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Proceedings of the XVI ECSMGEGeotechnical Engineering for Infrastructure and DevelopmentISBN 978-0-7277-6067-8

© The authors and ICE Publishing: All rights reserved, 2015doi:10.1680/ecsmge.60678

plinary project (geology, remote sensing, geotech-nics and geophysics) that investigates ongoing slope instability and debris flow hazard in an active Al-pine environment. Some parts of the open scree slopes in the Meretschibach catchment that are sub-ject to repeated and significant surface erosion, have been characterised and test pits have been excavated to install sensors to measure water content, suctions and temperature. In parallel, further monitoring is underway periodically from a ground-based porta-ble radar interferometer combined with ongoing me-teorological measurements and data from a range of sensors to determine the degree of saturation, suc-tion and temperature in the ground at various loca-tions on the slope.

Future plans for modelling and analysis are dis- cussed briefly, in view of the challenges in predict-ing the impact of extreme events and protecting in-habitants and infrastructure located in the valley be-low.

ACKNOWLEDGEMENTS

The authors are most grateful for funding from the SNF Project No. 200021_144326/1 and Canton Valais, together with the Councils of Agarn and Leuk. In particular, we wish to acknowledge the support from Pascal Stoebener and Ernst Bleiker.

REFERENCES

Askarinejad, A. 2012. Failure mechanisms in unsaturated silty sand slopes triggered by rainfall, ETH Dissertation Nr. 21423. Askarinejad, A. Casini, F. Bischof, P. Beck, A. & Springman, S.M. 2013. Rainfall induced instabilities: a field experiment on a silty sand slope in northern Switzerland. Rivista Italiana di Geotecnica 3/2012 (luglio - settembre 2012): 9-30. Askarinejad, A. & Springman, S.M. 2014. Centrifuge modelling of the effects of vegetation on the responses of a silty sand slope sub-jected to rainfall. 14th International Conference of the Internation-al Association for Computer Methods and Advances in Geome-chanics (14IACMAG), Kyoto, Japan. (Eds: Oka, Murakami, Uzuoka & Kimoto), 1339-1344. Taylor & Francis Group, London, ISBN 978-1-138-00148-0. Berger, C. McArdell, B.W. & Schlunegger, F. 2010. Sediment transfer patterns at the Illgraben catchment, Switzerland: Implica-tions for the time scales of debris flow activities, Geomorpholo-gy 125, 421–432. Berger, C. McArdell, B.W. & Schlunegger, F. 2011. Direct meas-urement of channel erosion by debris flows, Illgraben, Switzerland,

J Geophys Res 116, F01002, doi:10.1029/2010JF001722. Caduff, R., Schlunegger, F., Kos, A., 2011. Assessment of precise spatial & temporal slope deformation w. Gamma Portable Radar In-terferometer in the Illgraben debris flow catchment, Central Swiss Alps, Geophys Res Abs 13. Casini, F. Serri, V. & Springman, S.M. 2013. Hydromechanical be-haviour of a silty sand from a steep slope triggered by artificial rainfall: from unsaturated to saturated conditions. Canadian Ge-otechnical Journal 50(1): 28-40. http://dx.doi.org/10.1139/cgj-2012-0095. Fankhauser, K. 2014. Geophysical Slope Characterization Using GPR and ERT in an Active Debris Flow Catchment. Master The-sis, EEG, ETH Zurich. Gabus, J.H. Weidmann, M. Bugnon, P.C. Burri, M. Sartori, M. & Marthaler, M. 2008a. Feuille 1287 Sierre, Atlas géologique Suisse, 1:25’000, Carte 111. Hürlimann, M., Rickenmann, D. & Graf, C. 2003. Field & monitor-ing data of debris flow events in the Swiss Alps, Canadian Ge-otechnical Journal 40(1), 161-175. Kos, A. Strozzi, T. Wiesmann, A. Amann, F. Tomkinson, W. & Conforti, D. 2010a. Spatial monitoring of large rock slopes insta-bilities at high resolution using a portable radar interferometer. Proceedings,3rd Int Dis & Risk Conf, Davos. Kos, A. Lunghi, D. Conforti, D. Tomkinson, W. Strozzi, T. & Wiesmann, A. 2010b. Integration of terrestrial Lidar and ground- based radar interferometry for monitoring rock slopes following blast mitigation. Proceedings, IAEG, Auckland, NZ. Lucas, D. Askarinejad, A.A. Herzog, R. Bleiker, E. & Springman, S.M. 2015. Volumetric water content determination by TDR sen-sors and decagons in gravelly soils. ECSMGE, Edinburgh. McArdell, B.W. Bartelt, P. & Kowalski, J. 2007. Field observations of basal forces and fluid pore pressure in a debris flow, Geophys. Res. Lett. 34, L07406. Oggier, N. 2011. Modelling of debris flows using RAMMS at the Meretschibach, Agarn, Wallis. Master Thesis, IGT, ETH Zurich. Raetzo, H. 2012. FOEN, pers. comm. Schlunegger, F. Badoux, A. McArdell, B.W. Gwerder, C. Schnydrig, D. Rieke-Zapp, D. & Molnar, P. 2009. Limits of sedi-ment transfer in an alpine debris-flow catchment, Illgraben, Swit-zerland, Quat. Sci. Rev. 28, 1097-1105. Springman, S.M. Laue, J. Boyle, R.,White, J. & Zweidler, A. 2001. The ETH Zurich Geotechnical Drum Centrifuge, Int. Journal of Physical Modelling in Geotechnics 1(1): 59-70. Springman, S.M. Jommi, C. & Teysseire, P. 2003. Instabilities on moraine slopes induced by loss of suction: a case history. Géotechnique 53(1): 3-10. Szymczak, S. Bollschweiler, M. Stoffel, M. & Diku, R. 2010. De- bris-flow activity & snow avalanches in a steep watershed of Va-lais Alps (Switzerland): dendrogeomorphic event reconstruction & identification of triggers, Geomorphology 116, 107-114. Teysseire & Candolfi AG. 2006. Meretschibach unterhalb Ges- chiebefang, Bericht Nr. 1642a/3, Visp. Theler, D. 2010. Contribution à la cartographie géomorpholo-gique de la dynamique sédimentaire des petits bassins versants torrentiels, Thèse de doctorat, Université de Lausanne, Faculté des géosciences et de l'environnement.

Application of Markov chains in the prediction of rock slopes degradation

Application des chaines de Markov dans la prévision de la dégradation de talus rocheux

J.Tinoco*1, S. Sanches2 , T. Miranda1 , A. Neves2 , M. Pinheiro1 , A. Ferreira2 and A.G. Correia4

1 Institution for Sustainability and Innovation in Structural Engineering/School of Engineering/University of Minho, Guimarães, Portugal

2 Ascendi, Matosinhos, Portugal * Corresponding Author

ABSTRACT In this paper it is proposed a degradation model for rock slopes, which will integrate a global tool for sustainable manage-ment of road infrastructures, currently under development. The proposed model is based on a continue-time Markov chain and predicts theevolution of a Slope Quality Index (SQI) over time. Taking advantage of the memoryless property of Markov models, the proposed modelis able to predict the next state only based on the current state, not depending on the sequence of events that preceded it. In addition, the ef-fect of maintenance actions is also considered in the prediction of the state condition of a given slope. The achieved results, which representa first iteration towards the development of a predictive model for rock slope degradation processes during its exploration phase, show thatthe use of Markov chains can lead to reliable results.

RÉSUMÉ Dans cet article, il est proposé un modèle de dégradation des talus rocheux, qui intégrera un outil global et durable de gestiondes infrastructures routières, actuellement en développement. Le modèle proposé est basé dans une chaîne de Markov de temps continue etprévoit l'évolution d'un indice de la qualité du talus (SQI) avec le temps. Prenant en compte la propriété sans mémoire des chaines de Mar-kov, le modèle proposé est capable de prédire l'état suivant utilisant seulement son état actuel, il ne dépende pas de la séquence des événe-ments qui l'ont précédé. En plus, l'effet des actions de maintenance est également pris en compte dans la prédiction de l'état d’un talus. Lesrésultats obtenus, qui représentent une première itération à l'élaboration d'un modèle de prévision pour le processus de dégradation des talusrocheux au cours de sa phase d'exploration, montrent que l'utilisation des chaînes de Markov peut guider à des résultats fiables.

1 INTRODUCTION

Portugal, as most of the developed countries, has nowadays a fairly complete road network (Neves 2005). These transportation infrastructures, com-posed by a set of elements that can be classified as bridges, road pavements, retaining walls, slopes and electronic equipments, are necessary to ensure good mobility conditions. In order to guarantee all security and mobility conditions of the road network, it is im-portant to make available a set of tools to support the maintenance strategies of each one of these elements. Although for some of them maintenance and conser-vation strategies based on degradation models al-ready exist, for others, particularly for rock slopes, almost nothing has been done so far.

In order to overcome this drawback, namely in terms of degradation and maintenance models for rock slopes, in this paper a theoretical approach is proposed based on Markov chains (Knill 2009). The proposed model has the particularity of being able to be applied during the exploration phase of the slope and requires only basic parameters. The main reasons be-hind the choice of Markov chains for rock slope degradation modelling are, on one hand, due to its successful application in different knowledge do-mains, namely in the study of bridge degradation (Sobreiro 2011) and, on the other hand, due to some of its particular and useful properties, namely the memoryless property. According to this property the future behaviour of a given slope depends only upon the actual condition state and not on the sequence of

Geotechnical Engineering for Infrastructure and Development

1904

events that preceded it (Yang et al. 2009). Thus, given the lack of historical data about rock slopes degradation processes, this property is of great im-portance for this study. The proposed model is based on a slope quality index (SQI) that is determined dur-ing the exploration phase of the infrastructure (Pin-heiro et al. 2014), which range from 1 to 5, and pre-dicts its evolution over time. The proposed model will integrate a tool that is currently in development under the project SustIMS (Sustainable Infrastructure Management System), aiming a sustainable and geo-referenced management of road infrastructures. For the maintenance model it is considered the improve-ment in performance and the deterioration delay ef-fect, as well as the reduction effect in degradation ra-tio of different maintenance and repair actions currently applied to rock slopes.

It should be also highlighted the innovation of the proposed model for rock slopes degradation predic-tion over time, as well as the maintenance model able to incorporate the impact of different actions. Indeed, so far and within our knowledge, there are no models able to predict the condition state of a given rock slope over time during its exploration phase, particu-larly based on a SQI that only requires basic informa-tion.

2 MARKOV MODEL

As stated, in this work a degradation model for rock slopes is proposed based on a Markov process. A Markov chain is a particular stochastic process with discrete states (where the in-dependent parameter, usually time, can be discrete or continuous). Stochas-tic processes have been extensively applied for mod-elling the deterioration of infrastructures over time such as bridges, pavement or wastewater systems, due to the randomness that characterizes a degrada-tion process (Baik et al. 2006; Kobayashi et al. 2010). According to Mishalani & Madanat (2002) there are two degradation models in discrete state processes: time based models and state based models. In a state based model (adopted in the present work), it is defined the probability of a slope to stay in the same condition state or to move to the next state, in a predefined period of time ∆t. In the present work we adopted a continuous time Markov process since the

intervals between inspections ∆t are not regular or similar for all slopes.

Yang et al. (2009) proposed the definition of a transition intensity matrix Q, defined as:

)exp( tQPQPPt

(1)

The intensity matrix, Q, represents the instantane-ous probability of transition between state i and state j (independently of ∆t) and is directly correlated with any transition probabilities matrix P.

Keep in mind that the slope degradation is a natu-ral and continuous process, it is impossible to im-prove its condition state without any intervention. Therefore, if a classification system contemplates 5 different condition states, then the matrix Q presents the format as shown in equation 2, meaning that dur-ing the deterioration process, in each infinitesimal time interval, the slope can only advance between ad-jacent condition states.

00000000

000000000

44

33

22

11

Q

(2)

The degradation model here proposed for rock slopes was implemented in the statistic software R (Core Team 2009) using the msm pack-age (Jackson 2011).

3 SYNTHETIC DATA

A database containing information related to the con-dition state of different rock slopes over time is a key element for the development of a degradation model, namely following a Markovian approach. However, since currently data related to the condition state of the rock slopes over time does not exist, a synthetic database was compiled aiming to simulate its degra-dation process as well as possible. For this task it was considered all the experience of the team involved in the SustIMS project related to the maintenance and conservation activities of rock slopes. The different criteria adopted for the synthetic database compila-tion are listed below.

Number of slopes: 200;

Classification system for slope condition state: C1 → Very Good; C5 → Very Bad;

Slope construction year: between 01/01/1965 and 31/12/2005. The slope construction year was defined according to a lognormal distribu-tion ( 0 and 3.0 );

Initial condition state: it was defined that after its construction a rock slope is in a condition state C1, C2 or eventually in C3. Therefore, the initial condition state was defined according to a exponential distribution ( 1 );

Inspections frequency: for the number of in-spections that each slope was submitted during its lifetime, the following law was adopted:

ettNinsp 0871.0708.0005.0 2 , where t is the age of the slope (years) at the moment of the inspection and e is a random error that fol-lows a normal distribution )2,0(N .

Condition state over time: the condition state of the rock slopes over time was defined con-sidering that the mean time periods in states C1 to C4 are 14; 9; 7 and 4 years respectively. These values were affected by a random error according to a normal distribution )2,0(N .

Figure 1 illustrates the mean time period (and re-spective 95% level confidence intervals according to a t-student distribution) for states C1 to C4. The small deviations observed are due to the random error e that was introduced.

In order to perform a better characterization of the synthetic database, Table 1 summarizes the number of times that a state r is followed by a state s.

4 DEGRADATION MODEL

4.1 Model design

As previously mentioned, a markovian approach was adopted for the development of a degradation model for rock slopes that only takes into account its actual condition state. Moreover, considering the character-istics of the maintenance routines applied in this type of slopes, i.e., the fact that the inspections frequency is much higher than the time that these slopes take to move to the next state, it was defined that in a given

moment a rock slope can only move between two consecutive condition states or stay in the same state. This implies that the matrix Q will have the structure presented in equation 2, respecting this way the con-dition of continuous degradation.

C1

C2

C3

C4

Time (years)

0 2 4 6 8 10 12 14

0 2 4 6 8 10 12 14

13.937

10.169

6.278

5.490

Figure 1. Experimental mean time period rock slopes measured in the synthetic database.

Table 1. Summary of the number of transitions from state r to state s.

From/To C1 C2 C3 C4 C5 C1 1130 123 0 0 0 C2 0 942 180 0 0 C3 0 0 348 164 0 C4 0 0 0 228 134

An initial estimate of matrix Q was obtained

through the calculation of the θi values based on the Jackson (2011) approach:

i

iji t

n (3)

where nij represents the number of elements that moved from state i to j and ∆ti is the time between observations where the initial condition state is i. In the calculation of matrix Q using equation 3 it is as-sumed that the data represent the exact transition moment in the Markov process (Jackson 2011). However, this situation hardly happens. However, as-suming that the data represent the exact transition moment, the matrix Q calculated by equation 3 will

1905

events that preceded it (Yang et al. 2009). Thus, given the lack of historical data about rock slopes degradation processes, this property is of great im-portance for this study. The proposed model is based on a slope quality index (SQI) that is determined dur-ing the exploration phase of the infrastructure (Pin-heiro et al. 2014), which range from 1 to 5, and pre-dicts its evolution over time. The proposed model will integrate a tool that is currently in development under the project SustIMS (Sustainable Infrastructure Management System), aiming a sustainable and geo-referenced management of road infrastructures. For the maintenance model it is considered the improve-ment in performance and the deterioration delay ef-fect, as well as the reduction effect in degradation ra-tio of different maintenance and repair actions currently applied to rock slopes.

It should be also highlighted the innovation of the proposed model for rock slopes degradation predic-tion over time, as well as the maintenance model able to incorporate the impact of different actions. Indeed, so far and within our knowledge, there are no models able to predict the condition state of a given rock slope over time during its exploration phase, particu-larly based on a SQI that only requires basic informa-tion.

2 MARKOV MODEL

As stated, in this work a degradation model for rock slopes is proposed based on a Markov process. A Markov chain is a particular stochastic process with discrete states (where the in-dependent parameter, usually time, can be discrete or continuous). Stochas-tic processes have been extensively applied for mod-elling the deterioration of infrastructures over time such as bridges, pavement or wastewater systems, due to the randomness that characterizes a degrada-tion process (Baik et al. 2006; Kobayashi et al. 2010). According to Mishalani & Madanat (2002) there are two degradation models in discrete state processes: time based models and state based models. In a state based model (adopted in the present work), it is defined the probability of a slope to stay in the same condition state or to move to the next state, in a predefined period of time ∆t. In the present work we adopted a continuous time Markov process since the

intervals between inspections ∆t are not regular or similar for all slopes.

Yang et al. (2009) proposed the definition of a transition intensity matrix Q, defined as:

)exp( tQPQPPt

(1)

The intensity matrix, Q, represents the instantane-ous probability of transition between state i and state j (independently of ∆t) and is directly correlated with any transition probabilities matrix P.

Keep in mind that the slope degradation is a natu-ral and continuous process, it is impossible to im-prove its condition state without any intervention. Therefore, if a classification system contemplates 5 different condition states, then the matrix Q presents the format as shown in equation 2, meaning that dur-ing the deterioration process, in each infinitesimal time interval, the slope can only advance between ad-jacent condition states.

00000000

000000000

44

33

22

11

Q

(2)

The degradation model here proposed for rock slopes was implemented in the statistic software R (Core Team 2009) using the msm pack-age (Jackson 2011).

3 SYNTHETIC DATA

A database containing information related to the con-dition state of different rock slopes over time is a key element for the development of a degradation model, namely following a Markovian approach. However, since currently data related to the condition state of the rock slopes over time does not exist, a synthetic database was compiled aiming to simulate its degra-dation process as well as possible. For this task it was considered all the experience of the team involved in the SustIMS project related to the maintenance and conservation activities of rock slopes. The different criteria adopted for the synthetic database compila-tion are listed below.

Number of slopes: 200;

Classification system for slope condition state: C1 → Very Good; C5 → Very Bad;

Slope construction year: between 01/01/1965 and 31/12/2005. The slope construction year was defined according to a lognormal distribu-tion ( 0 and 3.0 );

Initial condition state: it was defined that after its construction a rock slope is in a condition state C1, C2 or eventually in C3. Therefore, the initial condition state was defined according to a exponential distribution ( 1 );

Inspections frequency: for the number of in-spections that each slope was submitted during its lifetime, the following law was adopted:

ettNinsp 0871.0708.0005.0 2 , where t is the age of the slope (years) at the moment of the inspection and e is a random error that fol-lows a normal distribution )2,0(N .

Condition state over time: the condition state of the rock slopes over time was defined con-sidering that the mean time periods in states C1 to C4 are 14; 9; 7 and 4 years respectively. These values were affected by a random error according to a normal distribution )2,0(N .

Figure 1 illustrates the mean time period (and re-spective 95% level confidence intervals according to a t-student distribution) for states C1 to C4. The small deviations observed are due to the random error e that was introduced.

In order to perform a better characterization of the synthetic database, Table 1 summarizes the number of times that a state r is followed by a state s.

4 DEGRADATION MODEL

4.1 Model design

As previously mentioned, a markovian approach was adopted for the development of a degradation model for rock slopes that only takes into account its actual condition state. Moreover, considering the character-istics of the maintenance routines applied in this type of slopes, i.e., the fact that the inspections frequency is much higher than the time that these slopes take to move to the next state, it was defined that in a given

moment a rock slope can only move between two consecutive condition states or stay in the same state. This implies that the matrix Q will have the structure presented in equation 2, respecting this way the con-dition of continuous degradation.

C1

C2

C3

C4

Time (years)

0 2 4 6 8 10 12 14

0 2 4 6 8 10 12 14

13.937

10.169

6.278

5.490

Figure 1. Experimental mean time period rock slopes measured in the synthetic database.

Table 1. Summary of the number of transitions from state r to state s.

From/To C1 C2 C3 C4 C5 C1 1130 123 0 0 0 C2 0 942 180 0 0 C3 0 0 348 164 0 C4 0 0 0 228 134

An initial estimate of matrix Q was obtained

through the calculation of the θi values based on the Jackson (2011) approach:

i

iji t

n (3)

where nij represents the number of elements that moved from state i to j and ∆ti is the time between observations where the initial condition state is i. In the calculation of matrix Q using equation 3 it is as-sumed that the data represent the exact transition moment in the Markov process (Jackson 2011). However, this situation hardly happens. However, as-suming that the data represent the exact transition moment, the matrix Q calculated by equation 3 will

Tinoco et al.

Geotechnical Engineering for Infrastructure and Development

1906

correspond to the exact maximum likelihood esti-mates V. Thus, through the application of equation 3 the initial values of the matrix Q were estimated, which was then optimized in order to consider that the transitions take place at unknown occasions in between the observation times. At the end, the devel-oped Markov model for rock slope degradation pre-diction is characterized by the following matrix Q:

000001679.01679.000001480.01480.000000997.00997.000000751.00751.0

Q

(4)

The calculation of the likelihood used during the optimization process of the matrix Q was defined through the following expression:

N

slope

M

transitionijPV

1 11471)ln( (5)

where M is the number of transitions observed in slope s, N is the number of analyzed slopes and Pij is the probability of occurrence of observed transition, as predicted by the Markov model, according to equation 1.

Matrix Q (see equation 4), which represents the proposed model for rock slopes degradation predic-tion, can now be used to quantify the transition prob-abilities for a given period of time. For instance, if we would like to estimate the transition probabilities for a period of 10 years, we will obtain the probabil-ity transition matrix presented in Table 2. From its analysis, we can see that for a slope that actually is in a condition state C1, 10 years from now the probabil-ity of that slope being in condition state C3 or higher is around 21%.

Table 2. Probability transition matrix for a period of 10 years based on the proposed model for rock slop degradation process.

From/To C1 C2 C3 C4 C5 C1 0.4721 0.3141 0.1290 0.0549 0.0299 C2 0 0.3690 0.2917 0.1865 0.1527 C3 0 0 0.2276 0.3054 0.4669 C4 0 0 0 0.1865 0.8135 C5 0 0 0 0 1.0000

4.2 Model assessment

Equation 6 compares the total number of different condition states between two consecutive observa-tions (observed and predicted by the model) (So-

breiro 2011). From its analysis, we can see that the observed and predicted values are very close, which means a good quality of the proposed model.

658389565

10661129

656392564

10651130

5

4

3

2

1

5

4

3

2

1

predobsCCCCC

CCCCC

(6)

Assuming now the prediction of the slope degra-dation as a classification problem, the calculation of the confusion matrix (Hastie et al. 2009) can give us an overview of the quality of the proposed model. The global precision of the model is 84%. Addition-ally to the global performance, some other statistics can be calculated for each one of the classes (condi-tion states), namely: Sensitivity (Sen) and Specificity (Spe) (Hastie et al. 2009). In a perfect model, both Sen and Spe will present a unit value. Table 3 sum-marizes the values of Sen and Spe for each one of the classes (condition states) showing once again the good performance of the proposed model.

Table 3. Summary of Sen and Spe values.

From/To C1 C2 C3 C4 C5 Sen 1.00 0.88 0.68 0.58 0.80 Spe 0.95 0.93 0.95 0.96 1.00

In order to illustrate graphically the performance

of the proposed model, Figure 2 shows the relation-ship between the experimental values of the condi-tions state and the predicted ones according to the proposed model. Additionally, it is also shown the mean value and the standard deviation for each one of the 5 condition states, from which can be seen that the proposed model has a very interesting precision. However, some dispersion is also observed, except for condition state C1.

The identification of the time period, according to the model is also an important aspect in model per-formance assessment. Therefore, the mean time pe-riod for each condition state (Ti) was calculated using equation 7.

Figure 3 shows the mean time period according to the proposed model. From its analysis we can see that there is a high proximity with the values ob-served in the database used in this work to fit the model (see Figure 1). This observation underlines

once again the good performance of the proposed model for rock slope degradation prediction. The main difference between Figures 1 and 3 is related with the higher dispersion observed in the last one.

1 2 3 4 5

12

34

5

Experimental condition state

Pred

icte

d co

nditi

on st

ate

Predicted valuesMean of predicted values

Figure 2. Relationship between experimental values of condition state and predicted.

iiT

1 (7)

C1

C2

C3

C4

Time (years)

0 5 10 15

0 5 10 15

13.323

10.032

6.757

5.954

Figure 3. Mean time period in each condition.

5 MAINTENANCE MODEL

After the development of the degradation model it is important to understand the impact in the degradation curve of the application of maintenance and repair actions. Accordingly, we started by compiling a list with the different maintenance and repair actions that currently are applied to rock slopes and grouped them into three groups: improvement actions, delay actions or reduction actions. Improvement actions are those that when applied improve the slope perform-ance, at the time of their application by a certain value. Delay actions holds the degradation process for a specified period of time, and reduction actions reduce the degradation ratio during a specific period of time. After that, we quantified their impact in the slope performance and modelled their effects over the degradation curve.

Let us assume that an action is to be applied at in-stant ti, that at this time the slope performance is within the range of applicability of such action and that the condition state of the element can be deter-mined through equation 1. At instant ti, just before the application of the action, the condition state is de-fined through the following condition vector, Ct:

54321 CCCCCCt (8) If, based on expert judgment, it is known that the

maintenance action improves the condition state to Ctarget, then the components of the updated condition state vector are given by:

ett

n

Cj ettjt

ettit

it

Ciif

CiifC

CiifC

Cett

arg

arg,

arg,

*,

0arg

(9)

If we are applying a delay action that causes a de-lay in deterioration for a period of time after its ap-plication, it is assumed that the performance remains unchanged. This can be modelled considering that, during the effect of the action, the transition intensity matrix Q is an identity matrix. If an action causes a reduction in the deterioration ratio for a period of time after its application, the new condition state dur-ing that period can be calculated using equation 10, where the degradation ratio is reduced by a factor (RDR):

))100(exp( RDRtQCC if (10)

1907

correspond to the exact maximum likelihood esti-mates V. Thus, through the application of equation 3 the initial values of the matrix Q were estimated, which was then optimized in order to consider that the transitions take place at unknown occasions in between the observation times. At the end, the devel-oped Markov model for rock slope degradation pre-diction is characterized by the following matrix Q:

000001679.01679.000001480.01480.000000997.00997.000000751.00751.0

Q

(4)

The calculation of the likelihood used during the optimization process of the matrix Q was defined through the following expression:

N

slope

M

transitionijPV

1 11471)ln( (5)

where M is the number of transitions observed in slope s, N is the number of analyzed slopes and Pij is the probability of occurrence of observed transition, as predicted by the Markov model, according to equation 1.

Matrix Q (see equation 4), which represents the proposed model for rock slopes degradation predic-tion, can now be used to quantify the transition prob-abilities for a given period of time. For instance, if we would like to estimate the transition probabilities for a period of 10 years, we will obtain the probabil-ity transition matrix presented in Table 2. From its analysis, we can see that for a slope that actually is in a condition state C1, 10 years from now the probabil-ity of that slope being in condition state C3 or higher is around 21%.

Table 2. Probability transition matrix for a period of 10 years based on the proposed model for rock slop degradation process.

From/To C1 C2 C3 C4 C5 C1 0.4721 0.3141 0.1290 0.0549 0.0299 C2 0 0.3690 0.2917 0.1865 0.1527 C3 0 0 0.2276 0.3054 0.4669 C4 0 0 0 0.1865 0.8135 C5 0 0 0 0 1.0000

4.2 Model assessment

Equation 6 compares the total number of different condition states between two consecutive observa-tions (observed and predicted by the model) (So-

breiro 2011). From its analysis, we can see that the observed and predicted values are very close, which means a good quality of the proposed model.

658389565

10661129

656392564

10651130

5

4

3

2

1

5

4

3

2

1

predobsCCCCC

CCCCC

(6)

Assuming now the prediction of the slope degra-dation as a classification problem, the calculation of the confusion matrix (Hastie et al. 2009) can give us an overview of the quality of the proposed model. The global precision of the model is 84%. Addition-ally to the global performance, some other statistics can be calculated for each one of the classes (condi-tion states), namely: Sensitivity (Sen) and Specificity (Spe) (Hastie et al. 2009). In a perfect model, both Sen and Spe will present a unit value. Table 3 sum-marizes the values of Sen and Spe for each one of the classes (condition states) showing once again the good performance of the proposed model.

Table 3. Summary of Sen and Spe values.

From/To C1 C2 C3 C4 C5 Sen 1.00 0.88 0.68 0.58 0.80 Spe 0.95 0.93 0.95 0.96 1.00

In order to illustrate graphically the performance

of the proposed model, Figure 2 shows the relation-ship between the experimental values of the condi-tions state and the predicted ones according to the proposed model. Additionally, it is also shown the mean value and the standard deviation for each one of the 5 condition states, from which can be seen that the proposed model has a very interesting precision. However, some dispersion is also observed, except for condition state C1.

The identification of the time period, according to the model is also an important aspect in model per-formance assessment. Therefore, the mean time pe-riod for each condition state (Ti) was calculated using equation 7.

Figure 3 shows the mean time period according to the proposed model. From its analysis we can see that there is a high proximity with the values ob-served in the database used in this work to fit the model (see Figure 1). This observation underlines

once again the good performance of the proposed model for rock slope degradation prediction. The main difference between Figures 1 and 3 is related with the higher dispersion observed in the last one.

1 2 3 4 5

12

34

5

Experimental condition state

Pred

icte

d co

nditi

on st

ate

Predicted valuesMean of predicted values

Figure 2. Relationship between experimental values of condition state and predicted.

iiT

1 (7)

C1

C2

C3

C4

Time (years)

0 5 10 15

0 5 10 15

13.323

10.032

6.757

5.954

Figure 3. Mean time period in each condition.

5 MAINTENANCE MODEL

After the development of the degradation model it is important to understand the impact in the degradation curve of the application of maintenance and repair actions. Accordingly, we started by compiling a list with the different maintenance and repair actions that currently are applied to rock slopes and grouped them into three groups: improvement actions, delay actions or reduction actions. Improvement actions are those that when applied improve the slope perform-ance, at the time of their application by a certain value. Delay actions holds the degradation process for a specified period of time, and reduction actions reduce the degradation ratio during a specific period of time. After that, we quantified their impact in the slope performance and modelled their effects over the degradation curve.

Let us assume that an action is to be applied at in-stant ti, that at this time the slope performance is within the range of applicability of such action and that the condition state of the element can be deter-mined through equation 1. At instant ti, just before the application of the action, the condition state is de-fined through the following condition vector, Ct:

54321 CCCCCCt (8) If, based on expert judgment, it is known that the

maintenance action improves the condition state to Ctarget, then the components of the updated condition state vector are given by:

ett

n

Cj ettjt

ettit

it

Ciif

CiifC

CiifC

Cett

arg

arg,

arg,

*,

0arg

(9)

If we are applying a delay action that causes a de-lay in deterioration for a period of time after its ap-plication, it is assumed that the performance remains unchanged. This can be modelled considering that, during the effect of the action, the transition intensity matrix Q is an identity matrix. If an action causes a reduction in the deterioration ratio for a period of time after its application, the new condition state dur-ing that period can be calculated using equation 10, where the degradation ratio is reduced by a factor (RDR):

))100(exp( RDRtQCC if (10)

Tinoco et al.

Geotechnical Engineering for Infrastructure and Development

1908

Figure 4 shows the impact in the degradation curve of applying a combination of these three type of actions (maintenance scenario), which are im-provement, delay and reduction actions. Particularly, it was applied a reduction action after 5 years of the slope construction and then periodically every 10 years during 15 years, followed by a delay action when the slope was 25 years old and finally an im-provement action after 40 years from the slope con-struction, which was then periodically applied every 10 years.

0 20 40 60 80 100

12

34

5

Time (years)

Con

ditio

n sta

te

Natural degradation curveReduction actionDelay actionMaintenance action

Figure 4. Impact in the degradation curve of applying a combina-tion of three types of actions.

6 CONCLUSIONS

In this work, a theoretical approach for rock slope degradation prediction was presented. The model is based on Markov chains and predicts the evolution of a slope quality index (SQI) over time, which is de-fined during the exploitation phase of the slope and only requires basic information.

Based on the obtained results, the global perform-ance of the proposed model is very good. Indeed, the model was able to predict with high accuracy the time period for each condition state observed in the database. Moreover, the proposed model showed a good capacity to predict correctly the slopes degrada-tion over time. However, it should be stressed that

the model was developed based on a synthetic data-base, which may be influencing its performance. Ad-ditionally, it was also proposed a maintenance model that is incorporated in the degradation model, par-ticularly the effect of improvement, delay and reduc-tion actions.

ACKNOWLEDGEMENT

The authors wish to thank to AdI Innovation Agency, for the financial support through the program POFC for the project R&D SustIMS Sustainable Infrastruc-ture Management Systems (FCOMP-01-0202-FEDER-023113).

REFERENCES

Baik, H. Jeong, H. & Abraham, D. 2006. Estimating transition probabilities in markov chain-based deterioration models for man-agement of wastewater systems, Journal of water resources plan-ning and management 132(1), 15–24. Core Team , R Development 2009. R: A language and environ-ment for statistical computing, R Foundation for Statistical Com-putting, Vienna, Austria Hastie, T. Tibshirani, R. & Friedman, J. 2009. The Elements of Statistical Learning: Data Mining, Inference, and Prediction – sec-ond ed., Springer-Verlag, New York. Jackson, C. 2011. Multi-state models for panel data: the msm package for R, Journal of Statistical Software 38(8), 1–29. Knill, O. 2009. Probability Theory and Stochastic Processes with Applications. Overseas Press. Kobayashi, K. Do, M. & Han, D. 2010. Estimation of markovian transition probabilities for pavement deterioration forecasting, KSCE Journal of Civil Engineering 14(3), 343–51. Mishalani, R. & Madanat, S. 2001. Computation of infrastructure transition probabilities using stochastic duration models, Journal of Infrastructure systems 8(4), 139–48. Neves, L. 2005. Life-cycle analysis of bridges considering condi-tion safety and maintenance cost interaction, Ph.D. Thesis, De-partment of Civil Engineering, University of Minho, Portugal. Pinheiro, M. Sanches, S. Miranda, T. Neves, A. Tinoco, J. Fer-reira, A. & Gomes Correia, A. 2014. Slope quality index (SQI): An empirical system for rock slopes stability analysis in exploita-tion phase. International Journal of Rock Mechanics and Mining Sciences. (under revision). Sobreiro, F. 2011. Deterioration prediction models for existing bridges: Markov processes, Master’s thesis; Faculty of Science and Technology, New University of Lisbon; Lisboa, Portugal. (in por-tuguese). Yang. Y. Pam, H. & Kumaraswamy, M. 2009. Framework devel-opment of performance prediction models for concrete bridges, Journal of Transportation Engineering 135(8), 545–54.

Sliding potential of tuff Le Potentiel de Glissement du Tufs

E.Togrol1*, F. Kıran2 and M. Sezerer3

1 Istanbul University Department of Civil Engineering/Istanbul University, Istanbul, Turkey2 STFA Temel Arastırma ve Sondaj A.S./ Istanbul, Turkey3 STFA Temel Arastırma ve Sondaj A.S./ Istanbul, Turkey

* Corresponding Author

ABSTRACT Landslides and potential landslides in argillized tuff layers encountered at western Anatolia near the Aegean Sea coast pro-vide an excellent example of soil characterization to thus evaluate of potential risks at such an area. The aim of this paper is thus to draw at-tention to some of the geological and geotechnical factors involved in the assessment of landslide risk at such a terrain. The values of liquid limits and plasticity indices of tuff samples are considered as a sign of the material's high swelling potential. Mineralogical properties of tuff samples indicate the existence of a high percentage of montmorillonit type of clay minerals in the tuff. Montmorillonit is also accom-panied by small amounts of kaolinit, halloysite and illite. One important observation made in from the study of borehole logs is the exist-ence of weak zones at various depths. The emergence of ·weak zones is possibly connected to changes in the stress field when the rock mass has been subjected to straining. Surface excavations and other causes might have resulted in stress relief in the soil mass, an outcome which is seen as the principal factor of the observed straining of the soil body. Such a development could possibly be related to the accumu-lation of negative pore pressures and finally dilation of those zones. Inclinometers were installed along the alignment of slopes. Probable slide surfaces detected at inclinometers coincided with the slope movements occurring along the surfaces connecting the weak zones ob-served in boreholes.

RÉSUMÉ Les glissements de terrain ainsi que les glissements de terrain potentiels dans les couches de tufs argilières près des côtes de la Mer Egée, fournissent un excellent exemple de caractérisation du sol et ainsi qu'un moyen d'évaluer les risques potentiels dans une telle ré-gion. Le but de ce communiqué est d'attirer l'attention à certains facteurs géologiques et géotechniques engagés dans l'estimation des risques de glissements dans de pareil s terrains. Les valeurs des limites liquides ainsi que des indices de plasticité des échantillons de tuf sont considérés comme une indication d'un haut potentiel de gonflement du matériau. Les propriétés minéralogiques des échantillons du tuf, pointent à l'existence d'un haut pourcentage de minéral d'argile du type montmorillonite dans le tuf, aussi accompagné de kaolinite, d'halloysite et d'illite en petites quantités. Une observation importante faite à partir de l'étude des pieux de trous de forage, est l'existence de zones faibles à différentes profondeur s. L'émergence de zones faibles est vraisemblablement due à des variations de champs de pres-sion, suite à des tensions subies par la masse rocheuse. Les excavations faites en surface ou d'autres causes peuvent être à l’ origine d'un re-lâchement de pression dans la masse de terre, entraînant l'acception de celle-ci comme étant le facteur principal ayant provoqué des ten-sions dans le corps de terre. Une telle évolution pourrait être attribuée à l'accumulation de pressions négative dans les pores, résultant en une dilatation de ces zones. Des inclinomètres furent installés le long de l'alignement des pentes. De probables surfaces de glissement dé-tectées par les inclinomètres coïncident avec les mouvements de pente se trouvant le long des surfaces connectant les zones faibles obser-vées dans les trous de forage.

1 INTRODUCTION

The slope instability of soft rock terrain ranges from deep-seated bedrock failure to the generation of translational slides that usually run parallel to the general slope of the ground. In that later context,

landslide prone and moisture sensitive strata carry especially high risk of sliding.

Landslides and potential landslides in argillized tuff layers encountered at western Anatolia near the Aegean Sea coast provide an excellent example of soil characterization to thus evaluate of potential