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QUANTITATIVE LANDSLIDE HAZARD ASSESSMENT IN REGIONAL SCALE
USING STATISTICAL MODELING TECHNIQUES
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Manouchehr Motamedi
August, 2013
QUANTITATIVE LANDSLIDE HAZARD ASSESSMENT IN REGIONAL SCALE
USING STATISTICAL MODELING TECHNIQUES
Manouchehr Motamedi
Dissertation
Approved:
Accepted:
______________________________
Advisor
______________________________
Department Chair
______________________________
Co-Advisor or Committee Member
Dr.Ala Abbas
______________________________
Dean of the College
Dr. George K. Haritos
______________________________
Committee Member
Dr. Lang Zhang
______________________________
Dean of the Graduate School
Dr. George R. Newkome
______________________________
Committee Member
Dr. Hamid Bahrami
______________________________
Date
Dr.Robert Y. Liang
Dr. Wieslaw K Binienda
______________________________
Committee Member
Dr. Ali Hajjafar
ii
ABSTRACT
In this research study, a new probabilistic methodology for landslide hazard assessment in
regional scale using Copula modeling technique is presented. In spite of the existing
approaches, this methodology takes the possibility of dependence between landslide hazard
components into account; and aims at creating a regional slope failure hazard map more
precisely. Copula modeling technique as a widely accepted statistical approach is integrated
with the hazard assessment concept to establish the dependence model between ―
landslide
magnitude‖, ―
landslide frequency‖ and ―
landslide location‖ elements. This model makes us
able to evaluate the conditional probability of occurrence of a landslide with a magnitude
larger than an arbitrarily amount within a specific time period and at a given location. Part of
the Seattle, WA area was selected to evaluate the competence of the presented method. Based
on the results, the mean success rate of the presented model in predicting landslide
occurrence is 90% on average; while the success rate is only 63% when these hazard
elements were treated as mutually independent.
Also, Seismic-induced landslides are one of threatening effects of earthquakes around the
world that damage structures, utilities, and cause human loss. Therefore, predicting the areas
where significant earthquake triggered hazard exists is a fundamental question that needs to
be addressed by seismic hazard assessment techniques. The current methods used to assess
seismic landslide hazard mostly ignore the uncertainty in the prediction of sliding
displacement, or lack the use of comprehensive field observations of landslide and
iii
earthquake records. Therefore, a new probabilistic method is proposed in which the
Newmark displacement index, the earthquake intensity, and the associated spatial factors are
integrated into a multivariate Copula-based probabilistic function. This model is capable of
predicting the sliding displacement index (
) that exceeds a threshold value for a specific
hazard level in a regional scale. A quadrangle in Northridge area in Northern California
having a large landslide database was selected as the study area. The final map indicates the
sliding displacements in mapping units for the hazard level of 10% probability of exceedance
in 50 years.
Furthermore, to reduce human losses and damages to properties due to debris flows runout
in many mountainous areas, a reliable prediction method is necessary. Since the existing
runout estimation approaches require initial parameters such as volume, depth of moving
mass and velocity that are involved with uncertainty and are often difficult to estimate,
development of a probabilistic methodology for preliminary runout estimate is precious.
Thus, we developed an empirical-statistical model that provides the runout distance
prediction based on the average slope angle of the flow path. This model was developed
within the corridor of the coastal bluffs along Puget Sound in Washington State. The
robustness of this model was tested by applying it to 76 debris-flow events not used in its
development. The obtained prediction rates of 92.2% for pre-occurred and 11.7% for nonoccurred debris flow locations showed that the model results are consistent with the real
debris-flow inventory database.
iv
DEDICATION
To “Mahdi” & “Simin”,
my parents, my honest friends and my life‘s giving trees, for their endless, pure and
unconditional love and support
To “Kamelia”,
a loyal friend and a lovely and beautiful partner
And to “……”
for giving me the joys and sorrows of ―
being‖, this everlasting journey
v
ACKNOWLEDGEMENTS
First of all, I would love to express my appreciation to my advisor, Professor Robert
Liang, for his guidance, vision, patience, and generous support. I have learned a lot from
this great man especially a ―
right perspective‖ that I will use for the rest of my life.
Thanks to all committee members including Professor Ala Abbas, Professor Lan
Zhang, Professor Ali Hajjafar and Professor Hamid Bahrami for valuable discussions,
comments and reviews of this dissertation.
I would love to express my deepest appreciation to my always dear siblings, Hessam
and Negin; my loved ones, Abbas and Zarrin; a lovely and insightful lady, Mariel Barron;
a generous friend, Leila Bolouri; a kind couple, Reza and Farnaz Noohi; a good humanbeing and friend, Alireza Shabani; kind buddies, Kiarash Kiantaj and Shahriar
being‖ in my life, continuous support, patience, love, joy and
Mirshahidi, for their ―
encouragement throughout the hard and frustrating days, months and years of my PhD.
Finally, I would like to acknowledge Ms. Kimberly Stone for her invaluable piece of
advice, Ms. Christina Christian for her great help and all staffs and colleagues for
cooperative and kindly environment. I would also like to acknowledge Majid Hosseini
and Ali Tabatabai for their help and support; and thanks to Ms. Lynn Highland, for
cooperative and valuable information in USGS.
vi
TABLE OF CONTENTS
Page
LIST OF TABLES……………………………………………………………………….Xi
LIST OF FIGURES…………………………………………………………………….Xiii
CHAPTER
I.
INTRODUCTION .....................................................................................................1
1.1 Problem Statement ...........................................................................................1
1.2 Objectives of the Study ....................................................................................8
1.3 Outline of the Dissertation ...............................................................................9
II.
LITERATURE REVIEWS AND BACKGROUNDS ............................................11
2.1 Overview .......................................................................................................11
2.2 Landslides and their Causal Factors .............................................................11
2.3 Landslide Mitigation and Prevention ............................................................13
2.3.1 Landslide Risk Management and Assessment ...........................................14
2.3.2 Hazard Evaluation ..................................................................................16
2.3.3 Landslide Susceptibility Approaches .....................................................17
2.3.4 Probability of Landslide Magnitude ......................................................23
vii
2.3.5 Probability of Landslides Frequency .....................................................25
2.3.6 Vulnerability ..........................................................................................28
2.3.7 Landslide Risk Management Strategies .................................................30
III. QUANTITATIVE LANDSLIDE HAZARD ASSESSMENT USING COPULA
MODELING TECHNIQUES ............................................................................................33
3.1 Overview .......................................................................................................35
3.2 The Proposed Methodology ...........................................................................38
3.3 Study Area and Data ......................................................................................44
3.4 Method of Analysis ........................................................................................46
3.4.1 Data Preparation....................................................................................48
3.4.2 Dependence Assessment ........................................................................53
3.4.3 Marginal Distribution of Variables ........................................................57
3.4.4 Model Selection and Parameter Estimation ..........................................62
3.4.5 Goodness-of-fit Testing……………………………………………... ..65
3.4.6 Copula-based Conditional Probability Density Function ......................67
3.4.7 Probability of Landslide Frequency .......................................................67
3.5 Validation and Comparison of the Results ......................................................72
3.6 Landslide Hazard Map .....................................................................................73
3.7 Discussion ........................................................................................................73
3.8 Summary and Conclusion ................................................................................77
viii
IV .SEISMICALLY-TRIGGERED LANDSLIDE HAZARD ASSESSMENT: A
PROBABILISTIC PREDICTIVE MODEL MODELING TECHNIQUES .....................79
4.1 Overview .........................................................................................................79
4.2 Literature review ............................................................................................80
4.3 Methodology ..................................................................................................86
4.4 Application of the Proposed Model ...............................................................91
4.4.1 Study area and Database ........................................................................91
4.4.2 Development of Seismic Hazard Model ...............................................95
4.5 Seismic Landslide Hazard Map .....................................................................105
4.6 Validation ........................................................................................................105
4.7 Summary and Conclusion ...............................................................................106
V. AN EMPIRICAL-STATISTICAL MODEL FOR DEBRIS-FLOW RUNOUT
PREDICTION IN REGIONAL SCALE .........................................................................109
5.1 Overview ........................................................................................................109
5.2 Literature review ............................................................................................110
5.3 Methodology ..................................................................................................113
5.4 Application of the Proposed Model ...............................................................116
5.4.1 Study area and Database ..........................................................................116
ix
5.4.2 Development of the Empirical-Statistical Model ...................................120
5.5 Debris-flow Runout Prediction Results ..........................................................127
5.6 Validation ........................................................................................................128
5.7 Summary and Conclusion ................................................................................128
VI. CONCLUSIONS AND RECOMMENDATIONS ....................................................131
6.1 Summary of Important Research Results ........................................................131
6.2 Recommendations for Future Research ............................................................133
REFFERENCES .............................................................................................................135
x
LIST OF TABLES
Table
2.1
Page
Pros and Cons of priority ranking systems for state transportation agencies (Huang
et al., 2009) ........................................................................................................................17
3.1
Rank correlation coefficients of the pairs (M, S), (M, T) and (S, T) ........................65
3.2
Maximum likelihood estimation (MLE) of the examined distribution parameters ..69
3.3
AIC values of the examined probability distributions ..............................................71
3.4
Kolmogorov–Smirnov (KS) test for the data after Box–Cox transformation ..........71
3.5
Parameter estimation and confidence interval of the Copulas ..................................73
3.6
Landslide hazard probability values obtained from Copula-based and
multiplication-based models for 79 failure events .............................................................86
4.1
Rank correlation coefficients of the pairs (ac, DN) and (Ia, DN ) ..........................114
4.2 Calculated Z-values for spatial autocorrelation significance test ...........................116
4.3
Performance of marginal distributions for random variables and selected
probability density functions............................................................................................119
4.4
Kolmogorov–Smirnov (KS) test for the data after Box–Cox transformation.......119
xi
4.5
Parameter estimation and Kendall‘s τ of the Copulas ..........................................122
4.6
The AIC values of different Copulas functions ....................................................122
5.1
Summary of the three debris flow data subsets ....................................................144
5.2
Rank correlation coefficients of the pairs of the three debris flow data subsets...146
5.3
Summary of the significance tests of the best fit regression equation ..................149
xii
LIST OF FIGURES
Figure
Page
2.1
Integrated risk management process (Lacasse et al., 2010) .....................................14
2.2
Example of multi-layered neural network in landslide susceptibility analysis........23
2.3
Classification of landslide susceptibility analysis approaches (Yiping 2007) .........24
2.4
Example of risk criterion recommendation (F-N curve)..........................................35
3.1
Current general approaches in quantitative landslide hazard assessment ................42
3.2
The proposed methodology for quantitative landslide hazard assessment using
Copula modeling technique ...............................................................................................48
3.3
The location of the study area and landslide events in part of Western Seattle, WA
area ....................................................................................................................................50
3.4
Temporal distribution of landslides from the landslide database versus the year of
occurrence ..........................................................................................................................55
3.5
Definition of slope failure height (h), slope angle (β) and length (L) in a shallow
landslide .............................................................................................................................57
3.6
a) Slope map; b) Geologic map; c) Landslide location index (S) map in the study
area .....................................................................................................................................62
3.7
Scatter plot of landslide hazard component indices: a) Location index versus
magnitude index, b) Frequency index versus magnitude index, c) Frequency index versus
location index ....................................................................................................................66
xiii
3.8
Marginal distribution fitting to: a) transformed location index, b) magnitude index
3.9
Simulated random sample of size 10,000 from 14 chosen families of Copulas; a)
Ali-Mikhail-Haq, b) Frank, c) Galambos, d) Gumbel-Hougard, e) BB2, f) BB3 : upon
transformation of the marginal distributions as per the selected models (whose pairs of
ranks are indicated by ―
white‖ points) along with the actual observations .......................70
3.10 Goodness-of-fit testing by comparison of nonparametric and parametric K(z) for
Copula models. ..................................................................................................................73
3.11 Cumulative Gumbel-Hougard joint probability density function of S and M indices;
dark points represents the 79 validation points in ―1
-CDF‖ form .....................................74
3.12
Points indicating the landslide locations in part of the study area and a counting
circle used for exceedance probability calculation in the mapping cell ―
A‖ .....................77
3.13 Example of landslide hazard map (10m×10m cell size) for 50 years for landslide
magnitudes, study area M ≥ 10,000 m^2 in the study area. The value in each map cell
gives the conditional probability of occurrence of one or more landslide within the
specific time in that location ..............................................................................................81
4.1
Infinite slope representation showing the sliding block and the parameters used to
define the critical acceleration ...........................................................................................85
4.2
Flow chart showing the common required steps for developing earthquake-induced
landslide hazard Map .........................................................................................................96
4.3 Flow chart showing the presented methodology in this study for developing
earthquake-induced landslide hazard map .......................................................................101
4.4 Location of the study area, limit of greatest landslide concentration and Northridge
earthquake epicenter ........................................................................................................105
xiv
4.5 a) Geologic map of the selected study area; (b) cohesion and (c) friction angle as the
components of shear shear strength assigned to the geologic units (Yerkes and Campbell
1993, 1995); (d) Shaded-relief digital elevation model (DEM) of the selected area .......108
4.6 The contour map of the Arias intensity (I_a) generated by the 1994 Northridge
earthquake in the selected quadrangle. Each displayed Intensity value is the average of
the two horizontal components ........................................................................................110
4.7 Goodness-of-fit testing by comparison of nonparametric and parametric K(z) for
Copula models .................................................................................................................112
4.8 Seismic landslide hazard maps (10m×10m cell size) indicating the displacements
levels in mapping units for a) λ=0.0021 1/yr and b) λ=0.0021 1/yr in the study area ....123
4.9 Validation of the predicting model using the Q-Q plots of landslide areal percentages
versus Newmark displacements .......................................................................................125
5.1 Flowchart showing different current runout models. .................................................127
5.2 Schematic cross section defining debris flow runout distance (L), and slope gradient
(α) used in debris flow runout prediction.........................................................................135
5.3 Flow chart showing the presented methodology in this study for debris flow hazard
assessment. .......................................................................................................................136
5.4 a) Example of debris flow near Woodway, Washington on January 17, 1997 (Harp et
al. 2006), b) Selected study area in this paper. ................................................................139
5.5 Cumulative frequency plot of runout distances for the 326 debris-flow runout lengths
mapped from north Seattle to Everett (Harp et al. 2006). ................................................141
5.6 Slope gradient map of the study area. ........................................................................143
5.7 a) Regressions of L versus α for the three field data subsets, b) the final developed
regression model. .............................................................................................................144
xv
5.8 Calibration of the predictive model using the debris flow inventory data .................151
5.9 Example of debris flow hazard (exceedance probability) map for critical runout
distance of (Lc= 80m) ......................................................................................................152
xvi
CHAPTER I
INTRODUCTION
1.1 Problem statement
Landslides, which are mass of rock, debris or earth moving down a slope, are one of
the most frequently occurring natural disasters in the world as they expose a lot of human
and economic losses annually in different geographic areas. Study of landslides,
understanding of their different types, the involving aspects and causal factors are
necessary, specifically in more susceptible locations. In recent decades, landslide risk
assessment and management have led to development of the land utilization regulations
and minimized the loss of lives and damage to property. Landslide risk assessment and
management includes the analysis of the level of potential risk, deciding whether or not it
is acceptable, and applying the appropriate treatments to reduce it when the risk level
cannot be accepted (Ho et al., 2000).
Landslide hazard analysis is one of the fundamental components of the landslide risk
management; and during the recent few decades many attempts have been made to
develop that using a variety of techniques. Among the existing techniques, quantitative
methods attempted to assess the landslide hazard more precisely. Quantitative methods
mainly include different approaches for predicting ―
where‖, ―
when‖ and/or ―
how
frequently‖ the landslide might happen in the future. Although some advanced studies
have been conducted to integrate the three components of location, time and frequency of
landslide for hazard modeling (Guzzetti et al. 2005, Jaiswal et al. 2010), their models
undertake simplifying and unrealistic assumptions. These assumptions which dictate the
independence between landslide hazard components are mainly due to lacking of a more
comprehensive methodology at hand. Therefore to overcome the existing shortcomings,
more study and research is required in this regard.
Landslides are among the most threatening effects of earthquakes all around the world.
In fact, damage from triggered landslides is sometimes more than direct loss related to
earthquakes; largest earthquakes have the capability to cause thousands of landslides
throughout areas of more than 100,000
(Keefer, 1984). For example, on May 12,
2008, a magnitude (Mw) of 7.9 Sichuan earthquake in China had triggered more than
11,000 landslides and these events have threatened 805,000 persons and damaged their
properties (Gorum, 2008).Hence predicting the magnitude and location of the strong
shakings which trigger landslides are the fundamental questions need to be addressed in
any regional seismic hazard assessment. Among different methods for assessment of
earthquake-triggered landslides, probabilistic seismic landslide displacement analysis
(PSLDA) offers the opportunity to an engineer to quantify the uncertainty in the
assessment of the performance of susceptible slopes during the seismic loading, and it has
been applied in different studies in recent years (e.g., Ghahraman and Yegian 1996;
Stewart et al. 2003; Bray and Travasarou, 2007; Rathje and Saygili 2009). However due
to the complexity of these methods, little progress in PSLDA has been made so far. The
2
recent availability of new earthquake records offer the opportunity to develop more
realistic probabilistic techniques require advancing the assessment of landslide hazard
during earthquake.
Debris flow, which is sometimes referred as mudslide, mudflow or debris avalanche is
defined in the literature as a mixture of unsorted substances with low plasticity including
everything from clay to cobbles (Varnes 1978; Lin et al. 2006). Debris flows which are a
common type of fast-moving landslides are produced by mass wasting processes. They
are one of the most frequently occurring natural phenomena that cause a lot of human
loss and damage to properties annually all around the world (Hungr et al. 1999;
Prochaska et al., 2008). For instance, in 1996 and 2003 destructive debris flows took
place in the Faucon catchment, causing significant damage to roads, bridges and property
(Hussin 2011). Debris flows also play an important role in channel aggradations, flooding
and reservoir siltation and also basin sediment yielding (Bathurst et al., 2003; Burton et
al., 1998). Therefore, evaluation of the potential debris flows is a very vital task in
landslide risk management and generally it helps to delimit the extension of the hazard
and scope of endangered zones.
The existing debris-flow runout approaches require estimation of the influencing
factors that control the flow travel such as runout distance, depth of deposits, damage
corridor width, depth of the moving mass, velocity, peak discharge and volume (Dai et al.
2001). However, accurate estimation of all of these initial parameters which are involved
3
with a lot of uncertainty is very difficult in practice (Prochaska et al. 2008). The main
need is to develope a reliable probabilistic methodology which could be simply based on
a single influencing factor. Therefore, such a methodology will be capable of considering
the uncertainty of the debris-flow parameter(s) without complexity of the most existing
models. This model can be used for preliminary estimation of triggered debris-flow
runout distance based on the slope gradient of the travel path in regional scale. It is
believed that such an approach is valuable, time saving and can be applied to any similar
debris-flow hazard evaluation in the future.
1.2 Objectives of the study
 Developing a general methodology for quantitative modeling of landslide hazard
in a regional scale inventory: In spite of the existing approaches, this
methodology takes the possibility of dependence between landslide hazard
components into account; and aims at creating a regional slope failure hazard
map more precisely. Copula modeling technique as a widely accepted statistical
approach is integrated with the hazard assessment concept to establish the
dependence model between ―
landslide magnitude‖, ―
landslide frequency‖ and
―
landslide location‖ elements. This model makes us able to evaluate the
conditional probability of occurrence of a landslide with a magnitude larger than
an arbitrarily amount within a specific time period and at a given location.
4

Developing an improved model for earthquake-induced landslide hazard
assessment methods: The proposed model is based on casual relationships
between landslide displacement index (D), preparatory variables and seismic
parameters using probabilistic perspective for application in the constructionplanning in landslide susceptible areas. The sub-objectives of this method are as
follows: (a) to find out a more realistic indication of well-known Newmark
displacement index (D) for each mapping unit; (b) to explore a more precise oneby-one relationship between Newmark displacement values and each of the
affecting parameters in regional scale; and (c) to compare the results of empiricalbased technique and the proposed probabilistic method given the same involving
parameters.

Developing a reliable probabilistic method for debris-flow runout distance to be
simply based on a single influencing factor: Such a methodology will be capable
of taking the uncertainty of the debris-flow parameter(s) into account without
complexity of the most existing models. Thus, the proposed model is used for
preliminary prediction of debris-flow runout distance based on the slope gradient
of the travel path in regional scale. This model is built upon a reliable regression
analysis and exceedance probability function. This methodology can be usefully
implemented for similar debris flow assessment and mapping purposes in
regional scale studies. The final resulting hazard maps can be updated with any
additional related information in the future.
5
1.3 Outline of the dissertation

Chapter I provides the problem statement to be addressed in this research,
together with the specific objectives to be accomplished and also the required
approach of each part of the study. The organization of the dissertation is outlined
in this chapter as well.

Chapter II provides a literature review and background of related research. The
basic understanding of different types of landslides and their casual factors
together with landslide mitigation requirement is presented. Also landslide risk
management and the most essential components of that including hazard analysis,
susceptibility estimation, frequency-magnitude relations, recurrence probability
and decision making strategies are reviewed in this chapter as well.

Chapter III presents a general approach for quantitative modeling of landslide
hazard in a regional scale. Western part of the Seattle, WA area is selected to
evaluate the competence of the method. 357 slope failure events occurred from
1997 to 1999 and their corresponding slope gradient map and geology data are
considered in the study area to establish and test the model. A comparison is also
performed between the Copula-based hazard model and the traditional
6
multiplication-based one. Finally the result of the presented model is illustrated as
a landslide hazard map in the selected study area.

Chapter IV proposes a new probabilistic method in which the Newmark
displacement index and corresponding earthquake intensity as well as spatial
variables are integrated into one multivariate conditional function. The 1994
Northridge, earthquake California is used as appropriate database in this study
which includes all of the data sets needed to perform a detailed regional
assessment of seismic-triggered landslide. To validate the probability model, the
probabilistic results is compared with the field performance; in that areal
extension of the landslides already occurred in the study area is used as field
performance in this study. A regional seismic-induced landslide hazard map is
created using the presented probabilistic model as well.
 Chapter V provides an empirical-statistical model that predicts the runout distance
based on the average slope angle of the flow path. This model is developed within
the corridor of the coastal bluffs along Puget Sound (from north Seattle to
Everett), in Washington state. 250 historic and recent debris-flow events are used
to build the model. Correlation between the debris flow distance and slope
gradient ( ) for debris flow events is examined using graphical and quantitative
measures. The final regression model is applied in the normal exceedance
probability function and it is calibrated based on Gauss error function. The
7
robustness of this model is tested by applying it to 76 debris-flow events not used
in its development. Using the obtained predicting model, the debris flow hazard
map of the study area is also created.
 Chapter VI provides a summary of work accomplished in this research and
presents recommendations for future research.
8
CHAPTER II
LITERATURE REVIEW AND BACKGROUND
2.1 Overview
This chapter provides a review of early works and commonly accepted definitions
relevant to the landslide hazard assessment and risk management. The classification of
landslides and the principles of slope stability are discussed. Different methods of
landslide mapping and monitoring, landslide susceptibility evaluation techniques, their
hazard assessment and vulnerability concepts and common decision making approaches
are reviewed as well.
2.2 Landslides and their causal factors
Landslide is one of the most frequently occurring natural disasters in the world. They
bring about a lot of human loss and damages to properties annually in different
geographic areas. We define ―
landslide‖ (or slope failure) as a mass of rock, debris or
earth when it moves down a slope (Cruden, 1991).
The main driving force in landslide is gravity and the amount of this force will be
proportional to the angle of slope (Case, 1996). Increasing the slope angel will reduce
the stability of the mass. Also in case of impact of a triggering factor such as rain or
9
earthquake, the resisting forces which prevent the mass from sliding down the slope can
be significantly reduced. Understanding of the existing triggering factors of landslides
helps figuring out the most suitable remediation strategies. Popescu (1994) and Dai et al.
(2002) have compiled a framework for understanding the various causal variables of
landslide which divides the causing factors into two categories as: preparatory variables
and triggering variables. Preparatory variables place the slope in a marginally stable state
without initiating the failure. These variables mainly include slope gradient and aspect,
geology, slope geotechnical properties, elevation, vegetation cover and weathering. On
the other hand, the landslide triggering factors are understood as rains, floods,
earthquakes, and other natural phenomena as well as human activities such as terrain
cutting, filling, grading and over-construction. These events change the slope state from a
stable to an unstable condition in a short duration (Costa and Baker, 1981).
Landslides can be classified based on the type of material of the slope and the type of
slope movement (Varnes, 1978). The landslide movements are categorized into five
classes of sliding, falling, toppling, spreading and flow as follows (Cruden and Varnes
1996). Sliding is understood as a downward movement of a soil or rock mass due to
intense shear strain along rupture surfaces. Rock falls is defined as detachment of rock
from a steep slope along a surface and then descending of particles mainly through the air
by falling, bouncing, or rolling (Cruden, 1991). Rock toppling is the rotation of rock units
about their base (below the gravitational center). Toppling happens due to gravity applied
by upper units and sometimes due to water (or ice) in joints of the displaced rock units
10
(Pensomboon, 2007). The term spreading is described as sudden movements on waterbearing seams of sand or silt overlain by homogeneous clays or loaded by fills (Terzaghi
and Peck, 1948). The dominant mode of such a movement is lateral extension
accompanied by shear or tensile fractures (USGS fact sheet, 2004). Fast debris flow (also
called mudslides, mudflows, or debris avalanches) is defined as a rapid mass movement
in which a combination of loose soil, rock, air and water mobilize as slurry and flows
downward. Debris flows occur during intense rainfall on saturated soil. They start on
steep and once started, however, debris flows can even travel over gently slopes. Canyon
bottoms, stream channels, areas near the outlets of canyons and slopes excavated for
buildings and road-cuts could be the most susceptible areas (Hungr et al., 2001).
2.3 Landslide mitigation and prevention
Landslides are of primary importance because they have caused huge human and
economic losses all around the world. Numerous attempts have been made to mitigate the
losses due to slope failures mainly due to increased urbanization and development in
landslide-susceptible areas as a result of rapid population growth, increased precipitation
due to changing climatic patterns and continued deforestation of different geographic
regions (Dai, et al. 2002).
11
Landslide effects are lessen mainly through preventive means, such as restricting
people from areas with landslides history and by installing early warning systems based
on the measurement of ground conditions such as slope movement, groundwater level
changes and strain in rocks and soils (Fell, 1994). There are also various direct methods
of mitigating landslides, including geometric methods that modify the slope geometry;
using chemical agents to strengthen the slope material, applying structures such as piles
and retaining walls, grouting rock joints and fissures, hydro-geological methods to lower
the groundwater level and rerouting surface and underwater drainage (Encyclopedia
Britannica Inc., 2011).
2.3.1 Landslide risk management and assessment
Although risk is an unavoidable component in all engineering processes and its
prediction with certainty is not an easy task, it cannot be totally neglected. Risk
management is a set of strategies required for solving or at least reducing the problem of
landslides such as human losses and economical damages. Three main structures of
landslide risk management have been suggested in the literature as: (a) risk analysis, (b)
risk assessment, and (c) risk management (Fell and Hartfort, 1997). The risk analysis is
aimed at providing a judgment basis to understand how safe a susceptible area is; and it
can be practiced in different ways of qualitative to quantitative approaches (Aleotti and
Chowdhury, 1999). The main aim of risk assessment is to decide whether to accept or to
deal with the risk, or to set the priorities in that regard. This decision on acceptable risk
12
involves the responsibility of the owner, client or law-maker, based on risk comparison,
treatment options, benefits, potential human and economic losses, etc. Finally the risk
management is defined as a process at which the decision-maker decides whether to
accept the risk or treat the existing risk (Dai et al., 2002) (see Figure 2.1).
In terms of conditional probability, landslide risk when defined as the annual probability
of economic loss (annual loss of property) value of a specific element may be calculated
as follows (Morgan et al., 1992):
(2.1)
Where R: Risk ($/year); H: Hazard (0–1/year); V: Vulnerability (0–1); and C: Element
value ($). In other words, risk analysis reveals that how much loss ($) we will have per
year in any specific location of the study area.
13
Risk management
Risk assessment
Vulneribilty and
elements of risk
Hazard analysis
Historical
data
Figure 2.1 Integrated risk management process (Lacasse et al., 2010)
There are different levels in landslide risk assessment, including site-specific, global and
distributed landslide risk. Site-specific landslide assessment map is made for evaluation
of the economic loss at a specific site or landslide. On the other hand, a global risk
assessment map is calculated by summing up site-specific risk of all landslides in the
study area. Finally distributed landslide risk assessment provides a risk map that shows
the economic loss or damage at different spots of a given area (Dai et al., 2002). The
spatial subdivision in this type of landslide risk mapping is obtained using GIS and
multiplication of spatial landslide probability, affected zones, land-use or spatial
distribution of population or property and vulnerability (Leone et al., 1996).
14
One empirical method to do risk assessment is a priority ranking system for choosing
the more critical landslide sites for more investigation which can systematically ranks
landslide spots in an order of priority based on their probability and the consequence of
failure occurrence. Such a priority system is based on an inventory database of all
potentially unstable slopes in the study area. These ranking systems have mostly been
developed for or by state transportation agencies in the U.S. and the concept of these
systems are primarily based on the method of Pierson et al. (1990). Summing or
multiplication of all individual category scores are two approaches that leads to the final
rating score in such priority systems. Each of these developed ranking systems has its
own advantages and disadvantages as summarized in Table 2.1.
2.3.2 Hazard evaluation
Varnes and the IAEG Commission on Landslides and other Mass-Movements (1984)
proposed that the definition adopted by the United Nations Disaster Relief Organization
(UNDRO) for all natural hazards be applied to the hazard posed by mass movements
)Guzzetti et al., 2005). Based on Varnes definition the landslide hazard is the probability
of landslide occurrence within a specified period of time and a given area. Guzzetti et al.
(1999) improved this definition and added the concept of landslide magnitude. In contrast
to earthquakes hazard, in which the magnitude is applied as a measure of released energy
during the event, measure of landslide magnitude is not uniform in the literature.
Landslide magnitude has been defined as destructiveness of landslide by Hungr (1997).
15
Cardinali et al. (2002) and Reichenbach et al, (2005) defined landslide destructiveness as
a function of the landslide volume and expected landslide velocity. However, evaluation
of landslide volume and velocity is difficult for large areas and thus making the method
impractical. Destructiveness can also be accessible from accurate inventory maps where
landslide movements are primarily slow earth-flows (Hungr, 1997; Guzzetti, 2005).
Therefore, based on the modified definition of landslide hazard assessment it includes
the concepts of magnitude, frequency and location. Equation below expresses landslide
hazard as the conditional probability of these three concepts.
(2.2)
Where
,
and
predict how large a landslide will occur, how frequently it will occur
and where it will be located respectively. The multiplication used in the equation above is
due to the independence assumption among the three concepts. Although, due to lack of
information regarding the landslide phenomenon, independence can be an approximation
that makes the hazard assessment analysis easier to do but this assumption may not hold
all the time and in every area (Guzzetti et al., 2005). For example in many areas it is seen
that the larger landslides occurs less frequently than the small ones. The third component
which is susceptibility of landslide was discussed earlier; therefore the other two
probabilities of size and time are reviewed as follows.
16
2.3.3 Landslide susceptibility approaches
To obtain the probability of occurrence for a landslide in a given geographic area,
spatial probability (or called susceptibility) approaches are used. Numerous methods to
analyze the probability of landslide occurrence are divided by Soeters and van Westen
(1996) and van Westen et al. (1997) into inventory, heuristic, deterministic and statistical
approaches.
An initial step to any study of landslide hazard and landslide susceptibility mapping is
to produce a landslide inventory. The basic information for assessment and decrease of
landslide hazards or risk on a regional or community scale include the state of activity,
certainty of identification, primary type of landslide, main direction of motion, thickness
of material involved in landslide and dates of landslide occurrences. Detailed landslide
inventory maps are able to provide these types of information (Wieczorek, 1984).
Inventory maps preparation is done by collecting historic data of landslide events or
using aerial photograph interpretation accompanied with field verification. Depending on
the nature of problem, inventory maps can show detailed features of the landslides and
sometimes they present only points representing locations of the landslides. In addition,
the frequency of landslides can be determined in an area. It should also be mentioned that
17
Table2.1 Pros and Cons of priority ranking systems for state transportation agencies (Huang et
al., 2009)
Ranking
system agency
ODOT I
ODOT II
OHDOT
NYSDOT
UDOT
WSDOT
TDOT
Pros
Cons
-Weak risk component
-Strong hazard rating system
-Lacking asset management
-Include asset management
-Weak hazard rating
-Use highway function class
-Does not include soil slopes, fill failures or frozen ground
-Includes rock slopes, soil slopes
-Complex review procedure
and embankments
-Does not include frozen ground
-Includes risk assessment
-Heavily weights ditch effectiveness
-Does not include soil slopes, fill failures or frozen ground
-Includes risk assessment with
-Heavily weights ditch effectiveness
adjustments for geologic factor
-Does not include soil slopes, fill failures or frozen ground
-Good risk and asset management
-Weak hazard rating
program
-Does not include soil slopes, fill failures or frozen ground
-Balanced hazard and risk rating
-Lacking asset management
-Does not include soil slopes, fill failures or frozen ground
-Balanced hazard and risk rating
MODOT
-Unique graphic relationship
between risk and consequence
-Strong hazard rating system
BCMOT
-Scaling factors for each category
-Lacking asset management
-Does not include soil slopes, fill failures or frozen ground
-Scaling factors increase low hazard and low risk potentialDoes not include soil slopes, fill failures or frozen ground
are not equal
-Includes cost estimates
18
the frequency-magnitude relations may be derived from landslide inventories that is
useful to realize landslide probabilities (Yesilnacar, 2005).
Heuristic approaches estimate landslide potential from data on preparatory variables
based on expert knowledge and opinions. The assumption is that the relationships
between landslide susceptibility and the preparatory variables are determined. Then a set
of variables is entered into the model to estimate susceptibility of landslide (Gupta and
Joshi, 1989). The heuristic models have some limitations such as they need long-term
information about the landslides, their triggering factors, the subjectivity of weightings
(and ratings of the variables) and mostly the required information for them is not
available.
Deterministic approaches deal with slope stability analyses and they are commonly
used in site-specific scale. To be able to apply these methods, the ground condition
should be uniform relatively across the study region and the landslide types should be
clear and fairly easy to analyze (Terlien et al., 1995; Wu and Sidle, 1995).
Statistical models determine the statistical combinations of influential variables in
landslide occurrence in the past. This determination is done for areas which are currently
free of landslides, but having conditions that are similar to those with landslide in the
past. In the literature, there are two major statistical method groups, bivariate and
multivariate methods (Wahono, 2010).
19
In bivariate statistical method, each variable influential in landslide is overlaid with
the landslide inventory map, and considering landslide densities weighting values are
calculated for each class. In order to calculate the weighting values, there are many
methods such as information value, weights of evidence and statistical index methods in
the literature (Gokceoglu et al., 2000; Zerger, 2002; Lee and Sambath, 2006; Kanungo,
2006). In multivariate statistical method, the weighting values are obtained by combining
of all influential variables. These values denote the contribution degree of each
combination for each influential variable to landslide susceptibility within each mapping
unit. This method is based on the presence or absence of landslide occurrence in the
mapping unit (Suzen and Doyuran, 2004; Gorum et al., 2008; Lamelas et al., 2008;
Tunusluoglu et al., 2008). It should be mentioned that one of the limitations of
multivariate statistical approaches is that it is probable that such statistical methods result
in very deceptive analysis result when used in a black-box manner (Ho et al., 2000). In
comparison with the bivariate method, the multivariate approach is to some extent costly
and time consuming in gathering and data analysis (Firman, Wahono, 2010). In the
following, some common bivariate and multivariate statistical approaches including
multiple linear regression, discriminate factor and logistic regression models are
discussed.
Multiple regression and discriminant analysis are such conventional multivariate
statistical methods in which the weights of influential variables in landslide occurrence
denote the relative contribution of each variable in landslide occurrence in the study area.
20
It is needed to use continuous data in these multivariate statistical models and also when
both methods takes only two values for influential variables (landslide occurs or not),
they present limited value and under these conditions, the assumptions required to test the
hypothesis in regression analysis are violated (Carrara, 1983). In such cases, another
multivariate technique, named logistic regression is implemented to estimate the
probability of landslide occurrence.
Logistic regression, which is a multivariate analysis technique, predicts the presence
or absence of a result based on values of a set of predictor variables. There are
advantages in using logistic regression over the other methods. First, variables can be
either continuous or discrete, or any combination of these two types and secondly the
distributions of variables do not necessarily have to be normal (Lee and Evangelista,
1991).
Among the recent methods for landslide susceptibility assessment, some studies
adopted fuzzy logic and artificial neural network models. Fuzzy set theory presented by
Zadeh (1973) is a powerful method to handle complicated problems in different
disciplines (Lee, 2006). A fuzzy set can be described as a set containing members that
have various degrees of membership in the set (Ross, 1995). In fact, membership of the
elements can take on any value in the range of (0, 1) reflecting the degree of certainty of
membership. For landslide susceptibility maps, generally, the attribute of interest like
landslide causal factors measured over discrete intervals, and the membership function
21
can be expressed as membership values (Lee, 2006). Given two or more maps with fuzzy
membership values for the same variable, a variety of operators can be applied to
combine the membership values (Zimmerman, 1996).
An artificial neural network (ANN) is a mechanism inspired by the structure of
biological neural networks which makes us able to acquire, represent and compute a
mapping from one multivariate area of information to another, given a set of data
representing that mapping. In other words, an artificial neural network creates a model to
generalize and predict outputs from inputs (Garrett, 1994).
The most frequently used neural network method is the back-propagation training
algorithm. Generally, this algorithm trains the network until the intended minimum error
is obtained between the actual and desirable output values (Pradhan and Lee, 2010). The
training process is done by a set of examples of associated input and output values. This
learning algorithm which is a multi-layered neural network is composed of different
layers including an input layer, hidden layers, and an output layer (see Figure 2.2).
Multiplying each input by a corresponding weight is done by the hidden and output layer.
After summing the product, they get processed using a nonlinear transfer function to
obtain the final outcome (Gurney, 1997).
The artificial neural network has many advantages over the other statistical
approaches. For instance in this method we do not need to use specific statistical
22
variables and also any specific statistical distribution for the data. In addition, integration
of GIS
Figure 2.2 Example of multi-layered neural network in landslide susceptibility analysis
data or remote sensing data is easier in artificial neural network because we are able to
define the target classes with much more consideration to their distribution in the
corresponding domain of each data source (Zhou, 1999). Also ANNs method give a more
optimistic assessment of landslide susceptibility than logistic regression analysis
(Nefeslioglu et al., 2008). The chart displayed in Figure 2.3 shows a classification for a
various of landslide susceptibility methods discussed.
23
2.3.4 Probability of landslide magnitude
To estimate the probability of landslide magnitude, different types of landslide
inventories were used (Malamud et al., 2004; Picarelli et al., 2005). The area, volume,
velocity or momentum of the landslide event were used as the proxy for the landslide
Figure 2.3 Classification of landslide susceptibility analysis approaches (Yiping 2007)
magnitude in different studies (e.g., Guzzetti, 2002; Marques, 2008; Jaiswal et al., 2010).
Assuming landslide size as its magnitude approximation, the first component of the
hazard assessment equation is estimated from the analysis of the frequency–area
distribution of landslide dataset (inventory map). Studies show that the landslide size
24
increases up to a maximum value as the number of landslides grows; then it decreases
rapidly along a power law (Malamud et al., 2004; Guthrie et al., 2004).
Landslide hazard researchers (Pelletier et al., 1997; Stark and Hovius, 2001; Guzzetti
et al., 2002; Guthrie and Evans, 2004; Malamud et al., 2004) have found that the
probability density function (PDF) of landslide size is compatible with a truncated
inverse gamma distribution very well. Using this distribution the probability density
function of landslide size (
) can be given as equation below.
(2.3)
Where
is the gamma function of
and
are
parameters of the distribution. By fitting equation above to the available inventories, the
distribution parameters can be found.
In another study of frequency–size (area) of landslides, double Pareto probability
distribution has been found in agreement with probability distribution function of
landslide size (Stark and Hovius, 2001). Using this distribution
is given as equation
below.
(2.4)
25
Where
and
These probability distributions are converted to cumulative distribution function (cdf) to
estimate the probability of occurrence of a landslide with a magnitude larger than a
specific minimum amount.
2.3.5 Probability of landslides frequency
To obtain the probability of landslide occurrence time, we should gain the exceedance
probability of occurrence of the event during time ‗t‘. To be able to model the landslide
occurrence time in this way, we need to make an assumption that landslides are
independent random point-events over time (Crovelli, 2000).
Two common probability models in this regard are the Poisson and the Binomial
models. The Poisson model shows the probability of a random-point events occurring in a
fixed period of time if these events occur with a known average rate and independently of
the time since the last event (Ahrens, 1982). Frequency analysis of natural phenomena
such as volcanic eruptions floods and landslides are modeled using the Poisson model
(Crovelli, 2000). The probability occurrence of ‗n‘ landslides during time ‗t‘ using the
Poisson model is given by equation below.
(2.5)
26
Where
is the average rate of landslide occurrence and corresponds to the reverse of
mean recurrence interval between successive events. These variables can be obtained
from a multi-temporal landslide inventory maps.
Also, the exceedance probability of landslide (the occurrence probability one or more
event during time ―
t‖ is given as:
(2.6)
There are some assumptions which should be taken into account in both prediction of
future landslide occurrences and applications of the results of the probability model.
These assumptions include: (1) the probability of a landslide event happening in a very
short time is proportional to the length of the time period; (2) the number of landslide
events in separate time periods are independent (3) the probability of more than one event
can be neglected in a short time period; (4) The past and future events observations have
the same mean recurrence time; and (5) The probability distribution of landslide events
having fixed time period length is the same (Crovelli, 2000, Guzzetti, 2005). These
assumptions should be considered for the binomial probability model as well which is
discussed as follows.
27
The second commonly used model is binomial model which consider the landslide
events as random-point events in time. In this model, time is divided into discrete
intervals of fixed length that within each time interval an event may or may not occur
(Neumann, 1966). The binomial model was adopted to study the temporal occurrence of
phenomena such as floods and landslides. The exceedance probability of landslide
occurrence during time ―
t‖ by employing the binomial probability model is given as
equation below.
(2.7)
Where P is the estimated probability of a landslide event in time ―
t‖, and its reverse (1/P)
is the estimated mean recurrence interval between successive landslide events. This
parameter(s) is obtained from landslide inventory maps the same as Poisson model.
The results of Poisson and binomial probability models are not the same for short
mean recurrence intervals while the binomial model overvalue the exceedance probability
of landslide events in the future. But for long intervals these two models result in the
compatible and similar outcomes (Crovelli, 2000).
2.3.6 Vulnerability
28
The International Society of Soil Mechanics and Geotechnical Engineering defines
―
vulnerability‖ as the degree of loss to a given element (or set of elements) within the
area affected by the landslide hazard. It is a non-dimensional value between 0 (no loss)
and 1 (total loss). For properties, the loss is expressed as (the value of the damage)/(the
value of the property); for persons, it is shown as the probability that a person will die,
given the person is hit by the landslide (Wong et al., 1997). Vulnerability can also be
used to predict the damage probability for properties and persons; In this case it will be
expressed as the probability of loss and not the degree of loss (Fell, 1994; Leone et al.,
1996).
There is no completely united methodology for quantitatively assessment of landslide
vulnerability in the literature. This estimation of vulnerability for landslides is qualitative,
subjective and based on expert knowledge most of the time. As an example, the
vulnerability of a property which is located at the base of a steep slope is more vulnerable
(higher vulnerability) than a property located further from the base and it is because of
the debris movement velocity (Dai et al., 2002). Although vulnerability is often defined
as a value varying between 0 and 1, it has been considered in some literature (e.g.
Remondo et al., 2008) as a value greater than 1 (i.e. (the value of the damage)/(the value
of the property) >1 or repair cost more than a new construction).
Vulnerability depends on both the landslide intensity ―
I‖ and the element susceptibility
―
S‖ (or reverse of element resistance at risk which is ―
1/R‖). The general equation can be
29
written in form of V = f (I, S) (Li et al., 2010). Various types of landslide have different
intensity levels and therefore impose different degree of vulnerability. For example based
on study of Amatruda et al. (2004), large rock falls intensity level in the open areas is
always high to very high, while slides display low to medium intensity most of the times.
In the meantime, if a person is buried by debris, (s)he will die most likely but if (s)he is
struck by a rockfall, only injury may happen (Li et.al., 2010).
In order to differentiate low, medium, and high intensity for block fall, rockfall, slides,
and flows, Heinimann (1999) determined the vulnerability to different landslide
intensities for different types. However, he mentioned the necessity of assumption for
required information as a main limitation of the method. Some ranges and values on
vulnerability of persons in exposed areas, in a vehicle or in a building have been
recommended based on historic records of Hong Kong by Finlay (1996). The
vulnerability matrices were provided by Leone et al., (1996) which can be used to obtain
the vulnerability of structure, person, facilities, etc. Duzgun and Lacasse (2005) proposed
a 3D conceptual framework with scale (S), magnitude (M) and elements at risk (E) to
assess vulnerability. Also Zêzere et al., (2008) provided the vulnerabilities of structures
and roads affected by different intensities of landslides their work were based on
experiences or historic databases while using the information of structure type, age, etc.
In addition, a univocal conceptual framework for quantitative assessment of physical
vulnerability to landslides was introduced by Uzielli et al. (2008). Kaynia et al. (2008)
studied the use of first-order second-moment method application in vulnerability
30
estimation. Some researcher (Li and Farrokh, 2010) proposed approximate functions for
vulnerability of properties and persons based on landslide intensity and the susceptibility
of elements at risk.
Generally speaking, it is very complex to assess vulnerability to landslides in contrast
to other natural phenomena such as floods and earthquakes. For example, the location,
the status of a person (sleeping or awake) within the affected property or the size of
windows in the building would influence significantly the amount of loss and therefore
vulnerability degree. Moreover, vulnerability to landslide is impacted by some other
factors such as mass velocity, impact angle and strength of the structure material; since it
is not easy to estimate exactly all of these factors involved in vulnerability, it is advised
to analyze the vulnerability in an average sense (Li, 2010).
2.3.7 Landslide risk management strategies
As discussed, the landslide risk management is a process on whether the obtained risks
by risk analysis are tolerable and whether the existing risk control methods are adequate.
If the control measures are not enough, this landslide assessment process tells us what
alternative approaches are required. These recommendations should be done by
consideration of the significance of the estimated risks and the related social,
environmental and economic aftermaths (Fell et al., 2005). Landslide risk management is
described as a whole process including susceptibility, hazard, vulnerability and risk
31
mapping done to assess the existing landslide threat in the study area (Lacasse et al.,
2010).
When the risk from a landslide in susceptible areas to landslide is recognized, the
relevant authorities have to select a solution form a variety of strategies to deal with it
and mitigate the risk. Dai et al., (2002) categorized these strategies into planning control,
engineering solution, acceptance criterion and monitoring and warning systems as
follows.
i) Planning control
Planning control tries to direct the development pattern of the susceptible (or affected)
area in a way that reduces the landslide risk and is very economical approach to choose.
This strategy can be performed by eliminating the current constructions and/or
discouraging new constructions in high risk areas to implement regulating in susceptible
areas (Kockelman, 1986).
ii) Engineering solution
Two engineering solutions are commonly applied in general to mitigate the landslide
risk in susceptible area. The first one is to modify the unstable slope to stop onset of
32
landslide and the second one is to control the landslide movement not to be too
damaging. It should be mentioned that this is the most expensive option to reduce
landslide risk (Dai et al., 2002).
iii) Acceptance option
When a landslide risk is well understood, it may be recommended to select the
acceptance option. It is one of the most difficult steps in landslide risk management
process and the selection of this option depends mainly on whether the benefits the
people receive in the area under the study by this option compensate the risks of landslide
or not (Bromhead, 1997). To select this strategy, acceptable risk criteria should be set up
first which is shown as F–N curve in theory and is established using the existing
observations for different events such as road traffic, plane crash, dam failure and
landslide as well. This curve is a plot relating the number of fatalities (N) to the
cumulative frequency (F) of N or more fatalities in log–log scale (see Figure 2.4). To
define the acceptable and tolerable risk levels the public concern is considered by the
authorities. Acceptable level is described as a level of risk in which no further reduction
is obtained. This is desirable for the society to reach. Tolerable risk criteria usually
require that maximum risk of mortality (death of a person) in a specific place should not
be higher than a pre-defined threshold (Bunce et al., 1995).
Acceptability of landslide risk depends upon factors such as the nature of the risk
(engineered or natural slope failure), media coverage about the incident and the public
33
expectation from the event (whether it is frequent or rare to occur) (Lacasse and Farrokh,
2010).
Figure 2.4 Example of risk criterion recommendation (F-N curve)
iv) Monitoring and warning systems
The main goal of a monitoring and warning system for a specified landslide is to
evaluate the current status of the slope in terms of stability. This system determines
34
whether the landslide is in active condition or not. In case it is active and risky, some
strategies as discussed earlier are at hand to choose. One can be to accept the
consequences of landslide occurrence and the other option is to stabilize the slope or any
other engineering solution (a monitoring system can be applied to confirm the
effectiveness of the engineering work) (Dai et al., 2002). Finally a monitoring and
warning system can be employed to warn people for emergency preparations or
evacuation. Therefore landslide monitoring and warning system would be an alternative
strategy to reduce landslide risk in case that engineering solutions were not practical and
economical to choose or they did not work (Kousteni et al., 1999).
A common used system for monitoring is inclinometer which detects small
movements and deformations. The other widely used device is down-hole slope
extensometer and although its sensitivity and precision are not the same and will not
present completely the same result, it is able to support large soil displacements.
Interferometric Synthetic Aperture Radar (InSAR) technique is a modern and powerful
tool for monitoring mass movements of unstable slopes. The InSAR methodology is that
using a couple of satellites on different orbits and based on the assessment of phase
difference of two SAR interferogram of the same area, InSAR makes 3D maps form the
terrain as discussed earlier (Kimura and Yamaguchi, 2000).
The choice between available options is dependent on the preference of the decisionmakers considering all possible outcomes of each of the options. After the probabilities
35
and consequences have been estimated, methods of decision analysis may be used to
arrive at management decisions by identifying the alternatives of actions, possible
outcomes, and respective consequences or cost for each scenario.
v) Decision making
Risk assessment has the principal objective of deciding whether to accept, or to treat the
risk, or to set the priorities. The decision on acceptable risk involves the responsibility of
the owner, client or regulator, based on risk comparison, treatment options, benefits,
tradeoff, potential loss of lives and properties, etc. Risk management is the final stage of
risk management, at which the decision-maker decides whether to accept the risk or
require the risk treatment. The risk treatment may include the following options (AGS,
2000): (i) accepting risk (ii) avoiding risk, (iii) reducing the likelihood, (iv) reducing
consequences, (v) monitoring and warning system, (vi) transfering the risk, and (vii)
postponing the decision (Pensomboon 2007).
The alternative with the least expected cost is usually chosen if the expected value is
the criterion for decision. Cost–benefit analysis is the most widely used method in the
process of decision making. It involves the identification and quantification of all
desirable and undesirable consequences of a particular mitigation measure. When
measuring the cost of risk, a monetary value is generally used. By identifying the various
available options together with relevant information for assigning design parameters, the
cost and benefit of each mitigation measure can be assessed (Dai et al., 2001).
36
CHAPTER III
QUANTITATIVE LANDSLIDE HAZARD ASSESSMENT USING COPULA
MODELING TECHNIQUES
3.1. Overview
Landslide is one of the most frequently occurring natural disasters on the earth. They
cause a lot of human loss and damages annually in different areas around the world.
Study of landslides and understanding of different involving aspects and their various
causes are urgent in today world specifically in more susceptible areas and developing
societies. In recent years, landslide hazard assessment has played an important role in
developing land utilization regulations aimed at minimizing the loss of lives and damage
to property. Vernes (1984) has defined landslide hazard as the probability of occurrence
of a potentially destructive landslide within a specified period of time and within a given
geographical area. Guzzetti et al. (1999) modified this definition and added the concept
of landslide magnitude to that. This new component of landslide hazard mostly denotes
37
the destructiveness extent (landslide area) or intensity of the landslide event however
there is no unique measure for that in the literature (Hungr 1997; Guzzetti 2002).
Equation 3.1 presents this currently used definition of landslide hazard.
H = P[M ≥ m in a specified time period & in a given location with specified preparatory factors]
(3.1)
Where in this equation H is the landslide hazard value (0-1)/year; ―
M‖ is the landslide
magnitude and ―
m‖ is a specific magnitude amount (to be considered as an arbitrary
minimum amount). In other words, Equation 3.1 is a conditional probability of
occurrence of a landslide with a magnitude larger than an arbitrary minimum amount
within a specific time period and in a given location. The ―
preparatory factors‖ in the
equation refer to all of the specifications of the area including geology, geometry and
geotechnical properties.
To evaluate landslide hazard quantitatively in regional scale, three probability
components related to the concepts of ―
magnitude‖, ―
frequency‖ and ―
location‖ need to
be estimated. a) To estimate the probability of landslide magnitude, the area, volume,
velocity or momentum of the landslide event have been used as the proxy for the
landslide magnitude in different studies (e.g., Guzzetti 2002; Marques 2008; Jaiswal et al.
2010). However, no single generally accepted measure of landslide magnitude exists
(Hungr 1997; Guzzetti 2005). In order to estimate the probability of landslide magnitude
38
in most studies, the relationship of the landslide magnitude and frequency is observed to
typically have a power law distribution with a flattening of curve at the lower amount
(e.g. Pelletier et al. 1997; Stark and Hovius 2001; Malamud et al. 2004; Guthrie and
Evans 2004). These probability distributions are converted to cumulative distribution
function (cdf) to estimate the probability of occurrence of a landslide with a magnitude
larger than a specific minimum amount. b) To assess the probability of landslide
recurrence (frequency) the first approximation is that landslides can be considered as
independent random point-events in time (Crovelli, 2000). Two common models named
the Poisson model and the binomial model are used to investigate the occurrence of slope
failures using this approximation (Coe et al. 2000; Onoz and Bayazit 2001; Guzzetti
2005). The Poisson model is a continuous model consisting of random-point events that
occur independently during time. Also the binomial model is a discrete model in which
time is divided into equal discrete intervals and a single point-event may or may not
occur within each time interval (Haight 1967). c) To estimate the probability of
occurrence a landslide in a given location, spatial probability (or called susceptibility)
approaches are applied. Generally speaking, landslide susceptibility methods are divided
into two categories of qualitative and quantitative approaches. Qualitative techniques
such as geomorphological mapping demonstrate the hazard levels in descriptive terms
(Yesilnacar 2005) which are subjective and based on expert‘s knowledge and experience.
On the other hand, quantitative techniques mainly include deterministic and statistical
methods which both guarantee less subjectivity.
39
After evaluation of the hazard component values for the study area, the common
quantitative approach is to multiply them assuming they are mutually independent (e.g.,
Guzzetti, 2005; Jaiswal et al., 2010). Chart displayed in Figure 1 presents the common
steps required for quantitative landslide hazard assessment based on the independence
assumption. In fact, in probabilistic terms landslide hazard is calculated as the joint
probability of three probabilities as equation below.
(3.2)
Where
is the probability of landslide magnitude,
is the probability of landslide
spatial probability‖ (susceptibility). The terms ―
spatial
recurrence (frequency) and S is ―
probability‖ and ―
susceptibility‖ are used in the literature interchangeably for describing
the probability related to the ―
location‖ of a landslide occurrence. Also ―
probability of
landslide recurrence‖ is used in this paper in a same meaning with ―
exceedance
probability‖ and ―
temporal probability‖ as maybe seen in other studies (Coe, 2004;
Guzzetti, 2005).
40
Magnitude-frequency
analysis
Probability of landslide
magnitude (Pm)
Triggering factors
Temporal probability models
Probability of landslide
recurrence (Pt)
Independent preparatory factors
Susceptibility analysis
Validation
Historical landslide
database
Spatial probability (S)
Landslide Hazard = Pm ˟ Pt ˟ S
Validation
Landslide hazard assessment/map
Figure 3.1 Current general approaches in quantitative landslide hazard assessment
3.2 The Proposed Methodology
The expression of landslide hazard in Equation 3.2 which is currently used for
quantitative probabilistic landslide hazard assessment at medium mapping scale
(1:25,000-1:50,000) could be argued (Jaiswal et al. 2010). This assumption that the
41
landslide hazard components including ―
magnitude‖, ―
frequency‖ and ―
location‖ of a
landslide are independent may not be valid always and everywhere (Guzzetti et al. 2005).
This currently used simplification may be due to lack of enough historical data in a study
area, lack of a suitable approach and also because of complexity of the landslide
phenomenon (Jaiswal et al. 2010). It can be seen in lot of areas that geographical settings
(―
location‖ component of hazard) such as lithology, gradient (slope angle), slope aspect
(the horizontal direction to which a mountain slope faces), elevation, soil moisture and
soil geotechnical properties impact the landslide magnitude (Dai et al. 2002). Also, more
frequent severe rainfalls (higher ―
frequency‖ of landslide) can trigger more destructive
(larger ―
magnitude‖) landslides; or as Jaiswal (2010) mentions small triggering events
will result in another landslide magnitude distribution than a large triggering event and
also the locations where landslides will occur may be substantially different.
Furthermore, in many areas we expect slope failures to be more frequent (higher
―
frequency‖ of landslide) where landslides are more abundant and landslide area is large
(Guzzetti et al. 2005). In fact, the dependence between the landslide hazard components
is neglected just for simplification which can notably reduce the accuracy of the landslide
hazard prediction. Other attempts have been done to overcome this simplifying
assumption by estimating landslide hazard by direct using the number of landslides per
year per mapping unit (Chau et al. 2004). However such approach requires a
comprehensive multi-temporal landslide database, which is rarely available in practice
(Jaiswal et al. 2010). To do a contribution in this regard, in this section we present a
general methodology to consider the possible dependence between the landslide hazard
42
elements. Inventing such a general procedure seems to be necessary for better forecasting
landslide hazard in regional scale. The required steps needed to quantitatively estimate
the landslide hazard under the assumption of dependence between the hazard elements is
explained as follows.
In terms of landslide hazard elements of ―
magnitude‖ and ―
frequency‖, we obtain
probability distribution for these two components if there is no noticeable dependence
between any of these two and the others. Otherwise, any dependent component is
evaluated as a ―
numerical index‖ and not a probability distribution value. The ―
numerical
index‖ for landslide magnitude could be the quantity of area, volume, momentum or
velocity of landslide depending on the data availability. Also, in terms of landslide
frequency, ―
mean recurrence interval‖ between successive failure events is implemented
as the ―
numerical index‖.
In terms of landslide location, the meaning of the susceptibility values changes
depending on the relationship they have with the other hazard components but their
values remain the same. In other words, since the susceptibility values are membership
indices (Lee and Sambath 2006) they could be both probability values to be employed
independently (in case there is no dependence between ―
location‖ and the other
components) and ―
numerical indices‖ to be applied in joint probability functions (when
there is a dependence relationship). Different spatial factors such as slope gradient,
aspect, curvature, distance from drainage, land use, soil parameters, altitude, etc are
43
selected by different authors depending on the study area and are combined using various
quantitative methods to produce these membership indices in each mapping unit (e.g.
Remondo et al. 2003; Santacana et al. 2003; Cevik and Topal 2003; Chung and Fabbri,
2003, 2005; Lee 2004; Saito et al. 2009).
In case we have a noticeable dependence between landslide hazard components, there
must be a joint probability function to be established and be applied for precise landslide
hazard prediction. Therefore, selection of a suitable joint multivariate probability function
is required. In the last years, different multivariate techniques were presented in different
fields especially hydrological and environmental applications as follows. Gaussian
distribution (normal) has been widely and easily applied in the literature but it has the
limiting feature that the marginal distributions have to be normal. To obtain such a
feature for Gaussian distribution, data transformation has been used in the literature
through Box–Cox‗s formulas (Box and Cox, 1964). However, these transformations are
not always useful and they could sometimes cause distortions of statistical properties of
the variables (Grimaldi and Serinaldi, 2005). To overcome such a shortcoming, bivariate
distributions with non-normal margins have been proposed: Gumbel bivariate
exponential model with exponential marginals has been applied in the literature (Gumbel,
1960; Bacchi et al., 1994); bivariate Gamma distribution has been suggested in flood
frequency analysis (Yue et al., 2001); bivariate Logistic and mixed models with Gumbel
margins have also been used (Yue and Wang, 2004). In addition application of
multivariate extreme value distributions have been presented and used by Tawn (1988,
44
1990) and Coles and Tawn (1991). Moreover, models such as multivariate Exponential,
Liouville, Dirichlet and Pareto has been completely described in detail by Kotz (2000).
Long and Krzysztofowicz (1992) studied the Farlie–Gumbel–Morgenstern and Farlie
polynomial bivariate distribution functions in hydrological frequency which are not
dependent of marginals. Also a general form of a bivariate probability distribution, using
the meta-Gaussian distribution introduced by Herr and Krzysztofowicz (2005) which
characterize the stochastic dependence between measured variables.
However, all these distributions and models have these shortcomings: (a) increasing
the number of involved variables gets the mathematical formulation complicated; (b)
although analyzed variables could show different margins in reality, all of the univariate
marginal distributions have to be the same type; (c) marginal and joint behavior of
studied variables could not be distinguished (Genest and Favre 2007). Fortunately, the
Copula modeling technique can overcome all of these drawbacks. In fact it is able to
formulate a variety types of dependence between variables (Nelsen 1999); it could allow
each of the random variables to follow its own distribution types independently from the
other variables; and it can model marginal distribution of each random variable unbiased
and not based on a pre-defined distribution type (Zhang et al. 2006); also marginal
properties and dependence structure of variables can be investigated separately using
Copula modeling technique (Grimaldi and Serinaldi 2005).
45
Dependence assessment using Copula model is widely applied in finance such as
credit scoring, asset allocation, financial risk estimation, default risk modeling and risk
management (Bouy et al. 2000; Embrechts et al. 2003; Cherubini et al. 2004; Yan 2006).
In biomedical studies, Copulas are applied in modeling correlated event times and
competing risks (Wang and Wells 2000; Escarela and Carri`ere 2003). In engineering,
Copulas are used in probabilistic analysis of mechanical and offshore facilities
(Meneguzzo, et al. 2003, Onken, et al. 2009) and also multivariate process control and
hydrological modeling (Yan 2006; Genest and Favre 2007). More specifically, in
hyrological engineering, bivariate Copula function (2D) has been used in order to model
rainfall intensity and duration (De Michele and Salvadori, 2003); bivariate Copula was
applied to describe the dependence between volume and flow peak (Favre et al., 2004);
trivariate Copula (3D) was employed in multivariate flood frequency analysis (Grimaldi
and Serinaldi, 2005); and also De Michele et al. (2005) employed 2D Archimedean
Copulas which is the most common form of Copula models to simulate flood peakvolume pairs to check the adequacy of dam spillways. In geotechnical engineering
studies, Copula modeling approach has not been applied yet so it is believed that this
presented work could pave the way of implementing such a powerful statistical tool in
this field.
Based on the proposed methodology three scenarios are possible: a) when there is no
significant dependence between the hazard components in a study area, obviously
―
multiplication‖ would be performed to obtain the hazard values. b) bivariate Copula
function would be applied when there are only two dependent hazard elements. And c)
46
when all of the three components are significantly dependent to each other, trivariate
Copula function would be employed (Fig.2).
Historical landslide
database
Triggering factors
Independent preparatory factors
Magnitude index (M)
Frequency index (T)
Location index (S)
Dependent?
No
H = Pm ˟ Pt ˟ S
Probability of landslide magnitude (Pm)
if M is independent
H = Pm ˟ P(t ∩ S)
Probability of landslide recurrence (Pt)
if T is independent
H = Pt ˟ P(m ∩ S)
Spatial probability (S)
if S is independent
H = P(m ∩ t) ˟ S
Otherwise
Validation
Landslide Hazard
Yes
H = P(m ∩ t ∩ S)
Landslide hazard
assessment/map
Figure 3.2 The proposed methodology for quantitative landslide hazard assessment using
Copula modeling technique
3.3 Study Area and Data
Part of Western Seattle area was selected as a suitable region to evaluate the proposed
methodology (Fig.3.3). The Seattle area has long been received damages and losses to
47
people and properties due to landslides. Shallow landslides which are the most common
type of slope failure in this area usually occur every year from October through April
within the wet season (Thorsen 1989; Baum et al. 2005). In 1998, the Federal Emergency
Management Agency (FEMA) launched a hazard management program called Project
Impact and Seattle was one of the pilot cities in this study. Shannon and Wilson, Inc. was
assigned to collect a digital landslide database dating from 1890 to 1999 (Laprade et al.
2000) and the landslide data became freely available to the USGS for use. This
comprehensive compiled database in Seattle, WA was an exceptional opportunity for
scientists for further study of landslide in Seattle area. For example, Coe (2004)
employed this historical database to conduct the probabilistic assessment of precipitationtriggered landslides using Poisson and bionomial probability models. He expressed the
results as maps that show landslide densities, mean recurrence intervals, and annual
exceedance probabilities in the area. Also Harp et al. (2006) using this database
performed an infinite-slope analysis to establish a reliable correlation between a slopestability measure (factor of safety, FS) and the locations of shallow slope failures in the
Seattle area. Most of these mentioned efforts including analysis results, maps and the
historical landslide database established an appropriate base for this presented study in
this section.
Seattle area was selected as the study area because it was believed that the number of
landslide events in the selected region was large enough (357 events) (Fig.3.3).
48
Figure 3.3 The location of the study area and landslide events in part of Western Seattle, WA area
(Coe et al. 2004)
This number of landslide record makes it possible to test the dependence between
hazard components with more accuracy and also to evaluate the presented model more
confidently. Further, the temporal length of the landslide database was long enough (87
years from 1912 to 1999) to make the expected validity time of the model longer.
Also as it was required to use one part of the landslide records for modeling and the
other part for validation, a division was required to be done. The number of landslide
events in each part was the basis for this categorization. In fact, it was needed to choose a
49
time duration (yearly basis) before the year of the last landslide record (1999) which the
landslide events within that be applied for validation part. The time duration of 19971999 that include 79 slope failure events was selected to verify how much the proposed
model is capable to predict the future landslide hazard. Although this selection was
subjective, it is believed that in any other way of data grouping, the total dataset would be
poorly divided due to the large number of landslide events available in 1996 (see
Fig.3.4). Furthermore, it should be noted that a shortcoming of the existing landslide
database is that all of the geographical locations of failures consist of point locations, not
polygons, which cause some level of approximation in the analysis (Harp et al. 2006).
3.4 Method of analysis
As previously mentioned, Copula is a type of distribution function used to describe
the dependence between random variables. In other words, Copulas are functions that
connect multivariate probability distributions to their one-dimensional marginal
probability distributions (Joe 1997). The Copula concept in dependence modeling goes
back to a representation theorem of Sklar (1959) as follows. Let assume that we have n
variables (or populations) as
variable as (
and N observations are drawn for each
are the marginal cumulative
; and also
distribution functions (CDFs) of the variables
the
non-normal
multivariate
distribution
Now in order to determine
of
the
variables,
denoted
as
Copula, C, could be used as an associated dependence function.
50
This function returns the joint probability of variables as a function of marginal
probabilities of their observations regardless of their dependence structure as below.
(3.3)
Where
if
( ) for i = 1, 2,.. n is the marginal distribution and C:
= Copula
is continuous (Grimaldi and Serinaldi 2005). Different families of Copulas have
been developed in last decades and the comprehensive descriptions of their properties
have been compiled by some authors (Genest and MacKay 1986; Joe 1997; Nelsen
1999). The most popular Copula family is Archimedean Copulas (Yan 2006).
Archimedean Copulas are frequently used because they allow modeling dependences
easily and very well with only one parameter. Archimedean Copula for two random
variables, X and Y, with their CDFs, respectively, as F(x) = U and F(y) = V is given by
(3.4)
Where
and
denote a specific value of U and V respectively; and also
Copula generator which is a convex decreasing function satisfying
is the
; and
when
Also the conditional joint distribution based on Copula is expressed as follows. Let X
and Y be random variables with their CDFs, respectively, as F(x) = U and F(y) = V.
51
The conditional distribution function of X given Y = y can be presented by the equation
below.
(3.5)
As discussed earlier, this paper presents three random variables of ―
S‖, ―
T‖ and ―
M‖ as
the ―
numerical index‖ for landslide location, landslide frequency and landslide
magnitude respectively for all 278 landslide events (i.e. observations). Considering what
discussed above regarding the Copula theory, equation below presents the mathematical
terminology used hereafter in this paper:
Variables: (S, T, M)
Observations: (
,(
Dependence modeling using Copula function:
(3.6)
Where depending on the degree of correlation between the variables, the Copula function
(C) could be in either bivariate or trivariate form.
52
3.4.1 Data preparation
Obviously as the first step toward the modeling, it was needed to prepare the input
data for all of the three landslide hazard components including landslide magnitude,
landslide frequency (recurrence) and landslide location. As discussed earlier, these
components could be either ―
numerical index‖ or probability distribution values
depending on their dependence relationship with each other. But in order to begin the
modeling, we assume that they are all numerical indices (Fig3.2). Using this assumption,
we would be able to test the mutual dependence to observe if it is required to keep each
hazard component values as numerical indices or to transform them to probability
distribution form. The calculation process for landslide magnitude index (M), landslide
frequency index (T) and landslide location index (S) is discussed as follows for 278 out of
357 landslide events from 1912 to 1996 (the rest of the dataset records will be applied for
validation part).
3.4.1.1 Landslide magnitude
Although different proxies have been used for landslide magnitude in different studies
as mentioned earlier, ―
area‖ (aerial extent) is recognized as one of the most common and
reasonable representatives for ―
landslide magnitude‖ by different authors (Stark and
Hovious 2001; Guthrie and Evans 2004; Malamud et al. 2004; Guzzetti 2005). In the
53
historical landslide database of West Seattle region the ―
landslide area‖ was not directly
recorded and the ―
slope failure height‖ was the only available dimension of the slope
failure events. The ―
slope failure height‖ (h) is defined as the approximate elevation
difference between the head-scarp and the slip toe (Fig3.5), and it was estimated from
historical records and also field verification (Shannon and Wilson Inc 2000).
Figure 3.4 Temporal distribution of landslides from the landslide database versus the year of
occurrence
To estimate the ―
area‖ of the each landslide from its ―
slope failure height‖ (h), this
height dimension was converted to the ―
landslide length‖ (L) assuming that the angle of
the slope and the slip gradient of the failure are the same for each landslide event (Eq3.7)
(Fig3.5). Furthermore, to estimate the failure area, a general relationship between the
landslide dimensions (length and width) was required. Such a relationship was needed to
54
be generic, geometry-based and independent from the geology of the displaced materials
in landslide; and therefore the ―
fractal theory‖ was the best tool to be used in this regard.
The fractal is a mathematical theory developed to describe the geometry of complicated
shapes or images in nature (Mandelbrot, 1982). The fractal character of landslides can be
described by a self-similar geometry and it was recommended as a valuable tool for
investigation of landslide in susceptible areas (Kubota, 1994; Yang and Lee 2006). Yokoi
(1996) states that landslide blocks in a huge landslide have a fractal character. He also
concluded that their fractal dimension with respect to width is 1.24 and with respect to
length is 1.44 on average and this fractal dimension in width (or length) for a large
landslide area is not dependent on the base rock geology. These findings were valuable
mathematical relationships (between length and width) to be applied in our landslide
database for estimating the ―
area‖. Assuming each landslide event as one block in our
dataset we estimated the width of the ―
failure area‖ from the available length (Eq3.8 and
Eq3.9).
(3.7)
L h/
D ∝ 1.24W and
A
L
D ∝ 1.44L
1.16
W ∝ 1.16 L
(3.8)
(3.9)
/
Where D is the landslide fractal dimension, W is the width of each landslide block, L is
the length of each landslide block,
is the slope gradient, h is the slope failure height and
55
A is the landslide ―
area‖ estimation (Fig3.5). This calculation was done for 278 landslide
events in the study area to estimate one magnitude index (M) for each record.
Figure 3.5 Definition of slope failure height (h), slope angle (
and length (L) in a shallow
landslide
3.4.1.2 Landslide frequency
In terms of landslide frequency, ―
mean recurrence interval‖ between successive
failure events was selected as the landslide frequency index (T) in the study region. The
―
mean recurrence interval‖ is simply a representative of landslide frequency; and easily
calculated by dividing the total landslide database duration over the number of events.
Every single landslide event (totally 278 events) is assigned either a ―
mean recurrence
56
interval‖ or its corresponding value from a ―
probability distribution‖ depending on the
dependence between hazard components.
3.4.1.3 Landslide location
As discussed earlier, landslide susceptibility values should be determined to obtain the
―
membership indices‖ for the landslide location component. These susceptibility values
could be estimated using different methods. Among all of the available approaches,
deterministic method was selected for this study. This method deals with slope stability
concept and can be used in both site-specific scale and regional-scale studies (Terlien et
al. 1995; Wu and Sidle 1995; Guzzetti et al. 1999, 2005; Nagarajan et al. 2000;
Ardizzone et al. 2002; Cardinali et al. 2002; Santacana et al. 2003). Deterministic
methods in regional-scale studies mostly calculate the factor of safety (FS) in the
mapping units (cell) of the study area. FS is the ratio of the forces resisting slope
movement to the forces driving it, i.e. the greater the FS value the more stable is the slope
in the mapping unit. Number of methods has been invented by different authors using
GIS analyses to calculate the FS values in different scales (Pack et al. 1999; Montgomery
and Dietrich, 1994; Hammond et al. 1992; 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 Seattle area was used as the required ―
membership indices‖ (Eq3.10). This
method is described as follows.
(3.10)
57
Where FS is the factor of safety, α is the slope angle and
is water unit weight, γ is the
unit weight of slope material, c′ is the effective cohesion of the slope material, υ′ is the
effective friction angle of the slope material, t is the normal slope thickness of the
potential failure block, and r is the proportion of the slope thickness that is saturated. It
was assumed that groundwater flow is parallel to the ground surface (with the condition
of complete saturation, that is r = 1) (Harp et al. 2006). The FS was then calculated by
inserting values for friction, cohesion and slope gradient in Equation 10 for each preassigned mapping cell in the area. In order to provide the initial values of the FS equation
(Eq3.10), geology and topography data was needed to be employed. The geology data
was obtained from the digital geology map of Seattle area at a scale of 1:12,000 created
by Troost (2005) (Fig3.6 a). Material properties (including friction angle and cohesion
values as shear-strength components) were selected based on an archived database of
shear-strength tests performed by geotechnical consultants. These properties were
assigned to each of the geologic units of the digital geology map. Also topography data
was acquired from the topography (slope gradient) map produced by Jibson (2000) in
Seattle area. This map was created by applying an algorithm modified in GIS format to
the used Digital Elevation Model (having 1.8-m cell) which compares the elevation of
adjacent cells in the region to calculate the steepest slope (Harp et al. 2006) (Fig3.6 b).
According to the probabilistic definition of location index, the values are basically
between (0, 1) however the results for FS show that there are some values of FS > 1.0
58
(Eq.10) (Harp et al. 2006). In order to use the FS value of each mapping cell as a
―
numerical index‖, it was needed to become normalized between (0, 1) and inverted.
Then each landslide point location was assigned the numerical index of its corresponding
mapping cell. Figure 3.6.c displays the landslide location (S) map in the study area.
3.4.2 Dependence assessment
In previous section three index values of frequency, location and magnitude were
obtained for each landslide record. In other words, there are three sets of data for the
whole study area including ―
T‖, ―
S‖ and ―
M‖ each having 278 members (Eq3.7). Since
the Copula modeling technique is used to model the dependence between random
variables, two critical questions need to be answered here first: 1) can we consider all of
the indices obtained in previous sections for 278 landslide events as ―
uncorrelated
random variable‖? 2) how is the dependence relationship between the three sets of ―
T‖,
―
S‖ and ―
M‖?
To respond to the first question, ―
random variable‖ concept is required to be defined
first. ―
Random variable‖ is a variable whose value is subject to variations due to chance.
In fact, a random variable does not have a pre-known amount and it can take different
values, each with an associated probability (Fristedt et al. 1996). Considering this
definition, three index values of frequency, location and magnitude of every single
59
landslide event in our study area are unknown and random by nature. In other words, it is
not difficult to understand that none of the three values (S, T, M) of a single landslide
event can be predicted when the event is considered separately from the other landslides.
Furthermore analysis of the historical record of landslides indicates that the 357 events
listed in the catalogue occurred at 322 different spots (in digital map), with only 28 sites
affected two or more times. In fact, the same spot was hit on average 1.1 times, indicating
a low rate of recurrence of events at the same place. All this concurs that for the Western
Seattle area landslides can be considered as ―un
correlated random‖ events in time.
To address the second question, visual and quantitative tools were used to check the
dependence between ―
T‖, ―
S‖ and ―
M‖ sets. To visually measure the correlation between
each two sets, a scatter plot of data could be a good graphical check (Genest and Favre
2007). This visual check was performed for 278 landslide events starting in 1912 and
ending in 1996 for all three potential dependence scenarios (Fig3.7). Scatter plots show
that although the correlation between ―
S‖ and ―
M‖ sets is significant enough the
correlation between the other sets is negligible in the study area. The trivial correlation
between ―
S‖ and ―
T‖ indices is also confirmed by the previous study conducted by Coe
(2004) in which concluded that there is no significant correlation between slope angle (as
the location parameter) and landslide frequency in the Seattle area.
60
(a)
(b)
(c)
Figure 3.6 a) Slope map; b) Geologic map; c) Landslide location index (S) map in the study area
61
In addition, to quantify the level of dependence, a standard test was required. In cases
such as ours where the association is non-linear (Fig3.7 a), the relationship can
sometimes be transformed into a linear one by using the ranks of the items rather than
their actual values (Everitt, 2002). For instance, instead of measuring the dependence
between (
and (
(
values the (
are employed where
and
stands for the rank of
stands for the rank of
among (
and
among (
. In this regard, well-known
rank correlation indicators such as Spearman's and Kendall's rank correlation
coefficient were used in this study (Spearman 1904, Kendall 1973). These rank
correlation coefficients were preferred to be applied here because they have been
developed to be more robust than traditional Pearson correlation and more sensitive to
nonlinear relationships (Croxton 1968). Spearman‘s rho (rank correlation coefficient) is
given by equation below.
=1–
Where
(3.11)
is the Spearman‘s rho (rank correlation coefficient) and
is the difference
in the ranks given to the two variable values. Also the Kendall's rank (empirical version)
is defined by
=
(3.12)
62
Where
is the Kendall's Tau (rank correlation coefficient); and
and
number of concordant and discordant pairs respectively. For instance, two pairs of (
and (
are said to be concordant when (
(
are
)
and discordant when
(Genest and Favre 2007).
Both of these rank correlation coefficients were computed for measuring dependence
between each two pairs of sets i.e. (M, S), (M, T) and (S, T) (Table 3.1). The findings
verify the negligible correlation between (M, T) and also (S, T). Further, it shows a
significant and positive correlation between location and magnitude indices (M, S) as
expected. Therefore, magnitude and location indices were considered as the only
significant correlated pair to be used for building the bivariate Copula model in the
following sections.
Table 3.1 Rank correlation coefficients of the pairs (M, S), (M, T) and (S, T)
Correlation
M-S
M-T
S-T
Spearmans’
0.653
-0.012
-0.043
Kendalls’
0.541
-0.008
-0.031
63
3.4.3 Marginal distribution of variables
To create the bivariate Copula model, it was required to obtain the best fitted marginal
probability distributions for both location (S) and magnitude indices (M). Different
marginal probability distributions can be examined when the initial values of the random
variables are not limited to a maximum value. Since the values of location indices (FS) are
limited to the range of (0, 1), data transformation needs to be performed first. This
transformation changes the range of the limited data to an unlimited positive range of [0, ∞)
theoretically. This unlimited positive range was (0, 452) for our transformed location
indices. These transformed location indices are labeled as ―
TS‖ hereafter for simplification.
To be able to find the best marginal distribution for the variables, both visual and
quantitative methods were employed. As a visual approach, histograms of TS and M
datasets were obtained, drawn and several probability density functions were fit to each to
find out which one is the best choice (Fig3.8). Table 3.2 shows the parameters of the
candidate distributions for the variables of both sets. Also as a quantitative method, AIC
criterion was used to choose the best distributions. AIC criterion is a measure of the
relative goodness of fit of a statistical model (Akaike 1974). ―
AIC‖ of each model is
calculated using its likelihood function (the probability density function of each distribution
with the estimated parameters) and the number of parameters in the model (Eq3.13). The
model having smaller AIC is the most appropriate one for the data.
64
(a)
(b)
(c)
Figure 3.7 Scatter plot of landslide hazard component indices: a) Location index versus
magnitude index, b) Frequency index versus magnitude index, c) Frequency index versus location
index
65
(3.13)
Where ―
L‖ is the maximized value of the likelihood function for the estimated model and
―
K‖ is the number of parameters in the model. The estimated AIC criterion for all the
distributions can be seen in Table 3.3. The Kolmogorov–Smirnov test is another goodnessof-fit statistics which can be used to determine if a random variable X could have the
hypothesized, continuous cumulative distribution function; it uses the largest vertical
difference between the theoretical and the empirical cumulative distribution function
(Yevjevich, 1972). As given in Table 3.4, the KS test at the 5% significance levels shows
that the Box–Cox transformed data are normally distributed.
Considering and comparing all the visual and quantitative results, Exponential
distribution (Eq3.14) was selected as the best fit to TS set. Exponential distribution is
defined on the interval of [0, ∞) and is given by
(3.14)
Where λ > 0 is the parameter of the distribution and it was calculated as λ = 2.71 here for
TS data. Moreover, it was recognized that M set is faithfully described by a lognormal
distribution (Eq3.15) with mean 3.753
and standard error 3.344
66
.
(3.15)
is the shape parameter and k is the scale parameter of the distribution.
Where
These two obtained marginal distributions were applied to create the bivariate Copula
function in the following sections.
Table 3.2 Maximum likelihood estimation (MLE) of the examined distribution parameters
Variable
Exponential
Extreme value
Gamma
Generalized Pareto
Rayleigh
k = 0.44
µ = 24.01
TS
a = 0.48
λ = 2.71
σ = 0.87
σ = 91.39
b = 5.60
ϴ
Variable
a = 16.09
=0
Weibull
Extreme value
Gamma
Lognormal
a = 66.23
µ = 69.98
a = 9.36
µ = 4.04
Rayleigh
a = 44.28
M
b =3.15
σ = 26.38
b = 6.38
67
σ = 0.36
3.4.4 Model selection and parameter estimation
In order to model the dependence between the ―
TS‖ and ―
M‖ sets and to have a variety
of choices from different families of Copula, fourteen Copula functions were considered
at first. These Copula functions can be classified into four broad categories: a)
Archimedean family including Frank, Clayton, Ali-Mikhail-Haq and Gumbel–Hougaard
functions (Nelsen, 1986; Genest 1986) and also the BB1–BB3 and BB6–BB7 classes
(Joe, 1997); b) Elliptical Copulas mainly including normal, student, Cauchy and t-Copula
functions (Fang et al., 2002); c) Extreme-value Copulas, including BB5, Hüsler-Reiss
and Galambos families (Galambos, 1975; Hüsler and Reiss 1989); d) Farlie–Gumbel–
Morgenstern family as a miscellaneous class of Copula (Genest and Favre, 2007). These
(a)
(b)
Figure 3.8 Marginal distribution fitting to: a) transformed location index, b) magnitude index
68
mentioned Copula functions were used to model the dependence between ―
TS‖ and ―
M‖
sets mostly using ―
R‖ which is a free software environment for statistical computing (R
Development Core Team 2007); and also MATLAB, that is a numerical programming
language (MathWorks Inc.).
To sieve through the applied Copula models, an informal graphical test was performed
as follows (Genest and Favre 2007). The margins of the 10,000 random pairs ( ,
)
from each of the 14 estimated Copula models were transformed back into the original
units using the already defined marginal distributions of Exponential and Lognormal
Table 3.3 AIC values of the examined probability distributions
Distribution
TS
Distribution
M
Exponential
11.187
Weibull
10.302
Extreme value
16.545
Gamma
14.501
Gamma
10. 647
12.091
Lognormal
10.225
11.633
Rayleigh
11.092
Generalized
Pareto
Rayleigh
Extreme
value
69
13.800
Table 3.4 Kolmogorov–Smirnov (KS) test for the data after Box–Cox transformation
Goodness-of-fit statistics
TS
M
P-value
0.82
0.92
KS statistics
0.07
0.06
models. Then the scatter plots of resulting pairs
=(
(
),
(
))
were created,
along with the actual observations for all families of Copulas. This graphical check made
it possible to generally judge the competency of the models and to select the best
contenders along with the actual observations. Therefore, six best Copula model were
selected (Fig3.9).
The next step was to estimate the parameters of all six selected models. This
estimation was done based on maximum pseudo-likelihood. Pseudo-likelihood is a
function of the data and parameters, which has properties similar to those of a likelihood
function. Pseudo-likelihood is primarily used to estimate the parameters in a statistical
model when the maximization of the likelihood function will be complicated (Arnold and
Strauss 1991). In order to do that, an estimating equation which is related to the logarithm
of the likelihood function is set to zero and maximized (Lindsay 1988). The obtained
parameter values of each Copula model using maximum pseudo-likelihood with 95%
confidence intervals are given in Table 3.5.
70
3.4.5 Goodness-of-fit testing
Now the question is that which of the six models should be used to obtain the joint
distribution function of variables. As the second attempt towards our model selection, the
generalized K-plot of each six Copulas was obtained and compared to each other. The
generalized K-plot indicates whether the quantiles of nonparametrically estimated
is in agreement with parametrically estimated
for each function.
is
basically defined as the equation below (Genest and Rivest, 1993).
(3.16)
Where
is derivative of
with respect to z; and z is the specific value of Z =
Z(x, y) as an intermediate random variable function. If the plot is along with a straight
line (at a 45° angle and passing through the origin) then the generating function is
satisfactory. Figure 3.10 depicts the K-plot constructed for all of the six Copula functions.
Considering the K-plots and the graphical check (Fig3.10 and Fig3.11), it is clear that
Gumbel-Hougaard distribution (Eq3.17) is the best fit to the data and should be used for
the rest of the analysis.
71
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.9 Simulated random sample of size 10,000 from 14 chosen families of Copulas; a) AliMikhail-Haq, b) Frank, c) Galambos, d) Gumbel-Hougard, e) BB2, f) BB3 : upon transformation
72
of the marginal distributions as per the selected models (whose pairs of ranks are indicated by
―whit
e‖ points) along with the actual observations.
Table 3.5 Parameter estimation and confidence interval of the Copulas
Copula
Estimate (ϴ)
95% confidence interval (CI)
Ali-Mikhail-Haq
ϴ
= 1.465
CI = [1.324 1.606]
Frank
ϴ
= 0.384
CI = [0.249 0.518]
Galambos
ϴ
= 3.037
CI = [2.270
3.805]
Gumbel-Hougard
ϴ
= 2.126
CI = [0.670
1.801]
BB2
ϴ
0.174, ϴ = 1.638
CI = [0.255 2.801] [0.260
3.316]
BB3
ϴ
1.424, ϴ = 1.053
CI = [1.512
1.404]
1.903] [0.134
3.4.6 Copula-based conditional probability density function
The presented analysis so far shows that Gumbel-Hougaard distribution is the best fitted
Copula function for modeling the dependence between transformed location indices (ST)
and magnitude indices (M) for the selected landslide database of West Seattle area. Since,
the concept of landslide hazard assessment in this methodology is ―c
onditional‖, the
obtained joint distribution also needs to be compatible to this definition. In other
73
74
Figure 3.10 Goodness-of-fit testing by comparison of nonparametric and parametric K(z) for
Copula model
(3.17)
Figure 3.11 Cumulative Gumbel-Hougard joint probability density function of S and M indices;
dark points represents the 79 validation points in ―1
-CDF‖ form
words, the probability of landslide hazard in this study is expressed as ―
the probability
of occurrence of a landslide having a magnitude larger than a specified value (
)
under the condition of having a specific spatial index in that location‖; and therefore the
obtained Copula function is required to be used in its conditional form. Considering
Eq3.5 and Eq3.17, the conditional Gumbel-Hougaard distribution is given by
75
(3.18)
By substituting the parameters estimated previously for the Gumel–Hougaard Copula
(Table 3.5) into the above equation, the desirable conditional joint distribution is obtained
for this analysis (Eq3.19). Since the transformed location indices (ST) was used to build
the model, the results of this function was transformed back to the initial location indices.
Moreover, this conditional cumulative density function was illustrated as can be seen in
Figure 3.11.
( ln )2.126]0.5)
(3.19)
3.4.7 Probability of landslide frequency (exceedence probability)
In previous sections, the dependence between two landslide hazard components
including ―
magnitude‖ and ―
location‖ was modeled using Copula theory. As explained
earlier, the concept of landslide hazard consists of three elements; hence, the third
component which is ―
frequency‖ needs to be taken into account as well. As the
dependence tests suggested earlier, there is no significant correlation between the indices
76
of frequency component and the ones of the other two hazard elements (Fig3.7 b & c).
Therefore, the landslide frequency was needed to be applied in form of ―
probability
distribution values‖ and not numerical indices. In order to do that, the ―
mean recurrence
interval‖ (µ) of each landslide event was required to be inserted into a temporal
probability model as follows.
One of the most popular probability models that are used to investigate the occurrence
of naturally occurring random point-events in time is Poisson distribution (Guzzetti
2005). Poisson distribution is a discrete model that expresses the probability of a number
of events occurring in a fixed period of time if these events occur with a known average
rate and independently of the time since the last event (Hu 2008). Poisson distribution has
been applied to temporal modeling of different natural phenomena such as floods (e.g.
Yevjevich, 1972; Onoz and Bayazit, 2001); volcanoes (e.g. Klein 1982; Connor and Hill
1995; Nathenson 2001) and landslides (e.g. Coe et al. 2000; Guzzetti 2005). Assuming
landslides as independent random events in time (Crovelli 2000) Poisson distribution is
used for the temporal modeling as below.
(3.20)
Equation 3.20 expresses the probability of experiencing ―
n‖ landslides during time ―
t‖;
where λ is the average rate of occurrence of landslides as 1/µ, which again µ is the
estimated mean recurrence interval between successive landslide events and is calculated
77
as D/N; where N is the total number of events (
and D is the total duration (year)
within all N landslides occurred. The special case of Equation 20, i.e. the probability of
occurrence of one or more landslides during time ―
t‖, is given by
(3.21)
exceedance probability‖ was applied to determine the
Equation 3.21 which is called ―
probability of landslide frequency for our landslide database. The process of such
analysis was adopted from the work done by Coe et al. (2004). The Seattle area was
digitally (in GIS format) overlain with a grid of 25
covering an area of 40,000
25-m cells. Next, a count circle
(4 ha) was digitally placed at the center of each cell, and
the number of landslides occurring within the circle was counted. Then, ―
mean
recurrence interval‖ (µ) was calculated by dividing the total database duration (D = 84
years; 1912-1996) over the number of landslides. Next, the ―
exceedance probability‖ was
obtained for every counting circle using Equation 3.21 and this value was assigned to the
corresponding mapping cell (Fig3.12). An example of such calculation for a mapping cell
is given below.
exceedance probability computation: t = 1, µ = D/N = 84/8 = 10.5 ,
0.095
78
=
=
This means that the annual probability of occurrence of one or more landslides in that
mapping cell is about %10.
As Coe et al (2004) described, the above method of calculating ―
mean recurrence
intervals‖ is more accurate than making average of sample recurrence intervals and there
are some explanations for that: a) map cells having one landslide event cannot be
calculated using mean sample interval method; b) using sample interval would be
inaccurate for some slope failures having occurrence date accuracy within plus or minus
1 year; c) sample interval approach is not able to consider the period between the last
event and the end of the database. Also it should be mentioned that the 4-ha counting
circle was used because it is roughly equivalent in ―
area‖ to the largest landslides that
have occurred in the region and also 25
25-m cell sizes was chosen because it is
approximately equivalent in ―
area‖ to an average-sized city lot (Coe et al. 2004).
79
Figure 3.12 Points indicating the landslide locations in part of the study area and a counting circle
used for exceedance probability calculation in the mapping cell ―A‖
Finally, we now have all the information to quantitatively determine landslide hazard
in the Western Seattle area. Therefore, three components of landslide hazard need to be
combined based on the proposed methodology (Fig3.2). Since the ―
frequency‖ element is
independent of the other two components, its corresponding obtained values will be
multiplied to the Copula function results as below.
(3.22)
3.5 Validation and comparison of the results
Validation is a necessary part of any model in hazard analysis to prove its capability in
forecasting landslides (Chung and Fabbri, 2003; Fell et al. 2008). The best way to check
the validity of a model is to employ landslides independent from the one used to obtain it
(Jaiswal et al. 2010). As mentioned earlier, we chose 79 landslide events from 1997 to
1999 to be used for validation part. Since the validation process was performed for the
already occurred landslides, the incident is certain i.e. the probability of their occurrence
is 100%. Therefore, the closer the predicted values of the landslide hazard model
(Eq3.22) to 100%, the more valid and reliable it would be. Table 3.6 presents location
80
indices (S), magnitude values and the result of landslide hazard computations for all 79
landslide events. Based on the result, the success rate of the Copula-based model in
prediction of landslide hazard is 90% on average.
81
Additionally, a comparison was performed to observe how the dependence modeling
of the hazard components in this study can enhance the forecasting accuracy of the
landslide hazard in practice. This comparison was conducted for the 79 landslide events
and probability of landslide hazard was computed this time under the assumption of
independence (Fig3.1). In fact, the probabilities of ―
location‖, ―
time‖ and ―
magnitude‖ of
landslide events multiplied to each other to calculate their hazard values. Based on the
results, the mean success rate for multiplication-based model is only 63% (Table 3.6).
This comparison shows that Copula-based model predicts landslide hazard much more
precisely than the traditional method does.
3.6 Landslide hazard map
In order to portray the model results conveniently as a landslide hazard map, the study
area was first digitally overlain with a grid of 1.83
1.83-m cells. This size of mapping
cells was selected because it was compatible with the smallest size of the mapping units
in the ―
FS‖ and ―
exceedance probability‖ maps (Fig3.6 b, and Fig3.12). Then, the
―
location‖ index (FS) of every mapping cell and an arbitrary ―
magnitude‖ value were
inserted into Equation 3.22. Lastly, the result was multiplied by the probability of
landslide frequency (exceedance probability) of the mapping unit. Figure 3.13 displays
the resulting map (in GIS format) for landslide magnitude of M ≥ 10,000
within a
time period of 50 years. Each mapping cell expresses the probability of occurrence of one
82
or more landslide larger than 10000
in that location. The arbitrary value of landslide
magnitude and time periods was selected only for illustration purpose.
3.7 Discussion
Some assumptions made in this chapter are reviewed and discussed as follows. Some
points are also made to help the future studies to applying the proposed methodology
properly.
a) To have valid FS values obtained from susceptibility analysis; the main assumption
was that the geological and also topographical features of the study area won‘t change in
an expected validity time-frame of the model. In fact, we assumed that landslides will
occur in the future under the same conditions and due to the same factors that triggered
them in the past. Such an assumption is a recognized hypothesis for all functional
statistically-based susceptibility analyses (Carrara et al. 1995; Hutchinson 1995; Aleotti
and Chowdhury 1999; Yesilnacar and Tropal 2005; Guzzetti et al. 2005; Ozdemir
2009). Therefore, depending on the selected validity time-frame for another study
using this methodology, geology and also topography need to be evaluated first.
This evaluation is required to make sure that these features won‘t change
significantly during the selected time frame.
83
Figure 3.13 Example of landslide hazard map (10m 10m cell size) for 50 years for landslide
magnitudes, M ≥ 10,000
in the study area. The value in each map cell gives the conditional
probability of occurrence of one or more landslide within the specific time in that location.
b) Considering the FS equation (Eq3.10), it was assumed that groundwater flow is
parallel to the ground surface (condition of complete saturation, that is r = 1). Harp et al.
(2006) mentioned in his work that changing the saturation value is not going to make any
difference in the correlation of the FS values with the actual failures from the data set.
Also previous studies show that regional groundwater conditions do not appear to
strongly affect the general distribution of Seattle landslides (Schulz et al. 2008). In fact,
historical landslides are equally distributed inside and outside of the region potentially
influenced by regional groundwater. Therefore, the accuracy of the calculated
susceptibility values (FS) by Harp et al. (2006) was not reduced by that assumption.
84
Table 3.6 Landslide hazard probability values obtained from Copula-based and multiplication-based
models for 79 failure events
Landslide
Year
S
1
1997
0.93
2
1997
3
M(
1 - Copula Conditional CDF
Multiplication
5.39
0.965
0.417
0.46
6.18
0.944
0.579
1997
0.96
4.32
0.954
0.668
4
1997
0.83
1.35
0.890
0.625
5
1997
0.21
2.51
0.912
0.664
6
1997
0.25
3.16
0.951
0.613
7
1997
0.89
4.18
0.923
0.652
8
1997
0.54
1.95
0.972
0.710
9
1997
0.36
5.22
0.890
0.630
10
1997
0.26
2.37
0.949
0.571
11
1997
0.36
7.11
0.845
0.284
12
1997
0.35
2.42
0.761
0.593
13
1997
0.08
3.07
0.804
0.526
14
1997
0.92
1.53
0.954
0.744
15
1997
0.95
2.42
0.791
0.586
16
1997
0.82
3.21
0.981
0.614
17
1997
0.27
1.58
0.989
0.680
18
1997
0.32
2.37
0.868
0.553
number
85
19
1997
0.89
6.32
0.802
0.664
20
1997
0.15
2.32
0.970
0.546
21
1997
0.82
3.08
0.803
0.529
22
1997
0.98
4.31
0.834
0.605
23
1997
0.12
2.53
0.941
0.513
24
1997
0.40
2.44
0.869
0.694
25
1997
0.23
1.49
0.980
0.631
26
1997
0.69
2.24
0.908
0.762
27
1997
0.03
2.41
0.899
0.474
28
1997
0.52
6.60
0.979
0.538
29
1997
0.92
2.46
0.812
0.537
30
1997
0.02
4.88
0.891
0.524
31
1997
0.53
3.16
0.926
0.713
32
1997
0.21
4.97
0.842
0.761
33
1997
0.56
2.18
0.981
0.521
34
1997
0.55
5.02
0.870
0.423
35
1997
0.82
2.55
0.975
0.780
36
1997
0.42
5.34
0.779
0.609
37
1997
0.10
4.78
0.670
0.383
38
1997
0.76
2.32
0.933
0.316
39
1997
0.54
6.64
0.694
0.460
40
1997
0.87
2.18
0.819
0.511
86
41
1997
0.26
3.16
0.838
0.527
42
1997
0.86
3.21
0.925
0.680
43
1997
0.97
5.06
0.891
0.530
44
1997
0.80
3.94
0.728
0.512
45
1997
0.27
5.34
0.951
0.673
46
1997
0.25
6.61
0.983
0.665
47
1997
0.69
3.91
0.882
0.554
48
1997
0.04
0.79
0.814
0.699
49
1997
22.8
2.00
0.922
0.516
50
1997
21.2
4.83
0.839
0.570
51
1997
11.3
6.61
0.840
0.518
52
1997
31.9
0.85
0.905
0.601
53
1997
27.5
1.24
0.881
0.562
54
1997
31.9
2.70
0.903
0.710
55
1997
30.0
2.74
0.878
0.685
56
1997
21.1
4.88
0.897
0.542
57
1997
25.0
5.71
0.876
0.513
58
1997
19.2
4.09
0.794
0.657
59
1997
26.9
4.83
0.990
0.747
60
1997
39.9
1.67
0.917
0.716
61
1997
24.8
2.33
0.767
0.235
62
1997
5.4
3.95
0.980
0.434
87
63
1997
7.6
2.28
0.924
0.520
64
1997
17.3
3.21
0.952
0.548
65
1997
9.0
6.60
0.879
0.756
66
1997
35.7
3.62
0.946
0.634
67
1997
16.2
2.34
0.971
0.601
68
1997
11.7
3.26
0.893
0.548
69
1997
33.0
2.55
0.834
0.622
70
1997
26.1
5.90
0.997
0.539
71
1997
22.6
3.95
0.968
0.623
72
1997
17.1
5.71
0.936
0.768
73
1998
10.8
3.99
0.947
0.526
74
1998
11.8
2.28
0.968
0.432
75
1998
15.6
3.11
0.946
0.533
76
1997
14.6
5.90
0.941
0.719
77
1999
42.6
1.77
0.891
0.980
78
1999
16.3
4.83
0.974
0.713
79
1999
17.6
5.03
0.894
0.529
Furthermore, in the FS equation (Eq3.10) average thickness of 2.4 m was used for ―
t” to
reflect the average thickness of landslides. This depth was a good estimate because most
landslides that occur in Western Seattle area are shallow; and the field observations and
88
measurements showed that the average thickness of shallow failures is 2.4 m (Harp et al.
2006).
c) In regard to FS map (Fig3.6 c), the stability of each mapping unit was assumed to be
independent from the surrounding mapping cells. The validity of such an assumption was
confirmed by testing the low rate of recurrence of events in previous sections.
d) Considering the temporal distribution of landslide events in the selected study area
(Fig 3.6), we assumed that the ―
mean recurrence interval‖ of landslide events will remain
the same in the expected validity time of the model in the future. The previous studies in
Seattle area indicate that most of the landslides in the region are rainfall-triggered (Baum
et al. 2005). It means that if the rate of occurrence of the meteorological events or the
intensity-duration of the triggering rainfalls does not change significantly over time, the
recurrence interval of landslides will almost remain the same. Therefore, the possible
change in the pattern of precipitations needs to be considered before applying this
methodology in any other study.
3.8 Summary and conclusions
The simplifying assumption of independence among the hazard elements which is
commonly used in the literature was addressed and discussed elaborately in this chapter.
89
A new quantitative methodology was presented to assess the landslide hazard in regional
scale probabilistically. This approach considers the possible dependence between the
components and attempts to forecast the future landslide using a reliable statistical tool
named Copula modeling technique. We tested the model in western part of the Seattle,
WA area. 357 slope-failure events and their corresponding slope gradient and geology
databases were considered to establish and test the model. The mutual correlation
between landslide hazard elements were assessed by standard dependence tests; and
Copula modeling technique was applied for building the probabilistic hazard function.
Based on the results, the mean success rate of Copula model in prediction of landslide
hazard is 90% on average, however this mean success rate for traditional multiplicationbased approach is only 63%. The results verify the superiority of the presented
methodology over the traditional one. Furthermore, an example of landslide hazard map
of the study region was generated using the proposed approach.
90
CHAPTER IV
SEISMICALLY-TRIGGERED LANDSLIDE HAZARD ASSESSMENT: A
PROBABILISTIC PREDICTIVE MODEL
4.1 Overview
Seismically-triggered landslides are one of the most important impacts of moderate
and large earthquakes. These landslides usually account for a significant portion of total
earthquake loss. Largest earthquakes have the capability to cause thousands of landslides
throughout areas of more than 100,000
(Keefer 1984). For instance, on May 2008, a
magnitude (Mw) of 7.9 Sichuan earthquake in China had triggered more than 11,000
landslides and these events have threatened 805,000 persons and damaged their
properties (Gorum 2008). Other examples include Chi-Chi and Northridge earthquakes
which had triggered widespread shallow and deep-seated slope failures and resulted in
extensive damage and human loss (Wang et al., 2008). Also failure of a solid-waste
landfill or an earth embankment can trigger significant human and financial losses and
severe environmental impact (Bray and Travasarou 2007). Therefore, assessment of the
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potential for earthquake-induced landslides is a very vital and important task associated
with earthquake hazard management initiatives.
Among different methods for assessment of earthquake-triggered landslides,
probabilistic methods offer unique capabilities for the engineers to quantify the
uncertainties in the assessment of the performance of susceptible slopes during the
seismic loading. Probabilistic seismic landslide displacement analysis takes into
consideration the uncertainties of seismic and preparatory parameters (related to
topography, geology, and hydrology). The probabilistic approach has been applied in
different studies in recent years (e.g., Del Gaudio and et al. 2003; Stewart et al. 2003;
Bray and Travasarou 2007; Rathje and Saygili 2008 and 2009). Seismic landslide hazard
maps, which are the main result of such studies, have been developed throughout
seismically active areas to identify zones where the hazard of earthquake-induced
landslides is high. Improvement in creating more precise seismic hazard maps help
making decisions regarding infrastructure development, hazard planning in the
susceptible areas, and providing the necessary tool for reducing human and financial loss.
The main objective of this paper is to propose a probabilistic method for use in
construction-planning in earthquake susceptible areas. This presented approach is based
on casual relationships between landslides displacement index (
), preparatory
variables and seismic parameters. In this method, the Newmark displacement index, the
earthquake intensity, and the associated spatial factors are integrated into a multivariate
Copula-based probabilistic function. This model is capable of predicting the sliding
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displacement index (
) that exceeds a threshold value for a specific hazard level in a
regional scale. The predicted results will be presented as an earthquake-induced landslide
hazard map in a regional-scale.
4.2 Literature Review
In the past, considerable effort has been devoted to improve earthquake-induced
landslide hazard assessment methods using a variety of approaches. Newmark‘s model
for modeling earthquake-induced landslide displacements provides the basis to forecast
the displacements resulting from seismic loading in a susceptible area. This method
models a landslide as a rigid block on an inclined plane and calculates cumulative
displacement of the block resulting from the application of an acceleration time record
(Newmark 1965). The main assumption is that the sliding boundary has a rigid plastic
behavior and that displacement can occur when the sum of driving forces exceeds the
resistance (Wang, 2008). In order to conduct the spatial analysis, an infinite slope model
is used to analyze regional static slope instability. This model assumes that the slope
failure is the result of translational sliding along a single plane of infinite length which is
parallel to the ground surface. The first step of the Newmark sliding block model is to
obtain the static factor of safety (FS) using the infinite slope model as shown in Figure.1:
(4.1)
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Where c' is the effective cohesion (kPa); ϕ' is the effective angle of friction; α is the slope
angle; m is the percentage of the sliding depth below the water table; z is the depth of
sliding block ( ); γ is the unit weight of soil (KN/
); and
is the unit weight of water.
Also, the following relationship was presented by Newmark (1965) to calculate the
critical acceleration in terms of gravity in a case of planar sliding:
(4.2)
Where g is the gravity acceleration,
is critical (yielding) acceleration in g unit, and α'
is the thrust angle of the landslide block which can be approximated by the slope angle
(in case of shallow landslides, one can assume α'=α). Although some other expressions
for
have been obtained with different assumptions for ground acceleration, their
computed
are only slightly different (Saygili and Rathje 2009).
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Figure 4.1 Infinite slope representation showing the sliding block and the parameters used to
define the critical acceleration (
)
To perform the conventional Newmark analysis, selection of a suitable earthquake
record and determination of the yielding (critical) acceleration of the selected slopes are
required. The conventional Newmark analysis calculates the cumulative displacement of
the friction block by double-integrating those parts of the horizontal earthquake timehistory that exceed the critical acceleration (Wilson and Keefer, 1983). The Newmark
calculation process can be represented as:
(4.3)
Where
is the Newmark displacement,
is the critical acceleration and a(t) is the
ground acceleration time history.
Such a conventional analysis involves computational complexity and difficulties of
selecting an appropriate earthquake time-history (Jibson et al. 1998). To estimate the
Newmark displacement (
) and avoiding such limitations, different approaches have
been presented previously as follows. The flow chart in Figure.2 shows the common
framework for integrating the critical acceleration and seismic parameters to estimate the
Newmark displacements which finally leads to developing an earthquake-induced
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landslide hazard map. Common approaches that are used in this framework are described
as follows.
Time-history technique presented by McCrink (2001) is based on a single ground
motion acceleration-time history. In this method, an acceleration-time history is selected
and used to compute displacement index (
) for several values of yielding acceleration.
The selection of the single acceleration-time history is based on the probabilistic seismic
hazard analysis (PSHA). Although some of the existing ground motion uncertainty was
considered by the PSHA-derived ground motion information, it has been shown that this
method does not consider the fact that different time-histories can produce significantly
different levels of displacement index (
) (Jibson 2007; Bray and Travasarou 2007;
Saygili and Rathje 2008).
The other approach for seismic-induced hazard assessment is to deterministically use
an empirical relationship that relates
to
as well as other ground motion parameters,
such as peak ground motion acceleration (PGA), Arias Intensity ( ) and peak ground
velocity (PGV). These deterministic models could be considered as pseudo-probabilistic
when the ground motion level is derived from PSHA (Rathje and Saygili 2008). Although
these models have been applied in many studies (e.g. Wilson and Keefer 1985; Jibson et
al. 1998; Miles and Hoxx 1999; Bray and Travasarou 2007; Rathje and Saygili 2009),
they generally do not account for the ground motion uncertainty. Also, as Rathje (2009)
described, the standard deviations in such empirical relationships are too large such that
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the range of predicted
for a specific yielding acceleration and ground motion level is
usually greater than one order of magnitude.
In contrast to the above deterministic or pseudo-probabilistic methods, expressing
slope instability within a fully probabilistic framework is theoretically capable of taking
all the uncertainty of the ground motions into account. Lin and Whitman (1983) analyzed
the probability of failure of sliding blocks using a rigid block modeling technique. They
considered the strong ground motion as a Gaussian stationary process and characterized
ground motions by several parameters including PGA, central frequency, and root mean
square acceleration. As a result, to obtain the annual probability of exceedance from a
specified displacement threshold, the seismic hazard for peak ground acceleration needs
to be evaluated. Also, Yegian et al. (1991) studied the probability of failure of sliding
blocks using the Newmark rigid block concept. In this study, seismic displacements were
normalized by the PGA, predominant period, and number of equivalent cycles of loading
of the ground motions acquired from 86 earthquake records. They developed a
relationship between the normalized seismic displacements and the ratio of yield
acceleration over PGA. Finally, since the seismic displacement is conditioned on more
than one ground motion parameter, computing the joint hazard of these parameters was
required. Another method proposed by Bray et al. (1998) for solid-waste landfills uses
seismic slope stability analysis to provide median and standard deviation estimates of
maximum horizontal equivalent acceleration and the normalized seismic displacement.
As a more recent probabilistic study, Bray and Travasarou (2007) presented a model that
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can be applied within both fully probabilistic and deterministic frameworks to evaluate
seismic-induced slope instability. Earthquake-triggered displacements in this method
were computed using 688 seismic records; further yield acceleration, the initial
fundamental period of the sliding mass, the earthquake magnitude and the spectral
acceleration were used as the model parameters. In addition, Rathje and Saygili (2008,
2009) recently presented two approaches named the scalar and vector probabilistic
models using over 2,000 acceleration time histories. The scalar model uses PGA and
magnitude to predict sliding displacements, while the vector approach implements PGA
and PGV to estimate seismic-induced displacements. Compared to the vector approach
that requires vector PSHA analysis or an estimate of the VPSHA distribution, the scalar
approach can be implemented using the results from probabilistic seismic hazard
assessment (PSHA).
Although these probabilistic models were developed using real datasets and advanced
techniques, as Kaynia et al (2010) described, there is still more than a magnitude
difference in their final prediction results. Also, most of the recent presented probabilistic
methods have applied the predictor variables such as peak ground acceleration (PGA) and
peak ground velocity (PGV), which only represent the instantaneous maximum values
regardless of the earthquake duration (Luco 2002; Rathje and Saygili 2008, 2009). The
other main drawback is that, none of these probabilistic models has been validated using
comprehensive field observations of seismic-triggered landslide and associated
earthquake records (Kaynia et al. 2010). For instance, Rathje and Saygili (2008, 2009)
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stated that their proposed models required additional future wok to assess accuracy of
their method. Moreover, the recent availability of new earthquake records and also more
realistic probabilistic techniques require the researchers to updated and advance the
existing probabilistic methods. This paper attempts to address the above mentioned
shortcomings by presenting a new probabilistic model using a rigorous statistical
technique. It is believed that the proposed methodology will present an accurate and
advanced method which can be verified by comprehensive field observations.
Figure 4.2 Flow chart showing the common required steps for developing earthquake-induced
landslide hazard map
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4.3 Methodology
The first step toward modeling the seismic hazard in this study is to apply a suitable
empirical relationship between Newmark displacement and selected predictor variables
(e.g., peak ground acceleration, Arias Intensity, peak ground velocity). Since Arias
intensity ( ) is considered the most relevant ground motion parameter for studies of
earthquake-induced landslides (Wang et al. 2011), the presented model in this paper
requires
as one of the predictor variables. Arias intensity (Arias, 1970) is defined as
the sum of all the squared acceleration values from a strong motion record; and it is a
measure of the energy dissipated at a site by shaking as below:
(4.4)
Where a(t) is the acceleration value from the record,
is the acceleration due to gravity and
is the duration of the shaking, g
is in m/s. The use of Arias intensity ( ) rather
than PGA was first suggested by Jibson (1993) to characterize the strong shaking. This
measure incorporates both amplitude and duration information, making it more capable
of representing the shaking intensity of ground motion than other parameters, such as
PGA and PGV, which only consider the instantaneous maximum in the shaking intensity.
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As can be seen in Equation.2, the use of yielding acceleration is affected by the
geological, hydrological and geometrical features of a seismic-susceptible location.
Hence, the other selected variable for building the hazard model in this paper is yielding
acceleration (
. Among a variety of regression equations previously suggested by
different authors for various sets of data (Ambraseys and Menu 1988; Yegian et al. 1991;
Jibson 1993; Jibson et al. 1998, 2000; Jibson 2007; Bray and Travasaro, 2007; Saygili
and Rathje 2008), the regression model suggested by Jibson et al (2007) was selected
(Eq.5). This model includes both Arias intensity and yielding acceleration and it was
regressed using 875 Newmark displacements extracted from a large data set. This model
is fit well enough (
71%) and has a very high level of statistical significance. As
Jibson et al (2007) described, this model has an improvement over the previous ones and
it is recommended for regional scale seismic landslide hazard mapping.
(4.5)
Where Dn is Newmark displacement in centimeters,
is Arias intensity in
and ac is
the critical acceleration in g’s.
Applying the regression equation using the yielding accelerations (
motion records ( ) of the study area, we can compute a
and ground-
value for each mapping unit.
The next step would be to calculate the probability of exceedance the computed
given
a seismic displacement threshold. This exceedance probability is called the seismic
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hazard and is conditioned on two previously mentioned predictor variables of
and
(Eq.6).
+
>
(4.6)
Assumption:
Where is mapping unit subscript,
represents the seismic hazard;
and
represent
the yield acceleration and Arias intensity, respectively in each selected mapping unit.
Also
represent the joint probability of
and
Newmark displacement exceedance and the predictor variables in a mapping unit. In the
equation above, it is assumed that yield acceleration and Arias Intensity represent two
random exhaustive events in a given location, which means that they cover the spectrum
of all possible events. In fact, we presume that the union of occurrence of critical
acceleration and Arias intensity creates the possibility of occurrence of seismic
displacement (
). Such an assumption has been undertaken in many different studies
(e.g., Miles and Keefer 2000, 2001; Del Gaudio et al. 2003; Murphy and Mankelow
2004; Haneberg 2006). The flow chart displayed in Figure.3 shows the presented
probabilistic methodology in this paper for developing earthquake-triggered landslide
hazard map. This flow chart shows how that Factor of safety (FS) obtained from
geospatial database results in critical acceleration, which in turn is implemented in an
empirical relationship to obtain the Newmark displacement values. These Newmark
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displacement values are applied in a probabilistic analysis using Copula method to
estimate the seismic-induced hazard of the landslides for every mapping unit.
To obtain the conditional probabilities of
and
in
our presented model, a multivariate joint probability function is required. In the past,
some multivariate joint functions were introduced for different applications; however
most of them are restricted by the following main assumptions: a) either the used random
variables have the same type of the marginal probability distribution, b) or all the applied
variables follow the normal distribution.
In practice, the random variables do not follow the same types of probability
distributions and they are not necessarily normal, Thus, the mentioned assumptions are
not always true (Zhang and Singh 2006). The Copula joint probability functions, which
do not require the above mentioned assumptions, offer more flexibility compared to the
other joint probability functions. Therefore we will use Copula joint probability function
in this paper.
In general, Copula is a type of distribution function used to describe
the dependence between random variables. In other words, Copulas are functions that
connect multivariate probability distributions to their one-dimensional marginal
probability distributions (Joe 1997). The Copula concept in dependence modeling goes
back to a representation theorem of Sklar (1959) as follows.
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Figure 4.3 Flow chart showing the presented methodology in this study for developing
earthquake-induced landslide hazard map
Let‘s assume that we have n variables (or populations) as
observations are drawn for each variable as (
and N
; and also
are the marginal cumulative distribution functions (CDFs) of the variables
Now in order to determine the non-normal multivariate distribution of
the variables, denoted as
Copula, C, could be used as an
associated dependence function. This function returns the joint probability of variables as
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a function of marginal probabilities of their observations regardless of their dependence
structure as below.
(4.7)
Where
if
( ) for i = 1, 2,.. n is the marginal distribution and C:
= Copula
is continuous (Grimaldi and Serinaldi 2005). Different families of Copulas have
been developed in last decades and the comprehensive descriptions of their properties
have been compiled by some authors (e.g., Genest and MacKay 1986; Joe 1997; Nelsen
1999). The most popular Copula family is Archimedean Copulas (Yan 2006).
4.4 Application of the Proposed Model
4.4.1 Study area and database
To demonstrate the application of the presented methodology, the inventory of 1994
Northridge earthquake occurred in Northern Los Angeles, California was used as a
comprehensive ground motion database in this study. Several studies (e.g. Jibson et al.
1998, 2000) applied this earthquake database to perform a detailed regional assessment of
seismic-triggered landslides. In fact the features that distinguish this dataset from the
other available databases include a) its extensive inventory of occurred slope failures; b)
about 200 ground motion records throughout the region; c) detailed geologic map and
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available engineering properties throughout the area (1:24,000-scale), d) and available
high-resolution digital elevation models (DEM) of the topography in the region. All of
the mentioned datasets have been digitized at 10-m grid spacing in the ArcGIS platform
in the selected quadrangle centered at 34.241° N, -118.592°W in Northridge area (Fig.4).
This quadrangle was selected based on greatest landslide concentration and the
availability of a significant amount of shear strength data in that region (Jibson et al.
1998). By integrating these data sets in the presented model based on Newmark‘s sliding
block concept and Copula modeling technique, the seismic-triggered landslide
displacement is estimated in each grid-cell throughout the study area as follows.
Figure 4.4 Location of the study area, limit of greatest landslide concentration and Northridge
earthquake epicenter (Celebi et al. 1994)
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To calculate yielding acceleration values for each mapping unit, it was required to use
Equation.1 to obtain the factor of safety (FS) first. Factor of safety (FS) values were
calculated by inserting available friction angle, cohesion and slope angle data into
Equation.1. Topographic and shear strength data for the quadrangle were mainly supplied
by Jibson et al. (1998) as follows.
Geology: A digital geologic map is essential to assign material properties throughout
the area (Fig.5a, b, c). For this purpose, the 1:24,000-scale digital geologic map of the
region originally created by Yerkes and Campbell (1993, 1995) was used. By using this
map, we assigned representative values of the frictional and cohesive components of
shear strength values to each geologic unit. As Jibson et al. (1998) indicated in his work:
these shear-strength parameters were initially selected based on compilation of numerous
local direct-shear test results and the judgment of experienced engineers in the region.
Digital elevation model: Using high-resolution scanning of the original USGS contour
plates of the 1:24,000-scale map, a 10-m digital elevation model (DEM) was created. As
Jibson et al. (1998) indicated that the10-m scanning resolution was used because too
many topographic irregularities are not considered in the more commonly used 30-m
elevation models.
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Figure 4.5 a) Geologic map of the selected study area; (b) cohesion and (c) friction angle as the
components of shear strength assigned to the geologic units (Yerkes and Campbell 1993, 1995);
(d) Shaded-relief digital elevation model (DEM) of the selected area
Topography: The slope map (Fig.5d) was created by using a simple algorithm to the
DEM which compares the elevations of adjacent mapping units and calculates the
maximum slope. This slope map might underestimate some of the slopes (steeper than
60°) due to not so good representation of the slopes on the original contour map. It should
be noted that because almost all of the landslides in the Northridge earthquake occurred
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in dry conditions, no pore water pressure was considered (m = 0) in Equation.1. Also as
Jibson et al. (1998) suggested a typical unit weight of 15.7 kN/m3 and sliding depth of 2.4
m were used for simplicity to be representative of Northridge shallow landslides.
Regarding the Arias intensity values, the method applied by Jibson et al. (1998) was
adopted. In this method, a ground-shaking grid from the Northridge earthquake was
produced. In fact, the average Arias intensity was plotted in a regularly-spaced grid for
each of 189 earthquake records using its two existing horizontal components. Then, the
kriging algorithm (Krige 1951) was applied to interpolate the intensity values across the
grid (Fig.6). Finally, Newmark displacements (
values from the Northridge
earthquake were computed for each mapping unit (10-m cell) of the selected quadrangle
by using Equation.2 , together with the corresponding grid values of critical acceleration
and Arias intensity.
The obtained Newmark displacements range from 0 to 2414 cm that represent seismic
landslide susceptibility values in the selected region. These computed displacements are
used in our probabilistic analysis as follows.
4.4.2 Development of seismic hazard model
Dependence assessment
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Figure 4.6 The contour map of the Arias intensity ( ) generated by the 1994 Northridge
earthquake in the selected quadrangle. Each displayed Intensity value is the average of the two
horizontal components
As the first step toward establishment of the probabilistic model (Eq.6), the
dependence between the involving variables has to be assessed. To check the linear
association between the pairs of
and
, Pearson‘s product-moment
correlation coefficient was computed as below:
(4.8)
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Where N is the sample size;
are sample variance of
and
are sample means of the variables; and
and
and . As given in Table.1, the findings verify the significant
and positive correlation between
and negative correlation between
.
Such a significant correlation between seismic parameters and Newmark displacement
was already confirmed by earlier studies in different landslide-earthquake inventories,
including 1994 Northridge, 1995 Kobe, 1999 Duzce and 1999 Kocaeli (Kao and Chen
2000; Rathje 2001; Jibson et al 2007; Hsieh and Lee 2011). It is not difficult to
understand that a larger
, a parameter representing the surrogate of strength properties
of a slope, means that the slope has a higher ability to withstand earthquake, resulting in a
smaller Newmark displacement. On the other hand,
represents the ground motion
intensity of an earthquake, and the same slope will result in a larger Newmark
displacement when subjected to a stronger ground motion intensity. These findings were
used for building the hazard prediction model and also for creating the regional landslide
hazard map as follows.
Table 4.1 Rank correlation coefficients of the pairs
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and
Spatial auto correlation test
Although, commonly used statistical methods including Copula technique often
assume that the measured outcomes are independent of each other, it might be the case
that some results exhibit spatial autocorrelation (Knegt et al. 2010). When spatial
autocorrelation exists, the relative result of two points is related to their distance; and it
violates the main assumption of using uncorrelated random variables. Therefore, it is
important to check for autocorrelation before performing the Copula modeling technique.
In order to do that, the well-known Moran‘s I autocorrelation index (Moran 1950) was
used to calculate the autocorrelation of the mapping units with their adjacent units using
the ArcMap software. This index is expressed as below:
(4.9)
Where N is the total number of units;
and
is the spatial adjacency between units i and j;
are the index values of the units i and j. It should be noted that the Moran index
values range from -1 to +1 indicating prefect negative and positive correlations
respectively. A zero value indicates a random spatial pattern for the units.
To calculate Moran's I, a matrix of inverse distance weights was created. In the
matrix, entries for pairs of units that were close together were higher than for pairs of
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units that were far apart. For simplicity, the latitude and longitude values were
considered on a plane rather than on a sphere in this chapter. Then the significance test
was performed for Z value (Eq.10).
(4.10)
Where
and
are the Expected Value and Standard Deviation in Moran's I
respectively. Using the 5% significance level and one-tailed test, calculated
were compared with
values
(the corresponding critical value to 5% significance level is
1.645). As given in Table.2, all the obtained Z values are smaller than the critical value,
indicating that spatial autocorrelation is not significant for our data. Therefore, the result
of spatial autocorrelation test suggests proceeding with the standard Copula modeling
technique.
Table 4.2 Calculated Z-values for spatial autocorrelation significance test
All the values are compared with
= 1.645.
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Marginal distribution of variables
To create the bivariate Copula model, it requires to obtain the best fitted marginal
probability distributions for yield acceleration (
displacement (
), Arias intensity ( ) and Newmark
). Therefore, Akaike information criterion (AIC), developed by Akaike
(1974), was used for identifying the appropriate probability distribution as given below.
(4.11)
Where ―
MSE” represents the mean square error of a distribution against its empirical
non-exceedance probability function. The empirical probability function used in this
study was Gringorten position-plotting formula (Gringorten 1963; Cunnane 1978); and
accordingly the best model is the one which has the minimum AIC (Table.3).
Considering the AIC values summarized in Table.3, Log-normal distribution is best fit to
( ), ( ) sets, while Gamma is the best model for (
) set. The parameters of these
distributions were estimated by the maximum likelihood method.
In addition, the Kolmogorov–Smirnov (KS) test is another goodness-of-fit statistics
which uses the maximum absolute difference between the empirical distribution and the
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hypothesized probability distribution (Yevjevich 1972). This test was used to determine
whether the data after applying the Box–Cox transformation (Box and Cox 1964) could
be considered as normally distributed. As given in Table.4, the KS test at the 5%
significance levels expresses that the Box–Cox transformed data are normally distributed.
Model selection and parameter estimation
In order to model the dependence between the pairs of
and
and to
have a variety of choices from different families of Copula, fourteen Copula functions
were considered from four broad categories: a) Archimedean family including Frank,
Clayton, Ali-Mikhail-Haq and Gumbel–Hougaard functions (Nelsen 1986; Genest 1986)
and also the BB1–BB3 and BB6–BB7 classes (Joe 1997); b) Elliptical Copulas mainly
including normal, student, Cauchy and t-Copula functions (Fang et al. 2002); c) Extremevalue Copulas, including BB5, Hüsler-Reiss and Galambos families (Galambos 1975;
Hüsler and Reiss 1989); d) Farlie–Gumbel-Morgenstern family as a miscellaneous class
of Copula (Genest and Favre 2007).
To sieve through the available Copula models, an informal graphical test (Genest and
Favre 2007; Reddy and Ganguli 2012) was performed as follows. The margins of the
10,000 random pairs ( ,
) from each of the fourteen estimated Copula models were
transformed back into the original units using the already defined marginal distributions
of Log-normal and Gamma functions. Then the scatter plots of resulting pairs
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=
(
( ),
( )) were created, along with the actual observations for all families of
Copulas. This graphical check makes it possible to generally judge the competency of the
functions and to select the best contenders along with the actual observations. Therefore,
three best Copula model were selected for each pair (Table 5). The next step was to
estimate the parameters of the three selected models for each pair. This estimation was
carried out over a random sample of bivariate observations
for the selected Copula models.
Table 4.3 Performance of marginal distributions for random variables and selected density functions
The bold values are the statistical models with the best estimate for the variables.
Table 4.4 Kolmogorov–Smirnov (KS) test for the data after Box–Cox transformation
116
P-value is the probability of the maximum KS statistic being greater than or equal to the critical value (0.05).
This estimation is based on the procedure described by Genest and Rivest (1993) as
follows:
a) Obtain the Kendalls‘τ from observations using equation below,
(4.12)
Where N = number of observations; sign = 1 if
< 0, i, j = 1, 2 … N; and
b) Determine the Copula parameter
> 0, sign = -1 if
= estimate of τ from the observations;
from the above value of τ for each Copula
function. As can be seen in Table.5, there are closed-form relations between
for the used Copula families in this paper. In each relation,
and
and τ
denote a specific
value of the marginal cumulative distribution functions (CDFs) of the variables.
The obtained parameter values of each Copula model ( ) and values of Kendall‘s are
given in Table.5. These values are used for further examination, narrowing down the
results and building the final hazard model as follows.
Goodness-of-fit testing
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Now we would like to determine which of the three models should be used for each
pair of
to obtain their final joint distribution function. As the
and
second step towards our model selection, AIC criterion (Eq.8) was applied. The values of
AIC were computed (Table 6) and the results show that Frank and Cook-Johnson Copula
models
have
the
smallest
AIC
value
for
the
correlated
pair
of
respectively.
and
Moreover, the generalized K-plot of each four Copulas was obtained and compared to
each other. The generalized K-plot indicates whether the quantiles of non-parametrically
estimated
is in agreement with parametrically estimated
(Genest and Rivest 1993).
for each function
is basically defined as the equation below:
(4.13)
Where
is derivative of
with respect to z; and z is the specific value of Z =
Z(x, y) as an intermediate random variable function. If the plot is along with a straight
line (at a 45° angle and passing through the origin) then the generating function is
satisfactory. Figure.7 depicts the K-plot constructed for all of the six Copula functions.
The result of K-plots is in agreement with the obtained AIC values that Frank and CookJohnson Copula models are the best-fitted functions. It should be noted that the Copula
model in the form of cumulative distribution functions produces the probability of nonexceeding values (
. However, the desired joint function should be able to predict
the exceedance Newmark displacement as it was discussed earlier (Eq.6). Hence, the (1118
CDF) forms of the selected Copulas were used to build the seismic-triggered landslide
hazard model as below:
Table 4.5 Parameter estimation and Kendall‘s of the Copulas
The reasom for ‗NA‘ is that Ali–Mikail–Haq Copula can only be used to simulate the correlated random variables
within Kendall‘s
belongs to roughly [-0.2, 0.3]
Table 4.6 The AIC values of different Copulas functions
The reason for ‗NA‘ is that Ali–Mikail–Haq Copula can only be used to simulate the correlated random variables
within Kendall‘s
belongs to roughly [-0.2, 0.3].
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Figure 4.7 Goodness-of-fit testing by comparison of nonparametric and parametric K(z) for
Copula models
120
(4.14)
Where
and
are the selected marginal distributions of critical acceleration, Newmark
displacement and Arias intensity respectively, as described earlier in Table.3.
4.5 Seismic landslide hazard map
Using the obtained predicting model, the seismic hazard map of the study area was
created in 10-m grid spacing in the ArcGIS platform (Fig 4.8). This size of mapping cells
was selected because it was compatible with the size of the mapping units in the applied
geology and topography maps. The obtained map represents the hazard level of 10%
) for the study area. The obtained
probability of exceedance in 50 years (
map indicates that the most significant hazard (having the largest displacements) is
associated with the steeper slopes in the northern and southeast areas of the quadrangle.
4.6 Validation
Newmark displacements do not necessarily represent the measurable slope movements
in the field; rather, these estimated displacements provide an index related to the field
121
performance. In order to validate the Newmark-based predictive models, calculated
displacements need to be correlated with a quantitative indicator in the field; such as
landslide displacement area (Jibson et al. 1998). Therefore, in this chapter the previous
landslides occurred in the study area were used (USGS, open report 1995) to examine
their relevancy to the obtained Newmark displacements.
Figure 4.8 Seismic landslide hazard maps (10m×10m cell size) indicating the displacements
and b)
levels in mapping units for a)
in the study area.
The areal percentage of seismic-triggered slope movements in every mapping cell was
digitally measured and plotted versus its corresponding Newmark displacement value
using a Q-Q plot (Fig.9). Generally speaking, the points in the Q-Q plot will
approximately lie on a straight line when the datasets are linearly related (Wilk and
122
Gnanadesikan 1968). As can be seen in Figure.9, the extremely good correlation (
shows how well the model results are related to the real landslide inventory data
in the field. This consistency proves the capability of the predictive model in seismicinduced landslide hazard assessment.
4.7 Summary and conclusion
Seismic landslide hazard maps are important in development and zoning decisions,
and provide a very useful tool for detailed studies of seismic-triggered landslides in
susceptible areas. In the last decade, considerable efforts have been devoted to improve
these hazard mapping techniques using a variety of different approaches. The current
procedures used to develop seismic landslide hazard maps have one major limitation in
that they mostly ignore the uncertainty in the prediction of sliding displacement.
Also, almost none of the existing probabilistic models have been validated using
comprehensive field observations of seismic-triggered landslide and associated
earthquake records (Kaynia et al. 2010).
In this chapter, a new probabilistic method was proposed in which the Newmark
displacement index, the earthquake intensity, and the associated spatial variables are
integrated into one multivariate conditional probabilistic function. This model is built
based on a powerful statistical approach: Copula modeling technique.
123
Figure 4.9 Validation of the predicting model using the Q-Q plots of landslide areal percentages
versus Newmark displacements
The presented approach was applied to a quadrangle in Northridge area in Northern
California having a large landslide database. The predictive hazard model was applied to
the study area and it was tested using standard goodness-of-fit procedures. Also, to
compare the model results with the field performance, areal extension of the landslide
events previously occurred in the region were used. The validation showed that the model
results are consistent with the real landslide inventory database. Finally, the presented
probabilistic model was used to develop a seismic landslide hazard map for the selected
124
area. The obtained regional hazard map provides the sliding displacement values for the
hazard level of 10% probability of exceedance in 50 years (
).
The probabilistic model proposed and demonstrated here represents a rational
approach to take the uncertainties of the commonly involved seismic and spatial variables
into account for assessing the earthquake-triggered landslide hazards. This methodology
can be implemented for similar seismic hazard assessment and mapping purposes in
regional-scale studies. The final resulting hazard maps can be updated with any
additional seismic and spatial information in the future.
125
CHAPTER VI
AN EMPIRICAL-STATISTICAL MODEL FOR DEBRIS-FLOW RUNOUT
PREDICTION IN REGIONAL SCALE
5.1 Overview
Debris flow which is sometimes referred as mudslide, mudflow or debris avalanche is
defined in the literature as a mixture of unsorted substances with low plasticity including
everything from clay to cobbles (Varnes 1978; Lin et al. 2006). Debris flows which are a
common type of fast-moving landslides are produced by mass wasting processes. They
are one of the most frequently occurring natural phenomena that cause a lot of human
loss and damage to properties annually all around the world (Hungr et al. 1999;
Prochaska et al., 2008). For instance, in 1996 and 2003 destructive debris flows took
place in the Faucon catchment, causing significant damage to roads, bridges and property
(Hussin 2011). Debris flows also play an important role in channel aggradations, flooding
and reservoir siltation and also basin sediment yielding (Bathurst et al., 2003; Burton et
al., 1998). Therefore, evaluation of the potential debris flows is a very vital task in
landslide risk management and generally it helps to delimit the extension of the hazard
and scope of endangered zones.
126
The existing debris-flow runout approaches require estimation of the influencing
factors that control the flow travel such as runout distance, depth of deposits, damage
corridor width, depth of the moving mass, velocity, peak discharge and volume (Dai et al.
2001). However, accurate estimation of all of these initial parameters which are involved
with a lot of uncertainty is very difficult in practice (Prochaska et al. 2008). The main
scope of this paper is to present a reliable probabilistic methodology which could be
simply based on a single influencing factor. Therefore, such a methodology will be
capable of considering the uncertainty of the debris-flow parameter(s) without
complexity of the most existing models. This model is used for preliminary estimation of
triggered debris-flow runout distance based on the slope gradient of the travel path in
regional scale. It is believed that such an approach is valuable, time saving and can be
applied to any similar debris-flow hazard evaluation in the future.
5.2 Literature Review
Current and past methods in studying runout behavior of a landslide are generally
classified into three categories of dynamic methods, physical scale modeling and
empirical approaches (Chen and Lee 2000; Hurlimann 2006).
Dynamic methods are physically based on momentum or energy conservation of the
flow (Rickenmann 2005). These models require two parameters named the flow velocity
and the frictional parameter which their exact estimate can be complicated (Cannon and
127
Savage 1988; Kang 1997). The dynamic modeling generally includes simplified
analytical method and numerical approach as follows: a) Simplified analytical methods
describe the physical behavior of debris flow movement using lumped mass models and
assuming the debris mass as a single point (Zahra 2010). Although different studies have
been performed in this regard (e.g. Hutchinson 1986; Skempton et al. 1989; Bracegirdle
et al. 1991; Takahashi 1991; Hungr 1995; Rickenmann 2005) the lumped mass models
are not able to consider the lateral confinement and spreading of the flow and the possible
changes in flow depth. Thus, these models should be applied only for comparing very
similar downhill paths which have the same geometry and material properties to some
extent (Dai et al. 2001); b) the numerical methods mainly include fluid mechanics models
and distinct element approaches (Naef et al. 2006). Continuum fluid mechanics models
employ the conservation equations of mass, momentum and energy and also a rheological
model which describe the dynamic motion and the material behavior of debris fellow
respectively (Hurlimann et al. 2008). A lot of studies have been conducted on runout
behavior of debris flow using numerical methods (e.g. Hungr 1995; Chen and Lee 2000;
Jakob et al. 2000; Hürlimann et al. 2003 and Revellino et al. 2004). Although these
models have generally the potential to provide accurate runout estimation results, they are
complex and their analysis is costly (Gonzalez et al. 2003; Prochaska et al. 2008).
The second group of the runout models is the physical scale modeling which uses field
and laboratory experiments to study debris flow mechanics. Debris flow flumes are
applied in these models to simulate an actual debris flow runout and also to assess the
flow with high-speed photography and videotaping (Iverson 1997). It should be noted
128
that these simulations can be costly to perform and also the geometric scale applied in
such models could be uncertain (Hussin 2011). Also as Dai et al. (2001) stated due to the
difference in scale and mechanics of the modeled outputs, using these methods to field
situations is not always recommended.
The third type of approaches is empirical models aimed at providing practical tools for
predicting the runout distance and distribution of landslide debris. Empirical methods are
one of the most common approaches to estimate the runout distance of debris flows by
establishing a relation between effective elements such as slope gradient, morphology
and volume rate using multivariate regression analysis (Hurlimann et al. 2008). There are
many various studies regarding this topic including Hsü (1975), Corominas, (1996),
Legros, (2002), Fannin and Wise (2001), Bathurst et al. (2003) and Crosta et al. (2003).
The empirical approaches can be classified into mass-change method and the angle of
reach technique as follows: a) The concept of mass-change method is that as the debris
flow moves down slope, the initial mass of the landslide changes and that the landslide
debris stops when the volume of the moving materials becomes negligible (Cannon
1988). In such a method, stepwise multivariate regression analysis is applied to establish
a relationship between the effecting factors (like gradient of the down slope path and
vegetation types) and the channel morphology on the volume loss-rate (Hungr et al. 1987;
Corominas 1996; VanDine 1996; Lo 2000); b) the other empirical method named the
angle of reach which this angle is defined as the angle of the line connecting the head of
the failure source to the distal margin of the mobilized debris mass. These approaches
usually develop a linear relationship between angle of reach of debris flow and the
129
average slope of the travel path (Prochaska 2008). Regression equations for obtaining the
angle of reach were developed by different authors (e.g. Scheidegger, 1973; Corominas
1996; Bathurst et al. 2003, Prochaska et al. 2008). Figure 1 displays the flowchart
summarizing all the described runout prediction approaches.
Generally speaking, empirical approaches are much more straightforward, common
and less time-consuming compared to the other techniques mentioned earlier. These
approaches have an advantage over the other techniques that prevent the use of uncertain
and variable initial parameters such as volume, velocity, and frictional parameters
(Bathurst et al. 1997; Prochaska et al., 2008). The initial information required by these
techniques is generally accessible and when the historical landslide database is available,
the empirical modeling can be easily developed (Dai et al. 2002). However, a common
problem with this kind of methods is that the scatter of the data is too large to allow the
reliable application of the results. Therefore, these techniques should be used only for
preliminary prediction of the runout travel distance (Dai et al., 2002). Also as Harp et al.
(2006) stated, no current empirical method is able to model runout distances except in
uniform materials with few irregular particles such as trees and other types of vegetation.
Moreover, the debris-flow estimation techniques are generally limited to deterministic
perspective. That is, existing methods including dynamic methods, physical scale
modeling and empirical approaches mostly ignore the uncertainties involved with the
influencing parameters in practice (Archetti and Lamberti, 2003; Lin et al., 2006).
Removing these mentioned limitations was the main motivation in this paper to develop a
new empirical approach for the estimation of debris-flow runout distances.
130
Figure 5.1 Flowchart showing different current runout models
5.3 Methodology
To compute the configuration of debris-flow travel in this study, deposit toe, the
runout travel distance, slope height and gradient need to be shown and defined first. In
this paper, runout distance (L) is the distance from the debris flow initiation point until
the point of complete sedimentation of the flow; and the slope failure height (h) is
defined as the approximate elevation difference between the head-scarp and the deposit
toe (see Fig 6.2). These simplified geometrical relationships between debris flow
parameters would help us to develop our predictive model in the following sections.
131
Figure 5.2 Schematic cross section defining debris flow runout distance (L), and slope gradient
( ) used in debris flow runout prediction
In order to assess the hazard due to debris flow events, it is needed to assign a
probabilistic perspective to the definition of this flow mobilization. In this paper, this
hazard (H) is the probability of the exceedance of the travel distance of the debris
materials from a critical distance in a location with a specific slope gradient. This
probability is mathematically expressed as below:
(5.1)
Where
represents the critical travel distance;
represents the safety margin as
is the debris flow travel distance and
. The Equation.1 can be further written
as
132
(5.2)
Where
variable;
is
the
standardized
normal
random
represents the reliability function which is the cumulative distribution
function of the standardized random variable ; and
standard deviation of
, respectively. The
and
are the mean and the
can be estimated as below:
(5.3)
Where
is the expectation operator, and
respectively. When
and
are independent,
and
are the means of
and ,
is given by
(5.4)
Where
and
are respectively the variance of
and . Since,
certain deterministic distance, it‘s only required to estimate
and
and
is set equal to a
to calculate
using the Equations 3 and 4.
On the other hand, the topography of a slope impacts the debris materials mobilization
so that the steeper slopes magnify the acceleration of the debris runout of an occurred
133
landslide (Dai et al. 2001; Chen and Lee 2000). Also another study performed by
Nicoletti and Valvo (1991) showed that the down slope morphology where debris from
rock avalanches mobilized influences significantly the runout distance. Therefore, the
slope gradient could be the main influencing factor which is related to the debris flow
distance. In other words, the debris travel distance should be related to the slope gradient
of the flow path as below:
(5.5)
Where
represents the slope gradient of the debris flow path. Such an empirical
relationship could be found separately for the debris flow database of interest. This
empirical relationship is applied in the exceedance probability function (Eq.2) and gets
calibrated to obtain the final model. The final predictive model would be capable of
estimating the probability of exceedance of the debris flow from a specific critical length
given a path having a specified slope gradient. The flow chart displayed in Figure.3
shows the presented method in this paper. This flow chart shows how debris flow runout
data are integrated into the predictive model to help assessing the debris flow hazard in a
study area.
5.4 Application of the Proposed Model
134
5.4.1 Study area and database
Bluffs and hillsides in Seattle WA in the Puget Sound area have long been exposed to
a significant amount of damage to people and properties due to landslides and debris
flows which occur almost every year during the wet season (Thorsen 1989).
Figure 5.3 Flow chart showing the presented methodology in this study for debris flow hazard
assessment
Debris flows is one of the main types of landslides in Seattle area (Baum et al. 2005).
These flows which move onto the mid bluff bench or to the base of the slope typically
involve colluviums derived from glacial till, silt and outwash sand. They can travel at
135
velocities as high as 60 km/hr which makes them one of the most hazardous types of
landslides (Harp et al. 2006, see Fig.4.a). In 1996, the U.S. Geological Survey (USGS)
started a program to study natural disasters in the five counties bordering on Puget
Sound. Also in 1998, the Federal Emergency Management Agency (FEMA) launched a
hazard management program called Project Impact and Seattle was one of the pilot cities
in this study. In this regard, Shannon and Wilson, Inc. was assigned to create a digital
landslide database from 1890 to the 2000 (Laprade et al. 2000). This comprehensive
database was a great opportunity for scientists for further study of landslide and debris
flows especially in Seattle area.
The debris flow database is available for the slope failures that mostly occurred during
two precipitation events on January and March in the winter of 1996/97 (Baum et al.
2000). Slope failures in the database were mostly the debris flows that were plotted on
1:24,000-scale USGS maps from stereo aerial photography acquired in April 1997.
Totally 326 debris flows recorded within the corridor of the coastal bluffs along Puget
Sound (from north Seattle to Everett), in Washington. This region was selected as the
study area in this paper (see Fig.4b). The horizontal runout length and also the elevations
of both the head and toe of the slide for each debris flow event were available in the
dataset (Baum et al. 2000). The distribution of runout distances from this data set is
shown in Figure.5. From 326 records of debris flow 250 records are selected for model
development and the remaining records are implemented for validation in this paper.
From 250 debris flow events, the minimum length from headscarp to toe is about 11.9 m,
the maximum length is around 174.1 m and the mean length is 89.3 m.
136
(a)
(b)
Figure 5.4 a) Example of debris flow near Woodway, Washington on January 17, 1997 (Harp et
al. 2006), b) Selected study area in this paper
137
In this study the slope map created by Jibson (2000) was applied to determine the
slope gradient values for each failure event (see Fig.6). This slope map was produced by
using an algorithm modified in GIS format to the DEM (1.83-m cell) which compares the
elevation of adjacent cells to calculate the steepest slope. To estimate the average
gradient of each slope in runout dataset, the mid-elevation from headscarp to toe of each
failure events from aerial photos were used (see Fig.2). This selected average reference
point which has a physical meaning was localized for the dataset in GIS format. Several
different reference points were examined before this one was decided upon. The use of
these other reference points was rejected due to subjectivity in identification of their
placement, sensitivity in their application, or lack of a physical meaning to their
locations. It should be noticed that there was no aerial photos available for some of
thefailure events in the dataset. For those cases, based on a method presented by Coe et
al. (2004), the approximate point location corresponding to each failure area was
considered to estimate the mean slope gradient.
5.4.2 Development of the empirical-statistical model
Statistical data
As the first step toward establishment of the predictive model in this paper, 250
selected debris flow records were divided into three subsets based on their triggering
factors reported by Shannon and Wilson Inc. (2000). This division was done in order to
examine the effect of triggering mechanism upon the relationship between runout lengths
138
and mean slope gradients. These three subsets include: subset1) debris flows triggered by
natural factors; subset2) debris flows induced by water ground/surface; and subset3)
debris flows triggered by cut and/or fill. The statistics of these subsets are summarized in
Table.1.
Figure5.5 Cumulative frequency plot of runout distances for the 326 debris-flow runout lengths
mapped from north Seattle to Everett (Harp et al. 2006)
139
Figure 5.6 Slope gradient map of the study area
Table 5.1 Summary of the three debris flow data subsets
140
Dependence assessment
The second step toward development of the model is to measure the dependence
between debris flow length (L) and slope gradient ( ) data for each data subset. To check
the linear association between each category of the pairs
, Pearson‘s product-
moment correlation coefficient was computed as below:
(5.6)
Where N is the sample size;
are sample variance of
and
are sample means of the variables; and
and
and . As given in Table.2, the findings verify the significant
and positive correlation between
Such significant correlations between geometric
parameters of debris flow was already confirmed by earlier studies in different landslidedebris flow inventories, including Swiss Alps debris flows, Canadian Cordillera debris
flows and Japan (Kamikamihori valley) debris flows (Schilling and Iverson 1997; Okuda
and Suwa 1981; Zimmermann 1996; Rickenmann 1999; Jakob 2000). Our dependence
assessment findings are used for building the hazard predictive model and also for
creating the regional debris flow hazard map in the following sections.
141
Table 5.2 Rank correlation coefficients of the pairs of the three debris flow data subsets
As a result, the three field data subsets were then combined into a single regression
equation, which is plotted on Fig.7.b:
(5.8)
The small intercept of the equation above makes a physical sense, so that debris
materials do not flow so long for flat basins with small or negligible
angles.
Figure 5.7 a) Regressions of L versus α for the three field data subsets, b) the final developed
regression model
142
Table 5.3 Summary of the significance tests of the best fit regression equation
Predictive model
The relationship between debris flows distance (L) and mean slope gradient (α) was
established in the previous section (Eq.8). Now it‘s required to apply this linear
relationship into the probability exceedance function of the debris flow distance as
discussed earlier. Using the calculated statistical data (mean and standard deviation) of
the total 250 records, and integrating the final regression equation (Eq.8) with the normal
cumulative exceedance probability function (Eq.2), the general form of the predictive
model is obtained as below:
(5.9)
143
Once calibrated, the equation above can be used to predict the hazard of the debris
flow as a function of the mean slope angle in that location and the selected critical debris
flow distance. In order to obtain the coefficients of A, B and C in the equation above, the
existing 250 debris flow events were used for calibration. The curve fits the data very
well,
87.1%, (see Fig.8). Therefore the final predictive model was obtained as
below:
(5.10)
Where ‗erf‘ represents the Gauss error function (Andrews 1997);
and
represent the
mean slope angle of the debris flow travel path, and the corrected critical debris flow
distance respectively.
5.5 Debris-flow runout prediction results
Using the obtained predicting model, the debris flow hazard map of the study area was
created in 10-m grid spacing in the ArcGIS platform (Fig.9). This size of mapping cells
was selected because it was compatible with the size of the mapping units in the applied
geology map. The obtained map represents the exceedance probability of a potential
debris flow from a critical arbitrary length (
144
) in each mapping unit with a
specific mean slope gradient. This critical value was selected because it‘s very closed the
mean of the debris flow distances and could be a good index for comparison. The
obtained map indicates that the most significant hazard is associated with the steeper
slopes in the northern and coastal regions of the study area.
Figure 5.8 Calibration of the predictive model using the debris flow inventory data
145
Figure 5.9 Example of debris flow hazard (exceedance probability) map for critical runout
distance of ( = 80m)
5.6 Validation
The best way to check the validity of a model is to employ landslides data independent
from the one used to develop it (Jaiswal et al. 2010). As mentioned earlier, we chose 76
debris flow events to be used for validation part. Since the validation process was
performed for the already occurred debris flow events, the incident is considered as
certain, i.e. the probability of their occurrence is 100%. Therefore, the closer the
predicted values of the hazard model (Eq.10) to 100%, the more valid and reliable it
146
would be. Based on the results, the prediction rate of the model predictions is 92.2%.
This prediction rate is calculated as the average amount of the hazard value of the 76
debris flow events. Additionally, a computation was performed to compute the hazard for
100 random locations throughout the area with zero debris flow record (probability of
their occurrence is considered as 0%). The mean prediction rate for this group of data is
only 11.7%.
5.7 Summary and conclusion
The current debris-flow runout methods require estimation of the initial parameters
that control the flow travel such as runout distance, depth of deposits, damage corridor
width, depth of the moving mass, velocity, peak discharge and volume (Dai et al. 2002).
However, accurate estimation of all of these influencing factors which are involved with
a lot of uncertainty is very difficult in practice (Prochaska et al. 2008).
The main purpose of this paper was to present a reliable probabilistic method which
could be simply based on a single influencing factor. Hence, such a methodology will be
capable of taking the uncertainty of the debris-flow parameter(s) into account without
complexity of the most existing models. Thus, the proposed model is used for
preliminary prediction of debris-flow runout distance based on the slope gradient of the
147
travel path in regional scale. This model is built upon a reliable regression analysis and
exceedance probability function.
The presented approach was applied to 326 debris flow events in Seattle area, WA.
The data was divided into three subcategories and regression analysis was performed on
them. The final regression equation was applied into the exceedance probability function;
and then it was calibrated using 250 debris flow events to obtain the predictive hazard
model. This model was applied to the study area and it was tested by measuring the
prediction rate of two groups of debris flow records. The obtained success rates of 92.2%
and 11.7% showed that the model results are consistent with the real landslide inventory
database very well. Finally, the presented probabilistic model was used to develop a
debris flow hazard map for the selected area. The obtained map represents the
exceedance probability of a potential debris flow from a critical arbitrary length (
) in each mapping unit with a specific mean slope gradient.
The model proposed and demonstrated here represents a rational approach to take the
uncertainties of the commonly involved geometric variables of the debris flow into
account; and also presents a preliminary technique to assess the debris flow hazards. It
should be mentioned that this method like the other topographic models have the
limitation of requiring knowledge of an initiation point for the failure on the slope to
obtain the average slope of travel path. However, this methodology can be usefully
implemented for similar debris flow assessment and mapping purposes in regional scale
148
studies. The final resulting hazard maps can become updated with any additional related
information in the future.
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
6.1 Summary of Important Research Results
In chapter III, the simplifying assumption of independence among the hazard elements,
which has been commonly used in the literature, was discussed. A new quantitative
methodology was presented to assess the landslide hazard probabilistically in a regional
scale. This approach considers the possible dependence between the hazard components and
attempts to forecast probabilistically the future landslide using a reliable statistical tool
named Copula modeling technique. We tested the model in western part of the Seattle, WA
area. A total of 357 slope-failure events and their corresponding slope gradient and geology
databases were used to build the model and to test its validity. The mutual correlations
between landslide hazard elements were considered; and Copula modeling technique was
applied for building the hazard function. Based on the comparison results, the mean success
rate of Copula model in prediction of landslide hazard is 90% on average; however, this
149
mean success rate of the traditional multiplication-based approach is only 63% which is due
to neglecting the existing dependency between hazard elements. The comparison results
validate the advantages of the presented methodology over the traditional one due to explicit
considerations of dependence among landslide hazard components.
In this chapter, considering the probability of landslide frequency, we assumed that the
―
mean recurrence interval‖ of landslide events will remain the same in the expected validity
time of the model in the future. The previous studies in Seattle area indicate that most of the
landslides in the region are rainfall-triggered (Baum et al. 2005). It means that if the rate of
occurrence of the meteorological events or the intensity-duration of the triggering rainfalls
does not change significantly over time, the recurrence interval of landslides will almost
remain the same. Therefore, the possible change in the pattern of precipitations needs to be
considered and checked for a study area before applying this methodology in the future.
In chapter IV, a new probabilistic method was proposed in which the Newmark
displacement index, the earthquake intensity, and the associated spatial variables are
integrated into one multivariate conditional probabilistic function. This model is built based
on a powerful statistical approach: Copula modeling technique. The presented approach was
applied to a quadrangle in Northridge area in Northern California having a large landslide
database. The predictive hazard model was applied to the study area and it was tested using
standard goodness-of-fit procedures. Also, to compare the model results with the field
performance, areal extension of the landslide events previously occurred in the region were
used. The validation showed that the model results are consistent with the real landslide
150
inventory database. Finally, the presented probabilistic model was used to develop a seismic
landslide hazard map for the selected area. The obtained regional hazard map provides the
sliding displacement values for the hazard level of 10% probability of exceedance in 50 years
(
).
The probabilistic model proposed and demonstrated in this chapter represents a rational
approach to take the uncertainties of the commonly involved seismic and spatial variables
into account for assessing the earthquake-triggered landslide hazards. This methodology can
be implemented for similar seismic hazard assessment and mapping purposes in regionalscale studies. The final resulting hazard maps can be updated with any additional seismic and
spatial information in the future.
The main purpose of chapter V was to present a reliable probabilistic method for
predicting debris-flow runout distance which could be simply based on a single influencing
factor. Hence, such a methodology will be capable of taking the uncertainty of the debrisflow parameter(s) into account without complexity of the most existing models. Thus, the
proposed model is used for preliminary prediction of debris-flow runout distance based on
the slope gradient of the travel path in regional scale. This model is built upon a reliable
regression analysis and exceedance probability function. The presented approach was applied
to 326 debris flow events in Seattle area, WA. The data was divided into three subcategories
and regression analysis was performed on them. The final regression equation was applied
into the exceedance probability function; and then it was calibrated using 250 debris flow
events to obtain the predictive hazard model. This model was applied to the study area and it
151
was tested by measuring the prediction rate of two groups of debris flow records. The
obtained success rates of 92.2% and 88.3% showed that the model results are consistent with
the real landslide inventory database. Finally, the presented probabilistic model was used to
develop a debris flow hazard map for the selected area. The obtained map represents the
exceedance probability of a potential debris flow from a critical arbitrary length (
)
in each mapping unit with a specific mean slope gradient.
The model proposed and demonstrated in chapter V represents a rational approach to take
the uncertainties of the commonly involved geometric variables of the debris flow into
account; and also presents a preliminary technique to assess the debris flow hazards.
6.2 Recommendations for Future Research
Areas for future research are recommended as follows.
a) To have valid FS values obtained from susceptibility analysis in chapter III; the main
assumption was that the geological and topographical features of the study area won‘t change
in an expected validity time-frame of the model. In fact, we assumed that landslides will
occur in the future under the same conditions and due to the same factors that triggered them
in the past. Such an assumption is a recognized hypothesis for all functional statisticallybased susceptibility analyses (Carrara et al. 1995; Hutchinson 1995; Aleotti and Chowdhury
1999; Yesilnacar and Tropal 2005; Guzzetti et al. 2005; Ozdemir 2009). Therefore, in the
future depending on the selected validity time-frame for any study using this methodology,
152
geology and also topography need to be evaluated first. This evaluation is required to make
sure that these features won‘t change significantly during the selected time frame.
b) The implemented geospatial data in chapter IV helped to estimate the seismic-induced
landslide hazard accurately. However, the more influencing data we involve in our analysis,
the more precise results can be obtained (Fig.4.3). It is recommended to consider a predictive
multivariable model in the future to be able to predict the seismic-triggered landslide hazard
even more precisely.
c) It should be mentioned that the presented method in chapter V, like the other
topographic models has the limitation of requiring knowledge of an initiation point for the
failure on the slope to obtain the average slope of travel path. However, this methodology
can be usefully implemented for similar debris flow assessment and mapping purposes in
regional scale studies. The final resulting hazard maps can be updated with any additional
related information in the future.
153
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