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Avalanche risk in Iceland Keylock, Christopher James 1996

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A V A L A N C H E RISK IN ICELAND by CHRISTOPHER JAMES K E Y L O C K B.A.(Hons), The University of Oxford, 1994 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR T H E D E G R E E OF MASTER OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Geography) We accept this thesis as conforming tt^ fche^xequir^ l standard t T H E UNIVERSITY OF BRITISH COLUMBIA September 1996 © Christopher James Keylock, 1996 X • r \ •1 / In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of &)to(j^^ The University of British Columbia Vancouver, Canada Date free £V<7/£ ABSTRACT In this thesis I present a probabilistic approach to modelling avalanche risk for settlements in Iceland. In particular, two simulation models are developed. These are used to calculate the probability of avalanches travelling a certain distance, and of the flow being a specific width. These two simulation models, in combination with knowledge of the average frequency of avalanche occurrence, permit the calculation of the probability of encountering an avalanche at any point in the terrain (the encounter probability). This may be further combined with knowledge of the proportion of time the location is occupied (the exposure), and the proportion of damage that the avalanche causes (vulnerability) to derive a value for risk. Following their development, the simulation models are validated against the records of avalanching in Iceland and are found to be able to represent the conditions upon many of the avalanche paths satisfactorily. However, it would appear that the models are best applied to those paths in the West Fjords which have a relatively high frequency of avalanching. Paths where the nature of avalanching cannot be adequately represented by the simulation models are identified. A sensitivity analysis shows that both models are fairly robust, being most sensitive to the avalanche sizes for which the data are of the highest quality. Consequently, it would appear that the error introduced from attempting to represent those sizes for which little or no data are available, is minimal. I conclude by providing some example risk simulations for paths in the West Fjords. It is to be hoped that the model outlined in this thesis provides a useful plamiing tool in Iceland. TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES vii LIST OF FIGURES ix ACKNOWLEDGEMENTS xiii CHAPTER 1: Introduction 1 1.1 Overview 1 1.2 The Study Area 2 1.3 Hazard And Risk Mapping 8 1.4 Some Definitions For Risk Analysis 9 1.5 Existing Approaches To Land-Use Planning In Avalanche Terrain 11 1.5.1 Formulation of hazard zones 11 1.5.2 Determining avalanche impact pressure for hazard zonation 12 1.5.3. The determination of avalanche return period for hazard zonation 14 1.5.4 Statistical approaches to runout estimation (Bovis and Mears, 1976) 15 1.5.5 The 'alpha-beta' model (Lied and Bakkelwri, 1980) 16 1.5.6 The runout ratio method (McClung and Lied, 1987) 18 1.6 Simulation Modelling 21 CHAPTER 2: The Development Of A Model For Avalanche Risk 22 2.1 The Components Of An Avalanche Risk Model 22 2.2 Avalanche Size And Frequency Of Occurrence 27 iii 2.2.1 Avalanche frequency and the Poisson distribution 27 2.2.2 The relative frequency of different Sized avalanches 28 2.2.3 Estimating avalanche size from the Icelandic record 31 2.2.4 Combining the Icelandic and Canadian data 32 2.3 Distributions For Runout Distance 34 2.3.1 Data collection 34 2.3.2 Discussion concerning the appropriate location of the beta point in Iceland 34 2.3.3 Distribution fitting to sizes three to four 40 2.3.4 Distribution estimation for other sizes 43 2.3.5 The runout simulation model 43 2.3.6. Separate model formulations for different classes of path terrain 50 2.4 Distributions For Avalanche Width 53 2.4.1. The nature of the Icelandic data 53 2.4.2. Fitting distributions to the width data from Iceland 55 2.4.3 The combined effect of the width distributions 61 2.4.4 Avalanche width and the Gumbel distribution 62 2.5 Deriving A Frequency Of Avalanching Upon a Specific Path 66 2.5.1. Fundamental problems with frequency evaluation from the historical record 66 2.5.2. A method for deriving a frequency estimate 67 2.5.3 Introducing avalanche frequency to the simulation model 68 2.5.4 Determining average avalanche frequency - an example 69 2.6 Vulnerability 72 2.6.1. Simplifying vulnerability to a manageable level 72 2.6.2. Establishing relations between avalanche size and vulnerability 72 2.6.3 Vulnerability as specific loss 74 2.6.4 Vulnerability as percentage fatalities 76 2.6.5 The effect of using stronger materials in construction 77 2.7 The Risk Model In Practice 79 CHAPTER 3: Model Validation And Sensitivity 81 3.1 Overview 81 3.2 Extreme Runout In Iceland 82 3.3 Validation Of The Runout Simulation Model 85 3.3.1 Comparing the parameters of the simulation model and path data 85 3.3.2 Outlier paths 87 3.3.3 Summary 90 3.4 Estimation Of Average Avalanche Frequency 91 3.5 Validation Of The Width Simulation Model 98 3.5.1 Comparison of the scale parameters for the model and the path data. 98 3.5.2 Avalanche frequency and location parameter validation 100 3.6 Sensitivity Of The Simulation Models 107 3.6.1 Overview 107 3.6.2 Sensitivity to the mean and standard deviation 109 3.6.3 Sensitivity to individual sizes 109 3.6.4 Sensitivity to the distributions obtained by extrapolation of parameter values 111 3.6.5 Changes to the relative frequency distribution.... I l l 3.6.6 Summary 114 3.7 Sensitivity Of The Risk Model 115 v CHAPTER 4: Conclusion 118 BIBLIOGRAPHY 124 APPENDIX 1: Avalanche Size Classifications 131 APPENDIX 2: The Poisson Distribution 133 APPENDIX 3: The Extreme Value Type I Distribution 136 APPENDIX 4: The Generation Of Random Variates 139 APPENDIX 5: Microsoft® Visual Basic® Macro For The Simulation Of Risk 145 vi LIST OF TABLES TABLE 2.1 : Canadian snow avalanche size classification and typical factors 23 TABLE 2.2 : The proportion of avalanches of a particular size at Revelstoke and Rogers' Pass, British Columbia, Canada . 29 TABLE 2.3 : Goodness-of-fit of extreme runout data to an extreme value distribution for various definitions of the beta point 38 TABLE 2.4 : Runout ratio statistics for avalanches from sizes 3 to 4, with an assessment of fit to the normal distribution 42 TABLE 2.5 : Derived parameters for normal distributions of the runout ratio 44 TABLE 2.6 : Deposit width statistics for avalanches from sizes 3 to 4, with an assessment of fit to the Gamma distribution 56 TABLE 2.7 : Estimated shape and scale parameters for gamma distributions of avalanche width 59 TABLE 2.8 : Comparison between several formulations of the model and the Icelandic data.... 59 TABLE 2.9 : Determination of the underlying relative size distribution for a width simulation at a runout ratio of 0.35 63 TABLE 2.10 : Extreme value distribution approximations to the width simulation model for various runout ratios 63 TABLE 2.11 : Degree of damage to buildings in Montenegro, 1979 75 TABLE 2.12 : Vulnerability expressed as specific loss or proportion of fatalities for two different construction materials 78 TABLE 2.13 : An example risk calculation for Eyrarhryggur at Flateyri 80 vii TABLE 3.1 : Parameters for Gumbel distributions fitted to extreme runout for various mountain ranges 84 TABLE 3.2 : Comparison of frequencies of avalanche occurrence derived from the model and the historical record 93 TABLE 3.3 : Parameters for width distributions fitted to 16 paths 105 TABLE 3.4 : Correction factors applied to the location parameter of the width model when adopting the frequency of avalanching appropriate for runout 105 TABLE 4.1 : Risk statistics for persons exposed voluntarily or involuntarily to various hazards (Reid, 1989) 123 viii LIST OF FIGURES FIGURE 1.1 : Iceland's location in the northern hemisphere 3 FIGURE 1.2 : Avalanche prone regions in Iceland 4 FIGURE 1.3 : The West Fjords of Iceland 5 FIGURE 1.4 : Sudavik from the air 6 FIGURE 1.5 : Definition of terms for the alpha-beta and runout ratio models 17 FIGURE 2.1 : Flow chart illustrating the structure of the risk model 26 FIGURE 2.2 : Cumulative distribution functions for the relative frequency of different sized avalanches at Revelstoke and Rogers' Pass, Canada 29 FIGURE 2.3 : A distribution for the relative frequency of different avalanche sizes derived from a combination of Canadian and Icelandic data 33 FIGURE 2.4 : Avalanche registration map for Hnifsdalur 35 FIGURE 2.5 : Extreme runout from Iceland for paths with at least thirty years of data. (The beta point is defined for a local slope angle of 22°) 39 FIGURE 2.6 : Extreme runout from Iceland for paths with at least thirty years of data. (The beta point is defined for a local slope angle of 14°) 39 FIGURE 2.7 : Histogram of observed runout ratios for size 3 avalanches with a fitted normal distribution.... 41 FIGURE 2.8 : Histogram of observed runout ratios for size 3.5 avalanches with a fitted normal distribution 41 FIGURE 2.9 : Histogram of observed runout ratios for size 4 avalanches with a fitted normal distribution 42 FIGURE 2.10 : Normal distributions of avalanche runout distances for different size classes 44 ix FIGURE 2.11 : Distributions for runout weighted by the relative frequency of occurrence of the different sizes 46 FIGURE 2.12 : Simulation model results for the percentage of avalanches attaining or exceeding a given runout ratio 46 FIGURE 2.13a : A simulation highhghting the region where runout ratios he between -0.2 and +0.2 48 FIGURE 2.13b : A complete set of simulated runout ratios 48 FIGURE 2.13c : Simulated runout ratios censored at a runout ratio of zero 49 FIGURE 2.14 : Plot of factor loadings for 50 avalanche paths 51 FIGURE 2.15 : Scatterplots illustrating the degree of association between the parameters for a Gumbel distribution fitted to the path data, and factor loadings derived from topographic variables 51 FIGURE 2.16 : Scatterplot of avalanche deposit widths as a function of size 54 FIGURE 2.17 : Histogram of observed widths for size 3 avalanches with a fitted Gamma distribution 56 FIGURE 2.18 : Histogram of observed widths for size 3.5 avalanches with a fitted Gamma distribution 56 FIGURE 2.19 : Histogram of observed widths for size 4 avalanches with a fitted Gamma distribution 57 FIGURE 2.20 : Gamma distributions for avalanche deposit width for different size classes 60 FIGURE 2.21 : Simulated extreme value distributions of avalanche width for three different runout ratios 64 FIGURE 2.22 : The Gumbel distribution parameters for the width simulation as a function of position along the path 64 FIGURE 2.23 : Example procedure for obtaining a value for the Poisson parameter 70 FIGURE 2.24 : Damage to a residence in Sudavik from the 1995 avalanche 73 FIGURE 2.25 : Damage to the school in Su6avik from the 1995 avalanche 73 FIGURE 3.1 : Extreme runout in Iceland 83 FIGURE 3.2 : Distribution of standardized residuals for the fitted distribution 83 FIGURE 3.3 : Scatterplot of Gumbel distribution parameters for the model (one event per year) and various paths 86 FIGURE 3.4 : Scatterplot of Gumbel distribution parameters excluding the outlier paths 86 FIGURE 3.5 : Profile of the ski-lift path at fsafjor6ur (path number 7) 88 FIGURE 3.6 : Scatterplot of observed avalanche frequencies versus the fitted scale parameter 88 FIGURE 3.7 : Runout for path 18 at Neskaupstadur with the fitted model (dashed line) 89 FIGURE 3.8 : Scatterplot illustrating the trend for the paths with more events to have a lower scale parameter for the fitted distribution 89 FIGURE 3.9 : Scatterplots illustrating the path runout data and the fitted model 94 FIGURE 3.10 : Box plots of the scale parameters obtained from path data and from application of the model to the median runout ratios for each path 99 FIGURE 3.11 : Comparison between the width data and the model for individual paths 101 FIGURE 3.12 : The effect of increasing the moments of the size-runout distribution by ten percent 108 FIGURE 3.13 : The effect of increasing the moments of the size-width distribution by ten percent 108 FIGURE 3.14 : Sensitivity of the runout model to a 10 % increase in the mean and standard deviation of each size-runout distribution 110 xi FIGURE 3.15 : Sensitivity of the width model to a 10 % increase in the mean and standard deviation of each size-width distribution 110 FIGURE 3.16 : Sensitivity of the runout simulation model to a perturbation introduced to all sizes except 3, 3.5 and 4 112 FIGURE 3.17 : Sensitivity of the width simulation model to perturbation introduced to all sizes except 3, 3.5 and 4 112 FIGURE 3.18 : Sensitivity of the runout simulation model to modifications of the relative frequency distribution 113 FIGURE 3.19 : Sensitivity of the width simulation model modifications of the relative frequency distribution 113 FIGURE 3.20 : Comparison of risk for the original and modified models 116 FIGURE 4.1 : Sample risk map for Eyrarhryggur and mnri-Baejarhryggur at Flateryi 121 FIGURE 4.2 : Sample risk map for Tra6argil at Hnifsdalur 122 xii ACKNOWLEDGEMENTS I would like to extend my thanks to everybody at the Icelandic Meteorological Office for putting up with me during the summer of 1995. In particular I must thank the director of the avalanche division, Magnus Mar Magnusson for providing the resources for this study, and together with Karen Young, supplying board and lodging for part of my stay. Jon Gunnar Egilsson, Svanbjorg Haraldsdottir and Thorsteinn Sa?mundsson should also be thanked for their assistance in translating and interpreting the avalanche data, and being marvellous companions in and around the office. I am indebted to Dr. David McClung at UBC for supervising this thesis. His insightful criticisms and ideas have proved invaluable in helping me to negotiate the slippery slope of avalanche research. Many of the concepts underlying this thesis are derived directly from Dave's own work. I am grateful to the Natural Sciences and Engineering Research Council of Canada for providing additional financial support for this project. Finally, I wish to dedicate this study to the family and friends of the 36 residents of the West Fjords who were killed by avalanches during 1995. Let us hope that such a tragic series of events is never repeated. xni CHAPTER 1 Introduction 1.1 Overview : In the early hours of Thursday, October 26th 1995, an avalanche devastated the village of Flateyri in the West Fjords of Iceland. Nineteen houses, inhabited by forty-five people were struck and twenty people from a population of 379 were killed. This was the second major tragedy to strike the region that year. On Monday, January 16th an avalanche damaged or destroyed 22 houses from a total of seventy in the village of Siidavik. This event caused the death of 14 people, including eight children. Iceland has a long history of avalanche calamity. In fact, the first recorded deaths date back to the year 1118. Since this time over 600 fatalities have been recorded, more than for any other type of natural hazard. The Siidavik avalanche was the first major disaster for twenty years. In 1974, 12 people were left dead in the town of Neskaupstadur in eastern Iceland when a suite of avalanche paths became active. The recent events have precipitated an increased awareness of avalanche risk and have resulted in calls for further measures to be taken in an effort to prevent similar occurrences in the future. A topic of obvious importance is the development of risk maps that can be used to delimit areas of relatively greater safety and hence act as a useful guideline for construction and evacuation work. This thesis outlines a probabilistic approach to avalanche risk mapping to form the basis for avalanche risk planning Iceland. 1 1.2 The Study Area Iceland is a country located in the northern Atlantic Ocean. The total area of the land mass is some 100 000 km2, (approximately 80 % of the area of England) and the nation is populated by some 250 000 people. The vast majority of these (some 170 000) inhabit the region in and around the capital of Reykjavik, located in the south-western region of the country. The West Fjords, the area that has been most at risk from avalanching in recent years, lies north and west of Reykjavik. Figures 1.1 to 1.3 provide some locational information. Firstly, Iceland is set in its international context, and then at a finer scale, the major areas in Iceland prone to avalanching are illustrated. Figure 1.3 provides a more detailed characterisation of the West Fjords. The major settlements in the northern West Fjords are the towns of Isafjdrdur and Bolungarvik. In 1994 these towns had populations of 3518 and 1139 respectively. This was from a total of 9453 for the West Fjords constituency. The population for Isafjdrdur includes the neighbouring village of Hnifsdalur. Of the other avalanche prone settlements, Flateyri has a population of 379, and Sudavik has 227 residents. Figure 1.4 is a view of the town of Sudavik taken from the air. It illustrates the situation of many of the settlements in the West Fjords region. The mountainous nature of the terrain means that settlement is restricted to the narrow strip of relatively flat land that divides the hills from the fiords. Consequently many settlements in this area can be classified as lying within avalanche hazard regions. Since lines of communication must also pass through this hazardous corridor between land and sea, towns and villages may often be cut off in winter. This makes avalanche evacuation and rescue much more problematic. During the Sudavik avalanche of 1995, rescue teams had to be transported by boat, increasing the time until help could arrive. 2 Figure 1.1 : Iceland's Locat ion in the Northern Hemisphere 3 Figure 1.2 : Avalanche Prone Regions in Iceland The West Fjords Legend • Reykjavik • Major settlements Settlements prone to avalanching Figure 1.3 : The West Fjords of Iceland Legend 5 Figure 1.4 : Sudavik from the air. 6 Recent expansion of settlement has resulted in the construction of residences in regions that are more susceptible to avalanching than previously settled areas. Until very recently there has been little appreciation of the degree of risk to which some of these locations are exposed. Some of the housing shown in figure 1.4 was struck by the event of January 16th 1995. However, a consultation of the historical record would have shown this was not a freak occurrence. In 1908 an avalanche from these slopes ran to the sea. While some confined avalanche paths exist in Iceland, the vast majority of avalanches occur upon open slopes. Thus it is impossible to delimit a contributing basin that supplies snow to the path. Consequently, studies such as that by Bovis and Mears (1976) that predict a correlation between avalanche runout distance and basin area are of limited apphcabihty in Iceland. 7 1.3 Hazard And Risk Mapping The production of risk maps for natural hazard assessment is a relatively new concept. The majority of work in the past has concentrated on the production of what Einstein (1988) calls* 'Danger maps'. These are also known as inventory or registration maps. The underlying principle is that past areas of instability have an increased chance of acting as the source of future instabilities. Therefore the accurate delimiting of past events is a useful exercise in itself, and acts as an indicator of hazard, without necessarily making the progression to the determination of risk. Registration maps may also be used in other ways. Walsh et al (1990), attempted to derive relations between avalanche path location and geomorphological / geological parameters by employing such maps in combination with a Geographic Information System (GIS). This permitted the identification of likely avalanche terrain and hence established useful criteria for quickly locating avalanche prone regions at a small (cartographic) scale. This work is therefore directly comparable to the GIS based landslide analysis of Gupta and Joshi (1990). The registration maps discussed above may be extremely useful in constructing a hazard line (see section 1.5). They are, however, limited in their ability to convey to planners the degree of risk to which the residents of a particular location are susceptible. This is because they provide no information on avalanche frequency along the path and the degree of damage is not incorporated explicitly. Before methods of constructing risk maps that utilize more rigorous criteria are discussed, it is necessary to define terms such as 'risk'. 8 1.4 Some Definitions For Risk Analysis With the growth of literature dealing with natural hazards and their effect on anthropogenic activity, the terms risk and hazard have been employed in varying ways by different authors. This can easily cause the reader a great deal of confusion. For example, compare the formal definitions outlined below with the opening statement of Schumm (1988) that, 'The word hazard refers to a potential danger or risk.' The problem of misinterpretation of the use of these words is particularly evident in dialogue between English speaking and francophone workers where the French word 'risque' translates as equivalent to the English 'hazard* as defined by Varnes (1984). In natural hazards research, the definitions employed conform to those from statistical decision theory, as opposed to insurance evaluation, which focuses upon the monetary component of risk. Briefly, risk is the probability of death or losses and is the product of three sub-components: Encounter probability is the chance of an avalanche reaching a certain position in the path. Exposure is defined as the proportion of time that the objects or people of concern are subject to the phenomenon under consideration. Vulnerability is the degree of damage to the elements of concern. The general form of this nomenclature is established in several disciplines (Carrara et al, 1991; Koridze (1988); National Research Council, 1991; van Westen, 1993), although several authors have introduced slight modifications (Einstein, 1988; Morgan et al, 1992; Fell, 1994; Gerath, 1995). The definitions outlined above are employed throughout this study. The risk to an individual from avalanching consists of many components. Primarily, it is a function of the encounter probability which is dependent upon avalanche event frequency and magnitude. Secondly there is the exposure term. If one is concerned solely with buildings, the 9 exposure can be taken to be equal to unity. If one is dealing with the risk to the occupants then exposure is determined by the fraction of time that the building is occupied. Vulnerability can be formulated to varying degrees of complexity. Salient factors include construction materials, building height, orientation and even the floor plan which influences the probability of occupancy of the rooms at greatest risk. The most important element of vulnerability is the magnitude of the avalanche in question. A sophisticated formulation of vulnerability is only possible for specific applications, in many cases a simplified system suffices. Risk is primarily driven by the encounter probability and consequently accurate characterization of the avalanches should be the primary concern of a general study. It may be noted that if one considers a general case where the concern is with individual buildings instead of their inhabitants, setting exposure to unity permits the generation of maps of potential risk in paths that have yet to be inhabited. This is obviously of concern when planning the location of new settlements or the enlargement of existing towns and villages. 10 1.5 Existing Approaches To Land-Use Planning In Avalanche Terrain 1.5.1 Formulation of Hazard Zones McClung and Schaerer (1993) discuss many of the issues surrounding the determination of hazard zones and the action that is taken to ameliorate the impact of avalanches in these regions. It must he noted that none of the major zonation schemes employ are based upon a rigorous analysis of risk. In North America, where long term records of avalanching rarely exist, a complex zoning scheme is inappropriate. Consequently, it is recommended that the hazard line, (the position of maximum avalanche runout) is employed. This line is determined from historical documentation, or through analysis of tree damage. Inside of this line, the need for protective measures must he investigated on a site by site basis. In Switzerland, guidelines for formulating avalanche hazard zones have been in place since 1961. The Swiss zoning scheme has been adopted (with certain modifications) by a number of countries including Austria, France and Italy. This approach defines four zones which are colour coded and represent regions of differing hazard: In the red zone where hazard is high, no new structures and buildings are permitted. For those buildings that are already in existence, an evacuation plan must be formulated and structural reinforcement or control structures should be employed to provide protection. For an area to be included in the red zone, it must meet one of two criteria. Either avalanches with an impact pressure of 30 kPa are expected with a return period of up to 300 years, or it is expected that avalanches will run this far once in every 30 years, regardless of impact pressures generated. The blue zone delimits areas of moderate hazard. Flowing avalanches are expected with impact pressures of less than 30 kPa and return periods between 30 and 300 years, or powder avalanches are expected to occur with impact pressures of less than 3 kPa and return periods of 11 30 years or less. In this region new construction is permitted, but these must not be buildings that attract large groups of people such as schools and ski-lift terminals. These buildings must be protected and again, an evacuation plan must exist. The area of low hazard is coded as yellow. Here one may expect rare flowing events with return periods greater than 300 years, or powder events with impact pressures of less than 3 kPa and return periods of more than 30 years. The final zone, the white zone, represents areas where it is believed that avalanches are not likely to create a hazard. No restrictions to development exist in this region. 1.5.2 Determining Avalanche Impact Pressure For Hazard Zonation To apply the Swiss methodology successfully, one must be able to characterise both avalanche impact pressure and event return period. To obtain accurate estimates of both of these factors is problematic. McClung (1992) discusses many of the pertinent issues. In this section, I focus upon the impact pressure component. The determination of return period is examined in section 1.5.3. Avalanche impact pressures are calculated as the product of flow density and the square of avalanche speed. Because there are no direct measurements of flow density in moving avalanches, impact pressure cannot be estimated accurately. This introduces an early source of error into the calculation. The best one can do is to obtain an approximate estimate by measuring impact pressures and then employing the density of particles in the deposit. This was done for sites in Rogers' Pass, British Columbia, by McClung and Schaerer (1985), but more measurements in different climates are required to eliminate this source of error. Errors also arise in the determination of avalanche speed, which is commonly estimated through the use of a mathematical model. There have been many attempts to formulate succinct 12 mathematical descriptions of avalanche motion in the Hterature, the first of which dates back to Voellmy (1955). Popular approaches since then include the models of Perla et al, (1980) and Salm(1966). More recent modelling efforts have drawn more heavily upon the developing hterature surrounding the motion of dense granular flows. Granular flow modelling is based upon work in molecular dynamics, in particular Enskog dense gas theory. Models include work by Jenkins and Savage (1983) and Lun et al (1984). As yet, these models employ a very simplified view of the collision of particles. No change in particle mass is incorporated, analysis is restricted to binary collisions and rarely is an attempt made to deal with particle spin. In an extensive review of this hterature, Hutter and Rajagopal (1994) note that it is highly problematic to extend current granular flow research in a rigorous manner to deal with a mixed flow regime such as an avalanche, where particle interactions may be nearly instantaneous or may occur over longer time periods. According to Savage (1989), attempts to do so have crudely patched together results from both flow regimes. An example of such a model is that of Norem et a/(1987). At the moment, models that explicitly attempt to engage with the micro-mechanical processes that take place within a flowing avalanche appear to be seriously flawed. This is due to their inabihty to deal adequately with mixed flow regimes and their highly simplified representation of the geotechnical properties of snow. They suffer further from the fact that since there are no measurements of the mechanical properties of flowing snow, model validation and verification cannot be based upon real data. At present it would appear that the best one can do is to stick with models formulated at the larger scale, but to use the results of granular flow theory and considerations of internal deformation to inform the process of model development. This approach is taken by McClung 13 (1990), and by Salm (1993) in an extension to the Guidelines model of Salm et al (1990), the model currently employed for zone determination in Switzerland. A major problem with employing many of the models for risk zoning applications is that the avalanche is treated as a point mass. When one is attempting to delimit hazard zones, one must be aware that runout distances are highly sensitive to whether one considers the tip of the avalanche debris or the centre of mass. Consequently substantial error may be introduced when one uses the models mentioned above for runout calculation. For this reason, McClung and Mears (1995) develop a leading-edge model for determination of runout distances that utilizes the scaling model of McClung (1990) as a means to determine mcoming avalanche speeds. This approach is more conservative than the Swiss Guidelines model. The variation in the output from the numerical models means that the estimation of impact pressures is fraught with uncertainties. Avalanches accelerate and decelerate rapidly, and it is probably reasonable to suggest that at present modelling is not sufficiently accurate to be able to effectively separate the point where impact pressures drop below 30 kPa from the final runout position. The Swiss approach to zoning therefore suggests a greater degree of precision than is attainable and the hazard line method may in fact be a preferable approach. 1.5.3. The Determination of Avalanche Return Period for Hazard Zonation The return period of an avalanche is defined as the average time interval between events that reach or exceed a given location. In the ideal case, this may be determined from long term records and measurements of avalanche runout. However, it is very rare for systematic records of high quality to exist. In fact, it is more common for no records to be available whatsoever. In Iceland, avalanche events have been documented that occurred as long ago as 1118 AD. 14 However, the events in the historical record tend to be those of a large magnitude that caused damage to property or loss of life. Due to the substantial gaps in the record it is difficult to estimate the return period of events smaller than these catastrophic occurrences. Even with these large events, it is hard to define the exact path from which the avalanche initiated because paths may overlap. It may also be true that larger, non-fatal events have occurred, but have not been documented. Consequently the long term historical record must be used with some caution. In North America, in the absence of any historical record, a common practice that is utilized is the dating of damaged vegetation in the runout zone. This may be by direct dendrochronological methods, or through estimating the age of regrowth in the forest stand. In Iceland this approach is impractical due to the complete absence of trees throughout much of the country. When one has no recourse to historical information, whether from written records or from vegetation, an estimate of extreme runout, (with perhaps an estimated return period of 50 to 300 years), is still useful. Due to the problems with the numerical models mentioned in section 1.5.2, empirical, topographically-oriented statistical techniques have evolved that attempt to estimate the runout of large avalanches. The most cited examples are discussed below. 1.5.4 Statistical Approaches to Runout Estimation (Bovis and Mears, 1976). The first attempt at correlating avalanche travel distance and path topography can be attributed to Bovis and Mears (1976). In their paper, the authors used a statistical approach after expressing concern about the estimation of the friction coefficients for the Voelhny model. A least-squares regression relationship was established employing three parameters: track gradient; 15 runout zone gradient; and starting zone area. It was found that the third variable alone explained 65 % of the variance in the 67 avalanche paths examined. This important result is of limited application in Iceland where the vast majority of avalanches initiate from unconfined slopes, making it impossible to define a starting zone in the traditional sense. 1.5.5 The 'Alpha-Beta' model (Lied and Bakkehd, 1980). Four years later Lied and Bakkehoi (1980) studied 111 paths in Norway employing some 26 variables in their regression analysis. Their final equation was in fact overfitted and the only significant predictor of maximum runout was the beta angle. This is defined as the angle from the point on the path where the slope angle first declines to 10° (the beta point), to the starting zone. The response variable is the alpha angle, which is measured from the limit of avalanche debris (the alpha point) to the starting zone. These parameters are shown in figure 1.5. Since this study, the 'alpha-beta' method has been widely adopted in the literature [e.g. Bakkeh0i et al (1983), Martinelli (1986), Mears (1988) and Mears (1989)]. A major advantage of this approach is its simplicity. It has the disadvantage that the fits to the regression relation are relatively. The original data from Western Norway obtained by Lied and Bakkehjai (1980) shows a high degree of fit, with a Pearson's Correlation Coefficient of 0.93. Such a regression line explains 86.5% of the variance in the data. For other areas the degree of fit is not so good. Mears (1988) provides values of the correlation coefficient for regression relationships from North America. Squaring these numbers as r2 gives values of 0.61, 0.59 and 0.50 for Coastal Alaska, Colorado and the Eastern Sierra respectively. 16 Figure 1.5 : Definition of terms for the alpha-beta and runout ratio models Starting Zone A x 17 Mears (1988) does not provide standard errors for the regression relations that are derived for North America, but a visual inspection of his scatterplots suggests the standard error is significantly greater than that for the Norwegian data. It would appear therefore that estimates of extreme runout using the 'alpha-beta' technique are most accurate for Norway. The regression equation derived by Lied and Bakkehoi (1980) is given by: a = 0 . 9 6 p - 1 . 7 ° (1.1) with a standard error of 1.4°. There are two further problems with the alpha-beta model. The most obvious restriction is that it still only permits one to determine maximum runout, there is no means of assigning return periods to different points in the terrain. The second is that the use of linear regression necessarily implies a normal distribution of errors about the regression line. By analogy to many other natural phenomena such as floods and earthquakes, one might expect avalanches to conform to a positively skewed distribution. Fohn and Meister (1981) suggest avalanche runout conforms to an extreme value type I distribution (Gumbel, 1958). If this is the case, the alpha-beta model is unlikely to provide conservative estimates of maximum runout. 1.5.6 The Runout Ratio Method (McClung and Lied, 1987). An alternative to the linear regression approach was first suggested by McClung and Lied (1987). This method (known here as the runout ratio approach), has subsequently been used in various mountain ranges by several workers (e.g. McClung et al, 1989; McClung and Mears, 1991; Nixon and McClung, 1993). The method fits information upon extreme runout to a specific distribution. From this, statements about probable runout distance may be formulated. 18 By considering the largest events upon a number of paths, the problem is somewhat analogous to the techniques in hydrology of fitting a distribution to a record of annual maximum floods upon a river (Chow, 1964). The whole mountain range takes the place of the river and different paths replace the record of annual maximum floods. If one is attempting to formulate a distribution of runout distances upon different paths, one needs some means of standardising these values to permit comparison between paths and to create a single population of values. Consequently, McClung and Lied (1987) define a dimensionless runout ratio: AA: _ tan/?-tana Xp tan a - tan 8 The alpha and beta angles conform to the Norwegian definition, the delta angle is the angle sighting from the alpha point to the beta point, Ax is the horizontal component of the runout distance from the stopping position (alpha point) to the beta point, and X$ is the horizontal distance from the starting zone to the beta point. All of these components are illustrated in figure 1.5. From the definition of the runout ratio, one can see an implicit incorporation of a simple model of runout zone terrain. From the beta point, the distance Ax will be sensitive to local topographic variation. The alternative, angle-based definition of the runout ratio illustrates this point more clearly. The delta angle represents the average gradient of the terrain in the runout zone. This is an useful advantage of the runout ratio model over the alpha-beta method because the final distance travelled by an avalanche will be highly dependent upon the topography in the runout zone, where the avalanche velocity is low. Having obtained runout distances in the form of runout ratios for a set of avalanche paths, McClung and Lied (1987) then showed that these values conform to an extreme value type I (or 19 Having obtained runout distances in the form of runout ratios for a set of avalanche paths, McClung and Lied (1987) then showed that these values conform to an extreme value type I (or Gumbel) distribution. The nature of this particular statistical distribution is discussed more thoroughly in appendix 3. The fits to this distribution are good, with values for the coefficient of determination (r ) of at worst, 0.97 for the Coastal Alaska dataset. Because the Gumbel distribution is positively skewed, estimates of runout are liable to be more conservative than those from the Norwegian model, which employs a Gaussian model of runout. None of these existing empirical approaches permit the derivation of the variable of primary interest in risk studies, the relation between runout distance and return period along a single path. Instead, they can be used to assign a probability to the location of an extreme event. For a thorough analysis of risk, one requires more information than just the probable maximum extent of avalanching. Ideally one needs to know the probability of an avalanche exceeding any point in an avalanche zone. This necessarily requires an investigation of avalanche widths as well as runout distances. The aim of this thesis is to produce such relations. The method used to do this is known as simulation modelling. 20 1.6 Simulation Modelling. Banks et al (1996) define a simulation as, 'The imitation of the operation of a real-world process or system over time.' They further state that, 'Simulation involves the generation of an artificial history of a system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system.' Thus, if one is able to simplify the real-world system into a set of mathematical or statistical relations, it is possible to develop a model to estimate properties of that system. By considering avalanche occurrence as a stochastic phenomenon, the Icelandic records can be employed to develop a statistical simulation model. By pooling data obtained from a large number of avalanche paths, sufficient information becomes available to permit avalanche runout and width to be represented by specific statistical distributions. Because the model is based upon real data, the results of a simulation should bear a resemblance to the situation on an 'average' avalanche path. The aim in this thesis is to derive simulation models for avalanche runout and width and to link them to a functional avalanche risk model. The simulation models are critically important because they may be used to assign encounter probabilities throughout the area subjected to avalanching. Chapter 2 describes the approach used to produce the model components. A discussion of some of the techniques and concepts that underlie the work in this chapter can be found in the appendices. Benjamin and Cornell (1970), and Walpole and Myers (1989) are two alternative sources of background information. Chapter 3 discusses the means in which the models described in chapter 2 may be validated against the original dataset, and also examines the sensitivity of the model to alterations of particular elements. I conclude with chapter 4, in which some example simulations are provided. 21 CHAPTER 2 The Development Of A Model For Avalanche Risk 2.1 The Components O f An Avalanche Risk Model As was noted in section 1.4, a risk model consists of the following elements: (1) The avalanche encounter probability; (2) The proportion of time that an element or mdividual is endangered by avalanches, (exposure); (3) The degree of damage or proportion of fatalities caused by the avalanche (vulnerability). There are further considerations besides these factors. It would be preferable for the model to be applicable in three dimensions. Consequently, while the deterrmnation of runout distance is paramount, the width of the avalanche deposit is also important. The vulnerability of a building depends on impact force and construction type. Impact force is largely a function of avalanche magnitude. It is therefore important to segregate avalanches on the basis of size and input these separately into the model. This gives the model greater flexibility, as well as permitting a separate evaluation of vulnerability for each size. In appendix 1, a review of various size classifications is provided. The Canadian size classification is used in this study. This system is based upon the potential destructive effects of avalanches. It is therefore directly connected to risk. The Canadian classification uses five sizes, although it is common for half sizes to also be utilized, (a practice that is adopted in this study). Table 2.1 provides a description of the various classes along with typical values of observable parameters. Throughout the rest of the thesis, avalanche sizes are defined according to the values presented in this table. Table 2.1 : Canadian snow avalanche size classification and typical factors Size Description Typical Mass (xl0 3kg) Typical Path Length (m) Typical Impact Pressures (kPa) 1 Relatively Harmless to people <10 10 1 2 Could bury, injure or kill a person 100 100 10 3 Could bury a car, destroy a small building, or break a few trees 1000 1000 100 4 Could destroy a railway car, large truck, several buildings, or a forest with an area up to 4 hectares 10 000 2000 500 5 Largest snow avalanches known; could destroy a village or a forest of 40 hectares 100 000 3000 1000 From M c C lung and Schaerer (1993) 23 If an attempt is made to incorporate all of the above considerations into a risk model, and a probabilistic approach is employed, the following statistical distributions suggest themselves: (1) A set of distributions (one for each size class), representing the frequency of occurrence of events upon a path. (2) A set of distributions that describe the probability of a particular runout distance being obtained for an avalanche of a certain size. In combination with the frequency of occurrence, these distributions permit one to evaluate the encounter probability along the profile. (3) A similar set of distributions that permit the calculation of the probability of an avalanche of a particular width for a specific avalanche size. The sets of distributions considered in (2) and (3) together define the area affected by an avalanche event and lead to the evaluation of the encounter probability in three dimensions. (4) A relationship between the size of the avalanche and the degree of damage it is liable to cause. This distribution allows one to determine vulnerability. These four sets of distributions, in combination with values for the typical proportion of time buildings are occupied, (to evaluate exposure), are sufficient to develop a risk model. However, when it came to dealing with the available data, there was not sufficient information to accurately construct separate avalanche frequency distributions for different size classes. Consequently, the first component listed above had to be modified. This was done by replacing the set of size-based distributions with two more general relations. The first of these gives the frequency of an avalanche occurrence irrespective of size. This distribution was then supported by a second detailing the relative frequency of different sized events. In combination these two 24 distributions permit the frequency of different sized avalanches to be obtained. Therefore the effectiveness of the model is not compromised by this initial data limitation. Figure 2.1 is a flow chart that summarizes the structure of the risk model. It incorporates all the components listed above and provides a framework for the rest of the material in this chapter. Each of the model elements is discussed in turn in the following sections of this chapter. 25 Figure 2.1 : Flow chart illustrating the structure of the risk model Cumulative Poisson Distribution of Avalanche Frequency (Section 2.2.1 and 2.5) 1H9HPMI Cumulative Relative Frequency Distribution For Avalanche Size (Section 2.2.4) • • .,..1. >. 08 '•3 , t T 0 6 ( ] „ ' O 02 : 0 Frequency of Each Size Class (Avalanches per Year) Avalanche Encounter Probability Vulnerability Functions For Reinforced (R) and Low Quality (LQ) Constructions; Fatalities (f) and Buildings (b) (Section 2.6) Exposure In this study the exposure for individuals is assumed to be 0.5, (occupants reside in the buildings for half of the time). The exposure for the buildings themselves is unity, (they are fixed objects). 2.2 Avalanche Size And Frequency Of Occurrence. 2.2.1 Avalanche Frequency and the Poisson Distribution In appendix 2 it is shown that there are sound theoretical reasons for expecting avalanche occurrences upon a path to conform to a Poisson distribution. Briefly, this is because avalanches may be represented as independent and discrete events, that result from a set of continually occurring Bernouilli trials. The Poisson distribution has just one parameter, (/I), this is equal to the mean and variance of avalanche frequency. The frequency of avalanche events appears to vary greatly between different climatic regions in Iceland. For example, in Neskaupstadur on the east coast, hardly any avalanches have been recorded since the major avalanche cycle of 1974, while in the town of Flateyri in north-west Iceland, eight large events have been recorded on one path since the beginning of 1990. It is undoubtedly true that the quality of avalanche observations is variable around the country. However, the range of different avalanche frequencies is such that it must reflect real differences between regions as well as between different paths. This intrinsic variability means that every effort should be made to formulate an average frequency of avalanche occurrence for each individual path. There are very few paths in Iceland with more than three or four avalanches recorded. A method is required to somehow 'fill-in' the missing events from the record, those that do not run far enough to be recorded, in order to permit a calculation of frequency. There are various possible means of doing this, all of which are dependent to varying degrees upon the other distributions discussed in this chapter. Consequently, a discussion of frequency determination will be left until section 2.5, when the other components of the simulation model have been reviewed. 27 2.2.2 The Relative Frequency of Different Sized Avalanches Given that an avalanche has occurred, one must have a means of rating its size. Examination of the available data in Iceland, revealed that due to the lack of systematic avalanche observation, this component of the model could not he completely specified from the existing records. The historical record of avalanching in Iceland contains a bias due to the emphasis upon events that caused damage to property and loss of life. These tend to he the larger avalanches. While the data from Iceland were considered appropriate for deterrnining the relative frequency of the very largest avalanches (size 4 and larger), as it was felt that events of this magnitude would generally run far enough down the path to be noted, an alternative means was required to derive the relative frequency of smaller avalanches. At Rogers' Pass and Revelstoke in British Columbia, there is a systematic record of avalanching. Avalanche observation is performed by the National Research Council and Parks Canada at Rogers' Pass, and The British Columbia Ministry of Transportation and Highways at Revelstoke. The detailed record includes an estimate of avalanche size with each recorded observation. Both the Revelstoke and Rogers' Pass datasets include over 5 000 avalanches. Because an avalanche control programme exists in these two areas, artificially triggered avalanches alter the natural distribution of events, causing an increase in the smaller and medium sized events at the expense of larger events. If artificially triggered avalanches are eliminated from the population one is left with a record of avalanching that permits one to determine the relative frequency of natural events. Figure 2.2 is a cumulative distribution function for the relative frequency of avalanches categorized by size at Revelstoke and Rogers' Pass. There is little difference between these two 28 Figure 2.2 : Cumulative distribution functions for the relative frequency of different sized avalanches at Revelstoke and Rogers' Pass, Canada. Table 2.2 : The proportion of avalanches of a particular size at Revelstoke and Rogers' Pass, British Columbia, Canada. Avalanche Size Percentage Frequency at Revelstoke Percentage Frequency at Roger's Pass Average Case (Percentage Frequency) 1 31.54 33.05 32.30 1.5 15.44 15.46 15.45 2 21.40 20.75 21.07 2.5 10.80 10.56 10.68 3 16.69 16.40 16.55 3.5 3.06 2.71 2.89 4 0.98 0.97 0.97 4.5 0.11 0.12 0.11 5 0 0 0 29 areas. Table 2.2 details the number of events per size category for both datasets as well as the average case. The fact that very few size 4 . 5 and size 5 events occur in these records is due to the management of the snowpack conditions. Avalanches will be artificially released by explosive before the snowpack conditions are such that large avalanches can be initiated. This does not cause too much of a problem for analysis because the Icelandic record is utilized for these larger events. However, it does mean that the percentage of larger sized avalanches in the distribution may be underestimated due to the increase in smaller events. This error is difficult to quantify because there is no means to determine the probable size of an avalanche if artificial release had not occurred. However, this was not felt to be too significant when compared to the overall error associated with using the Canadian record for representing the relative frequency of the smaller sizes. Use of the Canadian dataset could be justified because the quality of the record, coupled with the large sample size gives it a degree of authority that would not be obtainable from a less complete record obtained in a region with greater climatic similarities to Iceland. It is also true that the nature of avalanching varies across Iceland with areas such as Neskaupstadur tending to undergo major avalanche cycles less frequently than the West Fjords. With no data to support separate frequency distributions for different climatic regions of Iceland, a good 'average' distribution was considered advantageous, hence the use of the Canadian record for the smaller sizes. While the Canadian size classification is employed in British Columbia, this has not been the case in Iceland. Consequently, methods were devised for assigning Icelandic avalanches to a size class and hence to determine the relative frequency of different sized avalanches. 30 2.2.3 Estimating Avalanche Size From The Icelandic Record. The quality of the avalanche record in Iceland is variable. For the earliest avalanches one can only estimate maximum runout distance from historical documents. In the last decade a network of avalanche observers has been set up around the country and this has led to an increase in the amount of information about each event. For the more recent avalanches there are measurements of runout distance, deposit width and depth, deposit volume and fracture depth. It has not always been possible to record all of these variables due to a lack of visibility or because of snow pack stability concerns. However, where deposit dimensions have been measured, a 'typical' density value can be employed to elicit avalanche mass, thereby permitting avalanche size to be determined using the values quoted in table 2.1. McClung and Schaerer (1985) provide density measurements from avalanche deposits in Rogers' Pass. For dry avalanches the mean density was 330 kg/m3. Moist and wet avalanche deposits yielded higher densities. The present study is primarily concerned with dry avalanches as these have less frictional resistance with the bed surface and consequently travel longer distances. A density of 350 kg/m3 was felt to be representative of these events while not suggesting an unjustifiable level of precision. This value was used to convert avalanche volumes to mass. Because the Canadian size classification is based upon a logarithmic scale, any error introduced by using a single density value should not be significant. In many cases not all the deposit dimensions have been recorded. Commonly, the width and depth of the deposit have been measured, but deposit length has been unrecorded. To obtain an estimate of size for such events, a regression equation relating the product of width and depth with volume was employed. This was derived from the events where volume measurements had been taken. 31 2.2.4 Combining The Icelandic and Canadian Data Using the methods outlined in section 2.2.3, the Icelandic record yielded one size 5 avalanche, 8 size 4.5 events and 31 size 4 avalanche. As noted in section 2.2.2, size four was considered to be the lower bound at which one could expect the vast majority of avalanches to have been recorded. From the mean of the Rogers' Pass and Revelstoke distributions presented in table 2.2, one can see that 98.92 % of avalanches are size three and a half or less. Consequently the forty events from Iceland larger than size 3.5 make up 1.08 % of the record. Using this result, a distribution can be obtained that uses the Canadian record for the smaller sizes and the Icelandic data for size 4 and above. This is presented graphically in figure 2.3. The net effect of this distribution and the Poisson distribution for the frequency of avalanche occurrence is that once one has an estimate of the average number of avalanches a year upon a path, one can simulate a number of years of data, deterrnining how many avalanches occurred and the size of each event. The Poisson distribution gives the actual number of avalanche occurrences in a particular year, while the distribution illustrated in figure 2.3 permits a size to be assigned to these events. 32 Figure 2.3 : A distribution for the relative frequency of different avalanche sizes derived from a combination of Canadian and Icelandic data. 0.3 [• 0.2 1 ' ' ' 1 1 2 3 4 5 Avalanche Size 2.3 Distributions For Runout Distance 2.3.1 Data Collection Information on runout distance is crucial for the development of the risk model. Runout distance data, in combination with the frequency of occurrence of avalanches upon the path, permits a simulation model for runout to be produced. Data on runout distances and avalanche path profiles were obtained from a number of 1 : 5000 scale avalanche registration maps produced by Vedurstofa Islands (The Icelandic Meteorological Office). A simplified registration map for path number 2 at Fhnfsdalur (Tradargil), is shown in figure 2.4. The contour resolution upon this map increases for the lower part of the path where greater accuracy is required. An effort was made to construct the avalanche path profile in a manner that was appropriate for the motion of an avalanche downslope from the starting zone. Avalanches are likely to only be deflected by major terrain features such as hillocks or the sides of deep gullies, they are therefore less sensitive to the terrain than watercourses. This is because of the low friction of dry avalanches. Therefore, an appropriate profile is neither a straight line from the starting zone to the runout zone, nor one which precisely follows the sinuosities of streams initiating in the path, rather, it is a compromise between these extremes. From information on altitude and horizontal distance, a profile could be constructed. From the values for X p and Ax (see figure 1.5), it was possible to formulate a dimensionless runout ratio so that runout information for different paths could be compared. 2.3.2 Discussion Concerning the Appropriate Location of the Beta Point in Iceland. The most common value employed in the literature for the location of the beta point, is that place upon the avalanche path profile where the local slope angle attains ten degrees (Lied 34 and Bakkehei, 1980). This definition is used in figure 1.5. However, alternative values have also been suggested. McKittrick and Brown (1992) employ a value for the local slope angle of eighteen degrees. This was for the pragmatic reason that on the high frequency paths that they examined, it was rare for avalanches to attain the point in the path where the local slope angle was ten degrees. Their sensitivity analysis of slope angles from eighteen to twenty-six degrees suggested that of these angles, the eighteen degree point was the most appropriate. Butler and Malanson (1992) adopted a value of thirty-two degrees taken cautiously from the debris stream hterature (Hsu, 1975). When one is primarily concerned with large magnitude avalanches with high runout ratios, it makes sense to employ a beta point some way down the profile. As one moves down the profile, the correlation between the runout ratio model and the real data is likely to improve because the local topography in the region of the avalanche stopping point is being more accurately represented in the model, i.e. the delta angle (the angle between the alpha and beta points) is determined for increasingly shorter sections of terrain. For the beta point to be of physical significance, it should relate to that point upon the avalanche path where deceleration of the flow is initiated. This point should correspond in some general way with the angle of friction of the flow. However, Dent (1993) shows that if one assumes the avalanche to be a dense, rapidly sheared granular material, the dynamic coefficient of friction may vary with speed, particle-restitution coefficient and inter-particle friction. McClung (1990) and McClung and Mears (1995) suggest that for avalanche runout, the value of the dynamic friction coefficient may need to be varied along the runout zone. Therefore there is no unilateral means of determining a physically-based value for the location of the beta point. Consequently, the value selected should be chosen for the expedient reasons that it gives a high degree of fit to the Gumbel distribution and is compatible with existing studies. It also should 36 not be so far downslope that paths have to be excluded because the local slope angle is not attained before the avalanche reaches the sea. To determine the most appropriate positioning of the beta point for the Icelandic data, Gumbel distributions were fitted to avalanches with extreme runout from paths with at least 30 years of data in a similar manner to that of McClung and Lied (1987). (Appendix 3 provides some background to the Gumbel distribution). Local slope angles in two degree intervals from 6 to 22 degrees were used to define the beta point. Table 2.3 details the parameters, coefficients of determination and standard errors of these distributions. The data show the hypothesized trend of increased correlation with the Gumbel model as the local slope angle for the beta point declines. Figures 2.5 and 2.6 illustrate the best-fit distributions to the data for a twenty-two degree and fourteen degree beta point respectively. The improved fit with the lower beta point is evident. The assumption of a thirty-two degree angle as appropriate (after Butler and Malanson, 1992) would appear unlikely on the basis of this analysis. Instead it can be seen that from ten to eighteen degrees correlations remain high. A drop occurs at eight degrees, but at six degrees it has increased again, although not to the same level as the ten to fourteen degree region. The highest value for the coefficient of determination is obtained using an angle of fourteen degrees. However, the ten degree point provides an r2 value that is also high and also has a reduced standard error. The ten degree point is compatible with other studies, both those using the runout ratio approach [McClung et al (1989), Nixon and McClung (1993)] and those that adopt the alpha-beta model [Lied and Bakkelueti (1980)]. It also avoids the problem associated with lower slope angles, (that of paths running into the sea before the beta point is attained). For these reasons the ten degree point is chosen as the location of the beta point in this study. 37 Table 2.3 : Goodness-of-fit of extreme runout data to an extreme value distribution for various definitions of the beta point. Local Slope Coefficient of Angle Used to Determination Standard Error Scale Parameter Location Define the Beta (r2) Parameter Point (degrees) 22 0.753 0.547 0.635 0.736 20 0.819 0.353 0.578 0.546 18 0.950 0.067 0.225 0.420 16 0.956 0.060 0.215 0.352 14 0.981 0.034 0.189 0.244 12 0.977 0.030 0.153 0.192 10 0.973 0.029 0.136 0.123 8 0.927 0.044 0.120 0.074 6 0.957 0.035 0.127 0.020 38 Figure 2.6 : Extreme runout from Iceland for paths with at least thirty years of data. (The beta point is defined for a local slope angle of 14°). Reduced Variate 2.3.3 Distribution Fitting To Sizes Three to Four. Once all measurements of runout have been converted to runout ratios and a size has been allocated to each event, runout ratios can be grouped by size, and distributions fitted to each assemblage. As noted in section 2.2.4, only one size 5 and eight size 4.5 avalanches were recorded in Iceland, not a sufficient number of observations to permit a distribution to be fitted. This was also the case for events smaller than size 3. Seventeen size 2.5 avalanches were recorded along with seven size 2 and one size 1.5. No size one avalanches were recorded. The fact that there were only a sufficient number of avalanches to fit a distribution to three of the size classes, (sizes 3,3.5 and 4), was a major limitation of the final simulation model. However, it was fortunate that these were the three sizes available for accurate characterization because as section 3.6.3 shows, the simulation model is most sensitive to these avalanche size classes. This is because smaller events do not run far enough or generate enough force to pose a large threat to residences, while events larger than this are sufficiently rare that again they present a lesser risk. The best-fit distribution to the runout distances for these three size classes was a normal distribution, shown in figures 2.7, 2.8 and 2.9. The degree-of-fit is assessed by the chi-squared statistic where a low value ofp suggests an insignificant degree of congruence between the data and the Gaussian model. Skewness coefficients provided in table 2.4 show that the data are slightly negatively skewed. Values for the fourth moment of the distribution (kurtosis) are also provided in table 2.4. A normal distribution has a value for kurtosis of 3. Table 2.4 reveals that the size 3 and size 4 datasets are leptokurtic, while the size 3.5 data are slightly flatter than one would expect for the normal distribution. With no obvious trend in skewness or kurtosis across these three size classes, it was not felt necessary to modify more than the first two moments of the distribution when estimating the parameters for other sizes. 40 Figure 2.7 : Histogram of observed runout ratios for size 3 avalanches with a fitted normal distribution. 24 h -0.6 to-0.5 -0.4 to-0.3 -0.2 to-0.1 0 to 0.1 0.2 to 0.3 0.4 to 0.5 Runout Ratio Figure 2.8 : Histogram of observed runout ratios for size 3.5 avalanches with a fitted normal distribution. 18 <=-0.4 -0.3 to-0.2 -0.1 to 0 0.1 to 0.2 . 0.3 to 0.4 > 0.5 Runout Ratio 41 Figure 2.9 : Histogram of observed runout ratios for size 4 avalanches with a fitted normal distribution. 10 -0.5 t o - 0 . 4 -0.3 t o - 0 . 2 -0.1 to 0 0.1 to 0.2 0.3 to 0.4 0.5 to 0.6 0.7 to 0.8 Runout Ratio Table 2.4 : Runout ratio statistics for avalanches from sizes 3 to 4, with an assessment of fit to the normal distribution. Avalanche Size 3 3.5 4 Number of Events 69 61 31 Mean Runout Ratio 0.019 0.107 0.139 Standard Deviation 0.164 0.168 0.199 of Runout Ratios Momental Skewness -0.113 -0.113 -0.363 Kurtosis 4.733 2.754 5.140 Chi-Squared Value 9.40 5.03 3.89 /?-level 0.052 0.284 0.143 42 2.3.4 Distribution Estimation For Other Sizes. From the means and variances for these three fitted distributions, methods had to be devised to estimate parameters for other sizes where there were either no data or an insufficient quantity to permit distribution fitting. Observations at Rogers' Pass in Canada would seem to suggest that as size increases, runout distances are augmented in a logarithmic manner. Applying such a relationship to the mean values of the runout ratio (MR) and avalanche size (S) leads to equation 2.1. This has an r2 of 0.95. Application of this equation for all sizes yields the mean runout ratios given in table 2.5. MR = 0.969logS - 0.436 (2.1) For the second moment of the distribution, it was expected that the standard deviation would increase as the size increased and this is borne out by the values provided in table 2.4. A linear fit between standard deviation (SDR) and size had an r2 of 0.822. The equation obtained by least-squares regression was: SDR = 0.03451 + 0.057 (2.2) where SDR is the standard deviation of the runout ratios. The values derived from this relationship are also given in table 2.5. Figure 2.10 shows the distributions derived in this section. The increase in mean and variance with size is evident. 2.3.5 The Runout Simulation Model. The runout simulation model makes use the distributions listed in table 2.5, in combination with the relative frequency distribution that is given in figure 2.3, and the frequency of occurrence of avalanches upon the path. 43 Table 2.5 : Derived parameters for normal distributions of the runout ratio Avalanche Size Mean Runout Ratio Standard Deviation 1 -0.452 0.091 1.5 -0.276 0.108 2 -0.151 0.126 2.5 -0.054 0.143 3 0.025 0.160 3.5 0.092 0.177 4 0.150 0.194 4.5 0.201 0.211 5 0.247 0.228 Figure 2.10 : Normal distributions of avalanche runout distances for different size classes. (Sizes increase from left to right). Runout Ratio The procedure adopted was to determine the fraction of events that could attain a specific runout distance for each size class. These values were then multiplied by the proportion of all avalanches that were of this size. The frequency of occurrence was introduced as a final step. Figure 2.11 is a plot of the derived runout distributions. It is similar to figure 2.10 except the area of the distributions is now proportional to the relative frequency of the respective sizes. Because size 1 avalanches are some 1200 times as likely as size 5 events, the largest sizes are barely detectable upon the plot. However, large events are still very important. As one moves right along the x-axis of figure 2.11, (in the direction of increasing runout ratios), these larger avalanches make up an increasing proportion of the total number of events. Figure 2.12 shows how the percentage of avalanches reaching or exceeding a specific runout ratio varies. This plot combines the relative frequency and runout distributions, but it excludes the effect of frequency of occurrence. It can be seen that approximately 1% of avalanches exceed a runout ratio of approximately 0.33. For a path with a frequency of occurrence of one avalanche per year, 0.33 would consequently correspond to the one hundred year event; for a path with two avalanches a year, the same position becomes the fifty year event. Fdhn and Meister (1981) suggest that runout upon the Salzertobel path in Switzerland conforms to an extreme-value type I (or Gumbel) distribution. An inspection of their data and the profile for the Salezertobel avalanche path suggests that recorded avalanches upon, this path had runout ratios varying between approximately -0.2 and +0.2. One expects the simulation model for runout to demonstrate similar behaviour within this range of runout ratios. To test this hypothesis, five simulations were performed utilizing the size-runout distributions and the relative frequency distribution. Ten thousand avalanches were generated in each simulation. In each case avalanches with runout ratios between -0.2 and + 0.2 were separated and the parameters of the Gumbel distribution were determined by a regression 45 Figure 2.11 : Distributions for runout weighted by the relative frequency of occurrence of the different sizes. -0.8 -0.6 -0.2 0 Runout Ratio Figure 2.12 : Simulation model results for the percentage of avalanches attaining or exceeding a given runout ratio. 100 I I 5 Pi 0.01 46 procedure. The coefficient of determination was employed as a measure of goodness-of-fit. The mean r2 was 0.99 with a mean standard error of 0.01. Fdhn and Meister obtained an r2 of 0.93. Figure 2.13 illustrates a runout simulation for a general case, (without the introduction of a path-specific value for frequency of avalanche occurrence). With the x-axis scaled as a reduced variate, the Gumbel distribution should plot as a straight line, (see appendix 3 for explanation). Figure 13a highlights the region bounded by runout ratios of-0.2 and +0.2; while the calculated points plot as a slight curve, reasonable conformity to a Gumbel distribution is evident. Across all 10 000 values one can detect a more substantial curve (figure 2.13b). The steeper gradient of the events smaller than the zero runout ratio is similar to an effect noted by McClung and Mears (1991) for actual data of extreme runout. For land-use planning, one is usually concerned with avalanches that exceed the beta point because dense settlement occurs for slope angles less than 10°. Censoring the model simulation at a runout ratio of zero again leads to a set of data that may be accurately represented by an extreme value distribution (figure 2.13c). Thirty simulations each with 10 000 events yielded mean values for the scale and location parameters of the Gumbel distribution of 0.1028 and -0.1435 respectively when the data were censored at the zero runout ratio. The variability of parameters across the thirty simulations was low, having a standard deviation of 0.0026 for the scale parameter and 0.0056 for the location parameter. The mean r2 was 0.982, with a standard deviation of 0.0049. The mean values for the scale and location parameters are used to represent the Gumbel distribution for simulated runout throughout this thesis. 47 Figure 2.13a : A simulation highlighting the region where runout ratios he between -0.2 and +0.2. Reduced Variate Figure 2.13b : A complete set of simulated runout ratios. Reduced Variate Figure 2.13c : Simulated runout ratios censored at a runout ratio of zero. 2.3.6. Separate Model Formulations for Different Classes of Path Terrain. The runout simulation model derived above is a general case. Analysis in chapter 3 illustrates that the model provides a good general description of avalanche runout upon 'average' avalanche paths in Iceland. However, atypical paths exist and it would be advantageous to have separate formulations of the simulation model to represent these varying conditions, especially if one can isolate distinct path clusters within the dataset. The degree to which this can be attempted is dependent upon the underlying data. If one assumes that thirty data points are required to attempt to fit a statistical distribution, then with a maximum of 70 observations for one size class (size 3), perhaps no more than a single division is permissible (into long and short running paths). To perform such a separation, one needs to be able to establish a set of criteria to segregate the paths. Because this study is concerned with avalanche path terrain, topographic factors would be the obvious choice for these criteria. To this end, an attempt was made to correlate runout distances with four topographic factors in both a univariate and multivariate fashion. The parameters employed were: (1) The difference in altitude between the top of the profile and the foot (path height); (2) The average gradient of the profile (the angle from the alpha point to the starting zone); (3) The 2nd derivative of the best-fit parabola to the data as a measure of path shape (after Lied and Bakkeluai, 1980). (4) Path roughness as estimated by the goodness-of-fit of the best-fit parabola. Factor analysis was employed to amalgamate these four criteria into two factors that provided the maximum degree of cUscrimination between the data. Figure 2.14 is a plot of this factor phase space using the fifty paths for which analysis was possible. Instead of any distinct 50 Figure 2.14 : Plot of factor loadings for 50 avalanche paths. (Sites are coded by region and then path number, f = Flateyri; h = rlnifsdalur; i = Isafjdr ur; k = KirkjubbTshh ; n = Neskaupsta ur; p = Patreksfjordur; s = Sii avik; si = Siglufjbr ur. « l 0 -l -2 si9 si6 o si8 n2g •" h5 h4h6 h3 s2 ..pi.. o kS -2 -I 0 l Factor l Figure 2.15 : Scatterplots illustrating the degree of association between the parameters for a Gumbel distribution fitted to the path data, and factor loadings derived from topographic variables. Scale Parameter 0 Location Parameter -1 -0.5 0 0.5 Factor 1 Factor 2 51 groups, one appears to have a main group of paths with isolated exceptions spread about it. There does not appear to be any obvious basis for performing a separation of the data. Even if such groupings were to exist, one would still need to establish a relationship with runout for a division to be meaningful. Figure 2.15 provides scatterplots of the two factor loadings plotted against the scale and location parameters for Gumbel distributions fitted to runout data for sixteen paths. While some relationship between runout distance and topography is evident, at the most, topography was only able to explain 55 % of the total variance in the runout scale parameter. Fits to the location parameter were less successful. An examination of figure 2.15 shows that the plots are highly sensitive to outliers in the dataset. Prediction from such plots can only be performed with extreme caution. Accordingly it was felt that no legitimate segregation of the data could be performed on the basis of available evidence. Although somewhat disappointing, this result bears out the findings of Lied and Bakkeh0i (1980), who analysed a suite of topographic parameters and found that only the beta angle was able to predict runout adequately. The Icelandic data only permit a single simulation model of runout to be derived. For the next stage in the development of the risk model, this runout model must be combined with a simulation model for avalanche width to translate a profile-based analysis into three dimensions. 52 2.4 Distributions For Avalanche Width 2.4.1. The Nature of the Icelandic Data The approach used to develop the width simulation model was very similar to that used for the runout model. Distributions of avalanche deposit widths were derived for each size of avalanche. These were then combined with estimates of frequency to simulate widths. Avalanche paths in Iceland tend to be of an open nature, with very little confinement and with no trees in the runout zone. Avalanches are therefore free to spread out across the terrain. It was hypothesised that this degree of spread would increase with the size of the avalanche due to a greater volume of snow. Figure 2.16 shows that such a relation does indeed exist. During avalanche motion, the dimensions of the volume of flowing snow are in a state of flux. No existing model can accurately describe how avalanche width evolves and interacts with terrain. In this study the maximum width of the avalanche deposit was used to represent avalanche width. The width of the deposit can be determined with a high degree of precision compared to the width when the avalanche is in motion. It is also a variable that has been recorded in, or may be inferred from the historical record. Employing the maximum width of the deposit provides a conservative estimate of the area of terrain affected by the avalanche. The Icelandic record contains much more information on runout distance than on deposit width. From a description of which buildings have been affected it is possible to estimate the maximum width of the deposit, but this has a higher degree of uncertainty than runout distance analysis, where the maximum extent of the avalanche tends to be known fairly precisely. A comparison between the widths obtained from Iceland with those from the last ten years of records at Revelstoke in British Columbia reveals a notable difference. The median width at Revelstoke for a size 3 event is 33 m and is 80 m for size 3.5. For Iceland the comparable values 53 Figure 2.16 : Scatterplot of avalanche deposit width as a function of size. 1 1 C Y 6 v ft /• *v /• •\ l X 8 • < p 0 c 3 ? 8 j 9 ^ • 1.5 2.5 3 3.5 Avalanche Size 4.5 are 100 m and 150 m. For other sizes there are not enough recorded values in either the Iceland or Revelstoke dataset to permit a comparison. This discrepancy in recorded widths may be explained by the different terrain in the two areas. Avalanches at Revelstoke tend to occur in clearly demarcated avalanche tracks. Avalanches also tend to 'channel' through breaks in the forest cover. In Iceland, avalanches occur on open, unforested slopes and are free to spread. The median deposit depths at Revelstoke were 2.5 m for size 3 events and 4 m for size 3.5. For the Icelandic data, the median depths were substantially less, 1 m and 1.75 m respectively. Thus, the increase in deposit width in Iceland is compensated by a reduction in depth. 2.4.2. Fitting Distributions To The Width Data From Iceland. Instead of a normal distribution, the gamma distribution yielded the highest degree of fit to the data for sizes 3 to 4, (the three sizes with enough data to permit an attempt at distribution fitting). This probability density function of this distribution is: e~xxa~1 / ( * ) = f T T ^ x>0;a>0. (2.3) r ( a ) where a is the shape parameter for the distribution and T(a) is the gamma function. The properties of this distribution, including the evaluation of the gamma function are considered in appendix 4. The mean of the gamma distribution is given by a 10, where 0 is the scale parameter of the distribution. Similarly, the variance is given by a I 02. The degree-of-fit of the gamma distribution to the width data given in table 2.6 along with the distribution parameters. Figures 2.17 to 2.19 allow goodness-of-fit to be appraised subjectively. The degree-of-fit is heavily influenced by the fact that many widths can only be estimated to one significant figure. This effect is particularly noticeable for the size 4 data where 55 Table 2.6 : Deposit width statistics for avalanches from sizes 3 to 4, with an assessment of fit to the Gamma distribution. Avalanche Size 3 3.5 4 Number of Events 71 61 31 Mean Width 98.54 150.73 224.5 Standard Deviation of Widths 37.92 48.06 65.82 Momental Skewness 0.502 -0.345 0.332 Kurtosis 3.203 2.331 3.010 Shape Parameter 6.094 7.979 11.472 Scale parameter 0.0618 0.0529 0.0511 Chi-Squared Value 2.97 9.40 5.40 p-level 0.563 0.052 0.020 Figure 2.17 : Histogram of observed widths for size 3 avalanches with fitted gamma distribution. 56 Figure 2.18 : Histogram of observed widths for size 3.5 avalanches with fitted gamma distribution. 16 14 \-Maximum Width of Avalanche Deposit (m) Figure 2.19 : Histogram of observed widths for size 4 avalanches with fitted gamma distribution. the modal size incorporates the 200 metre wide events and secondary peaks exist for the 300 and 100 metre data classes. This effect significantly perturbs the skewness coefficient for size 3.5 avalanches. This lack of precision in the data results in a distortion that is reflected in the relatively poor fits to the larger sizes where width measurements are less accurately recorded. Both an exponential curve and an order 2 polynomial could be fitted perfectly to the values for the shape parameter (a) of the gamma distribution presented in table 2.6. An appropriate fit to the scale parameter (0) was more difficult to discern. A linear equation had an r2 of 0.874, but when combined with values for a was found to dramatically overpredict avalanche width means and standard deviations for the smallest sizes (see table 2.8). Data from Revelstoke in British Columbia yield a mean width for a size 1 avalanche of 6 metres and 10 metres for size 1.5. Use of the linear fit to 6 gave values of 50 and 55 metres respectively. For this reason it was decided to utilize an exponential fit to the values for the means and to then derive 9 by dividing the mean by a. This yielded values that were much more physically reasonable when compared to the Revelstoke and Icelandic data. There was little difference between employing the polynomial and the exponential relation to represent a. From table 2.8, it can be seen that if one utilizes the polynomial fit, larger widths can be derived for sizes 4.5 and 5 due to an increase in the variance. This leads to more conservative estimate of width for events with long return periods and consequently the polynomial fit was favoured. Equations 2.4 and 2.5 present the regression relations used to derive values for the shape parameter and the mean width (MW) respectively. As in equations 2.1 and 2.2, S represents avalanche size. Table 2.7 gives the derived values for a and 9, while figure 2.20 presents the distributions graphically. Table 2.8 allows one to compare width estimates derived using the 58 Table 2.7 : Estimated shape and scale parameters for gamma distributions of avalanche width. Size 1 1.5 2 2.5 3 3.5 4 4.5 5 Shape 14.63 10.08 7.15 5.82 6.09 7.98 11.47 16.57 23.28 Scale 3.621 0.590 0.201 0.094 0.062 0.053 0.051 0.050 0.049 Table 2.8 : Comparison between several formulations of the model and the Icelandic data. Favoured Distributions Distributions Icelandic Data Procedure for With Exponential With Straight Deriving the Fit To Shape Line Fit To Scale Distributions Parameter Parameter size mean sd* mean sd mean sd mean sd 1 - - 4.04 1.06 4.04 2.00 49.56 24.56 1.5 8.00 - 17.09 5.38 17.09 8.31 55.11 26.79 2 13.86 7.36 35.55 13.30 35.55 16.71 63.41 29.80 2.5 43.68 15.11 61.65 25.56 61.65 27.36 76.88 34.12 3 98.54 37.92 98.55 39.92 98.55 39.92 100.46 40.69 3.5 150.73 48.06 150.73 53.36 150.73 53.36 144.31 51.09 4 224.50 65.82 224.50 66.28 224.50 66.28 229.79 67.84 4.5 383.75 76.33 328.82 80.77 328.82 77.63 402.73 95.07 5 625.00 - 476.31 98.72 476.31 87.06 763.88 139.62 sd is the standard deviation 59 Figure 2.20 : Gamma distributions for avalanche deposit width for different size classes. (Sizes increase as one moves to the right). Avalanche Deposit Width (metres) 60 favoured form of the model with: the Icelandic data; the estimates obtained employing a linear fit to the scale parameter; and those from an exponential fit to the shape parameter. a = 28.541 - 17.1275 + 3.21552 (2.4) MW = -27.493 + exp[2.758 + (0.69351)] (2.5) 2.4.3 The Combined Effect of the Width Distributions. In section 2.3.5, the relative frequency distribution derived in section in 2.2.4 was combined with the runout distributions to produce a simulation model that gave percentage exceedance as a function of runout ratio. This simple approach is not possible for the width estimates because runout distance and avalanche width are correlated. As one moves down the avalanche path profile, the probability of observing an avalanche with a small width value is reduced since few small events have long travel distances. Consequently, the nine width distributions had to be amalgamated with the combined effect of the runout distributions and the relative frequency distribution in order to produce a simulation model for width. For a given runout ratio, the procedure used was to: (1) Determine the proportion of events from each of the nine runout distributions that yield avalanches with a runout distance at least equal to the prescribed runout ratio. (2) Multiply this figure by the frequency for each size of avalanche (i.e. employ the respective values of the relative frequency distribution from figure 2.3). (3) Find the sum of these nine products and use this value to re-standardize the new distribution. This gives the runout specific relative frequency distribution. (4) Couple the width distributions with this new relative frequency distribution using the same method adopted for runout in 2.3.5 to give the percentage exceedance for a particular avalanche width at a certain location down the profile. 61 A worked example is provided in table 2.9 for a runout ratio of+0.35. Avalanches smaller than size 2 make a negligible contribution to risk at this runout ratio. For the other sizes, the contributing proportion of the distribution is given in the second column of table 2.9. The third column lists the appropriate values of the relative frequency distribution, and the product of columns 2 and 3 is given in column 4. From the sum of these values it can be seen that on average, 0.77 % of avalanches reach this runout ratio. By employing this value to standardize the data, one can see from column 5 that approximately 45 % of the avalanches at this point in the path are of size 3 and 1 % are size 5. The new distribution in column 5 can be employed to allocate the proportion of different sized avalanches for a width simulation at a runout ratio of 0.35. 2.4.4 Avalanche Width and the Gumbel Distribution. The width model also conforms to extreme value statistics, although the degree-of-fit varies with position on the path. This result is expected from the fundamental model of extremes (Gumbel, 1958) and is therefore theoretically appealing. Table 2.10 summarizes the results of fitting Gumbel distributions to width simulations performed at various points along the avalanche path profile. Ten simulations were ran at each location. Figure 2.21 gives example simulations for three runout ratios. Figure 2.22 illustrates how the Gumbel distribution parameters are modified as runout ratio changes. From figure 2.21 it is evident that the conformity to a Gumbel distribution is not good at the very top of the path. This is due to the fact that the size 1 and 1.5 avalanches are very tightly distributed with low variances. However, this part of the path is not of concern for land-use planning. 62 Table 2.9 : Determination of the underlying relative size distribution for a width simulation at a runout ratio of 0.35 Avalanche Size Fraction of Distribution That Contributes to a Runout Ratio of 0.35 Relative Frequency of the Size Class Product of Columns 2 & 3 New Distribution 1 0 0.3229 0 0 1.5 0 0.1545 0 0 2 0.00003 0.2108 0.0006 0.0008 2.5 0.0023 0.1066 0.0245 0.0318 3 0.0212 0.1654 0.3506 0.4539 3.5 0.0721 0.0290 0.2088 0.2704 4 0.1515 0.0084 0.1271 0.1646 4.5 0.2389 0.0022 0.0518 0.0671 5 0.3264 0.0003 0.0088 0.0114 sum - 1 0.7722 1 Table 2.10 : Extreme value distribution approximations to the width simulation model for various runout ratios. (Means and standard deviations are derived from 10 simulations of 10 000 events). Runout Ratio Scale Parameter Location Parameter Coefficient of Determination Standard Error mean sd* mean sd mean sd mean sd -1 37.25 0.275 19.73 0.069 0.925 0.002 13.65 0.310 -1 (censored at the mean) 48.30 0.561 -0.56 0.654 0.993 0.002 4.49 0.627 -0.2 42.60 0.362 47.38 0.276 0.969 0.003 9.69 0.594 -0.1 45.04 0.215 56.82 0.248 0.978 0.002 8.72 0.344 0 48.33 0.517 67.43 0.236 0.979 0.003 9.02 0.813 0.25 61.62 0.470 95.36 0.432 0.983 0.001 10.47 0.422 0.5 86.57 0.478 148.06 0.325 0.994 0.001 8.53 0.410 0.7 93.98 0.576 225.82 0.452 0.991 0.001 11.66 0.458 1 87.57 0.763 334.81 1.047 0.984 0.001 14.39 0.542 sd is the standard deviation 63 Figure 2.21 : Simulated extreme value distributions of avalanche width for three different runout ratios. Figure 2.22 : The Gumbel distribution parameters for the width simulation as a function of position along the path. 1 • 1 • 1 + | • • • / -I + u. 1 1 • 1 + r , H • • . . . . . . . . . 100 9 0 80 70 60 eter e 50 « es PH 4 0 _o 13 o 3 0 20 10 0 -1.2 -1 -0.8 -0 .6 -0 .4 -0.2 0 0.2 0.4 0.6 0.8 Runout Ratio + Location Parameter • Scale Parameter As one moves down the path the fit of the uncensored model to the extreme value distribution increases, obtaining a peak at runout ratios of approximately 0.5. There is a decline in the degree of fit beyond this point as the simulation model becomes reliant upon fewer size classes, and hence tends towards the underlying gamma distribution. Of the 224 measured runout ratios for avalanches in Iceland, 188 he between runout ratios of -0.1 and 0.5. Table 2.10 and figure 2.21 lend support to the view that over this range of values, the extreme value approximation to the width simulation model appears to be valid. 65 2.5 Deriving A Frequency Of Avalanching Upon a Specific Path 2.5.1. Fundamental Problems With Frequency Evaluation From The Historical Record. Now that the simulation models have been derived, it is possible to return to the issue of avalanche frequency discussed in section 2.2.1. In particular, values for A, (the Poisson parameter for avalanche frequency) can be obtained. To do this, one needs an estimate of the total number of avalanches of all sizes that occurred upon a path in a given time period. Until very recently only major events were recorded in Iceland, leaving a record effectively censored at an arbitrary runout ratio resulting from the limit of settlement. It is clearly not appropriate to take the number of avalanches from this censored population as equivalent to the total number of avalanches upon the path. If it is assumed all avalanches greater than a certain size have been recorded, the remaining record can be simply filled in from knowledge of the relative frequency of different sized events (figure 2.3). However, it is runout position which has determined whether or not an avalanche has done damage and has consequently been recorded. Close to settlements, it is probably the case that all size 4 and larger avalanches have been recorded, but this is certainly not true for more remote paths. If the complete runout simulation model is used to estimate frequency, the existing model structure is being used to derive a component which should be independently determined. This circular logic may result in problems when validating the model. If frequency is derived from existing model elements, the model may appear to perform well and be internally consistent, but the physical basis will be weakened because the frequency is being optimized to best-fit the model. 66 2.5.2. A Method for Deriving a Frequency Estimate This issue of circular reasoning cannot be avoided altogether as the record must be extended somehow and the only means to do this is by using some of the model components. The aim must be to utilize the model as little as possible, and to only use those elements of the simulation model that can be validated in isolation of the frequency term. Separate validation ensures that conformation with real data when the two are used together is not just a property of model structure, but has real, physical meaning. The Gumbel approximation to the runout simulation model when the data are censored at a runout ratio of zero is dependent upon the underlying distributions, but may also be readily compared to available records of the distribution of avalanche runout upon actual paths. It is shown below that this distribution can be validated independently. This is not the case for the mdividual model components, (such as the normal distribution for the runout of size 1 avalanches), which have no direct basis upon data and consequently cannot be tested against reality. The mean values for the parameters of the Gumbel distribution for runout were given in section 2.3.5 as 0.1028 and -0.1435 for the scale and location terms respectively. As is discussed in chapter 3 and shown in figure 3.3, this value for the scale parameter appears to be a reasonable, but conservative estimate when compared to many of the avalanche paths where adequate records exist. These parameters of the simulation model can be considered to be appropriate for a path where 1 avalanche occurs each year. For paths with different frequencies, only the location parameter needs to be modified. This result is proven below. A means to derive estimates of avalanche frequency emerges as a consequence of this. 67 2.5.3 Introducing Avalanche Frequency to the Simulation Model. A conventional exceedance probability (pe) is simply the inverse of the return period (RP). The one hundred year flood has a probability of occurrence of 0.01 in a given year. This definition incorporates time. This is not the case for the simulation model where recurrence intervals are in terms of number of events, not years. To translate from one to the other while maintaining consistent dimensions, a frequency factor (X) is required with units of events per year. Pe(years) ~ j^p ~ Pe(events) • ^ (2-6) Since the runout ratios .conffom to a Gumbel distribution, a double negative-logarithmic transformation of the non-exceedance probabilities produces a reduced variate (RV) that is linearly related to the runout ratio (see appendix 3): RV = -ln[-lll(l- pe(years))] (2 7) RR = b.RV + u (2.8) The Gumbel distributions fitted to runout information for individual paths employ conventional exceedance probabilities as appear in equation 2.7. To compare the simulation model to the path data, the Poisson parameter (X) must be introduced to the model so that the exceedance probabilities are consistent (from equation 2.6). This gives the form of the model shown in on the left-hand side of 2.9. The right-hand side of this equation shows the model form with X incorporated into the distribution parameters. 0.1028 -In -In f v ^ ^ re(years) J 0.1435 = 0.1028{-ln[-ln(l - Pe(years))]} + « (2.9) The value for the scale parameter does not change because the Poisson parameter affects all exceedance probabilities to the same degree and the reduced variate is linearly related to the 68 The double negative logarithmic transformation of the probabilities gives a reduced variate that is linearly related to the runout ratio. A change in the value of X causes the reduced variate to be modified, but the new estimate of the runout ratio will be altered in proportion and thus no change in slope results. Therefore in equation 2.9, b = 0.1028 and the effect of a change in the value of X is wholly reflected in the location parameter. This result is of importance in that it permits vahdation of the simulation model independent of frequency. Because the scale parameter is not altered when translating from event-based exceedance probabilities to conventional exceedance probabilities, its value may be compared directly to the scale parameter for distributions fitted to path data. 2.5.4 Deterrning Average Avalanche Frequency - An Example. Consider a path where two events have been recorded, with runout ratios of 0.2 and 0.22, and where the larger occurred fifty years ago and the smaller more recently. Approximate return periods for the two events are fifty and twenty-five years, which yields values for the reduced variate of 3.902 and 3.199 respectively. These points can be plotted on an appropriate diagram (figure 2.23). Lines constructed from these points, with gradients equal to the scale parameter for the model (0.1028), permits values for u to be obtained, (-0.181 and -0.129 in this example). Rearrangement of equation 2.9 allows values for X to be derived: X = - (2.10) "(H+0.14353))] 0.1028 J l-ey For this hypothetical path, the fifty year avalanche yields an average frequency of 0.70 avalanches per year. The twenty five year event gives a value of 1.15 avalanches per year. In this 69 Figure 2.23 : Example procedure for obtaining a value for the Poisson parameter. 0.5 r 0.4 -0.3 -Reduced Variate case one would probably choose to work with the larger, more conservative estimate of frequency. This technique is used for deriving frequencies for different paths in chapter 3. Use of the runout model was favoured over the width model for two main reasons: The degree of precision of the width data was much less than the runout data. The stopping position of avalanches is known to within a few metres for many events, while quoted widths are often estimates to one significant figure. Such an error will be passed through to the estimates of u, and X. Secondly, because both the location and scale parameters of the Gumbel approximation to the width model vary, checking the vahdity of this model against data is less accurate than for the runout model. In the case of runout, a single value for the scale parameter is sufficient to describe the data. The scale parameter of the width model is dependent upon position and this makes validation problematic. Strictly speaking, each event should be represented by a separate distribution, with different values for both parameters. To circumvent this problem, in chapter 3, a model is formulated for the median runout ratio of the avalanches that occurred upon the path. While not strictly correct, this does allow a rough (and independent) means of checking that the values for X derived with the runout model are reasonable. The simulation models for width and runout, in combination with the Poisson distribution for avalanche frequency allow the encounter probability to be specified. The final stage in the development of the risk model is to provide values for exposure and vulnerabuity. Exposure is taken to be 1 when analysing damage to buildings because buildings are fixed objects. When concerned with the occupants of these houses, the exposure value used is 0.5. This is a rough estimate of the proportion of time that a person spends at home. 71 2.6 Vulnerability 2.6.1. Simplifying Vulnerability to a Manageable Level. Many different factors contribute to vulnerability and a generalized study cannot take into account all such elements. For specific applications, the engineer may wish to employ an avalanche dynamics model to calculate the impact pressure upon a building and then calculate through a knowledge of the construction materials, height, orientation and floor plan of the building a detailed vulnerability value. This level of detail is not appropriate for a study such as this where the aim is to characterise encounter probability and to provide general estimates of risk. The simulation models developed in the previous sections may be combined with a detailed vulnerability term at a later date if required. The most important part of the vulnerability term, (apart from the avalanche magnitude), is the nature of the materials used in construction. Figure 2.24 and 2.25 are photographs of two buildings struck by the January 16th 1995 avalanche at SuSavik. The first is a residence built of low quality materials without reinforcement. Two walls and the contents of the house were deposited some ten metres downslope. The second picture is of the local primary school, a reinforced concrete construction. It is located some 25 metres to the north-east of the building in figure 2.24, at a similar distance from the starting zone. Although the building was damaged, its structure remained essentially sound. The difference that particular construction methods introduce to the risk at a site is readily apparent from these photographs. 2.6.2. Establishing Relations Between Avalanche Size and Vulnerability. The simulation models outlined earlier in this chapter operate with avalanches segregated by size. This makes it relatively simple to incorporate avalanche magnitude into the vulnerability calculation. All that is required is a relation between the degree of damage and avalanche size. 72 Figure 2.24 : Damage to a residence in Sudavik from the 1995 avalanche. Figure 2.25 : Damage to the school in Sudavikfrom the 1995 avalanche. 73 This may be expressed in terms of damage to the building or as percentage lives lost. These relations can then be modified for different construction materials. Unfortunately, the amount of detailed information available on how avalanches of particular sizes affect structures is very limited. This is partly because size classifications are not widely adopted, and also because little basic research has been performed. Risk analysis is more widely developed in the study of earthquakes, where a number of studies provide example vulnerability calculations. However, even here many simplifications are routinely made. For example, Fournier D'Albe (1988) assumes that there is a simple power relation between specific loss (cost of repair expressed as a proportion of cost of replacement) and degree of damage (defined by the classes outlined in table 2.11). There are obvious differences in the nature of the stresses and strains that avalanches and earthquakes induce upon structures. It is also difficult to translate between earthquake intensity and avalanche size. Owing to these limitations one can only hope to provide a rough estimate of vulnerability. However, by using the information from Iceland on the degree of damage caused by avalanches, it is hoped that reasonable vulnerability values can be derived. 2.6.3 Vulnerability as Specific Loss For earthquakes in Romania, Fournier D'Albe (1988) uses a relation between degree of damage (DD) and specific loss (SL) of: SL = ±™L (2.1) 100 In other words, the cost of repairing a property that sustains heavy (class 4) damage is 64 % of the value of the residence. He then provides distributions that give the percentage frequency of 74 Table 2.11 : Degree of damage to buildings in Montenegro, 1979. Degree of Damage Phenomena Observed 1 (none) No visible damage to the structural elements; possible fine cracks in walls and ceiling mortar; barely visible non-structural and structural damage. 2 (slight) Cracks in wall and ceiling mortar; falling of large patches of mortar from wall and ceiling surface; considerable cracks in or partial failure of chimneys, attics and gable walls; disturbance, partial sliding, sliding and collapse of roof coverings; cracks in structural members. 3 (moderate) Diagonal or other cracks in structural walls, walls between windows and similar structural elements; large cracks in reinforced concrete structural members (columns, beams, reinforced concrete walls); partially failed or failed chimneys, attics or gable walls; disturbance, sliding and collapse of roof covering. 4 (heavy) Large cracks with or without detachment of walls, with crushing of materials; large cracks with crushed wall material between windows and similar elements of structural walls; large cracks with slight dislocation of reinforced concrete structural elements (columns, beams and reinforced concrete walls); slight dislocation of structural elements and the whole building. 5 (severe) Structural members and their connections undergo extreme damage and dislocation; many crushed structural elements; substantial dislocation of the entire building and damage to roof structure; partial or complete failure From Fournier D'Albe (1988) 75 events in each damage class for earthquakes of different magnitudes. This permits an overall average value of loss to be obtained for each size of seismic event. Employing a similar approach to the Suoavik avalanche of 1995, it was estimated that of the 22 houses struck, four could be classified as undergoing class 5 damage. Four more were allocated to class 4, three to class 3, four to class 2 and seven to class 1. The weighted average specific loss from this allocation of events was 39 %. A similar estimate for the 1995 Flateyri event gave a figure of 66 %. As table 2.1 shows, impact pressures of 30 kPa, (sufficient to destroy a house), do not arise until avalanches are of size 2.5 or more. If one assumes that for this size there is a simple negative exponential relation between degree of damage and percentage of buildings, the average specific loss is 12 %. Size 5 avalanches represent the opposite extreme. If a simple positive exponential relation is considered appropriate for this case, the average specific loss is 82%. This value may be considered a reasonable maximum value for the specific loss. Furthermore, if a size 1 avalanche is considered to be of insufficient force to cause any damage to a structure, a best-fit equation can be fitted to these 5 points to fill-in values for the other sizes. This leads to the average values for specific loss provided in the second column of table 2.12. 2.6.4 Vulnerability as Percentage Fatalities Before the 1995 avalanche, the 210 persons living in SuQavik occupied 70 houses. Therefore it is reasonable to suggest 66 people inhabited the 22 houses struck by the avalanche. In total 14 persons or 21 % of the inhabitants of the houses were killed. The fatalities were obviously concentrated in the few homes that bore the full force of the avalanche, but this is a useful average figure that may be validly employed for large-scale risk assessment. 76 In the case of the 1995 Flateyri event, a size 4.5 avalanche killed 20 of the 45 inhabitants of 19 buildings severely damaged by the avalanche. Altogether, some 26 residential buildings were struck by the event, suggesting that the figure for the percentage of fatalities from all buildings was some 33 %. Because avalanches smaller than size 2.5 are not sufficiently powerful to destroy a residence, it was assumed that for non-reinforced masonry buildings, no fatalities would arise for size 2 avalanches or smaller. The values given in the third column of table 2.12 were obtained by fitting a relationship to these fatality estimates. 2.6.5 The Effect of Using Stronger Materials in Construction. To allow for the fact that improved construction standards may be implemented in avalanche prone areas in Iceland, an attempt was made to calculate vulnerabihty functions for two types of building material. The values obtained in 2.6.3 and 2.6.4 are appropriate for Tow quality' materials. Reinforced concrete structures will obviously be more resistant to damage than such buildings. Fournier D'Albe (1988) provides data from Sandi and Vasilescu (1982) concerning the difference in vulnerabihty to earthquakes between reinforced concrete and low quality constructions in Bucharest. The data that is provided is comparable for three earthquake intensities. Converting the given values for degree of damage into specific losses reveals that on average, the loss values for the concrete are 60 % of those for the low quality buildings, with no obvious trend across the three intensities. This correction is used in this study to convert the vulnerabihty values for low quality buildings to those for reinforced concrete structures. The correction is applied to both the figures for specific loss and those for percentage fatalities. 77 Table 2.12 : Vulnerability expressed as specific loss or proportion of fatalities for two different construction materials. Low Quality Constructions Reinforced Concrete Structures Avalanche Size Specific Loss Fatalities Specific Loss Fatalities (Percentage) (Percentage) (Percentage) (Percentage) 1 0 0 0 0 1.5 3 0 2 0 2 7 0 4 0 2.5 12 3 7 2 3 20 7 12 4 3.5 30 13 18 8 4 39 21 24 13 4.5 66 33 40 20 5 82 50 50 30 78 2.7 The Risk Model In Practice ' ^ In this chapter simulation models for runout and avalanche width have been derived. The combined impact of runout, width and frequency of occurrence gives the encounter probability. Vulnerability values are given in table 2.12. When dealing with the risk to humans an exposure term (other than unity) is needed. The product of hazard, exposure and vulnerabihty gives a value for risk. Table 2.13 outlines the means by which the risk calculation proceeds. The path used is path 3 at Flateyri (Eyrarhryggur). The example calculation is for the specific loss to low quality constructions at a runout ratio of 0.20 and at 50 m from the centre of the path (a 100 m wide avalanche). The derived risk value of 0.00295 is roughly half the value for an avalanche of any width at this runout ratio (a risk of 0.00582). The risk model developed in this chapter must be thoroughly tested before it may be employed in practice. In the next chapter the model structure is vahdated and the sensitivity of the model components is tested. 79 Table 2.13 : An example risk calculation for Eyrarhryggur at Flateyri. (Runout ratio 0.20; width 100 m; average frequency of 1.802 avalanches per year; vulnerability as specific loss to low quality constructions). Column 2 Column 3 Column 4 Column 5 Column 5 Column 6 Avalanche Size Proportion of Each Size Exceeding a 0.20 Runout Ratio Relative Frequency of the Individual Size Classes Product of Columns 2 and 3 Proportion of Each Size Exceeding . a 100 metre Width General Encounter Probability (Product of Columns 4 and 5) Path Specific Encounter Probability (Product of Column 5 and 1,802) 1 0.0 0.32287 0.0 0.0 0.0 0.0 1.5 0.0 0.15453 0.0 0.0 0.0 0.0 2 0.0026 0.21082 5.44 x 10"4 0.0003 1.53 x 10"7 2.76 x 10"7 2.5 0.0374 0.10663 3.99 x 10-3 0.0805 3.21 x 10"4 5.78 x 10-4 3 0.1367 0.16536 2.26 x 10"2 0.4319 9.76 x 10"3 1.76 x 10"2 3.5 0.2707 0.02896 7.84 x 10"3 0.8322 6.52 x 10"3 1.12 x 10"2 4 0.3983 0.00839 3.34 x 10"3 0.9896 3.31 x 10"3 5.96 x 10-3 4.5 0.5022 0.00217 1.09 x lO - 3 1.0 1.09 x 10"3 1.96 x 10"3 5 0.5815 0.00027 1.57 x 1CT4 1.0 L57 x lO - 4 2.83 x W4 Sum 1 Column 7 Column 8 Column 9 Avalanche Size Vulnerability (Specific Loss to Low Quality Structures) Exposure for Buildings Risk (Product of Columns 6,7 and 8) 1 0.0 1 0.0 1.5 0.03 1 0.0 2 0.07 1 1.93 x 10"8 2.5 0.12 1 6.94 xlO" 5 3 0.20 1 3.52 x 10"3 3.5 0.30 1 3.53 x 10"3 4 0.39 1 2.32 x 10"3 4.5 0.66 1 1.30 x 10-3 5 0.82 1 2.32 x 10-4 Sum 0.011 80 CHAPTER 3 Model Validation And Sensitivity 3.1 Overview In the previous chapter the components of the risk model were derived. It is now necessary to test the relationship between the model and actual avalanche events. Because the simulation models were derived from an amalgamated dataset, they represent the situation upon an 'average' path. If all paths behave in a similar manner then the models may be applicable for all cases. However, it is more than likely that there will be certain paths which cannot be adequately simulated. These must be identified, and an attempt made to detennine the reasons for this atypical behaviour. Developing the model required a number of assumptions in order to surmount problems with the available dataset. A robust model will be relatively insensitive to perturbations in the model structure caused by alterations to these assumptions. If a small change to a distribution causes a disproportionately large modification to the overall simulation then the model may need to be reformulated to reduce this sensitivity. From validating the model against actual paths, estimates of the Poisson parameter (the mean frequency of avalanche occurrence) are derived. After examining the behaviour of extreme runout in Iceland, I discuss the vahdation and sensitivity of the two simulation models. Vahdation was performed by fitting a Gumbel distribution to the runout and width data from a path and comparing this to the distribution for the model. Model sensitivity was examined by adjusting the components underlying the simulation models and exploring how these changes perturbed the output. 81 3.2 Extreme Runout In Iceland Thanks to the work of Lied and Bakkehoi (1980), McClung et al (1989) and Nixon and McClung (1993) there exists a dataset of extreme runout for a number of different mountain ranges in the northern hemisphere. In all cases extreme runout appears to obey Gumbel statistics. Owing to differences in climate and terrain, the specific nature of the fitted Gumbel distribution varies for different regions. A similar effect is seen in the alpha-beta model with the regression parameters varying between mountain ranges. It was hypothesized a priori that the data from Iceland would fit a distribution with parameters most similar to those for the Coastal Alaska dataset (McClung and Mears, 1991), due to similarities in topography, climate and latitude between these two locations. Figure 3.1 illustrates the fit of an extreme value distribution to extreme runout in Iceland. Figure 3.2 is a histogram plot of the residuals. While there is a good fit between the data and the Gumbel distribution, and the residuals are normally distributed, as in all rank-order formulations, there is a significant degree of serial correlation in the residuals. The value of the Durbin- Watson (D) statistic is 1.065 - less than the tabulated lower bound on the statistic for a sample size of 45 at the 5 % significance level (1.48). This small D value means that successive residuals have values more similar than one would expect for a situation where no autocorrelation exists. While serial correlation introduces problems when estabhshing confidence limits upon a regression, the regression coefficients (the distribution parameters) are still unbiased estimators. Because it is the parameter values which are of interest here, serial correlations were not considered a significant influence on the results. Table 3.1 lists values for the scale and location parameters derived for Iceland as well as for other mountain ranges. It is evident that Coastal Alaska is the closest approximation to the Icelandic situation, showing that in a general way the historical runout data behave as expected. 82 Figure 3.1: Extreme runout in Iceland. Figure 3.2 : Distribution of standardized residuals for the fitted distribution. Table 3.1: Parameters for Gumbel distributions fitted to extreme runout from various mountain ranges Mountain Range Scale Parameter Location Parameter Canadian Rockies and Purcells (censored at /?=e_1) 0.07 0.079 Western Norway 0.077 0.142 British Columbia Coast Mountains 0.088 0.107 Coastal Alaska 0.108 0.185 Iceland 0.135 0.128 Colorado Rockies 0.202 0.288 Sierra Nevada 0.205 0.375 84 3.3 Validation of the Runout Simulation Model 3.3.1 Comparing the Parameters of the Simulation Model and Path Data. In chapter 2 it is noted that the simulation model for runout can he represented by a Gumbel distribution. For the simulation model for runout to be validated successfully, the scale parameter for the simulation model must He close to that for an 'average' path in Iceland, and value for the location parameter must be reasonable. The comparison of the Gumbel distribution parameters for the simulation model and the path data is shown in figures 3.3 and 3.4. Because the maximum number of events recorded upon a single path in Iceland is eleven, distributions fitted to the path data may have significant errors. To reduce this, the paths used in the validation process are those with at least 4 recorded avalanches and a fit to the Gumbel distribution of r2 > 0.8. The exception to this is path 2 at Sudavik, which is included due to the interest in this site following the avalanche of January 1995. Across figures 3.3 and 3.4, three runout ratio isolines are plotted. These represent the modelled runout ratio of the one hundred year event for avalanche frequencies of 4,1 and 0.1 events per year. On the basis of recent records, 4 events per year is the maximiun average frequency of avalanching upon Icelandic paths. Hence the solid line on figures 3.3 and 3.4 defines the limit of prediction of the model for the 100 year avalanche. The model parameters are such that reasonable, but conservative estimates of runout can be derived for most paths. The scale parameter of the model (0.103) is greater than the median for the seventeen paths (0.08). Hence, for an unit increase in return period, the model predicts a greater increase in the values of the runout ratio than occurs upon the average path. A conservative model was seen as advantageous because it is frequently true that the return periods of concern in risk studies are greater than the period of observation upon a path. 85 Figure 3.3 : Scatterplot of Gumbel distribution parameters for the model (one event per year) and various paths. Figure 3.4 : Scatterplot of Gumbel distribution parameters excluding the outlier paths. 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 * h i * >• + >^ s i . ^ * " . + • i l + + L5 » •n odei v-c : ^ + > • * "ji3 i 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 Scale Parameter Path Codes f - Flateyri h - Hnifsdalur i - Isafjordur k - Kirkjubolshlid n - Neskaupstadur s - Sudavik si - Siglufjordur 100 year event 1 event per year 100 year event 4 events per year 100 year event 0.1 events per year 86 3.3.2 Outlier Paths. It is clear from figure 3.3 that the model can provide conservative estimates of runout for all but three paths. The ski-lift path in Isafjordur (Isafjordur path 7) has already been noted to behave unusually due to the exceptional shape of its profile, (see figure 3.5). The events that exceed the bench in the profile and run very long distances increase the scale parameter of the fitted distribution. It is less obvious why path 18 at Neskaupstadur should behave unusually. However, avalanche frequency upon the east coast is much lower than on the west coast. McClung et al (1989) suggest that over a sufficiently long period (such as 100 years), the largest events upon paths with similar terrain subject to dissimilar climatic regimes, are inclined to travel comparable distances. Over short time intervals, because more avalanches occur upon high frequency paths, it is expected that the runout distance of the furthest travelling event would be greater than for a low frequency path. Therefore, the rate of change of runout distance with return period must be greater for the low frequency path. This is equivalent to stating that the scale parameter for low frequency paths is greater. It can be seen from figure 3.6 that there is a tendency for lower frequency paths to have larger scale parameter values. This effect is enhanced for path 18 at Neskaupstadur because it is located above the town centre and observation of small avalanches occurs more frequently. While the three lowest recorded runout ratios for the other 19 paths in this area are -0.36, -0.30 and -0.26, for path 18, the three smallest events have runout ratios of -0.59, -0.48 and -0.29. This occasional inclusion of very small avalanches distorts the fitted model. In figure 3.7 it can be seen that a fit to those avalanches with a runout ratio greater than zero, gives a distribution with parameters that are less unusual (a scale parameter of 0.21 and a location parameter of -0.34). 87 Figure 3.5 : Profile of the ski-lift path at Isafjor ur (path number 7). Figure 3.6 : Scatterplot of observed avalanche frequencies versus the fitted scale parameter § s> •q CD i3 O o s I i « OH CD s Q to •B 0 1 P i 0.16 0.12 0.08 0.04 Observed Frequency of Avalanching (Avalanches per Year) 88 Figure 3.7 : Runout for path 18 at Neskaupsta ur with the fitted model (dashed line). Figure 3.8 : Scatterplot illustrating the trend for the paths with more events to have a lower scale parameter for the fitted distribution. h 6 si6 + i6 n l 3 + h5 + £3 + s i + f4 + i3 + i2 + i l + h 2 + h i + k5 + 3 4 5 6 7 8 9 10 11 12 Number of Avalanches Observed Upon The Path Path Codes f - Flateyri h - Hnifsdalur i - Isafjordur k - Kirkjubolshlid n - Neskaupstadur s - Sudavik si - Siglufjordur Avalanche Path Path number 2 at Sudavik is also an outlier. The value for the location parameter is very high, meaning that avalanches with a relatively short return period run long distances. Observations by members of the avalanche division of the Icelandic Meteorological Office explain this unusual pattern. Magniisson (personal communication) reports that major avalanches occur upon this path when snow accumulation overflows a bench in the starting zone and triggers a release upon the slopes below. This loading from above causes runout distances to be drastically increased. If major avalanches result from the interaction of such rare phenomena, it is perhaps not surprising that the simple topographic analysis performed in section 2.3.6 in an attempt to discriminate between short and long running paths was unsuccessful. 3.3.3 Summary. The parameters for the runout simulation model appear reasonable from this initial analysis. With three notable exceptions, the 100 year runout can be represented using a realistic avalanche frequency . Figure 3.8 shows that there is a trend towards paths with more data (and therefore better quality records) yielding a lower scale parameter value. Hence, the scale parameter for the model gives a conservative estimate of extreme runout for those paths in which there is the greatest confidence in the fitted distribution. However, it is important to note that fourteen of the paths shown in figure 4 are from the West Fjords where avalanche frequency is high. While higher frequency paths tend to be of the greatest concern, they form a relatively small proportion of all paths. The parameters for the model may not be so appropriate in other areas of Iceland. Evidence for this comes from the fact that Siglufidrdur 6, Neskaupstadur 13 and Neskaupstadur 18 he outside the main cluster of points in figure 3.3. 90 3.4 Estimation of Average Avalanche Frequency. The runout model can be further validated by checking that avalanche frequencies derived from the model correspond in a general way with the observed frequencies. The method used to derive an estimate of the Poisson parameter for the frequency distribution of avalanching is outlined in 2.5.2. It was assumed for the West Fjords that observation since 1989 is of sufficient quality to record most size 2 or larger avalanches. Because these events form approximately 52 % of the complete population, (from section 2.2.4), the observed frequencies should be approximately half the derived frequencies. For example, at Flateyri, the derived /lvalues for paths 3 and 4 are 1.80 and 3.63 events per year respectively. Therefore, one would hope to see approximately 0.9 events per year for path 3 (Eyrarhryggur), and 1.8 for path 4 (Innri-Bagarhryggur). The observed frequency upon Innri-Bagarhryggur in recent years has been 1.6 events per year. Upon the adjacent path, only 2 events have been recorded in the last 9 years. There are three explanations for the low number of observed events upon this path compared to the number required to fit the model in an appropriate manner. The first is that due to the bowl-shaped starting zone for the Eyrarhryggur path, small events are more likely to he concealed from the observer than for hniri-Bagarhryggur. Another possible reason is that due to the inherent stochasticity of avalanche occurrence, few size 2 or larger events have occurred purely by chance. For a X value of 1.802, there is approximately a 39 % chance of no size 2 or larger avalanche in a given year. From the binomial theorem, this means there is a 2 % chance of 2 or fewer observable events in 9 years. Thirdly it may well be the magnitude-frequency relationship for avalanching upon this path differs significantly from the 'average' relations that the model is based upon. Avalanches upon 91 this path are less frequent, but travel further than normal. The fit between the model and the data (shown in figure 3.9i) is good, and thus accurate risk modelling is not affected by the use of a frequency that may not be strictly physically based. Table 3.2 shows that for many of the paths, the derived X values are in general agreement with the observed values. In six cases the observed frequency for size 2 and larger avalanches is greater than the estimated value. Nevertheless, considering the errors due to the different observation quality for different paths, and the problems of estimating the proportion of avalanches observed, only for path 2 at Isafjordur, (known as Hrafhagil) is this difference highly significant. The derived frequency was 0.374 events per year, but observations suggest this is too low. Figure 3.9f shows that applying the observed frequency of 0.857 events per year gives runout ratio estimates greater than those found in the historical record by about 0.1, a reasonable margin of safety. Use of the observed frequency may be justified on the basis that this path lies close to the house of the avalanche observer, and consequently a high percentage of events are recorded. Figure 3.9 shows the model fitted to the sixteen paths from figure 3.3 for which width and runout data were available. The model appears to provide a reasonable, yet conservative estimate of runout upon the majority of the paths in the West Fjords. Again, the three outlier paths from figure 3.3 (Isafjordur 7, Neskaupstadur 18 and Sudavik 2) are problematic, as is the path from Kirkjubolshlid . This latter path is the fifth steepest of the 78 paths analysed. Consequently, small avalanches travel relatively long travel distances, giving a small scale parameter for the fitted distribution. 92 Table 3.2 : Comparison of frequencies of avalanche occurrence derived from the model and the historical record. Derived Frequency Of Derived Frequency For Observed Frequency Path Name Avalanching Avalanches Of Size 2 For The Last Seven (events per year) Or Larger Years F4 3.630 1.888 1.60 SI 2.562 1.332 0.571 H2 1.904 0.990 0.857 F3 1.802 0.940 0.220 H6 1.740 0.905 0.143 11 1.602 0.833 0.857 HI 1.253 0.652 0.714 H5 0.731 0.380 0.429 13 0.60 0.312 0.143 Si6 0.548 0.285 0 K5 0.466 0.242 0.286 12 0.374 0.194 0.857 16 0.159 0.083 0.143 N13 0.143 0.143 0.074 93 a. a D m >. a< £ o* i s § i § 2 O 2 O 13 •: ^ \ * i \ V * ' ' : ' " 1" *\": M M X •a O o <N i - i O O O . - H o o o o I i " i . X . i ! ; ; ; ' ; . • m r s i—i o o o O 1-H o o <N CO o o o •c I oi oij^a i n o n t r g a a. •2 S ~ S -3 8 o ^ 13 ft // •a i 95 I l l I I I • • I I I I I o «-> O -5 o o o © 97 3.5 Validation of the Width Simulation Model 3.5.1 Comparison of the Scale Parameters for the Model and the Path Data. The parameters for a Gumbel distribution fitted to the simulated events for a width simulation vary downslope, because width is dependent upon position. This prevents a direct comparison of the width model parameters with path data as was performed in figures 3.3 and 3.4. However, if it is assumed that the median runout ratio is representative of the runout ratios recorded upon a path, it is still possible to fit a Gumbel distribution to the width data and compare the parameters with those for the model fitted to the median runout distance. Figure 3.10 presents three box-plots of scale parameter values. The first is derived from the least-squares fitting of Gumbel distribution parameters to the width data for the seventeen paths with the most extensive records. The second is similar, but excludes data from the three outlier paths of section 3.3.2. The final box-plot (the model) was obtained by using figure 2.22 to assign appropriate parameters for the width simulation model to the median runout ratio for each path. The change of parameter values downslope in the model represent the shift in the distribution that occurs upon an 'average' path with varying runout distance. In the case of the data, variability is introduced from both changing runout distances and the varying nature of each path. Consequently, the data show much greater variability in values for the scale parameters. As such, only the median values upon these plots may be compared. These are indeed similar^  suggesting that the width model is reasonable. However, a complete validation requires the location parameter to also be inspected. If the width and runout models are consistent with one another, in the majority of cases, the same frequency should be applicable for both the runout and width simulation models. 98 Figure 3.10 : Box-plots of the scale parameters obtained from path data and from apphcation of the model to the median runout ratios for each path. CD re OH re o 1/3 From Data From Model Data (No Outliers) Maximum Minimum • Median; 75% & 25% 99 3.5.2 Avalanche Frequency and Location Parameter Vahdation. Figure 3.11 presents graphs for 16 paths, with in each case the data and the best-fit distribution shown, along with a distribution derived from the width simulation model. The model parameters are set at values appropriate for the median runout ratio in the data, and the X values given in table 3.2. For path 2 at Siifiavik and path 7 at Isafjbrdur a frequency of 10 events per year is employed. Table 3.3 lists the parameter values for the distributions shown in these plots. The width simulation model does not seem to represent the data as well as the runout model because widths tend to be recorded to just one significant figure, resulting in lower quality data than for the runout simulation model. Also, when the path data contains a range of runout ratios, the median runout ratio will not yield width model parameters appropriate for all events. However, in general the frequency values derived for the runout model provide a conservative estimate of avalanche width, suggesting that the two simulation models are compatible. Figures 3.1 lb, 3.1 l i and 3.111 are cases where the frequency value that is appropriate for the runout model appears to be too great for the width model. A correction factor was introduced to account for the fact that these paths tend to produce avalanches that are narrower than expected for an avalanche of a specific runout distance. These correction factors are listed in table 3.4. These adjustments are shown as dotted lines upon figure 3.11. For path 6 at Isafjordur, avalanches do not tend to run very far, or occur very often, but do tend to be wider than expected. This is because avalanches here initiate from an unconfihed slope and consequently are not channelled by topography. A positive correction factor appears to be necessary for this path. The three paths that are outliers in the runout analysis also appear to be problematic with respect to their width distributions, this is particularly true for path 2 at Sudavik and path 18 at Neskaupstadur. This suggests that either the events upon these paths do not conform to a Gumbel 100 cn Q . " T O -g > T J C CD " O O E xz T3 a TO TO TO " O SZ > CD C CD CD CD _Q C o to TO CL E o O o o CD rs O ) Li-ra Q o > o o 5 ° a •— CO ej O O » 2 2< / / / •c > 3 •o (ui) i u p i M 3 J •a o _ § o II o c s o o ~ - CO o •"3-H -a — o E 2 / / / •a 3 i ' f 03 •c a > 04 (ui) tppiM o f—• d T3 o 2 / •c > 3 -a (ui) inp!A\ (ui) inpiM 102 103 S e3 O <U tL, CU — t/J V. s 3 0 3 Q o T "8 z E 2w / > 3 T3 C U—» c o O CO <D c n (ui) m p i M ~ o £ 2 3 CO Q o H 3 O oo c •3 3 "o X /// 3 oo o T O II O > 3 Di 3 « Q o / o O 2 II o ex 3 ca u 2 \ V -\ \ \ 3 -o Di o o o o o o o o (ui) irjpiyYv o c <3 (— >-•• O « OH o. I 2«SJ / o 00 d II r~ o ^ ts ° 2 cO g E > T3 CD O 3 T3 <U Di O O o o o o o o o o o o o o <i~i m rs ~ (m) m p i M (ui) mpi/A Table 3.3 : Parameters for width distributions fitted to 16 paths. Distribution Fitted to the Data Model Distribution Path Name Scale Parameter Location Parameter Scale Parameter Location Parameter Flateyri 3 114.8 -109.4 57.0 121.1 Flateyri 4 66.0 25.1 54.9 154.8 Hnifsdalur 1 34.2 76.0 55.2 97.0 Hnifsdalur 2 40.7 18.5 52.0 111.5 Hnifsdalur 5 46.5 28.0 50.3 57.6 Hnifsdalur 6 26.4 35.4 61.4 129.0 Isafjdr ur 1 73.7 -61.3 47.5 87.8 f il Isafjor ur 2 43.9 -4.8 48.3 19.9 r * Isafjor ur 3 14.16 148.1 55.3 56.5 I fjdr r 6 103.8 -157.5 46.0 -24.8 Isafjor ur 7 112.3 112.3 56.8 217.9 Kkkjubolshli 5 57.2 -153.8 47.4 28.8 Neskaupsta ur 13 55.9 -91.0 47.0 -27.8 Neskaupsta ur 18 140.8 -102.3 43.7 97.9 Sii avik 1 80.9 -16.9 54.2 133.6 Suavik2 153.9 -55.1 60.5 232.8 Table 3.4 : Correction factors applied to the location parameter of the width model when adopting the frequency of avalanching appropriate for runout. Path Name Correction Factor Introduced To The Width Model Flateyri 4 - 50 metres Hnifsdalur 1 -75 metres Hnifsdalur 2 -100 metres Hnifsdalur 6 -100 metres Isafjdr ur 2 -50 metres Isafjor ur 6 +150 metres Kirkjubolshli 5 -100 metres 105 distribution, or that the extreme events, (whether largest or smallest) in the record have true return periods substantially different from those which could be estimated from the data, (significantly greater for the large events and considerably reduced for the smaller events). Such an effect can be seen for path 1 at Sudavik (figure 3.1 lm) as well as Eyrarhryggur at Flateyri (figure 3.1 li). In the case of the former, if the largest event (which appears to be an outlier) is excluded, the scale and location parameters are adjusted from 81 and -17 to 58 and 21 respectively. This latter value for the scale parameter lies close to the value of 54 for the fitted model. At Eyrarhryggur, exclusion of the smallest recorded width modifies the parameters from 115 and -109 to 88 and -22. These alternate fits to the data are shown as dashed lines in figures 3 . l l i and 3.11m. If the correction factors are introduced to account for some paths producing wider or narrower avalanches than the 'average' path, then with the exception of the three notable outlier cases and path three from Isafjordur (where only two different widths have been recorded), all the test paths can be modelled in an acceptable manner. 106 3.6 Sensitivity of the Simulation Models 3.6.1 Overview. To explore the sensitivity of the simulation models to perturbations in the underlying distributions, several types of adjustments to the size-width, size-runout and relative frequency distributions were introduced: (1) Increasing the mean and standard deviations of the size-runout (and size-width) distributions by 10 %, both separately and together. These changes allow one to detennine the relative sensitivity of the model to the moments of the Gamma distribution. (2) Increasing the first two moments of each individual size-runout or size-width distribution by 10 %, to determine the sensitivity of the overall model to different sizes. (3) The distributions for sizes 3, 3.5 and 4 are based upon data and thus there is little justification for altering then moments until more data are available. Therefore, 10 % and 25 % changes were introduced to the other six size-runout or size-width distributions to evaluate the sensitivity of the model to the methods used in sections 2.3.4 and 2.4.2 to derive these distributions. (4) Two types of adjustment were also made to the relative frequency distribution. In the first, the proportion of avalanches of size 2.5 or smaller was reduced by 10 % while the proportion of the larger sizes was increased by a similar margin. The inverse set of changes were performed to give the second modified distribution. The effect of these four types of changes are discussed below. For the analysis of each type of perturbation, 30 simulations of 10 000 events were used to produce stable results. In all cases, fit to the Gumbel distribution remained high at T2 = 0.97 or 0.98, and the variance about the mean parameter values for the thirty simulations was low. 107 Figure 3.12 : The effect of increasing the moments of the size-runout distributions by ten percent. 2 3 4 5 6 7 standard deviation Reduced Variate Figure 3.13: The effect of increasing the moments of the size-width distributions by ten percent. 400 Reduced Variate 108 The precise form of the width simulation varies downslope because the relative frequency distribution is a function of position. To eliminate this source of variability, throughout this section analysis of the width model is performed at a zero runout ratio. 3.6.2 Sensitivity to the Mean and Standard Deviation. Figures 3.12 and 3.13 illustrate the effect of a 10 % increase in the mean and standard deviations of the size-runout and size-width distributions respectively. The runout simulation model is most sensitive to changes in the standard deviation, while the width model is most sensitive to changes in the mean values. However, both models appear to show similar levels of sensitivity. For the runout model, increasing the mean by 10 % causes a 1 % increase in runout for the 1 : 200 event, while the 10 % perturbation to the standard deviation results in an 8 % increase in runout. When the two changes are combined, runout increases by 9.5 %. Analysis of the width model for the 1 : 200 event shows that a 10 % increase to the mean causes a 6.8 % increase in the width, while a similar perturbation to the standard deviation causes a 2 % change. The combined impact of the two modifications is a 9.4 % increase. 3.6.3 Sensitivity to Individual Sizes. From figures 3.14 and 3.15 it is evident that the model is most sensitive to those sizes for which data are available, a very reassuring result. In fact, with the exception of size 4.5 avalanches for the width model, other sizes cannot be distinguished from the original model for the range of return periods shown (a reduced variate of 4.5 represents the 1 : 90 event, while a reduced variate of 7 is the 1 : 1100 event). 109 Figure 3.14: Sensitivity of the runout model to a 10 % increase in the mean and standard deviation of each size-runout distribution. 0.6 4.5 5 5.5 6 6.5 7 ^ S i z e 4 Reduced Variate Figure 3.15: Sensitivity of the width model to a 10 % increase in the mean and standard deviation of each size-width distribution. 450 i • • • • • - i 400 S 350 300 250 4.5 5 5.5 6 Reduced Variate 6.5 \ Original Model •-. Size 3 \ Size 3.5 Size 4 size 4.5 Both models are most sensitive to size 3 avalanches, followed by size 3.5 and then size 4. However, the runout model is especially sensitive to size 3 events, while the width model shows more similar levels of sensitivity to each of the three sizes. Figure 3.14 shows that for high return period events, the runout ratio of size 3 events is increased by 0.025 with alO % change to the parameters. With a typical horizontal distance from the starting zone to the beta point (ATp) of 1000 meters, this translates to a 25 metro displacement in the simulation, a relatively low perturbation and certainly within the range of a reasonable margin of safety. 3.6.4 Sensitivity to The Distributions Obtained By Extrapolation of Parameter Values. Figure 3.16 shows three simulations where the distributions for sizes smaller than size 3 and greater than size 4, have been perturbed by ten and twenty five percent. For a 25 % increase to the two moments, the runout ratio is augmented by just 0.01, (some 10 meters for a typical path). The runout model is consequently insensitive to the regression procedure used to arrive at these distributions. A similar conclusion can be arrived at for the width model. Figure 3.17 shows that increasing the moments of the extrapolated distributions by 25 % causes a change in the simulation of less than 15 meters. 3.6.5 Changes to the Relative Frequency Distribution. Because the relative frequency distribution was formed by grafting the Icelandic data onto that from Canada in a simplistic manner, it was felt that substantial error might be introduced from this element of the simulation models. However, as is shown in figure 3.18, the runout model appears to be relatively insensitive to modifications of the relative frequency distribution. Increasing the relative frequency of the larger sizes by 10 % results in a model that yields runout 111 Figure 3.16 : Sensitivity of the runout simulation model to a perturbation introduced to all sizes excluding 3, 3.5 and 4. Figure 3.17: Sensitivity of the width simulation model to a perturbation introduced to all sizes excluding 3, 3.5 and 4. 450 400 S 350 o P 5 5.5 6 Reduced Variate Original Model Model plus 10 % Model plus 25 % 112 Figure 3.18: Sensitivity of the runout simulation model to modifications of the relative frequency distribution. Figure 3.19: Sensitivity of the width simulation model to modifications of the relative frequency distribution. 450 4 5 5 5 5 g 6 5 7 sizes increased Reduced Variate ratios that are larger by approximately 0.012 (or roughly 12 meters on the ground), compared to the original model. The width model (figure 3.19), is less sensitive to an increase in the relative frequency of the larger sizes, with just a 0.7 % increase in the width model for the 1 : 200 event (reduced variate of 5.3). The same modification induces a 3 % increase in the runout model for the 1 : 200 event. 3.6.6 Summary. In conclusion it would appear that both simulation models are relatively insensitive to perturbations in their underlying components. Sensitivity is greatest for the avalanche sizes with available data. The major difference between the models is that the runout model is more sensitive to the variance of the size-runout distributions, while for the width model, there is greater sensitivity to the mean of the size-width distributions. The form of both models is such that there is very little sensitivity to those sizes for which the distributions were extrapolated by regression techniques. This is very reassuring as it was felt a priori that this would be a major source of error. While both models are most sensitive to size 3 avalanches, it was necessary to use the Canadian data to determine the relative frequency for this size. The magnitude of the correction resulting from this assumption is difficult to discern. To help eliminate this assumption, it is advisable that in the future accurate records of avalanching in Iceland extend to the measurement of all avalanches of size 2.5 and larger. An improved dataset may also permit the determination of different relative frequency distributions for the east and west coasts, which should improve the ability of the model to represent paths in places such as Neskaupstadur. 114 3.7 Sensitivity of the Risk Model Modifications to the simulation models are obviously transmitted to the final risk model. In this section I explore the combined impact of perturbations to the runout and width models upon the estimates of risk. The vulnerabihty and exposure terms are the same as for the original model, to isolate the changes to the simulation models. Two types of changes were introduced to both models: a 25 % increase to the means and standard deviations of the extrapolated runout and width distributions (see section 3.6.4); and the adjustment to the relative frequency distribution that caused an increase in the estimates of runout and width (see section 3.6.5). Such changes yield risk values that are less conservative than those obtained from the original model. Figures 3.20 illustrates the difference between the original and modified risk models for runout ratios from 0 to 0.5, and deposit widths from 0 to 400 m. The two types of constructions behave in a similar manner because the vulnerabihty values for reinforced buildings were derived by a simple linear scaling of those values for non-reinforced structures. Therefore only graphs for the non-reinforced buildings are presented here. From this diagram, it is clear that the difference between the models increases as the runout distance and deposit width increase. The probability of an avalanche occurring with a runout ratio of 0.5 and a width of 400 m is 0.006 %. For such an event, the risk values differ by a factor of 4.00 for specific loss and 3.81 for fatalities. An event with a runout ratio of zero and a width of 100 m is more common, occurring once in every 15 events on average. For this avalanche, the altered model increases the risk by a factor of 1.45 for specific loss and 1.40 for fatalities. 115 < <! 5 8 <a -o o £ o 6 C '51b •c o <u J3 •c O c 0 •c 1 O C J O CN CD < <3 S m II 0 1 8 2 £ 2 £ 3 — CN 3 O c o T 3 CD u, CD CD T3 O co CD X ) o u c I CD >> CD a, CD .s S •C >> O fe II & J O o ' o CD a. 116 Thus, it appears that even for very rare events, the significant perturbations introduced to the altered model are insufficient to adjust the risk values by an order of magnitude. It may be concluded that the risk model is relatively insensitive to the underlying model structure. 117 CHAPTER 4 Conclusion In this thesis a model has been developed for determining avalanche risk in Iceland. Many of the paths in the West Fjords can be modelled accurately. Paths where the model does not appear appropriate have been duly noted. In the case of the factory path at Su avik, (path 2), and the ski-lift path at isafjdr ur, (path 7), recent avalanches have reached the sea and the base of the valley respectively. Consequently it is recommended that no constructions should be located in these paths. Avalanche tracks upon the east coast, (such as path 18 at Neskaupsta ur), are interesting cases. The historical record shows that few avalanches occur for extended periods. However, occasional major avalanche cycles, (such as the 1974 events) cause major damage. Thus, while the year-to-year frequency is relatively low, large magnitude events have similar return periods as those in the West Fjords. This leads to large scale parameters for the Gumbel distributions fitted to the data for these paths. It is these major avalanche cycles which are of concern for risk studies upon the east coast. Because the return period of these events is similar to that for avalanches in the north-west, fitting the model to these extreme events is likely to provide a reasonable representation of high return period avalanches. However, the low magnitude avalanches will not be described accurately and the derived avalanche frequency will not be physically based. Ideally, intensive observation of avalanches upon the north and east coasts should eventually permit a separate model to be derived for these areas. Validation of the model has necessarily focussed upon the paths where the most data are available. For the other paths, it is suggested that if the observed avalanche frequency since the 118 late 1980s is doubled, a reasonable frequency value can be obtained. If no avalanches have been recorded in this time, or observations have not been performed, a frequency of one event per year is likely to provide a safe, conservative estimate of risk. In the following pages, risk maps are presented for three paths in the West Fjords. The risk shown is for fatalities in buildings constructed from low quality materials. Figures 4.1 and 4.2 show three paths, Eyrarhryggur and Innri-Baejarhryggur at Flateyri, (paths 3 and 4), and Tra argil at Hnifsdalur, (path 2). Avalanches upon a particular path do not all follow the same trajectory. This can easily be seen from an avalanche registration map such as figure 2.4. Thus it is advisable to only apply the width model towards the flanks of the path and to define the risk at the centre of the path as equal to that derived solely from the runout simulation model. In order to produce the risk contours shown in figures 4.1 and 4.2, a computer program was written in Microsoft® Excel Visual Basic® macro language. This routine is listed in appendix 6. The program defines two limit runout ratios of -1 and 2 and a risk value is determined for each, as well as for the runout ratio lying halfway between them (0.5). The computer then iteratively searches for the appropriate runout ratio for the specified risk level and width. In figure 4.1 it is evident that both the paths converge upon the town of Flateyri. Therefore the actual risk is the sum of the risk values determined for the individual paths. The risk levels that the residents of these towns are exposed to are unacceptable. Fell (1994) quotes risk values provided by Reid (1989) for a number of occupations and natural hazards. Some of these are listed in table 4.1. Risk in this table is in terms of the proportion of fatalities for exposed persons. It is therefore directly comparable to the values in figures 4.1 and 4.2. Residents of some of the houses in the West Fjords appear to be exposed to levels of risk that are comparable to that for people participating in offshore oil and gas exploration, and sports such as parachuting. 119 Fell (1994) states that in Australia, communities appear willing to accept a voluntary risk (e.g. a risk to their homes) in the order of 10"3. If this value is also considered acceptable by governments and public bodies, then it is evident that fairly widespread measures must be taken in north-west Iceland for protecting residences. These actions may take the form of house relocation (proposed for path 2 at Su avik), or the construction of defence structures, (proposed for Flateyri). While such procedures are costly, the financial expense can be clearly justified on the basis of the risk values from the model developed in this study. 120 Figure 4.1 : Sample risk map for Eyrarhryggur and Innri-Baejarhryggur at Flateryi • Risk = 0.01 N • • • • R i s k = 0.001 121 Figure 4.2 : Sample risk map for Tradargil at Hnifsdalur 600 10 — — — - Risk Contours 122 Table 4.1 : Risk statistics for persons exposed voluntarily or ^voluntarily to various hazards (Reid 1989). Cause Risk (x IO"3) Building Hazards: Structural failure of building (UK.) 0.00014 Natural Hazards: Hurricane (U.S.A) 0.0004 Lightning (U.S.A) 0.0005 Earthquake (California) 0.002 General Accidents: Road Accidents (U.S.A.) 0.3 Occupations: Offshore Oil and Gas Exploration (U.K.) 1.65 Deep-sea fishing (U.K.) 2.8 Sports: Scuba Diving (U.S.A.) 0.42 Parachuting (U.S.A.) 1.9 All Causes: Whole Population (U.K.) 12 Woman aged 30 (U.K.) 0.6 Man aged 30 (U.K.) 1 123 BIBLIOGRAPHY Ahrens J.H. and Dieter U. 1974. 'Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions.' Computing 12 223-246. Bakkehjai S., Domaas U. and Lied K. 1983. 'Calculation of snow avalanche runout distance.' Annals of Glaciology 4 24-29. Banks X, Carson J.S. and Nelson B.L. 1996. Discrete-Event System Simulation. 2nd ed.(New Jersey: Prentice-Hall). 548 pp. Benjamin J.R. and Cornell C. A. 1970. Probability, Statistics and Decision for Civil Engineers. (New York: McGraw-Hill). 673 pp. Blom G. 1958. 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Walsh S.J., Butler D.R., Brown D.G. , and Bian L. 1990. 'Cartographic Modeling of Snow Avalanche Path Location Within Glacier National Park, Montana.' Photogrammetric Engineering And Remote Sensing 56 5 615-621. Watt W., Lathem K.W., Neill C.R., Richards T.L. and Rouselle J. 1989. Hydrology of Floods in Canada: A guide to planning and design. (National Research Council of Canada) 245 pp. 130 Appendix 1 : Avalanche Size Classifications Various methods for determining avalanche size have been suggested in different parts of the world. These are briefly reviewed below: The United States : Avalanches are classified into five sizes, but this classification is performed relative to the path in question. This makes comparison between paths problematic and as such has hmited utility in a study such as this, where avalanche records from different paths must be compared. Switzerland : Fdhn et al (1977) propose a three class system with categories of small sluffs, medium events, and large avalanches covering an area greater than 50 000 m 2. McClung and Schaerer (1993) question whether this method yields enough classes for effective Discrimination and also express uncertainty about the use of area covered by the avalanche as the critical variable for size detennination. Japan: Several systems have been proposed in Japan, but none are used operationally. A common theme in the Japanese classifications is the use of a logarithmic scale, the detemrining variable varying with the selected system, be it avalanche potential energy (Shoda, 1965); or mass, or kinetic energy (Shimizu, 1967). 131 Canada: The Canadian system is used in this study. It is directly concerned with the impact of avalanches upon their surroundings and thus easily integrated with risk. Five sizes are used, and with the addition of half sizes, there are sufficient classes to discriminate effectively between avalanches with different destructive potential. Because this classification is used operationally, datasets that use this system are readily available. Table 2.1 details the definitions of the various classes. 132 Appendix 2 : The Poisson Distribution If one considers a sequence of n Bernoulli trials where the outcomes from an experiment are binary and mutually independent, and the probability of a successful outcome remains unchanged during the experiment, the probability mass function (PMF) of the total number of successes Y is given by: PY (y) yi(n-y)i -py(i-p) n-y y = 0,l,2,...,n (al) Where p is the probability of a successful outcome and y represents a specific number of successes. The PMF outlined here is known as the binomial distribution. If there is a negligible probability of two events occurring in the same time period t, and the probability of one event occurring in the period t is given by p, the total number of successes Y in n = t trials is given by the binomial distribution. As t tends to zero, n tends to infinity and p tends to zero. Because the total number of successes 7 is unchanged, the product oip and n must remain constant. If one represents this constant by v, the PMF of the binomial distribution may be rewritten as: PY 00 = fv\y( yi(n-yV-l -\n. 1-V nJ (a2) nl K n) (n-y)\ny<\-v/y v / n' yi V V n n(n -1)(« - 2) . . . (« - y +1) n(l-v/)Y The (1 - vin )" term simplifies to e'y in the limit and the term in braces will reduce to unity due to the fact that it contains^ terms in both the numerator and denominator and these terms 133 approach n for large values of n. This result yields a new PMF that represents the situation where one cannot identify discrete trials, it is just known that many such trials are occurring. This is known as the Poisson distribution after S.D. Poisson who provided a derivation of the result in 1837. The probability mass function is given by: vye~y pr(y) = —— (a3) y • This distribution has just a single parameter that is equal to both the mean and the variance. This makes for a very versatile distribution which is very simple to apply. Due to the dependence of the values of n and p on the chosen time interval as shown above, there are advantages to writing the PMF as: Py(y) = ^—L-, (a4) yi where X is known as the Poisson parameter. A Poisson process is a stochastic process, the outcomes of which conform to the Poisson distribution. The physical mechanisms that underscore the incidents with which one is concerned, must satisfy three assumptions: 1. Independence. The number of incidents in a time interval is independent of the number of incidents that occur in any other non-overlapping time interval. 2. Stationarity. The probability of an event in a short interval from time t to time t + h is approximately Xh, no matter the value of t. 3. Nonmultiplicity. The probability of two or more events in a short period of time is considered negligible in comparison to Xh. 134 These assumptions can be considered to hold for the determination of the frequency of occurrence of avalanches upon a path, consequently the Poisson model provides a legitimate representation of this phenomenon. The average rate of arrival of a Poisson process is given by the value for X. This parameter will vary in the value it takes for different avalanche paths, as it is a function of many factors including path gradient, path shape, climate and aspect. The fact that frequency of occurrence can be considered as a Poisson process provides the basis for distribution choice in section 2.2. 135 Appendix 3 : The Extreme Value Type I Distribution Early work on the statistics of extremes may be attributed to Frechet (1927), Fisher and Tippett (1928) and von Mises (1936). Since the 1940s a major contribution to this field has been made by Gumbel (1958). The extreme value type I, or Gumbel distribution has been found to be of much use in the investigation of hydrological phenomena. Discussions of its use appear in many standard hydrological texts such as Chow (1964), Haan (1977) and Watt et al (1989). The use of this distribution in avalanche studies has been much more limited. However, two main types of application can be detected. The first and most common may be attributed to McClung and Lied (1987). They characterise extreme avalanche runout in a mountain range, by formulating statistics based on the furthest running events on a number of different avalanche paths. A number of subsequent studies have used this method to derive fits to the extreme value distribution for several mountain ranges (McClung et al, 1989; McClung and Mears, 1991; McKittrick and Brown, 1992). The alternative use of this distribution may be attributed to Fdhn and Meister (1981) who fit the record of avalanche runout along a single path to this distribution. The Gumbel distribution occurs when any distribution of an exponential type, (such as a normal or log-normal distribution), converges to an exponential function as the value of the random variable X increases. The probability density function (PDF) of this distribution is given by: b (a5) 136 where u and b are parameters of the distribution. The u term represents the mode of the distribution, while b provides a measure of dispersion. These parameters may therefore be interpreted as location and scale parameters respectively. The cumulative distribution function (CDF) of the extreme value type I distribution is given by: (—) F(x) = ee- * (a6) The mean of this distribution is given by u + b.y, where y is Euler's constant (0.57721...). The standard deviation equates to (71 / 60'5) b. If one formulates a term known as the reduced variate (7) with a value for u of 0 and for b of unity, the CDF of the reduced variate is simply: F(Y) = e~'-r (a7) From this expression it can be seen that a specific value for the reduced variate is equal to - In [- In (p)], where p is a particular quantile of the distribution. Consequently, the reduced variate linearizes the extreme value distribution: X p =u- b[- In(- In p)] = u + b. Yp (a8) To make use of this distribution, one needs some means of obtaining values for the two parameters. There are several approaches that one may use to do this. A common method is resolution through method of moments (Chow, 1964; Ghiocel and Lungu, 1975). However, a simple, stable method that is the most used in the avalanche literature is a procedure which is based upon the fitting of a least-squares regression line to the relationship given in equation a8; (McClung and Lied, 1987). For this method to be applied successfully, a means of relating the probability of occurrence to the variable of concern is required. This problem amounts to attempting to estimate F(x,) for a set of n finite, ranked observations i. The most common plotting formula that is used to 137 do this is the Gumbel or Weibull plotting position F(x,) = i / (n + 1). This was shown by Gumbel (1958) to have general applicability regardless of the distribution of the variate. Another early formula is that attributed to Hazen (1930) [F(x,) = (i + 0.5) / «]. Several other formulae include that of Gringorten (1963): [F(x,) = (/ - 0.44) / (n + 0.12)], and that of Kimball (1960): [F(x,) = (i -0.375) I (n + 0.25)]. All these formulae, and indeed most that are used in the hterature, follow the form identified by Blom (1958), which is indexed by a parameter a. This formula is given as F(x,) = (j! - a) I (n + 1 - 2a). A thorough analysis of the issues embodied in this debate are given by Cunnane (1978). The paper concludes that the best estimator for F(x,) depends upon the distribution under consideration. For the extreme value type I distribution, the Gringorten plotting position is considered advantageous. This is all very well when dealing with large or uncensored samples, but McClung and Mears (1991) show that this method is not necessarily the most appropriate when dealing with smaller samples. Their analysis yielded a new expression that was nearly independent of n and was superior to other plotting formulae. This expression [F(x,) = (/' - 0.4) / n] is used in this study for determining the distribution parameters for extreme runout and extreme width. 138 Appendix 4 : The Generation of Random Variates Overview: A random variate is a pseudo-random number, that has a probability of being produced that conforms to the probability density function of a specific statistical distribution. To produce a random variate, it is presumed that there is an available source of uniformly distributed random or pseudo-random numbers. The 'pseudo' prefix indicates that if one is employing a known algorithm to create these numbers, the results are theoretically replicable and consequently are not truly random. However, to simplify discussion, the 'pseudo' prefix is rarely employed in this dissertation. It will be tacitly assumed that the widely accepted methods that are discussed here, do produce number sequences that may be considered to be effectively random. Uniformly Distributed Random Numbers: There is a large literature that discusses the acceptable methods of producing a sequence of numbers that appear to be independent, with no autocorrelation, and where each number has the same likelihood of occurrence (hence the conformity to a uniform distribution). Law and Kelton (1991) provide a review of these techniques. The approach that appears to be the most widely employed is the linear congruential method proposed by Lehmer (1951). It may be expressed as a sequence of integers X\, X2,... according to: XM = (kXi + c) modm i = 0,1,2,... (a9) Where H s a constant multiplier, c determines the increment, m is the modulus that determines the range of random numbers, and X0 is known as the seed. A division of Xx by m produces a uniformly distributed random number (R\) between 0 and 1. 139 A potential problem with this approach is that since each X, is an integer in the range { 0 ,1,2 m -1}, each R\ value is discrete across the interval from 0 to 1. However, one requires a continuous set of values. This problem appears to be avoided by setting m to a very large integer. A common value for m is 231-1. This is a prime number and induces a period to the chain of random numbers of m - 1 values before the sequence is recycled. Such a large periodicity may be required for the testing of very stable systems, where many numbers must be generated to adequately explore the states of the system Once a sequence of uiiiformly distributed random numbers are obtained, one may progress to the introduction of the distribution of concern. The manner in which this is done is dependent upon the nature of the distribution to which the random numbers must conform. Two methods are discussed here, the direct transformation for the normal distribution and the acceptance-rejection technique for the Gamma distribution. The discussion of the direct transformation necessitates a preliminary engagement with a further method, the inverse transform technique. The Inverse Transform Technique: This technique is the most simple method for sampling from the exponential distribution and is also of utility for the triangular and Weibull distributions. It is not necessarily efficient computationally. If the CDF of the required random variable X\s set equal to a uniformly distributed random variable U, through simple rearrangement one may obtain an expression for X in terms of U which may be used to derive the required sequence of random numbers. Taking the exponential distribution as an example, the CDF of this distribution is given as: F(x)=l-e~*x ,X>0 (alO) 140 where X is the Poisson parameter discussed in appendix 1. Setting alO equal to U and rearranging gives: X . = ^ - / / i ( l - t f , ) (all) A, Consequently one may generate a set off' random numbers that conform to an exponential distribution by applying this transformation to / uniformly distributed random numbers. The Direct Transformation For The Normal Distribution: The cumulative distribution function for the standard normal distribution is given by: * 1 — F(x)=^—=e2dt - o o < x < o o (al2) !2TV Box and Muller (1958) describe a direct transformation that derives an independent pair of normal variates with a mean of zero and a variance of 1 for this distribution. If the values of two standard normal variables N\ and N 2 are employed as Cartesian co-ordinates to create a point A in an N\-N2 space, the point may be assigned polar co-ordinates such that: TV, =J? cost? (al3a) N2=Bsm6 (al3b) Where B is the distance of the point from the origin O, and 0 is the angle between the Ni axis and OA. The formula B2 = N 2 + N 2 2 has a chi-square distribution with 2 degrees of freedom. This is equivalent to an exponential distribution with a mean of 2. Random variates that conform to the exponential distribution may be simply obtained using the inverse transform technique. These exponentially distributed variates may then be further transformed to fit a normal distribution. 141 If the uniformly distributed random variable R is set equal to the CDF of the exponential distribution that describes the random variable B2, rearrangement expresses B2 in terms of R. R = l - e 1 B 2 (al4a) B 2 = - - ln(l-R) (al4b) Ifi? is uniformly distributed between 0 and 1, it follow that this is also true for 1 - R. Therefore exponentially distributed variates B\ can be obtained by replacing 1 - R with R and taking the square root. Bi=(-liaRiy-' (al5) The angle 0 is uniformly distributed between 0 and 2n radians due to the symmetry of the normal distribution. Because B and 0 are mutually independent, equation al3 may be re-expressed as Nt = (-2 In R,)"5 cos (2TVR2) (al6a) N2 = (-2 In R,)°S sin (2TCRZ) (al6b) Where the variates are derived from two independent random numbers R\ and R2. Normal variates X; with a mean p, and a variance a 2 may then be obtained by application of: Xi=n + aZ. (al7) This technique for deriving normally distributed random variates is employed in this study when modelling the runout relations for avalanches of different sizes. Acceptance-Rejection And The Gamma Distribution: The acceptance-rejection procedure is used to condition the distribution of a generated variable in such a way that it conforms to a specific distribution. Boolean operators are used to 142 ascertain if the variable is acceptable or not. The technique is only efficient when the number of rejections per acceptance is relatively low. The method used in this study to produce random numbers that conform to a gamma distribution is the algorithm GC of Ahrens and Dieter (1974), and it appears to be widely employed in the literature (e.g. Knuth, 1981; Press et al, 1992). This algorithm operates as an acceptance-rejection process and produces values that conform to a standard gamma distribution with the probability density function: e~xxa " 1 /(*) = Try- x>0;a>0. (al8) r ( o ) Where a is the shape parameter for the distribution and T(a) is the gamma function, which is evaluated as (a - 1)! if a is an integer, or more generally by the definite integral QO T ( f l ) = J e u u a l d u (al9) 0 The mean of the gamma distribution is given by a 10, where 0 is the scale parameter of the distribution, initially taken to be equal to 1. The variance of the distribution is a 102. Non-standard distributions, those with a scale parameter equal to a value other than unity, are determined by dividing each number generated from the standard distribution by the scale parameter. To use this algorithm, one requires two uniformly distributed random numbers R\ and R2. One then establishes two parameters J and K. J= t a n ^ . T r ) (a20) K = (jyl2a-l) + a-l (a21) 143 If the value of the parameter K is less than zero, one restarts, employing new values for Ri and Rj. If this first stage is passed successfully, a new expression H is formulated: H--(J>+1)J^^') 0*2, If// is greater than the second uniform variate R2 the value is accepted. Otherwise this run of the procedure is rejected , the uniform variates are re-derived and one returns to the start of the procedure. The acceptance and rejection criteria are such that this procedure leads to a number series that conforms to the gamma distribution. 144 Appendix 5 : Microsoft® Visual Basic® Macro For The Simulation of Risk (Subroutine names are given in italics; comments on the code are in bold type;) Definition of Variables (Statements proceed as 'Dim variable-name' on separate lines). 'Dim' statements are defined for the following variables: RRvalue; widthvalue; exposure; vulnl; vulnl5; vuln2; vuln25; vuln3; vuln35; vuln4; vuln45; vuln5; propRR; propRRl; propRR15; propRR2; propRR25; propRR3; propRR35; propRR4; propRR45; propRR5; propwl; propwl5; propw2; propw25; propw3; propw35; propw4; propw45; propw5; epl; epl5; ep2; ep25; ep3; ep35; ep4; ep45; ep5; riskl; riskl5; risk2; risk25; risk3; risk35; risk4; risk45; risk5; totalrisk; positionl; position2; position3; position4; position5; position6; RunoutRatio; depositWidth; simulatedwidth; displaywidth; loopl; targetrisk; newmarker; marker 1; marker2 Four constants which take values specified by the user. Frequency specifies the Poisson parameter for the frequency distribution vulnerabilitytype 1 = specific loss to low quality constructions vulnerabilitytype 2 = proportion of fatalities in low quality constructions vulnerabilitytype 3 = specific loss to concrete constructions vulnerabilitytype 4 = proportion of fatalities in concrete constructions adjustment factor = values specified in table 3.4 Xbeta is the horizontal distance to the beta point. It allows runout ratios to be translated into real distances. Const Frequency = 1.9044 Const vulnerabihtytype = 1 Const adjustmentfactor = 0 Const Xbeta = 1006 Sub RisksimulationlQ These two parameters are defined by the user, they represent: the target value for the risk; and the simulated value for the deposit width targetrisk = 0.01 depositWidth = 100 The initial values for the markers are set to extreme values for runout marker 1 = -1 marker2 — 2 The variable "RunoutRatio" is set to the lower of the marker values RunoutRatio = marker 1 Set the value of the width to display in the final output. displaywidth = depositWidth 145 Introduce the adjustment factor to the width calculation. depositWidth = depositWidth + adjustmentfactor If this gives a negative width then reset to zero width. If depositWidth < 0 Then simulatedwidth = 0 Else simulatedwidth = depositWidth End If Risk Calculation vulnerability proportionrunout relativefrequency widths encounterprobability risk Show error message if this risk value cannot be simulated. If totalrisk <= targetrisk Then message The variable "RunoutRatio" is set to the higher of the markers. RunoutRatio = marker2 Risk Calculation vulnerability proportionrunout relativefrequency widths encounterprobability risk Show error message if this risk value cannot be simulated. If totalrisk > targetrisk Then message Search for the runout ratio corresponding to the target risk value. searchengine End Sub Sub vulnerabilityQ This subroutine assigns the values from table 2.12 to each size class and also establishes the value for exposure. If vulnerabihtytype = 1 Then vulnl = 0: vulnl5 = 0.03: vuln2 = 0.07: vuln25 = 0.12: vuln3 = 0.2: vuln35 = 0.3 vuln4 = 0.39: vuln45 = 0.66: vuln5 = 0.82: exposure = 1 Elself vulnerabihtytype = 2 Then vulnl = 0: vulnl5 = 0: vuln2 = 0: vuln25 = 0.03: vuln3 = 0.07: vuln35 = 0.13: vuln4 = 0.21: vuln45 = 0:33: vuln5 = 0.5: exposure = 0.5 Elself vulnerabihtytype = 3 Then vulnl = 0: vulnl5 = 0.02: vuln2 = 0.04: vuln25 = 0.07: vuln3 = 0.12: vuln35 = 0. vuln4 = 0.24: vuln45 = 0.4: vuln5 = 0.5: exposure = 1 Elself vulnerabihtytype = 4 Then vulnl = 0: vulnl5 = 0: vuln2 = 0: vuln25 = 0.02: vuln3 = 0.04: vuln35 = 0.08: vuln4 = 0.13: vuln45 = 0.2: vuln5 = 0.3: exposure = 0.5 End If End Sub Sub proportionrunoutQ In this subroutine the proportion of avalanches of each size that attain the current value for the runout ratio are determined. Set RRvalue = Worksheets( 1). Cells( 1,1) RRvalue. Value = RunoutRatio Set propRRl =Worksheets(l).Cells(2, 1) propRRl.Formula = , ,=l-NORMDIST(al,-0.45207,0.091328,TRUE) , , SetpropRR15 = Worksheets(l).Cells(3, 1) propRRl 5.Formula = , ,=l-NORMDIST(al,-0.27597,0.10842,TRUE)" Set propRR2 = Worksheets(l).Cells(4, 1) propRR2.Formula = "=l-NORMDIST(al,-0.15103,0.125511,TRUE)" Set propRR25 = Worksheets(l).Cells(5, 1) propRR25.Formula = "=l-NORMDIST(al,-0.05411,0.142603,TRUE)" SetpropRR3 =Worksheets(l).Cells(6, 1) propRR3.Formula = "=l-NORMDIST(al,0.025072,0.159694,TRUE)" SetpropRR35 = Worksheets(l).Cells(7, 1) propRR3 5.Formula = ,,= l-NORMDIST(al,0.092022,0.176785,TRUE)" Set propRR4 = Worksheets(l).Cells(8, 1) propRR4.Formula = "=l-NORMDIST(al,0.150016,0.193877,TRUE)" SetpropRR45 = Worksheet^ l).Cells(9, 1) propRR45.Formula = "=l-NORMDIST(al,0.201171,0.210968,TRUE)" SetpropRR5 =Worksheets(l).Cells(10, 1) propRR5.Formula = , ,=l-NORMDIST(al,0.24693,0.22806,TRUE) ,' End Sub Sub relativefrequencyQ The proportions obtained above are then multiplied by the relative frequency of occurrence of the different sizes. propRRl.Value = propRRl.Value * 0.32287 propRR15.Value = propRRl5.Value * 0.15453 propRR2.Value = propRR2. Value * 0.21082 propRR25.Value = propRR25.Value * 0.10663 propRR3.Value = propRR3.Value * 0.16536 propRR35.Value = propRR35.Value * 0.02896 propRR4.Value = propRR4.Value * 0.00839 propRR45.Value - propRR45.Value * 0.00217 propRR5.Value = propRR5.Value * 0.00027 Set propRR = Worksheets(l).Cells(l 1, 1) propRR. Value = propRRl.Value + propRR15.Value + propRR2. Value + propRR25.Value + propRR3.Value + propRR35.Value propRR. Value = propRR. Value + propRR4.Value + propRR45.Value + propRR5. Value End Sub Sub widthsQ The proportion of avalanches that attain the current width value are determined independently from the runout distance. Set widthvalue = Worksheets(l).Cells(l, 2) widthvalue. Value = simulatedwidth Set propwl = Worksheets(l).Cells(2, 2) propwl.Formula = "=l-GAMMADIST(bl,14.6289,0.276188,TRUE) , , Set propwl5 = Worksheets(l).Cells(3, 2) propwl5.Formula = , ,=l-GAMMADIST(bl,10.084,1.695145,TRUE)" Set propw2 = Worksheets(l).Cells(4, 2) propw2.Formula = •,=l-GA]VlMADIST(bl,7.1466,4.974526,TRUE),, Set propw25 = Worksheets(l).Cells(5, 2) propw25.Formula = , ,=l-GAMMADIST(bl,5.8167,10.59854,TRUE)M Set propw3 = Worksheets(l).Cells(6, 2) propw3.Formula = , ,-l-GA]V[MADIST(bl,6.0943,16.17074,TRUE) , , Set propw35 = Worksheets(l).Cells(7, 2) propw35.Formula = "=l-GAMMADIST(bl,7.9794,18.88934JRUE)" Set propw4 = Worksheets(l).Cells(8, 2) propw4.Formula = ''=l-GAMMADIST(bl,11.472,19.56943,TRUE)" Set propw45 = Worksheets(l).Cells(9, 2) propw45.Formula = , ,=l-GAMMADIST(bl,16.5721,19.84149,TRUE) , , Set propw5 = Worksheets(l).Cells(10, 2) propw5.Formula = ,,= l-GAMMADIST(bl,23.2797,20.46039,TRUE) , , End Sub Sub encounterprobability^) Values for encounter probability are determined as the product of the two proportion values and the average frequency of avalanching. epl = propwl.Value * propRRl.Value * Frequency epl5 = propwl5.Value * propRRl5.Value * Frequency ep2 = propw2.Value * propRR2. Value * Frequency ep25 — propw25.Value * propRR25.Value * Frequency ep3 = propw3.Value * propRR3.Value * Frequency ep35 = propw35.Value * propRR35.Value * Frequency ep4 = propw4. Value * propRR4. Value * Frequency ep45 = propw45.Value * propRR45.Value * Frequency ep5 = propw5.Value * propRR5.Value * Frequency End Sub Sub riskQ Risk is the product of encounter probability, exposure and vulnerability. Total Risk is the sum of the risk values derived for each size class. riskl = epl * exposure * vulnl riskl5 = epl5 * exposure * vulnl5 risk2 = ep2 * exposure * vuln2 risk25 = ep25 * exposure * vuln25 risk3 = ep3 * exposure * vuln3 risk35 = ep35 * exposure * vuln35 risk4 = ep4 * exposure * vuln4 risk45 = ep45 * exposure * vuln45 risk5 = ep5 * exposure * vuln5 totalrisk = riskl + riskl5 + risk2 + risk25 + risk3 + risk35 + risk4 + risk45 + risk5 End Sub Sub messageQ IS the specified risk value is too large or small to be observed upon a path an error message is produced and the macro halts. Msg = "The designated target risk value lies outside " Msg = Msg & "a reasonable range of values for the runout ratio. " Msg - Msg & "Hit OK to exit macro." Dialogstyle = vbOKOnly Title = "Application Error" response = MsgBox(Msg, Dialogstyle, Title) If response = vbOK Then End End Sub Sub searchengineQ The program loops fifteen times, upon each loop the width band of runout ratios is halved. This is done by allocating the half way value 'newmarker' to one of the limit values ('markerl' or 'marker2 ' ) . For loopl = 1 To 15 newmarker = (markerl + ((marker2 - markerl) / 2)) RunoutRatio = newmarker vulnerability proportionrunout relativefrequency widths encounterprobability risk If totalrisk < targetrisk Then test] If totalrisk = targetrisk Then resultsplot If totalrisk > targetrisk Then test2 Next loop 1 resultsplot End Sub Sub testlQ marker2 = newmarker End Sub Sub test2Q markerl = newmarker End Sub 149 Sub resultsplotQ The resulting runout distance is displayed along with other parameters. Set positionl = Worksheets(2).Cells(2, 2) Set position2 = Worksheets(2).Cells(2, 3) Set position3 = Worksheets(2).Cells(2, 4) Set position4 = Worksheets(2).Cells(2, 5) Set position5 = Worksheets(2).Cells(2, 6) Setposition6 = Worksheets(2).Cells(2, 7) positionl.Value = RunoutRatio position2.Value = displaywidth position3. Value = Frequency position4. Value = vulnerabilitytype position5. Value = totalrisk position6. Value = (Xbeta + (RunoutRatio * Xbeta)) columnheadings End Sub Sub columnheadingsQ Column headings are produced to help in the interpretation of the results. Set positionl = Worksheets(2).Cells(l, 2) Set position2 = Worksheets(2).Cells(l, 3) Set position3 = Worksheets(2).Cells(l, 4) Set position4 = Worksheets(2).Cells(l, 5) Set position5 = Worksheets(2).Cells(l, 6) Set position6 = Worksheets(2).Cells(l, 7) positionl. Value = "Runout Ratio" position2.Value = "Deposit Width" position3. Value = "Frequency" position4.Value = "Vulnerability Type" position5.Value = "Total Risk" position6. Value = "Runout Distance" End End Sub 

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