Landslides, Rockfalls and Sandpiles - Environmental Modelling: Finding Simplicity in Complexity - page 259

approaches a quasi-steady state with a mean occupation
of about 2.1 grains per site. This state has critical
properties in the sense of an infinite correlation length,
which means that any additional grain may finally
destabilize any site of the grid, no matter how far away
it is located. Quantitatively, this property is reflected by
a power-law distribution of the avalanche sizes.
Soon after SOC was discovered, it was recognized
in some other computer models, too. The earliest
and still most important models in the context are
the Drossel-Schwabl forest fire model (Drossel and
Schwabl, 1992) and the Olami-Feder-Christensen
earthquake model (Olami et al ., 1992). Although both
models are very simple - and perhaps not self-organized
critical in the strict sense - they both turned out to be
very influential. The Olami-Feder-Christensen model
explains not only the power-law distribution manifested
in the Gutenberg-Richter law, but also more complex
features such as foreshocks and aftershocks (Hergarten
and Neugebauer, 2002; Helmstetter et al ., 2004).
Recently, a significant statistical difference between
lightning-induced and anthropic forest fires could be
explained by a modification of the Drossel-Schwabl
model (Krenn and Hergarten, 2009).
If SOC is the unifying theoretical concept behind the
power-law distributions in several natural hazards, there
should be appropriate models for landslides and rockfalls,
too. Let us first revisit the BTWmodel. Figure 16.3 shows
the avalanche sizes obtained from simulating the BTW
model on grids of different sizes. Here, avalanche sizes are
measured in terms of the number of relaxations where
each site that relaxes more than once is counted multiply.

Analyzing cluster sizes - counting each relaxed site only
once - leads to essentially the same result with a slightly
different power-law exponent.
For large grid sizes, the probability density of the
avalanche sizes follows a power law with an exponent
β =

15. This result almost exactly matches the observed
exponent of rockfalls β = 1 . 07. As the BTW model is
often entitled the sandpile model, are rockfalls nothing
but sandpile avalanches?
Although this concept seems to be perfect at first
sight, one problem arises. If we recall the model rules
of the BTW model, we immediately recognize that its
relationship to sandpile dynamics is rather vague. The
stability of a sandpile mainly depends on the local slope
gradient, but not on the absolute number of grains at
any location as assumed in the BTW model. To get
around this fundamental problem, one may be tempted
to skip the idea that the variable in the BTW model
represents a number of grains, but interpret it as an
abstract property that is somehow related to the slope of
a sandpile. However, the attempt to relate this variable
to slopes quantitatively failed (Hergarten, 2002, 2003). So
the BTW model provides a fundamental description of
avalanche propagation on a rather abstract level, but a
physically consistent relation to sandpile dynamics or any
type of gravity-driven mass movements is not apparent.
However, the simplest apparently realistic sandpile
model looks somewhat similar to the BTW model. Let
us still assume that grains are randomly distributed on a
square lattice, but assume that a site becomes unstable if
its height exceeds the height of any neighbour site by two
or more grains. In this case, one grain topples from the
unstable site to its lowest neighbouring site. In principle,
the threshold value is arbitrary, only a value of one grain
makes no sense as no topography could be formed then.
Obviously, this criterion of instability comes much closer
to the slope of the sandpile than that of the BTWmodel.
Figure 16.4 shows the evolution of a sandpile in this
simple model, starting from a flat surface. The resulting
probability densities of the avalanche sizes after the final
state has been reached are displayed in Figure 16.5. By
analogy with the BTW model, avalanche sizes are mea-
sured in terms of the number of relaxations. While the
shape of the sandpile looks reasonable, the event-size
distributions are far away from a power law, and there are
no large avalanches at all. The model is self-organizing,
but obviously not critical.
What is the reason for this disappointing result? We
first recognize in Figure 16.4 that the avalanches are,
in principle, one dimensional. Taking a closer look at

1 .

10 8

10 6

256 x 256

10 4

1024 x 1024

10 2

64 x 64

10 0

10 − 2

10 − 4

10 0 10 2 10 4

Event size

10 6 10 8

Figure 16.3 Probability density of the avalanches in the BTW
model on grids of different sizes.