Geoscience Reference
In-Depth Information
are caused by either long slopes or upslope contour convergence, and low slope angles. Points with the
same value of the index are predicted as having the same hydrological responses. The topographic index
approach was developed into a complete rainfall-runoff model by Beven and Kirkby (1979) and has been
generalised since to allow for differences in soil characteristics within the catchment (see below and Box
6.1). The assumptions are similar to those used in the development of the “wetness” index developed
independently by O'Loughlin (1981, 1986) and used in the model of Moore
et al.
(1986). In passing, it
is worth mentioning that Horton (1936) had already used a very similar concept, though in his case he
considered only the maximum water table that might be supported by input at the infiltration capacity of
the soil (see Beven, 2004c, 2006d).
TOPMODEL in its original form takes advantage of the mathematical simplifications allowed by a
third assumption: that the distribution of downslope transmissivity with depth is an exponential function
of storage deficit or depth to the water table:
T
o
e
−
D/m
T
=
(6.2)
where
T
o
is the lateral (downslope) transmissivity when the soil is just saturated [L
2
/T],
D
is a local
storage deficit below saturation expressed as a depth of water [L] and
m
is a model parameter controlling
the rate of decline of transmissivity in the soil profile, also with dimensions of length [L]. A physical
interpretation of the decay parameter
m
is that it controls the effective depth or active storage of the
catchment soil profile. A larger value of
m
effectively increases the active storage of the soil profile. A
small value generates a shallow effective soil, but with a pronounced transmissivity decay.
Given this exponential transmissivity assumption, it can be shown that the appropriate index of
similarity is ln(
a/
tan
β
) or, if the value of
T
o
is a
llo
wed to vary in space, ln(
a/T
o
tan
β
) such that,
given a mean storage deficit over a catchment area
D
, a local deficit at any point can be calculated as
(see Box 6.1):
m
γ
−
ln(
a/T
o
tan
β
)
D
i
=
D
+
(6.3)
where
γ
is the mean value of the index over the catchment area. Thus, every point having the same
soil/topographic index value (
a/T
o
tan
β
) behaves functionally in an identical manner. The (
a/T
o
tan
β
)
variable is therefore an index of hydrological similarity. Other forms of transmissivity profile assumption
lead to different forms for the index and local deficit calculation (see Box 6.1). Of particular interest
are points in the catchment for which the local deficit is predicted as being zero at any time step. These
points, or fraction o
f t
he catchment, represent the saturated contributing area that expands and contracts
with the change in
D
as the catchment wets and dries (Figure 6.4). This simplest form of estimate for
the local deficit
D
i
also assumes that the theoretical negative deficits (steady-state oversaturation on the
contributing area) can be included in the calculation of the mean catchment deficit. Saulnier and Datin
(2004) show how this can be modified to allow for dynamic changes in the contributing area and account
only for the deficit on the non-saturated area. The equations can also be derived in terms of water table
depth rather than storage deficit but this introduces at least one additional effective storage parameter
(Beven
et al.
, 1995). In each case, there is a relationship between the transmissivity profile assumed and
the form of the recession curve at the catchment scale produced by soil drainage. For the exponential
transmissivity assumption, the derived recession curve function is given by:
Q
o
e
−
D/m
Q
b
=
(6.4)
where
Q
o
=
Ae
−
γ
for a catchment area of
A
. This equation (and equivalent forms for different trans-
missivity assumptions) is derived under the assumption that the effective hydraulic gradients for the
subsurface flow do not change with time, as would be predicted by a more complete analysis.
The calculation of the index for every point in the catchment requires knowledge of the local slope
angle, the area draining through that point and the transmissivity at saturation. The spatial distribution
