a physically based model for the hydrologic control on...

A physically based model for the hydrologic control on shallow

landsliding

Renzo Rosso,1 Maria Cristina Rulli,1 and Giovanni Vannucchi2

Received 17 June 2005; revised 29 November 2005; accepted 6 March 2006; published 20 June 2006.

[1] Both rainfall intensity and duration take part in determining the hydrologic conditionsfavorable to the occurrence of shallow landslides. Hydrogeomorphic models of slopestability generally account for the dependence of landsliding on soil mechanical andtopographic factors, while the role of rainfall duration is seldom considered within aprocess-based approach. To investigate the effect of different climate drivers on slopestability, we developed a modeling framework that accounts for the variability of extremerainfall rate with the duration of rainfall events. The slope stability component includes thekey characteristics of the soil mantle, i.e., angle of shearing resistance, void ratio, andspecific gravity of solids. Hillslope hydrology is modeled by coupling the conservation ofmass of soil water with the Darcy’s law used to describe seepage flow. This yields a simpleanalytical model capable of describing combined effect of duration and intensity of aprecipitation episode in triggering shallow landslides. Dimensionless variables areintroduced to investigate model sensitivity. Finally, coupling this model with the simplescaling model for the frequency of storm precipitation can help in understanding theclimate control on landscape evolution. This leads to predict the temporal scale of hillslopeevolution associated with the occurrence of shallow landslides. Model application isshown for the Mettman Ridge study area in Oregon, United States.

Citation: Rosso, R., M. C. Rulli, and G. Vannucchi (2006), A physically based model for the hydrologic control on shallow

landsliding, Water Resour. Res., 42, W06410, doi:10.1029/2005WR004369.

1. Introduction

[2] Mass wasting is the major landform shaping processin mountainous and steep terrain. Many landslides resultfrom infrequent meteorological or seismic events thatinduce unstable conditions on otherwise stable slopes oraccelerate movements on unstable slopes. Thus the delicateequilibrium between the resistance of the soil to failure andthe gravitational forces tending to move the soil downslopecan be easily upset by external factors, such as rainstorms,snowmelt, and vegetation management. The major trigger-ing mechanism for slope failures is the build-up of soil porewater pressure. This can occur at the contact between thesoil mantle and the bedrock [Pierson, 1977; Sidle andSwanston, 1982; Megahan, 1983] or at the discontinuitysurface determined by the wetting front during heavyrainfall events. The control factors of landslide susceptibil-ity in a given area may be subdivided into two categories:quasi-static and dynamic. The quasi-static variables dealwith geology, geotechnical properties, slope gradient,aspect, and long-term drainage patterns. The dynamicvariables deal with hydrological processes and humanactivities, which trigger mass movement in an area of givensusceptibility.

[3] Landslides hazard assessment is based on a variety ofapproaches and models. Most rely on either multivariatecorrelation between mapped (observed) landslides andlandscape attributes [e.g., Carrara et al., 1991, 1995;Carrara, 1983; Chung et al., 1995], or general associationsof landslides hazard from rankings based on slope lithol-ogy, landform or geological structure [e.g., Campbell,1975; Hollingsworth and Kovacs, 1981; Lanyon and Hall,1983; De Graff and Canuti, 1988; Seely and West, 1990;Montgomery et al., 1991; Neimann and Howes, 1991;Derbyshire et al., 1995]. Antecedent precipitation amounts[e.g., Campbell, 1975;Wieczorek, 1987; Canuti et al., 1985]and daily rainfall rate [e.g., Crozier, 1999; Glade et al.,2000] were further introduced as triggering factors ofshallow landsliding. The statistical approach can providean insight of multifaceted processes involved in shallowlandsliding occurrence, and useful assessments of suscepti-bility to shallow landslide hazard in large areas. However,the results are very sensitive to the data set used in theanalysis, and it is not straightforward to derive the hazard(i.e., probability of occurrence) from susceptibility.[4] Other scientists [e.g., Caine, 1980; Cancelli and

Nova, 1985; Cannon and Ellen, 1985; Keefer et al., 1987;Wieczorek, 1987; Wieczorek et al., 2000] analyzed theintensity and duration of rainfalls triggering landslides.They built the critical rainfall threshold curves, defined asenvelope curves of all rainfall triggering landslides eventsfor a certain geographic area. Because of the lack of aprocess-based analysis, this method is unable to assess thestability of a particular slope with respect to certain stormcharacteristics and it does not predict the return period of

1Hydrology Section, Dipartimento di Ingegneria Idraulica, Ambientale,Infrastrutture Viarie, Rilevamento, Politecnico di Milano, Milan, Italy.

2Geomechanics Section, Department of Civil Engineering, University ofFlorence, Florence, Italy.

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the landslide-triggering precipitation [D’Odorico et al.,2005].[5] Another approach deals with spatially distributed and

physically based models coupling slope stability equationwith a hillslope hydrological model. This can provide aninsight of triggering processes of shallow landslides at thebasin scale, also accounting for the spatial variability ofthe involved parameters. Some of these models consider thebuildup of soil pore water pressure deriving uniquely fromthe increase of a saturated layer above a predefined criticalslip surface approaching subsurface flow in different way,i.e., steady state or dynamic [e.g., Okimura and Ichikawa,1985;Montgomery and Dietrich, 1994;Dietrich et al., 1995;Pack et al., 1998; Wu and Sidle, 1995; Casadei et al., 2003;Iida, 1999, 2004]. Montgomery and Dietrich [1994],Dietrich et al. [1992, 1993, 1995, 2001], and Pack et al.[1998] treat the subsurface flow as steady state analyzing thetopographic control on the pore pressure. Using the porepressure in the slope stability equation permits to estimateslope stability and produce maps of relative potential ofshallow landslides [Montgomery et al., 2000; Dietrich etal., 2001].[6] Montgomery and Dietrich [1994] developed a simple

model for the topographic influence on shallow landslidesinitiation by coupling digital terrain data with near-surfacethroughflow and slope stability models. The hydrologicalmodel is based on the flow tube approach [see, e.g.,O’Loughlin, 1986; Dawes and Short, 1994; Moore, 1988;Moore and Grayson, 1991]. It predicts the degree of soilsaturation in response to steady state rainfall for topographicelements defined by the intersection of contours and flowboundaries. The slope stability component uses this relativesoil saturation to analyze the stability of each topographicelement for the case of cohesionless soils and of spatiallyconstant thickness and saturated conductivity. Furtherdevelopments of this model include SHALSTAB, a free-ware software application to evaluate slope instabilityassociated with the potential occurrence of shallow land-sliding (W. E. Dietrich and D. R. Montgomery, SHALSTAB:A digital terrain model for mapping shallow landslidepotential, available at http://ist-socrates.berkeley.edu/�geomorph/shalstab/). Several applications show this ap-proach to be capable of capturing the spatial variability ofshallow landslides hazard, because of the essential role oftopographic control in shallow landsliding. This approachpermits uncalibrated predictions and has proven reasonablysuccessful, though there is a tendency for overpredictionto occur, depending on the quality of topographic data[Dietrich et al., 2001].[7] The approach by Montgomery and Dietrich [1994]

does not account for transient movement of soil water. Itonly accounts for density and friction angle of the soil inanalyzing slope stability, neglecting other soil character-istics and the moisture content in the soil layer above thegroundwater table. This simplification can affect modelcapability of predicting shallow landslide potential becausethe steady flow condition can be unrealistic in the course ofrainstorm.[8] Unsteady flow was approached by Okimura and

Ichikawa [1985] modeling shallow subsurface flow withfinite difference model, Wu and Sidle [1995] introducing acontour based distributed physically based model that

couples the infinite slope approach to slope stability witha groundwater kinematic wave model, also accounting forchanging vegetation root strength in time. Casadei et al.[2003] linked a dynamic spatially distributed shallow sub-surface runoff model accounting for evapotranspiration andunsaturated zone storage to an infinite slope model. Iida[2004] presented a hydrogeomorphological model consid-ering both the stochastic character of rainfall intensity andduration and the deterministic aspects controlling slopestability. The unsteady subsurface flow producing saturatedsoil depth was investigated using a simplified conceptualmodel.[9] These models provide an insight of the triggering

mechanisms of shallow landslides, but they include acomplex parameterization of hillslope properties and drain-age patterns, so requiring detailed field data analysis. Thebuildup of pore water pressure as generated by the advanceof the wetting front was investigated by Pradel and Raad[1993] using the Green-Ampt infiltration model to analyzethe critical wetting front position triggering failure. Rulli etal. [1999] developed a distributed model coupling the Greenand Ampt infiltration model, kinematic subsurface flow andinfinite slope stability model in order to investigate shallowlandslides in areas where Hortonian runoff generation ispredominant. Iverson [2000] provided an insight of physicalmechanism underlying landslide triggering by rain infiltra-tion by solving the Richard’s equation. The model linksslope failure and landslide movement to groundwater pres-sure heads that change in response to rainfall. These modelsprovide an insight of the triggering mechanisms of shallowlandslides, but they do not emphasize the link betweenrainfall duration and intensity and shallow landslidetriggering.[10] Reid [1994] considered both rainfall intensity and

rainfall duration in the stability analysis. D’Odorico et al.[2005] enhanced the Reid’s approach by coupling the short-term infiltration model by Iverson [2000] and the long-termsteady state topography driven subsurface flow [Montgomeryand Dietrich, 1994] and analyzed the return period oflandslide triggering precipitation using hyetograph atdifferent shapes. They assumed that the pore pressure tran-sient observable in the course of a rainfall is due to theunsteady vertical flow through the soil profile, while slope-parallel subsurface flow is assumed to at a longer timescaleand to determine the pre storm wetness conditions.[11] The present paper improves the pioneering approach

by Montgomery and Dietrich [1994] indicating that simplemodels coupling soil mechanics with hydrology can providean insight of shallow landslide initiation useful for mappingthe potential hazard of landslide occurrence. Accordingly,the topographic description of hillslope elements is basedon the flow tube approach. The hillslope stability modelaccounts for the key characteristics of the soil mantle, i.e.,angle of shearing resistance, void ratio and specific gravityof solids (see Section 2). Hillslope hydrology is modeled bycoupling the conservation of mass of soil water with theDarcy’s law used to describe seepage flow (see section 3).This yields a simple model capable of accounting for thecombined effect of storm duration and intensity in thetriggering mechanism of shallow landslides (see section 4).Model application is then reported in section 5 for theMettman Ridge study catchment in Oregon in order to

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W06410 ROSSO ET AL.: HYDROLOGIC CONTROL ON SHALLOW LANDSLIDING W06410

compare model performance with that descending fromapplication of the original approach by Montgomery andDietrich [1994]. Finally, coupling of the model for thehydrologic control on shallow landsliding with the simplescaling model for the frequency of storm precipitation[Burlando and Rosso, 1996] can help understanding theclimate control on landscape evolution associated with theoccurrence of shallow landslides (see Section 6).

2. Slope Stability

[12] In mountainous and hilly areas, the surface of theslope is quite often underlain by a plane of weakness lyingparallel to it. This potential failure surface generally lies at adepth z below the surface, and this depth is small ifcompared with the length of the slope. Because the thick-ness of the soil mantle is much smaller then the length of theslope, one can generally assume that edging effects arenegligible, so one can determine the safety factor of theslope against slip, FS, from the analysis of a wedge or sliceof material of unit width and unit thickness [Skempton andDeLory, 1957].[13] Let q denote the slope angle to the horizontal, and z

the depth of the potential failure plane, as shown in Figure 1.The water table is taken to be parallel to the slope at a heighth = w z above the failure plane, with 0 � w � 1. Steadyseepage is assumed to occur in the direction parallel to theslope. The side forces for any vertical slice are equal andopposite, and the stress conditions are the same at any pointon the failure surface. One also assumes that the rigidperfectly plastic rheologic model holds for the soil, that is,there is null strain until failure and shear strength is constantafter the failure independently on strain.[14] The shear strength of the soil along the potential

failure plane is

tf ¼ c0 þ s� uð Þ tanf0 ð1Þ

with c0 denoting soil cohesion, s the normal total stress, uthe pore water pressure, and f0 the angle of shearingresistance of the soil mantle. If one denotes with t the shearstress, the safety factor is

FS ¼ tft: ð2Þ

If one denotes with g the average bulk unit weight of soilabove the groundwater level, and with gsat the saturated unitweight of soil under the groundwater level, the expressionsfor s, t and u are

s ¼ 1� wð Þ gþ w gsat½ � z cos2 q; ð3aÞ

t ¼ 1� wð Þ gþ w gsat½ � z sin q cos q; ð3bÞ

u ¼ w z gw cos2 q: ð3cÞ

[15] Substituting equations (3a)–(3c) for s, t and u inequations (1) and (2) yields the general expression for thesafety factor in the form

FS ¼ c0 þ 1� wð Þ gþ w g0½ � z cos2 q tanf0

1� wð Þ gþ w gsat½ � z sin q cosJ : ð4Þ

where g0 = gsat – gw is the submerged unit weight of soil.[16] For cohesionless soils and normally consolidated

clays cohesion is negligible, so one can take c0 = 0 inpractice. If c0 = 0 in equation (4) the safety factor isindependent of thickness of soil mantle z, that is

FS ¼ 1� wð Þ gþ w g0½ �1� wð Þ gþ w gsat½ �

tanf0

tan q: ð5Þ

Let denote with Gs = gs/gw the specific gravity of solids,Sr = Vw/Vv the average degree of saturation and e = Vv/Vs theaverage void ratio above the groundwater table (being Vv,Vs, and Vw the volume of voids, that of solids and that ofwater in the control volume, respectively). Thus

g

gw¼ Gs þ e Sr

1þ e; ð6aÞ

gsat

gw¼ Gs þ e

1þ e; ð6bÞ

g0

gw¼ Gs � 1

1þ e: ð6cÞ

These are substituted for g, gsat and g0 in equation (5) toobtain

FS ¼ Gs þ e Sr � w 1þ e Srð Þ½ �Gs þ e Sr þ w e 1� Srð Þ½ �

tanf0

tan qð7Þ

[17] The following special cases are of interest. If w = 0,i.e., the groundwater level lies at the potential failuresurface,

FS ¼ tanf0

tan q: ð8Þ

Figure 1. One-dimensional sketch for slope equilibrium.

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If w = 1, i.e., the water table lies at the surface of the slope,

FS ¼ Gs � 1ð ÞGs þ eð Þ

tanf0

tan q¼ g0

gsat tanf

0

tan q: ð9Þ

One notes that the value of FS descending from equation (9)is about one half of that computed from equation (8)because g0 ffi gsat/2 in practice.[18] If 0 � w � 1, i.e., the water table lies between the

potential failure surface and the slope surface, the limitequilibrium condition (FS = 1) occurs when the groundwa-ter level index w assumes a critical value, say, wCR. Thismeans that the slope is stable for w not exceeding wCR. Thisis given by

wCR ¼Gs þ e Srð Þ 1� tan q

tanf0

� �

1þ e� e 1� Srð Þ 1� tan qtanf0

� � ð10Þ

[19] For given Gs and e, equations (8), (9) and (10) yieldfour states depending on actual slope and groundwaterlevel index. (1) If tanq/tanf0 � (Gs � 1)/(Gs + e), theslope is unconditionally stable. (2) If tanq/tanf0 1 theslope is unconditionally unstable. (3) If (Gs � 1)/(Gs + e)< tanq/tanf0 < 1 and w < wCR, the slope is stable. (4) If(Gs � 1)/(Gs + e) < tanq/tanf0 < 1 and, w > wCR theslope is unstable. The present approach improves that

introduced by Montgomery and Dietrich [1994] whoneglected the effect of the void ratio and the degree ofsoil saturation above the groundwater table.[20] Figure 2 shows these four states for soils with

specific gravity Gs = 2.7 and different values of averagevoid ratio and average saturation degree of soil above thegroundwater table. One notes that Sr has a negligibleinfluence on slope stability. Figure 3 shows model sensitiv-ity to void ratio for different average saturation degree ofsoil above the groundwater table. One notes that Sr can beclose to unity in practice, so the influence of void ratio canbe moderate.

3. Hillslope Hydrology

[21] The approach assumes that overland flow is gener-ated by saturation excess. A hillslope is divided in topo-graphic elements defined by intersection of contour andflow tube boundaries orthogonal to the contours, as shownin Figure 4.[22] Let consider a control volume defined as the subba-

sin closed at one specific topographic element. It is assumedto occupy the entire flow domain by integrating out thespatial dependence of flux terms. Thus all flux terms arelocated on the boundaries of the flow domain and they canbe grouped into bulk inflow rates and outflow rates. Letdenote with p the net rainfall, i.e., precipitation less evapo-transpiration and deep drainage into the bedrock, and a theupslope contributing area (i.e., the cumulative drainage area

Figure 2. Soil mantle equilibrium for varying relative slope ratio, tanq/tanf0, versus relativegroundwater depth, h/z, with specific gravity Gs = 2.7 and void index e of (a) 0.5, (b) 1.0, (c) 1.5,and (d) 2.0.

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of all topographic elements draining into the examinedelement) draining across b, the contour length of the lowerbound to each element. Under the assumptions of (1) nullsoil volumetric strain (i.e., average void ratio is constant)and (2) constant average soil saturation degree above thegroundwater table (i.e., Sr is constant), conservation of massyields

ap� q ¼ dS

dt0¼ a e

1þ e 1� Srð Þ dh

dt0; for h � z; ð11Þ

and

ap� q� r ¼ 0; for h > z; ð12Þ

with t0 denoting time after the beginning of the storm, S thewater storage in the element, and r the overland flowdischarge occurring when soil is saturated (i.e., Sr = 1).[23] The Darcy’s law provides the seepage flow in the

groundwater table. Thus

q ¼ bh cos qð ÞK tan q ¼ bhK sin q; ð13Þ

where K is the saturated conductivity of the soil, and tanq isthe head gradient assumed to be parallel to the local groundslope.

Figure 3. Model sensitivity to void ratio, e, for different degree of soil saturation above the groundwatertable: (a) dry, (b and c) intermediate, and (d) saturated.

Figure 4. Sketch of an elementary drainage area: (a)planar view and (b) hydrological fluxes.


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[24] Substituting equation (13) into (11) and integratingthe differential equation for the initial condition of stablepiezometric at the depth of hi(0) = hi, yields

h ¼ apz

Tb sin q1� exp � 1þ e

e� eSr

Tb sin qaz

t

� ��

þ hi exp � 1þ e

e� eSr

Tb sin qaz

t

� �; for

ap

Tb sin q> 1; ð14Þ

for a rainfall episode with duration t, with T = Kz denotingthe hydraulic transmissivity, i.e., the vertical integral alongthe soil depth of saturated conductivity of soil.[25] By considering the simple case of the initial condi-

tion of stable piezometric at depth of h(0) = 0 and intro-ducing the saturation precipitation rate, only depending ongeometric and hydrologic properties of the hillslope and notfrom rainfall,

p* ¼ Tb sin qa

ð15Þ

yields

h ¼ p

p*z 1� exp � 1

A1

p*

zt

� �� ; for p=p* � 1; ð16Þ

with

A1 ¼e

1þ e 1� Srð Þ ð17Þ

denoting a soil dimensionless factor.[26] For ap/Tb sin q > 1, i.e., p/p* > 1, one obtains

h ¼p

p*z 1� exp � 1

A1

p*

zt

� �� ;

if t � t*

z; if t > t*

8><>: ð18Þ

and

t* ¼ �A1

z

p* ln 1� p*

p

� �: ð19Þ

One notes that the characteristic time t* depends on threefactors, i.e., the dimensionless soil factor A1 describingactual soil physics, z/p* combining hillslope geomorphol-ogy with soil mantle characteristics, and p/p*, that describesthe relative precipitation rate to the critical value.[27] Figures 5 and 6 show the variability of h and q + r,

respectively, with storm duration for different values of thetopographic ratio a/b. For given slope, soil thickness, voidratio, degree of saturation of the soil above the groundwatertable, soil transmissivity, drainage area and net rainfall rate,an increase of the topographic ratio a/b yields the steadystate thickness of the groundwater table to increase. How-ever, the rate of increase of h in time is much higher forlarge a/b than that characterizing small values of a/b. Thuslarge values of a/b yield runoff production to rapidlyachieve steady state conditions, while much more time isneeded for elementary areas with small values of a/b toachieve steady state runoff production.[28] Introducing the groundwater level index, w = h/z,

yields

w ¼ p

p*1� exp � 1

A1

x� ��

; for p=p* � 1; ð20Þ

and, for p/p* > 1,

w ¼

p

p*1� exp � 1

A1

x� ��

;if x � �A1 ln 1� p*

p

� �

1; if x > �A1 ln 1� p*

p

� �:

8>><>>: ð21Þ

with x = (p*/z)t denoting a dimensionless time. Figure 7shows the variability of the groundwater level index, w, withdimensionless time, x, for different values of dimensionlessprecipitation rate p/p*.

Figure 5. Height of the groundwater table versus stormduration for different a/b under specified conditions.

Figure 6. Runoff production versus storm duration fordifferent a/b under specified conditions.

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[29] Montgomery and Dietrich [1994] assumed that asteady state hydrologic model can mimic the effect oftransient rainstorm with effective rainfall greater than asteady state value. Here one finds that the combined effectof rain rate and duration on hillslope stability can besignificant. Also, the model is capable of describing thethreshold effect occurring when full saturation occurs in thesoil mantle, so triggering overland flow. This occurs, e.g.,for p/p* equal to 5 in Figure 7.

4. Precipitation Threshold for Slope Instability

[30] Coupling hillslope hydrology with geomechanicsyields landslide triggering by precipitation. This is obtainedby substituting the left-hand side of equation (10) with theright-hand side of equation (14) expressed in terms ofgroundwater level index, w. One obtains

ap

bT sin q1� exp � 1þ e

e� eSr

bK sin qa

t

� ��

þ hi

zexp � 1þ e

e� eSr

Tb sin qaz

t

� �

¼Gs þ e Srð Þ 1� tan q

tanf0

� �


� � ; ð22Þ

Solving equation (22) for p one obtains the rainfall ratethreshold causing instability in the soil mantle underanalysis, that is

pCR tð Þ ¼

Tbasin q

Gs þ e Srð Þ 1� tan qtanf0

� �

1þ e � e 1� Srð Þ 1� tan qtanf0

� �� hi

zexp � 1þ e

e� eSr

Tb sin qaz

t

� �2664

3775

1� exp � 1þ e

e� eSr

Tb sin qaz

t

� � ;

with sub CR indicating the critical conditions for landslideinitiation. Note that the precipitation threshold pCR given byequation (23) is a function of measurable quantitiesdescribing the physical properties of the hillslope.[31] It is seen that the value taken by pCR strongly

depends on duration t of precipitation occurring at theconstant rate pCR, as displayed, e.g., in the examples shownin Figure 8 highlighting that the diffusive character of theequations smooth sudden peaks in rainfall intensity. There-fore the steady state approach by Montgomery and Dietrich[1994] can overestimate landslide triggering conditionsespecially for storms with short duration, typically of lessthan 1 or 2 days. This also depends on initial moistureconditions, as one can see by comparing, e.g., the thresholds

Figure 7. Groundwater level index versus dimensionless time for different values of dimensionlessprecipitation rate, p/p*, and soil dimensionless factor, A1.

ð23Þ


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shown in Figure 8d (Sr = 0.75) with those shown in theother ones (Sr = 0.5). Finally, one can observe that thetopographic index (a/b) still remains a fundamental factor indetermining the rainfall threshold for shallow landslideinitiation, as shown, e.g., in Figure 8. The above approachfor obtaining the rainfall rate threshold causing instabilitycan be generalized to the case of nonconstant hyetograph.The simple procedure considers the hyetograph subdividedin n time steps. For each time step the model computes thecorresponding h considering as initial condition hi theh calculated at the previous step. Finally Pcr will be obtainedby substituting in equation (23) as hi the hn–1.[32] Equation (23) can be written in dimensionless form

as

h ¼ pCR

p*¼

A2 1� tan qtanf0

� �

A3�A4 1� tan qtanf0

� �� hiz

exp � 1

A1

x� �

1� exp � 1

A1

x� � ; ð24Þ

with

A2 ¼ Gs þ eSr; ð25aÞ

A3 ¼ 1þ e; ð25bÞ

A4 ¼ e 1þ Srð Þ; ð25cÞ

denoting dimensionless soil factors depending on specificgravity of soil particles, void ratio and saturation degree ofsoil above the groundwater table. The dimensionless criticalprecipitation rate h = pCR/p* is plotted against dimension-less time x = (p*/z)t in Figure 9 for different values ofrelative slope ratio (tanq/tanf0).

5. Application

5.1. Study Area

[33] The Mettman Ridge study site in Oregon was inves-tigated in order to compare the present approach with thatby Montgomery and Dietrich [1994] who reported this casestudy in their paper. The available data set includes high-

Figure 8. Relationship between critical rainfall rate and storm duration for different a/b under specifiedconditions (a = 0.02 km2, f0 = 42, e = 1, Gs = 2.65, z = 1 m) and different combination of values takenby q, Sr and T.

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resolution digital elevation data (topography was createdusing airborne laser altimetry with a data density of about2.5 m (Shalstab tutorial [Dietrich et al., 2001; W. E.Dietrich and D. R. Montgomery, SHALSTAB: A digitalterrain model for mapping shallow landslide potential,available at http://ist-socrates.berkeley.edu/�geomorph/shalstab/]) field tests for hydrologic and geomechanicparameterization of soil, observations of hydrologic varia-bles and field surveys for shallow landslide mapping[Montgomery, 1991; Montgomery et al., 1997; Torres etal., 1998; Yee and Harr, 1977; Schroeder and Alto, 1983;Burroughs at al., 1985]. Following previous analysis byMontgomery and Dietrich [1994], soil and hydrologicparameters are treated as spatially uniform.[34] The study site consists of a 0.3 km2 drainage basin

along Mettman ridge in the Coast Range, north of CoosBay, Oregon. The area is highly dissected and characterizedby narrow ridge tops and steep slopes. Many shallowlandslides and debris flows were observed occurred in thisarea from 1987 to 1992. The storm events observed totrigger shallow landsliding in this catchment had an esti-mated return period ranging from one to two years, with24 hour rainfall rate ranging from 50 to 75 mm/d. Digitalelevation data with 5 m grid resolution were used.

[35] The colluvial soil in the study area is silty sandranging in thickness from roughly 0.1–0.5 m on topographicnoses to over 2 m in topographic hollows [Montgomery,1991]. Where the slope exceeds 45, bedrock crops out.Saturated hydraulic soil conductivity of colluvial soildeclines from about 10�3 m/s at the ground surface to about10�4 m/s at the depth of 2m [Montgomery, 1991]. Follow-ing Montgomery and Dietrich [1994] and based on thesedata, hydraulic soil transmissivity is taken to equal 65 m2/d.Saturated soil bulk density is taken to equal 1600 kg/m3

[Torres et al., 1998]. Several authors [Yee and Harr, 1977;Schroeder and Alto, 1983; Burroughs et al., 1985] reportvalues of the friction angle for soil developed on sandstonesin the Oregon Coast Range varying from about 35 to 44.Following Montgomery and Dietrich [1994] the effect ofroots on shear stress resistance was taken into account byconsidering an angle of shearing resistance f0 of 45, i.e.,increasing the soil strength as much as 1.4 times the actualshear strength. Other model parameters are T = 65 m2/d,Gs = 2.60 and z = 0.75 m.

5.2. Simulation

[36] The same basin partitioning used by Montgomeryand Dietrich [1994] is adopted here for model benchmark-

Figure 9. Dimensionless critical rainfall rate versus dimensionless time for different values of relativeslope ratio and specified soil parameters e, Sr and Gs.


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ing. Accordingly, the catchment is divided into topographicelements defined by the intersection of contours and flowtubes boundaries orthogonal to contours.[37] The approach by Montgomery and Dietrich [1994]

is remodeled using the same watershed partitioning pro-cedure (i.e., flow tube approach), the same equations forslope stability analysis and for hydrologic fluxes and thesame initial conditions (i.e., h(0) = 0). Shallow landslidinginstability is represented in terms of log(q/T), withT denoting soil transmissivity and q steady state precipi-tation rate. In Figure 10 the resulting map of MettmanRidge catchment is shown where the shallow landslidingprone areas predicted by Montgomery and Dietrich modelare delineated in terms of steady state rainfall intensity[mm/d] necessary for slope instability in each topographicelement. These are also compared with the observedshallow landslides that occurred from 1987 to 1992. Thisindicates a rather satisfactory model performance in repro-ducing the observed landslides, thus indicating a strongtopographic control on shallow landsliding, but some ofthe observed landslides are poorly reproduced. Also, thereare large areas with high instability where no landslidingoccurred during the period of observation. Application of theMontgomery and Dietrich approach yields to overestimatethe extent of instable areas, and to slightly underestimate thestable ones.[38] Application of the present model requires to compute

net rainfall, that yields shallow subsurface flow according toequations (11) and (12). This is further routed throughoutflow tubes, so computing the local flow throughout eachtopographic element using equation (13).[39] One can run the model in different modes. Element

stability can be simulated for a given rainfall intensity

pattern, not necessary steady state and for given initialmoisture conditions. This is the diagnostic mode, useful toinvestigate hillslope response to a given precipitationinput. Alternatively, the critical rainfall needed to causeinstability for each topographic element can be determined.This is the predictive mode, useful to assess the probabilityof occurrence of a shallow landslide riggered by stormprecipitation. Because the first mode is useful to testmodel performance and to investigate possible improve-ments of previous work, it is used here. The predictivemode is useful for hazard assessment, and it will used insection 6.[40] Running the model with the same input data used

by Montgomery and Dietrich [1994] considering rainfallduration equal to one day, yields the instability patternshown in Figure 11. One can compare the predictedpatterns of shallow landsliding instability with the ob-served shallow landslides that occurred from 1987 to1992. It is seen that observed landslide spots are wellreproduced by the model. Overlapping of predicted insta-bility thresholds with observed landslides shows the modelto be capable of capturing the hydrologic control onshallow landsliding occurrence. Also, the predicted insta-bility areas where landslides did not occur during theperiod of observation are minor. This indicates that thepresent approach can preserve the capability of that byMontgomery and Dietrich [1994] in reproducing topo-graphic control, but it improves model capability ofreproducing hydrologic control, because transient hydro-logic conditions play an important role in landslide trig-gering. Table 1 shows model comparisons in terms ofunstable and stable cells. Note that a cell is here anytopographic element into which the Mettman Ridge basin

Figure 10. Map of Mettman Ridge catchment showing shallow landsliding prone areas predicted fromthe application of Montgomery and Dietrich model delineated in terms of steady state rainfall intensity[mm/d] necessary for slope instability. Model parameters are f0 = 45, T = 65 m2/d, and rs = 1600 kg/m3.Observed shallow landslides that occurred from 1987 to 1992 are also displayed.

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is partitioned. By considering five indexes, defined asfollows,

I1 ¼ UR=UO

I2 ¼ UR=UC

I3 ¼ SR=SO

I4 ¼ SR=SC

ITOT ¼ Av ðI1 þ I2 þ I3 þ I4Þ

where

UO observed unstable cells;SO observed stable cells;UC simulated unstable cells;UR rightly simulated unstable cells;

UW wrongly simulated unstable cells;SC simulated stable cells;SR rightly simulated stable cells;SW wrongly simulated stable cells;

it is shown how the RRV model preserves the capability ofthe MD model in predicting shallow landslides reducingfalse positives.

6. Temporal Scale of Hillslope Evolution

[41] The intensity-duration-frequency relationship (hereaf-ter referred to as IDF) for station precipitation provides theprobability that a given rainfall rate (time average) is exceededover a specified duration of the precipitation episode. GuptaandWaymire [1990] introducedtheconceptofscale invarianceor ‘‘simple scaling’’ for temporal precipitation. Under theassumption of scale invariance, Burlando and Rosso [1996]showed the rainfall rate p, exceeded with a given proba-

Figure 11. Map of Mettman Ridge catchment showing shallow landsliding prone areas in terms ofsteady state rainfall intensity [mm/d] necessary for slope instability as predicted from the application ofthe present model. Model parameters are f0 = 45, T = 65 m2/d, rs = 1600 kg/m3 and Gs = 2.60. Observedshallow landslides that occurred from 1987 to 1992 are also displayed.

Table 1. Comparison Between the Montgomery and Dietrich Model and the Present Model in Terms of Model Performance Indexesa

Description Rosso-Rulli-Vannucchi Montgomery-Dietrich

N total cells of watershed 698029 698029UO observed unstable cells 9812 9812SO observed stable cells 688217 688217UC simulated unstable cells 158596 118290UR rightly simulated unstable cells 6991 5506UW wrongly simulated unstable cells 151605 112784SC simulated stable cells 539433 579739SR rightly simulated stable cells 536612 575433SW wrongly simulated stable cells 2821 4306I1 = UR/UO rightly simulated unstable cells/observed unstable cells 71.2% 56.1%I2 = UR/UC rightly simulated unstable cells/simulated unstable cells 4.4% 4.7%I3 = SR/SO rightly simulated stable cells/observed stable cells 78.0% 83.6%I4 = SR/SC rightly simulated stable cells/simulated stable cells 99.5% 99.3%ITOT 63.3% 60.9%

aA rainfall event of 100 mm/d causing instability is considered.


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bility, to scale with rainfall duration t as a power functionof t. The scaling exponent n of this function is the climaticsignature of the site under exam [see, e.g., De Michele etal., 2001]. The rainfall rate pF that can be exceeded with aprobability of (1 – F) in a year can be thus determined as

pF tð Þ ¼ m1xF t� 1�nð Þ; ð26Þ

with m1 denoting the expected value of annual maximumrainfall depth for the unit duration, n the scaling exponent,with 0 < n < 0.6, and xF the F quantile of the renormalizedvariate, that is, the ratio between rainfall depth and itsexpectation, independent of duration because of simplescaling. Equation (26) provides the general form of the IDFcurves independent of the underlying probability distribu-tion, with xF only depending on the parent distribution usedto fit extreme value data. Therefore application requires tospecify a probability distribution model to extreme values ofthe observed rainfall rates. For the general extreme value(GEV) distribution one has

xT ¼ eþ ak

1� e�ky� �

; ð27Þ

with k, a, and e denoting the shape, scale and locationparametersof theGEV,respectively,andy theGumbel reducedvariate [see, e.g., Kottegoda and Rosso, 1997, pp. 472–473]. This only depends on the return period, TR, that is

y ¼ � ln � lnFð Þ ¼ � ln lnTR

TR � 1

� �; ð28Þ

with TR = 1 / (1 – F).[42] Because equation (23) gives the critical rainfall depth

as a function of storm duration, i.e., pCR = pCR(t), couplingof equations (23) and (26) yields

Tb

asin q

Gs þ e Srð Þ 1� tan qtanf0

� �

1þ e � e 1� Srð Þ 1� tan qtanf0

� �� hi

zexp � 1þ e

e� eSr

Tb sin qaz

t

� �2664

3775

1� exp � 1þ e

e� eSr

Tb sin qaz

t

� �

¼ m1xF t� 1�nð Þ:

This provides the critical frequency for initiation of hillslopeinstability, as represented by the value of xF that satisfiesequation (23) for a given rainfall duration t. One notes thatthe critical duration tCR that yields the minimum returnperiod for slope instability can be determined by the furthercondition @pF(t)/@t = @pCR(t)/@t, for t = tCR. This yields

Tbasin q Gs þ e Srð Þ 1� tan q

tanf0

� ��


� �2664

þ hi

z

1þ e

e� eSr

Tb sin qaz

exp � 1þ e

e� eSr

Tb sin qaz

t

� �#

1 þ e

e � eSr

Tb sin qaz

exp � 1þ e

e� eSr

Tb sin qaz

t

� �

1� exp � 1þ e

e� eSr

Tb sin qaz

t

� �� 2

¼ � 1� nð Þm1xF t� 1�nð Þ�1

The solution is given by that value of F or T yielding pF(t) =pCR(t) for t = tCR, being pF(t) < pCR(t) for any t 6¼ tCR, that is@pF(t)/@t = @pCR(t)/@t, for t = tCR. One notes that the solutionis unique, because from equation (23) @2pCR/@t

2 < 0, for t >0; and, conversely, from equation (26) one gets @2pF/@t

2 > 0for t > 0. Therefore the duration tCR has the physicalmeaning of critical duration of the precipitation that triggersshallow landsliding.[43] The example of Figure 12 shows that the topographic

index (a/b) strongly controls the return period of expectedhillslope instability, but is also has a not negligible influenceon the critical duration of precipitation triggering shallowlandslides. It is seen that increasing of (a/b) yields the returnperiod of potential failure to decrease, and the criticalduration of precipitation for hillslope instability to increase.In fact for a/b = 200 the return period of potential failure isfifteen years for the critical rainfall duration of about twodays, while for a/b = 100 the return period of potentialfailure is 75 years for the critical rainfall duration of aboutone day and for a/b = 50 the return period of potentialfailure is six hundred years for the critical rainfall durationof about half of a day.[44] Model application in predictive mode is carried out

for the Mettman Ridge study site in Oregon. The IDF curvesfor this site are estimated from 22 years of station precip-itation data recorded at Allegany, Oregon, using the GEVdistribution. The resulting IDF curves parameters are k =0.141, a = 0.3, e = 0.864, m1 = 17.945 mm and n = 0.55(see Figure 13). Figure 14 shows the resulting map ofMettman Ridge catchment where the return period ofrainfall causing potential failure is reported consideringinitial condition of stable piezometric at the depth ofbedrock, h(0) = 0. Table 2 shows the percentage of unstabletopographic elements for different duration and rainfalldepth. By considering different initial conditions, e.g.,stable piezometric located at depth of 0.15 m from the

Figure 12. Coupling of the relationship between criticalrainfall rate and duration of precipitation with the IDFcurves for the failure return period under specified hillslope(q = 35, a = 0.02 km2, f0 = 42, e = 1, Sr = 0.5, Gs = 2.65,T = 40 m2/d and z = 1 m) and climate conditions (k = �0.05,a = 0.30, e = 0.80, m1 = 68 mm and n = 0.45). Differentvalues of topographic index a/b and return period TR areconsidered.

(30)

ð29Þ

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bedrock, the results shown in Figure 15 and in Table 3 areobtained. One notes that estimating of the return period ofpotential shallow landsliding in a catchment can provideuseful information on the temporal scales of the hydrologiccontrol on landscape evolution. By comparing Figures 14and 15 it is shown how the return period of shallowlandsliding potential decrease at increasing the depth ofgroundwater table.[45] One notes that soil cohesion and the effects of

vegetation on the stability of the aggregate are not deeplyanalyzed in these exercises, so these simplifications canhighly affect the results in terms of return period ofrainfall triggered landslides. According to Montgomery andDietrich [1994], the effect of roots on shear stress resistancewas taken into account by increasing of 40% the value ofthe shear stress resistance angle. Schmidt et al. [2001] and

D’Odorico and Fagherazzi [2003] showed that soil and rootcohesion are inherently necessary to build up soil in steephollows, otherwise landslides would occur even for lightrainfall.

7. Conclusion

[46] An enhanced approach was developed here toimprove the representation of soil mechanics and hillslopehydrology in the pioneering model by Montgomery andDietrich [1994] of shallow landslide initiation, preservingoriginal model simplicity and straightforward application topredict rain triggered landslide hazard. Coupling of soilmechanics with hillslope hydrology yields a simple analyt-ical model capable of accounting for the combined effectof storm duration and intensity in the triggering mechanismof shallow landslides. The model preserves the capability ofthe approach by Montgomery and Dietrich [1994] inexplaining the topographic control on shallow landsliding,but it also provides an insight of hydrologic control onshallow landsliding. The approach to slope stabilityaccounts for the slope angle and the key characteristics ofthe soil mantle, i.e., angle of shearing resistance, void ratioand specific gravity of solids and the average degree ofsaturation of soil above the groundwater table. Therefore itaccounts for the effect of initial soil moisture conditionsconsidering both the depth of the groundwater table and thedegree of saturation above the groundwater table. Fourstates of slope stability descend from this model. Two statesare independent of the water content, i.e., unconditionallystable or instable slope, and two states depend on watercontent, i.e., stable or instable slope controlled by thethickness of the water table. It is seen that the role of soilparameters missed in the pioneering model by Montgomeryand Dietrich [1994] can be important in the characterizationof hillslope stability.

Figure 13. IDF curves for Allegany rain gauge station fordifferent values of return period TR.. Twenty-two years ofthe hourly rainfall data set are considered.

Figure 14. Map of Mettman Ridge catchment showing shallow landsliding prone areas as predictedfrom the application of the present model in terms of return period of potential failure considering aninitial condition of stable piezometric at the depth of bedrock, h(0) = 0.


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[47] Hillslope hydrology is modeled by coupling theconservation of mass of soil water with the Darcy’s lawused to describe seepage flow. This also yields a verysimple runoff production model, capable of properlyaccounting for topography, soil transmissivity, and initialsoil moisture. The model can describe transient precipitationand the threshold effect associated with the achievement offull saturation of the soil mantle, that were missed in thesteady state approach by Montgomery and Dietrich [1994].Model sensitivity was further investigated by introducingdimensionless variables, so one can investigate the effectsof topography, soil transmissivity and soil moisture abovethe groundwater table under unified approach.[48] These are the dimensionless critical precipitation

rate h = pCRa/(bT sinq) and duration x = (bT sinq)t/(az).

The relationship between h versus x is given in equation

(24) as parameterized by the combined characteristics oftopography and soil mantle.[49] Finally, the model for the hydrologic control on

shallow landsliding is coupled with the simple scalingmodel of storm precipitation in the frequency domain. Thiscan help understanding the climate control on landscapeevolution associated with rain triggered landslides. Modelapplication for the Mettman Ridge study area in Oregon

shows a satisfactory model performance in both diagnosticand predictive modes of operation.[50] Further research is needed (1) to assess the diagnos-

tic performance of the model under different geographicconditions, (2) to assess model limitations associated withthe present simplified approach by comparison with moredetailed geomechanical models, (3) to include more appro-priately the effects of roots and vegetation cover [see, e.g.,Schmidt et al., 2001; D’Odorico and Fagherazzi, 2003], (4)to assess the role of local discontinuities associated with,e.g., roads and other engineering works, and (5) to assessthe predictive performance of the model as compared to thatof traditional statistical approach.

Notation

c0 cohesion in effective stress terms.e void ratio.

FS safety factor.Gs specific gravity of solids.h height of the groundwater level above the failure

plane.hw hydrostatic head at midpoint of base of slice.Sr degree of saturation.

Table 2. Percentage of Unstable Topographic Elements for Different Duration and Rainfall Height Considering Initial Condition of

Stable Piezometric at the Depth of Bedrock, h(0) = 0

Stability Classes Montgomery-Dietrich Model

Rosso-Rulli-Vannucchi Model

Rainfall Duration, hours

6 12 18 24 36 48

Unconditionally stable 7.1% 7.1% 7.1% 7.1% 7.1% 7.1% 7.1%>200 50.3% 65.5% 57.3% 51.7% 49.9% 49.0% 48.7%100–200 11.1% 8.4% 13.4% 16.4% 16.1% 13.4% 12.4%50–100 7.5% 3.4% 5.4% 7.1% 8.4% 10.9% 10.9%0–50 10.6% 2.2% 3.5% 4.3% 5.0% 6.2% 7.5%Unconditionally unstable 13.3% 13.3% 13.3% 13.3% 13.3% 13.3% 13.3%

Figure 15. Map of Mettman Ridge catchment showing shallow landsliding prone areas as predictedfrom the application of the present model in terms of return period of potential failure considering aninitial condition of stable piezometric at the depth of hi, with hi(0) = 0.15 m.

14 of 16


u pore water pressure.z depth of the potential failure plane.f0 angle of shearing resistance.g bulk unit weight.g0 buoyant unit weight.

gsat saturated unit weight.gw unit weight of water.q angle of slope.s normal total stress.tf shear strength at failure.t shear stress.

w = h/z groundwater level index (0 � w � 1).wCR critical groundwater level index.

[51] Acknowledgments. This work was supported by EuropeanUnion through UE grant IRASMOS-018412. The authors thank W.E.Dietrich, who provided some of the data necessary for the analysis. Wealso thank J. Roering, the other two anonymous reviewers, and theAssociate Editor P. D’Odorico for scrupulous and thoughtful reviews thatgreatly improved the manuscript.

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Table 3. Percentage of Unstable Topographic Elements for Different Duration and Rainfall Height Considering Initial Condition of

Stable Piezometric at the Depth of hi, With hi(0) = 0.15 m

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��R. Rosso and M. C. Rulli, DIIAR, Politecnico di Milano, Piazza

Leonardo da Vinci 32, I-20133 Milano, Italy. ([email protected];[email protected])

G. Vannucchi, DIC, University of Florence, Via di Santa Marta 3,I-50139 Firenze, Italy. ([email protected])

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