Environmental Engineering Reference
In-Depth Information
somewhat disappointing. The clear power-law distribu-
tion of avalanche sizes was lost, and if we recognized a
power law at all, it was only appropriate for landslides
instead of rockfalls.
What is the fundamental difference between landslides
and rockfalls? Clearly, effects of inertia will have a strong
effect on the event-size distribution as they facilitate
propagation of large avalanches. In case of slow landslides
they may be negligible, but not for rockfalls reaching high
velocities. Under this aspect, we should not even try to
apply such a simple model to phenomena such as debris
flows where effects of inertia are even stronger.
Furthermore, we always assess our models with the
help of real-world data, but can we be sure that the
statistics of landslide and rockfall sizes are in fact so
different? The largest event in the inventories shown in
Figure 16.2 involved a volume of about 10 6 m 3 , which
is not very large. It is in the lower range of the events
predicted by applying our sandpile model to the Alps.
In contrast, the largest rockfall recorded in the Alps, the
Flimser Bergsturz in the Rhine valley, involved a volume
of about 10 km 3 or 10 10 m 3 . So it was four orders of
magnitude larger than the biggest events found in the
inventories from Figure 16.2 and still two orders larger
than the biggest events in our simulation. Can the power
law from Figure 16.2, f ( V )
10 4
Threshold 1
Threshold 1.4
Threshold 2
Threshold
10 2
3
Threshold
5
10 0
10 2
10 4
10 6
10 4
10 5
Area [m 2 ]
10 6
10 2
Threshold
1
Threshold 1.4
Threshold 2
Threshold
10 0
3
Threshold
5
10 2
10 4
10 6
10 V 1 . 07 where V is taken
in m 3 and f ( V )inm 3 , persist up to this size? If so, the
number of events in each of the three inventories with
volumes between 1 km 3 and 10 km 3 can be obtained by
integrating the frequency density:
=
10 8
10 5
10 6
10 7
10 8
Volume [m 3 ]
Figure 16.9 Probability density of the avalanches obtained
from applying the sandpile model to the recent topography of
the Alps. The dashed lines show power laws with
10 10
β =
4.
N
=
f ( V ) dV
10 9
found that the event-size distribution is more robust
against changing the model rules than the properties
of the critical state itself are (Hergarten, 2002). Even if
landslides and rockfalls are close to SOC as well as our
sandpile model, this does not mean that their critical
states are the same. Self-organization may compensate
the differences between model and reality, so that the
event-size distribution finally fits. In contrast, we force
the model towards a 'wrong' state without allowing self-
organization and thus cannot expect a reasonable event-
size distribution.
Finally, did we find the simplicity behind the com-
plexity of gravity-driven mass movements? Starting from
rockfalls and the BTW model it looked very good. How-
ever, the model is far off from avalanche dynamics in
its spirit. Our attempts to make it more realistic were
10
0 . 07 [(10 10 ) 0 . 07
(10 9 ) 0 . 07 ]
=
5
(16.4)
Thus, each of the inventories should include at least five
events involving a volume of more than 1 km 3 ,whichis
obviously not the case. Thus, a power law with such a low
exponent cannot be valid anymore at large event sizes,
and there must be some kind of cutoff. So the difference
between rockfalls and landslides might be not as big as it
seemed first.
Thus, it appears that neither the statistical data nor
the recent knowledge on the processes behind mass
movements are sufficient to decide which models are
appropriate and will finally enable us to make reliable
predictions.
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