LandslidesDOI 10.1007/s10346-013-0399-zReceived: 28 September 2012Accepted: 19 March 2013© Springer-Verlag Berlin Heidelberg 2013

Manouchehr Motamedi I Robert Y. Liang

Probabilistic landslide hazard assessment using Copulamodeling technique

Abstract A new probabilistic methodology for landslide hazard as-sessment in regional scale using Copula modeling technique ispresented. The current probabilistic landslide hazard analyses areperformed under the assumption that landslide hazard elements, suchas magnitude, frequency, and location, are independent. In this paper,a general approach is proposed to consider the possible dependenceamong hazard elements. Part of the Seattle, WA area was selected toevaluate the competence of the presented method. A total of 357 slopefailure events and their corresponding topography and geology datawere included in the study to develop and test themodel. Based on theresults, the mean success rates of the presented model in predictinglandslide occurrence are 90% in hazardous area and 12% in safelocations on average, while these success rates are 63 and 44% whenthese hazard elements were treated as mutually independent.

Keywords Landslide hazard assessment . Copula . Probability .

Seattle . WA

IntroductionLandslide is one of the most frequently occurring natural di-sasters on the earth. In recent years, landslide hazard assessmenthas played an important role in developing land utilization reg-ulations aimed at minimizing the loss of lives and damage toproperty. Varnes (1984) has defined landslide hazard as the prob-ability of occurrence of a potentially destructive landslide withina specified period of time and within a given geographical area. Amodified definition that embodies landslide magnitude was re-cently proposed by Guzzetti et al. (1999) as expressed by Eq. 1:

H ¼ P½M � m in a specified time period and in a given

location with specified preparatory factors�ð1Þ

where H is the landslide hazard value (0–1)/year, M is the land-slide magnitude, and m is a specified magnitude. In other words,Eq. 1 is a conditional probability of occurrence of a landslide witha magnitude larger than an arbitrarily chosen reference (or min-imum) amount within a specific time period and at a givenlocation. The “preparatory factors” in the equation refer to allof the specifics of the area, including geology, geometry, andgeotechnical properties. To evaluate landslide hazard quantita-tively in a regional scale, three probability components related tothe concepts of “magnitude,” “frequency,” and “location” need tobe defined and estimated.

After evaluation of the hazard component values for thestudy area, the common quantitative approach is to multiplytheir probabilities by assuming that they are mutually indepen-dent (e.g., Guzzetti et al. 2005; Jaiswal et al. 2010). In fact, inprobabilistic terms, landslide hazard is calculated as the jointprobability of three probabilities by Eq. 2.

H ¼ Pm � Pt � S ð2Þ

where Pm is the probability of landslide magnitude, Pt is the proba-bility of landslide recurrence (frequency), and S is spatial probability(susceptibility). The terms “spatial probability” and “susceptibility”are used in the literature interchangeably for describing the proba-bility related to the location of a landslide occurrence. Also, “prob-ability of landslide recurrence” is used in this paper with the samemeaning as “exceedance probability” and “temporal probability” asmay be seen in other studies (Coe et al. 2004; Guzzetti et al. 2005).

The proposed methodologyThe expression of landslide hazard in Eq. 2 could be argued for itsvalidity (Jaiswal et al. 2010). This definition, which is currently usedfor quantitative-probabilistic landslide hazard assessment at a me-dium mapping scale (1:25,000–1:50,000), embodies a simplified as-sumption of independence between landslide hazard components.Although it has been attempted to overcome this simplifying as-sumption by directly using the number of landslides per year permapping unit (Chau et al. 2004), such approach requires a compre-hensive multi-temporal landslide database, which is rarely availablein practice (Jaiswal et al. 2010). In this paper, we present a generalmethodology to consider the possible dependency between the land-slide hazard elements.

Regarding the magnitude and frequency, we obtain their prob-ability distributions in the case where there is no noticeable depen-dence between any of these two components and the other hazardelements. Otherwise, their “numerical indices” rather than theirprobability distributions will be used. The “numerical index” forlandslide magnitude could be the quantity of area, volume, momen-tum, or velocity of landslide. Also, in terms of landslide frequency,“mean recurrence interval” between successive failure events isadopted as the numerical index. Regarding the landslide location,the meaning of the susceptibility values changes in this paperdepending on the relationship they have with the other hazardcomponents but their values remain the same. In other words, sincethe susceptibility values are membership indices (Lee and Sambath2006), they could be both probability values (to be employed inde-pendently, in case there is no dependence between location and theother two components) and numerical indices (to be applied in jointprobability functions, when there is a dependence relationship be-tween location and the other two components). To produce thesemembership indices, different spatial factors such as slope gradient,aspect, curvature, distance from drainage, land use, soil parameters,altitude, etc. can be used depending on the study area and can becombined using various quantitative methods (e.g., Remondo et al.2003; Santacana et al. 2003; Cevik and Topal 2003; Chung and Fabbri2005; Lee 2004; Saito et al. 2009). Therefore, based on the proposedmethodology, three scenarios are possible: (a) when there isno significant dependency between the hazard components ina study area, Eq. 2 involving “multiplication” would be ap-plied to obtain the hazard values; (b) when there are only twodependent hazard elements, bivariate Copula function wouldbe used; and (c) when all of the three components are

Landslides

Original Paper

significantly dependent on each other, trivariate Copula func-tion would be employed. The proposed methodology forquantitative landslide hazard assessment using Copula model-ing technique is depicted in Fig. 1.

Study area and landslide inventoryPart of the Western Seattle area was selected as a suitable region toevaluate the proposed methodology (Fig. 2). Shallow landslides,the most common type of slope failure in this area, usually occurevery year from October through April within the wet season(Thorsen 1989; Baum et al. 2005). In 1998, the Federal EmergencyManagement Agency launched a hazard management programcalled Project Impact, and Seattle was one of the pilot cities inthis study. Shannon and Wilson, Inc. was assigned to collect adigital landslide database dating from 1890 to 1999 (Laprade et al.2000) and the landslide data became freely available to the USGSfor use. Past studies using this set of data include those by Coe et al.(2004) and Harp et al. (2006). The landslide record of 278 out of 357landslide events from 1912 to 1996 was used for model developmentand the remaining landslide record between 1997 and 1999 wasused for validation.

Method of landslide hazard assessmentAs previously mentioned, Copula is a type of distribution functionused to describe the dependence between random variables. Inother words, Copulas are functions that connect multivariateprobability distributions to their one-dimensional marginal prob-ability distributions (Joe 1997). The Copula concept in dependencemodeling goes back to a representation theorem of Sklar (1959)which is described as follows: Let us assume that we have n vari-ables (or populations) as X1; X2 . . . ; Xnð Þ; and N observations aredrawn for each variable as x1; x2; . . . ; xNð Þ and also FXi xið Þ; i ¼1; 2; . . . ; n are the marginal cumulative distribution functions(CDFs) of the variables Xi ; i ¼ 1; 2 . . . ; n: Now, in order to determinethe non-normal multivariate distribution of the variables, denotedas HX1X2 ...Xn x1; x2; . . . ; xnð Þ; Copula, C, could be used as an associateddependence function. This function returns the joint probability of

variables as a function of marginal probabilities of their observa-tions regardless of their dependence structure as shown below:

H x1; x2; . . . ; xnð Þ ¼ C F1 x1ð Þ; F2 x2ð Þ; . . . ; Fn xnð Þð Þ ð3Þ

where Fi xið Þ for i01, 2,…, n is the marginal distribution and C:0; 1½ �n ! 0; 1½ �0Copula if Fi is continuous (Grimaldi et al. 2005).Different families of Copulas have been developed in the last de-cades and the comprehensive descriptions of their properties havebeen compiled by some authors (e.g., Genest and MacKay 1986a, b;Joe 1997; Nelsen 1999). The most popular Copula family is Archi-medean Copulas (Yan 2006).

The conditional joint distribution based on Copula is expressedas follows: Let X and Y be random variables with their CDFs,respectively, as F(x)0U and F(y)0V. The conditional distributionfunction of X given Y0y can be presented by the equation below:

H X � xj Y ¼ yð Þ ¼ C ujV ¼ vð Þ ¼ lim4V!0C u; vþ4vð Þ � Cðu; vÞ

4v

¼ @

@vCðu; vÞjV ¼ v:

ð4Þ

As discussed earlier, this paper presents three randomvariables of S, T, and M as the numerical index for landslidelocation, landslide frequency, and landslide magnitude, respec-tively, for all 278 landslide events (i.e., observations).

Considering what was discussed above regarding the Cop-ula theory, equation below presents the mathematical termi-nology used hereafter in this paper:

Variables: (S, T, M)Observations: S1; S2; . . . ; S278ð Þ; T1; T2; . . . ; T278ð Þ; M1; M2; . . . ; M278ð ÞDependence modeling using Copula function:

H xS; xT ; xMð Þ ¼ C FS xSð Þ; FT xTð Þ; FM xMð Þð Þ ð5Þ

where depending on the degree of correlation between the variables,the Copula function (C) could be in either bivariate or trivariate form.

Fig. 1 The proposed methodology for quantitative landslide hazard assessment using Copula modeling technique

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Landslides

Landslide magnitudeAlthough different proxies have been used for landslide magnitude,“area” (aerial extent) is recognized as the most common and rea-sonable representation for “landslide magnitude” (Stark andHovious 2001; Guthrie and Evans 2004; Malamud et al. 2004;Guzzetti et al. 2005). In the historical landslide database of the WestSeattle region, the “landslide area” was not directly recorded; how-ever, the “slope failure height” was the available dimension of theslope failure events. The slope failure height (h) is defined as theapproximate elevation difference between the head scarp and the sliptoe, as depicted in Fig. 3 (Shannon and Wilson Inc 2000).

To estimate the area of the each landslide using its slope failureheight (h), this height dimension was converted to the “landslidelength” (L) assuming that the angle of the slope and the slip gradientof the failure surface are the same for each landslide event using Eq. 6(Fig. 3). Furthermore, to estimate the failure area, a general relation-ship between the landslide dimensions (length and width) was re-quired. Such a relationship needs to be generic, geometry-based, andindependent from the geology of the displacedmaterials in landslide.Therefore, the “fractal theory”was the best tool for this purpose. Thefractal is a mathematical theory developed to describe the geometryof complicated shapes or images in nature (Benoît 1982). The fractalcharacter of landslides can be described by a self-similar geometry(Kubota 1994; Yang and Lee 2006). Yokoi et al. (1996) stated thatlandslide blocks in a huge landslide have a fractal character. He alsoconcluded that their fractal dimension with respect to width is 1.24and with respect to length is 1.44 on average; and this fractal dimen-sion in width (or length) for a large landslide area was not dependent

on the base rock geology. These findings were valuable mathematicalrelationships (between length and width) to be applied in our land-slide database for estimating the area. Assuming that each landslideevent as one block in our dataset, we estimated the width of the“failure area” from the available length (Eqs. 7 and 8).

L � h sin2= bð Þ ð6Þ

ð7Þ

A � L�W � 1:16h2 sin2= bð Þ ð8Þ

where D is the landslide fractal dimension, W is the width of eachlandslide block, L is the length of each landslide block, β is theslope gradient, h is the slope failure height, and A is the landslidearea estimation. This calculation was done for 278 landslide eventsin the study area to estimate the magnitude index (M) for eachlandslide record.

Landslide frequencyIn terms of landslide frequency, mean recurrence interval betweensuccessive failure events in the study area was selected as thenumerical index (T). The mean recurrence interval is simply arepresentation of landslide frequency and is easily calculated bydividing the total landslide database duration over the number ofevents. Therefore, as discussed earlier, every single landslide event(a total of 278 events) is assigned either a mean recurrence intervalor its corresponding value from a “probability distribution”

D 1.24W and D 1.44L W 1.16 L

Fig. 2 The location of the study area and landslide events in the Western Seattle, WA area. (Coe et al. 2004)

Fig. 3 Definition of slope failure height (h), slope angle (β), and length (L) in a shallow landslide

Landslides

depending on the dependency between the landslide frequencyand the other two hazard components.

Landslide locationAs discussed earlier, landslide susceptibility values are determinedas the “membership indices” for the landslide location compo-nent. These susceptibility values could be estimated using differentmethods. Among all of the available approaches, a deterministicmethod was selected for this study. Deterministic methods dealwith slope stability concept and can be used in both site-specificscale and regional-scale studies (Terlien et al. 1995; Guzzetti et al.1999, 2005; Nagarajan et al. 2000; Cardinali et al. 2002; Santacanaet al. 2003). These methods in regional-scale studies mostly calcu-late the factor of safety (FS) in the mapping units (cell) of the studyarea. A number of methods have been developed using geographicinformation system (GIS) analysis techniques to calculate the FSvalues in different scales (Hammond et al. 1992; Pack et al. 1999;Iverson 2000; Baum et al. 2002; Savage et al. 2004). In this paper,the FS values of susceptibility analysis performed by Harp et al.(2006) in the Seattle area were used (Eq. 9):

FS ¼ C0

gt sinaþ tan 8

0

tan a� rgw tan 8

0

g tanað9Þ

where FS is the factor of safety, a is the slope angle, and gw is theunit weight of water, g is the unit weight of slope material, c′ is theeffective cohesion of the slope material, 8 ′ is the effective frictionangle of the slope material, t is the normal slope thickness of thepotential failure block, and r is the proportion of the slope thick-ness that is saturated. It was assumed that groundwater flow isparallel to the ground surface with the condition of completesaturation, that is, r01 (Harp et al. 2006). The required geologydata were obtained from the digital geology map of the Seattle areacreated by Troost et al. (2005) at a scale of 1:12,000 (Fig. 4a). Theseproperties were assigned to each of the geologic units of the digitalgeology map. In addition, topography data were acquired from thetopography (slope gradient) map produced by Jibson et al. (2000)in the Seattle area (see Fig. 4b).

According to the definition of location index (S), these valuesare basically between 0 and 1; however, the obtained results showedthat some of the factor-of-safety values are FS>1.0. Thus, in order touse the FS value of each mapping cell as a numerical index, it wasnecessary to normalize these values to be between 0 and 1. Finally,each landslide point location was assigned the obtained numericalindex of its corresponding mapping cell (Fig. 4c).

Dependence assessmentIn a related analysis conducted by the authors (not published yet),it was found that there is a negligible correlation between the pairsof (M, T) and (S, T). However, the results showed that there is asignificant and positive correlation between location and magni-tude indices (M, S). Therefore, magnitude and location indiceswere considered for building the bivariate Copula model in thefollowing sections.

Marginal distribution of variablesTo create the bivariate Copula model, it was required to obtain thebest fitted marginal probability distributions for both location (S)and magnitude indices (M) when these random variables are not

limited to a maximum value. Since the values of location indices(S) are limited to the range of (0, 1), data transformation needs tobe performed first and then different marginal probability distri-butions are examined on the data. This transformation changesthe range of the limited data to a theoretically unlimited positiverange of [0, ∞). The unlimited positive range was (0, 452) for ourtransformed location indices. These transformed location indicesare labeled as TS hereafter for simplification.

Fig. 4 a Geologic map, b slope map, and c location index (S) map in the study area

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Landslides

Therefore, histograms of TS and M datasets were fit toseveral probability density functions to find out which one isthe best choice. Comparing all the density functions, exponen-tial distribution (Eq. 10) was selected as the best fit to TS set(Fig. 5a). Exponential distribution is defined on the interval of[0, ∞) and is given by

f x; 1ð Þ ¼ 0; xj < 01e�1x; xj � 0

�ð10Þ

where l>0 is the parameter of the distribution and it is l02.71for TS data. Moreover, it was recognized that M set can bedescribed well by a lognormal distribution (Eq. 11) with a meanof 3.753×104 m2 and a standard error of 3.344×103 (Fig. 5b):

fX x; k; σð Þ ¼ 1xσ

p2p

e�lnx�kð Þ22σ2 ; x > 0 ð11Þ

where σ2 is the shape parameter and k is the scale parameterof the distribution.

Model selection and parameter estimationIn order to model the dependence between the TS andM sets and tohave a variety of choices from different families of Copula, 14 Copulafunctions were considered from four broad categories: (a) Archime-dean family including Frank, Clayton, Ali-Mikhail-Haq, andGumbel–Hougaard functions (Nelsen 1986; Genest and MacKay1986a, b) and also the BB1–BB3 and BB6–BB7 classes (Joe 1997); (b)elliptical Copulas mainly including normal, student, Cauchy, andt-Copula functions (Fang et al. 2002); (c) extreme-value Copulas,including BB5, Hüsler–Reiss, andGalambos families (Galambos 1975;Hüsler and Reiss 1989); and (d) Farlie–Gumbel–Morgenstern familyas a miscellaneous class of Copula (Genest and Favre 2007).

To sieve through the applied Copula models, an informal graph-ical test was performed as follows (Genest and Favre 2007): Themargins of the 10,000 random pairs (Ui,Uj) from each of the 14estimated Copula models were transformed back into the originalunits using the already definedmarginal distributions of exponentialand lognormal models. Then, the scatter plots of resulting pairs(Xi,Yi)0(W

-1(Ui),L-1(Uj)) were created, along with the actual obser-

vations for all families of Copulas. This graphical check made itpossible to generally judge the competency of the models and toselect the best contenders along with the actual observations. There-fore, six best Copula models were selected (Fig. 6). The next step wasto estimate the parameters of all six selected models. This estimationwas done based on maximum pseudo-likelihood which is a functionof the data and parameters having properties similar to those oflikelihood function (Arnold and Strauss 1991). In order to do that, anestimating equation which is related to the logarithm of the likeli-hood function is set to zero andmaximized. The obtained parametervalues of each Copula model using maximum pseudo-likelihoodwith 95 % confidence intervals are given in Table 1. These valuesare used for comparison of different functions, narrowing down theresults and building the model.

Goodness-of-fit testingNow, the question is that which of the six models should be used toobtain the joint distribution function of variables. As the secondattempt towards our model selection, the generalized K-plot of eachof the six Copulas was obtained and compared to each other. ThegeneralizedK-plot indicates whether the quantiles of nonparametricallyestimated KN(z) are in agreement with parametrically estimated KN(z)for each function. KN(z) is basically defined as the equation below(Genest and Rivest 1993):

K zð Þ ¼ z � 8ðzÞ8 0 ðzÞ ð12Þ

where 8 ′( ) is the derivative of 8( ) with respect to z and z is the specificvalue of Z0Z(x, y) as an intermediate random variable function. If theplot is along a straight line (at a 45° angle and passing through theorigin), then the generating function is satisfactory. Figure 7 depicts theK-plot constructed for all of the six Copula functions.

Considering the K-plots and the graphical check (Figs. 6 and 7),it is clear that the Gumbel–Hougaard distribution (Eq. 13) is the bestfit to the data and should be used for the rest of the analysis:

C u; vð Þ ¼ C FXðxÞ; FYðyÞ½ � ¼ HX; Y x; yð Þ

¼ exp � �lnuð Þθ þ �lnvð Þθh i 1

θ

� �θ 2 1; 1½ Þ: ð13ÞFig. 5 Marginal distribution fitting to a transformed location index and b

magnitude index

Landslides

Copula-based conditional probability density functionThe presented analysis so far showed that the Gumbel-Hougaarddistribution was the best fitted Copula function for modeling thedependence between transformed location indices (TS) and mag-nitude indices (M).

Since the concept of landslide hazard assessment in this method-ology is “conditional,” the obtained joint distribution needs to becompatible with this definition. In other words, the probability oflandslide hazard in this study is expressed as “the probability ofoccurrence of a landslide having a magnitude larger than a specifiedvalue (M≥m) under the condition of having a specific spatial index inthat location”; therefore, the obtained Copula function needs to beused in its conditional form. Considering Eqs. 4 and 13, the conditionalGumbel–Hougaard distribution substituted by previously estimatedparameters is given by

C ujV ¼ vð Þ ¼ 1v ðexp � �lnuð Þ2:126 þ �lnvð Þ2:126� �0:47n o

� �lnuð Þ2:126 þ �lnvð Þ2:126� �0:5Þ: ð14Þ

Figure 8 illustrates the obtained results of the conditional func-tion in its cumulative form (CDF). In fact, every point in thisfunction expresses the joint probability of occurrence of a landslidehaving the given magnitude and location indices. However, as thecumulative density function defines the nonexceedance probabilityfor a specific value, the results of the Copula function were needed toconvert to (1–CDF) for the validation purpose which will bediscussed later. Moreover, the results were converted back from thetransformed location indices (TS) to the original location indices.

Probability of landslide frequencyIn previous sections, the dependence between two landslide haz-ard components including magnitude and location was modeledusing Copula theory. As what the dependence tests (performed bythe authors and not published yet) suggested, there was no signif-icant correlation between the indices of frequency component andthe other two hazard elements. Therefore, the landslide frequencyneeds to be applied in a form of “probability distribution values”and not numerical indices. In order to do that, the mean recur-rence interval (μ) of each landslide event needs to be inserted intoa temporal probability model.

One of themost popular probabilitymodels for investigating theoccurrence of naturally occurring random point-events in time isPoisson distribution (Guzzetti et al. 2005). Poisson distribution is adiscrete model that expresses the probability of a number of eventsoccurring in a fixed period of time if these events occur with a knownaverage rate and independently of the time since the last event (Hu2008). Assuming landslides as independent random events in time(Crovelli 2000), Poisson distribution is used as below:

P NðtÞ � 1½ � ¼ 1� exp �1tð Þ ¼ 1� exp � tμ

� �: ð15Þ

Fig. 6 Simulated random sample of size 10,000 from 14 chosen families of Copulas: a Ali-Mikhail-Haq, b Frank, c Galambos, d Gumbel–Hougaard, e BB2, and f BB3, upontransformation of the marginal distributions as per the selected models (whose pairs of ranks are indicated by white points) along with the actual observations

Table 1 Parameter estimation and confidence interval of the Copulas

Copula Estimate (θ) 95 % confidence interval (CI)

Ali-Mikhail-Haq θ01.465 CI0 [1.324, 1.606]

Frank θ00.384 CI0 [0.249, 0.518]

Galambos θ03.037 CI0 [2.270, 3.805]

Gumbel–Hougaard θ02.126 CI0 [0.670, 1.801]

BB2 θ100.174,θ201.638

CI0 [0.255, 2.801]×0.260, 3.316]

BB3 θ101.424,θ201.053

CI0 [1.512, 1.903]×[0.134, 1.404]

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Landslides

Equation 15 expresses the probability of experiencing one ormore landslides during time t, where l is the average rate of occur-rence of landslides as 1/μ, which again μ is the estimated meanrecurrence interval between successive landslide events and is cal-culated asD/N,whereN is the total number of events (N≠n) andD isthe total duration (in years) within which all N landslides occurred.

Equation 15, which is also called exceedance probability, wasapplied to determine the probability of landslide frequency for ourlandslide database. The process of such analysis was adopted fromthe work done by Coe et al. (2004). The study area was digitallyoverlain with a grid of 25×25-m cells. Next, a count circle covering anarea of 40,000m2(4 ha) was digitally placed at the center of each cell,and the number of landslides occurring within the circle was count-ed. Next, mean recurrence interval (μ) was calculated by dividing thetotal database duration (D084 years, 1912–1996) over the number oflandslides. Then, the exceedance probability was obtained for everycounting circle using Eq. 15 and this value was assigned to thecorresponding mapping cell.

We now have all the information to quantitatively determinelandslide hazard in the Western Seattle area. Therefore, three com-ponents of landslide hazard can be combined based on the proposedmethodology (Fig. 1) as below:

landslide hazard : H ¼ PT � P M \ Sð Þf g: ð16Þ

Model validationThe best way to check the validity of a model is to employlandslide data independent from the one used to develop it(Jaiswal et al. 2010). As mentioned earlier, we chose 79 landslideevents from 1997 to 1999 to be used for the validation part. Since

Fig. 7 Goodness-of-fit testing by comparison of nonparametric and parametric K(z) for Copula models

Fig. 8 Cumulative Gumbel–Hougaard joint probability density function of S andM indices; dark points represent the 79 validation points in “1–CDF” form

Fig. 9 Example of landslide hazard map (10×10-m cell size) for 50 years forlandslide magnitudes M≥10,000 m2 in the study area. The value in each map cellgives the conditional probability of occurrence of one or more landslide within thespecific time in that location

Landslides

the validation process was performed for the already occurred land-slides, the incident is certain, i.e., the probability of their occurrenceis 100 %. Therefore, the closer the predicted values of the landslidehazard model (Eq. 16) to 100 %, the more valid and reliable it wouldbe. Based on the results, the success rate of the Copula-based modelin the prediction of landslide hazard is 90 %. This success rate iscalculated as the average amount of the hazard value of the 79landslide events. These 79 computed probability values are alsographically shown in their exceedance cumulative form (1–CDF) asdiscussed earlier (Fig. 8). Additionally, a computation was performedto compute the landslide hazard under the assumption of indepen-dence between the components (Fig. 1). The mean success rate forthis multiplication-based model is only 63 %. Moreover, as a furthereffort toward model verification, the landslide hazard of 79 random“safe” locations (no landslide) were calculated in the study areawhere the landslide hazards are expected to be 0 %. The arbitrarymagnitude value of 10,000 m2 was considered to be applied in bothCopula-based and also multiplication-based models. Based on theresults, the mean success rate of Copula model in the prediction oflandslide hazard is only 12 % on average; however, this mean successrate of the multiplication-based approach is 44 %.

Landslide hazard mapIn order to portray the model results conveniently as a landslidehazard map, the study area was first digitally overlain with a grid of1.83×1.83-m cells. This size of mapping cells was selected because itwas compatible with the smallest size of the mapping units in the“FS” and exceedance probability maps. Then, the location index (S)of every mapping cell and an arbitrary magnitude value wereinserted into Eq. 14. Then, the result was converted to the exceedanceform of “1–CDF,” and lastly, it was multiplied by the probability oflandslide frequency of the mapping unit (see Eq. 16). Figure 9 dis-plays an example of the resulting map (in GIS format) for anarbitrary landslide magnitude within the time period of 50 years.

Summary and conclusionsThe simplifying assumption of independence among the hazardelements, which has been commonly used in the literature, wasdiscussed in this paper. A new quantitative methodology waspresented to assess the landslide hazard probabilistically in a region-al scale. This approach considers the possible dependence betweenthe hazard components and attempts to forecast probabilistically thefuture landslide using a reliable statistical tool named Copulamodel-ing technique. We tested the model in the western part of the Seattle,WA area. A total of 357 slope-failure events and their correspondingslope gradient and geology databases were used to build the modeland to test its validity. The mutual correlations between landslidehazard elements were considered, and Copula modeling techniquewas applied for building the hazard function. Based on the validationresults for hazardous areas, the mean success rate of Copula modelin the prediction of landslide hazard is 90 % on average; however,this mean success rate of the traditional multiplication-based ap-proach is only 63 % which is due to neglecting the existing depen-dency between hazard elements. Also, the hazard prediction resultsin safe locations for Copula-based and multiplication-based modelsare 12 and 44 %, respectively. The comparison results validate theadvantages of the presented methodology over the traditional onedue to explicit considerations of dependence among landslide haz-ard components.

AcknowledgmentsThe authors are very thankful to Professor L. Zhang for her criticalcomments and useful review. We acknowledge Rex L. Baum, JeffreyA. Coe, Jonathan Godt, Edwin Harp, and Lynn Highland of USGeological Survey for providing us with the Seattle area databasesas well as for their valuable comments.

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M. Motamedi ())Geotechnical Engineering,University of Akron,Akron, OH, USAe-mail: alm.motamedi@gmail.com

R. Y. LiangCivil Engineering Department,University of Akron,Akron, OH, USAe-mail: rliang@uakron.edu

Landslides