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.