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tial variations in rainfall input, there are no vari-
lief in the watershed, which is very small. Figure
ations in soil moisture according to this model.
8.18a shows a grayscale image of the soil mois-
As pointed out by Rodriguez-Iturbe et al. (1995),
ture at Washita on June 17, 1992. White spaces
the variability at the small scale of the Washita
indicate areas where the watershed is interrupted
watershed is not likely to be the result of spatial
by roads or lakes. Figure 8.18b is a synthetic
variations in rainfall.
two-dimensional fractional Gaussian noise with
β = 1.8 constructed using the Fourier-¬ltering An alternative approach to this problem
which will generate spatial variations in soil
method. The mean and variance of the synthetic
moisture at these small scales assumes that the
moisture ¬eld have been chosen to match those
evapotranspiration rate is a random function in
of the observed image. The synthetic image re-
space and time. This models spatial and temporal
produces the correlated structure of the real soil
variations in evapotranspiration resulting from
moisture image.
variable atmospheric conditions and heterogene-
Entekhabi and Rodriguez-Iturbe (1994) pro-
ity in soil, topography, and vegetation character-
posed a partial differential equation for the dy-
istics. The resulting equation is
namics of the soil moisture ¬eld s (x, t):

‚s ‚s
= D ∇ 2 s ’ ·s + ξ (x, t) = D ∇ 2 s ’ ·(x, y, t)s
(8.32) (8.33)
‚t ‚t
8.8 THE FREQUENCY-SIZE DISTRIBUTION OF LANDSLIDES 207


same distribution that was observed in real soil
moisture ¬elds is obtained.
Landslides occur when the shear stress ex-
102 ceeds a threshold value given approximately by
N ∝ A’0.8 the Mohr--Coulomb failure criterion (Terzaghi,
N (>A)
1962):
„ f = „0 + (σ ’ u) tan φ (8.35)
101
where „0 is the cohesive strength of the soil, σ
is the normal stress on the slip plane, u is the
pore pressure, and φ is the angle of internal
100
100 102
101 103 friction. Landslides are initiated in places where
A (km2)
„ f is greater than a threshold value. The move-
ment of the soil at the point of instability in-
Fig 8.19 Cumulative frequency-size distribution of patches
creases the shear stress in adjacent points on
of soil moisture larger than a threshold value for the synthetic
soil moisture ¬eld with power spectrum S(k) ∝ k ’1.8 . the hillslope causing failure of a connected do-
Modi¬ed from Pelletier et al. (1997). Reproduced with main with shear stress larger than the thresh-
permission of Elsevier Limited.
old value. To model the landslide instability and,
in particular, the frequency-size distribution of
landslides, it is therefore necessary to model
This equation is a variant of the Kardar--Parisi--
the spatial variations of „0 , σ , u, and φ. The
Zhang equation (Kardar et al., 1986), which has
variables „0 and u are primarily dependent on
received a great deal of attention in the physics
soil moisture for a homogeneous lithology and
literature. This equation is nonlinear and can
a slip plane of constant depth. The dependence
only be solved numerically.
of each of these variables on soil moisture has
Amar and Family (1989) have solved this equa-
been approximated using power-law functions
tion and have found that the solutions have tran-
(Johnson, 1984). The shear stress and normal
sects with power spectra that have a power-law
stress are linearly proportional to soil moisture
dependence on wave number with exponent β =
through the weight of water in the soil. The
1.8. The similarity between the ¬eld generated by
shear stress and normal stress are also trigono-
Eq. (8.33) and the soil moisture observed in the
metric functions of the local slope. Based on the
Washita images suggests that Eq. (8.33) may be
results given earlier, the soil moisture will be
capturing much of the essential dynamics of soil
modeled as a two-dimensional fractional Brow-
moisture at these scales.
nian walk with β = 1.8. Power spectral analyses
Rodriguez-Iturbe et al. (1995) quanti¬ed the
of one-dimensional transects of topography have
scale-invariance of soil moisture variations by
shown topography to have a power-law power
computing the cumulative frequency-size distri-
spectrum with β = 2 (e.g. Huang and Turcotte,
bution of patches of soil with moisture levels
1989). This behavior is applicable only over a cer-
higher than a prescribed value. They found that
tain range of scales, however. At scales smaller
the cumulative frequency-size distribution had a
than the inverse of the drainage density, the to-
power-law function on area with an exponent of
pography is controlled by hillslope processes and
approximately ’0.8. The number of soil patches
does not exhibit fractal behavior. To represent the
N depended on the area according to
small-scale smoothness of hillslope topography
N (> A) ∝ A ’0.8 as well as the fractal behavior at larger scales,
(8.34)
we can ¬rst construct a two-dimensional surface
with one-dimensional power spectrum S(k) ∝ k ’2
This is a power-law or fractal relation. We have
determined the equivalent distribution for the with the Fourier-¬ltering method. At small scales,
synthetic soil moisture ¬eld illustrated in Fig- however, the synthetic topography is linearly in-
ure 8.18b. The result is plotted in Figure 8.19. The terpolated. A shaded-relief example of the model
208 STOCHASTIC PROCESSES


shows that there are no contours smaller than
(a)
8 — 8 pixels.
The shear stress necessary for failure is a com-
plex function of soil moisture and slope. How-
ever, to show how landslide areas may be asso-
ciated with areas of simultaneously high levels
of soil moisture and steep slopes, we assume
a threshold shear stress criterion proportional
to the product of the soil moisture and the
slope. In addition, we assume that slope and
soil moisture are uncorrelated. A grid of syn-
thetic soil moisture and topography of 512 —
(b) 512 grid points was constructed according to the
models described above. The domains where the
product of the soil moisture and the topogra-
phy were above a threshold value are shown in
Figure 8.21a. The threshold value was chosen
such that only a small fraction of the region
was above the threshold. Figure 8.21b shows the
cumulative frequency-size distribution of the re-
gions above threshold, our model landslides. It
can be seen that at large areas a power-law dis-
tribution with an exponent of ’1.6 exists. Actual
Fig 8.20 (a) Synthetic model of topography used in the
landslide distributions, discussed below, show a
stochastic landslide model. The topography is a scale-invariant
similar trend. Distributions obtained with differ-
function with S(k) ∝ k ’2 above a scale of eight lattice sites
ent realizations of the model yielded power-law
and is planarly interpolated below that scale. (b) Slope model
exponents of 1.6 ± 0.1 when ¬t to the landslides
corresponding to the topographic model of (a). The slopes
above A = 10. This form was independent of the
are white noise above a scale of 8 lattice points and constant
threshold value chosen for landslide initiation, as
below that scale. Modi¬ed from Pelletier et al. (1997).
Reproduced with permission of Elsevier Limited. long as the threshold was chosen such that only
a small fraction of the lattice is above threshold.
The exponent of this distribution is more neg-
ative than that of the soil moisture patches of
topography is illustrated in Figure 8.20a along Figure 8.19. This results from the less correlated
with its contour plot. The plot has 128 — 128 slope ¬eld ˜˜breaking up™™ some of the large soil
grid points with interpolation below a scale of 8 moisture patches. The effect of smooth topogra-
pixels. The result of the interpolation is clearly phy at the hillslope scales is to create a rolloff
identi¬ed as piecewise linear segments in the in the frequency-area distribution for small land-
transects along the boundaries of the plot. The slide areas. This is a consequence of the fact that
slope corresponding to this model of topography fractal scaling breaks down at hillslope scales,
is illustrated in Figure 8.20b. Below a scale of 8 and hence slopes tend to fail as a unit rather
pixels, the slopes are constant. Above this scale, than triggering many small, isolated landslides
the slopes are a two-dimensional Gaussian white on the same slope. The effect of strong ground
noise. This follows from the fact that our model motion from earthquakes is to lower the shear
for topography at large scales, a Brownian walk stress necessary for failure. This does not alter
with S(k) ∝ k ’2 , can be de¬ned as the summation the frequency-size distribution of landslides ac-
of a Gaussian white noise time series. White noise cording to this model since the form of the
means that adjacent values are totally uncorre- distribution is independent of the value of the
lated. The contour map of the slope function threshold.
8.8 THE FREQUENCY-SIZE DISTRIBUTION OF LANDSLIDES 209




N ∝ A’1.6
102
N (>A)


101


(a) (b)

100
100 101 102
A (km2)
were triggered by heavy rainfall. Large landslides
Fig 8.21 (a) Contour map of the product of the synthetic
in the two areas match power-law distributions
soil moisture ¬eld with the synthetic slope function. Areas
from Eq. (8.36) with exponents ’1.6 and ’2. Fig-
inside the contour loops represent model landslides. (b)
Cumulative frequency-size distribution of model landslides. ure 8.22c presents the cumulative frequency-area
The distribution compares favorably to the distributions of distribution of more than 11 000 landslides trig-
real landslides. Modi¬ed from Pelletier et al. (1997). gered by the 1994 Northridge earthquake over an
Reproduced with permission of Elsevier Limited.
area of 10 000 km2 (Harp and Jibson, 1995). Work-
ing with high-resolution aerial photography ac-
quired the day after the earthquake, Harp and
Do actual landslide distributions produce a
Jibson estimate that their catalog of landslides
similar distribution? Whitehouse and Grif¬ths
is complete for landslides larger that 25 m2 . At
(1983), Ohmori and Hirano (1988), and Hovius
smaller sizes, a signi¬cant number of landslides
et al. (1997), among others, have all presented
may have been missed. Large landslides in this
evidence that landslide frequency-size distribu-
dataset have a power-law dependence on area
tions are power-law functions of area for large
with an exponent of approximately ’1.6. As in
landslide areas. Figure 8.22 presents cumula-
the Japan and Bolivia datasets there is a roll
tive frequency-size distributions of landslide area
off in the power-law distribution for small areas.
from three areas ¬rst analyzed in Pelletier et al.
These results suggest that cumulative frequency-
(1997). In Figure 8.22a, the number of landslides
size distributions are remarkably similar despite
with an area greater than A are plotted as a func-
different triggering mechanisms, and that they
tion of A for seven lithologic units from a dataset
match the predictions predicted by a stochastic
of 3 424 landslides with areas larger than 104 m2
model for landslides triggered by soil moisture
in the Akaishi Ranges, central Japan (Ohmori and
and topography.
Sugai, 1995). For large landslides, the distribution
Some studies have suggested that rolloffs at
is well characterized by a power law with an ex-
ponent of approximately ’2, that is the small end of landslide distributions are the
result of an incomplete catalog (Stark and Hov-
N (> A) ∝ A ’2 ius, 2001). While it is certainly true that power-
(8.36)
law relationships describe only the upper tail of
the landslide size distribution, it is not true that
In this region landslides occur as a result of both
breaks in scaling at the lower end of the dis-
heavy rainfall and strong seismicity. Figure 8.22b
tribution are primarily due to incomplete sam-
presents two sets of landslide areas mapped in
pling. Consider the Northridge dataset shown in
adjacent watersheds in the Yungas region of the
Figure 8.22c. If power-law scaling held down to
Eastern Cordillera, Bolivia. All of these landslides
210 STOCHASTIC PROCESSES



103
(a) 103 (b)
N∝A
N (>A)
N (>A) N∝A
102
N∝A 102


101
101


100 100
100
10
10 10
10
10 10
A (km2) A (km2)
104
(c)

N∝A
103
N (>A)
102


101

100
10 10
10 10
10
A (km2)

landslides clearly shows that there is some phys-
Fig 8.22 (a) Cumulative frequency-size distribution of
ical mechanism for limiting these small land-
landslides in six lithologic zones in Japan. The distributions are
well characterized by N(> A ) ∝ A ’2 above approximately slides. In the model of this section, the break
0.1 km2 . (b) Cumulative frequency-size distribution of in topographic scaling at the hillslope scale is
landslides in two areas in Bolivia. (c) Cumulative frequency-
the cause of the rolloff in the landslide size dis-
size distribution of landslides triggered by the 1994
tribution. The planarity of slopes at small scales
Northridge, California earthquake. This dataset is
means that when slides are triggered, large sec-
characterized by N(> A ) ∝ A ’1.6 above approximately
tions of the slope tend to fail, with fewer isolated
0.01 km2 .
sections of slope failure relative to a power-law
distribution.
landslide areas as small as 10’4 km2 , for exam-
ple, the power-law extrapolation in Figure 8.22c
would predict approximately 1 million landslides 8.9 Coherence resonance and the
larger than 10’4 km2 . Working with detailed
timing of ice ages
aerial photographs, Harp and Jibson found that
they could resolve landslides as small as 25 m2 .
Therefore, if the break in power-law scaling is The behavior of Earth™s climate system in the last
due to undersampling, this implies that Harp and 2 Myr is a consequence of both deterministic and
Jibson missed approximately 990 000 landslides stochastic forces. Signi¬cant periodicities exist in
at least four times larger than the minimum area time series of Late-Pleistocene global climate (i.e.
they could resolve in their images. The fact that proxies for temperature and ice volume) near
so few small landslides occur in the Northridge 29 and 41 kyr. These periodicities are controlled
case relative to the power-law trend for larger by the precession and obliquity of Earth™s orbit.
8.9 COHERENCE RESONANCE AND THE TIMING OF ICE AGES 211


There is little agreement, however, on which pro- can make a randomly kicked ball move back and
cesses and feedbacks control other characteristics forth periodically between the two hills if a de-
of Late-Pleistocene climate, including the 100-kyr layed feedback is introduced into the system. In
cycle and the asymmetry of glacial-interglacial this climate system, the growth and decay of ice
transitions. In this section we explore a simple sheets and the motion of the lithosphere beneath
model of Earth™s climate system in the Pleis- them provide that delayed feedback.
tocene as an example of how stochastic processes There are two time scales in this model sys-
can amplify nonlinear system behavior. Although tem: the delay time and the average time that
not strictly limited to surface processes, the cli- the ball spends in each valley before switching
mate system over these time scales is strongly over to the other valley. This latter time scale
controlled by both ice sheets and the lithospheric is controlled by the height of the hill and the
response to ice-sheet loading. As such, the cli- magnitude of the random variability. If these two
mate system involves key aspects of surface pro- time scales are close in value, a resonance occurs
cesses we have discussed in earlier chapters. in which the random switching between valleys
Many models have been introduced over the is made more regular by the delayed feedback,
past few decades that reproduce one or more of which encourages a repetition in the history of
the key features of Earth™s climate in the Pleis- the system. For example, if a series of random
tocene. Several of the most successful models kicks move the ball over the hill, the delay feed-
have focused on nonlinear feedbacks between back will act to push the ball up over the hill in
continental ice-sheet growth, radiation balance, the same manner at a later time by providing sys-
and lithospheric de¬‚ection (e.g. Oerlemans, tematic kicks in the uphill direction. This behav-
1980b; Pollard, 1983). In this section we will ex- ior is strongly self-reinforcing: the more regularly
plore a simple model of Late-Pleistocene climate a ball makes its way up over the hill, the more
that reproduces the dominant 100-kyr oscillation regularly will the delay feedback work to system-
of ice ages. The mechanism for producing this atically push it up the hill at the later time.
oscillation process is ˜˜coherence resonance.™™ Co- The standard equation that displays coher-
herence resonance has been investigated by sta- ence resonance is given by
tistical physicists for several years (e.g. Masoller,
Tn
= T n ’ T n3 + T n’„ + D ·n
2002). The necessary components of a coherently- (8.37)
t
resonating system are (1) a system with two stable
states, (2) a delayed feedback, and (3) random vari- where T is the system variable (e.g. elevation in
ability (Tsimring and Pikovsky, 2001). A number the case of a rolling ball, or temperature in the
climate system), n is the time step, „ is the delay
of physical, chemical, and biological systems have
time, and · is a white noise with standard de-
these components and exhibit coherence reso-
nance. A simple example of a system with two viation D . The ¬rst two terms on the right side
stable states is a ball that rolls back and forth be- of Eq. (8.37) represent bistability in the system.
tween two valleys when pushed over a hill. Ran- The delay feedback and noise are represented by
dom noise may be introduced into this system to the third and fourth terms, respectively. Systems
produce random ˜˜kicks™™ that push the ball over with both positive and negative feedback may oc-
the hill and into the other valley if enough kicks cur depending on whether is positive or nega-
work in the same direction. If the kicks are ran- tive. Coherence resonance occurs in both cases,
dom, they generally tend to cancel each other out but with a different periodicity.
to keep the ball trapped in one valley. From time An example of model output from Eq. (8.37) is
to time, however, a series of kicks in the same di- plotted in Figure 8.23a, with the corresponding
rection will occur that move the ball over the hill. power spectrum plotted in Figure 8.23b. Model
parameters for this example are „ = 600, =
Delayed feedback occurs if an additional force is
’0.1, and D = 0.1. The power spectrum in Fig-
exerted on the ball that is not random, but de-
pends on the elevation of the ball at a previous ure 8.23b was computed by averaging the spec-
time. The basic idea is that coherence resonance tra of 1000 independent model time series. The
212 STOCHASTIC PROCESSES


spectrum in Figure 8.23b is constant for low fre-
(a) quencies and proportional to f ’2 above a thresh-
1.0 old frequency. This is called a Lorentzian spec-
trum and is a common feature of stochastic
T
models with a restoring or negative feedback. A
Lorentzian spectral signature was documented in
0.0
Late-Pleistocene paleoclimatic time-series data by
Komintz and Pisias (1979), suggesting a stochastic
element to the climate system limited by a neg-
ative feedback mechanism at long time scales.
This observation was the basis for many stochas-
1000 2000
0 tic climate models including Hasselman (1971),
t
North et al. (1981), Nicolis and Nicolis (1982), and
(b) Pelletier (1997), among others.
10
The climate model we consider here is a single
equation for global temperature analogous to Eq.

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