Landslides, Rockfalls and Sandpiles - Environmental Modelling: Finding Simplicity in Complexity - page 261

somewhere. The height will then be lower than originally,
so that the higher neighbours may become unstable, too.
In the BTW model, four grains topple if a site becomes
unstable due to adding one grain. The simplest way to
modify our simple sandpile model in this direction is
to assume a higher threshold. Let us assume that a site
becomes unstable if its height exceeds the height of any
neighbour by at least n where n

have a lower height afterwards, the lowest of these sites
is included into the relaxation, so that three sites are
brought to the same height by replacing their heights with
their mean height. This procedure is extended until there
is no lower neighbour any more.
Although anisotropy may be reduced by the eight-
neighbour topology and by distributing unstable neigh-
bours among several sites, any systematic relationship
between slope direction and lattice orientation should
still be avoided.We therefore change the quadratic region,
which will still generate a pyramid into a circular model
domain. This approach can easily be implemented by
keeping all sites outside a given circle at zero height.
Figure 16.6 shows the evolution of the topography in
this improved sandpile model. A circular cone evolves
instead of a pyramid, so that the slope is parallel to a
lattice axis only in parts of the region. Furthermore, the
surface is not smooth any more as a consequence of
the modified relaxation rule. Finally, many but not all
avalanches reach the boundary.
Figure 16.7 shows the resulting probability density of
the avalanches after the surface has reached a quasi-steady
state. As we already found that the number of relaxations
is not a goodmeasure of avalanche size, we nowdetermine
the displaced volume by adding the height changes of all
sites whose height was lowered during the avalanche.
In contrast to our original approach, this model yields
a considerable number of large events. Nevertheless, the
curves look strange at first sight and are still not very
close to a power law. The most striking property is a
bump in the distribution at volumes of about l 6 on grids
of diameter l . Since its position moves linearly with l ,
it presumably describes one-dimensional avalanches as
they may occur where the slope is parallel to a grid axis.
In the simulation of the largest grid ( l

n

2 grains
topple towards this neighbour then. The latter assumption
means that grains topple until the two sites involved have
arrived at the same height, which is essentially the same
as in our first model.
As the absolute height values are not important, we
can rescale the height by dividing all height values by the
parameter n . As a consequence, instability now occurs
if the slope (in this case the height difference) towards
any neighbour becomes greater than or equal to one.
Conversely, adding a grain increases the height only by an
amount

> 2, and that

1

100; larger
values mainly increase computing time without having a
significant effect on the results.
However, there is still room for improvement. A square
lattice with nearest neighbour connections (four neigh-
bours per site) is known to be highly anisotropic. If the
slope gradient is parallel to an axis of the lattice, displacing
grains will strongly increase the slope of the downward
neighbour (where the material moves to) and the upward
neighbour. In contrast, the two other neighbours will not
be affected strongly, so that straight, one-dimensional
avalanches will still be preferred. A hexagonal topology
where each site is connected to six direct neighbours
is less anisotropic. Alternatively, the square grid can be
improved by including diagonal connections, so that each
site has eight neighbours. As this topology is more con-
venient when digital elevation models are used instead
of artificial surfaces, we use this version in the following.
The distance towards nearest and diagonal neighbours is
different now, so we cannot use the height difference as
a criterion for instability, but have to divide the height
difference by the dist an ce between the sites considered,

n . In the following, we assume n

=

1024) we may
recognize a power law over a limited range of volumes
between 3

=

10 5 . However, its range of
validity is less than two orders of magnitude and thus
not comparable to that found for rockfalls in nature.
Furthermore, the power-law exponent

×

10 3 and 2 ×

β = 1 . 9isfar

which is either 1 or √ 2.
Next, why should all unstable grains move towards the
same site? If the slopes towards two or more neighbouring
sites are similar, moving grains should be distributed
among these sites, which can easily be implemented.
In a first step, the heights of the unstable site and its
lowest neighbour are replaced with their mean value.
This approach corresponds to the original rule where
so many grains are displaced that the two involved sites
arrive at the same height. If any remaining neighbours

from the value 07 of rockfalls. Thus, the improved

sandpile model strongly underestimates the number of
large rockfalls.
Conversely, the exponent

β = 1 .

9 perfectly matches
the observed distribution of landslide sizes if it is trans-
formed from area to volume. Furthermore, landslides
show power-law behaviour only over a small range of
event sizes. In our model, the power law holds over about
two orders of magnitude on volume, which should be
about three orders of magnitude in area. This is more

β = 1 .