Incorporating the Vadose Zone into the Budyko Framework

The Budyko framework provides a quantitative description of long-term average annual evapotranspiration at catchment scales in terms of macro-climatic variables. This framework, however, makes no reference to the vadose zone because it neglects changes in subsurface storage in the catchment water balance. Recent studies have shown clearly that vadose-zone water storage cannot be neglected at sub-catchment or sub-annual space and time scales, resulting in numerous model attempts to extend the original Budyko framework to incorporate the full water balance equation. Here we apply the approach taken in a companion paper on the original Budyko framework to show that it can be generalized rigorously to include changes in vadose-zone water storage in a manner that is both parsimonious in hypotheses and broad in scope.

Equation ( 1) is subject to the boundary conditions [2][3][4]: ET → ET 0 as P ↑ ∞ (energy supply is limiting) ( ET → P as ET 0 ↑ ∞ (water supply is limiting) As discussed by Budyko ([2] pp.322-323), Equation (2) states that ET will take on its potential value in the limit of an infinitely abundant precipitation input, while Equation (3) indicates that ET will consume all precipitation in the limit of an infinitely abundant energy input.
The Budyko framework was developed under the assumption of steady-state water balance, which equates precipitation to the sum of evapotranspiration and runoff, thus implying negligible changes in subsurface water storage [2][3][4].This assumption has been validated in observations of long-term average annual ET from large catchments, but it is known to fail at smaller temporal and spatial scales [3].For example, the boundary condition in Equation (3) can be violated by annual ET from an arid sub-catchment if ET exceeds P because of contributions from water stored in the vadose zone [5,6].Therefore, in order to describe ET under non-steady-state conditions, the assumption of negligible changes in subsurface water storage must be relaxed and Equation (1) must be accordingly generalized [1,3,5]: ET = f(P,ET 0 ,∆S) where ∆S designates a change in subsurface water storage, either a gain or a loss.In a recent comprehensive review of the Budyko framework, Wang et al. [3] have discussed numerous attempts to extend the Budyko framework based on Equation (4), noting their use of adjustable parameters with no a priori relationship to catchment properties and their tendency to conflate climatic variability with catchment variability.They concluded that additional research was needed to "advance the Budyko framework with respect to model parsimony and physical meaning" [3] (p. 17).The present paper is a response to this challenge.It will be shown that, by making a single physically-relevant hypothesis about the functional relationship in Equation ( 4), the Budkyo framework can be extended to include changes in vadose-zone water storage in a manner broad enough to include all extant models [1,3,6] as special cases, while offering a rigorous approach to the specific modeling of Equation ( 4) that does not require adjustable parameters.

The Homogeneity Postulate
The Budyko framework can be extended to include changes in vadose-zone water storage as an independent variable in a manner consistent with the original framework [2,4] by postulating that f(P,ET 0 ,∆S) in Equation ( 4) is a homogeneous function of its arguments; meaning that if the independent variables are each multiplied by an arbitrary scale factor, λ, the value of f(P,ET 0 ,∆S) is multiplied by the same scale factor, λ [4,7]: Sposito [4] demonstrated in a companion to the present paper that the homogeneity hypothesis applied to the original Budyko framework, Equation (1), is sufficiently broad to include all extant models of long-term average annual catchment ET, parametric or non-parametric, as special cases.The same will be shown in the present paper for the extended Budkyko framework, Equation (4).
Following Hankey and Stanley [7], one can prove that the assumed property of homogeneity is both necessary and sufficient to express Equation (4) in two equivalent forms: or Equation ( 6) constitutes the "Budyko picture" of ET in the "Budyko space", i.e., ET/P as a function of ET 0 /P and ∆S/P, while Equation (7) constitutes the "Turc picture" of ET in the "Turc space", i.e., ET/ET 0 as a function of P/ET 0 and ∆S/ET 0 [1,8].In the present paper, only the Budyko picture will be utilized.For a detailed comparison of the Budyko and Turc pictures, see [1,8].
Equation (6) implies that, with respect to ET, two terrestrial regions at different geographic locations, or the same region at two different time periods, are scaled versions of one another.Thus, any two regions, or the same region at different times, exhibiting the same values of ET 0 /P and ∆S/P are in corresponding ET states.This "law of corresponding states" [4], an example of hydrologic similarity/space-time symmetry [9], is demonstrated experimentally by the close clustering of ET data when plotted according to Equation (6).Such a test was first performed systematically by Budyko [2] (pp.325-327) in the original framework (i.e., neglecting ∆S/P) using long-term hydrologic and climatological data for more than 1000 catchments and thereafter has been tested in a large number of careful studies [10][11][12][13][14].The study of Ye et al. [13], which confirms the law of corresponding states in the original framework with data collected daily over half a century in more than 250 catchments located in a variety of climatic zones and physiographic regions, is exemplary.

Boundary Conditions
In the original Budyko framework, the well-known boundary conditions on ET/P corresponding to Equations ( 2) and ( 3) are [2][3][4]15]: ET P ↓ 0 as ET 0 P ↓ 0 (energy-supply limit) ET P ↑ 1 as ET 0 P ↑ ∞ (water-supply limit) However, if ∆S = 0, Equations ( 8) and ( 9) are not in general applicable because stored vadose-zone water can serve either as an additional source of ET or as a sink for P. If ∆S < 0, the minimum value of ET/P in the energy-supply limit is not 0, as in Equation ( 8), because there are now two sources of ET, precipitation and stored vadose-zone water, so the minimum value of ET/P in the energy-supply limit would correspond to ET derived wholly from stored vadose-zone water [1,16], while P is transformed solely into runoff, in which case ET = −∆S.Since by definition ET = ET 0 in the energy-supply limit, Equation ( 8) becomes [1]: On the other hand, if ∆S > 0, the minimum value of ET/P remains at 0, which corresponds to P transformed either into runoff or water stored in the vadose zone [1,17].Therefore, Equation ( 8) remains valid.
In the water-supply limit, the maximum value of ET/P is not equal to 1, as in Equation ( 9), but instead equals [1 − (∆S/P)], which corresponds to stored water serving either as an additional source of ET (∆S < 0), thus increasing it, or as a sink for P that reduces ET (∆S > 0) [1].Therefore, Equation ( 9) becomes [1]: Graphical examples of Equations ( 10) and (11) are presented in [1,5,16].

The Euler Relation
In addition to the scaling relationship in Equation ( 5), all homogeneous functions satisfy the Euler relation, as discussed in detail for the original Budyko framework in [4].The Euler relation following from the assumption of homogeneity for f(P,ET 0 ,∆S) is: which is derived from Equation ( 5) by differentiating both sides with respect to λ then setting λ = 1 and noting Equation (4).As in the original Budyko framework [4,18], the partial derivatives in Equation ( 12) are subject to boundary conditions following from those in Equations ( 10) and ( 11): The substitution of the six boundary conditions in Equations ( 13)-( 18) into Equation ( 12) reproduces the boundary conditions in Equations ( 10) and (11).Graphs illustrating the boundary conditions in Equations ( 13)-( 16) in the original Budyko framework are given in Figure 6 of [18] based on a parametric model of Equation ( 6) for the case ∆S = 0. See also Figure 6 in [14] and Figure 3 in [11] for illustrative plots of (∂ET/∂P) versus ET 0 /P in the original Budyko framework.
The physical interpretation of these boundary conditions is as follows (see also [19]).Equations ( 13) and ( 15) reflect the fact that, when the water supply is not limiting, ET becomes insensitive to changes in P and approaches its maximum value, ET 0 .Equations ( 14) and ( 16) reflect the opposite tendency, i.e., the water supply is limiting and all available P is consumed by ET, which then becomes insensitive to changes in ET 0 .Equations ( 17) and ( 18) are unique to the extended Budyko framework.They are, however, analogous to Equations ( 13) and ( 14): ET becomes insensitive to changes in ∆S when the water supply is not limiting, akin to its insensitivity to P, whereas when the water supply is limiting, ET becomes very sensitive to changes in ∆S for the same reason that it is very sensitive to changes in P. Thus Equations ( 13) and (17) and Equations ( 14) and ( 18) express complementary water-balance effects on ET.The difference in sign between the right sides of Equations ( 14) and ( 18) is because ET increases as P increases, whereas ET decreases as ∆S increases.
Lastly, as shown in detail in [4], only two of the partial derivatives in the Euler relation, Equation (12), can be specified independently because of the constraint imposed by the Gibbs-Duhem equation, which follows from calculating the total differential of ET using the Euler relation then comparing the result with the total differential of ET based on Equation (4).The two independent partial derivatives are conveniently chosen to be the climate elasticities, ∂ET ∂P ET 0 ,∆S and ∂ET ∂ET 0 P,∆S , which have often been modeled to investigate the effects of climate variability on ET [3,4,14].

Basic Postulates
Moussa and Lhomme [1] have recently formulated an extended Budyko framework which is a special case of the general framework presented here.The underlying physical motivation for their formulation lies in the observations that (1) while the original Budyko framework postulates the entirety of ET as derived from P, in the extended Budyko framework it is only the portion (ET + ∆S) that is derived from P if ∆S < 0 and (2) while the entirety of P is available for ET in the original Budyko framework, only the portion (P − ∆S) is available for ET in the extended Budyko framework if ∆S > 0. This line of reasoning, based solely on the complete water balance, can be summarized in the following two hypotheses: (1) For ∆S < 0, the portion of ET that is derived from precipitation, (ET + ∆S), has the same functional relationship to (ET 0 + ∆S) in the extended Budyko in the extended framework, where B( ET 0 + ∆S P ) has the same mathematical form as the model function, B(ET 0 /P), found suitable for representing ET/P in the original framework, but with ET 0 /P in B(ET 0 /P) replaced by (ET 0 + ∆S)/P in B( ET 0 + ∆S P ).
(2) For ∆S > 0, ET/(P − ∆S), where (P − ∆S) is the portion of P that is available for transformation into ET, has the same functional relationship to ET 0 /(P − ∆S) as ET/P does to ET 0 /P in the original Budyko framework, i.e., ET P − ∆S = B( ET 0 P − ∆S ) where B( ET 0 P − ∆S ) has the same mathematical form as B(ET 0 /P) in the original Budyko framework but with ET 0 /P in B(ET 0 /P) replaced by ET 0 /(P − ∆S) in B( ET 0 P − ∆S ).
Under these two hypotheses, any model of ET/P = B(ET 0 /P) found suitable for application in the original Budyko framework to represent ET/P is automatically a candidate for use in the Moussa-Lhomme (ML) extended Budyko framework.Moussa and Lhomme [1] have tabulated several such candidate functions from the literature based on both parametric and non-parametric models of ET/P.
The Euler relation, Equation ( 12), takes on a simplified form in the ML extended Budyko framework because of the two hypotheses above, noting that, given the constraint in Equation ( 19),

Example: Extending the Schreiber Model
The ML formulation can be illustrated in perhaps the simplest manner with the Schreiber model, cited by Budyko [2] (p.323) as the earliest published mathematical expression of the relationship ET/P = B(ET 0 /P).The Schreiber model is non-parametric, conforming to the boundary conditions in Equations ( 8) and ( 9) while interpolating smoothly between them.It is often used as a basis for comparison with parametric models [15].
As noted above, if models of the climate elasticities are available, they can be substituted into the Euler relation to derive an expression for ET/P.In the Schreiber model, the climate elasticities are: where ϕ = ET 0 /P is the well-known aridity index [3,11].The partial derivatives in Equations ( 24) and ( 25) conform to the boundary conditions in Equations ( 13)-( 16), with ∆S = 0.The physical basis for Equation (24) is that, for a given ET 0 , the sensitivity of ET to P increases with ET 0 , a measure of the energy that can be potentially transported to the atmosphere if P is not limiting, whereas it decreases with increasing P as the water supply for ET becomes more plentiful.The physical basis for Equation ( 25) is that, for a given P, the sensitivity of ET to ET 0 increases with P, the amount of water that potentially can be transported to the atmosphere if ET 0 is not limiting, whereas it decreases with increasing ET 0 as the energy supply for ET becomes more plentiful.Upon the substitution of Equations ( 24) and ( 25) into the Euler relation, Equation (12) with ∆S = 0, the Schreiber model is derived in the original Budyko framework [2] (p.323): Equation ( 26) is readily seen to conform to the boundary conditions in Equations ( 8) and ( 9).According to the ML formulation, the Schreiber model, when extended to include ∆S, should become: for ∆S < 0, where ϕ' = ϕ + ∆S/P is a modified aridity index.For ∆S > 0, the extended Schreiber model becomes: where ϕ* = ϕ/(1 − ∆S/P) is another modified aridity index.Equation ( 27) conforms to the boundary conditions in Equations ( 10) and ( 11), while Equation (28) conforms to those in Equations ( 8) and (11).Both model equations reduce to Equation ( 26) when ∆S = 0. Sankarasubramanian and Vogel [20,21] have shown that a model incorporating subsurface water storage and having the same mathematical form as Equation (28) provides a quantitative description of ET/P calculated using databases on climate and runoff for more than 1000 catchments in the United States.Their model expression has the form: where γ = b/P and the parameter b > 0 represents the water-holding capacity of the upper vadose zone as defined in the abcd model of monthly catchment water balance [20][21][22].If Equations ( 28) and (29) were physically equivalent, then the identification γ = [1 − (∆S/P)] = (P − ∆S)/P, with ∆S > 0, could be made.Although this identification is consistent with the fact that γ < 1 is overwhelmingly observed among catchments in the United States [22], the abcd model parameter b represents a water-holding capacity in soil [20][21][22] that cannot be equated with (P − ∆S), the portion of precipitation that has not been used to increase vadose-zone water storage and, therefore, is available for ET and runoff.This example illustrates how goodness-of-fit to ET/P data is not sufficient to discriminate among proposed evapotranspiration models that differ in their underlying hydrologic basis.The only valid way to do a comparison is to test the physical assumptions underlying the models, which, in the present case, is a comparison of the abcd model as applied in [20,21] with ML hypothesis (2) above, which led to Equation (28).

Discussion
The two principal elements of the extended Budyko framework under the assumption of homogeneity are: Equation ( 6), the extended Budyko equation, which relates ET/P to the scaled variables, ET 0 /P and ∆S/P and thus implies hydrologic similarity/space-time symmetry [9], and Equation (12) , because the constraint imposed by the Gibbs-Duhem equation [4] makes modeling the third partial derivative in Equation ( 12),
The ML formulation of the extended Budyko framework is a special case of the general approach described in the present paper.It follows from two physical hypotheses: (1) For ∆S < 0, (ET + ∆S) plays the same role in water balance in the extended framework as ET does in the original framework, and, therefore, (ET + ∆S)/P can be modeled by any function found suitable for modeling ET/P in the original framework, with (ET 0 + ∆S)/P replacing ET 0 /P as the independent variable.
(2) For ∆S > 0, (P − ∆S) plays the same role in water balance in the extended framework as P does in the original framework, and, therefore, ET/(P− ∆S) can be modeled by any function found suitable for representing ET/P in the original framework, with ET 0 /(P − ∆S) replacing ET 0 /P as the independent variable.
The Schreiber model of ET/P [2] was used in Section 3.2 to demonstrate the application of these hypotheses, but more realistic models developed in the original Budyko framework can be transformed in the same way, as shown in [1].For example, according to the ML hypothesis (1), the Fu model [19], which has been widely applied in the original Budyko framework [3,4], in the ML formulation becomes, for the case ∆S < 0: ET Equation ( 31) is discussed further in the Appendix A in relation to recent efforts to extend the Fu model by Moussa and Lhomme [1] and Greve et al. [6].
As noted above, the ML formulation can be applied to generalize any model function, B(ET 0 /P), found suitable for use in the original Budyko framework, independently of whether the model contains an adjustable parameter such as w [1].In light of this parameter independence, recent studies carried out in the original Budyko framework [3,15,[23][24][25], which present statistical correlations between the Fu model parameter w and non-climatic catchment characteristics, may be premature, pending investigation of which catchment characteristics are already incorporated implicitly by considering the complete catchment water balance and including the independent variable, ∆S.Equation (28) offers a case in point, since varying ∆S/P in this equation has the same shape-changing effect on the ET/P versus ET 0 /P curve as does varying w in Equation (30), which neglects vadose-zone water storage.
It must be noted in passing that the ML formulation has not been tested extensively against experimental data, although, as Moussa and Lhomme [1] have pointed out, the behavior of model equations in their formulation as functions of ET 0 /P is consistent with experimental data on catchment ET/P reported by others whose successful models of these data are special cases of the ML formulation [5,6,16,26].The ML formulation is also consistent with the compilation of ET/P data for catchments in the United States reported by Sankarasubramanian and Vogel [20,21], as discussed above.Thus, further testing of the ML formulation with large data sets, particularly those that have been used in establishing statistical correlations between a model parameter such as w and catchment characteristics, is warranted and likely to be fruitful.The introduction of Equations (A4) and (A5) into the Euler relation, Equation (12), then produces the extended Fu model in the ML formulation [1]: which is in agreement with Equation (A3) and which, after rearrangement to solve for ET/P and factor ET 0 /P from the term ET 0 +∆S P inside the square brackets, is in agreement with Equation (31) as well.Thus, the model of Greve et al. [6] is a special case of the ML formulation, as was also concluded by Moussa and Lhomme [1].However, Moussa and Lhomme [1] used the erroneous result for ET/P reported by Greve et al. [6] in making a comparison between the two models, leading them to present a spurious non-linear relationship between the parameter y 0 and −∆S/ET 0 .

0 )
expressed as a function of the climate elasticities.It is straightforward to demonstrate by the direct incorporation of Hypotheses (1) and (2) into Equation (12) with the third term neglected that, in the ML formulation, the Euler relation becomes: ET = −∆S + ( ∂ET ∂P ) ET 0 ,∆S P + ( ∂ET ∂ET
, the Euler relation, which implies that the calculation of ET/P requires only a