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content ( α
= 3 + 2 λ in Equation (B5.4.2), see Box 5.4) and the total profile moisture content is η
=
θ r ) and the coefficient
( θ s
LK s tan β
( θ s
C =
(6.9)
θ r ) α L α
At a point, the continuity equation may then be written as
∂η
∂t
∂q
∂x +
C ∂η
=−
p
=−
∂x +
p
(6.10)
where x is plan distance and p is the rainfall rate. This is a kinematic wave equation for the change in
total profile soil moisture with time. The solution to Equation (6.10) is greatly simplified if it assumed
that the rate of change of η is everywhere constant in space. This allows an integration to the basin scale
to derive an expression for the rate of change of the total storage volume with time as
C α
α
dV
dt =−
+ 1
αNx
V α
+
Nxp
(6.11)
where
N
k N
α
α
+
1
α
+
1
N
α
α
1
C
=
(6.12)
C 1
i
i
=
1
where the summation is taken over all the N pixels in the catchment, k represents the total number of pixels
contributing to point i and V is the total storage in the catchment. In the earliest versions of TOPKAPI
(Ciarapica, 1998; Ciarapica and Todini, 2002), the equations were solved numerically but later Liu and
Todini (2002) developed analytical solutions for the storage at successive times. The theory, then, allows
the calculation of local storage given the total storage on the catchment and local values of the 1 / C index
in a similar way to the original TOPMODEL formulation. Ciarapica (1998) has applied the TOPKAPI
model to the Montano del Reno basin in Italy and the Can Vila basin in Spain with comparisons to
the ARNO and SHE models. Liu and Todini (2002) report on an application to the large Arno basin
at over 8000 km 2 , using grid elements of 1 km 2 . They suggest that the TOPKAPI theory ensures that
parameters retain their physical significance, even when used at larger scales. Liu et al. (2005) have
used TOPKAPI for flood forecasting at even larger scales, on the 10 000 km 2 Upper Xixian catchment
in China. These newer versions use integrations of the equations over the grid square and the addition of
layers in the vertical to allow for interception and deeper groundwater fluxes. Sinclair and Pegram (2010)
have used TOPKAPI to estimate soil moisture for 1 km scale cells across South Africa, for comparison
with satellite-derived estimates.
6.6 Semi-Distributed Hydrological Response Unit (HRU) Models
As noted in the introduction to this chapter, one method of relating hydrological responses to charac-
teristics of the landscape has arisen naturally out of the use of GIS in hydrological modelling. A GIS
is commonly used to store data derived from soil maps, geological maps, a digital elevation map and
a vegetation classification. They can also be used to organise remote sensing and other types of image.
These different maps do not generally provide information of direct use in hydrological modelling but
they certainly provide information that is relevant to hydrological modelling. By overlaying the different
types of information, a classification of different elements of the landscape into hydrological response
 
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