Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 125

z

r

H = z − r

y
Figure 4.13. Schematic representation of a soil-mantled slope.

of mass-wasting events. We first consider the case in which regolith thickness H
does not affect the depth of soil removed by a landslide other than through the bound
h

≤

H . Thus state dependency in the random driver arises only from this bound, i.e.,
we assume that each landslide removes an exponentially distributed random amount
h of soil or the whole soil column H , whichever is less. The overall dynamics of soil
development can be expressed by ( 4.9 ) with g ( H )

1 and a truncated distribution
p ( h )of h ,asinFig. 4.14 .If H max is the maximum thickness of the regolith above
which the rates of soil production are negligible, we normalize H with respect to

=−

p h

0.8

1

α e h α

p h

0.6

e h α 0.4 P h h

0.2

h

h

Figure 4.14. Probability distribution p ( h )of h truncated at h ∗ ( H ). The distribution
is exponential for 0

=
h ∗ ( H ) equal to the probability that h exceeds h ∗ ( H ) in the nontruncated exponential
distribution (shaded area in this figure). p ( h ) is clearly state dependent.

< <

h ∗ ( H ) and has a probability mass (or “atom”) P ∗ at h h