averaging soil hydraulic diffusivity and transpiration weights are different at different stages of soil moisture profile evolution.

Thus, for the steady state, the mixing of two different soil types always yields an overestimation of the evapotranspiration rate. This conclusion is also valid for averaging two vegetation extraction weights if they are associated with a sandy soil. However, no such straightforward conclusions can be drawn for vegetation growing on clay soil. In that case, directly averaging the transpiration weights of the two types of vegetation gives either an overestimation or an underestimation, depending on which two are mixed. During the transient period, it is even more difficult to make a general conclusion for averaging both vegetation and soil properties. The dynamic situations are rooted in the evolution of the soil moisture profiles.

Only the normal ranges of values of common vegetation and soil types are discussed in this paper, and the evolution time scale should not be taken too literally (it is an estimation using a simplified analytical model and should not be applied out of context). In addition, in the analytical model, vegetation plays a more direct and active role than soil type. This also explains why it seems that the percentage errors caused by averaging two different vegetation types are more significant than averaging two different soil types.

The results shown in this study are not definitive due primarily to its omission of the nonlinear feedbacks that defy simplified treatments, but they can serve as a reference for further research in this direction. For example, the feedback between transpiration and soil moisture can eventually change the sensitivity to a linear averaging of soil properties. Further research will be done to enrich the dynamic framework and better represent reality by including biological controls through stomatal conductance, leaf area index, hydraulic lift (Caldwell et al. 1998; Ren et al. 2004), and a nonlinear feedback on the moisture extraction weight. In a certain sense, the statistical approach taken by Li and Avissar (Li and Avissar 1994) and Rodriguez-Camino and Avissar (Rodriguez-Camino and Avissar 1999) in order to investigate the positive/negative errors coming from the aggregation step serves a similar purpose.

As such, scaling issues derived from the land surface heterogeneity is a very relevant problem in land surface modeling and merits further research because land surface heterogeneity occurs basically on every scale and can easily exceed the processing capability of the available computing resources, especially for online coupling with other physical processes. This paper contributes to the issue by solving analytically a simplified version of the soil moisture transport equation. This equation still maintains the basic features of the more complex numerical models and allows straightforward interpretation of the results. Once high-resolution data for land surface properties are available for a region (e.g., from remote sensing platforms), analyses such as those described here can be undertaken with benefits for numerous applications.

## Acknowledgments

We thank Dr. Robbins M. Church for carefully reading through the manuscript and giving useful suggestions. The first author also thanks Drs. David Karoly and Lance Leslie for their encouragement in finishing this work. We sincerely thank one anonymous reviewer for his valuable comments that helped in the overall improvement of the quality of this paper.

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Basic soil properties and value ranges of *K _{D}* and Θ for different soil types.

Average potential evaporation rate (mm day^{−1}) for different climatic regions (Allen et al. 1998).

Numerical experiments performed for *P _{K}* (

*K*

_{1},

*K*

_{2}).

Numerical experiments performed for *P _{W}* (

*W*

_{1},

*W*

_{2}).