Geoscience Reference
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gradient in the catchment stays constant). Lamb et al. (1998b) have shown how data on the
spatial distribution of water table depth in a catchment can be used to modify the index
distribution to reflect heterogeneity of effective transmissivity values.
Recent attempts to reformulate the index theory include the modified index of Saulnier
and Datin (2004), which avoids the assumption that a steady recharge rate and transmissivity
function also apply to water ponded on the surface by averaging the local soil moisture deficits
only over the non-saturated part of the catchment. Thus Equation (B6.1.5) becomes:
D i =− m t
a
T o tan ˇ
D
ln
(B6.1.23)
A A t i
where D is now the average deficit over the non-saturated area at time t ,
t =
1
A i
ln a
T o tan ˇ , A t is the contributing area at time t , A i is the area of a pixel (or group of similar
pixels) as before, and the summation is now taken only over unsaturated pixels.
Since, by definition, the local deficit on the saturated area is everywhere zero then:
A A t
A
D t
D t =
(B6.1.24)
and, combining Equations (B6.1.23) and (B6.1.24):
A
A A t
D D i =− m t
a
T o tan ˇ
ln
(B6.1.25)
with D i =
0 at the critical index value:
ln
A
A A t
D
m
a
T o tan ˇ
c = t +
(B6.1.26)
This can be rearranged to define a function G ( A t ) that can be calculated from the cumulative
distribution function of the soil-topographic index such that:
D
m
G ( A t )
=
(B6.1.27)
where
ln
c t A
a
T o tan ˇ
G ( A t )
=
(B6.1.28)
A A t
An example of the function G ( A c ) is shown as G ( ) in Figure B6.1.3. H ( ) is the original
topographic index function. This function is specific to a catchment but may be c alc ulated
during the topographic analysis prior to a simulation. At any time step, knowledge of D can be
used to calculate the functi on G ( A c ) and thence the contributing area and the average deficit
over the non-saturated area D . Figure B6.1.3 also shows the relative correction to the original
TOPMODEL function. Thus, this turns out to be quite a simple modification and results in
a more satisfactory representation of the time variation in the average (non-saturated) deficit
 
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