although it does not represent transient effects

Lake is located at the western edge of the Mo-

of moisture in the days to weeks following rare

jave National Preserve just south of Baker, Cal-

precipitation events.

ifornia. CLIM-MET data collected from 1999 to

Wind speeds measured 2 m above the ground

present by a team from the US Geological Sur-

were converted to friction velocities using Eq.

vey are available for these areas. The water table

(8.23) with z0 = 0.005 m. This value was obtained

beneath Soda Lake is at or near the ground sur-

from Marticorena et al.™s (1997) relationship z0 =

face along the western edge of the playa, deepen-

0.005h c , together with a canopy height of h c =

ing to the east to 20 m and lower beneath the

1 m. Friction velocities inferred from the mea-

alluvial fans on the eastern side of the basin.

sured wind speed data were ¬t to the two-

Water-table depths indicated in Figure 8.13 were

parameter Weibull distribution. The best-¬t pa-

obtained from the US Geological Survey National

rameters obtained for the Balch and Crucero sta-

Groundwater Database.

tions are β = 0.398 and 0.405, and γ = 2.17 and

Figure 8.14 illustrates the CLIM-MET station

2.15, respectively. Based on these numbers, we

data for hourly rainfall, soil moisture, peak wind

chose β = 0.40 and γ = 2.15 as representative val-

speed, and the number of particles in saltation

ues for the meteorological conditions at Soda

measured at the Balch and Crucero stations near

Lake. The dry threshold friction velocities var-

the southern playa margin. Figures 8.15b and

ied from 0.5 to 0.8 m/s at the three CLIM-MET

8.15e illustrate that the soil moisture at both

202 STOCHASTIC PROCESSES

(a) (e)

(f)

q

q

(b)

(g)

(c)

(h)

(d)

(— 104)

(— 104)

tively, for a range of water table depths up to

Fig 8.14 Hourly CLIM-MET data from (a)“(d) Crucero and

10 m. Figure 8.15c indicates that saltation (and

(e)“(h) Balch stations. (a) and (e) Rainfall, (b) and (f)

volumetric soil moisture, (c) and (g) peak wind speed at 2 m hence dust emission) is essentially absent for wa-

above ground, (d) and (h) number of particles in saltation, as ter tables within 4 m of the surface for this set of

measured by SENSIT instruments. Modi¬ed from Pelletier

parameters. As the water table depth increases,

(2006). Reproduced with permission of Elsevier Limited.

saltation ¬‚ux also increases, asymptotically ap-

proaching a maximum value of nearly 6 g/m2 /s.

stations. We chose a value of u—td = 0.5 for our For water tables deeper that 10 m, the surface is

dry enough that soil moisture plays a relatively

calculations, but clearly a spatially distributed

minor role. The shape of the saltation curve de-

model that resolves variations in threshold fric-

pends primarily on the value of the wilting point

tion velocity would be the most accurate ap-

moisture θw . Figure 8.15d, for example, illustrates

proach for characterizing the playa basin as a

¬‚ux curves for representative end-member values

whole.

of 0.2 and 0.3. A larger wilting-point moisture

Soil-hydrologic parameters were estimated

means a smaller role for soil moisture in sup-

from values given in Gardner and Fireman (1958)

pressing emissions. The θw = 0.3 case, therefore,

and Rawls et al. (1992). Gardner and Fireman

has saltation initiated at a shallower water ta-

(1958) studied soils with ¬ne sandy loam and

ble depth (2 m instead of 4 m), and rises more

clay textures, and obtained b values of 0.04 and

0.055 m2 , respectively. Rawls et al. (1992) provided rapidly to its asymptotic value as a function of

estimates of ± and » appropriate for ¬ne-grained water table depth. In contrast, saltation curves

are relatively insensitive to the values of b, ± and

soils. Representative values of ± = 0.3 m’1 , » =

». Varying each of these parameters by 10%, for

0.2, and θw = 0.2 were chosen.

example, results in ¬‚ux changes of only a few

Figures 8.15a, 8.15b, and 8.15c illustrate the

percent for any particular water table depth.

solution to Eqs. (8.19), (8.26), and (8.25), respec-

8.7 ESTIMATING TOTAL FLUX BASED ON A STATISTICAL DISTRIBUTION OF EVENTS 203

104 10

(a)

(a) <q>

(g/m2/s)

102

yz=d 5

u*td = 0.45

(m)

u*td = 0.5

100

0

2.0 10

(b)

(b) <q>

u*t 1.5 (g/m2/s)

b = 0.45

(m/s) 5

1.0

b = 0.4

0

0.5

10

10 (c)

(c) <q>

<q>

(g/m2/s)

(g/m2/s)

5

5

g = 1.95

g = 2.15

0

0 4

2

0 6 8 10

d (m)

10

(d)

<q>

Fig 8.16 Sensitivity of saltation ¬‚uxes to changes in (a) dry

(g/m2/s)

threshold friction velocity, (b) Weibull scale factor, and (c)

qw = 0.3

5

Weibull shape factor, for a range of water table depths. From

qw = 0.2 Pelletier (2006). Reproduced with permission of Elsevier

0 Limited.

10

8

4

2 6

d (m)

Fig 8.15 Solutions to (a) Eq. (8.19), (b) Eq. (8.26), and (c)

dependence on saltation ¬‚ux on friction velocity.

Eq. (8.25) for the model parameters. (d) Sensitivity to the

A 10% change in the Weibull shape factor γ leads

value of the wilting-point moisture. Modi¬ed from Pelletier

to a small but signi¬cant ¬‚ux response (≈ 20%)

(2006). Reproduced with permission of Elsevier Limited.

for deep water tables.

Field measurements of saltation activity in-

Figure 8.16 illustrates the sensitivity of the dicate that threshold friction velocities are vari-

saltation curve to changes in the dry threshold able in time (e.g. Gillette et al., 1980, 1997). This

friction velocity and the mean and variability of variability can occur for many reasons, including

wind speeds. Although likely future changes in limited availability of transportable material,

regional wind speeds cannot be easily quanti¬ed, short-term bursts in near-surface turbulence,

many global climate models suggest that wind temporal changes in microsur¬cial characteris-

speeds will become more variable in the future. tics (i.e. formation and disturbance of protec-

This type of scenario should be considered as a tive crusts), and the selective entrainment of ¬ne

possibility within future air-quality management particles and subsequent surface armoring with

plans. Figure 8.16a illustrates the saltation ¬‚ux coarse lag deposits. In the absence of detailed

as a function of water-table depth corresponding quantitative observations of controlling sur¬cial

to a 10% decrease in the dry threshold friction characteristics, a stochastic model component

velocity. This 10% decrease causes a proportion- is necessary to represent this variability. Here

ate increase in saltation ¬‚ux for all water table we generalize the model equations to include a

depths. A 10% increase in the Weibull scale factor range of threshold wind velocities.

β (closely related to mean wind speed), shown in Figures 8.17a--8.17c illustrate the variability

Figure 8.16b, results in a near doubling of salta- in threshold velocities in the CLIM-MET station

tion ¬‚ux for most water-table depths. This result data. In this ¬gure, soil moisture is plotted as a

is not surprising given the nonlinear, threshold function of wind speed, with saltating conditions

204 STOCHASTIC PROCESSES

50

1.0

North Soda Lake

40

0.8

(a) (d)

30 0.6

no saltation

q

p

saltation

(%)

20 0.4

10 0.2

North Soda Lake

0

0

50

1.0

Balch

Chepil™s model

40

0.8

(b) (e)

30 0.6

q

p

(%)

20 0.4

10 0.2

Balch

0

0

50

1.0

Crucero

40

0.8

(c) (f)

30 0.6

q

p

(%)

20 0.4

10 0.2

Crucero

0 0

15

10 20

16 18

20 14

0 8 10 12

5

peak wind speed peak wind speed

(m/s) (m/s)

indicated by black dots and non-saltating condi-

Fig 8.17 (a)“(c) Control on saltation activity by wind speed

tions indicated by gray dots. These data illustrate

and soil moisture at (a) North Soda Lake, (b) Balch, and (c)

Crucero CLIM-MET stations. Saltation activity is indicated that high wind speeds occasionally fail to pro-

with black dots; gray dots indicate no saltation. Chepil™s duce saltating conditions. Conversely, low wind

quadratic relationship between soil moisture and threshold

speeds can sometimes trigger saltation. Subset-

friction velocity is also shown. (d)“(f) The frequency of

ting the CLIM-MET data at time intervals of a few

saltation activity at a given wind speed shows a linear increase

months does not signi¬cantly reduce this over-

from a minimum threshold velocity to a maximum threshold

lap. The range of threshold friction velocities at

velocity. Modi¬ed from Pelletier (2006). Reproduced with

these locations, therefore, appears to be related

permission of Elsevier Limited.

to microsur¬cial changes that occur over intra-

annual time scales.

8.8 THE FREQUENCY-SIZE DISTRIBUTION OF LANDSLIDES 205

The CLIM-MET station data support a linear

8.8 The frequency-size distribution

relationship between excess friction velocity and

of landslides

the probability of saltation, as shown in Fig-

ures 8.17d--8.17f. Mathematically, this can be ex-

pressed as a linear increase in the probability Landslides occur in regions of steep slopes and

of saltation p from a minimum value of 0 at are often triggered by intense rainfall, rapid

u = u—td,min to a maximum value of 1 at u—td,max : snowmelt, and ground motion from earthquakes

u— ’ u—td,min and volcanic eruptions. Given the spatial and

p= (8.29)

u—td,max ’ u—td,min temporal complexity of these triggering mecha-

nisms and the underlying topography, it is not

where u—td,min and u—td,max de¬ne a range

surprising that stochastic models play a promi-

of threshold friction velocities. For a surface

nent role in our understanding of landslide pop-

that experiences minor microsur¬cial variabil-

ulations. Although our ability to forecast land-

ity through time, the difference between u—td,min

slides is limited, we can gain understanding of

and u—td,max will be small. Conversely, large sur-

landslide population statistics by combining de-

¬cial changes will result in a correspondingly

terministic models of landslide occurrence with

large range of values. Among the three CLIM-MET

stochastic models of the underlying spatial and

stations, North Soda Lake and Crucero (Figures

temporal quantities such as topographic slope

8.17d and 8.17f) exhibit a relatively small range

and soil moisture. This section illustrates that

of threshold velocities, while Balch (Figure 8.17e)

approach.

exhibits a relatively large range.

In this section we investigate the frequency-

The range of threshold friction velocities can

size distribution of landslides in areas where dif-

be included in the model by integrating Eq. (8.29)

ferent triggering mechanisms dominate. We ar-

in a piecewise manner:

gue that fractional Gaussian noises are useful

ρ u—td,max

du— p(u— ) fw (u— )u— (u2 ’ u2 ) for modeling the variations in soil moisture and

<q >= — —t

g topography that trigger landslide instability. A

u—td,min

number of authors have examined the frequency-

∞

+ du— fw (u— )u— (u— ’ u—t ) (8.30) size distribution of landslides (Whitehouse and

2 2

u—td,max

Grif¬ths, 1983; Ohmori and Hirano, 1988;

Hovius et al., 1997). These authors have consis-

to obtain

tently noted that the distribution is power-law

ρβ

<q >≈ 6β 2 ’ u2 —td,max above some threshold size. This is precisely anal-

g γ2

ogous to the Gutenberg--Richter law for earth-

+ 2(u—td,max ’ u—td,min )2

quakes, which states that the cumulative fre-

(8.31)

quency of events with a seismic moment greater

The incomplete gamma functions have been than M o is a power-law function of M o .

Studies suggest that a threshold dependence

approximated as unity in Eq. (8.31) to yield an

expression analogous to Eq. (8.28). The additional on soil moisture is appropriate for slope insta-

terms in Eq. (8.28) were found to have a minor blity since landslides often occur when the prod-

effect on the dependence of saltation on water- uct of rainfall duration and intensity exceeds a

table depth (e.g. Figures 8.15 and 8.16) because threshold value (Caine, 1980; Wieczoreck, 1987).

the additional terms in Eq. (8.31) are related di- In situ monitoring of pore pressure has shown

rectly to wind speed rather than soil moisture. that increases in pore pressure resulting from

Equation (8.31) does, however, provide a more ac- heavy precipitation are coincident with landsli-

curate estimation for the absolute value of the des (Johnson and Sitar, 1990). Several authors

saltation ¬‚ux than Eq. (8.28), because it explicitly have shown that variations in the frequency of oc-

represents the range of threshold wind velocities currence of landslides respond to climatic chan-

ges, with higher rates of landslides associated

observed in CLIM-MET station data.

206 STOCHASTIC PROCESSES

with wetter climates (Grove, 1972; Innes, 1983; (a)

Pitts, 1983; Brooks and Richards, 1994). Further

evidence for correlations between landslides and

soil moisture is the observation that soil drainage

is the most successful method of landslide pre-

vention in the United States. Seismic triggering of

landslides has also been studied by many authors

(b)

(Keefer, 1984; Jibson, 1996). Correlations between

landslides and earthquakes can be inferred in a

number of ways. Many earthquakes have been

known to trigger large numbers of landslides and

landslide densities have a strong correlation with

active seismic belts.

The Washita experimental station has pro-

vided very useful information on the evolution Fig 8.18 (a) Grayscale map of microwave remotely sensed

of soil moisture in space and time. By character- estimates of soil moisture on June 17, 1992 at the Washita

experimental watershed. (b) Grayscale map of a synthetic soil

izing the spatial variability of soil moisture us-

moisture ¬eld with the power spectrum S(k) ∝ k ’1.8 with

ing observed data sets, we can construct statis-

the same mean and variance as the remotely sensed data.

tical models that match the observed statistics.

Modi¬ed from Pelletier et al. (1997). Reproduced with

For this purpose, microwave remotely-sensed soil

permission of Elsevier Limited.

moisture data collected from June 10 to June 18,

1992 at the Washita experimental watershed can

be used (Jackson, 1993). The watershed received

In this equation, the evapotranspiration rate ·

heavy rainfall preceding the experiment but did

is assumed to be constant in space and time.

not receive rainfall during the experiment. The

Soil moisture disperses diffusively in the soil,

data are gridded estimates of soil moisture in the

and rainfall input ξ (x, t) is modeled as a random

top ¬ve centimeters of soil calculated using the

function in space and time. Variations in rain-

algorithm of Jackson and Le Vine (1996). Each

pixel represents an area of 200 m — 200 m and fall from place to place cause spatial variations

in soil moisture that are damped by the effects

the total area considered is 45.6 km by 18.6 km.

of diffusion and evapotranspiration. Without spa-

The soil moisture values do not correlate with re-