a framework for scaling of hydrologic conceptualizations based on a disaggregation–aggregation...

HYDROLOGICAL PROCESSESHydrol. Process. 18, 1395–1408 (2004)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.1419

A framework for scaling of hydrologic conceptualizationsbased on a disaggregation–aggregation approach

Neil R. Viney1* and Murugesu Sivapalan2

1 CSIRO Land and Water, Private Bag No. 5, Wembley,WA 6913, Australia

2 Centre for Water Research, Department of Environmental Engineering, University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia

Abstract:

A fundamental question in scale research is how to scale up descriptions of hydrological responses from the smallscales at which they are developed to the larger scales at which predictions are required or can be validated, in thepresence of spatial heterogeneity of soils, vegetation and topography, and of space–time variability of climatic inputs.One scaling approach is the disaggregation–aggregation approach, which involves disaggregating catchment-scale statevariables to point-scale distributions, applying a suitable point-scale physical model using these state variables, andaggregating the resulting point-scale responses to yield a catchment-scale response. Thus, the approach provides a wayof linking the catchment-scale state variables with the catchment-scale responses and thereby permits the developmentof empirical large-scale models that still retain some essence of the small-scale physics. In this paper we illustrate thedisaggregation–aggregation approach with two examples that deal with developing catchment-scale conceptualizationsof (a) infiltration excess runoff and (b) evapotranspiration, using detailed process-based models available at the smallscale. These conceptualizations became the building blocks of the large-scale catchment model, LASCAM. Copyright 2004 John Wiley & Sons, Ltd.

KEY WORDS scaling up; aggregation; disaggregation; model conceptualizations; sub-grid variability

INTRODUCTION

Large-scale conceptualizations of hydrological processes are needed for the development of catchment-scalerainfall-runoff models and associated water-quality models, and also for linking surface hydrology to globalclimate models. Ideally, these models should implicitly account for sub-grid variability in the hydrologicalresponses. Although point-scale hydrological processes are well understood and readily represented inphysically based models, it is now well recognized that, for most hydrological processes, the use of effectiveparameters—let alone average parameters—is not appropriate owing to the strong non-linearity of the processdescriptions. In the past, hydrologists have usually resorted to conceptual models, often based on functionschosen arbitrarily, and whose parameters must be estimated by calibration. Because these parameters areestimated without reference to process-based descriptions, the conceptual models are not physically based,in that knowledge of the small-scale landscape properties (e.g. soil hydraulic properties) cannot be used toestimate, or in any way be related to, these parameters.

In the next section, we outline a disaggregation–aggregation approach for linking catchment-scale statevariables with catchment-scale responses, thus enabling the development of physically realistic large-scaleconceptual parameterizations. The ensuing sections, through two examples, illustrate how this approach mightbe used to translate knowledge of small-scale physics into catchment-scale models of infiltration excess runoffand evapotranspiration.

* Correspondence to: Neil R. Viney, CSIRO Land and Water, Private Bag No. 5, Wembley, WA 6913, Australia. E-mail: [email protected]

Received 15 July 2002Copyright 2004 John Wiley & Sons, Ltd. Accepted 3 March 2003

1396 N. R. VINEY AND M. SIVAPALAN

DEVELOPMENT OF A SCALING FRAMEWORK

Let us denote the small-scale hydrological responses by f(s, q, I) and the corresponding large-scale responsesby F(S, 2, I), where s and S denote state variables (e.g. soil moisture), i and I represent climatic inputs(rainfall, potential evaporation) and q and 2 are vectors representing landscape properties (i.e. parameters). Ingeneral, s and i may vary in both space and time, whereas q varies in space only. If the function f is linearthen the scaling up problem is trivial, since f will remain invariant, and model parameters at the larger scalecan be estimated by some kind of averaging of the small-scale values.

The problem in large-scale hydrological modelling is to obtain some representation of the large-scaleresponse F in terms of the large-scale variables �S, 2, I�. There is, therefore, a real need to estimate notonly the larger scale parameter values, but also to derive the corresponding, as yet unknown, functionalforms �f ! F�, along with the scaling connections between the state variables �s $ S� and parameter values�q $ 2�.

To overcome some of these shortcomings, Sivapalan (1993) proposed a modelling framework based ona disaggregation–aggregation approach that implicitly incorporates knowledge of point-scale processes intolarge-scale conceptualizations. The approach involves three stages (Figure 1). Firstly, disaggregation operatorsare developed to scale the state variables from large to point scale (S ! s). Secondly, we use the known point-scale physics, together with known point-scale parameters q to evaluate f at each point in the domain. Finally,we aggregate the point-scale responses across the catchment to derive a catchment-scale response F. Thisapproach now gives us a way of connecting S with F (the dashed line in Figure 1). We can then investigate thefunctional dependence of F on S, and develop empirical relationships describing this dependence. Furthermore,we can also use this approach to infer relationships between q and 2. The two examples presented in thispaper assume that the climatic inputs are constant in space, but clearly the modelling approach can be extendedby incorporating downscaling schemes for the climatic variables (e.g. Charles et al., 1999). We note that theinverse aggregation functions are usually trivial.

Sivapalan (1993) addressed only the first of the stages described above. He developed steady-state scalingrelationships between S and s that are conditioned on topography, climate and vegetation cover. We use theserelationships here in the first example: predicting catchment-scale infiltration capacities and infiltration-excessrunoff in terms of catchment-scale storages. The second example involves the development of equationsfor predicting and partitioning evaporation among three conceptual stores. These two examples are intendedprimarily as demonstrations of the scaling framework, rather than as fully fledged models in their own right.As such, they may, in certain circumstances, appear somewhat limiting in their assumptions. None of this,however, should detract from the applicability of the framework itself.

(S, Θ, I) F(S, Θ, I)

f(s, θ, i)(s, θ, i)

CatchmentScale

aggregation

PointScale point-scale

conceptualisation

catchment scaleconceptualisation

disaggregation

Input variables Responses

Figure 1. Functional relationships between input variables and responses at large and small scales

Copyright 2004 John Wiley & Sons, Ltd. Hydrol. Process. 18, 1395–1408 (2004)

SCALING OF HYDROLOGIC CONCEPTUALIZATION 1397

EXAMPLE 1: INFILTRATION-EXCESS RUNOFF

Model development

Although great strides have been made in the modelling of infiltration and runoff generation in the pastfour decades, even the best existing solutions for the infiltration problem have only been obtained for one-dimensional infiltration and for the simplest of boundary and initial conditions. Unfortunately, these solutionsdo not translate well to the real world, where moisture content is variable in space and time and the boundaryconditions are quite complex and variable. To date, too little research has been carried out on investigatingthe linkages between the micro-scale, process-based descriptions of infiltration and possible macro-scaleconceptualizations.

In this section we describe a methodology for deriving lumped, physically based parameterizations ofinfiltration and runoff generation in heterogeneous catchments. Model development builds upon a procedureoutlined by Beven (1984) and Robinson and Sivapalan (1995).

At any point on the surface of the catchment, the local infiltration capacity and infiltration rate are modelledusing an extension of the Green and Ampt (1911) equation, modified to allow for a non-uniform initialmoisture content in equilibrium with a water table at finite depth and the exponential decrease of saturatedhydraulic conductivity with depth. The catchment is subjected to a number of simulated rain storms of varyingintensity and duration. By applying the point infiltration model for a large number of points on the catchment,catchment averages of infiltration capacity, infiltration rate and cumulative infiltration volume are computedfor these storm events. Based on this simulation approach, empirical relationships are established between thecatchment-scale infiltration capacity and the corresponding catchment-scale cumulative infiltration volume fordifferent antecedent conditions.

Infiltration rates through the soil surface are strongly influenced by the antecedent moisture content andsaturated hydraulic conductivity of the soil, and by their variations with depth. At equilibrium and in thepresence of a water table at finite depth, soil moisture content tends to increase with depth reaching thesaturation value near the level of the water table. For simplicity, the initial moisture content profile �0�z� isassumed in this model to be that corresponding to zero vertical drainage. Furthermore, the unsaturated soilhydraulic properties, hydraulic conductivity K, suction head , and moisture content �, are assumed to followthe characteristic relations proposed by Brooks and Corey (1964), namely:

K�z, � D Ks�z�

( c

)2C3υ

and ��z, � D �r C ��s � �r�

( c

)υ

for > c �1a�

K�z, � D Ks�z� and ��z, � D �s for � c �1b�

where Ks�z� is the saturated hydraulic conductivity at depth z, c is the depth of the capillary fringe, �s isthe saturation moisture content, �r is the residual moisture content and υ is a constant.

Following Beven (1982) and Sivapalan et al. (1987), the saturated hydraulic conductivity is assumed todecrease exponentially with depth according to

Ks�z� D Ks�0� exp��fz�where Ks�0� is the surface value and f is a decay parameter. We assume that all other soil hydraulic propertiesremain constant with depth. This is consistent with many field observations (Beven, 1982, 1984), which haveindicated rapid decreases of the saturated hydraulic conductivity with depth compared with the more modestvariations in the water retention parameters �s and �r. Beven (1982) suggested that this may be due to thefact that the saturated hydraulic conductivity of field soils may be dominated by larger non-capillary pores,which generally decrease in size and frequency with depth.

Traditional Hortonian models of infiltration make the assumption that the initial moisture content is constantacross the catchment. In fact, even on small catchments, antecedent moisture content exhibits tremendous



spatial heterogeneity. Many field studies have indicated, however, that there may be an underlying regularityto the apparent spatial heterogeneity displayed by soil moisture and runoff generation responses. Barling et al.(1994) attribute this heterogeneity to variations in the spatial distributions of four factors: soil characteristics,topography (including aspect and shading), vegetation and weather. There are also further indications, yet tobe quantified in any meaningful way, of close correlations between observed patterns of soils and vegetation,and topography. All of these suggest that it might be possible to utilize topographic data, which is usuallyreadily available, to characterize the spatial variability of antecedent soil moisture.

To describe the spatial variability of antecedent soil moisture, we use the TOPMODEL formulation ofBeven and Kirkby (1979), as modified by Sivapalan et al. (1990). This model is based upon the fundamentalassumption that, under quasi-steady conditions, the difference between the local pre-storm water tabledepth Di and its catchment-wide average D is linearly related to the corresponding difference between thetopography–soil index, ln[aTe/�Ti tanˇ�], and its catchment average �. This relationship can be expressed by

Di D D� 1

f

[ln

(aiTe

Ti tan ˇ

)� �

]�2�

where ai is the area draining through location i per unit contour length, Ti is a local transmissivity, tanˇis the local slope and Te is an effective catchment mean of Ti (Sivapalan et al., 1987). Given D and thespatial pattern of values of the topography–soil index, Equation (2) enables the prediction of the local initialwater table depth Di for all points within the catchment. Thus, as Sivapalan (1993) pointed out, Equation (2)represents the linkage between the state variables representing the permanent groundwater table system at thelocal and catchment scales.

Prior to the storm, those parts of the catchment that are saturated are those where the value of thetopography–soil index is such that the local depth to groundwater is less than the thickness of the capillaryfringe c. These regions expand and contract both during storms and seasonally, and their areal extent andlocations can also be predicted by the model (Robinson and Sivapalan, 1995). If it can be assumed that thewater table depth Di and the soil moisture deficit Si are uniquely related by a function Di D h�Si�, then surfacesaturation will occur at those points where Gi�t�, the volume of infiltrated water at any given time, exceedsthe local initial soil moisture deficit. This condition may be expressed by

ln(

aiTe

Ti tan ˇ

)½ �C f[D� h�Gi�t��] �3�

Now, using Equations (1a), (1b) and (3), the total volumetric moisture deficit in the vertical soil profilemay be obtained by integrating �s � �0�z� between the ground surface (z D 0) and the top of the capillaryfringe (z D Di � c) to yield

Si�Di� D ��s � �r�

{Di � c � 1

1 � υ

[( c

Di

)υ

Di � c

]}if Di > c �4�

with Si�Di� D 0 otherwise. The total moisture deficit in the entire catchment SB can be expressed in terms ofthe average water table depth D by integrating Si over all extant values of the topography–soil index.

Following the traditional Green and Ampt (1911) formulation, we assume the existence of a sharp,rectangular wetting front. The initial moisture content �0 at the level of the wetting front increases as thewetting front depth increases, reaching the saturation value when the wetting front depth becomes equal toDi � c. The suction at the wetting front 0, which is an important parameter of the Green and Ampt model,decreases with increasing �0. In the case of a water table at finite depth, this causes an additional reductionin the infiltration capacity with increasing wetting front depth. A further reduction is caused by the assumedexponential decrease of saturated hydraulic conductivity with depth. When the wetting front depth becomesequal to Di � c, the infiltration capacity drops abruptly to zero because the soil moisture deficit Si has



now been completely filled by the infiltrated water. This condition is often adopted as the definition of thesaturation-excess mechanism of runoff generation.

The volume of infiltrated water Gi�t� when the wetting front has reached an average depth of Lf is given by

Gi�Lf� D Si�Di�� Si�Di � Lf� for Lf < Di � c

This equation may be inverted numerically to give the depth to the wetting front Lf�Gi� as a function ofthe cumulative infiltration volume Gi. The infiltration capacity of the soil at any point on the ground surfacewhen the wetting front is at depth Lf is then given by (Beven, 1984)

gi D gi�Gi� D Lf�Gi�[1 C0�Lf�Gi��/Lf�Gi�]∫ Lf�Gi�

0K�1

s �z� dz

�5�

where 0, the suction at the wetting front, can be expressed in terms of the soil hydraulic characteristics as(Brutsaert, 1976)

0 D � 1

Ks

∫ 1

0sεK

d

dsds �6�

In Equation (6), s D �� 0�/��s � �0� is the reduced moisture content and ε is a weighting factor. Brutsaert(1976) suggested a value of ε D 0Ð5 for most soils. The term �K d /ds represents a normalized soil waterdiffusivity term. With the substitution of Equations (1a) and (1b), Equation (6) becomes

0 D c

{1 C 1 � s0

υ

∫ 1

0sε[s0 C �1 � s0�s]

�ds}

�7�

where � D �1 C 2υ�/υ and s0 D ��0 � �r�/��s � �r� is the initial fractional moisture content. No stableanalytical solution exists for Equation (7) across the range of s for non-integer exponents ε and � . Equation (7)must, therefore, be integrated numerically.

The fact that 0 is independent of Ks means that Equation (5) can be used to estimate g for anytype of variation of Ks with depth. For the special case where Ks decreases exponentially with depth,Equation (5) becomes

gi�Gi� D Ks�0�fLf�Gi�

exp�fLf�Gi�� 1

[1 C0�Lf�Gi��/Lf�Gi�

]�8�

Knowing gi�Gi�, the actual infiltration rate gi�t� at any time t during a rainfall event is calculated as thelesser of gi and the local, instantaneous rainfall intensity pi�t�. The rate of infiltration-excess runoff is thengiven by qi�t� D pi�t�� gi�t�. Clearly, to scale these instantaneous, point predictions of gi�t� and qi�t� up tothe scale of a catchment and the duration of an event, we must integrate them over space and time.

Model parameterization

The runoff prediction model is applied to four small catchments in southwestern Western Australia: Wights�0Ð94 km2�, Salmon �0Ð82 km2), Jackitup Creek �0Ð38 km2� and Minjin �0Ð41 km2�. Owing to the very highpermeabilities of the gravelly surface soils of these catchments, almost all the rainfall reaching the groundinfiltrates and rapidly percolates down to the level of the less permeable clayey layer at 2–5 m depth.Subsurface runoff generated at this interface then flows downslope and contributes to the formation ofa perched groundwater table. The volume of this subsurface infiltration-excess runoff is affected by theantecedent soil moisture at the surface of the clay horizon, which, in turn, is influenced by the local depth tothe permanent groundwater table.

For the four catchments, the distributions of the topographic index are derived from digital elevation mapsand used with Equation (2) to disaggregate an assumed mean water table depth to determine the local water



table depths Di at every point in the catchment. The local soil moisture deficits are then calculated usingEquation (4), which, on integration, yields the catchment-scale soil moisture deficit SB. The initial contributingarea is given by Equation (3), with h D c, and indicates those parts of the catchment on which all rainfallimmediately runs off. As the storm progresses, this area expands as water infiltrates the unsaturated locationsand begins to fill the soil moisture deficit. After the beginning of the storm, some of the rain that falls on thosepoints for which Si > 0 infiltrates, thereby filling a portion of the soil moisture deficit. The surface becomessaturated when Si is completely filled by the infiltrated water. The infiltration rates and volumes are calculatedusing Equation (8), which updates the infiltration capacity as a function of the volume of infiltrated water.

We can now compute the catchment-scale averages of infiltration capacity g and cumulative depth ofinfiltration G, by linearly combining the point-scale values of all the grid points. The objective, then, is tolook for and to develop invariant relationships that can be used to predict runoff generation under a varietyof conditions on a daily time scale.

This model was used to simulate infiltration and runoff generation on the four catchments under a varietyof conditions involving different combinations of the soil hydraulic properties (Ks�0� and f) and antecedentwetness (D), and with storms of varying depth, duration and frequency. Since the catchments are small, weassume spatial homogeneity of rainfall, but rainfall intensities are allowed to vary temporally within the stormduration. For the purposes of this example, we also assume that the volume of water leaving the infiltrationstore during the storm is negligible.

We first sought to establish whether the infiltration capacity at the catchment scale could be expressed asa function of the cumulative volume of infiltrated water, and be independent of storm characteristics. If thisfunction can be demonstrated to be generally independent of rainfall history (depth, duration and temporalpattern), then it could form the basis for the specification of a catchment-scale runoff prediction equation thatis appropriate for conceptual rainfall-runoff models with a daily time step.

Firstly, both g and G were cast in dimensionless form in a similar manner to that used by Larsen et al.(1994) and Robinson and Sivapalan (1995), with the respective normalized quantities being

gŁ D g

Ks�0�

and

GŁ D G

c��s � �r�

The variations of gŁ with GŁ for a variety of storms on one catchment, but with all other parameters keptconstant (Figure 2), clearly support the adoption of a unique gŁ versus GŁ relationship that is independent ofstorm depth, duration and temporal pattern. Although there is some curvature at large values of cumulativeinfiltration, the straight lines of the log–log graph in Figure 2 suggest a relationship of the form

g D aG�b

�9�

where a and b are regression constants that are likely to depend on the catchment attributes D,f and Ks�0�.The validity of this relationship over a range of catchment topographies was tested by performing

simulations on different catchments, but with all catchment attributes other than the topographic indexdistribution being constant. The results (Figure 3) show slight differences in response between catchments,which may be interpreted as being caused solely by differences in catchment topography. The general natureof the relationships adds further confirmation to the form of Equation (9).

To test the dependence of g on G further, a series of simulations was performed by varying D while keepingthe other parameters constant. The results (Figure 4) show a set of almost parallel lines on a log–log graph,thus suggesting that the slope parameter b is independent of D, but that the intercept parameter a can bepostulated as a function of D (or SB).



10010-110-210-310-1

100

101

102

103

Cumulative infiltration (non-dimensionalised)

Infil

trat

ion

capa

city

(no

n-di

men

sion

alis

ed)

Storm 1Storm 2Storm 3Storm 4

Figure 2. The relationship between gŁ and GŁ for four storms of varying intensity and duration at Jackitup catchment. The solid line depictsthe linear regression of the data points

10-2 10-1 10010-1

100

101

102 WightsSalmonJackitupMinjin


Infil

trat

ion

capa

city

(no

n-di

men

sion

alis

ed)

Figure 3. The relationship between gŁ and GŁ for different catchments

A similar dependency test carried out for Ks�0� (not shown) indicates an invariance of the scaled gŁ versusGŁ relationship for a variety of storms. Thus, we postulate an empirical equation for the intercept parametera as

a D Ks�0��˛1 C ˇ1D� �10�



1 m

14 m

10010-110-210-3

10-1

100

101

102

103


Infil

trat

ion

capa

city

(no

n-di

men

sion

alis

ed)

Figure 4. The relationship between gŁ and GŁ for different values of D ranging between 1 and 14m

0.5 m -1

1 m-1

25 m-1

10-2 10-1 100 101

10-1

100

101

102


Infil

trat

ion

capa

city

(no

n-di

men

sion

alis

ed)

Figure 5. The relationship between gŁ and GŁ for different values of f ranging between 0Ð5 and 25m�1

where ˛1 and ˇ1 are constants. Note that Equation (10) could equally well have been expressed in terms ofSB rather than D.

When the exponential decay parameter f is varied, with all other parameters remaining constant, the slopeof the gŁ versus GŁ relationship increases with increasing values of f (Figure 5). Based on this systematic



investigation, involving a large number of simulations, we assume a linear relationship between b and f, andcan then postulate a functional form for the catchment-scale infiltration capacity as

g D Ks�0��˛1 C ˇ1D�G��˛2Cˇ2f� �11�

We may now use Equation (11) to predict the catchment-scale infiltration capacity g as a function of thecumulative infiltration volume G and the state of the permanent groundwater system (water table depth D orstorage deficit SB). Importantly, Equation (11) is independent of storm type; so, once calibrated for a particularcatchment, it can be used for all events. Provided both the permanent groundwater deficit SB and the state ofan intermediate store representing the volume of infiltrated water are continuously updated, then, on any givenday, the infiltration capacity can be computed from Equation (11). Given the volume of rainfall p on this day,the infiltration-excess runoff generated will be the larger of fp� g, 0g and the volume of actual infiltrationwill be the smaller of fp, gg. The volume of infiltrated water will then be used to update the intermediateinfiltration store. The total runoff generated will be composed of both the infiltration-excess runoff calculatedby Equation (11) and the saturation-excess runoff given by Equation (3).

The simulations reported here have concentrated on only three parameters, namely D,Ks�0� and f. Thisis because these are the parameters that have been shown to have the greatest impact on the volume ofrunoff generation (Sivapalan et al., 1987). However, it is a trivial exercise to extend the simulations tohelp parameterize the remaining constants ˛1, ˛2, ˇ1 and ˇ2 in Equation (11) in terms of physically based,measurable catchment attributes.

The usefulness of equations like Equation (11) is dependent on the extent to which the underlying point-scale formulations represent reality and include all relevant processes. For example, the model presentedhere assumes that soil moisture patterns are topographically driven and that all moisture movement duringinfiltration occurs along vertical soil columns and neglects downslope redistribution of moisture. Western et al.(1999) have shown that soil moisture patterns can exhibit a high degree of spatial organization during wetperiods, but that in dry conditions there can be little organization, thus suggesting that, in dry environments,a model based solely on topography might have limited applicability. The model presented here also neglectsreinfiltration and the potential effects of crusting, hydrophobicity and preferential flow. Limitations in thesteady-state assumptions of topography-based formulations have also been presented in the literature (e.g.Barling et al., 1994). Despite these potential limitations in the example presented here, none of these caveatsinvalidates the applicability of the scaling framework. Should any of these non-topography processes besignificant, then their effects could be incorporated in the point-scale representations and possibly also in thedisaggregation and aggregation methods, and model development could then proceed in a similar manner tothat outlined in the example.

A simplified, two-parameter version of Equation (11) was implemented in the LASCAM hydrological model(Sivapalan et al., 1996) to describe catchment-scale subsurface infiltration capacity in terms of infiltrationstorage and the catchment-scale water deficit.

EXAMPLE 2: EVAPOTRANSPIRATION

Model development

The physics of most aspects of evapotranspiration and plant water uptake at the point scale is wellestablished. However, large-scale hydrological and global climate models require parameterizations ofevapotranspiration that integrate the point-scale responses over a large spatial domain. In general, it is notfeasible to apply point-scale equations directly to large areas. This is because of the non-linearity of thephysically based equations and the unknown spatial or vertical distributions of important environmentalfactors (e.g. root density, soil hydraulic properties), and because it is computationally impractical. Therefore,we need some practical way to ‘scale up’ the physically based point-scale models to give catchment-scaleparameterizations.



The large-scale evapotranspiration model described here is designed for the large-catchment hydrologicalmodel, LASCAM (Sivapalan et al., 1996), and is thus developed with reference to the arrangement of storespertaining to that model. However, it is envisaged that the technique, if not the actual parameterization, willbe equally suitable for other applications.

LASCAM models a catchment in terms of three inter-dependent conceptual soil water stores (Figure 6)representing a near-stream, seasonal perched aquifer (the A store), the deep permanent groundwater system(B store) and an intermediate unsaturated infiltration zone (F store). The F store corresponds to the infiltrationstore G of the previous section, and the B store is an inverse of the storage deficit SB.

Evapotranspiration water may be sourced from any of these three stores, provided there are roots presentat suitable depths. It is assumed that, where the A store occurs in the landscape, evapotranspiration isdrawn first from that store, since it is near the surface and represents a saturated source. Elsewhere inthe landscape, or where the A store cannot satisfy demand, transpiration is drawn from the F store beforeit is drawn from the underlying B store. Consequently, we need an evaporation model that will allow us topartition the total evaporation among the three stores. Ideally, the resulting model should be conceptually andmathematically simple.

The physical models used for this analysis are based on those of Whisler et al. (1970) and Hoogland et al.(1981). Water uptake depends on saturated hydraulic conductivity, root length and radius (distributed withdepth), soil bubbling pressure, the Brooks and Corey parameter , leaf area, potential evaporation, and thedistribution of soil water potential with depth.

In building the model, we assume that the catchment can be collapsed into a single, two-dimensionalhillslope with simple idealized geometry. We then assume random, but realistic, values of the input variables(i.e. those controlling water uptake and those describing the hillslope and water-table geometry) in thephysically based point-scale model. The notional soil moisture at all points on the hillslope is distributed

Infiltration-excessrunoff

Groundwaterdischarge

Saturation-excessrunoff

Deeppercolation

Recharge

Evaporation

Evaporation

Throughfall

Infiltration

InterflowThroughfall

Throughfall

Baseflow

Recharge

B store

A store

I II III

F store

Figure 6. Conceptual hillslope of the LASCAM water balance model, showing the relationship of the three water storages. The transects, I,II and III, show the locations of the profiles depicted in Figure 7



DB

DB

DB

III

z

θ θ θθF θF

I II

θsBat θA

sat θBsat θF θA

sat θBsat

Figure 7. Notional soil moisture profiles for the three transects, I, II and III, in Figure 6. Here, DB is the depth to the B store, and �F, �satA

and �satB are respectively the moisture contents of the F, A and B stores

according to the levels of the three soil water stores. Here, we assume that the A and B stores are saturatedand that, in the F store, moisture content increases with store volume. Notional soil moisture profiles forselected locations in the landscape are shown in Figure 7.

The point-scale model is applied at a large number of regularly spaced points on the hillslope, and theevaporation responses are integrated to yield catchment-scale extractions from each store. We repeat thisprocedure for different values of the store levels and of the physical variables.

Model parameterization

The model simulations based on the simple hillslope geometry confirm that the evaporation rate from theA store may be determined analytically in terms of leaf area and a geometric parameter that depends oncatchment slope and A horizon depth. As such, it is not reliant on the disaggregation–aggregation approach,and will not be discussed here.

The generalized evapotranspiration responses from the B and F stores cannot be derived analytically. The Fstore evaporation curve typically takes the sigmoidal shape shown in Figure 8. Although this response curvecould have been modelled better by a more complicated function (e.g. cubic), we chose for simplicity tomodel only the upper part of the curve (above the inflection point), as it, too, appears linear on a log–loggraph. This suggests a catchment-scale relationship of the form

eF D ˛FepL

(1 � F

Fmax

)ˇF

�12�

where F is the level of the F store, Fmax is an F store scaling factor, ep is the potential evaporation remainingafter evaporation has been extracted from the A store, L is the leaf area index, and ˛F and ˇF are optimizableparameters that incorporate the effects of root distribution and soil properties. Equation (12) thus neglects thatamount of F store evaporation that is associated with small values of F for each curve, but the effect of thisunderprediction is small. Examples of the F store evaporation response for eight representative sets of inputvariables, together with the fitted catchment-scale model, are shown in Figure 8.

The B store evaporation curve typically takes an exponential shape, but may be approximated by a straightline on a log–log graph. This suggests that the response can be described by

eB D ˛BepL

(B

Bmax

)ˇB

�13�



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Scaled F store

Sca

led

F s

tore

eva

pora

tion

Point–scale model

Catchment–scale model

Figure 8. Relationship between the F store volume (scaled by Fmax) and its evaporation response (scaled by Lep) for the aggregatedpoint-scale model and for the catchment-scale model given by Equation (12)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Scaled B store

Sca

led

B s

tore

eva

pora

tion

Point–scale modelCatchment–scale model

Figure 9. Relationship between the B store volume (scaled by Bmax) and its evaporation response (scaled by Lep) for the aggregatedpoint-scale model and for the catchment-scale model given by Equation (13)

where B is the level of the B store, Bmax is a B store scaling factor and ˛B and ˇB are optimizable parameters.Here, ep is the potential evaporation remaining after any evaporation has been removed from the A and Fstores. Examples of this B store evaporation response for five representative sets of input variables, togetherwith the fitted catchment-scale model, are shown in Figure 9.



Thus, Equations (12) and (13) may be used to express catchment-scale evapotranspiration fluxes from theunsaturated store and from the deep groundwater store in terms of the respective catchment-scale storagelevels. The optimizable parameters (˛F, ˇF, ˛B and ˇB) are expected to depend on catchment hydraulic andbiophysical attributes. Both Equations (12) and (13) have been incorporated into the LASCAM model forprediction of daily evapotranspiration fluxes. As such, they are strictly appropriate only for that model, andare not necessarily transferable to, or comparable with, models with different storage structures. However,there is no reason why similar analyses could not be carried out for different model structures.

CONCLUSIONS

This paper has described a methodology to develop catchment-scale conceptualizations by integratingpoint-scale, process-based descriptions. A key aspect has been the ability to link the state variablesrepresenting the antecedent soil moisture state of the catchment between the catchment and local scales.This approach was illustrated by two examples, in which catchment-scale models of infiltration-excess runoffand evapotranspiration were developed. In the former case, the methodology involved both the temporaldisaggregation of daily inputs and the spatial disaggregation of the state variables (moisture content). In bothcases, spatial disaggregation took advantage of the natural (topographic) organization of moisture contentwithin a catchment, but the approach followed is also amenable the use of other forms of moisture organizationor randomness.

All conceptual rainfall runoff models necessarily have at least one component that models the volume ofrunoff generation, and most also include models of evapotranspiration. These models are typically expressedin terms of the status of one or more conceptual stores. In the past, the functional forms of these constitutiverelations were often assumed arbitrarily and their parameters estimated by calibration. Typically, the physicalbases of these functions have not been investigated. This paper represents an attempt to develop such linksbetween the conceptual model parameterizations and the underlying process-based, small-scale descriptions.In the case of the infiltration model (and potentially for the evapotranspiration model), the sensitivity analyseshave given considerable insights into the physical meaning of the lumped model parameters. As a result, themeasurability of the catchment-scale parameterizations has been substantially enhanced.

It is important to emphasize that the disaggregation–aggregation approach scales model output, rather thanactual catchment behaviour. The resulting catchment-scale conceptualizations are only reliable and effectivein so far as the point formulations they are derived from do indeed represent reality and encompass all thefactors that have a significant influence on hydrological variability.

The relations developed in this paper are intended for use in lumped catchment models operating ata daily time step. They have been integral to the development of the long-term water balance model,LASCAM (Sivapalan et al., 1996). The disaggregation–aggregation approach outlined here has been usedto conceptualize many of the hydrological fluxes in LASCAM (e.g. subsurface stormflow) and has potentialapplication in the development of other conceptual hydrological models.

ACKNOWLEDGEMENTS

The authors are indebted to Charles Jeevaraj and Jens Larsen for their assistance in developing the infiltrationand evapotranspiration models.

REFERENCES

Barling RD, Moore ID, Grayson RB. 1994. A quasi-dynamic wetness index for characterizing the spatial distribution of zones of surfacesaturation and soil water content. Water Resources Research 30: 1029–1044.

Beven KJ. 1982. On subsurface stormflow: an analysis of response times. Hydrological Sciences Journal 27: 505–521.



Beven KJ. 1984. Infiltration into a class of vertically non-uniform soils. Hydrological Sciences Journal 29: 425–434.Beven KJ, Kirkby MJ. 1979. A physically-based variable contributing area model of basin hydrology. Hydrological Science Bulletin 24:

43–69.Brooks RH, Corey AT. 1964. Hydraulic properties of porous media. Hydrology Papers 3, Colorado State University, Fort Collins, United

States.Brutsaert W. 1976. The concise formulation of diffusive sorption of water in dry soil. Water Resources Research 12: 1118–1124.Charles SP, Bates BC, Hughes JP. 1999. A spatiotemporal model for downscaling precipitation occurrence and amounts. Journal of

Geophysical Research 104: 31 657–31 669.Green WH, Ampt GA. 1911. Studies on soil physics, part I. The flow of air and water through soils. Journal of Agricultural Science 4:

1–24.Hoogland JC, Feddes RA, Belmans C. 1981. Root water uptake model depending on soil water pressure head and maximum extraction rate.

Acta Horticulturae 119: 123–135.Larsen JE, Sivapalan M, Coles NA, Linnet PE. 1994. Similarity analysis of runoff generation processes in real-world catchments. Water

Resources Research 30: 1641–1652.Robinson JS, Sivapalan M. 1995. Catchment-scale runoff generation model by aggregation and similarity analyses. Hydrological Processes

9: 555–574.Sivapalan M. 1993. Linking hydrologic parameterizations across a range of spatial scales: hillslope to catchment to region. In Exchange

Processes at the land surface for a Range of Space and Time Scales , Bolle H-J, Feelds RA, Kalma JD (eds). IAHS Publication No. 212.IAHS Press: Wallingford; 115–123.

Sivapalan M, Beven K, Wood EF. 1987. On hydrologic similarity. 2. A scaled model of storm runoff production. Water Resources Research23: 2266–2278.

Sivapalan M, Wood EF, Beven K. 1990. On hydrologic similarity. 3. A dimensionless flood frequency model using a generalizedgeomorphologic unit hydrograph and partial area runoff generation. Water Resources Research 26: 43–58.

Sivapalan M, Ruprecht JK, Viney NR. 1996. Water and salt balance modelling to predict the effects of land use changes in forestedcatchments. 1. Small catchment water balance model. Hydrological Processes 10: 393–411.

Western AW, Grayson RB, Bloschl G, Willgoose G, McMahon TA. 1999. Observed spatial organization of soil moisture and its relation toterrain indices. Water Resources Research 35: 797–810.

Whisler FD, Klute A, Millington RJ. 1970. Analysis of radial, steady-state solution, and solute flow. Soil Science Society of AmericaProceedings 34: 382–387.


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