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mate for long-term-average moisture conditions,
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-

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