www.elsevier.com/locate/compgeo

Computers and Geotechnics 31 (2004) 529–536

Uncertainty analysis in a slope hydrology and stability model usingprobabilistic and imprecise information

Eva Rubio a, Jim W. Hall b,*, Malcolm G Anderson c

a Instituto de Desarrollo Regional, University of Castilla-La Mancha, Campus Universitario, 02071 Albacete, Spainb Department of Civil Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UK

c School of Geographical Sciences, University of Bristol, University Road, Bristol BS8 1SS, UK

Received 5 April 2004; received in revised form 17 August 2004; accepted 6 September 2004

Abstract

In practical geotechnical analysis information may appear in a range of formats, including inexact measurements, probability

distributions, linguistic classification and expert judgements. These information formats also appear in the literature, from where

modellers may wish to obtain prior information about uncertain soil parameters. Conventional probabilistic uncertainty analysis

requires that all uncertain information be expressed as precise probability distributions, regardless of the (often non-probabilistic)

format of the original information. The theory of random sets provides a general mechanism for handling information in the form of

intervals, sets of intervals or fuzzy sets, as well as (discrete) probability distributions. Relevant theory is developed for constructing

random relations describing soil properties, aggregating information from different sources and propagating it through geotechnical

models. The theory is applied to the analysis of the stability with respect to rainfall-induced landsliding using a combined slope

hydrology and stability model. In the example the soil properties determining slope hydrology are described by joint probability

distributions whilst the main geotechnical parameters are represented as sets of intervals. The methodology is readily extended

to other combinations of probabilistic and interval-valued information.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Uncertainty; Random sets; Imprecise probability; Hydrology; Slope stability

1. Introduction

It is widely recognised that slope stability analysis is

characterised by numerous uncertainties due to limited

sampling, discrepancy between different methods of lab-

oratory and in situ strength testing, and uncertainties in

soil models [1–4]. There is the ever-present possibility ofunderestimating these uncertainties. For example, Obe-

rguggenberger and Fellin [5,6] demonstrate the very

large range of estimated probability of failure that alter-

0266-352X/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2004.09.002

* Corresponding author. Tel: +44 117 928 9763; fax: +44 117 928

7783.

E-mail address: jim.hall@bristol.ac.uk (J.W. Hall).

native probabilistic methods can generate, even when a

relatively large number of soil samples are available.

Near the outset of the site investigation process all

that may be available is classification of soils at the site.

Scarce site-specific measurement will have to be supple-

mented with other information such as published data

and measurements from broader-scale assessments.Prior information about soil properties appears in a

range of formats including probability distributions [7–

9]. Rackwitz [10] reports interval ranges for the means

and standard deviations of soil properties for a range

of cohesive and non-cohesive soils. Widely used engi-

neering manuals [11–14] present information on soil

properties in terms of intervals. Expert assessments are

often expressed in linguistic or imprecise terms [15,16].

530 E. Rubio et al. / Computers and Geotechnics 31 (2004) 529–536

Given severe information scarcity, which is the usual

state of affairs, it is undesirable to exclude relevant infor-

mation on the grounds of its format, whilst by trans-

forming existing imprecise information to a

probabilistic format the amount of information may,

unwittingly, be overstated. An approach that admitsimprecise information (including interval values) as well

as probabilistic information is therefore required.

A more fundamental argument is that because of

imperfections in laboratory and field tests, some soil

properties should be regarded as being fuzzy quantities

[17–21]. It is unrealistic to suppose that even at a point

location all soil properties can ever be precisely known

with certainty. Similarly, it is argued that model uncer-tainties may be more appropriately dealt with by fuzzy

representation rather than being represented as proba-

bility distributions [22–24].

In this paper the conventional probabilistic approach

to uncertainty analysis is extended to include uncertain

information in a variety of formats including probability

distributions, intervals and sets of intervals. This gener-

alised approach to probabilistic and imprecise measure-ments is handled through the theory of random sets. It

enables available information on parameter values to

be used in the format it appears. The paper begins by

describing the random set theory on which the method-

ology is based, first explaining how uncertain variables

can be described by a random relation and then explain-

ing how a random relation can be propagated through a

function using the random set extension principle. Thefunction in this case is the CHASM combined slope

hydrology/stability model, which is then described.

These aspects are combined in a numerical methodology

for uncertainty analysis, which is illustrated with an

example of slope stability analysis.

2. Random set theory

Random set theory provides a general mechanism for

handling interval-based measurement, fuzzy sets and

probability distributions. Following Dubois and Prade

[25,26], a finite support random set on a universal set

X is a pair ðI;mÞ, where I ¼ fAiji ¼ 1; . . . ; tg;Ai 2 PðX Þ 8i ¼ 1; . . . ; t; where P(X) is the power set of

X and a mass assignment, m is a mapping

m : I ! ½0; 1� ð1Þsuch that m(Ø) = 0 andXA2I

mðAÞ ¼ 1: ð2Þ

Each set A 2 I contains possible values of a variable

x 2 X, and m(A) can be viewed as the probability thatx 2 A but does not belong to any special subset or sup-

erset of A. Therefore, random set can be seen as a gen-

eralisation of a random variable. Every A 2 P(X), where

P(X) is the power set of X, for which m(A) 6¼ 0 is re-

ferred to as a focal element. Given a random set

ðI;mÞ, a belief function Bel [27] can be defined as the

following set function:

8B 2 P ðX Þ; BelðBÞ ¼XA�B

mðAÞ: ð3Þ

Its dual plausibility function Pl is defined by

8B 2 P ðX Þ; PlðBÞ ¼ 1� Belð�BÞ ¼XA\B6¼;

mðAÞ; ð4Þ

Bel(B) can be viewed as the lower bound on a family of

probability measures and Pl(B) as the upper bound,

although the converse is not true, i.e., lower and upper

probability functions are more general than belief and

plausibility functions.When I is a set of nested sets then Bel is a necessity

measure g and Pl is a possibility measure p [28], and the

random set is said to be consonant. Consider universal

set X = {x1,x2, . . . ,xs}, where the focal elements Ai are

nested, e.g., A1�A2�� � ��At, then the membership

lF(x) of an element x 2 AinAi + 1 in a fuzzy set F on X

can be defined as follows:

lF ðxÞ ¼Xt

k¼i

mðAkÞ ¼ PlðfxgÞ ¼ pðfxgÞ: ð5Þ

When I contains only singletons Bel = Pl is a probabil-

ity measure (with finite support). Random set thereby

provides a coherent generalisation of probability and

fuzzy set theory, at least according to one interpretation

of fuzzy sets [29].

Suppose that a closed interval [x1,xs + 1] is partitioned

into disjoint sub-intervals [x1,x2], (x2,x3], . . . , (xs � 1,xs],(xs,xs + 1] labelled A1, A2, . . . ,As, respectively. A set of

intervals {Ai, . . . ,Aj} i < � � � < j is labelled {Ai,j + 1}, i.e.,

according to its extreme lower and upper limits. The

lower and upper cumulative probability distribution

functions, F*(x) and F*(x), respectively, at some point

x can be obtained as follows:

F �ðxÞ ¼XxPxj

mðfAi;jgÞ ð6Þ

and

F �ðxÞ ¼XxPxi

mðfAi;jgÞ: ð7Þ

2.1. Averaging of random sets

If more than one random set is available relating to

some uncertain quantity, a mechanism is required tocombine the various sources. The methods for aggregat-

ing information represented as random or fuzzy sets or

belief functions that have been proposed [30] include

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 15 20 25 30 35 40

Cum

ulat

ive

prob

abili

ty

Effective friction angle f'

Fig. 2. Lower and upper cumulative probability distributions corre-

sponding to the normal distribution with interval mean [22.8�, 28.8�]

E. Rubio et al. / Computers and Geotechnics 31 (2004) 529–536 531

conjunctive [27,31] disjunctive (for example Dubois and

Prade�s disjunctive consensus rule [32]) and averaging

methods. In situations where only one source of infor-

mation is believed to be correct but it is not known

which one, averaging procedures have been justified on

the grounds that they provide an unbiased combinationof information. Baldwin et al. [33] apply a voting anal-

ogy as justification for equal weighting of information

sources. Suppose there are q alternative random sets

ðIi;miÞ : i ¼ 1; . . . ; q, describing some variable x, each

one corresponding to an independent source of informa-

tion and each one defined on P(X). ðI;mÞ is the averagerandom set with I ¼

SiIi and

mðAÞ ¼ 1

q

Xq

i¼1

miðAÞ; miðAÞ ¼ 0 if A 62 Ii: ð8Þ

and SD [3.4�, 5.7�].

2.2. Example of constructing a random set from multiple

information sources

Consider the case in which information on a geotech-

nical property of interest, for example effective friction

angle / 0, is available from five different sources asfollows:

1. an interval: [31.5�,36.7�],2. a set of intervals: [31.1�,39.2�], [33.3�,38.0�],

[32.5�,37.8�], [31.9�,33.0�],3. a fuzzy set as illustrated in Fig. 1,

4. a lognormal probability distribution: ln (/ 0) �N(3.5,0.08),

5. normal probability distribution with interval mean

[22.8�,28.8�] and interval standard deviation

[3.4�,5.7�], as illustrated in Fig. 2.

Each of these information sources can be representedas a random set. In each case the focal elements and

mass assignment are listed in Table 1.

1. The interval estimate can be regarded as a random set

with a single focal element with a mass m1 = 1.

0

0.2

0.4

0.6

0.8

1

25 30 35 40 45

Fuz

zy m

embe

rshi

p m

Effective friction angle ff'

Fig. 1. Fuzzy set for effective friction angle / 0.

2. The set of intervals can be regarded as a random set

with focal elements corresponding to each interval

estimate and a mass of m2 = 0.25 assigned to each

focal element.3. The fuzzy set can be regarded as a consonant random

set. Suppose five a-cut levels of the fuzzy set are con-

sidered, each corresponding to a focal element, with a

mass of m3 = 0.2 associated with each focal element.

The random set is a discrete outer approximation to

the fuzzy set.

4. The normal probability distribution can be treated as

a random set by calculating the probability on, say,10 discrete intervals. In doing so it is necessary to

impose lower and upper bounds on the distribution.

In the example given in Table 1 the bounds have been

set at the 1 and 99 percentiles of the distribution.

5. In the case of lower and upper cumulative probability

distributions, a random set has to be constructed with

corresponding belief and plausibilities that are outer

approximations to the lower and upper probabilitydistributions implied by the interval parameters. In

the case when a cumulative distribution is required,

the approach based on the method of Williamson

and Downs [34] involves dividing the vertical axis in

Fig. 1 into, say, 10 discrete levels and identifying

the interval that bounds the lower and upper proba-

bilities for all of that discrete level. Hall and Lawry

[35] present a more general, but computationallymore expensive, methodology for constructing ran-

dom set approximations to lower and upper probabil-

ity distributions.

The random sets listed in Table 1 have been com-

bined according to Eq. (8) on the basis that each of

the information sources carries equal weight. Since the

focal elements, A, from the five different sources arenot in coincidence, then Eq. (8) reduces to

Table 1

Random sets from different information formats

Source 1 Source 2 Source 3 Source 4 Source 5

Focal elements m1 m Focal elements m2 m Focal elements m3 m Focal elements m4 m Focal elements m5 m

[31.5�,36.7�] 1 0.2 [31.1�,39.2�] 0.25 0.05 [30.2�,39.2�] 0.2 0.04 [27.5�,28.7�] 0.038 0.008 [17.9�,27.8�] 0.1 0.02

[33.3�, 38.0�] 0.25 0.05 [31.0�,38.4�] 0.2 0.04 [28.7�,30.0�] 0.068 0.014 [20.1�,28.1�] 0.1 0.02

[32.5�,37.8�] 0.25 0.05 [31.8�,37.5�] 0.2 0.04 [30.0�,31.2�] 0.123 0.025 [21.0�,28.4�] 0.1 0.02

[31.9�,33.0�] 0.25 0.05 [32.5�,36.7�] 0.2 0.04 [31.2�,32.5�] 0.170 0.034 [21.7�,28.6�] 0.1 0.02

[33.3�,35.8�] 0.2 0.04 [32.5�,33.7�] 0.185 0.037 [22.3�,28.8�] 0.1 0.02

[33.7�,34.9�] 0.162 0.032 [22.8�,29.3�] 0.1 0.02

[34.9�,36.2�] 0.117 0.023 [23.0�,29.9�] 0.1 0.02

[36.2�,37.4�] 0.081 0.014 [23.2�,30.6�] 0.1 0.02

[37.4�,38.6�] 0.037 0.007 [23.5�,31.5�] 0.1 0.02

[38.6�,39.9�] 0.027 0.005 [23.8�,33.7�] 0.1 0.02

532 E. Rubio et al. / Computers and Geotechnics 31 (2004) 529–536

m(A) = 0.2mi(A) for i = 1, . . . , 5. The mass assignment m

on the combined random set, is also given in Table 1.

The cumulative lower and upper probabilities associated

with the resulting random set (Eqs. (6) and (7)) are plot-

ted in Fig. 3.

2.3. Random set extension principle

Besides providing a convenient mechanism for com-

bining probabilistic and set-based information, it is

straightforward to extend random sets through a func-

tion [36]. Let g be a mapping X1 ·� � �· Xr ! Y. Let

x1, . . . ,xr be variables whose values are incompletely

known. The incomplete knowledge about

x = (x1, . . . ,xr), including their dependency, can be ex-

pressed as a random relation, which is a random setðI;mÞ on the Cartesian product X1 ·� � �· Xr. The ran-

dom set ðR; qÞ, which is the image of ðI;mÞ through

g, is given by [37]:

R ¼ fgðB1Þ; . . . ; gðBtÞg; where gðBÞ ¼ fgðxÞ; x 2 Bg;ð9Þ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Effective friction angle f'

Cum

ulat

ive

prob

abili

ty

Fig. 3. Lower and upper cumulative probabilities of / 0 exceeding a

given value.

qðRÞ ¼X

Bi :R¼gðBiÞmðBiÞ for R 2 R: ð10Þ

The summation in Eq. (10) accounts for the fact thatmore than one focal element Ai may yield the same R

on Y. Special cases of Eqs. (9) and (10) for (i) set-valued

variables (ii) consonant random Cartesian products (iii)

stochastically decomposable Cartesian products and (iv)

joint probability distributions were addressed by Dubois

and Prade [26]. In the case of consonant random sets

Eqs. (9) and (10) yield the min–max extension principle

for fuzzy sets [38] and the image of the random relationcan be constructed from the images of the level cuts.

When the marginal random sets are independent

(according to random set independence [39]) then the

mass assignment in the joint space can be obtained as

the product of the masses mi of the marginal random

sets:

mðA1 � � � � � ArÞ ¼Yri¼1

miðAiÞ; A1 � � � � � Ar 2 I:

ð11ÞIn general to compute Eqs. (9) and (10) involves calcu-

lating the image R of each focal element B 2 I by apply-

ing twice the techniques of global optimisation. If thefocal elements of I are connected compact sets and g

is a continuous function,

gðBÞ ¼ ½l; r�; ð12Þwhere

l ¼ minx2B

gðxÞ; ð13Þ

r ¼ maxx2B

gðxÞ: ð14Þ

When each variable xi is specified by a marginal random

set, whose focal elements are each an interval [l,u], then

methods of interval analysis [40] are applicable.

Under certain special conditions the Vertex method[41] applies and can be used to greatly reduce computa-

tional expense. Suppose each focal element B of the ran-

dom set ðI;mÞ is an r-dimensional box, whose 2r vertices

E. Rubio et al. / Computers and Geotechnics 31 (2004) 529–536 533

are indicated as vj, j = 1, . . . , 2r. If the extreme points are

in the vertices, then

gðBÞ¼ minjfgðvjÞ : j¼ 1; . . .2rg; max

jfgðvjÞ : j¼ 1; . . .2rg

� �:

ð15ÞThus function g has to be evaluated 2r times for each fo-

cal element B. This computational burden can be further

reduced if g is a strictly monotonic function with respect

to each parameter xi, in which case vj and vk can be iden-

tified merely by consideration of the direction of in-

crease of g. Thus g has to be calculated only twice for

each focal element B [37]. The function g in the analysis

described in this paper is the combined slope hydrology/stability model called CHASM, which is now briefly

described.

Table 2

Parameters used in CHASM

Parameter Symbol

Saturated hydraulic conductivity Ks

Saturated soil moisture content hsa coefficient: Van Genuchten suction–moisture curve an coefficient: Van Genuchten suction–moisture curve n

Residual soil moisture content:

Van Genuchten suction–moisture

hr

Saturated bulk density (unit weight) csUnsaturated bulk density cEffective cohesion c 0

Effective friction angle / 0

3. Combined slope hydrology/stability model (CHASM)

CHASM is a physically based combined soil hydrol-

ogy and slope stability model that simulates of changesin pore water pressures in response to rainfall, and

considers their role in maintaining slope stability. The

model comprises fully integrated hydrology, surface

cover (vegetation) and stability analysis. In the current

analysis CHASM has been used to estimate the slope

factor of safety (FOS) against rotational failure (and

uncertainty therein) in a specified rainfall event. The

model has previously been used to estimate the FOS ina distribution of rainfall events and also in a continuous

simulation of an extended time series of rainfall. The

model has been described fully elsewhere [42] and so

only a brief outline of the key features is required here.

The hydrological system is modelled using a forward

explicit finite difference scheme. The model simulates

detention storage, infiltration, evaporation, and unsatu-

rated and saturated flow regimes. Unsaturated verticalflow in a series of adjacent columns through the soil pro-

file is computed using the Richards� equation [43], solved

in explicit form, with the unsaturated conductivity de-

fined by the Millington Quirk [44] procedure. Flow be-

tween columns is modelled using the Darcy equation

[45] for saturated flow, adopting the Dupuit–Forcheimer

[46] assumption for the bottom boundary condition.

The integration of the unsaturated and saturated flowregimes allows determination of the pressure head field

within the slope material and subsequent input into

the stability analysis.

The stability assessment techniques used in CHASM

are Bishop�s simplified circular method [47] and Janbu�snon-circular method [48]. In the analysis described here

Bishop�s method has been adopted. At each time step of

the simulation pore pressures, both negative and posi-tive, are incorporated directly into the effective stress

determination of the Mohr–Coulomb equation for soil

shear strength [49]. The Fredlund et al. [50] criterion is

used for the unsaturated portion. This provides input

into the limit equilibrium technique for derivation of

the minimum FOS, with temporal variations arising

from hydrodynamic responses and changes in the posi-tion of the critical slip surface [51]. The variable param-

eters used in CHASM are listed in Table 2.

4. Example of slope stability analysis

The slope chosen for example application is a 39�slope in deposits of colluvium. The slope geometrywas assumed to be deterministic. Textural classifica-

tion according to the USDA classification system indi-

cated that the soil was sandy loam, providing access to

published marginal distributions and cross-correlations

for the variables of Ks, hs, a, n, hr [52] (Tables 3 and

4). A Cholesky decomposition was used to simulate a

large number of points (�106) points from the joint

distribution using given in Tables 3 and 4. The rangeof Ks, hs, a, n, hr was partitioned between recom-

mended limits [52], into a number of discrete intervals.

The mass mh of each hypercube (focal element) in the

joint space of Ks, hs, a, n, hr was estimated from the

relative frequency of simulated points located in each

hypercube.

Five results from laboratory tests were available pro-

viding imprecise evidence for c 0 and / 0. Measurementsof /b, the angle of shearing resistance with respect to

matric suction [53,54], were not available. Therefore,

based on previous research with low maintained suc-

tions [55], the suction was used directly in the effective

stress equation. c and cs were found to be in the ranges

[17.4,19.8] and [18.3,20.2], respectively. From each soil

test the range of possible values of c 0 and / 0 was ex-

pressed in terms of a random relation with a single focalelement. The five focal elements on the Cartesian prod-

uct of c 0 and / 0 are illustrated in Fig. 4. Note how the

dependency between c 0 and / 0 is reflected in the shape

Table 3

Marginal parameter distributions for hydrological properties for sandy loam [52]

Parameter Distribution Distribution parameters Lower limit Upper limit

hs Normal (l,r) l = 0.410 r = 0.0899 0.132 0.688

hr Beta (q,r) q = 2.885 r = 2.304 0.0173 0.102

a (cm�1) Beta (q,r) q = 1.816 r = 3.412 0.00872 0.202

n Lognormal (c,f) c = 0.634 f = 0.0818 1.46 2.43

Ks (cm/s) Lognormal (c,f) c = �7.46 f = 1.330 9.6 · 10�6 0.0347

Table 4

Cross-correlations for hydrological properties for sandy loam [52]

hs hr a n Ks

hs 1 0 0.01 0 0.01

hr 1 0.14 �0.79 �0.22

a 1 0.36 0.82

n 1 0.6

Ks 1

29

30

31

32

33

34

35

36

37

38

39

0 2 4 6 8 10

c'

f'

Fig. 4. Random relation between c 0 and / 0.

Fig. 5. Bounds on probability distribution of FOS.

534 E. Rubio et al. / Computers and Geotechnics 31 (2004) 529–536

of the focal elements and also in the distribution of focal

elements in the joint space.

An 8 hour rainfall event at a return period of 100

years with a total rainfall of 395 mm was used for the

CHASM simulation. Zero seepage was assumed at the

up-slope boundary, but the boundary was demonstrated

to be sufficiently remote not to influence the analysis re-

sults. An initial surface suction of 2 m was applied. Theminimum FOS was generally achieved in less than 10 h

so a 24-h simulation was performed to ensure that the

minimum FOS following the rainfall event was ob-

tained. The water table was initially at 29% of the slope

height.

Whilst not a necessary condition for the analysis,

considerable computational advantage is achievable if

it can be demonstrated that the FOS generated byCHASM is a monotonic function of the model�s input

variables over their range of possible values. Further

computational savings are achievable by exploiting

knowledge of the direction of increase of FOS with re-

spect to each variable.

The image in g (i.e., the corresponding interval range

of FOS values from CHASM) of every combination of

focal elements in the random relation of cs, c, c 0, / 0,

Ks, hs, a, n, hr was calculated using CHASM. At thepoints corresponding to the lower and upper bounds

on the FOS for each focal element, the location of the

slip surface is recorded for later analysis.

Fig. 5 shows the bounds on the cumulative probabil-

ity distribution of the FOS. The method provides a di-

rect indication of contributions of imprecision and

probabilistic variability. The stepped form of the lower

probability curve is a consequence of the curve for eachfocal element being steep, but located at a different FOS.

The bounds on the probability of slope failure are ob-

tained at a FOS of unity and are [0, 0.005].

Fig. 6 shows the lower probability distribution for the

location of the slip surface, in terms of the cumulative

lower probability that at a given point in the domain

the slip surface will be lower than that point. The graph

can be used to target further investigation or remedialworks on the basis of information on the location of

the slip surface that takes account of some of the uncer-

tainties in the available soil data.

Fig. 6. Lower cumulative probability distribution for the location of

the slip surface at the minimum FOS (lighter shade indicates higher

probability of slip surface being below given level, on a scale from zero

(black) to one (white)).

E. Rubio et al. / Computers and Geotechnics 31 (2004) 529–536 535

5. Conclusions

Probabilistic analysis of slope stability is now quite

commonplace, justified by the considerable uncertain-

ties involved in soil modelling. However, probabilistic

analysis may underestimate uncertainties, particularlywhen information available to the soil modeller does

not naturally appear in a probabilistic format. Impre-

cise information, such as interval measurements or lin-

guistic statements, tends to contain less information

than probability distributions. If an analysis is to be

faithful to the uncertainty in the available information,

then the available information sources must be pre-

served in the format in which they appear. It has beendemonstrated how this can be achieved with the theory

of random sets.

The averaging approach to aggregating information

from multiple sources has been demonstrated. A ran-

dom relation is a random set on the Cartesian product

of a vector of uncertain variables. The analysis in this

paper has addressed the cases of independent sets, joint

discrete probability distributions and random relationsconstructed from joint imprecise measurements.

Extension of a random relation through a numerical

model involves calculating the image in the numerical

model of each focal element of the random relation. In

general this involves solving an optimisation problem,

but it has been illustrated how the computational effort

can be greatly reduced by exploiting any monotonic

properties of the numerical model.The proposed approach has been demonstrated by

application to slope stability analysis. A numerical rou-

tine has been described, dealing with the situation where

the parameters determining slope hydrology are availa-

ble as probability distributions (with cross correlation

matrices) whilst the geotechnical parameters are availa-

ble as sets of intervals. The method generates interval

probability distributions on the FOS and the location

of the landslide slip surface. Besides indicating the mag-

nitude of uncertainty in predictions these interval prob-

ability distributions illustrate the relative contributions

of random variability and imprecision to the overalluncertainty in the analysis. The slope stability example

is typical of the situation at the outset of an infrastruc-

ture project when there is rather limited information

about the soils at the site. However, the proposed ap-

proach lends itself to iterative refinement of the esti-

mates of slope stability, in an imprecise Bayesian

updating mode, as more information is acquired in

terms of increasingly precise site-specific measurementsof soil properties [56].

Acknowledgements

Dr. Rubio�s work has been supported by the Spanish

Ministerio de Ciencia y Tecnologıa through a Post-Doc-

toral Research Fellowship and through a contract in theRamon y Cajal 2001 program. Dr. Hall�s research is

funded by a Royal Academy of Engineering Post-Doc-

toral Research Fellowship.

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