THE TOPKAPI MODEL by E. TODINI and L. CIARAPICA Department of Earth and Geo-Environmental Sciences University of Bologna, Bologna, Italy ABSTRACT: The paper introduces a new distributed rainfallrunoff model derived upon the assumption that the horizontal flow at a point in the soil and over the surface can be approximated by means of a kinematic wave model. The point assumption is then integrated up to a finite pixel dimension, thus converting the original differential equation into a non-linear reservoir equation based upon physically meaningful parameters, which solution can be found numerically. The catchment behaviour is finally obtained by aggregating the non-linear reservoirs into three cascades, representing the soil, the surface and the drainage network, following the topographic and geomorphologic elements of the catchment. The main advantage of this approach lies in the possibility of deriving the model structure and the parameters on the basis of digital elevation maps, soil maps, land use maps etc.. and in its capability of being applied at increasing spatial scales without loosing model and parameters physical interpretation. These properties make the model suitable for land-use and climatic change impact assessment; for extreme value analysis, given the possibility of its extension to ungauged catchments; and last but not least as a promising tool for the General Circulation Models (GCMs).
1. INTRODUCTION The idea of developing the new rainfall-runoff model arises from the experience gained in years of applied research as well as from the need of finding a model which could be easy to calibrate from available maps and valid at different spatial scales. Due to the generally large computer time requirements of distributed models, in the past, lumped and semi-distributed (a cascade of lumped) models (Todini, 1989) have been widely
preferred for representing the rainfall-runoff process at scales ranging from few square kilometres to several hundreds. Unfortunately, given the lack of direct physical interpretation of most of their parameters, they need "sufficiently long" meteorological and hydrological records for their calibration which may not always be available; their calibration involves curve fitting making physical interpretation of the fitted parameter values very difficult; predicting effects of land use change by changing parameter values cannot therefore be done with confidence since such models do not generally make use of data such as topography, soil type and patterns and changes of vegetation types. Physically based distributed models, on the other hand "can in principle overcome many of the above deficiencies through their use of parameters which have a physical interpretation and through their representation of spatial variability in the parameter values" (Abbott et al., 1986; Beven, 1985; Beven, 1989). Given their capability of representing the internal spatial hydrological pattern, the major areas of application of distributed models can be found in forecasting the effects of land-use change, the effects of spatially variable inputs and outputs, the movement of pollutants and sediments, and the hydrologic response of ungauged catchments (Beven and O'Connell, 1982). Recent advances in remote sensing, geographic information systems, and computer technology make in fact distributed hydrologic models approach attractive to flow simulation and prediction. The linkage of a distributed hydrologic model with the spatial data contained in geo-morphological maps and digital elevation models (DEM) can be dealt with by means of a Geographic Information System (GIS) and offers the advantage given by the full utilisation of spatially distributed data for the analysis of hydrologic processes (Orlandini and Rosso, 1996). The complex heterogeneity of the land surface through soils, vegetation and topography, all of which have different length scales, and their interaction with meteorological inputs that vary with space and time, results in energy and water fluxes whose scaling properties are unknown. Research into land-atmospheric
interactions suggests a strong coupling between land surface hydrological processes and climate (Charney et al., 1977; Walker and Rowntree, 1977; Shukla and Mintz, 1982; Sud et al., 1990). Due to this coupling, the issue of "scale interaction" for land surface-atmospheric processes has emerged as one of the critical unresolved problems for the parameterisation of climate models (Wood, 1995). From the above considerations, the need for creating a new generation of lumped models with parameters directly related to physically meaningful quantities to be derived from the appropriate distributed scales of representation, clearly emerges, thus combining the advantages of a physically based parameterisation with a suitable lumped operational model to be used at larger scales (Todini and D端menil, 1999). The present work takes the lead from a critical analysis of two well known and widely used semi-distributed hydrological models, namely the ARNO and the TOPMODEL. The ARNO model (Todini, 1996; Todini, 2001), following the earlier work of Zhao (1977), is a variable contributing area semidistributed conceptual model driven by the total soil moisture storage, which is functionally related, by means of simple analytical expressions, to the directly contributing areas, the evapo-transpiration, the drainage and the percolation. This makes the model extremely useful in evaluating the total amount of soil moisture available for evapo-transpiration, which is one of the major requirements for inclusion in GCMs (Todini and D端menil, 1999, Todini, 1995). The major disadvantage of the ARNO model in this respect is the lack of physical grounds for establishing some of the parameters, and in particular those relevant to the "drainage" namely the horizontal movement in the unsaturated zone, which have to be estimated on the basis of the available precipitation and runoff data; this is not particularly critical in hydrological applications but it becomes so when the model is used in GCMs, where direct input/output observations are not really available (Todini, 1995; Todini and D端menil, 1999).
The TOPMODEL (Beven and Kirkby, 1979; Sivapalan et al., 1987) is a variable contributing area lumped conceptual model in which the predominant factors determining the formation of runoff are represented by the topography of the basin and a negative exponential law linking the transmissivity of the soil with the distance of the saturated zone below ground surface. The TOPMODEL is attractive in that it can be considered as a "simple physically-based conceptual model" (Sivapalan et al., 1987), in the sense that its parameters can be measured directly, or more realistically in the sense that its parameters can be physically interpreted. Despite the caution in accepting the definition of physically-based model, the inclusion of the effects of variability of topography on contributing area dynamics represents a major advance over previous models based on "point" hydrological responses assumed to apply at the catchment scale. The TOPMODEL represents a first attempt to combine the computational and parametric efficiency of a lumped approach with the link to physical theory and possibilities for more rigorous evaluation offered by a distributed model (Beven et al., 1995). The major disadvantage of TOPMODEL lies in the steady state assumption at a point, which is advocated in order to derive the model integral equation. This assumption becomes unrealistic for cells of the order of magnitude of hundreds of meters, which implies the need for parameter values far beyond their physical limits (Franchini et al., 1996). According to the above sketched analysis, the TOPKAPI, the new rainfall-runoff model, which is introduced in this paper, aims at: -
exploiting the potentiality of distributed models based upon physically meaningful parameters (Abbott et al. 1986; Beven 1989a);
-
overcoming the scale dependency of parameters as in TOPMODEL (Franchini et al., 1996);
-
incorporating the possibility of obtaining a lumped version of the same model, by integration of the processes over
increasing size domains, without need of re-calibration (Todini,1995); -
allowing for, as a consequence of the previous points, the application of the model at increasing spatial scale, from hillslope to catchment scales and, in perspective, to the GCMs one (Todini and D端menil, 1999; Todini, 1995).
2. THE DISTRIBUTED TOPKAPI APPROACH TOPKAPI is the acronym of: TOPographic Kinematic Approximation and Integration (Todini, 1995). Seeking to reconcile the need for physical meaning to the need for relative simplicity and parsimony in parameterisation, the TOPKAPI approach aims at developing a physically based model at a point, which will preserve its average physical meaning across scales, over pixels of increasing sizes, up to the catchment scale. As pointed out, two versions of the TOPKAPI model are available. The first one, the distributed version, is aimed at reproducing the internal spatial variability and at affording a clearer understanding of the effects of scale on hydrometeorological data, on physical phenomena and on the parameters for the development of a saturated area vs soil moisture storage volume relationship (similar to the ARNO model curve) to be used as the basic tool for the derivation of the lumped TOPKAPI version. The TOPKAPI model is based on the idea of combining the kinematic approach with the topography of the basin described by means of a lattice of square cells, generally increasing in size with the scale of the problem, over which the model equations are integrated. Each cell representing a computational node for the physical characteristics of the model, namely the mass balance and the momentum balance. The flow paths and slopes are evaluated from the DEM, according to a neighbourhood relationship based on the principle of minimum energy, namely the maximum elevation difference which takes into account the links between the active cell and the four surrounding cells connected along the edges; the active cell is assumed to be
connected downstream with a sole cell, while it can receive upstream contributions from up to three cells (Band, 1986). At present, the proposed model is structured around three modules which represent the soil component, the overland flow component and flow through the drainage network respectively. The present version of the TOPKAPI model does not account for water percolation towards the deeper soil layers and for their contribution to the discharges; this will be introduced as an additional model layer in the future. The horizontal flow in the unsaturated zone is assumed to occur in a layer of limited thickness and with generally high hydraulic conductivity due to the preferential flow paths and macro-porosity. This mechanism plays a fundamental role in the model both as a direct contribution to the flow into the drainage network, and most of all as a factor regulating the soil moisture balance, particularly with regard to the dynamics of the saturated areas. Overland flow is generated by the excess rainfall on the different saturated cells while the total runoff (surface + subsurface) is then drained by the drainage network. Evapo-transpiration is taken into account as water loss, subtracted from the soil’s water balance. This loss is rarely available as a measured quantity, generally being calculated using hydro-meteorological data and available topographic, geographic and climatic information; in the present development of TOPKAPI evapo-transpiration is computed by the same methodology used in the ARNO model (Todini,1996). In the following sections only the soil component, will be described in detail, while only the resulting equations and the main features of the overland flow and drainage network components will be presented, due to the structural similarity to the soil component equations.
2.1. Basic assumptions for the soil component The assumptions on which the soil model is based, are described in the sequel.
1) Precipitation is assumed to be constant over the integration domain (namely the single cell), by means of suitable interpolation of raingauge data, such as Thiessen polygons techniques, Kriging or others. 2) All the precipitation falling on the soil infiltrates, unless the soil has reached saturation; this is equivalent to adopting, at the scale of the pixel, the saturation excess or Dunne mechanism (Dunne, 1978) instead of the possible infiltration excess or Hortonian mechanism; this decision is justified by the fact that the infiltration excess mechanism is characteristic at a point, whereas the saturation excess mechanism, being linked to a cumulative phenomenon and conditioned by a lateral redistribution movement of the water in the soil, becomes dominant as the scale of the model increases (Blรถschl and Sivapalan, 1995) and already reasonably true at scales of tenths of meters. 3) The slope of the water table is assumed to coincide with the slope of the ground, unless the latter is very small (less than 0.1%); this roughly constitutes the fundamental assumption of the approximation for the kinematic wave, and it implies the adoption of a kinematic wave propagation model with regard to horizontal flow, in the unsaturated area (Henderson and Wooding, 1964; Beven, 1981, 1982; Borah et al., 1980; Sloan and Moore, 1984; Hurley and Pantelis, 1985; Stagnitti et al., 1986; Steenhuis et al., 1988). 4) Hydraulic conductivity at saturation is taken constant with depth in a surface soil layer but much larger than that of deeper layers; 5) The integral of the hydraulic conductivity over the vertical in the unsaturated zone, an extension of the concept of transmissivity, together with the horizontal flow, can be reasonably expressed as a function of the total water content of the soil, i.e. on the basis of the integral of the water content profile in the vertical direction; 6) During the transition phase (the filling of the unsaturated zone) the variation over time of the water content is not set to
zero as in TOPMODEL, but assumed to be constant over the elementary cell. 2.2. The vertical lumping Given the vertical soil moisture profile, a detailed description of the vertical infiltration can be performed by means of Richards equation, which account for gravity and capillary pressure. Nonetheless, due to the high conductivity value caused by macropores in the top layer of soil (Beven and Germann, 1982), gravity will be the dominant mechanism driving water from the top of the soil to an impermeable or semi-impermeable lower boundary which constitutes the bottom of a sort of thin perched aquifer where a non negligible horizontal flow, also involving unsaturated flow, will occur (Todini, 1995). The depth of this high conductive soil (from few centimetres up to one or two meters) will be negligible with respect to the horizontal grid dimensions (generally hundreds of meters). For this reason, within the range of reasonable errors, the use of Richards equation for the description of the soil vertical infiltration (Todini, 1995) can be avoided by assuming that at the cell scale or at larger scales the water will always infiltrate until the saturation is reached, which is consistent with the Dunnian hypothesis. An analogue of the transmissivity, which is normally defined for saturated aquifers, can be introduced for the unsaturated soil layer:
∫( ) L
~ T = k ϑ ( z ) dz
(1)
0
where L is the thickness of the layer affected by the horizontal ~ ~ flow, z is the vertical direction, k (ϑ (z )) = k sϑ (z )a is the hydraulic conductivity given by Brooks and Corey (1964), for nonsaturated condition, as a function of the reduced water content ~ ϑ − ϑr , ϑ r is the residual soil moisture content, i.e. water ϑ= ϑ s − ϑr
which cannot be removed by capillarity or gravity, ϑ s is the saturated soil moisture content and ϑ the water content in the soil. According to hypotheses 4) and 5) introduced in the previous section the transmissivity defined by equation (1) does not strongly differ from the one estimated in terms of the total soil moisture content integrated along the vertical profile (Benning, 1994), as in equation (2):
( )
~ ~ T Θ = k s LΘ α
(2) ~
where k s is the saturated hydraulic conductivity, Θ =
L
1 ~ ϑ ( z )dz L0
∫
is
the mean value along the vertical profile of the reduced soil moisture content and α is a parameter which depends on the soil characteristics (Benning, 1994; Todini, 1995). This approximation was validated by means of a number of numeric simulations. A number of families of hypothetical vertical soil moisture profiles, all of them with the same depth, was generated. Each family was characterised by a different ~ value of Θ and different vertical profiles. For all the profiles the horizontal flux was calculated from the actual profile, by means of the Brooks and Corey's formula, to give: L
~ q = tan (β )k sϑ a dz
∫
(3)
0
~
The horizontal flux was also calculated for each value of Θ by means of the approximated formula (2): ~ q = tan (β )k s LΘ α
(4)
Comparing the two results in Figure 1 (Benning, 1994), when one is interested in a finite domain overall behaviour, given the uncertainty in the saturated conductivity values which vary of orders of magnitude from one point to another, the approximation appears more than acceptable thus allowing to eliminate the vertical dimension from the original problem. Benning (1994) also analysed the use of the Van Genuchten
Transmissivity (%)
equation (1980), for which similar results can be obtained analytically.
Average soil moisture content (%)
Fig. 1. Comparison between the estimate of horizontal transmissivity in unsaturated soil (as a percentage of that at saturation) estimated as a function of the total water content (solid line) or according to the actual vertical humidity profile (different symbols relate to different profiles)
2.3. The model for the single cell The model of the single cell can be derived from the continuity of mass and an approximated momentum equation, expressed after the vertical lumping, namely: ~ ∂Θ ∂q + =r (ϑ s − ϑ r )L ∂t ∂x q = tan (β )k LΘ~ α s
where:
(5)
x t q r
is the main direction of flow along the horizontal dimension of the cell; is the time co-ordinate; is the horizontal flow in the soil as a discharge per unit of width, and is the precipitation intensity.
The model is written in just one direction since it is assumed that the flow along the slopes is characterised by a strong preferential direction, which can be described as the direction of maximum slope. Equations (5) can be combined and rewritten in terms of the total soil moisture content over the vertical profile: ~ η = (ϑ s − ϑ r )LΘ
(6)
∂q ∂η ∂t = r − ∂x Lk s tan (β ) α q = η = Cη α (ϑ s − ϑ r )α Lα
(7)
to give:
as a function of a local conveyance C=
Lk s tan (β )
(8)
(ϑ s − ϑr )α Lα
which depends on soil parameters and is directly proportional to physically meaningful quantities such as hydraulic conductivity and slope, and inversely proportional to the storage capacity. Combining the two equations (7) the following kinematic type equation is finally obtained:
(
∂η ∂q ∂ Cη α =r− =r− ∂t ∂x ∂x
)
(9)
that can be integrated in the soil over the ith cell (a pixel of the DEM) to give: ∂ V si ∂t
(
= rx − C si η sαi s − C si−1η αsi−s1
)
(10)
where Vsi is the volume per unit of width stored in the ith cell while the last term in equation (10) represents the inflow and outflow balance. A footer s has been introduced to distinguish this equation from the ones relevant to the overland and the drainage network flows and will be kept from now on. It is quite evident that the coefficients C s are no longer the physically measurable quantities which are defined at a point, but rather they represent integral average values for the entire cell, which nonetheless are still strongly related to the measurable quantities.
2.4. The distributed model The cells in TOPKAPI are connected together by a tree shaped network; the distributed model is based on the idea of moving along this tree shaped cell network starting from the initial cells (without upstream contributing areas) representing the "sources", and moving downstream toward the outlet. According to this procedure, and assuming that in each cell the variation of the vertical water content η si along the cell is negligible, the volume of water stored into each cell (per unit width) can be related to the total water content η si by means of the following simple expression: Vsi = xη si
(11)
Substituting for η si in equation (10) and writing it for the “source” cells namely the uppermost cells in each branch, the following non-linear reservoir equation is obtained: ∂Vsi ∂t
= ri x −
C si xα s
Vsαi s
(12)
Similarly a non-linear reservoir equation can be written for any generic cell, given the total inflow to the cell: ∂Vsi ∂t
(
)
= ri x + qoui + q sui −
C si xα s
Vsαi s
(13)
where q oui
q sui
is the discharge entering the active cell as overland flow from the upstream surface contributing area; is the discharge entering the active cell as interflow from the upstream contributing area.
Although, as it will be seen in the sequel, the TOPKAPI model strongly differ from the TOPMODEL, an analogy exists up to this point between the basic soil equations in TOPKAPI and the ones used in TOPMODEL. Let us consider the equation that describes the discharge per unit width used in TOPMODEL (Beven et al., 1995): q =
K0 tan (β ) exp(− fz i ) f
(14)
where z i is the depth into the soil moisture profile directed downwards. Given that in TOPMODEL the subsurface flow is assumed to take place only in the saturated portion of the profile, the variable η used in TOPKAPI can be expressed in terms of z i ~ by assuming Θ equal to one: η = (ϑ s − ϑ r )(L − z i )
(15)
The basic TOPMODEL assumption is that at the scale of a cell, the steady state is immediately reached, which allows for writing (Beven et al., 1995): ai r =
K0 tan (β ) exp(− fz i ) f
(16)
where r is the spatially homogeneous recharge rate and a i is the area per unit contour length drained by the location i. Taking into account the time variation of the volume of water stored per unit width ∂Vi ∂t , which is set to zero under the steady state assumption, equation (16) can be written as: ∂Vi K = a i r − 0 tan (β ) exp(− fz i ) f ∂t
(17)
Substituting for z i , given by equation (15), into equation (17) and bearing in mind equation (11), one obtains: K fVi ∂Vi = a i r − 0 tan (β ) exp(− fL) exp ∂t f a i (ϑ s − ϑ r )
(18)
The similarity between equations (12) and (18) is evident since the exponential in equation (18) does not essentially differ from the power function in equation (12). This is the only similarity that can be found in TOPMODEL and TOPKAPI which also includes an overland flow and a drainage network flow models; moreover, the lumpig of TOPKAPI is obtained by first demonstrating, under less restrictive hypotheses, the invariance of the basic equation at a change of scale and successively showing the possibility of lumping the model parameters in the form of a non linear weighted average. Up to this point it was implicitly assumed that the entire outflow from a cell flows into the immediately downstream cell. However this is not entirely true; in fact the cells can be divided into two groups on the basis of the minimum drained area (threshold) suggested by O'Callaghan and Mark (1984), with a first group where the overland flow is dominant and an other one where a channel flow is also present. For the latter cells the outflow is still evaluated by means of the equation (15) but it is then partitioned between the channel and the downstream cell according to a gradient based upon the average slope of the four surrounding cells. This allows to determine the amount of interflow feeding the drainage channel network.
2.5. The distributed overland and drainage network models In TOPKAPI, of the overland and drainage network model components are based upon a classical kinematic approximation (Henderson and Wooding, 1964; Wooding, 1965) and result in equations quite similar to that of the soil component.
In order to represent the overland flow, similar assumptions to those made for the subsurface component are introduced; in particular, it is assumed that the conditions for the application of the kinematic model hold at least in its integral form over a model pixel and over a channel reach of the length of a pixel. Since it is assumed that all the precipitation infiltrates into the ground until the soil reaches saturation, the input to the overland flow model is the precipitation excess resulting from the saturation of the soil layer. In analogy to what was done for the soil, the overland flow model is now derived by adopting the Manning friction law and assuming that the water depth is constant over a cell. This allows for integrating the kinematic equation over the longitudinal dimension to give the following non-linear reservoir model: ∂Voi ∂t
= eoi −
Coi x
αo
α
Vo o i
(19)
where Voi
is the volume of the water on the surface;
eoi
is either the saturation excess (positive) resulting from the solution of the soil moisture balance equation (13); is a coefficient relevant to the Manning formula, formally equivalent to the term C si of the soil component; is the surface slope of the ith cell; an exponent deriving from the combination of Chezy and Manning formulae and is the Manning coefficient for the surface roughness.
C oi = tan (β i )
12
tan (β i )
αo = 5 3 noi
n oi
Similar considerations also apply to the drainage network, which is assumed to be tree shaped with reaches of wide rectangular cross sections. In this case the surface width is not constant but it is assumed to be increasing towards the catchment
outlet. Under these assumptions, the following expression can be written for the generic reach: ∂Vni ∂t
(
)
= rsi + roi + qnui −
Cn i
(xwi )α
n
Vnαi n
(20)
where is the volume of water stored in the network reach;
Vni roi = rsi =
C oi xα o C si xα s
Voαi o
is the overland runoff reaching the ith reach;
Vsαi s
is the soil drainage reaching the ith reach;
q nui Cni = S 01i 2 n n
is the discharge inflow from the upstream reaches; is the coefficient relevant to the Manning
i
formula; equivalent to the previously seen Coi ; S 01i 2
is the bed slope, assumed to be equal to the
αn = 5 3 nni wi
cell slope tan (β i ) ; is an exponent which derives from using Chezy and Manning formulae; is the Manning friction coefficient for the channel roughness; is the surface width of the generic channel reach, which is taken to increase as a function of the area drained by the i th cell, on the basis of geo-morphological considerations.
It must be pointed out that in the case of the channel network, due to the longitudinal variation in surface width, the term Vni in equation (20) is taken to actually represent a volume, while in the previous equations, (13) and (19), the terms Vsi and Voi
correspond to volumes per unit width.
2.6. The computational algorithm As described above, the TOPKAPI model formulation leads to three tree shaped cascades of non-linear reservoirs, each of which is described by an ordinary differential equation (ODE) to be solved in time. This formulation, which is fully consistent with the kinematic equation hypothesis, is very convenient in that the original set of partial differential equations to be numerically solved simultaneously have been transformed, in the finite domains represented by the cells, into a set of ODE that can be solved in a cascade mode. Nonetheless, the number of reservoirs generally varies according to the problem from a few thousand to hundreds of thousands. Therefore an extremely efficient computer program had to be written in order to numerically and efficiently solve the differential equations in the appropriate succession, following the tree topology and connectivity. An original algorithm was devised, which only requires three vectors not larger than the number of pixels, and allows for the browsing of each tree in the appropriate succession starting from each source pixel on the boundary of the catchment contributing area (the divide) towards the outlet. Several algorithms were tested for the solution of the ODE for each single reservoir representing the soil, the surface and the channel network: the analytical solution of a linearised and of a second order approximation was compared with the results provided by various Runge-Kutta algorithms. The final choice, which produced the fastest and more reliable integration results, was introduced in the present version of TOPKAPI and is based upon a variable step fifth order Runge-Kutta algorithm due to Cash and Carp (1990).
3. THE LUMPED TOPKAPI APPROACH The basic concept in "lumping" lays in the possibility of obtaining a model at a larger scale by averaging all the smaller scale processes. Unfortunately the averaging operator, which
implies integration over the domain of interest, is a linear operator that does not commute with non-linear process models. This implies that the average model response will rarely coincides with that of the point model computed on the average parameter values. For example, the effect of Alps (with peaks over 4000 m a.m.s.l.) cannot be described by the mesh average (which may result into some 500 m a.m.s.l.) in a meteorological model describing the atmosphere by integrating a set of partial differential equations over meshes of 50x50 km2. Therefore, in order to obtain the lumped version of TOPKAPI, the process equation, namely the kinematic wave equation, must be integrated over the entire system of cells describing the basin. This is done first by computing the total volume stored in the soil, on the surface or in the channel network by adding up the single cell volumes as a function of the geo-morphology and topology of the catchment. Taking account of this aggregation and integrating the kinematic wave equation over the whole basin, under the following simplifying assumption, a zero-dimensional non-linear reservoir equation can be obtained for representing the basin as a whole. Assuming that the difference between the inflow to a cell and the time variation of the water elevation do not significantly vary in space, one can integrate equation (9) over the domain of one cell to give: xd
∂η q d = qu + r − dξ = qu + xq x ∂t xu
∫
(21)
ξ
where: is the inflow at the upstream end of the cell of co-ordinate xu ; qd is the outflow at the downstream end of the cell of co-ordinate xd ; qx is the specific increment of flow per unit width and unit length. Using the second of equations (7) the local water content can now be expressed as a function of q x qu
q η = d C
1α
q + qx x = u C
1α
(22)
which implies that the volume per unit width stored in a cell of longitudinal dimension x can be expressed as: V=
α +1 α +1 αC −1 α (qu + q x x ) α − (qu ) α (α + 1)q x
(23)
If one starts from the cells on the edge where qu is equal zero and assuming q x constant over the entire domain, it is not difficult to show (Todini, 1995; Ciarapica and Todini, 1998) that the following result holds for a contributing area of N cells: α
∂VT α + 1 α = Nxr − C VT ∂t αNx
with:
α +1 α +1 α α − 1 j j − N 1 N N = C i1 α C i =1
(24)
α
∑
( 25)
where: i j N Ci VT
is the index of a generic cell; is the of cells drained by the ith cell; is the total number of cells in the upstream contributing area; is the local conveyance for the ith cell and is the volume stored in the upstream contributing area .
The term 1 C incorporates in an aggregate way the topography and physical properties of the soil, via the terms Ci , and the topology of the links between the cells, via the α +1
summation of the terms of the type ( j N ) α . It represents a global conveyance coefficient for the entire contributing area.
The total volume stored and local water content for the system’s outlet cell are linked by the following relation: α +1 C ηN = αxN C N
1α
VT
(26)
It is possible to write an equation similar to equation (25) for any cell in the catchment; the integration of the equation (24) gives, at each time-step, the total volume stored in the whole system and, when this volume is known, it is possible to determine the local water content η stored in any cell in the basin together with the discharge draining from it. Equation (24) corresponds to a non-linear reservoir model and represents the lumped dynamics of the water stored in the soil. The same type of equation can be written for the overland flow and for the drainage network, thus transforming the distributed TOPKAPI model into a three non-linear reservoirs lumped model. This formulation enlightens the basic property of the TOPKAPI for which the shape of the governing non-linear reservoir equations is preserved across scales and the parameters of the lumped models can be directly obtained via a non-linear combination of the local parameters, that although taken as constant over the single cell, may vary extensively from one cell to another. This property qualifies the lumped TOPKAPI with respect to other models, such as ARNO and TOPMODEL, where the parameters are assumed constant in space while deriving the lumped catchment model with the consequence that the new aggregated parameters must now be estimated from the observations. Unfortunately, although this formulation can be written for the soil and for the overland and channel flows, it is not sufficient for separating, in lumped form, the infiltration from the overland runoff at the catchment scale. In order to obtain this separation, a relationship between the extent of saturated areas and the volume stored in the catchment, must be introduced, similarly to what is done in the Xinanjiang (Zhao,1977), in the Probability Distributed Soil Capacity model, Moore and Clarke (1981) and in the ARNO (Todini, 1996).
Nonetheless, given the availability of the distributed TOPKAPI version, this relationship can be obtained by means of extensive simulation. At each step in time the number of saturated cells is compared to the total volume of water stored in the soil over the entire catchment and the following two parameter Beta Distribution Function is adapted to the data: B r , s (ϕ ) =
Γ (r + s ) r −1 ϕ (1 − ϕ ) s −1 ; Γ (r )Γ (s )
ϕ ∈ [0,1]
(27)
x>0
(28)
with Γ (x ) the Gamma function defined as: +∞
Γ ( x ) = ξ x −1e −ξ dξ ;
∫ 0
instead of the classical power law used in the Xinanjiang, in the Probability Distributed Soil Capacity and in the ARNO models. The Beta Distribution Function is thus used in TOPKAPI to compute, in the lumped form, the overall inflow to the soil reservoir and the overall saturation excess which will be directly entered into the overland flow reservoir. 3.1. The effect of the drainage network In the aggregation model illustrated in the previous section it was implicitly assumed that the entire outflow from each cell flows into the respective downstream cell. However this representation, required in the first instance in order to understand how the aggregation is performed, is not entirely correct since account has to be taken of the depletion effected by the drainage network. The drainage network is schematised by means of rectangular section reaches located inside the cells subtending a drained area. Under these assumptions, the resulting aggregated structure actually used in the model is the following: α
∂VsT + 1 α 1 C V αs s = Nxr − s ∂t α s x N −1 N −1 T sT + 1 f ∏ m l∑ =1 m =l
(29)
N 1 = CsT i =1
∑
α s +1 α s +1 j −1 j −1 α s j −1 j −1 α s fm fm 1 + l =1 m = l l =1 m = l − N −1 N −1 N −1 N −1 + 1 1 + f f m m l =1 m = l l =1 m = l
∑∏
∑∏
η sN
∑∏
∑∏
C sT α +1 1 = s VsT αsx N −1 N −1 C s N f m + 1 l =1 m = l
∑∏
αs
1 αs Csi
(30)
1 αs
(31)
where j is the number of cells drained by the generic cell i. In equations (29), (30) and (31) the terms f m which appear below the multiplication symbols Π represent the fraction of the total outflow from the cell m which flows towards the downstream cell, while the fraction (1 − f m ) flows into the drainage network; obviously, f m = 1 in the cells where the drainage network is not present; these fractions are determined on the basis of the DEM, and identifying the two fractions proportionally to the average slope of the cell m towards the cell situated immediately downstream and towards the drainage network. In this way it is also possible to calculate the portion of the flow which arrives at the drainage network via interflow, namely: αs
N RsT = i =1
i −1
j −1
∑ ∑ ∏ f l =1
m =l
m
αs +1 C sT αs x
+ 1(1 − f m ) N −1 l =1
N −1
∑∏ m =l
f m + 1
α s +1
VsαT s
(32)
3.2 The overland and drainage network component. Similarly, by accounting for the effect of the drainage network, a non-linear storage model is also obtained which represents in a lumped form the overland flow:
αo
∂VoT ∂t N 1 = CoT i =1
∑
+ α 1 1 C V αo o − oT oT α o x N −1 N −1 + 1 f ∑ ∏ m l =1 m =l
= EoT
α o +1 α o +1 j −1 j −1 α o j −1 j −1 α o + 1 f f m m l =1 m = l l =1 m = l − N −1 N −1 N −1 N −1 fm +1 fm +1 1 l m l l m l = = = = 1
∑∏
∑∏
∑∏
∑∏
(33)
αo
Co1iα o
(34)
αo
N RoT = i =1
i −1 l =1
j −1
∑∑∏ m =l
αo +1 CoT αo x
f m + 1 (1 − f m ) Vα α s +1 oT N −1 N −1 f m + 1 l =1 m = l
(35)
∑∏
where: VoT is the total volume of the water stored on the surface,
EoT is the saturation excess feeding the surface component which
is estimated by means of the Beta function representing the extent of saturated area as a function of the total water volume stored in the soil; RoT is the contribution which arrives at the drainage network via
surface flow. The same derivation can be applied to the drainage network, bearing in mind that the width of the network reaches is now variable from one cell to another and that the outflow from each reach does not need to be subdivided into two direction flow; this gives: ∂V nT ∂t
= R nT
α n +1 − N α x W i n i =1
∑
αn
αn C nT V nT
(36)
N 1 = C nT i =1
∑
α n +1 αn
j
∑ w l
l =1 N
∑ l =1
wl
−
α n +1 αn
j −1
∑ w l
l =1 N
∑ l =1
w1i α n Cn1iα n
wl
αn
(37)
where VnT is the total water volume stored in the network, RnT = RsT + RoT
is the total contribution which arrives at the drainage network via subsurface and surface flow while
Rn N
αn +1 = N α x W i n i =1
∑
αn
αn αn C nT VnT = C n N Vn N
(38)
represents the total discharge from the closing section of the catchment. This approximation, which has proven to provide a successful reproduction of soil and surface runoff, has shown to be insufficient for correctly representing the network dynamics in lumped form. As a matter of fact, while the slow motion and the limited thickness of the water allows to consider the soil and the overland flow phenomena lumped at catchment scale mostly as the filling and emptying of a reservoir, the formation of flood waves in the channel network requires the additional notion of "delay", which is a characteristics of long reservoirs and cannot be resolved by the classical reservoir equation (Singh, 1988).
As it will be enlightened in the example, experimental results have shown that a proportionality do not exist between the total volume stored in the network VnT and the volume in the last cell VnN taken at the same time as equation (38) would suggest, but rather between delayed values, namely: Vn N (t ) ∝ VnT (t − τ )
(39)
where τ is a time shift that varies with the amount of the inflowing runoff, as a consequence of the TOPKAPI distributed model non-linear behaviour. Although further research is in progress in order to produce a more appropriate overall computational scheme for the network, on the lines of what proposed by Dooge for the uniform cascade of non-linear reservoirs (1967a,b), in the present TOPKAPI version, a constant time shift is applied in equation (37). The time shift is obtained by approximately reducing to a line the scatter plot of the two quantities of equation (39) obtained via simulation with the distributed version.
4. EXAMPLE OF APPLICATION Although TOPKAPI has already been applied to several catchments, for the sake of clarity, in the present paper only one application will be presented. The example described in this paper, the application of TOPKAPI to the Reno river, is mainly directed at clarifying the data and calibration requirements together the aggregation capabilities from the distributed to the lumped versions of TOPKAPI. Specific tests on the model were performed within the frame of project VAHMPIRE (1999) can be found in Ciarapica and Todini (1998). 4.1. The upper Reno catchment The Upper Reno basin closed at Casalecchio (Italy) has a surface area of 1051 km2, and comprises primarily clayey and marly soil, as well as alluvial deposits in its terminal section. In this
application, the basin is represented by a DEM based on a grid measuring 400 m per side, for a total of 6325 cells (Fig. 2).
0:200m 201:400m 401:600m 601:800m 801:1000m 1001:1200m 1201:1400m 1401:1600m 1601:1800m 1801:2000m
Fig. 2. The Digital Elevation Model (DEM) of the Upper Reno River
The drainage network was extracted automatically from the DEM by setting a threshold for the drained area at 1% of the total area (Fig. 3); the minimum and maximum width for the drainage network is set at 10 m and 100 m respectively with a variation related to the drainage area; hourly values for 24 raingauges and 10 thermometers were used in this case study, while discharges were computed from hourly levels by means of a well verified rating curve available at Casalecchio. The calibration of the model was performed using the hydrological data available for the year 1990 since in the calibration period, from November 24th to 26th, a relatively large flood event, with a flood peak discharge of 1317 m3/sec was present; all the 1991 available data were then used for verification.
Fig. 3. The drainage network extracted from the DEM
Class a
Soil Types Arenaceous turbidites, calcarenites
b
Arenaceous turbidites, marly limestone turbidites
c
Marly arenaceous turbidites
d
Marly arenaceous turbidites
e f
Marly limestone and calcarenite turbidites, clays and marly limestones Clays, sands and conglomerates
g
Alluvial deposits Table 1. Soil types in the Upper Reno river as in Fig. 4
An initial guess for the model parameters, was derived from the available broad description of soils types given in Fig. 4 and Tab. 1, by assuming parameter values taken from the literature. Adjustment of parameters was performed manually and, at the end, the values given in Tabs. 2 and 3 were retained.
a b c d e f g
Fig. 4. Soil types in the Upper Reno river
α ϑ s − ϑr k s (m/s) n o (m-1/3 s-1) L (m) a 0.18 2.5 2.50E-04 0.085 3.30 b 0.18 2.5 2.20E-04 0.085 3.20 c 0.18 2.5 1.50E-04 0.090 0.60 d 0.18 2.5 2.30E-04 0.085 3.00 e 0.15 2.5 9.00E-04 0.080 3.50 f 0.30 2.5 1.20E-03 0.080 5.50 g 0.32 2.5 1.40E-03 0.080 7.00 Table 2. Calibration parameters for the different soil classes Class
Order (Strahler) I II III IV
n n (m-1/3 s-1) 0.045 0.040 0.035 0.035
Table 3. Calibration parameters for the different network orders
1
Reno - BETA distribution
total vol./saturation vol.
0,9 0,8 0,7 0,6
experimental results (calibration)
0,5 0,4
experimental results (validation)
0,3 0,2
Beta (r=0,16 ; s=0,23)
0,1 0 0
0,2
0,4 0,6 0,8 fraction of total area at saturation
1
Fig. 5. The Beta Distribution Function fitted to the distributed TOPKAPI simulated values of water volume stored in the soil and area at saturation
Figure 5 shows the relation, obtained via simulation using of the total volume stored in the soil, between the areas at saturation and the total volume of water stored in the soil. It can be observed how the behaviour is preserved in the verification period. This relation is then fitted by a Beta Distribution Function with parameters r=0.16 and s=0.23, showing an interesting behaviour in the upper portion, namely the fact that, due to the extremely large storage in the alluvium, the conoid of the river where it drains in the water table, the curve reaches saturation only for extremely large stored water volumes. Figure 6 shows a comparison of the total runoff, namely the sum of the total soil drainage and the total overland flow, reaching the river network as computed by the distributed and by the lumped versions, during the largest flood period in the record (November 24th to 26th 1990). As one can see the two quantities are practically identical although the first one is computed by integrating a cascade of 6325 non linear reservoirs representing the soil and 6325 non linear reservoirs representing the overland flow, while the second one is computed by means of the Beta
Distribution Function given by equation (27) plus one reservoir representing the soil, as for equation (29), and one reservoir representing the overland flow, as for equation (33), with an enormous reduction in computer time consumption. 1800 1600
calculated runoff feeding the channel network (distributed)
3
runoff (m /sec)
1400 calculated runoff feeding the channel network (lumped)
1200 1000 800 600 400
17/12/90 0.00
12/12/90 0.00
time (hours)
07/12/90 0.00
02/12/90 0.00
27/11/90 0.00
0
22/11/90 0.00
200
Fig. 6. Comparison between the total runoff (Surface+Soil) entering the drainage network compurted using the distributed version (6325 cells) and the lumped version (1 cell).
5.0E+07
Total volume stored in network
4.5E+07 4.0E+07 3.5E+07 3.0E+07 2.5E+07 2.0E+07 1.5E+07 1.0E+07 5.0E+06 0.0E+00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
6.0E+05
7.0E+05
8.0E+05
9.0E+05
Volume stored in the outlet stretch
Fig. 7. Comparison of the total water volume stored in the network and the volume of the last stretch showing a marked loop effect.
Total volume stored in network
5.0E+07 4.5E+07 4.0E+07 3.5E+07 3.0E+07 2.5E+07 2.0E+07 1.5E+07 1.0E+07 5.0E+06 0.0E+00 0.0E+00 1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05 6.0E+05 7.0E+05 8.0E+05 9.0E+05 Three hours delayed volume stored in the outlet stretch
Fig. 8. Comparison of the total water volume stored in the network and the volume of the last stretch taken with a delay of 3 hours showing that most of the loop effect has almost completely disappeared. 1600 observed discharge 1400 calculated discharge (distributed)
3
discharge (m /sec)
1200
calculated discharge (lumped)
1000 800 600 400
17/12/90 0.00
12/12/90 0.00
time (hours)
07/12/90 0.00
02/12/90 0.00
27/11/90 0.00
0
22/11/90 0.00
200
Fig. 9. Comparison of observed discharges at Casalecchio and computed with the distributed (solid line) and lumped (dashed line) models
Similar behaviour was not observed for the river network, which lumped version given by equation (36) produced a flood wave generally anticipated with respect to the measured. Comparison of the volume stored in the last reach with the total volume stored in the network (Fig. 7), as produced by the distributed version, showed a marked looping behaviour. This loop almost disappeared when the volume of the lowermost cell was delayed by 3 hours (Fig. 8). Nevertheless from the same
figure it is quite evident that the delay is not constant: the small loops need larger delays to disappear, which is consistent with the non-linear behaviour of unsteady flow with travel times reducing with the dimension of the event. Figure 9 shows the final comparison of the observed discharges and the ones computed at Casalecchio using the distributed and the lumped TOPKAPI models when using equation (36) in which a delay of 3 hours is applied to the lowermost volume with respect to the total volume stored in the network.
5. CONCLUSIONS AND PERSPECTIVES From the tests performed, at the wider catchment scales, the soil horizontal flow appears to have a smaller quantitative influence on inflow to the network than it might be expected from the analyses performed at smaller scales; this suggests that the redistribution effect due to the interflow along the slopes, starting from areas at higher elevations, has less influence than the formation of saturated zones in the proximity of the drainage network. This justifies the assumption that, at scales larger than the hillslope scale, overland flow generated by saturation provides the largest contribution to the formation of floods, and that the adoption of a saturated areas vs stored volumes law as a representative function for the catchment dynamics is legitimate. Nevertheless, the role of interflow remains fundamental in governing the soil’s response in terms of storage capacity, and therefore in activating the overland flow formation mechanism. The encouraging results of the first development phase of the TOPKAPI model make a good case for seeking to refine the representation of the physical phenomena described. There are certainly many aspects to be improved, and it is expected that others will emerge during the model’s future development. However the possible steps required to obtain a more accurate overall representation can be delineated.
The lumped version of TOPKAPI is already usable for reproducing the overall runoff that reaches the river network generated by the distributed model, while the channel network luped equation requires the introduction of a variable time delay. The present version, which assumes constant time delay may be considered satisfactory, but further work is needed to fully account for the non-linearity present in the phenomenon. Percolation to deeper soil layers must be introduced; this was ignored in the initial stage since it was not important in the basins to which the model was originally applied. This objective may be pursued by the schematisation of a second soil layer with different characteristics from the upper layer, and involving water movement in a vertical direction feeding into the aquifer. The results obtained seem to confirm the objectives outlined at the onset of this study can be summarised in the following points: construction of a rainfall-runoff model based on a representation applicable to increasing spatial scales; development of meaningful parameterisation, which cuts calibration requirements to a minimum and allows a physical interpretation of the values assumed by the parameters and of their influence on the simulation of the real behaviour of a given system. Capitalising on the model and parameters physical interpretation, several interesting applications of the TOPKAPI model can be foreseen ranging from the analysis of land use and climate changes to the analysis of extreme values and the possibility of extending the results to ungauged catchments. Last but not least an appealing possibility exists in the future of deriving model representations from the available world 1x1 km2 carthography (such as for instance GOTOPO30 produced by the USGS) to be lumped at the 50x50 or 20x20 km2 meshes of the Mesoscale or Limited Area Meteorological Models in order to better reproduce the soil atmosphere exchanges.
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