Geoscience Reference
In-Depth Information
Box 6.1 The Theory Underlying TOPMODEL
B6.1.1 Fundamental Assumptions of TOPMODEL
The development of the TOPMODEL theory presented here is based on the three assumptions
outlined in the main text:
A1 There is a saturated zone that takes up a configuration as if it was in equilibrium with a
steady recharge rate over an upslope contributing area
a
equivalent to the local subsurface
discharge at that point.
A2 The water table is near to parallel to the surface such that the effective hydraulic gradient
is equal to the local surface slope, tan
ˇ
.
A3 The transmissivity profile may be described by an exponential function of storage deficit,
with a value of
T
o
when the soil is just saturated to the surface (zero deficit).
B6.1.2 Steady Flow in the Saturated Zone and the Topographic Index
Under these assumptions, at any point
i
on a hillslope the downslope saturated subsurface
flow rate,
q
i
, per unit contour length (m
2
/h) may be described by the equation:
q
i
=
T
o
tan
ˇ
exp(
−
D
i
/m
)
(B6.1.1)
where
D
i
is the local storage deficit,
m
is a parameter controlling the rate of decline of trans-
missivity with increasing storage deficit, and
T
o
and tan
ˇ
are local values at point
i
. Note that
tan
ˇ
is used to represent the hydraulic gradient because the slope is calculated on the basis
of elevation change per unit distance in plan (rather than along the hillslope).
Then, under the assumption that, at any time step, quasi-steady-state flow exists throughout
the soil, so that local subsurface discharge can be assumed equivalent to a spatially homoge-
neous recharge rate
r
(m/h) entering the water table, the subsurface downslope flow per unit
contour length
q
i
is given by:
q
i
=
ra
(B6.1.2)
where a is the area of the hillslope per unit contour length (m
2
) that drains through point
i
.
By combining (B6.1.1) and (B6.1.2) it is possible to derive a formula for any point relating
local water table depth to the topographic index ln (
a/
tan
ˇ
) at that point, the parameter
m
,
the local saturated transmissivity,
T
o
,
and the effective recharge rate,
r
:
D
i
=−
m
ln
ra
T
o
tan
ˇ
(B6.1.3)
Note that when the soil is saturated, the local deficit is zero; as the soil dries and the water
table falls, numerical values of st
or
age deficit get larger. An expression for the catchment
lumped, or mean, storage deficit (
D
) may be obtained by integrating (B6.1.3) over the entire
area of the catchment (
A
) that contributes to the water table. In what follows, we express this
areal averaging in terms of a summation over all points (or pixels) within the catchment:
A
i
A
i
1
ra
T
o
tan
ˇ
D
=
−
m
ln
(B6.1.4)
where
A
i
is the area associated with the
i
point (or group of points with the same characterisitcs).
In spatially integrating the whole catchment, it is also implicitly required that (B6.1.4) holds
even at locations where water is ponded on the surface (
D
i
<
0). This assumption can be
