Geoscience Reference
In-Depth Information
deviation of the elevations in a local area, such as a 3-by-3 array around the point of interest. Thus:
D
=
1
.
13589
+
0
.
08452 ln
σ
(6.5)
Then if the local steepest gradient
G
steepest
is calculated at the coarsest resolution, the equivalent value
at the target resolution can be defined by
αd
(1
−
D
)
G
steepest
=
(6.6)
where
d
is the distance between grid points (either in the principal or diagonal direction of the grid,
depending on the direction of greatest slope) and
α
can be determined at the coarse grid scale by inverting
this equation since
G
steepest
,
d
and
D
are known. Under unifractal scaling,
α
can be assumed constant
with change in scale, so that a scale's steepest slope at the target scale
g
steepest
can be calculated as
(1
−
D
)
g
steepest
=
αd
∗
(6.7)
where
d
is the equivalent distance between points in the same flow direction at the target grid scale.
Note that no additional points are being added here. It is assumed that data are only available at
the coarse resolution, but that this might not adequately represent the hillslope forms so that scaling
down to a finer target resolution might lead to better simulations. The importance of this scaling is
demonstrated by Pradhan
et al.
(2006). They show the dependence of the topographic index distribution
on grid resolutions from 50 m to 1000 m. They show how a good estimate of the distribution at the
50 m resolution can be obtained if only 1000 m data are available. They also show how, without this
adjustment, different parameter values are required in TOPMODEL for the different grid resolutions, but
if all the resolutions are scaled to a target resolution of 50 m and the calibrated parameter values for that
scale are used to make predictions, only slightly pooorer simulations are obtained than those calibrated
specifically for the 50 m resolution grid. Thus it seems, despite the gross simplifying assumptions that are
made, a lot of the scale dependence effect on parameter values can be eliminated. It is also worth noting,
however, that this analysis has been somewhat overtaken by events. As seen in Chapter 3, satellite-derived
topographic data at 30 m resolution are now available for almost the whole globe in the ASTER global
digital elevation model.
It is worth noting that a parameterisation of the (
a/
tan
β
) distribution may sometimes be useful.
Sivapalan
et al.
(1990) introduced the use of a gamma distribution in their scaled version of TOPMODEL.
Wolock (1993) also gives details of a gamma distribution version for continuous simulation. An analogy
with the statistical PDM model of Section 6.2 becomes apparent in this form. The advantage of using
an analysis of topography to define the index distribution beforehand is that there are then no additional
parameters to estimate. This is only an advantage, however, where the topographic analysis allows
a realistic representation of the similarity of hydrological responses in the catchment, which clearly
depends on the validity of the simplifying assumptions that underlie the index.
∗
6.3.3 Applications of TOPMODEL
Simulation of humid catchment responses
TOPMODEL was originally developed to simulate small catchments in the UK (Beven and Kirkby, 1979;
Beven
et al.
, 1984). These studies showed that it was possible to get reasonable results with a minimum
of calibration of parameter values. A summary table of applications is given by Beven (1997). More
recent applications include Franks
et al.
(1998) and Saulnier
et al.
(1998) in France; Lamb
et al.
(1998a,
1998b) in Norway (see Section 6.4); Quinn
et al.
(1998) in Alaska; Cameron
et al.
(1999) in Wales;
Dietterick
et al.
(1999) in the USA; Donnelly-Makowecki and Moore (1999) in Canada; and Guntner
et al.
(1999) in Germany (Figure 6.6). In most of these cases, it has been found that, after calibration
