Transient deterministic shallow landslide modeling: Requirements for susceptibility and hazard assessments in a GIS framework

Engineering Geology 102 (2008) 214–226

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Engineering Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o

Transient deterministic shallow landslide modeling: Requirements for susceptibility and hazard assessments in a GIS framework J.W. Godt a,⁎, R.L. Baum a, W.Z. Savage a, D. Salciarini b, W.H. Schulz a, E.L. Harp a a b

U.S. Geological Survey, Box 25046, MS 966, Denver, CO 80225, United States Department of Civil and Environmental Engineering, University of Perugia, Perugia, Italy

a r t i c l e

i n f o

Article history: Accepted 4 March 2008 Available online 31 July 2008 Keywords: Landslide Susceptibility Infiltration Rainfall GIS

a b s t r a c t Application of transient deterministic shallow landslide models over broad regions for hazard and susceptibility assessments requires information on rainfall, topography and the distribution and properties of hillside materials. We survey techniques for generating the spatial and temporal input data for such models and present an example using a transient deterministic model that combines an analytic solution to assess the pore-pressure response to rainfall infiltration with an infinite-slope stability calculation. Pore-pressures and factors of safety are computed on a cell-by-cell basis and can be displayed or manipulated in a grid-based GIS. Input data are high-resolution (1.8 m) topographic information derived from LiDAR data and simple descriptions of initial pore-pressure distribution and boundary conditions for a study area north of Seattle, Washington. Rainfall information is taken from a previously defined empirical rainfall intensity–duration threshold and material strength and hydraulic properties were measured both in the field and laboratory. Results are tested by comparison with a shallow landslide inventory. Comparison of results with those from static infinite-slope stability analyses assuming fixed water-table heights shows that the spatial prediction of shallow landslide susceptibility is improved using the transient analyses; moreover, results can be depicted in terms of the rainfall intensity and duration known to trigger shallow landslides in the study area. Published by Elsevier B.V.

1. Introduction Managing hazards associated with shallow landslides requires an understanding of where and when such landslides may occur, and how widespread a potential shallow landslide event might be. The wide availability of Geographic Information Systems (GIS) and digital topographic data has led to the development of various analytic methods for estimating potential hazards from shallow, rainfalltriggered, landslides over large geographic areas. These include empirical probabilistic methods based on historical records and deterministic methods. Empirically based probabilistic methods (e.g. Coe et al., 2004) use historical records of landslide occurrence to predict the temporal and spatial probability of future landslides. The empirically based statistical approach (e.g. Guzzetti et al., 1999; Carrara et al., 1999; Cannon et al., 2004) relies on multivariate statistical correlation between locations of failures on steep slopes and factors such as slope angle, slope curvature, bedrock lithology, soil type, and basin morphology. The results of the empirically based analyses are typically portrayed using GIS. Deterministic methods typically rely on application of simple models of groundwater flow combined with infinite-slope stability analyses to estimate the potential or relative instability of slopes over a broad ⁎ Corresponding author. Tel./fax: +1 303 273 8626. E-mail address: [email protected] (J.W. Godt). 0013-7952/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.enggeo.2008.03.019

region. They may involve one, two or three-dimensional steady or transient groundwater flow models and analytical, numerical, and hybrid mathematical schemes and, usually, an infinite-slope factor of safety equation. Some early attempts to develop and apply “physically based” shallow landslide initiation models over large areas combined steady or quasi-steady groundwater flow models with the assumption that flow is exclusively parallel to the slope and flux at the ground surface (i.e. rainfall) only acts to modify steady or quasi-steady porepressure distributions (e.g. Montgomery and Dietrich, 1994; Wu and Sidle, 1995). However, slope parallel seepage can only be in equilibrium with rainfall under a restrictive and unrealistic set of conditions in which 1) the rainfall is of very low intensity and 2) long duration, 3) the depth to the failure surface is small with respect to the square root of the contributing area, and 4) the saturated hydraulic conductivity of the soil is strongly anisotropic with the slope-parallel component greatly exceeding that in the vertical direction (Iverson, 2000). Such conditions are rarely met in nature. Because the steady and quasi-steady groundwater models do not account for the transient pore-water response resulting from infiltration, results cannot be used to examine changing rainfall and groundwater conditions that cause shallow landslides to develop during a rainfall event. To assess the effects of infiltration on near-surface pore-pressure distributions and consequent slope stability over broad regions requires transient solutions for groundwater flow. Both numerical (e.g. Simoni et al., 2008) and analytical (Morrissey et al., 2001; Crosta

J.W. Godt et al. / Engineering Geology 102 (2008) 214–226

and Frattini, 2003; Savage et al., 2003; Morrissey et al., 2004; D'Odorico et al., 2005; Salciarini et al., 2006; Godt et al., 2008) techniques have been applied with GIS-based slope stability models to map areas that are potentially unstable during rainstorms. This paper will use the model TRIGRS (Transient Rainfall Infiltration and Gridbased Regional Slope-stability analysis) which was developed to account for the transient effects of rainfall on shallow landslide initiation and combines an analytical solution for groundwater flow in one vertical dimension with an infinite-slope stability calculation (Baum et al., 2002; Savage et al., 2003). Application of models such as TRIGRS in a GIS context for landslide hazard assessment requires digital spatial topographic, geologic, and hydrologic information. In addition, inventories of previous rainfallinduced landslide events are required to test the model results. Digital topography (i.e. Digital Elevation Models — DEMs) of appropriate scale and resolution, areal distribution and depth of susceptible surficial materials and their hydrologic and strength properties, near-surface groundwater conditions, and spatial and temporal distributions of rainfall are required. Finally, these methods for estimating potential hazards from shallow, rainfall-triggered landslides require skilled users with an understanding of landslide mechanics and triggering processes. We consider each of these subjects in this order and in greater detail below and using an example application to a study area of landslide-prone coastal bluffs north of Seattle, Washington. We then conclude with a summary and discussion of the requirements for integrating deterministic shallow landslide models with GIS for hazard and susceptibility assessments. 2. Stability of infinite slopes Deterministic modeling of slope stability over broad regions typically relies on one-dimensional infinite-slope stability analyses applied over digital topography (Montgomery and Dietrich, 1994; Terlien et al., 1995; Wu and Sidle, 1995; Borga et al., 1998; Baum et al., 2002; Casadei et al., 2003; Savage et al., 2003; Salciarini et al., 2006, Harp et al., 2006, Godt et al., 2008). Infinite-slope stability analysis assumes that landslides are infinitely long and is most appropriate for analysis of landslides with planar failure surfaces that have a small landslide depth compared to their length and width, and are destabilized by widespread areas of positive pore-water pressure (Iverson et al., 1997) or loss of soil suction (Anderson and Howes, 1985; Collins and Znidarcic, 2004). Generalized for hillslope materials with cohesion, c′ under the Mohr–Coulomb failure criteria (Das, 2000), and to account for the influence of groundwater, the factor of safety for an infinite slope is FS ¼

tan/V c V−ψγw tan/V þ tanβ γs dlb sinβ cosβ

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Pack et al., 1998) and constant relative water-table position (e.g. Harp et al., 2006) are often made in application of Eq. (1) to map shallow landslide susceptibility over broad regions. 3. Transient vertical groundwater flow The characteristic time scales for lateral and slope-normal porepressure transmission are A / D0, and H2 / D0, respectively, where D0 is the saturated hydraulic diffusivity, H is the depth of the failure surface in the slope-normal direction, and A is the upslope contributing area above a given location. (Iverson, 2000). Areas of the landscape where the vertical component of pore-pressure response to rainfall dominates over lateral transmission can be identified by considering a ratio, ε, of the two characteristic length scales, H e ¼ pffiffiffi A:

ð2Þ

For areas of the landscape where ε ≪ 1 long and short-term pressure-head responses to rainfall is 1) dominated by vertical flow and 2) these areas are typical of shallow landslide locations (Iverson, 2000). This implies that pore-pressure variation in response to rainfall in initially wet materials can be adequately described by simplified, one-dimensional forms of the Richards equation. The Richards equation describes unsaturated Darcian flow in response to infiltration at the ground surface. In one vertical dimension for a sloping surface it can be written as    Aψ A Aψ C ðψÞ ¼ K ðψÞ − sin β At AZ AZ

ð3Þ

where ψ, is pressure head, K(ψ) the pressure-head dependent hydraulic conductivity, C(ψ) is the specific moisture capacity, C(ψ) = ∂θ / ∂ψ, θ is the volumetric water content, β is the slope angle of the ground surface, and Z is the vertical depth (Philip, 1991). Because of the nonlinear dependence of hydraulic conductivity and moisture content on pressure head, solutions to Eq. (3) generally require specialized numerical techniques (Rubin and Steinhardt, 1963; Freeze, 1969). Several analytical solutions to one-dimensional forms of the Richards equation are available that make use of discrete assumptions to simplify the analysis (Philip, 1957; Parlange, 1972; Philip, 1991; Srivastava and Yeh, 1991) and one has been applied to boundary conditions suitable for slope stability analysis (Savage et al., 2004, Baum et al., 2008).

ð1Þ

where ψ is the groundwater pressure head, γw, is the unit weight of water, γs, the unit weight of soil, and dlb is the depth of the failure surface (Fig. 1). The primes next to the cohesion, c′, and friction angle, ϕ′ indicate that these material strength parameters are determined for effective stress (Terzaghi, 1943), which accounts for the influence of pore-water on the stress field in the material and is defined as the total normal stress minus the pore-water pressure. This analysis assumes that hillside materials are saturated at the time of failure and neglects the influence of suction on the shear strength of soils (Fredlund et al., 1978). For shallow landslides triggered by rainfall and over short timescales such as that of a rainstorm, the pressure head, ψ, is the only time-dependent quantity in Eq. (1). Although vertical pore-pressure gradients in the saturated zone in infinite slopes vary as a function of the flux at the ground surface and the anisotropy of the hydraulic conductivity (Iverson, 1990) assumptions of spatially invariant failure depth (e.g. Montgomery and Dietrich, 1994; Jibson et al., 2000) steady slope parallel flow (e.g.

Fig. 1. Sketch showing the coordinate system and groundwater conditions in hillslopes above an impermeable lower boundary at dlb below the ground surface. The depth of the water table is dwt, and the slope angle is β. The vertical, Z, and slope-normal, z, coordinates are also shown.

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For the study presented here, we rely on solutions to Eq. (3) presented by Iverson (2000) assuming that the zone above the water table is tension-saturated to the ground surface (Fig. 1). For this endmember case the pressure dependent terms, K(ψ) and C(ψ) become constant values, KZ and C0. Eq. (3) then reduces to a linear diffusion equation Aψ A2 ψ ¼ D1 2 At AZ

ð4Þ

where D1 = D0 / cos2 β, and the hydraulic diffusivity, D0 equals KZ / C0, where C0 is the minimum slope of the soil–water characteristic curve. Iverson's (2000) original solutions for pore-pressure response in an infinitely deep half-space have been applied in a GIS framework (Morrissey et al., 2001; Crosta and Frattini, 2003). The solution to Eq. (4), given a time-varying specified flux boundary condition (as suggested by Iverson, 2000) at the ground surface and an impermeable basal boundary at a finite depth dlb, given by Savage et al. (2003) and implemented in TRIGRS (Baum et al., 2002), is:

4.2. Digital topography

ψðZ; t Þ ¼ ½Z−dλcosβ N

þ2 ∑

n¼1

N

−2 ∑

n¼1

1 ∞ InZ Hðt−tn Þ½D1 ðt−tn Þ2 ∑ KZ m¼1

inventories are typically compiled from aerial photography (e.g. Bucknam et al., 2001) using photogrammetric techniques (e.g. Godt and Coe, 2003) or eye-hand transfer of landslide features to topographic maps (e.g. Harp and Jibson, 1995). Triggering conditions are not easily determined from aerial photographs and their definition requires nearby measurement of rainfall and observations of soilmoisture conditions. Landslide inventory maps are typically either scanned or digitized manually to create a digital dataset compatible with GIS. The accuracy of the landslide inventory is a function of the original image quality, the scale and quality of the topographic base, and the skill of the geologist. The increasing availability of highresolution digital imagery and soft-copy photogrammetry promises to reduce the error introduced in the digital conversion of maps by allowing the geologist to compile the inventory digitally (e.g. Stock et al., 2006). For testing shallow landslide initiation models such as TRIGRS, landslide sources, tracks, and deposits should be mapped separately, but photograph quality and scale or resolution, landslide size, and the scale of the topographic base make this impossible in many cases.

(

∞ 1 InZ H ðt−tnþ1 Þ½D1 ðt−tnþ1 Þ2 ∑ KZ m¼1

" ierfc

# ð2m−1Þdlb −ðdlb −Z Þ

(

1

2½D1 ðt−tn Þ2 "

ierfc

" þ ierfc

# ð2m−1Þdlb −ðdlb −Z Þ 1

2½D1 ðt−tnþ1 Þ2

#) ð2m−1Þdlb þ ðdlb −Z Þ

þ ierfc

1

2½D1 ðt−tn Þ2

" #) ð2m−1Þdlb þ ðdlb −Z Þ 1

2½D1 ðt−tnþ1 Þ2

ð5Þ where ψ is pressure head, t is time, d is the depth of the steady-state water table measured in the z-direction, and where λ = cosβ − (IZ / KZ). Here IZ is the long-term (steady-state) rainfall flux at the ground surface in the Z-direction, KZ is the hydraulic conductivity in the Zdirection, InZ is the surface flux for the nth time interval, and dlb is the vertical depth to the lower boundary, N, is the total number of time intervals, and H(t − tn + 1) is the Heavyside step function. The function ierfc(η) is equivalent to 1 / √π exp(−η2) − ηerfc(η), where erfc(η) is the complementary error function. The first term on the right-hand side of Eq. (5) describes a steady-state pressure-head distribution in the Zdirection for a homogeneous soil in response to long-term rainfall, IZ. The other terms on the right-hand side describes the transient porepressure distribution in response to a time-varying flux at the ground surface in a layer with a finite depth, dlb, in the Z-direction. The first few terms of the infinite series converge to an acceptable degree of accuracy so that the computation of Eq. (5) is very efficient and can be applied over broad areas on a cell-by-cell basis in a GIS framework yielding a time-varying pore-pressure and factor of safety for each grid cell as a function of depth in response to rainfall (Baum et al., 2002). The TRIGRS model imposes an additional physical limitation that at any depth Z the maximum pressure head under downward gravitydriven flow cannot exceed that which would result from having the water table at the ground surface and maintaining the original flow direction and hydraulic gradient (Iverson, 2000; Baum et al., 2002) given by ψðZ; t ÞV Zλcosβ:

ð6Þ

4. Input and test data requirements 4.1. Landslide inventories Landslide inventories should include information on where and when the landslides occurred, the type, size and the regional extent of the landslides, as well as information on triggering conditions if they are to be used to test results from deterministic models. Landslide

Digital Elevation Models (DEMs) are regularly-spaced arrays of elevation values that have become essential for regional landslide hazard analysis. Other digital data structures exist to represent topography (e.g. Triangular Irregular Networks — TINs), however, we restrict our discussion here to DEMs because of their wide availability and use. The spatial resolution of a DEM is typically described by the distance on the ground represented by the array spacing. The accuracy of the DEM is a function of the accuracy and spacing of the original source data and the accuracy of the interpolation of those data to a regularly-spaced grid. DEMs are generated from a variety of original topographic data sources including: photogrammetrically generated contour maps, ground-based surveys, and remotely-sensed data. At this time, DEMs interpolated from topographic contour data are probably the most commonly used for landslide hazard mapping mainly because largescale topographic maps are widely available for many localities. However, elevation data from both airborne and spaceborne sensors are increasingly available and have been used in a variety of landslide applications. Of the remote-sensing technologies, Light Detection and Ranging (LiDAR) has arguably had the most impact on landslide hazard mapping and modeling (e.g. McKean and Roering, 2004; Schulz, 2007). LiDAR-derived DEMs are typically of very high spatial resolution (1 to 5 m) with low elevation (Z) errors (typically b20 cm in unvegetated, low-slope landscapes). In vegetated steeplands, LiDAR errors are typically much greater (e.g. Haneberg, 2008). Because tens or hundreds of thousands of laser pulses per second are made during a LiDAR survey, data processing algorithms have been designed to discriminate between returns from vegetation and those from the ground surface (e.g. Haugerud et al., 2003). Thus, LiDAR can potentially provide DEMs with a much more accurate depiction of the topographic surface than DEMs derived from photogrammetrically mapped contours, even in heavily vegetated areas. Deterministic shallow landslide susceptibility modeling requires DEMs of adequate resolution to capture landslide features in a given study area. Since most of these study areas are likely to be in highly dissected terrain with high relief, high-resolution data (5–10 m) are typically required (Zhang and Montgomery, 1994). Slope angle calculations and other elevation derivatives such as curvature and contributing area are dependent on the scale of the source elevation data and the grid-cell spacing of the DEM (Garbrecht and Martz, 1994; Zhang and Montgomery, 1994; Thieken et al., 1999; Claessens et al., 2005). Finer grid spacing typically produces steeper slope angles and at very fine spacing (e.g. b5 m) large local variability of curvature results from small-scale topographic features such as animal burrows

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and mounds surrounding vegetation (Heimsath et al., 1999). Elevation errors in DEMs also have an effect on the calculation of topographic derivatives such as slope and contributing area (e.g. Holmes et al., 2000). Haneberg (2006, 2008) shows that elevation errors in a 1-m LiDAR and 10-m USGS DEMs lead to errors in the calculation of topographic slope that may be as large as ±3°. The effect of the errors on the slope stability calculations is mitigated somewhat because the absolute error tends to decrease with increasing slope angle for slopes less than about 45°. Haneberg (2006) estimates that the net effect of the elevation error is uncertainty in the final results that is comparable to the uncertainty introduced by the spatial variability of the strength parameters of the hillslope materials. 4.3. Soil thickness and shallow landslide depth 4.3.1. Field observations Maps of the soil depth on steep hillsides are required for deterministic shallow landslide models that include the effects of infiltration or soil cohesion (Dietrich et al., 1995; Baum et al., 2002). However, since collecting sufficient measurements to map soil thickness compatible with the scale of high-resolution DEMs is a practical impossibility, deterministic modeling efforts have typically relied on empirical or theoretical models to create soil depth maps (e.g. Dietrich et al., 1995; DeRose et al., 1996; Casadei et al., 2003; Salciarini et al., 2006; Godt et al., 2008). Field observations of soil depth in landslide-prone areas indicate that colluvium tends to collect in areas of topographic convergence (hollows) and is periodically removed by shallow landsliding during heavy rainfall (Dietrich and Dunne, 1978; Dietrich et al., 1986; Reneau and Dietrich, 1987; Dengler and Montgomery, 1989; Reneau et al., 1990; DeRose, 1991). Attempts to correlate field measurements of soil depth with topographic attributes such as total topographic curvature and topographic slope have met with varying success and provide somewhat contradictory results. Topographic curvature was shown to be positively correlated with the thickness of colluvial soils in areas of topographic divergence (noses) on low gradient (0–25°) slopes in both Marin County, California (Heimsath et al., 1999) and the eastern Australian escarpment (Heimsath et al., 2000); however, little or no correlation with curvature or other topographic attributes was identified on divergent topography in the generally steeper terrain of the Oregon Coast Range (Roering et al., 1999; Heimsath et al., 2001). In convergent, steep (generally greater than about 20°) landslide source areas in the central California Coast Ranges, Reneau et al. (1990) reported data indicating that colluvial depth is poorly correlated with topographic slope. However, in the eastern Taranaki hill country of New Zealand where shallow landslides and debris flows dominate erosion processes on steep hillslopes (Trustrum and DeRose, 1988), DeRose et al. (1991) showed that at the scale of shallow landslides typical of the area, soil thickness in hollows steeper than 20° decreases exponentially with slope. Subsequent study of this area showed that topographic slope explains about 50% of the variation in mean soil depth for both convergent and divergent hillsides with slopes ranging from 5 to about 60° (DeRose, 1996). 4.3.2. Soil mantle evolution theory Dietrich et al. (1995) proposed a model, later developed by Heimsath et al. (1997, 1999), for vertical soil depth in temperate, soilmantled landscapes with well-developed dendritic drainage patterns assuming that 1) biogenic activity and moisture content variation is responsible for the production of colluvial soils on hillsides, 2) any mass loss due to solution processes is negligible, and 3) that the net downslope transport of colluvium is proportional to the local slope of the ground. For low-slope environments the net downslope flux of soil is often assumed to be a linear function of slope (e.g. Culling, 1960; McKean et al., 1993) and analogous to a diffusion process. However, for steep hillsides where shallow landslides are more likely to occur, the

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net downslope flux of colluvial material has been shown to be a nonlinear function of topographic slope (Andrews and Bucknam, 1987; Roering et al., 2001a,b). The nonlinear model does not explicitly account for sediment transport by landslides and debris flows but experimental results indicate that it may be adequate for areas where sediment flux is dominated by shallow landslides (Roering et al., 2001a). The rate of soil production is calibrated empirically and has been shown to decline exponentially with depth. Numerical solutions to produce maps of soil thickness (Dietrich et al., 1995; Heimsath et al., 1999; Casadei et al., 2003) require estimates of soil production rates from cosmogenic nuclide or other dating techniques (e.g. Heimsath et al., 1999). Assuming soil thickness ultimately achieves a local steadystate, numerical results agree well with field observations on noses with slopes less than about 25°. Similar models have been used to estimate the spatial variability of soil thickness for models of regional landslide susceptibility in temperate, unglaciated environments (e.g. Dietrich et al., 1995; Casadei et al., 2003). 4.3.3. Empirical models To estimate colluvial thickness in the southwestern part of Seattle for an assessment of rainfall-induced landslide susceptibility, Godt et al. (2008) relied on a set of site-specific correlations between colluvial depth and hillside morphology developed by Schulz et al. (2008). A systematic variation of colluvium thickness among three hillslope landforms (escarpment, midslopes, and footslope) was identified. These landforms were mapped using shaded-relief images of a LiDAR DEM and the height and topographic slope of each landform was calculated. A GIS database of borehole locations was created based on the compilation of geotechnical exploration logs (Troost and Booth, 2008). Ground-surface elevation and subsurface attributes (depth of colluvium, depth of water table, date of waterlevel measurement) for each borehole location were determined based on the geological and geotechnical properties provided in the logs, findings at nearby exploration locations, and the geologic and topographic setting. Empirical relations to fit a surface to the base of the colluvium were developed using the geotechnical borehole database locations. Four topographic parameters were used as input data in the model: 1) topographic slope angle of the ground surface, 2) slope angle of the escarpment, 3) height of the escarpment, and 4) distance downslope from the escarpment (Godt et al., 2008; Schulz et al., 2008). Topographic slope angle was used by DeRose (1996) in an exponential relation with soil thickness described previously to produce map depicting shallow landslide susceptibility. Salciarini et al. (2006) also used an exponential function of topographic slope to map the lower boundary depth for a parametric study of shallow landslide susceptibility in central Italy. The selection of an approach to model soil thickness for GIS-based hazard assessment will likely be driven by the available data. For welldissected landscapes in temperate environments where soil production rates can be estimated, numerical soil diffusion models may yield accurate estimates of soil thickness. However, site-specific empirical relations between soil thickness and topography may be more appropriate for formerly glaciated landscapes (e.g. Alpine environments) where non-diffusive processes such as rock fall dominate colluvial transport and deposition. 4.4. Initial and boundary conditions 4.4.1. Initial groundwater conditions Groundwater flow models are generally very sensitive to initial conditions and applications to simulate actual landslide events require some knowledge of initial water-table depths and the moisture conditions of the unsaturated zone. Godt et al. (2008) relied on a database of geotechnical borings and regression techniques similar to those described for colluvial thickness in the previous section to map average winter-season water-table depths for a TRIGRS application in

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Seattle (Schulz et al., 2008). Where detailed information on groundwater conditions is unavailable, parametric studies assuming a range of initial conditions may provide insights into the conditions that have caused shallow landslides in the past (Crosta and Frattini, 2003; Salciarini et al., 2006). Additional research is needed to establish efficient and reliable methods to define initial conditions for transient, deterministic, shallow slope stability models. 4.4.2. Rainfall flux The spatial and temporal scales of rainfall data required for transient modeling of infiltration and shallow slope stability is dictated by the purpose of the study. Attempts to recreate conditions for past landslide events will likely benefit from rainfall data with high spatial (kilometer scale) and temporal (hourly) resolution. Rainfall estimates from weather radars have been shown to improve model comparisons with landslide inventories both spatially and temporally (Crosta and Frattini, 2003). However, most studies will be likely forced to rely on available rainfall information from gauge networks (Godt et al., 2008). Empirical rainfall intensity–duration thresholds can be used in applications designed to estimate the likelihood of landslide occurrence for a given set of initial and rainfall conditions. 4.5. Hydraulic and strength properties of hillside materials The strength and hydraulic properties of hillside materials and an estimate of their spatial distribution are required for models such as TRIGRS. Coulomb strength parameters (angle of internal friction and cohesion) can be obtained from standard geotechnical tests (e.g. Das, 2000; Savage and Baum, 2005). Plant roots are thought to impart significant strength to hillside soils (e.g. Wu et al., 1979; Schmidt et al., 2001; Sidle and Ochiai, 2006); however, the resisting forces imparted by plant roots are typically dependent on failure depth and do not necessarily parallel the ground surface. Lumping this contribution with cohesion in infinite-slope stability analysis (Eq. (1)) is physically incorrect. Three-dimensional solutions are needed (e.g. Burroughs, 1985; Dietrich et al., 2006) to accurately represent lateral resisting forces typically associated with vegetation roots. Hydraulic properties of hillside materials that are typically required for analyses include the saturated hydraulic conductivity and saturated hydraulic diffusivity. Unlike material strength properties, the saturated hydraulic conductivity of soils with similar textures derived from the same parent material can vary over several orders of magnitude (Freeze and Cherry, 1979). Laboratory tests using either constant or falling head instruments are typically used for measuring saturated conductivity. Because diffusion solutions to groundwater infiltration are sensitive to the diffusivity term, some understanding of the unsaturated hydraulic characteristics of hillside materials is needed to accurately estimate this parameter and to define the range of soil-moisture conditions for which the approximate solution (e.g. Eq. (5)) can be applied. Loose, coarse-grained colluvial soils typically involved in shallow landslides exhibit pronounced hysteresis among the relations between moisture content, pressure head and hydraulic conductivity (Freeze and Cherry, 1979). Laboratory tests to determine the moisture content–pressure-head relation or soil–water characteristic curve (SWCC) and hydraulic conductivity function (HCF) should be obtained using a wetting process to simulate rainfall infiltration. Examples include the so-called Bruce and Klute experiments for measuring soil–water diffusivity in which the water is allowed to imbibe into a horizontal column (Bruce and Klute, 1956; Clothier and Scotter, 2002). These tests are typically performed on repacked materials and because bulk density has a significant influence on soil hydraulic properties, care should be taken to replicate field densities or account for variation. Other laboratory tests include those using constant-flow permeameters that can be used to determine the SWCC and HCF for either the wetting or drying process (Wildenschild et al., 1997; Lu et al., 2006).

In-situ tests on hillside materials in field areas probably provide the most representative estimates of material properties at the scale of shallow landslides. Data from well and ring permeameter tests of unsaturated materials can be used to estimate the field saturated hydraulic conductivity (Reynolds et al., 2002). The term field saturated is often used for these types of tests because air is usually entrapped in the soil by infiltrating water, which is typically the case during natural rainfall. Permeameter data can also be reduced to estimate unsaturated-zone parameters as well. Disc permeameters provide measurements of hydraulic conductivity at small negative pressures and data can be reduced to estimate soil–water characteristic curves and diffusivity (Clothier and Scotter, 2002). Prediction of SWCCs from more easily measured soil parameters such as particle-size distributions has been performed using both empirical and theoretical approaches (Brakensiek et al., 1981; Haverkamp and Parlange, 1986). The theoretical or physically-based models (e.g. Arya and Paris, 1981) typically link the cumulative particle-size distribution and other properties such as bulk density with the SWCC. Empirical pedotransfer functions (PTFs) are used to predict soil hydraulic properties from readily available soil information such as bulk density and soil texture using regression or neural network techniques (Leij et al., 2002). Where soils mapping is available, this approach, combined with field and laboratory analysis may yield reasonable estimates of the spatial variability of material parameters for model input. However, it is important to note that while empirical models are available to account for hysteresis (e.g. Kool and Parker, 1987) most predictive models and PTFs are designed for drying curves; empirical data for wetting curves are virtually nonexistent (Leij et al., 2002). More detailed information on empirical, field, and laboratory techniques to determine soil hydraulic properties can be found in the relevant chapters of Dane and Topp (2002) and Lu and Likos (2004). In the following section we describe an example application of a transient infiltration slope stability model to a landslide-prone study area north of Seattle, Washington and compare results with those

Fig. 2. Map showing the location of the study area and Edmonds field site north of Seattle, Washington.

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those of the study area) are now protected from wave attack by shoreline protection structures and engineered fill. The climate of the Seattle area is characterized by a pronounced seasonal precipitation regime with a wintertime maximum; 75% of the annual precipitation total of 963 mm falls between November and April (Church, 1974). Storms that trigger shallow landslides in Seattle are generally of long duration (more than 24 h) and of relatively low intensity compared to landslide-causing rainfall in other parts of North America (Godt et al., 2006). During the winter season of 1997–1998 heavy rainfall and the melting of a deep snowpack triggered numerous shallow landslides on the bluffs along Puget Sound that periodically disrupted rail service on the Seattle to Everett corridor (Baum et al., 2000). Fig. 3 shows part of a landslide inventory for an area along the Puget Sound north of Seattle, Washington, compiled primarily from the interpretation of 1:12,000 color aerial photography taken in May of 1998 (Baum et al., 2000). The inventory shows the outlines of shallow landslide scars and deposits in the study area. Shallow landslides in the Seattle area commonly involve the thin (less than 3 m thick) colluvium that mantles the steep hillslopes (Tubbs,1974; Galster and Laprade,1991; Baum et al., 2000; Schulz et al., 2008). The colluvium is largely the product of mechanical weathering of the underlying glacial and non-glacial parent sediments, landsliding, and other mass-movement processes. The particle-size distribution of the colluvium is generally dominated by sand, but

Fig. 3. Map showing landslides that occurred in the study area during the winter season of 1996–1997 (modified from Baum et al., 2000).

from a static infinite-slope stability analysis assuming fixed watertable positions. 5. Example application 5.1. Physiographic and geologic setting of the study area The 3 km2 study area is in the Puget Lowland about 15 km north of Seattle, Washington (Fig. 2). The Puget Lowland is a broad basin located between the Olympic Mountains to the west and the active volcanic arc of the Cascade Range to the east. The topography of the Puget Lowland consists of rolling north–south oriented ridges and valleys left by the retreat of glacial ice about 16,400 years ago (Booth, 1987; Booth et al., 2005). Steep coastal bluffs line the shores of Puget Sound and other bodies of surface water in the area and were formed by deposition of materials prior to and during glacial advance, and their subsequent erosion by fluvial and hillslope processes and wave erosion of bluff toes following glacial retreat (Booth, 1987; Shipman, 2004). Glacial isostatic rebound and glacial tectonic uplift are responsible for the relative elevation of the coastal bluff uplands above sea level. Since glacial retreat, the coastal bluffs have retreated by an estimated 150 to 900 m, primarily by landslide processes and wave action (Galster and Laprade, 1991; Shipman, 2004; Schulz, 2007). Many of the bluffs (including

Fig. 4. Rainfall (A), volumetric water content (B), and pressure head (C) measured at various depths at the Edmonds field site during the winter wet season. Arrows indicate the timing of landslides near the monitoring site.

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ranges from boulders to silt and clay (Galster and Laprade, 1991; Godt and McKenna, 2008). Gravel and cobble-sized material is generally supplied by the Vashon till (Qvt), a highly-variable, basal-lodgement till that discontinuously caps many of the uplands in the Seattle area. An extensive water-bearing unit, the Vashon advance outwash sand (Qva) supplies mainly sand-sized material. Finer-grained material is supplied by the generally well-consolidated sediments of Pleistocene and older Quaternary age that underlie Qva and in the study area are mapped as transition beds (Qtb) by Minard (1982). 5.2. Field monitoring site A system to monitor the shallow groundwater response to rainfall infiltration operated for 4 winter seasons on an area of steep (N40°) landslide-prone coastal bluffs (Baum et al., 2000) about 3 km south of the study area near Edmonds, Washington (Fig. 2) and was part of a U.S. Geological Survey near-real-time landslide-monitoring project along the Seattle — Everett, Washington rail corridor (Baum et al., 2005: Godt and McKenna, 2008). The 50-m-high bluff is capped by Vashon-age glacial till (Qvt) and is underlain by subhorizontally bedded glacial advance outwash sand overlying glaciolacustrine silt deposits (Minard, 1982). The monitoring instruments were installed in the dense, uniform, medium outwash sand about 30 m above sea level in Puget Sound. A thin (10 to 20 cm), loose, sandy colluvium produced by mechanical weathering of the outwash sand covers much of the hillside in the area where the instruments were installed. Downslope from the instrument installations, the colluvium is thicker (1 m). In January of 2006 a shallow landslide destroyed most of the instrument array. Details on the instruments and monitoring data are available in Baum et al. (2005) and in Godt and McKenna (2008). Monitoring data (Fig. 4) collected in a landslide-prone hillside near Seattle show that soil moisture is strongly seasonal (Baum et al., 2005). Soil moisture begins to increase near the ground surface with

the onset of rainy weather in the late fall and early winter. At depths greater than 1 m the soil moisture begins to increase after the first few rainstorms and typically stays elevated until later March or early April with the cessation of consistently wet weather. At the timescale of individual storms, the pore-water response to rainfall varies as a function of the water content of the soil; when soils are initially wet, infiltrating rainfall causes wetting fronts to rapidly propagate through the upper 2 m of hillside materials, however, the magnitude of the response is attenuated with depth and rainfall of high intensity and short duration has a limited effect at depth. Monitoring observations prior to shallow landslides occurring near the field sites indicate that pressure heads in hillside materials were near zero (Fig. 4). 5.3. Input data For the application presented here, slope angles (Fig. 5A) were calculated from a 1.83-m (6 ft) DEM derived from remotely-sensed LiDAR elevation data (Haugerud et al., 2003). These data have been processed to provide a detailed depiction of the ground surface beneath vegetation. We follow DeRose (1996) and Salciarini et al. (2006) and estimate the depth to the lower boundary, dlb, (Fig. 5B) as an exponential function of slope, β, where dlb = 7.72 e− 0.04β so that dlb is at a minimum of about 0.5 m on the steepest slopes, about 1.4 m on 40° slopes, and about 2.0 m on 30° slopes consistent with field observations of shallow landslide failure depths in the study area (e.g. Baum et al., 2000; Baum et al., 2006). The assumption of tensionsaturated materials in the infiltration model of TRIGRS (Baum et al., 2002) is consistent with field monitoring observations described above and for this application we assume that the initial water table is coincident with the lower boundary. Three property zones (Fig. 5C) were defined based on geologic mapping of the area (Minard, 1982) with an assumption that the fine-grained fraction of the colluvium increases with decreasing elevation (Godt et al., 2008). The saturated

Fig. 5. Maps showing topographic slope (A), depth to the lower boundary (B), and simplified geology (C) for the study area. Geology modified from Minard (1982). Geologic map units include: Qtb = transition beds including the Lawton Clay, Qva = Advance outwash sand, Qvt = till.

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Table 1 Material strength and hydraulic properties for the three geologic units in Fig. 5C Unit

Qtbf Qvag Qvth a b c d e f g h

I Za

K Sb

D0c

(m/s)

(m/s)

(m2/s)

(°)

(kPa)

1.5 × 10− 7 1.5 × 10− 7 1.5 × 10− 7

1.0 × 10− 5 5.0 × 10− 5 1.0 × 10− 6

5.0 × 10− 5 1.0 × 10− 4 1.0 × 10− 5

33.6 33.6 33.6

4.6 4.6 6.0

ϕ′d

c′e

IZ = long-term rainfall flux. KS = saturated hydraulic conductivity. D0 = hydraulic diffusivity. ϕ′ = minimum angle of internal friction for effective stress. c′ = cohesion for effective stress. Qtb = transition beds including the Lawton Clay. Qva = advance outwash sand. Qvt = till.

hydraulic conductivity (Table 1) was estimated for the three units based on slug tests performed at the field monitoring site and wellpermeameter tests performed in the study area and elsewhere in the Seattle area in similar geologic formations and hillside deposits. Hydraulic diffusivity (Table 1) was estimated using results from opentube capillary rise tests performed to determine the SWCC for the wetting process of colluvial soils in the study area and comparison of TRIGRS results with field monitoring data and 1-D numerical solutions for vertical flow in the unsaturated zone (Godt et al., 2008; Godt and McKenna, 2008). Material strength parameters were taken from direct shear tests on colluvial soil (Godt and McKenna, in press). We apply a constant rainfall flux of 1.5 mm/h for 36 h, which just exceeds the empirical threshold (Fig. 6) defined for the Seattle area (Godt et al., 2006). 5.4. Interpretation and testing of model results Pore-pressures and factors of safety, FS, are calculated at every timestep for each 1.8 m grid cell over the approximately 5 × 105 cells. Because these grid cells are much smaller than typical shallow landslides in the study area, we aggregate the FS results to a 9 × 9 m grid. Nine meters is approximately the lateral dimension of the smallest landslide mapped in the study area. To create the aggregated grid each small (1.83 m) grid cell is assigned either a 1 or 0 to indicate that it is either stable (FS N 1.0) or unstable (FS = 1.0), respectively. The value then assigned to the large (9 m) grid is based on the value of the majority of the 25 small grid cells. If 13 or more of the small grid cells have FS = 1.0

Fig. 6. Rainfall intensity–duration threshold above which the occurrence of widespread shallow landslides can be expected in the Seattle area. Threshold can be represented by I = 82.73D− 1.13 where I is the mean rainfall intensity (in mm/h) and D is the storm duration (in h) (modified from Godt et al., 2006).

Fig. 7. Example contingency table (also known as a confusion matrix) for a two class problem and performance metrics calculated from it. Each output grid cell from the three models can be mapped to one of the four elements of the matrix. See text for an explanation of criteria for membership in each of the elements and performance metrics (modified from Fawcett, 2006).

then the aggregated cell is unstable. The operation is not performed in a moving-window fashion, but by moving a non-overlapping 9 × 9 m block across the digital landscape. The aggregated stability grid can then be compared to the landslide inventory at roughly similar scales. The success of regional landslide susceptibility models has been typically evaluated by comparing the location of known landslides with model results (e.g. Montgomery et al., 1998, 2001; Godt et al., 2008). The least critical test is to count a success when a single grid cell with a FS = 1.0 falls within a mapped landslide polygon. More critical tests involve some assessment 1) the model's ability to correctly identify mapped landslides (true positive), 2) the error when mapped landslides are not correctly identified (false negative), and 3) over prediction (false positive). An ideal landslide susceptibility map simultaneously maximizes the agreement between known and predicted landslide locations and minimizes the area outside the known landslides predicted to be unstable (over prediction). Originally developed for use in signal detection theory and applied in diverse fields such as epidemiology, weather forecasting, machine learning, and landslide susceptibility mapping, receiver operating characteristics (ROC) graphs are a technique to assess the performance of models for which results can be assigned to one of two classes or states (Swets, 1988; Fawcett, 2006; Van Den Eeckhaut et al., 2006). For limit-equilibrium slope stability models, the states are stable or unstable, and for comparison of grid-based predictions of slope stability with a landslide inventory, four outcomes are possible for each grid

Fig. 8. Pore-pressure response (dashed line) and resultant factor of safety (solid line) simulated with TRIGRS for a single cell with a depth of 1.0 m an initial water-table depth of 1.0 m, and a slope angle of 40° for a constant rainfall of 1.5 mm/h.

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cell. If the grid cell is modeled to be unstable and is coincident with a mapped landslide in the inventory it is counted as a true positive; if it falls outside a mapped landslide it is counted as a false positive. If the grid cell is modeled to be stable and it falls outside a mapped slide it is counted as a true negative; if it falls within a mapped slide it is counted as a false negative. The outcome counts can be entered in a 2 × 2 contingency table (so-called confusion matrix, Fig. 7) from which several measures of model performance can be calculated (Fawcett, 2006). The true positive rate is the ratio of the number of true positives to the total number of positives (landslide cells). The false positive rate is ratio of the number of false positives to the to the total number of negatives (non-landslide cells). Areas with slopes ≤15° were excluded from the analysis because: 1) the mapped landslides include shallow landslide sources as well as deposits and the deposits tend to be on flat slopes and the models do not account for travel or deposition, and 2) for the material strengths (Table 1) and soil depths (Fig. 5B) used in this study, slopes less than about 17° are stable under any hydrologic condition resulting from infiltration. 6. Results Fig. 8 shows the pore-pressure response and resultant factor of safety at the lower boundary for an example grid cell with a 40° slope and outwash sand material properties (Table 1). The depth to the lower boundary and initial water-table depth were both 1.0 m. Pressure head increases linearly after the first hour in response to constant rainfall of 1.5 mm/h, which results in a linear decrease in the factor of safety with time. Failure is predicted after about 35 h of rainfall. Pressure head and factors of safety are calculated in the same manner for about 5 × 105 cells in the study area. After 36 h of rainfall, pressure heads have increased at the base of each cell from the initial condition of zero to greater than 0.4 m; the spatial variability in the pressure head increase is a function of the

slope, depth, and hydraulic properties of the hillside materials (Fig. 9A). The resultant factor of safety reflects the pattern of these same variables since we assume the material strength properties are the same throughout the study area (Fig. 9B). Factors of safety equal to 1.0 are located in the steepest topography along the bluff crests and at the toe of the bluffs along the Puget Sound. Fig. 10C shows the aggregated factor of safety; the main effect of aggregation is to eliminate isolated cells with FS = 1.0. 6.1. Comparison of TRIGRS results with static factor of safety maps Fig. 10 compares the results from the TRIGRS model that accounts for the transient effects of rainfall infiltration on pressure head with results from a static infinite-slope analysis assuming two water-table conditions. All other factors are equal or held constant between the three analyses. The static maps were generated by applying Eq. (1) on a cell-by-cell basis for the material strength properties in Table 1 and depths shown in Fig. 5B. Pore-pressures, ψ, were calculated with the water table located at half the depth of the lower boundary (Fig. 10A) and at the ground surface (Fig. 10B) assuming slope parallel flow over an impermeable lower boundary ψ ¼ mdlb cos2 β

ð7Þ

where m is the vertical height of the water table expressed as a fraction of the total depth, dlb, and β is the slope angle. Both static infinite-slope analyses capture more of the mapped landslide area and predict more of the study area to be unstable compared to that predicted using the transient analysis. Fig. 11 is a ROC graph comparing the results from the static infiniteslope stability calculations (m = 0.5 and m = 1.0) with those from the TRIGRS model and depicts the relative tradeoff between success (true positives) and over prediction (false positives). Model results that plot

Fig. 9. Modeled pressure head (A), factor of safety (B), and aggregated factor of safety (C) after 36 h of rainfall. Pressure heads and factors of safety are not shown for cells with slopes ≤15°.

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223

Fig. 10. Extent of unstable area for static infinite-slope stability calculation assuming the water-table depth is half the depth to the lower boundary (A) (m = 0.5), at the ground surface (m = 1.0) (B), and for the transient solution in the TRIGRS model after 36 h of rainfall (C).

towards the upper left of the graph are generally considered superior. The origin of graph (0,0) represents a model result with no false positive errors because it predicts no instability. The converse, a model result where the entire area is predicted to be unstable would be located at the upper right (1,1). A perfect prediction would be located at the upper left (0,1). A model result that falls along the dashed line is considered random in that it predicts instability falling within and outside the mapped landslides at the same rate (Fawcett, 2006). Comparison of the three models shows that while the TRIGRS model

tends to under predict the spatial extent of landslides relative to the static factor of safety calculations, its over prediction is much less severe (Table 2). The lower rate of over prediction of the TRIGRS model yields much greater estimates of accuracy and precision compared to the static models. Calibration of material strength parameters and water-table depths to generate a best fit to the landslide inventory could be performed and would likely improve the spatial prediction of the static susceptibility map (Harp et al., 2006). However, our results show that the transient analysis yields an improved prediction for this study area using material strength and hydraulic properties measured in the field and laboratory and for pore-pressure distributions that are physically linked with rainfall infiltration. 7. Concluding discussion Accounting for the transient effects of rainfall infiltration on porewater response and consequent effects on slope stability improves the effectiveness of regional shallow landslide hazard maps. Because the transient model physically links rainfall with pore-water response at Table 2 Comparison of model performance between static infinite-slope stability calculations for two water-table positions and results from the transient TRIGRS model after 36 h of rainfall Model

True positive rate

Static factor of safety 0.95 with m = 1.0 Static factor of safety 0.63 with m = 0.5 TRIGRS 36 h 0.42 Fig. 11. ROC graph comparing the results from each of the three models. Grid cells with slopes ≤15° were excluded from the analysis.

False positive rate

False negative rate

Accuracy Precision

0.86

0.05

0.22

0.11

0.33

0.37

0.66

0.18

0.16

0.58

0.80

0.23

Grid cells with slopes ≤ 15° were excluded from the analysis. See text and Fig. 11 for explanation of performance metrics.

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depth, results can be depicted in terms of historical, real-time, or forecasted rainfall. This greatly expands the range of landslide hazard issues and questions that can be explored compared to what can be examined using static analysis. However, transient modeling of infiltration and landslide processes over broad regions requires understanding of initial and boundary conditions as well as material strength and hydraulic properties and some estimate of their spatial variability. High-resolution (≤10 m) topographic data are required for the assessment of shallow landslide susceptibility and the increasing availability of LiDAR data promises to provide accurate topographic information even in heavily vegetated landscapes. In areas where abundant geotechnical information is available, site-specific, empirical models of soil thickness and water-table depth can be used to provide initial and boundary conditions such as used in the example presented here. Most applications, however, are likely to be in areas where such information is lacking and process-based theoretical models may provide a way to characterize the spatial variability of soil thickness and groundwater conditions over broad areas. Additional work is needed to show that the theoretical soil-thickness models are appropriate for landscapes in which shallow landslides and debris flows dominate sediment transport. Field and laboratory methods developed for soil science applications can be adapted to determine the hydraulic properties of hillside materials; but, appropriate statistical relations between soil hydraulic properties and more easily measured parameters such as grain-size distributions (so-called Pedotransfer Functions) await the collection of adequate field and laboratory data for the wetting process. Additional research is needed to identify the most effective ways to describe boundary and initial conditions and determine and map hillside material properties before transient landslide susceptibility models such as TRIGRS (Baum et al., 2002) can be applied as a hazard management tool. The example presented here, with simple assumptions of slope dependent boundary and initial conditions and material strength and hydraulic properties for hillside materials measured in the field and laboratory, yielded an improved spatial prediction of shallow landslide occurrence compared to static infinite-slope stability analyses with fixed water-table depths. Results from the static analyses with the water table located at the ground surface can be interpreted as the most conservative; however such assessments risk being underutilized because they tend to depict all steep slopes as equally hazardous. These results from the transient analysis suggest that the spatial and temporal accuracy of the shallow landslide susceptibility model can be further improved with accurate characterization of the initial and boundary conditions and spatial distribution of material properties. Incorporation of the effects of the unsaturated zone on infiltration (Savage et al., 2004; Baum et al., 2008) and slope stability (Fredlund et al., 1978; Collins and Znidarcic, 2004; Lu and Likos, 2004; Baum et al., 2006) may also improve model results. Acknowledgements Brian Collins, Jason Kean, Paolo Frattini, and an anonymous reviewer provided thoughtful and constructive reviews of this paper. Paolo Frattini also suggested the application of the ROC technique for comparing the static and transient analyses. References Anderson, M.G., Howes, S., 1985. Development of a combined soil water-slope stability model. Quarterly Journal of Engineering Geology 18, 225–236. Andrews, D.J., Bucknam, R.C., 1987. Fitting degradation of shoreline scarps by a nonlinear diffusion model. Journal of Geophysical Research 92, 12,857–12,867. Arya, L.M., Paris, J.F., 1981. A physico-empirical model to predict the soil moisture characteristic from particle size distribution and bulk density data. Soil Science Society of America Journal 45, 1023–1030. Baum, R.L., Harp, E.L., Hultman, W.A., 2000. Map showing recent and historic landslide activity on coastal bluffs of Puget Sound between Shilshole Bay and Everett, Washington. U.S. Geological Survey Miscellaneous Field Studies Map MF-2346.

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