A Rational Probabilistic Method for Spatially
Distributed Landslide Hazard Assessment
WILLIAM C. HANEBERG
Haneberg Geoscience, 10208 39th Avenue SW, Seattle, WA 98146
Key Terms: Geologic Hazards, Landslides, Probabilistic, Computer Applications, West Virginia
ABSTRACT
First-order, second-moment (FOSM) approximations of limit equilibrium slope stability equations can
be combined with digital elevation models to perform
spatially distributed probabilistic landslide hazard
analyses. This is most easily accomplished using the
infinite slope idealization, which is the basis of many
published reconnaissance-level slope stability assessments. Comparisons of FOSM and Monte Carlo
results show that FOSM approximations yield accurate results when input distributions are symmetric, or
nearly symmetric, probability density functions.
Contrary to the assumptions of previous authors,
however, the Monte Carlo results suggest that factors
of safety may be better represented by log-normal
distributions than by normal distributions. A 3- 3 2km area near Wheeling, West Virginia, covered by
a pre-existing landslide hazard map was used to
illustrate the application of the spatially distributed
FOSM approach. This area was chosen specifically
because it includes active translational landslides as
well as several map units that likely violate the infinite
slope idealization to one degree or another: large
dormant landslides, actively moving cove landforms,
and areas deemed susceptible to sliding by virtue of
underlying bedrock lithology. Using a 50 percent
probability of sliding threshold to delineate unstable
areas, the FOSM model predicted 74 percent of the
mapped active landslide area to be unstable and 77
percent of the area without mapped slope hazards to
be stable (both on a raster-by-raster basis). The
overall degree of correspondence for all hazard map
units was 54 percent if dormant landslides were
considered to be unstable and 65 percent if considered
to be stable. The degree of correspondence varies as
a function of the threshold probability but is similar to
values reported for pairs of landslide inventory maps
prepared by different geologists.
INTRODUCTION
Spatially distributed or grid-based landslide hazard
assessment using digital elevation models (DEMs)
augmented by geotechnical information has the potential
to become an important tool for geologists engaged in
reconnaissance-level watershed assessment, transportation corridor selection, and regional geologic hazard
analysis. This is particularly so for rational, or processbased, models that are based on an understanding of the
mechanics of slope instability rather than on empirical
correlations among variables (see Haneberg, 2000, for
a discussion of the differences between rational and
empirical models).
One of the weaknesses of spatially distributed rational
models is a real or perceived inability to incorporate the
effects of parameter variability and uncertainty. Most
applications of the distributed slope stability model
SHALSTAB (Montgomery and Dietrich, 1994), for
example, assume that parameters such as the angle of
internal friction, cohesion, density, and transmissivity are
constant across entire watersheds (Dietrich et al., 2001).
In other cases, spatial variability (but not uncertainty) has
been incorporated by assigning different values of
cohesion, angle of internal friction, and other geotechnical properties to different geologic map units
(e.g., McCalpin, 1997; Jibson et al., 1998; and Miles
and Keefer, 2000). Wu and Sidle (1995) as well as
McCalpin (1997) used the results of multiple scenarios to
show that parameter uncertainty can have a significant
effect on modeled landslide volumes. Although the
results of multiple-scenario modeling using best-case
versus worst-case or wet-season versus dry-season input
can be instructive, they also can be unwieldy, because
end users are forced to comprehend the results of many
model runs.
Previous attempts to mathematically incorporate the
effects of parameter uncertainty into spatially distributed
slope stability models have included use of a conservative
slope stability index based on the minima and maxima of
uniformly distributed input variables (Pack et al., 1998)
and first-order, second-moment (FOSM) rational probabilistic approaches that considered only normally distributed
input and output (van Westen and Terlien, 1996;
Mankelow and Murphy, 1998). Hammond and others
(1992) described a semi-distributed probabilistic approach
to landslide hazard assessment in which Monte Carlo
simulations are used to calculate single probabilities of
sliding for entire geomorphic map units. Although their
approach was not truly spatially distributed, it did allow for
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
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a wide variety of input probability distributions and yielded
probabilities of landsliding using calculated rather than
inferred probability distributions. Sidle and Wu (1999)
performed a spatially distributed Monte Carlo simulation
in which precipitation was assumed to be exponentially
distributed, soil cohesion and angle of internal friction were
assumed to be uniformly distributed, and the resulting
factor of safety was assumed to be normally distributed.
This paper expands on previously published work in
three ways. First, it explains the development of a general
FOSM probabilistic approach to spatially distributed
slope stability modeling that is not dependent on the
particular input or output probability distributions
assumed by other authors. Any symmetric, or nearly
symmetric, input distribution can be used with confidence. Second, it compares the results of the FOSM
approximation to those from more complicated Monte
Carlo simulations and shows that the factor of safety
against sliding is generally better represented by a lognormal probability distribution than the normal probability distribution assumed by other authors. Third, it
describes the application of the FOSM approach to an
area near Wheeling, West Virginia, that is characterized
by several different slope hazards and explores in detail
the relationships between the FOSM results and a preexisting landslide hazard map.
THEORETICAL BACKGROUND
Governing Equations
Almost all existing spatially distributed rational slope
stability models are based on variations of the infinite
slope idealization (e.g., Montgomery and Dietrich, 1994;
Wu and Sidle, 1995; van Westen and Terlien, 1996;
McCalpin, 1997; Jibson et al., 1998; Mankelow and
Murphy, 1998; Pack et al., 1998; and Miles and Keefer,
2000). A notable exception was described by Reid and
others (2000), who used circular slip surfaces in their
spatially distributed analysis of the stability of volcanic
edifices. Although no slope perfectly satisfies the
assumptions of the infinite slope model, many—if not
most—natural landslides have relatively low thicknessto-length ratios and are predominantly translational.
Therefore, although it is not suited for detailed designlevel investigations, the infinite slope idealization provides a useful reconnaissance-level approximation that
can help to identify areas in which more detailed field
investigations and office calculations are warranted.
The factor of safety against sliding for a forested
infinite slope (Hammond et al., 1992) is
FS ¼
28
cr þ cs þ½qt þcm D þðcsat cw cm ÞHw Dcos2 btan/
½qt þcm D þðcsat cm ÞHw Dsinbcosb
ð1Þ
in which
cr ¼ cohesive strength contributed by tree roots
(force/area)
cs ¼ cohesive strength of soil (force/area)
qt ¼ uniform surcharge caused by weight of vegetation (force/area)
cm ¼ unit weight of moist soil above the phreatic
surface (wt./vol.)
csat ¼ unit weight of saturated soil below the phreatic
surface (wt./vol.)
cw ¼ unit weight of water (9,810 N/m3)
D ¼ thickness of soil above the slip surface (length)
Hw ¼ height of phreatic surface above the slip surface,
normalized relative to soil thickness (dimensionless)
b ¼ slope angle (degrees)
/ ¼ angle of internal friction (degrees)
The influence of groundwater is incorporated using
a slope-parallel phreatic surface so that the pore-water
pressure is the pressure exerted by a column of water
equal in height to that of the phreatic surface above
a potential slip surface. This is a common (but not
necessary) assumption for infinite slope analyses (Iverson,
1990, 2000; Duncan, 1996). It is, however, reasonable for
cases in which a relatively permeable surficial deposit is
underlain by less permeable bedrock. The variable Hw
represents a normalized phreatic surface height that has
a range of 0 to 1 for non-artesian conditions.
The approach described in this paper does not
explicitly consider the mechanisms of pore-water pressure changes that might be responsible for landslide
initiation. Instead, it treats Hw as a random variable in
space and time. Alternative and potentially fruitful
approaches include incorporation of the temporal exceedance probability for threshold pore-water pressures using
extreme value statistics (Miller, 1988), a probabilistic
implementation of a kinematic wave groundwater flow
model that simulates time-variant pore-pressure distributions resulting from rainfall (Wu and Sidle, 1995; Sidle
and Wu, 1999), and probabilistic implementations of
process-based transient precipitation response models
(e.g., Iverson and Major, 1987; Haneberg, 1991a, 1991b;
Haneberg and Gökce, 1994; Reid, 1994; Iverson 2000;
and Baum et al., 2002). Future work will involve the
incorporation of methods such as these into the FOSM
approach described in this paper. The existing slopestability analysis models SHALSTAB and SINMAP use
pore-pressure distributions derived from steady-state
groundwater flow models (Montgomery and Dietrich,
1994; Pack et al., 1998). Although steady-state simulations might be useful in some applications, they
generally are not appropriate for simulations of precipitation-triggered landslides and are not likely to
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
Spatially Distributed Landslide Hazard Assessment
Figure 1. Schematic illustration of rational probabilistic slope stability analysis. Variables such as angle of internal friction (/), cohesive strength (c),
soil thickness (D), and dimensionless phreatic surface height (Hw) are specified as probability density functions rather than single values and used as
input to a limit-equilibrium factor of safety equation. Results are obtained in terms of a factor of safety (FS) probability distribution, the mean and
variance of FS, or a non-parametric slope reliability index.
produce better results than other methods of landslide
hazard assessment (Iverson, 2000; Reid et al., 2001).
Another assumption commonly incorporated into
slope stability calculations is that the angle of internal
friction is independent of the effective normal stress.
Laboratory studies, however, have shown that peak and
residual angles of internal friction can increase as the
effective normal stress decreases (Anderson and Sitar,
1993; Stark and Eid, 1994; and Watry and Ehlig, 1994).
This is particularly true for low values of effective normal
stress corresponding to soil depths of 5 m or less and for
clayey soils with high liquid limits. The implication for
slope stability analysis is that laboratory test results
obtained for high effective normal stresses may significantly underestimate the angle of internal friction and
overestimate the cohesive strength of thin soils in which
the effective normal stress is low. This kind of uncertainty can be incorporated into FOSM calculations by
letting the soil shear strength parameters vary over
physically realistic ranges.
FOSM Analysis
The use of FOSM approximations is well established
in the geotechnical, hydrological, and geographical
information system literature (e.g., Burrough and
McDonnell, 1998; Harr, 1996; Wolff, 1996; and Wu
et al., 1996) (Figure 1). A mean value of FS is first
calculated using the mean values of each of the
independent variables, or
FS ¼ FSðxÞ
ð2Þ
For uncorrelated independent variables, the variance (or
second moment about the mean) of FS can then be
estimated by
s2F
X@FS2
¼
s2xi
@x
i
x
i
ð3Þ
in which s2xi is the variance of the i-th independent
variable. The terms in parentheses are evaluated using
mean values for each of the independent variables. An
assumption inherent in Eq. (3) is that each derivative is
a constant that can be accurately evaluated using mean
values for all the variables. To obtain the results
presented here, the derivatives in Eq. (3) were symbolically and numerically evaluated using the computer
program Mathematica (Wolfram, 1999).
Correlation among independent variables in Eq. (1)
also can be incorporated (Wu et al., 1996; Burrough and
McDonnell, 1998), but the resulting variance expression
is considerably more complicated. An obstacle that
probably is more significant than increased mathematical
complexity is that the covariance between each pair of
correlated variables would have to be known or inferred.
As will be discussed in a subsequent section, the effect of
neglecting correlation among variables may be generally
insignificant for slope stability problems of the kind
addressed in this paper.
Once the mean and variance of the factor of safety
have been calculated, results can be expressed in terms of
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29
Haneberg
Table 1. Input for comparison of FOSM and LISA models.
Variable
Minimum
Maximum
Apex
Units
PDF
D
b
qT
cr
cs
/
cm
csat
Hw
1.68
19.0
0.72
0.24
0.24
17.0
16.2
19.2
0.3
4.42
33.0
2.15
1.68
1.68
47.0
21.3
22.0
0.7
—
—
—
—
—
—
—
—
0.5
m
Degrees
kPa
kPa
kPa
Degrees
kN/m3
kN/m3
—
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Triangular
LISA ¼ Level I Stability Analysis; FOSM ¼ first-order, secondmoment. Data from Hall and others (1994). PDF ¼ probability density
function.
the probability of sliding or a slope reliability index. The
former requires an assumption about the underlying
probability distribution of the factor of safety, whereas
the latter does not. As will be shown in the next section,
the results of numerical experiments using Monte Carlo
simulations suggest that the factor of safety distribution
generally is described more faithfully by a log-normal
distribution than the normal distribution assumed by
authors such as van Westen and Terlien (1996) and
Mankelow and Murphy (1998). The probability of sliding
is obtained from the cumulative distribution function for
a specified probability distribution having the calculated
mean and variance and evaluated at the critical value of
FS ¼ 1, or
ProbfFS 1g ¼ CDFð1Þ
ð4Þ
in which CDF(1) is the cumulative distribution function
for FS evaluated at the limiting value of unity. Equation
(4) can be evaluated for any cumulative distribution
function defined by a mean and variance, so it is not
necessary to assume that the results are normally
distributed. The probability of stability is the complement
of the probability of sliding, or 1 Prob fFS 1g.
The time-independent probability given in Eq. (4) is
the probability that the factor of safety is less than unity
given all possible values of the variables used in the
analysis; it is not the probability that a landslide will
occur over a given time period or as the result of a given
storm. Hammond and others (1992) discussed the nature
of this time-independent probability of failure, writing
that it can be used to make stability comparisons between
different areas or map units, to delineate critical areas in
need of further investigation, and as the basis for
expected monetary value decision analysis. They also
suggested that the time-independent probability of failure
can be used to estimate the percentage of an area that
might be expected to slide (e.g., 60 percent of the map
area having a 60 percent probability of sliding might be
30
expected to be unstable at one time or another).
Calculation of a time-dependent probability would
require a different procedure based on more detailed
field observations and a specification of how variables
such as pore-water pressure (e.g., Miller, 1988; Hammond et al., 1992) and tree root strength (e.g., Sidle and
Wu, 1999) vary over time. The development of timedependent probabilities of sliding is a promising research
topic but well beyond the scope of the present paper.
An alternative to the probability of sliding—and one
that does not require an a priori assumption about the
form of the output probability density function—is the
reliability index (Harr, 1996; Wu et al., 1996):
RI ¼
FS 1
sF
ð5Þ
in which sF is the standard deviation (SD) of the factor of
safety and unity is the limiting state value of the factor of
safety (FS ¼ 1). A value of RI ¼ 2, for example, would
indicate that the calculated mean factor of safety lies 2
standard deviations below the critical value of FS ¼ 1.
Values near zero indicate that stability or instability is
inferred with little confidence.
Comparison of FOSM and Monte Carlo Results
The utility and robustness of the FOSM approximation
can be evaluated by comparing its output to that from the
probabilistic slope stability program LISA (Level I
Stability Analysis), which incorporates weak correlation
among some variables and is based on a numerical Monte
Carlo algorithm (Hammond et al., 1992). The values
listed in Table 1 were used as input for FOSM and LISA
calculations that produced the results shown in Figure 2.
For cases in which input parameters are normally
distributed, the minima and maxima given in Table 1
were assumed to represent values of 63 standard
deviations from the mean. The FOSM and LISA results
are virtually indistinguishable if the probability of sliding
is assumed to be log-normally distributed in the FOSM
calculations. This is a significant result, because previous
authors have assumed that FS is normally distributed (van
Westen and Terlien, 1996; Mankelow and Murphy,
1998). In this case, the assumption of a normally
distributed FS would lead to a noticeable overestimation
of the probability of sliding. Also of interest is that the
use of uncorrelated variables in the FOSM calculations
did not produce results significantly different than those
obtained using weakly correlated variables in LISA.
Although the importance of correlated variables deserves
additional attention, these preliminary results suggest that
the use of uncorrelated geotechnical variables may be
a very reasonable first approximation that does not
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Spatially Distributed Landslide Hazard Assessment
Table 2. Input for comparison of FOSM results using different
probability density functions.
Variable
Minimum
Maximum
Units
D
b
qT
cr
cs
/
cm
csat
Hw
1.5
20.0
0.7
0.2
0.2
20.0
16.0
19.0
0.0
4.5
35.0
2.1
1.5
1.7
30.0
21.0
22.0
1.0
m
Degrees
kPa
kPa
kPa
Degrees
kN/m3
kN/m3
—
FOSM ¼ first-order, second-moment.
Figure 2. Comparison of analytical FOSM and numerical Monte Carlo
factor of safety results for the input data in Table 1. The FOSM
probabilities of sliding are calculated for both normally and lognormally distributed results. The Monte Carlo probability of sliding
was calculated as the percentage of factor of safety realizations with
values of FS 1.
produce errors that are unacceptably large. Similar results
were obtained for other input values as long as the input
probability density functions were symmetric (or nearly
so) and could be represented by theoretical probability
distributions.
The influence of different input probability density
functions on model results can be evaluated in a similar
manner. Table 2 contains a set of input minima and
maxima for FOSM and Monte Carlo calculations
performed assuming that all the input values were
uniformly, normally, and log-normally distributed. The
results, which are given in Figure 3, show good
agreement between the FOSM and Monte Carlo results
for uniformly and normally distributed input. The
agreement is worse for log-normally distributed input.
This is to be expected, however, because the asymmetry
of a log-normal distribution is conveyed by its third and
fourth moments (skewness and kurtosis) and not by its
first or second moments (mean and variance). Even so,
the FOSM method produces a useful estimate of the
probability of sliding and reliability index for the lognormally distributed input.
APPLICATION TO THE WHEELING AREA,
WEST VIRGINIA
Geologic Setting
The Wheeling, West Virginia, area (population
153,000) is nestled along the Ohio River and adjacent
tributary valleys. Bedrock consists of nearly flat-lying
sedimentary rocks (sandstone, shale, limestone, and coal)
of the Pennsylvanian Conemaugh and Monongahela
Groups and the Permian-Pennsylvanian Dunkard Group,
into which are incised colluvium- and landslide-mantled
valleys (Berryhill, 1963; Davies and Ohlmacher, 1978;
and Streib, 1980). Colluvium is thickest near the toes of
slopes, where it can approach 10 m. Maximum relief in
the area is approximately 200 m, and 67 percent of the
quadrangle is underlain by slopes that fall into the
landslide-prone range of 10 to 40 percent slope grade as
defined by Lessing and others (1976). Field evidence of
landsliding and accelerated creep (pistol-butt and jackstrawed trees, seeps and springs, and hummocky
topography, relic scarps, and toes) is common throughout
the area.
The location chosen for this study is a 2- 3 3-km area
straddling Wheeling Creek, 5 to 6 km southeast of the
city center (Figure 4). The area contains narrow and
broad valleys and straightforward bedrock geology. A
variety of landslide forms are present, but the area does
not contain failures of coal mine spoil that are common
throughout the region. It was specifically selected
because it contains a variety of landslide or potential
landslide types, including some that significantly violate
the infinite slope model assumptions, to better evaluate
the applicability of the FOSM model results to an area
with several different modes of slope instability.
With the exception of the Wheeling Creek valley
bottom, which is underlain by Quaternary alluvium and
Pleistocene stream terrace deposits, the surficial geology
of the study area consists of colluvium and landslide
deposits derived from the underlying bedrock. Soils in
the study area consist of the Westmoreland-GuernseyClarksburg Association on middle to upper slopes and
narrow valley bottoms (Ellyson et al., 1974). The
Huntington-Clarksburg-Monongahela Association occurs
along the Wheeling Creek valley bottom and adjacent
foot slopes in the northern half of the study area. Soils in
these associations generally fall into the ML and CL
categories of the Unified Soil Classification System.
Guernsey and Westmoreland soils are described as slide
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
31
Haneberg
Figure 3. Comparison of FOSM results for data having the same minima and maxima but different probability density functions as listed in Table 2.
The minima and maxima were assumed to represent values of 63 SD in the normal and log-normal distributions. As in Figure 2, FOSM probabilities
of sliding are calculated for both normal and log-normal FS distributions.
prone, whereas seasonal high water levels are commonly
within several decimeters of the ground surface in
Clarksburg, Guernsey, and Monongahela soils. Laboratory tests of a limited number of soil samples determined
32
that the clay-size particle fraction (0.002 mm) ranged
from 23 to 45 percent by weight (Ellyson et al., 1974).
The landslide hazard categories of Davies and
Ohlmacher (1978) that occur within the study area
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
Spatially Distributed Landslide Hazard Assessment
Figure 4. Map of the Wheeling, West Virginia, study area. Slope hazard map units correspond to those used by Davies and Ohlmacher (1978), and
topographic contours were obtained from the 30-m DEM of the Wheeling OH-WV 7.59 quadrangle. Areas for which no slope hazards are shown
constitute the ‘no slide’ map category discussed in the text.
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
33
Haneberg
include active landslides, dormant landslides, cove landforms, and areas susceptible to movement.
1998). For slope angles measured in degrees, the finite
difference expression for the mean maximum slope angle
at each DEM grid point is
Active Landslide
The active landslide category contains earth flows,
debris slides, earth and rock slumps that are identified on
the basis of historical records and field evidence. These
areas were described by Davies and Ohlmacher (1978) as
being extremely unstable.
Dormant Landslide
The dormant landslide category contains extensive
areas of hummocky ground caused by earth flows, earth
slumps, or rock slumps with no clear evidence of active
landsliding. These areas were interpreted by Davies and
Ohlmacher (1978) to be relatively stable if undisturbed.
Upslope boundaries are generally defined by scarps, but
downslope boundaries may be poorly defined or
gradational. This unit corresponds to the pre-historic
landslide category of Davies and Ohlmacher (1978), but
the name was changed for this study to reflect a lack of
information regarding absolute age and the possibility of
reactivation (Ohlmacher, 2000).
br;c ¼
180
p
2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
ðzr;cþ1 zr;c1 Þ2 þðzrþ1;c zr1;c Þ2 5
3 arctan4
2s
ð6Þ
in which the z values represent DEM elevation data, the r
and c subscripts denote the row and column of the
elevation value z within the DEM grid, and s ¼ 30 m is
the DEM grid spacing in this example.
Slope Angle Variances
If the variance of the elevations, s2z , is uniform across
the entire quadrangle, then the variance of the slope angle
at each point in the grid can be estimated using the FOSM
approximation:
s2b
"
¼
Cove Landform
The cove landform category contains U-shaped valleys
(coves) underlain by clayey soils that form coherent slabs.
Water at the base of the clays is typically under 0.6 to
2.5 m of artesian head. According to Davies and
Ohlmacher (1978), the slabs move at a rate of approximately 0.3 m/year unless accelerated by human activity.
Model Input
Mean Slope Angles
A grid of 6,868 slope angle values (68 rows and 101
columns) was calculated using 30-m digital elevation
model (DEM) data from the U.S. Geological Survey
(USGS) Wheeling 7.59 quadrangle (10-m DEM coverage
is not available for the quadrangle). The mean value for
the maximum slope at each DEM data point was
calculated using a second-order accurate central difference approximation (e.g., Burrough and McDonnell,
34
ð7Þ
or, substituting Eq. (6) into Eq. (7) and differentiating,
s2b ¼
2
180
p
Areas Susceptible to Movement
The category of areas susceptible to movement contains soil and rock similar to those involved in landslides
elsewhere in the map area but with no evidence of current
or past landsliding, covering primarily areas underlain by
mudrocks.
@br;c 2
@br;c 2
þ
@zrþ1;c
@zr1;c
2 #
@br;c
@br;c 2 2
þ
þ
sz
@zr;cþ1
@zr;c1
3
8ðsÞ2 s2z
4ðsÞ2 þ ðzrþ1;c zr1;c Þ2 þðzr;cþ1 zr;c1 Þ2
2
ð8Þ
Equation (8) has units of degrees squared and shows that,
all other things being equal, the uncertainty associated with
slope angle calculations will be inversely proportional to
the slope angle. It does not take into account factors such as
the presence or absence of forests or brush in different
areas, which may affect the reliability of DEMs. The
elevation uncertainty of USGS DEMs is given as a rootmean-square (RMS) error, which is identical to the
standard deviation of the errors, based on a comparison
between DEM and topographic map elevations at a minimum of 28 points per quadrangle. Thirty points were used
to calculate an RMS error of 5 m for the Wheeling DEM,
implying an elevation variance of 25 m2. The locations of
the control points used to calculate the RMS errors for
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Spatially Distributed Landslide Hazard Assessment
Table 3. Geotechnical soil properties for FOSM slope stability analysis
of the Wheeling area.
Variable
Minimum
Maximum
Units
PDF
/
cm
csat
Hw
17
19
19
0
36
20
20
1
Degrees
kN/m3
kN/m3
—
Uniform
Uniform
Uniform
Uniform
FOSM ¼ first-order, second-moment. PDF ¼ probability density
function.
USGS DEMs are not provided to users, so it generally is not
possible to determine whether the errors are spatially
correlated. Holmes and others (2000), however, analyzed
the uncertainty in a 30-m DEM from a mountainous area in
California and constructed variograms to show that the
elevation errors in their study area were spatially
correlated. Thus, use of a variance based on the spatially
uncorrelated RMS error for the quadrangle would tend to
overestimate the variability of slope angles calculated
using elevations spaced only 60 m apart. Based on the
findings of Holmes and others (2000), the elevation
variance used for slope angle variance calculations was
assumed to be 5 m2, which is one-fifth that which would
have been used had spatial correlation not been considered.
Shear Strength
Residual shear strength parameters were assumed on
the basis of widespread landsliding and accelerated creep
of colluvium throughout the region. The minimum and
maximum values in Table 3 were selected on the basis of
unpublished geotechnical reports supplemented by soil
survey data (Ellyson et al., 1974), published relationships
between / and clay content (Skempton, 1964, 1985), and
published relationships between geotechnical properties
and Unified Soil Classification System categories (Hammond et al., 1992). A plot of direct shear test results from
an unpublished geotechnical report for a geologically
similar site near (but not within) the study area suggests
that the residual shear strength envelope of colluvium in
the area may be nonlinear (Figure 5). The data are from
tests conducted at effective normal stresses of 48, 96, and
144 kPa on samples recovered from depths of 2.0 to
2.5 m. Simple linear regression yields shear strength
parameters of c ¼ 14.5 kPa and / ¼ 18 degrees and
produces a reasonably good fit to the data but yields
a cohesive strength estimate that is unreasonably high for
residual conditions. A regression line forced through the
origin in order to obtain c ¼ 0, which typically is assumed
for residual shear strength, yields / ¼ 24 degrees and
a noticeably worse fit. A third option is to assume that the
data points have no errors and interpolate a polynomial
passing through all three of the points and the origin,
which produces a nonlinear envelope with c ¼ 0 and,
Figure 5. Direct shear test results for samples of colluvium from an
unpublished geotechnical report for a confidential location near (but
not in) the study area. The boxed values show the estimated vertical
stresses in saturated and unsaturated alluvium that correspond to the
effective normal stresses for each of the three trials. The three lines
show alternative interpretations of the data in terms of c and /,
including the possibility of a nonlinear shear strength envelope.
depending on the normal stress, 11 degrees / 36
degrees. These limited data are not definitive, but they do
suggest that the possibility of nonlinear shear strength
envelopes needs to be considered when evaluating
shallow landslide hazards in the region.
Phreatic Surface Height
Dimensionless phreatic surface heights were allowed
to vary between 0 and 1, because piezometric data from
the region are extremely limited. Therefore, it was not
possible to define a narrower distribution. The result of
using such a wide distribution is increased uncertainty in
the FS results relative to those that would have been
obtained using a narrower distribution, which in turn
decreases the probability of sliding if FS , 1 and
increases the probability of sliding if FS . 1.
Tree Root Strength and Surcharge
Neither tree surcharge nor tree root strength were
included in the calculations. Realistic values of vegetation surcharge have only a small effect on slope stability,
and this effect is indistinguishable from that of a slightly
increased soil unit weight. Research along other portions
of the Ohio River Valley has suggested that tree roots
help to stabilize colluvium only up to a meter or so in
depth (Kochenderfer, 1973; Riestenberg, 1994), which is
less than the typical soil thicknesses in the study area.
Note that D drops out of Eq. (1) in the absence of soil
cohesion, root cohesion, and vegetation surcharge;
therefore, it was not necessary to estimate soil thickness
values for the present study.
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35
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Uniform Probability Distributions
Uniform probability density functions were chosen for
all independent variables other than the slope angles
because, although it was possible to define realistic
minima and maxima, no compelling evidence suggested
that any variable was drawn from a probability distribution with a strong central tendency. The net effect of
assuming uniformly distributed input parameters is
a larger degree of uncertainty in the calculated results
(compared to those that would be obtained using
distributions with central tendencies), which is an
appropriate reflection of the state of knowledge regarding
the input parameters. Likewise, the probability distributions for geotechnical and hydrologic parameters were
not allowed to vary in space, because landslides in the
region most commonly occur in colluvium covering the
hillsides. At present, no maps or geotechnical data are
available that might be used to delineate different
probability distributions for colluvium developed over
different bedrock units in the study area. The approach
taken in this paper honors the scarcity of data by
assuming, first, that the geotechnical properties at any
point are characterized by a very broad range of possible
values and, second, that a value at any point has the
possibility of being either anomalously high or low. For
cases in which detailed geologic maps and large amounts
of geotechnical data are available (e.g., van Westen and
Terlien, 1996; McCalpin, 1997; Jibson et al., 1998; and
Miles and Keefer, 2000), it may be possible to identify
distinct geotechnical probability distributions for different
bedrock or surficial units (or, perhaps, to show that
geologically different units are statistically indistinguishable with regard to their geotechnical properties).
Uniform Distribution Mean and Variance
The mean and variance of a uniform distribution are
calculated from minimum and maximum values (Haneberg, 2000) using
xmin þ xmax
2
ð9Þ
ðxmax xmin Þ2
12
ð10Þ
x ¼
and
s2x ¼
Computational Details
Because FS approaches infinity as b vanishes,
a computational concession had to be made. Therefore,
36
a constant value of 0.1 degree was added to all slope
angles before they were used in stability calculations.
This increment ensures the calculation of a finite value for
FS even if b ¼ 0, but it is small enough to have only an
insignificant effect on the calculated value of FS. An
alternative would have been not to calculate FS for cells
in which b falls below a threshold value, but this has the
disadvantage of producing an output grid composed of
mixed numerical and non-numerical values.
RESULTS
Figure 6 shows contour maps of the DEM topography,
calculated mean slope angle, and calculated slope angle
variance for the study area. The slope angle variance is
inversely proportional to the slope angle magnitude,
because a given magnitude of elevation error will have
a larger impact on small elevation differences than on
large elevation differences. Figure 7 contains a series of
four contour plots derived from the slope results shown in
Figure 6. The calculated mean factor of safety map
(Figure 7A) closely resembles the slope map, which is to
be expected in cases of uniform geotechnical properties.
In such situations, the effect of the slope stability
calculations is to establish a threshold slope angle above
which sliding is likely to occur. The factor of safety
variance, like the slope variance, is inversely proportional
to slope angle (Figure 7B). This is only partly
a consequence of the slope angle variance, however,
because similar results are obtained if the DEM is
assumed to be error free and the calculated slope angles
have zero variance. The results of the FOSM model show
that, in essence, less uncertainty exists about the stability
of steep slopes than about gentle slopes, regardless of the
slope angle variance. This is because tan b appears in the
numerator of Eq. (1) and its derivatives in Eq. (3), forcing
both FS and its variance to decrease as b increases. Figure
7C and D shows the slope reliability index and lognormal probability of landsliding. Most of the reliability
index values fall within a range of 2 RI þ2, which
indicates uncertainty about the calculated stability or
instability of slopes in the study area. Only small areas
are calculated to be reliably stable (RI . 2) or reliably
unstable (RI , 2), and a large proportion of the study area
is characterized by calculated factors of safety that fall
less than 1 standard deviation above or below the critical
value of FS ¼ 1 (i.e., 1 RI þ1). The probability of
sliding map, calculated under the assumption that FS is
log-normally distributed, shows more readily identifiable
areas of potentially unstable slopes, with a large proportion of the map having Prob fFS 1g , 0.2. These
low probability areas consist primarily of valley bottoms
and ridge tops.
Some readers may notice an apparent contradiction
between Figure 7B, which shows that gentle slopes have
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
Spatially Distributed Landslide Hazard Assessment
Figure 6. Topography and slope angle maps of the Wheeling, West Virginia, study area. (A) DEM topographic contours with a contour interval of
10 m. (B) Mean slope angles calculated using the second-order accurate finite difference approximation discussed in the text. (C) Slope angle
variance calculated using the FOSM approximation discussed in the text.
greater uncertainty than steep slopes, and Figure 7C,
which shows that gentle slopes are more reliably stable
than steep slopes (and vice versa). This can be understood
by inspecting Eq. 5, which shows that the increased
uncertainty associated with gentle slopes is more than
compensated for by the significantly higher mean factor
of safety calculated for the gentle slopes. The net result is
a higher reliability index.
Comparison with the Inventory Map
One of the primary objectives of this work was to learn
how the results of a simple probabilistic slope stability
model constrained with limited data compare to a more
traditional landslide inventory map. The amount of data
available to constrain the model, although certainly less
than desirable, probably is representative of many
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37
Haneberg
38
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
Spatially Distributed Landslide Hazard Assessment
Table 4. Unit-by-unit correspondence of FOSM results and Davies
and Ohlmacher (1978) landslide hazard map categories.
FOSM vs. Hazard
Map Correspondence (%)
Hazard map
category
Portion of Map
Area Covered (%)
Case I
Case II
No slide
Active slide
Dormant slide
Cove
Susceptible
All units
42
2
41
5
10
100
77
74
36
31
34
54
77
74
64
31
34
65
FOSM ¼ first-order, second-moment.
practical situations in which a FOSM model might be
applied as part of a reconnaissance-level slope stability
analysis. Table 4 as well as Figures 8 and 9 contain the
results of a detailed, unit-by-unit comparison of the
Davies and Ohlmacher (1978) map and the FOSM model
results. Figure 8 shows the distribution of calculated
probability of sliding values for each hazard map unit.
The active landslide and no landslide results fall into
distinct, asymmetric probability distributions that are
skewed in opposite directions and for which the Prob fFS
1g ¼ 0.5 threshold was a good (albeit coincidental)
boundary. Results for the dormant landslide, susceptible,
and cove map units define broad probability distributions
that fall between the active landslide and no landslide
distributions. In the case of the cove map unit, the wide
range of results likely was obtained because the geometry
of the cove landforms departs significantly from that
assumed in the infinite slope model. The situation is not
as clear for the dormant landslide and susceptible map
units. Some portions of these units (e.g., the heads and
mid-sections of dormant landslides) are characterized by
low slope angles and probably are stable under current
conditions. Other areas (e.g., the toes of dormant
landslides) may be steep and marginally stable to
unstable. The susceptible map unit was delineated on
the basis of bedrock lithology without regard to slope
angle or geotechnical parameters, and therefore can
include areas of low slopes in which translation sliding is
not likely to occur.
Results of FOSM model runs using probability of
sliding thresholds ranging over 0.1 Prob fFS 1g 0.9 show definite trends in degree of correspondence
Figure 7. FOSM slope stability model results. (A) Natural logarithm of
the factor of safety against sliding (FS). (B) Natural logarithm of the
variance of FS. (C) Non-parametric slope reliability index. (D)
Probability of sliding based on the assumption of a log-normal FS
distribution.
Figure 8. FOSM results for Davies and Ohlmacher (1978) landslide
hazard map units. The box-and-whisker symbols show, from bottom to
top, the minimum value; the 25th, 50th, and 75th cumulative
percentiles; and the maximum value for each map unit.
between FOSM results and the Davies and Ohlmacher
(1978) landslide hazard map (Figure 9). The degree of
correspondence for each hazard map unit was calculated
by comparing FOSM and hazard map results on a rasterby-raster basis and then dividing the result by the number
of rasters covered by each hazard map unit. Areas
covered by the active landslide map unit were considered to be unstable, and areas for which no landslide
hazards were mapped were considered to be stable. Similarly, FOSM result rasters with calculated values of
Prob fFS 1g , 0.5 were considered to be stable, and
rasters with Prob fFS 1g . 0.5 were considered to be
unstable. Note that this approach is more conservative
than those used by authors who consider an entire
landslide to have been successfully predicted if one or
more rasters within the landslide boundary or a buffer
zone are calculated to be unstable (e.g., Shaw and
Vaugeois, 1999). Therefore, comparisons of hazard
assessment models based on the percentage of landslides
claimed to have been predicted successfully need to be
made with due regard for the methods used to obtain the
results.
The degree of correspondence between the FOSM
results and the Davies and Ohlmacher (1978) map
depends, in part, on whether dormant landslides are
considered to be stable or unstable. Both possibilities can
be compared. As illustrated in Figure 9, a low probability
of sliding thresholds produces extremely high levels of
correspondence for the active landslide map unit and
moderate levels for areas in which Davies and Ohlmacher
(1978) mapped no slope hazards. The active landslide
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39
Haneberg
weighted by the map unit areas, depends on the assumed
stability or instability of areas mapped as dormant
landslides. If the dormant landslides are considered to
be unstable, the overall degree of correspondence decreases from 65 percent to 43 percent as the threshold
increases from 0.1 to 0.9. If, however, the dormant
landslides are considered to be stable, then the degree of
correspondence increases from 47 percent to 82 percent
as the threshold increases.
The degree of correspondence between the FOSM
model and the Davies and Ohlmacher (1978) hazard map
also can be compared with the degree of correspondence
that might be expected between landslide inventory or
hazard maps of an area prepared by different geologists.
Wills and McCrink (2002) compared the percentages of
areas shown as landslides on different landslide inventory
maps of a part of the Santa Cruz Mountains, California,
and reported degrees of correspondence ranging from 48
percent to 65 percent. Similarly, Ardizzone and others
(2002) compared landslide inventory maps of an area in
the northern Apennines of Italy produced by three
independent groups of geomorphologists and found the
correspondence between all three maps to be only 20
percent. Correspondence between pairs of inventory
maps ranged from 35 percent to 45 percent. Thus, the
degree of correspondence between the FOSM results and
the Davies and Ohlmacher (1978) landslide hazard map
appears to be no worse—and probably better—than one
might expect between two hazard maps prepared by
different geologists.
Figure 9. Plots showing the sensitivity of calculated degrees of FOSM
versus hazard map correspondence for different probability of sliding (Prob fFS 1g) thresholds. Case I considers dormant landslides
to be unstable ground that corresponds to FOSM results only if
Prob fFS 1g , 0.5, whereas case II considers dormant landslides
to be stable ground that corresponds to FOSM results only if Prob
fFS 1g . 0.5.
degree of correspondence falls significantly as the probability of sliding threshold value increases, especially for
threshold values greater than 0.5, whereas the no landslide correspondence shows the opposite trend.
Because the stability of ground in the dormant
landslide category is questionable, two sets of correspondence curves were calculated, one assuming the dormant
landslides to be stable and the other assuming the
dormant landslides to be unstable. Although it is not done
in this paper, one could make a similar argument
regarding the ambiguous stability of areas mapped as
susceptible by Davies and Ohlmacher (1978) but that
present no evidence of past or current instability. The
overall degree of correspondence, expressed as the sum
of the individual map unit degrees of correspondence
40
DISCUSSION
The FOSM approximations are a useful and computationally simple way of probabilistically assessing slope
stability for cases in which the input probability
distributions are symmetric (or nearly so). Comparison
of FOSM results with those from Monte Carlo simulations show that the resulting factor of safety probability
distributions are more faithfully represented by lognormal distributions than the normal distributions
assumed by previous authors (van Westen and Terlien,
1996; Mankelow and Murphy, 1998). Given only
a calculated mean (FS) and variance for the factor of
safety, the use of a log- normal rather than a normal
distribution will produce a slightly lower estimate of the
probability of sliding (Prob fFS 1g) if FS , 1 and
a slightly higher estimate if FS . 1. Depending on the
variance and the asymmetry of the FS probability density
function, the difference in calculated probability of
sliding for normal and log-normal distributions may, or
may not, be significant. Assuming variables to be
uncorrelated did not appear to weaken the FOSM results,
although such correlations could be incorporated if
necessary.
Environmental & Engineering Geoscience, Vol. X, No. 1, February 2004, pp. 27–43
Spatially Distributed Landslide Hazard Assessment
Figure 10. Composite landslide hazard map combining FOSM results and information from the Davies and Ohlmacher (1978) landslide hazard map.
Areas in which both the FOSM results and the hazard map predict instability are classified as unstable (white). Areas in which either the FOSM model
or the hazard map predict instability are classified as uncertain (gray). Areas in which the FOSM results and the hazard map predict no instability are
classified as stable (black).
The choice of an appropriate probability of sliding
threshold to distinguish between stable and potentially
unstable areas will depend on the reason for the
distinction. For example, an active landslide degree of
correspondence approaching 100 percent can be achieved
by conservatively using a low threshold value, but only at
the cost of misidentifying approximately half of the noslide map unit as being unstable. Conversely, nearly all
areas in which no slope hazards were mapped can be
modeled as stable by optimistically using a high threshold
value, but at cost of modeling almost none of the active
landslide map unit as unstable. The Prob fFS 1g ¼ 0.5
therefore represents a useful compromise between the
two extreme cases. For situations in which it may not be
desirable—or even possible—to specify a threshold
probability, image-processing techniques can be used to
objectively delineate zones of high probability or slope
reliability gradients that separate different landslide
hazard classes.
The degree of correspondence between the FOSM
model and the Davies and Ohlmacher (1978) landslide
hazard map is higher than that reported for pairs of
landslide inventory maps of other areas prepared by
different geologists. It may be that areas of disagreement
between the FOSM model and the inventory more likely
result from geological subjectivity or mapping errors than
from FOSM model errors. Some areas mapped by Davies
and Ohlmacher (1978) as dormant landslides or as
susceptible to landsliding are stable under present
conditions and, therefore, were correctly assigned low
probabilities of landsliding. It is likewise possible that
some areas not mapped as unstable or potentially unstable
by Davies and Ohlmacher (1978) are unstable and
correctly identified as such by the FOSM model.
The most constructive approach to landslide hazard
assessment is to combine the strengths of empirical
landslide hazard maps with those of quantitative landslide
hazard models, particularly those that are able to account
for the parameter uncertainty and variability inherent in
any application of a mathematical model. A conservative
option is to combine a binary version (i.e., mapped areas
are either stable or unstable) of the Davies and Ohlmacher
(1978) hazard map with a threshold probability version of
the FOSM results. The binary map is divided between
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41
Haneberg
areas in which Davies and Ohlmacher (1978) mapped no
landslides and inferred no susceptibility (stable) and those
in which either landslides were mapped or susceptibility
was inferred (unstable). The result, shown in Figure 10,
contains three hazard categories: 1) areas in which neither
the model nor the empirical hazard map predict instability
(stable), 2) areas in which either the model or the empirical
hazard map predict instability (uncertain), and 3) areas in
which both the model and the empirical hazard map predict
instability (unstable). A corresponding three-tiered land
management or zoning response to such a composite map
might be to permit development in stable areas; to require
detailed, site-specific geotechnical investigations before
allowing development in uncertain areas; and to limit
development in unstable areas. Other layers of information
(e.g., geomorphometric measures such as slope curvature)
could easily be added when and where appropriate.
ACKNOWLEDGMENTS
Wolfram Research, Inc., provided the Mathematica
software used in the present research. Comments,
criticisms, and suggestions by Greg Ohlmacher, Mark
Reid, Jeff Keaton, and three anonymous reviewers of
a previous version of this paper, as well as many people
who have heard verbal presentations of this work, are
greatly appreciated.
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