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Advancement and calibration of a 3D numerical model for landslide runout analysis Aaron, Jordan Balfour 2017

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ADVANCEMENT AND CALIBRATION OF A 3D NUMERICAL MODEL FOR LANDSLIDE RUNOUT ANALYSIS by Jordan Balfour Aaron B.A.Sc, Queen’s University, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate and Postdoctoral Studies (Geological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2017 © Jordan Balfour Aaron, 2017 ii  Abstract Rapid landslides pose a significant hazard worldwide, and there is currently no routine way of predicting the impact area and velocities of these catastrophic events.  Increased development in marginal areas is changing the landslide risk in many parts of the world.  There is an urgent need for practical methods to predict the motion of these tragic events to cope with this changing risk.  Practical methods currently in use rely on simplified landslide statistics that have a high degree of uncertainty, and are often unable to predict landslide velocities.  The focus of this thesis is on developing practical methods to reliably predict the motion of rapid landslides so that public safety in landslide prone areas can be improved. This thesis makes extensive use of runout modelling in order to analyse the motion of rock avalanches, debris avalanches and flowslides.  The work presented here can be broadly divided into two categories; the development of new tools and techniques to model flow-like landslide motion, and the compilation and analysis of a database of case histories.  The new tools include: 1)  A new rheology appropriate for the simulation of liquefied materials; 2) A new dynamic model to simulate the initially-coherent motion of some rock and debris avalanches; 3) Two new calibration methodologies.   These techniques were then applied to a database of rock avalanches, debris avalanches and flowslide case histories in order to infer movement mechanisms and give guidance for forward prediction.  The main findings include: 1) The character of the path materials is a plausible explanation for the mechanism governing rock avalanche motion.  Based on this, a probabilistic framework to predict rock avalanche motion was suggested; 2) A back-analysis of a fatal debris avalanche that occurred in British Columbia in 2012 revealed that this flow was likely moving in iii  an undrained condition, which had significant implications for the analysis of its motion; 3) It was found that flowslides can occur in fine grained colluvium, and this material should be recognized as potentially liquefiable.        iv  Lay Summary Extremely rapid, flow-like landslides are a hazard that can devastate large areas.  In order to protect people and infrastructure threatened by these hazards it is necessary to predict their velocity, thickness and impact area before an event occurs.  One promising technique to predict these quantities is computer based numerical runout models.  Many examples of these models exist, however they are not in routine use due to a lack of understanding of the mechanisms that control extremely rapid, flow-like landslides, as well as an absence of examples of successful applications.  This thesis, through the development of a new numerical runout model, as well as new techniques to apply these models, addresses these two knowledge gaps.            v  Preface Section 2.3 has been published in Aaron, J & Hungr, O. (2016) Dynamic Simulation of the Motion of Partially-Coherent Landslides.  Engineering Geology, 205: 1-11, and reprinted with permission.  A portion of Section 2.5 has been previously published in Aaron, J., McDougall, S., Moore, J., Coe, J., Hungr, O (2017) The role of initial coherence and path materials in the dynamics of three rock avalanche case histories (Invited Paper)  Geoenvironmental Disasters, 4(5): 15p, and reprinted here under the Creative Commons License 4.0.  I wrote both sections of the relevant manuscripts.  A version of Chapter 3 has been published in Aaron, J & Hungr, O. (2016) Dynamic Simulation of the Motion of Partially-Coherent Landslides.  Engineering Geology, 205: 1-11, and reprinted here with permission.  I conducted all the analysis and wrote most of the manuscript.   Chapter 4 was conducted in collaboration with Natalia Nolde in the UBC Department of Statistics.  I was responsible for the derivation and implementation of the methodology.  Section 5.6.3 in Chapter 5 has been published in Aaron, J., McDougall, S., Moore, J., Coe, J., Hungr, O (2017) The role of initial coherence and path materials in the dynamics of three rock avalanche case histories (Invited Paper)  Geoenvironmental Disasters, 4(5): 15p, reprinted here under Creative Commons Licesnse 4.0.  I conducted the analysis and wrote most of the manuscript.  This section has been reprinted with permission. A version of Chapter 6 has been published in Aaron, J., Hungr, O., McDougall, S. (2016) Re-examination of the dynamics of the 2012 Johnsons Landing debris avalanche.  GeoVancouver, Vancouver, Canada, 3-5 October.  I conducted the analysis and wrote most of the manuscript. vi  Section 7.8 in Chapter 7 has been published Aaron, J., Hungr, O., Stark, T., Baghdady, A. (2017) Oso Landslide of March 22, 2014 in Washington – Dynamic Analysis. ASCE Journal of Geotechnical and Geoenvironmental Engineering, and reprinted here with permission.  I conducted the analysis and wrote most of the manuscript. Section A.10 was conducted in collaboration with Jessica Castleton, Jeffrey Moore, Marcus Christl and Susan Ivy-Ochs.  I was responsible for the creation of the dynamic model used in the work, and performed model calibration using newly developed techniques.   Section A.11 was conducted in collaboration with Lorenz M Grämiger, Jeffrey R. Moore,  Christof Vockenhuber, Irka Hajdas and  Susan Ivy-Ochs.  I created the numerical model used in the analysis, and provided feedback on the model calibration. Section  A.12 was conducted in collaboration with Lorenz M Grämiger, Jeffrey R. Moore,  Christof Vockenhuber, Irka Hajdas and  Susan Ivy-Ochs.  I created the numerical model used in the analysis, performed model calibration using newly developed techniques and assisted with the interpretation of results.   Section  A.13 has been published in Aaron, J., McDougall, S., Moore, J., Coe, J., Hungr, O (2017) The role of initial coherence and path materials in the dynamics of three rock avalanche case histories (Invited Paper)  Geoenvironmental Disasters, 4(5): 15p, reprinted here under Creative Commons License 4.0.  I conducted the analysis and wrote most of the manuscript.  This section has been reprinted with permission. Section  A.19 has been published in Aaron, J., McDougall, S., Moore, J., Coe, J., Hungr, O (2017) The role of initial coherence and path materials in the dynamics of three rock avalanche vii  case histories (Invited Paper)  Geoenvironmental Disasters, 4(5): 15p, reprinted here under Creative Commons License 4.0.  I conducted the analysis and wrote most of the manuscript.  This section has been reprinted with permission. Section A.22 was conducted in collaboration with Pengfei Si, Scott McDougall, Ji Lu, Xiping Yu, Nick Roberts, and John Clague.  I assisted in the development of the numerical model used in the analysis, model calibration and interpretation of results.   viii  Table of Contents  Abstract .......................................................................................................................................... ii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ....................................................................................................................... viii List of Tables ................................................................................................................................xv List of Figures ............................................................................................................................ xvii Acknowledgements .................................................................................................................. xxix Chapter 1: Introduction ................................................................................................................1 1.1 Research Objectives and Thesis Structure ...................................................................... 2 1.1.1 Accepted Journal Papers ............................................................................................. 4 1.1.2 Peer Reviewed Conference Papers ............................................................................. 6 1.2 Landslide Classification .................................................................................................. 6 1.3 Landslide Risk Assessment........................................................................................... 10 Chapter 2: Overview of Dynamic Model and Rheologies ........................................................12 2.1 Introduction ................................................................................................................... 12 2.2 Overview of Empirical Methods ................................................................................... 12 2.3 Overview of Analytical Methods .................................................................................. 12 2.4 Equivalent Fluid Framework ........................................................................................ 15 2.5 DanW and Dan3D Overview ........................................................................................ 16 2.5.1 Savage-Hutter Method .............................................................................................. 19 ix  2.5.2 Basal Rheologies ....................................................................................................... 20 2.5.3 New Liquefied Rheology .......................................................................................... 22 2.5.4 Behaviour of DanW and Dan3D ............................................................................... 24 2.6 Summary ....................................................................................................................... 25 Chapter 3: Flexible Block Model ................................................................................................27 3.1 Initially Coherent Rock Avalanches ............................................................................. 27 3.1.1 Goldau Rock Avalanche ........................................................................................... 29 3.1.2 Mystery Creek Rock Avalanche ............................................................................... 33 3.2 Model Objectives .......................................................................................................... 34 3.3 Model Derivation .......................................................................................................... 36 3.3.1 Model Assumptions .................................................................................................. 36 3.3.2 Governing Equations ................................................................................................ 37 3.4 Model Implementation .................................................................................................. 46 3.4.1 Fluidization of the Sliding Mass ............................................................................... 47 3.5 Model Verification ........................................................................................................ 48 3.6 Back Analyses ............................................................................................................... 49 3.6.1 Goldau Rock Avalanche ........................................................................................... 50 3.6.1.1 Initially Rigid Simulations ................................................................................ 50 3.6.2 Mystery Creek Rock Avalanche ............................................................................... 52 3.7 Selection of Rigid Motion Distance .............................................................................. 56 3.8 Conclusions ................................................................................................................... 56 Chapter 4: Calibration of a Runout Model ...............................................................................58 4.1 Dan3D Calibration Example ......................................................................................... 59 x  4.1.1 Mt. Meager Rock Avalanche .................................................................................... 59 4.2 Theoretical Background ................................................................................................ 65 4.2.1 Mathematical Framework ......................................................................................... 65 4.2.2 Stochastic Model for Error Vector ............................................................................ 68 4.3 Quantification of Simulation Constraints ..................................................................... 73 4.3.1 Landslide Impact Area .............................................................................................. 74 4.3.2 Landslide Deposit Distribution ................................................................................. 76 4.3.3 Landslide Deposit Depth Estimates .......................................................................... 77 4.3.4 Landslide Velocities.................................................................................................. 77 4.3.5 Selection of Standard Deviation of a Measurement ................................................. 77 4.4 Sensitivity Analyses ...................................................................................................... 79 4.4.1 Posterior Analysis of Calibrated Parameters ............................................................ 80 4.4.2 Limitations of the Sensitivity Analysis Approach .................................................... 82 4.5 Optimization Approach - The Gauss-Marquart-Levenberg Algorithm ........................ 83 4.6 Implementation ............................................................................................................. 87 4.7 Example Case History- Zymoetz River Rock Avalanche ............................................. 88 4.7.1.1 Description of the Event ................................................................................... 88 4.7.1.2 Simulation Constraints ...................................................................................... 90 4.7.1.3 ZRRA - Sensitivity Analysis ............................................................................ 90 4.7.2 Application of the GML Algorithm .......................................................................... 95 4.8 Discussion and Conclusions ......................................................................................... 97 Chapter 5: Rock Avalanches.......................................................................................................99 5.1 Introduction ................................................................................................................... 99 xi  5.2 Background ................................................................................................................. 100 5.3 Back-Analysis Methodology ...................................................................................... 106 5.4 Case Histories ............................................................................................................. 110 5.5 Back-Analysis Results ................................................................................................ 115 5.6 Discussion ................................................................................................................... 118 5.6.1 Comparison of Two Extremely Large Volume Rock Avalanches ......................... 123 5.6.2 Two Rock Avalanches in the Bernese Alps ............................................................ 125 5.6.3 The West Salt Creek Rock Avalanche .................................................................... 127 5.6.4 Conclusions about Mobility Mechanisms ............................................................... 134 5.7 Disintegration Process ................................................................................................ 136 5.8 Proposed Probabilistic Runout Analysis Framework ................................................. 136 5.8.1 Bayesian Parameter Estimation Framework ........................................................... 141 5.8.2 Example Runout Forecast ....................................................................................... 146 5.8.2.1 Turnoff Creek.................................................................................................. 146 5.9 Deficiencies in the Database ....................................................................................... 150 5.10 Complications with Applying Proposed Prediction Framework ................................ 152 5.10.1 Location of Rheology Change ............................................................................ 152 5.10.2 Selection of Case Weighting ............................................................................... 154 5.11 Conclusions ................................................................................................................. 155 Chapter 6: Analysis of an Unusual Debris Avalanche ...........................................................156 6.1 Introduction ................................................................................................................. 156 6.2 Background ................................................................................................................. 156 6.3 Johnsons Landing Debris Avalanche .......................................................................... 160 xii  6.3.1 Event Description.................................................................................................... 162 6.3.2 Previous Dynamic Analysis .................................................................................... 163 6.4 Methodology ............................................................................................................... 165 6.5 Model Rheology.......................................................................................................... 165 6.6 Quantifying the Role of Trees on the Bench .............................................................. 166 6.7 Model Input ................................................................................................................. 170 6.8 Dynamic Analysis Results .......................................................................................... 172 6.8.1 DanW ...................................................................................................................... 172 6.8.2 Dan3D ..................................................................................................................... 173 6.9 Dynamic Analysis Discussion .................................................................................... 176 6.10 Summary – Johnsons Landing .................................................................................... 178 6.11 Appropriate Rheology for Debris Avalanches ............................................................ 179 Chapter 7: Flowslides ................................................................................................................181 7.1 Introduction ................................................................................................................. 181 7.2 Background ................................................................................................................. 181 7.2.1 Liquefaction of Granular Materials ........................................................................ 182 7.2.2 Liquefaction of Quick Clay .................................................................................... 185 7.2.3 Summary of Liquefaction Mechanisms .................................................................. 187 7.3 Coal Mine Waste Dump Failure ................................................................................. 188 7.4 Flowslides in Overconsolidated Silt and Clay ............................................................ 193 7.5 Methodology ............................................................................................................... 194 7.6 Model Rheology.......................................................................................................... 195 7.7 Attachie ....................................................................................................................... 195 xiii  7.8 Oso .............................................................................................................................. 201 7.8.1 Data Sources ........................................................................................................... 203 7.8.2 Morphology of the Debris Field ............................................................................. 203 7.8.3 Rupture Surface Reconstruction ............................................................................. 210 7.8.4 Numerical Simulation Results ................................................................................ 213 7.8.5 2D Simulations........................................................................................................ 214 7.8.6 3D Simulations........................................................................................................ 215 7.9 Discussion and Conclusions ....................................................................................... 221 Chapter 8: Conclusions .............................................................................................................225 8.1 Introduction ................................................................................................................. 225 8.2 Background Information ............................................................................................. 225 8.3 New Techniques to Analyse Extremely Rapid, Flow-like Landslides ....................... 226 8.4 Rock Avalanche Movement Mechanisms and Prediction .......................................... 228 8.5 Analysis of the Johnsons Landing Debris Avalanche ................................................. 229 8.6 Analysis of Two Flowslides in Fine Grained Colluvium ........................................... 230 8.7 Recommendations for Future Work............................................................................ 230 8.8 Conclusion .................................................................................................................. 233 References ...................................................................................................................................234 Appendix .....................................................................................................................................260 Appendix A Rock Avalanche Case Histories ......................................................................... 260 A.1 Crammont ............................................................................................................... 260 A.2 Huascaran ................................................................................................................ 265 A.3 Madison Canyon ..................................................................................................... 270 xiv  A.4 McAuley ................................................................................................................. 274 A.5 Mt. Meager Rock Avalanche .................................................................................. 277 A.6 Mt Steele ................................................................................................................. 281 A.7 Nomash ................................................................................................................... 284 A.8 Six des Eaux Froides ............................................................................................... 286 A.9 Val Pola ................................................................................................................... 290 A.10 Zion National Park – Sentinel Rock Avalanche ..................................................... 293 A.11 Daubensee ............................................................................................................... 295 A.12 Rinderhorn .............................................................................................................. 299 A.13 Bingham Canyon .................................................................................................... 302 A.14 Goldau ..................................................................................................................... 305 A.15 Thurweisser ............................................................................................................. 307 A.16 Avalanche Lake ...................................................................................................... 308 A.17 Mystery Creek ......................................................................................................... 311 A.18 Frank ....................................................................................................................... 313 A.19 Rautispitz ................................................................................................................ 316 A.20 Platten ..................................................................................................................... 320 A.21 Guinsaugon ............................................................................................................. 322 A.22 Chehalis................................................................................................................... 325 A.23 West Salt Creek....................................................................................................... 327 A.24 Zymoetz River ........................................................................................................ 327  xv  List of Tables Table 1-1: Landslide velocity scale from WP/WLI (1995) updated with human response by Hungr et al. (2014). ......................................................................................................................... 8 Table 1-2: Landslide type definitions (Hungr et al. 2014) ............................................................ 10 Table 3-1:  Basal resistance parameters used in the simulation of the two rock avalanche case histories.  Both rock avalanches were simulated using the frictional rheology in the source zone and the Voellmy rheology along the path.  For the spatial locations of the basal rheology switch see Figure 3-3 and Figure 3-4. ...................................................................................................... 31 Table 4-1: Value of the objective function and likelihood function for four different parameter combinations.  The parameter combination of 𝒇 = 𝟎. 𝟎𝟓, 𝛏 = 𝟓𝟎𝟎 𝐦𝐬𝟐 (shown in bold) results in the lowest value of the objective function, and therefore the highest value of the likelihood function, indicating that this parameter combination is the best fit of the four combinations tested. ............................................................................................................................................ 72 Table 4-2: Standard deviation of measurements used to normalize the residual values. ............. 91 Table 4-3: Parameter correlation matrix calculated at the conclusion of the inverse model. The moderate correlation between the two parameters reflects the fact that multiple parameter combinations can give similar fitness results................................................................................ 96 Table 5-1: Summary of the 24 back-analysed rock avalanche case histories. ............................ 112 Table 6-1: Landslide debris volume simulated in the deposit zones .......................................... 175 Table A-1: Standard deviation for the trimline fitness metric used in the Crammont case history...................................................................................................................................................... 264 Table A- 2: Standard deviation used to derive the PDF for Madison Canyon. .......................... 273 Table A-3: Standard deviations used to derive the PDF for McAuley Creek. ........................... 276 Table A-4: Standard deviations used for Mt. Meager simulations. ............................................ 279 Table A-5: Standard deviations used for Nomash River back-analysis. .................................... 285 Table A-6: Standard deviations used for the Six des Eaux Froides back-analysis. .................... 289 Table A-7: Standard deviation used to derive the posterior density for the Val Pola case history...................................................................................................................................................... 292 Table A-8: Standard deviations used for Sentinel Rock Avalanche. .......................................... 294 xvi  Table A-9: Standard deviation of the trimline fitness metric used for the Rinderhorn case history...................................................................................................................................................... 300 Table A-10: Standard deviations used for the Goldau Rock Avalanche. ................................... 306 Table A-11: Standard deviation used for the Avalanche Lake Rock Avalanche. ...................... 310 Table A-12:  Standard deviation used for the Mystery Creek Rock Avalanche. ........................ 312 Table A-13: Frank Slide: trimline standard deviation used to derive the posterior density for the path materials. ............................................................................................................................. 315 Table A- 14: Rautispitz standard deviations used to derive the posterior density function for the path materials. The deposit distribution standard deviation is the lowest mean square error that results from comparing 28 estimates of simulated and observed deposit depths. ...................... 319 Table A-15: Standard deviation used for the Platten Rock Avalanche ...................................... 321 Table A-16: Best-fit parameters determined for the Guinsaugon case history. .......................... 325  xvii  List of Figures Figure 1-1: Hungr et al. (2014) landslide classification based on movement type.  Landslide types highlighted in yellow show those that are very rapid to extremely rapid. ............................ 9 Figure 2-1: Conceptual free body diagram of a slice of material oriented in the direction of motion in Dan3D.  W is the weight, T is the basal resistance, P is the internal force due to free surface gradients, and E is the inertial resistance due to entrainment. ......................................... 18 Figure 2-2: Mohr circle of stress for a 2D element simultaneously undergoing internal shear failure and basal slip (after Hungr, 2008). .................................................................................... 19 Figure 3-1: North Nahanni Rock Avalanche.  A large coherent block can be seen on the rupture surface near the centre of the photo, indicating a phase of coherent motion during the event. Photo: O. Hungr. ........................................................................................................................... 28 Figure 3-2: Photo of the Goldau source area.  Photo: O Hungr. ................................................... 30 Figure 3-3: A) Initial conditions used to model the Goldau Rock Avalanche.  The frictional Rheology is used to calculate basal resistance north of the red line, and a Voellmy rheology is used south of the red line.  The parameters used in each of these rheologies are shown in Table 3-1.  B)  Final deposit shape and trimline of fully fluid simulation.  Dashed line shows the true trimline (after Fitze, 2010) and the grey area is the predicted trimline.  A cutoff of 0.3 m is necessary due to the solution method used by Dan3D.  The red vectors indicate instantaneous velocity vectors of the particles after 1 second.  The particles are moving laterally as well as downslope, contrary to the observed trimline. .............................................................................. 32 Figure 3-4: A) Model setup for the simulation of the Mystery Creek Rock Avalanche.  A frictional rheology is used to calculate basal resistance in the source zone (zone 1) and a Voellmy rheology is used to calculate basal resistance outside the source zone (zone 2).  The parameters used in each of these rheologies are shown in Table 3-1.  B)  Final deposit shape predicted by fully fluid solution for the Mystery Creek Rock Avalanche.  The dashed line shows the true trimline (after Nichol et al., 2002).  Excessive lateral spreading is predicted when the mass is fluidized instantly.......................................................................................................................... 34 Figure 3-5: Forces acting on a column when internal strength is neglected. G denotes the downslope component of gravity, Fn the component of gravity normal to the slope and F_basal is the resistance force.  Note Fn (blue) is used to calculate F_basal.  F_basal is not collinear with G as it acts in the direction of motion. .............................................................................................. 41 Figure 3-6:  Example of a block, discretized as a set of interconnected columns, is simulated to move down an inclined plane.  In this example there is a net torque in the clockwise direction.  The net force causes the centre of mass of the column assembly to translate down the inclined plane, and the net torque rotates the column assemblies.  All the columns experience the same displacement and rotation so the column geometry is maintained and columns do not interpenetrate.  The forces and torques acting on each individual column contribute to the net xviii  force and torque experienced by the column assembly.  Figure 3-5 shows the forces acting on each of the individual columns. .................................................................................................... 45 Figure 3-7: Two dimensional section showing column distortion in the vertical direction for a column assembly moving over a bilinear rupture surface.  Free vertical motion of the columns is permitted in order to ensure the failed mass remains on the sliding surface.  Since it is assumed that inter-column forces are negligible the new model is not suitable to simulate failures over strongly compound rupture surfaces. ............................................................................................ 46 Figure 3-8: Test configuration for the verification of the rotation algorithm.  The end is pinned and a constant force is applied in the y direction.  The rod rotates about the pinned end in a manner similar to a pendulum....................................................................................................... 49 Figure 3-9: Comparison of centre of mass location predicted by the analytical solution and Dan3D flex at various times ......................................................................................................... 49 Figure 3-10:   final deposit shape predicted by Dan3D Flex for the Goldau Rock Avalanche.  The red outline shows the location where the user specified fluidisation criteria was met. ................ 51 Figure 3-11: Maximum nodal velocities predicted by Dan3D flex for the Goldau Rock Avalanche. .................................................................................................................................... 51 Figure 3-12: Sensitivity of the predicted trimline to rigid motion distance for the Goldau Rock Avalanche.  The blue trimline represents model results of the mass is fluidized at the proximal end of the green zone, and the red trimline shows the result if the mass is fluidized at the distal end of the green zone.  For rigid motion distances in between these two extremes the predicted trimline would be in between the two results. .............................................................................. 52 Figure 3-13: Mystery Creek final deposit predicted by Dan3D Flex.  The red outline shows where the mass is fluidized. .......................................................................................................... 54 Figure 3-14: Mystery Creek maximum nodal velocities predicted by Dan3D Flex ..................... 55 Figure 3-15: Sensitivity of simulated trimline to rigid motion distance for Mystery Creek.  The blue trimline represents the simulated trimline if the mass is simulated to fluidize at the proximal end of the green zone, and the red trimline shows the result of the mass is simulated to fluidize at the distal end of the green zone.  Intermediate rigid motion distance would plot between these two trimlines. ................................................................................................................................ 55 Figure 4-1: Overview of the Mt. Meager rock avalanche.  The area of the initial rock slope failure (labelled ‘source zone’), impact area, as well as the locations of two superelevation measurements are shown.  Image: Google Earth 2017: Digital Globe. ........................................ 61 Figure 4-2: McDougall (2016) calibration of the Mt. Meager rock avalanche.  Based on the assumption that much of the apparent impact area of the event is due to post- rock avalanche flooding, McDougall (2016) determined best fit parameters of f = 0.05, turbulence = 500 m/s2.  Note that the ‘turbulence parameter’ labelled in the above Figure is referred to as the turbulence xix  coefficient throughout this thesis.  Figure from McDougall (2016), © 2008 Canadian Science Publishing or its licensors. Reproduced with permission. ............................................................ 64 Figure 4-3: An example of a trimline grid file created based on a field investigation. This grid has a value of one in locations where the field investigation indicated landslide impact and a value of zero where the landslide did not impact. Values of one are coloured dark grey and zeros are coloured light grey. ................................................................................................................. 75 Figure 4-4: Example of a maximum thickness gridfile output by Dan3D.  Thickness values are reported in meters. ........................................................................................................................ 75 Figure 4-5: Steps to run a sensitivity analysis. All parameter combinations are run in Dan3D, which then passes the simulation output to the post-processor. The post-processor accepts the simulation constraints and the Dan3D output as input in order to calculate fitness metrics. The fitness metrics are then combined using the least squares error function. The resulting value of the fitness function is then plotted. Once the dimensionless fitness values for all parameter combinations have been calculated the results are contoured. ..................................................... 80 Figure 4-6: Overview of the Zymoetz River rock avalanche.  The simulation constraints used are the volume of the deposit near the 'C' on the figure (100,000 m3), the velocity of the flow at the cross (17 m/s) and the impact area (extending from 'A' to 'D').  The rheology change was implemented in the channel, downstream of the location labelled ‘B’. Image modified from McDougall et al. (2006).  Image: Province of British Columbia, Copyright © Province of British Columbia. ...................................................................................................................................... 89 Figure 4-7: Normalized trimline fitness residual for all parameter combinations in the parameter space. ............................................................................................................................................. 92 Figure 4-8: Normalized residual of volume deposited in the channel.  One standard deviation around the mean is highlighted in red. .......................................................................................... 93 Figure 4-9: Normalized residual of velocity simulated at the bend in the channel.  One standard deviation around the mean is highlighted in red. .......................................................................... 93 Figure 4-10: Contours of the objective function (Equation [4.5]) for the Zymoetz River Rock Avalanche, calculated based on normalized simulation constraints. ............................................ 94 Figure 4-11: Zymoetz River Rock Avalanche posterior PDF.  High probabilities are assigned to parameter combinations that well reproduce the field constraints. ............................................... 94 Figure 4-12: Optimization steps taken by the GML algorithm for the Zymoetz River Rock Avalanche, overlaid on the contours of the objective function.  The red crosses show the steps taken by the GML algorithm, obtained by iteratively solving Equation [4.16]. ........................... 95 Figure 5-1: Explanation of the H/L ratio.  The dashed grey line represents the line connecting the top of the back-scarp to the distal end of the deposit.  The inclination of this line from horizontal, defined above as the angle of reach, is tan-1(H/L).  The dashed red line shows the line connecting xx  the center of mass of the source to that of the deposit.  For a frictional material, the inclination of this line from horizontal is the average friction coefficient experienced by the moving mass (Heim, 1932). .............................................................................................................................. 101 Figure 5-2: Volume vs. H/L of cases in the database of rock avalanche case histories.  For comparison, H/L data collected by Scheidegger (1973), Li (1983) and an unpublished database of Canadian rock avalanches (Brideau, M.A, BGC Engineering, unpublished data) are shown.  The cases analysed in this chapter appear to be more mobile than this background data. Cases are sorted by path material.  Descriptions and back-analyses of the majority of these case histories are summarized in Appendix A.  For cases not back-analyzed in this thesis, references to DanW and/or Dan3D analyses are provided.  Labels: 1. Zymoetz, 2. Crammont, 3. Six des Eaux Froides, 4. Huascaran, 5. Kolka (Huggel et al., 2005; McDougall, 2006; Evans et al., 2009b), 6. Mt. Meager, 7. Mt. Steele, 8. Nomash River, 9. Sherman Glacier (Sosio et al., 2012), 10. Thurweiser, 11. McAuley Creek, 12. Val Pola, 13. Avalanche Lake, 14. Goldau, 15. Mystery Creek, 16. Turnoff Creek, 17. Madison Canyon, 18. Chisca, 19. Hope, 20. Pandemonium Creek (Evans et al., 1989), 21. West Salt Creek, 22. Frank, 23. Guinsaugon, 24. Bingham Canyon, 25. Sentinel, 26. Daubensee, 27. Rinderhorn, 28. Rautispitz, 29. Platten, 30. Chehalis. ................. 102 Figure 5-3: Photo of the McAuley Creek rock avalanche.  Thick deposits at the toe of the source slope can be observed.  Image: Google Earth, Province of British Columbia. ........................... 107 Figure 5-4: Back-analysed friction angles in the source zone.  As mentioned in Section 5.3, these values are constrained by observations of the deposit distribution.  There is a reduction in source zone friction angle with increasing volume possibly due to a reduction in frictional strength due to shearing under high normal stresses.  See Figure 5-2 for case names that correspond to the case numbers. .............................................................................................................................. 116 Figure 5-5: Best fit path material parameter zones for each of the cases analyzed with the Voellmy rheology.  Each polygon represents the 70% credibility region for one case, the case names corresponding to the case numbers can be found in the caption of Figure 5-2.  The cases are coloured based on volume.  The strength in the path materials appears to be independent of volume.  For all cases except 4, 8 and 25 (Huascaran, Mt. Meager and Sentinel, respectively) the frictional rheology was used in the source zone. ........................................................................ 116 Figure 5-6: Best fit Voellmy parameters colored by path material.  Each polygon represents one case, case names can be found in the caption of Figure 5-2.  For all cases except Mt Meager (8), Huascaran (4) and Sentinel (25) the frictional rheology was used in the source zone, with a friction angle shown on Figure 5-4. ............................................................................................ 117 Figure 5-7: Best fit friction angles for cases that overran bedrock.  Each point represents one case.  Case names can be found in the caption of Figure 5-2. .................................................... 117 Figure 5-8: Comparison of the Bingham Canyon and Nomash River Rock Avalanches.  For Bingham Canyon, low strength in the source zone lead to no material depositing on the basal fault, which is inclined at 23°.  A lack of runup at the toe of the landslide indicates high basal resistance along the path.  Conversely, at Nomash River, low strength along the path lead to xxi  runup features, as well as long runout on a shallow slope.  This distribution of shear strengths is supported by the back-analyses of these two case histories.  Bingham canyon image is from Aaron et al. (2017b), who modified an image by Pankow et al. (2014) , Nomash River image was taken by Dana Ayotte, and published in McDougall & Hungr (2005), © 2008 Canadian Science Publishing or its licensors. Reproduced here with permission. .................................................. 121 Figure 5-9: Comparison of the Avalanche Lake and Sentinel rock avalanches.  At Avalanche Lake, the failed mass rapidly descended the source slope, and ran up a 640 m high adverse slope, creating the largest rock avalanche runup feature ever documented.  The spectacularly energetic mass also spread out along the valley floor, creating the main deposit and south lobe.  The Sentinel rock avalanche, which is similar in volume to Avalanche Lake, also impacted an adverse slope.  However, it only created a moderate runup against this slope, and only displayed limited spreading along the valley floor.  Avalanche Lake photo: Oldrich Hungr, Zion Canyon photo: Google Earth, USDA Farm Service Agency. .................................................................. 124 Figure 5-10: Comparison of the Daubensee and Rinderhorn rock avalanches.  The Daubensee rock avalanche overran glacially-abraded bedrock, which limited its mobility.  The Daubensee rock avalanche initially overran bedrock, but after descending its source slope it overrode fluvial sediments, which greatly enhanced its mobility.  After Grämiger et al. (2016), reprinted with permission. .................................................................................................................................. 126 Figure 5-11: Failure sequence for the West Salt Creek Rock Avalanche hypothesized by Coe et al. (2016).  The geometry after the slump (green line on panel b and c) was derived from a Dan3D-Flex analysis.  The section line is shown on Figure 5-13. ............................................. 129 Figure 5-12: Overview of the West Salt Creek Rock Avalanche (Photo: J Coe). ...................... 130 Figure 5-13: West Salt Creek Rock Avalanche accumulation and depletion map. Coe et al. (2016) noted that the estimated vertical error of the digital elevation data is ± 4.72 m.  The section line refers to Figure 9...................................................................................................... 131 Figure 5-14: Final deposit depth and predicted impact area when basal resistance is parameterized with the Voellmy rheology.  The red outline shows the observed impact area.  A minimum deposit depth value of 0.3 m is necessary due to the solution method used by Dan3D...................................................................................................................................................... 132 Figure 5-15: Results of the sensitivity analysis used to determine the best-fit Bingham parameters for the West Salt Creek rock avalanche.  Impact area fitness is calculated using a dimensionless number that measures the misfit between a user specified impact area and the simulated impact area (Section 4.3.1).  Lower numbers indicate better fitness (a value of zero indicates perfect agreement between observed and simulated impact area).  A good compromise between simulating the observed impact area and deposit distribution is found for τyield =32 KPa and μbingham = 7 KPa*s.  Volume is in m3. .................................................................................. 133 Figure 5-16: Predicted impact area and simulated deposit depths when basal resistance is parameterized with the best fit Bingham rheology.  The red outline shows the observed impact xxii  area.  A minimum deposit depth value of 0.3 m is necessary due to the solution method used by Dan3D. ........................................................................................................................................ 134 Figure 5-17: Decision tree to guide the selection of parameters for the prediction of rock avalanche runout using Dan3D ................................................................................................... 139 Figure 5-18: Comparison of histogram (left) and kernel density estimator (right) for the same data.  The kernel density estimator allows for variance in the measured values of ‘x’ to be accounted for.  In most Dan3D back-analyses, a zone of best fit parameters is supported by the data.  Selecting a single best fit parameter set, as is required to construct a histogram, cannot capture this source of variance.  A kernel density estimator, through the use of a probability density function associated with each data point, captures variance in the parameter values.  Image: Drleft at English Wikipedia, via Wikimedia Commons. ................................................ 143 Figure 5-19: Probability density function derived from combining the best fit parameters from all the case histories.  High probability zones correspond to parameter ranges that fit a large number of case histories (compare to Figure 5-6). .................................................................................. 145 Figure 5-20: Overview of Turnoff Creek.  Image: Google Earth, Digital Globe. ...................... 147 Figure 5-21: Probability density function for the Turnoff Creek path material parameters derived based on evaluating Equation [5.2].  Since Mt. Meager was excluded, and Nomash River assigned a low similarity probability, the probability of the parameters having a friction coefficient less than 0.08 are lower than in Figure 5-19. ............................................................ 149 Figure 5-22: Results of a parametric analysis using 25 different parameter combinations drawn from the low strength PDF.  The contours on the image are exceedance probabilities.  The extremely long runout represented by the 0.05 exceedance probability is due to the inclusion of Nomash River when evaluating Equation [5.2]. ......................................................................... 150 Figure 6-1: Schematic of the undrained loading process described by Sassa & Wang (2005).  Panel C and D show the shear and normal stress acting on the soil element highlighted in panels A and B.  Before being overridden, panel C shows that the soil element (red dot in C and D) is stable.  When the soil element is overridden (B and D), both the shear and normal stress acting on the element increase.  If drainage is restricted but no contraction occurs, shear stress increases until the column fails, while effective normal stress remains constant (panel (B) point A).  If the overridden mass is loose and liquefiable, then shear stress will decrease to a very low value.  ϕi is the internal friction angle of the path material. ........................................................................... 158 Figure 6-2:  Overview of the Johnsons Landing debris avalanche.  Image: Province of British Columbia, Copyright © Province of British Columbia. ............................................................. 161 Figure 6-3: Model basal resistance parameterisation used for simulations that hypothesize that trees dramatically increase basal resistance.  The black area shows areas of low resistance, and the grey area shows areas of high resistance. ............................................................................. 164 xxiii  Figure 6-4: Johnsons Landing simulation results when a two rheology simulation is used.  The deposit on the debris field matches the observed deposit, however, too much material is simulated to deposit in the mid-channel, and not enough in the upper channel.  A channel obstruction was assumed to limit the amount of debris that deposited in the section labelled “lower channel”. ......................................................................................................................... 164 Figure 6-5: Derivation of resultant force applied to the debris flow front due to tree breakage. 168 Figure 6-6: Close up of the location where the material avulsed from the channel.  For this section the topography has not been smoothed, however in the dynamic analyses that follow smoothed topography was used.  The radius of curvature at this location is approximately 160 m, indicating centripetal accelerations of 7 m/s2 (based on velocity estimates from Nicol et al. (2013) to 10 m/s2 (based on modelled velocities presented below).  This indicates that centripetal accelerations approximately double the normal stress. ............................................................... 171 Figure 6-7: Final deposit depths and velocities predicted by DanW using the liquefied Voellmy rheology.  When interaction with trees is explicitly accounted for, the simulated runout distance is reduced by 45 m. ..................................................................................................................... 173 Figure 6-8: Sensitivity analysis to compare the undrained Voellmy rheology vs. the frictional Voellmy rheology.  Trimline fit is assessed with the dimensionless fitness number described in Section 4.3.1.  Bench volume is in cubic metres.  The white dot on each image shows the best-fit parameter combination................................................................................................................ 174 Figure 6-9: Comparison of simulated deposit depths (top two panels) and velocities (bottom two panels) when the liquefied Voellmy rheology is used (left) and when the frictional Voellmy rheology is used (right).  The inset into the top left panel shows the deposit derived from LiDAR...................................................................................................................................................... 175 Figure 6-10: Still from video of surge of debris on the day following the main debris avalanche at Johnsons Landing.  The high proportion of timber within this surge provides indirect evidence of a channel obstruction composed of timber that may have failed, leading to this surge.  Video: Global News: https://www.youtube.com/watch?v=n1cCs-S5EKc. ........................................... 177 Figure 6-11: Final deposit depths predicted by Dan3D when a channel obstruction is assumed to be present in the channel downstream of the 70° bend. .............................................................. 178 Figure 7-1: Schematic showing the undrained response of a loose, granular material subject to static overstressing (after Olson (2001) and Wang (2008)).  Prior to loading the stress state of the material is at point ‘A’.  As the material is loaded, the stress state reaches point ‘B’, the peak shear strength.  Loading beyond the peak shear strength results in soil structure collapse, and liquefaction.  The strength reduces to point ‘C’, which corresponds to the liquefied strength. . 183 Figure 7-2: Top: drained shearing of a loose granular material.  The particles rearrange into a denser packing, and no excess pore pressures develop.  Contraction occurs and porewater is drained from the soil skeleton.  Bottom: Undrained shearing of a loose granular material.  xxiv  Because drainage is restricted, the particles cannot densify.  This results in a transfer of load from the particles to the pore fluid (grey arrows), and an excess porewater pressure develops. 184 Figure 7-3: Overview image of the Lemieux quick clay flowslide.  Photo: S.G Evans, with permission. .................................................................................................................................. 186 Figure 7-4: Schematic of a quick clay failure.  When the pore fluid has a high salt content, the cardhouse structure is favoured, and when the pore fluid has a low salt content, the dispersed structure is favoured.  If the clay is formed in a high-saline (e.g. marine) environment, a cardhouse structure will form.  Leaching of the pore fluid will result in this structure being metastable.  A disturbance of the clay will collapse the cardhouse structure and disperse the clay particles into the pore fluid, resulting in a change of state from a solid to a viscous fluid. ........ 187 Figure 7-5: Three images of the coal mine waste dump failure.  ‘A’ shows an image of the source zone, ‘B’ shows an image of the superelevation feature used to estimate flow velocity, and ‘C’ shows an image of the highly mobile, channelized portion of the flow.  Photos: O. Hungr. ..... 190 Figure 7-6: Coal mine waste flowslide simulation constraints. .................................................. 191 Figure 7-7: Top: Deposit depths and impact area simulated by the model.  The deposit in the source zone, as well as the distal runout distance, is well reproduced.  Bottom: Maximum velocities simulated by the model.  The velocity at the superelevation is reproduced. .............. 192 Figure 7-8: Pre-event image of the Attachie flowslide. (Airphoto credit: Province of British Columbia.  Airphoto BC7279-70, 1970, Copyright © Province of British Columbia.  Reprinted with permission.)......................................................................................................................... 196 Figure 7-9: Post event image of the Attachie flowslide . (Airphoto credit: Province of British Columbia.  Airphoto BC5529-75, 1973, Copyright © Province of British Columbia.  Reprinted with permission).......................................................................................................................... 197 Figure 7-10: Cross section through the Attachie rupture surface showing the stepped rupture surface and the material boundaries used in the dynamic analysis. ............................................ 198 Figure 7-11:  Constraints used to calibrate the Attachie runout analysis.  The constraints include the deposit thickness, deposit distribution and impact area. ....................................................... 199 Figure 7-12: Back-analysis results of the Attachie flowslide.  The red dashed line is the observed impact area.  The thick deposits in the source zone are well simulated, as is the average thickness of debris along the valley floor.  The leading edge of the debris is not simulated to deflect downstream.  This deflection can be attributed to factors not simulated by the model (see text)...................................................................................................................................................... 200 Figure 7-13: Comparison of model results to post event LiDAR.  The simulated deposit thicknesses are similar to those derived from the LiDAR data.  The section line is shown on Figure 7-11. ................................................................................................................................. 200 xxv  Figure 7-14: Landslide zones for dynamic analyses: the dashed orange line outlines the splash zone, the dash-dot red line outlines the valley floor deposit, and the solid black line outlines the source zone.  The locations of the boreholes used to constrain the rupture surface are shown (Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation). ........................................................................................................................... 204 Figure 7-15: Polygons of intact forest blocks in 2014 slide mass used to constrain the origin of different features of the debris.  Polygons 1 and 2 have a combined area of 59,000 m2, corresponding to the pre-failure area of trees on the Whitman Bench (see Figure 7-16).  Polygons 3 through 15 have a combined area of 50,000 m2, corresponding to the area of the trees in front of the Whitman Bench (see Figure 7-16).  Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation. ............................................... 206 Figure 7-16: Measurement of tree areas on pre 2014 ortho-photograph where Polygon 16 and 17 have a measured area of 59,000 m2 and 54,000 m2, respectively (Background image from National Agriculture Imagery Program, data available from the U.S Geological Survey). ....... 207 Figure 7-17: Post event image of 2014 Oso flowslide source zone with intact blocks that appear to have undergone extension while traveling over an inclined surface (see square on left photo).  The photo on the right shows a side view of the dropped down blocks (Photos: J. Aaron). ...... 208 Figure 7-18: Accumulation and depletion zones of 2014 Oso landslide that shows deposits in the splash zone are thin.  The thin deposits extending to the north-east are due to post landslide flooding (Keaton et al. 2014). ..................................................................................................... 209 Figure 7-19: Simplified stratigraphic section through the 2014 Oso landslide showing the borehole results presented by Badger (2016).  The materials comprising phases ‘A’ and ‘B’ are indicated. The dashed red line shows the proposed rupture surface, which agrees with available borehole data and surface observations.  The locations of the two intermediate scarps correspond to observed pre-failure scarps, and the depth of the rupture surface was determined from borehole data.  The location of the boreholes and section line are shown on Figure 7-14. ........ 212 Figure 7-20: Final deposit depths predicted by DanW when the two phases are simulated to occur at different times.  The final deposit surface agrees well with the 2014 LiDAR. ............. 215 Figure 7-21: Final deposit predicted by DanW when the two phases occur at the same time.  The results are similar to those presented on Figure 7-20. ................................................................ 215 Figure 7-22: Results of Dan3D simulation of first phase of movement with impact area and deposit distribution in agreement with field observations. No effort was made to reproduce the splash zone (Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation) ................................................................................................... 217 Figure 7-23: Results of Dan3D simulations of Phase B overlain on post 2014 event ortho-photograph (Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation. .................................................................................................... 218 xxvi  Figure 7-24: Accumulation and depletion zones predicted by Dan3D.  The splash zone was not modelled, so the impact area in the western section is less than that observed. ......................... 219 Figure 7-25: Model simulated intensity index compared to building damage.  The units of intensity index are m3s-2.  According to Jakob et al. 2011 values of the intensity index between 1 m3s-2 and 100 m3s-2 correspond to some/major structural damage, from 100 m3s-2 to 1000 m3s-2 correspond to major structural damage/complete destruction and greater than 1000 m3s-2 indicates complete destruction (Background image from National Agriculture Imagery Program, data available from the U.S Geological Survey). ....................................................................... 220 Figure 7-26: Representative cross section through the Dan3D results showing the simulated deposit depths.  The results are in good agreement with the deposit surface derived from the 2014 LiDAR................................................................................................................................ 221 Figure 7-27: Sections through the Dan3D Phase A simulation at ten second intervals showing slide debris piling up at the slope toe before spreading over valley floor. ................................. 221 Figure A-1: Overview of the Crammont Rock Avalanche.  Figure from Deline et al. (2011), distributed under Creative Commons Attribution 3.0 License. .................................................. 260 Figure A-2: Crammont fitness number plot; the best fit parameters are friction coefficient of 0.17 and turbulence parameter of 300 m/s2. ....................................................................................... 263 Figure A-3: Best fit Crammont results. ....................................................................................... 264 Figure A-4: Posterior probability density for the Crammont case history.................................. 265 Figure A-5: Overview of the Huascaran Rock Avalanche.  This overview is based on the interpretation provided by Evans et al. (2009a). ......................................................................... 266 Figure A-6: Right: Final volume predicted in Yungay in m3, Bottom: Trimline fitness.  Only low friction and high turbulence lead to simulation of the Yungay lobe........................................... 268 Figure A-7: Final deposit predicted by Dan3D using the best fit simulation parameters........... 269 Figure A-8: Left: Trimline fitness, Right: Similarity of final deposit with the cross sections presented in Hadley (1978). ........................................................................................................ 272 Figure A-9: Final deposit depth and predicted trimline using the best fit simulation parameters for Madison Canyon. .................................................................................................................. 273 Figure A-10: Posterior density function derived for the Madison Canyon rock avalanche. ...... 274 Figure A-11: McAuley Creek posterior distribution for the path materials. .............................. 276 Figure A-12: Best fit simulation result for the McAuley Creek rock avalanche ........................ 277 xxvii  Figure A-13: Posterior density function for the Mt. Meager case history. ................................. 280 Figure A-14: Mt Meager simulation results for best fit parameters ........................................... 280 Figure A-15: Trimline fitness number for the Mt. Steele sensitivity analysis. ........................... 283 Figure A-16: Mt Steele Dan3D simulation results using the best fit parameters determined from the sensitivity analysis.  The black outline is the observed trimline, the grey outline is the simulated trimline, and simulated deposit depths are contoured.  The distal end of the trimline appears to be a dust cloud (see Lipovsky et al. (2008)). ............................................................. 283 Figure A-17: Nomash fitness plot. .............................................................................................. 286 Figure A-18: Nomash final deposit depths and impact area predicted by best fit parameters. .. 286 Figure A-19: Final deposit depth predicted using best fit parameters.  The grey outline shows the predicted impact area.  Simulation constraints are shown. ......................................................... 289 Figure A-20: Six des Eaux Froides posterior probability ........................................................... 290 Figure A-21: Val Pola posterior density for the parameters. ...................................................... 292 Figure A-22: Val Pola best fit simulation results........................................................................ 293 Figure A-23: Posterior parameter density derived for the Sentinel Rock Avalanche................. 295 Figure A-24: Sentinel Rock Avalanche best fit simulation results. ............................................ 295 Figure A-25: Overview of the Daubensee rock avalanche.  Figure from Grämiger et al. (2016), reprinted here with permission.................................................................................................... 297 Figure A-26: Overview of Dan3D results of Daubensee rock avalanche.  Figure from Grämiger et al. (2016), reprinted here with permission. ............................................................................. 298 Figure A-27: Overview of the Rinderhorn rock avalanche.  Figure from Grämiger et al. (2016), reprinted here with permission.................................................................................................... 300 Figure A-28: Posterior probability of Rinderhorn path material parameters. ............................. 301 Figure A-29: Rinderhorn best fit model results. ......................................................................... 301 Figure A-30: Simulation results for the two rock avalanche phases of the Bingham Canyon slide (a). Simulation results for the first rock avalanche when the flexible block model is not used (b);  here a large amount of material spills out to the north of the source zone, which is inaccurate.  The red outline shows the source zone, the light blue outline shows the final deposit extent and the dark blue line shows the observed impact area.  The minimum deposit depth value was xxviii  specified as 1 m (a minimum deposit depth is necessary due to the solution method used by Dan3D). ....................................................................................................................................... 304 Figure A-31: Posterior parameter density derived for the Goldau case history. ........................ 307 Figure A-32: Avalanche lake posterior density function for the path material parameters. ....... 310 Figure A-33: Avalanche Lake best fit simulation results.  Figure from Aaron & Hungr (2016a), reprinted here with permission.................................................................................................... 311 Figure A-34: Best fit simulation result for the Mystery Creek rock avalanche .......................... 312 Figure A-35: Mystery Creek rock avalanche posterior parameter density function for the path material parameters. .................................................................................................................... 313 Figure A-36: Posterior probability density function derived for the Frank Slide. ...................... 315 Figure A-37: Best fit simulation results for the Frank Slide.  The blue line shows the simulated impact area, the purple line shows the best fit results when Dan3D-Flex is not used.  The black outline shows the observed impact area...................................................................................... 316 Figure A-38: Mapped release and deposit area extents for the Rautispitz rock avalanche (Nagelisen et al. 2015); inset photo from Rautispitz summit looking north over deposits near Lake Obersee. Coordinates are in meters of the Swiss grid system; map grid interval = 1 km. 317 Figure A-39: Rautispitz calibrated results using Dan3D-Flex.  The use of Dan3D-Flex enabled a model parameterization that could be anticipated before the event happened.  The black outline shows the observed impact area, and a minimum deposit depth of 5 m was used. .................... 318 Figure A-40: Posterior density function for the Rautispitz valley floor parameters................... 319 Figure A-41: Posterior density for the Platten path materials. ................................................... 321 Figure A-42: Platten best fit simulation results .......................................................................... 322 Figure A-43: Overview of the Guinsaugon rock avalanche.  The source zone, hummocky deposits and debris on the saturated paddy field are visible.  Images from Evans et al. (2007) licensed under Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License. ....... 323 Figure A-44: Guinsaugon best-fit simulation results. ................................................................. 325 Figure A-45: Overview of the Chehalis Lake rock avalanche and resulting landslide generated tsunami.  A: Image of the source area and bedrock slope that much of the debris deposited on.  B) Side image of the event showing runout into Chehalis Lake.  C) Destruction at a campsite located along the shore of Chehalis Lake.  D) Line of destruction caused by the landslide generated wave in Chehalis Lake.  Images courtesy of M. Brideau, reprinted here with permission. .................................................................................................................................. 326 xxix  Acknowledgements I was very fortunate to work closely with my advisor Oldrich Hungr.  I was constantly amazed by his ability to distil complex technical concepts into practical tools that are useful to practicing engineers.  I couldn’t have asked for a better supervisor, and I will miss him very much. Big thanks to my wife Sarah Partanen, for her support throughout this degree.  You’ve encouraged me every step of the way, and I’m grateful to have someone I can lean on.  Also, thanks to both of our parents and my siblings for your support and friendship throughout my degree.  I’m thankful for the opportunity to work with Scott McDougall, who has been a truly wonderful advisor.  Thanks for all your patience, support and guidance throughout my degree.  You’ve challenged me to ensure my work remains relevant, and have taught me a tremendous amount.   Thanks to my thesis and examining committee: Erik Eberhardt, Eldad Haber, Roger Beckie, Scott Burns, Davide Elmo and Brett Eaton.  Your comments and guidance are greatly appreciated. I would like to thank my office mates Afshin Amini, Wes Ashwood, Geidy Baldeon, Christina Brueckman, Graham Dick, Negar Ghahramani, Valentin Gischig, Steve Mak, Andrew Mitchell, Giona Preisig, Masoud Rahjoo, Erika Schmidt, John Whittall, Siobhan Whadcoat, Neda Zangeneh and Sophia Zubrycky for their great ideas and friendship.  Throughout my degree I’ve had amazing opportunities to collaborate with some truly wonderful colleagues.  These include Jeffrey Moore, Natalia Nolde, Andrea Wolter Tim Stark, Lorenz Grämiger, Jeffrey Coe, Pengfei Si, Marc-Andre Brideau, Marten Geertsema and Nick Roberts.  Thanks to all the people who xxx  provided data, including Philip Deline, Kiminori Araiba, Rejean Couture, Giovanni Crosta, Steve Evans and Tom Badger.  This work was funded by the Natural Sciences and Engineering Research Council of Canada, as well as scholarships given by The Department of Earth, Ocean and Atmospheric Sciences.1  Chapter 1: Introduction Extremely rapid, flow-like landslides (defined in Section 1.2) are a significant hazard worldwide, and landslide risk management is one of the premier challenges faced by geotechnical engineers and engineering geologists.  Despite decades of research into managing landslide hazards, the recent fifty million dollar settlement that resulted from the tragic Oso landslide in 2014 highlights the need to better assess and quantify the risk associated with these hazards (Bernton, 2016).  For flow-like landslides, quantifying the potential destructiveness of an event involves an analysis of the motion of the landslide.  When predicting landslide motion, it is necessary to anticipate the impact area, velocity and deposit depth of an event before it occurs.  A wide variety of empirical and numerical techniques have been developed in order to predict these quantities.  Predictions made with empirical techniques are typically imprecise, and provide limited information on flow depths or velocity.  For this reason, numerical techniques to predict landslide motion are often used for detailed assessments.   Numerical techniques to predict landslide motion can be broadly divided into those based on fundamental principles (e.g. Iverson & George, 2014), and those that are based on the “equivalent fluid principle”, formally defined by Hungr (1995).  The application of the equivalent fluid principle involves replacing the complex and heterogeneous landslide mass with an ‘equivalent fluid’ whose behaviour is governed by simple internal and basal rheologies.  The rheological parameters that govern these rheologies are not true material properties and can only be determined through calibration via back-analysis.  The equivalent fluid principle greatly 2  simplifies the analysis of landslide motion, although forward prediction becomes challenging due to the need to calibrate the models.  Model calibration for forward prediction is generally achieved by using back-analysed parameters of cases similar to the case of interest.  Applying this approach relies on an understanding of whether the mechanisms governing one case history will be the same for a future case of interest.   As summarized in Chapter 2, there have been a wide variety of equivalent fluid models developed to simulate the motion of extremely rapid, flow-like landslides.  However, forward predictions using these models are not routine.  There are a variety of reasons for this.  Some types of flow-like landslide exhibit coherent behaviour before turning flow-like, a feature not captured in previous equivalent fluid models.  Model calibration is typically performed using trial-and-error back-analysis, which can lead to subjectivity in the back-analysed parameter values.  Finally, the movement mechanisms governing many types of flow-like landslides are not well understood, and this limits the ability to parameterize models for forward analysis. 1.1 Research Objectives and Thesis Structure The objective of this thesis is to address the deficiencies noted above.  These are: 1. Develop a new rheology appropriate for the simulation of liquefied material. 2. Develop a new numerical model to simulate the motion of initially coherent landslides. 3. Develop a new calibration methodology that removes some of the subjectivity associated with trial-and-error calibration. 3  4. Compile and analyse a database of rock avalanche case histories in order to explore rock avalanche movement mechanisms.   5. Develop a new methodology that uses the rock avalanche database to perform probabilistic forward analysis. 6. Back-analyse the Johnsons Landing debris avalanche to infer its movement mechanisms, and comment on the implication of this case to the analysis of debris avalanche motion. 7. Back-analyse two flowslides in overconsolidated, glaciolacustrine material in order to infer movement mechanisms. Only three types of extremely rapid, flow-like landslides have been considered in this work: rock avalanches, debris avalanches and flowslides (see Section 1.2 for definitions of these landslide types); some of the techniques developed should be applicable to a wider variety of landslides.  Much work needs to be done in order to adapt these techniques to other flow-like landslides, such as debris flows.  However, a framework for calibration and probabilistic forward analysis, as well as a methodology to explore mechanisms controlling movement, has been developed in this work and should be broadly applicable. The thesis is structured as follows.  The rest of this chapter is devoted to a discussion of landslide classification and landslide risk analysis.  Chapter 2 provides an overview of numerical modelling of extremely rapid, flow-like landslides, and introduces a new rheology that is appropriate for simulating liquefied material.  Chapter 3 describes a new model that can be used to simulate semi-coherent landslides.  Chapter 4 summarizes a new calibration methodology that removes much of the subjectivity associated with trial-and-error calibration.  Chapter 5 discusses 4  the back-analysis of thirty rock avalanche case histories.  The results of this calibration exercise are used to infer movement mechanisms governing rock avalanche motion.  Additionally, a procedure to make probabilistic predictions based on this database is developed.  Chapter 6 presents a back-analysis of the Jonhsons Landing debris avalanche, and discusses its implications for the prediction of debris avalanche motion.  Chapter 7 provides an analysis to explore the mechanisms governing the movement of two unusual flowslides in overconsolidated, glaciolacustrine clay.  Chapter 8 presents conclusions and recommendations for future work. To date, the results from this thesis have been presented in the following peer-reviewed publications: 1.1.1 Accepted Journal Papers 1. Si, P., Aaron, J., McDougall, S., Lu, J., Yu, X., Roberts, N. J., & Clague, J. J. (2017). A non-hydrostatic model for the numerical study of landslide-generated waves. Landslides.  2. Aaron, J., Hungr, O., Stark, T., Baghdady, A. (2017a) Oso Landslide of March 22, 2014 in Washington – Dynamic Analysis. ASCE Journal of Geotechnical and Geoenvironmental Engineering, 143(9). 3. Moore, J., Pankow, K., Ford, S., Koper, K., Hale, M., Aaron, J., Larsen, C. (2017) Dynamics of the Bingham Canyon rock avalanches, Utah, USA. Journal of Geophysical Research: Earth Surface, 122: 615–640. 4. Aaron, J., McDougall, S., Moore, J., Coe, J., Hungr, O (2017b) The role of initial coherence and path materials in the dynamics of three rock avalanche case histories (Invited Paper)  Geoenvironmental Disasters, 4(5): 15p. 5  5. Stark, T., Baghdady, A., Hungr, O., Aaron, J. (2017) Oso Landslide of 22 March 2014 – Material Properties and Failure Mechanism.  ASCE Journal of Geotechnical and Geoenvironmental Engineering, 143(5).  6. Aaron, J & Hungr, O. (2016b) Dynamic Simulation of the Motion of Partially-Coherent Landslides.  Engineering Geology, 205: 1-11. 7. Aaron, J & Hungr, O. (2016a) Dynamic Analysis of an Extraordinarily Mobile Rock Avalanche in the Northwest Territories, Canada.  Canadian Geotechnical Journal, 53(6): 899-908 8. Castleton, J., Moore, J., Aaron, J., Christl, M., Ivy-Ochs, S. (2016). Dynamics and legacy of 4.8 ka rock avalanche that dammed Zion Canyon, Utah.  GSA Today, 26(6): 4-9 9. Grämiger, L., Moore, J., Vockenhuber, C., Aaron, J., Hajdas, I., Ivy-Ochs, S. (2016) Two early Holocene rock avalanches in the Bernese Alps (Rinderhorn, Switzerland).  Geomorphology, 268: 207-221.  10. Hungr, O., Aaron, J. (2013) Stability and failure behaviour of the Vajont Landslide.  Italian Journal of Engineering Geology and Environment,6: 51-62.  6  1.1.2 Peer Reviewed Conference Papers 1. McDougall, S.,  Aaron, J. (2017) Probabilistic forecasting of landslide runout.  (Invited Paper)  North American Symposium on Landslides, The United States of America, Roanoke, 4-8 June  2. Aaron, J., Hungr, O., McDougall, S. (2016b) Re-examination of the dynamics of the  2012 Johnsons Landing debris avalanche.  GeoVancouver, Vancouver, Canada, 3-5 October 3.  Aaron, J., Hungr, O., McDougall, S. (2016a) Development of a systematic approach to calibrate equivalent fluid runout models. International Symposium on Landslides, Naples, Italy, 12-19 June 4. Marinelli, G., Aaron, J., Borgatti, L., Jordan, P., & Hungr, O. (2015). Back Analysis of Johnsons Landing 2012 Landslide Using Two Dynamic Analysis Models. In G. Lollino, D. Giordan, G. B. Crosta, J. Corominas, R. Azzam, J. Wasowski, & N. Sciarra (Eds.), Engineering Geology for Society and Territory - Volume 2: Landslide Processes (pp. 1267–1270).  5. Aaron, J., Hungr, O (2014) Simulation of the motion of semi-coherent landslides.  Geohazards 6, Kingston Ontario, 15 June 1.2 Landslide Classification The term “landslide” refers to a diverse set of gravitational mass movements, from the failure of a single block of rock to the failure of hundreds of millions of cubic metres of debris.  The failure 7  velocity of landslides varies over approximately 10 orders of magnitude, from 16 mm/yr to over 5 m/s (Hungr et al., 2014).  All landslides have the potential to cause loss of life and/or economic losses, however, the tools used to analyse the risk of these events vary greatly depending on the type of landslide being assessed.   One convenient way to classify landslides is based on their failure velocity, as the mitigation strategy used for a specific landslide varies depending on this characteristic.  A table showing the velocity classes proposed by the International Geotechnical Societies’ Unesco Working Party on World Landslide Inventory (WP/WLI, 1995), and modified to include the human response (defined as the best course of action that can be taken to mitigate the hazard) assessed by Hungr (1981) is shown in Table 1-1.  As can be seen in this table, the human response to very rapid and extremely rapid landslides is nil, indicating that risk mitigation of these landslides must involve a prediction of their dynamics before they occur.      8  Table 1-1: Landslide velocity scale from WP/WLI (1995) updated with human response by Hungr et al. (2014).   Velocity Class Description Velocity (mm/s) Typical Velocity Response* 7 Extremely rapid 5.00E+03 5 m/s Nil 6 Very rapid 5.00E+01 3 m/min Nil 5 Rapid 5.00E-01 1.8 m/h Evacuation 4 Moderate 5.00E-03 13 m/month Evacuation 3 Slow 5.00E-05 1.6 m/year Maintenance 2 Very Slow 5.00E-07 16 mm/year Maintenance 1 Extremely Slow   Nil *Based on Hungr (1981) Another useful way to classify landslides is based on movement type.  Hungr et al. (2014) identified six different movement types, summarized in Figure 1-1.  The techniques used to model the various landslide movement types are different, as the dominant physical processes driving the movement are different.  Combining the velocity and movement type classification schemes provides a useful division when developing tools to predict the characteristics of landslides relevant to risk analysis.  Models can be developed that are applicable to landslides that share velocity and movement type classes.  The focus of this work is on very rapid to extremely rapid, flow-like landslides.   9  LandslideFall Topple Slide Spread FlowSlope Deformation Rock/Ice Fall Boulder/Debris/Silt Fall Rock Block Topple Rock Flexural Topple Gravel/Sand/Silt Topple Rock Rotational Slide Rock Planar Slide Rock Wedge Slide Rock Compound Slide Rock Irregular Slide Clay/Silt Rotational Slide Clay/Silt Planar Slide Gravel/Sand/Debris Slide Clay/Silt Compound Slide Rock Slope Spread Sand/Silt Liquefaction Spread Sensitive Clay Spread Rock/Ice Avalanche Sand/Silt/Debris Dry Flow Sand/Silt/Debris Flowslide Sensitive Clay Flowslide Debris Flow Mud Flow Debris Avalanche Debris Flood Earth Flow Peat Flow Mountain Slope Deformation Rock Slope Deformation Soil Slope Deformation Soil Creep Solifluction Figure 1-1: Hungr et al. (2014) landslide classification based on movement type.  Landslide types highlighted in yellow show those that are very rapid to extremely rapid. Very rapid to extremely rapid, flow-like landslides include six major classes of mass movements (Hungr et al., 2014).  These are debris and mud flows, debris floods, debris avalanches, flowslides and rock avalanches.  Formal definitions of these landslide types, as proposed by Hungr et al. (2014), are shown in Table 1-2.  Risk analysis for this subset of landslides is discussed in the following section.   10  Table 1-2: Landslide type definitions (Hungr et al. 2014) Landslide Type Hungr et al. (2014) definition Debris Flow “Very rapid to extremely rapid surging flows of saturated debris in a steep channel.  Strong entrainment of material and water from the flow path” Debris Flood “Very rapid flow of water, heavily charged with debris, in a steep channel.  Peak discharge comparable to that of a water flood” Mud Flow “Very rapid to extremely rapid surging flow of saturated plastic soil in a steep channel…strong entrainment of material and water from the flow path” Debris Avalanche “Very rapid to extremely rapid shallow flow of partially or fully saturated debris on a steep slope, without confinement of an established channel.  Occurs at all scales” Flowslide “Very rapid to extremely rapid flows…involving excess pore pressure or liquefaction of material originating from the landslide source.”   Rock Avalanche “Extremely rapid, flow-like motion of fragmented rock from a large rock slide or rock fall” 1.3 Landslide Risk Assessment Landslide risk is typically quantified using Equation [ 1.1 ] (e.g. Fell et al., 2005; APEGBC, 2010; Porter & Morgenstern, 2013; Corominas et al., 2014).   𝑅𝑖𝑠𝑘 = ∑ 𝑃(𝐻)𝑖𝑃(𝑆|𝐻)𝑖𝑃(𝑇|𝑆)𝑖𝑛𝑖=1𝑉𝑖𝐸𝑖 [ 1.1 ]  Where:  i is the landslide scenario number.  For example, this could be a specific landslide type and volume range. 11   𝑃(𝐻) is the annual frequency of an event occurring.  This is usually assessed as a frequency of events per year, for example, number of debris flows of a given volume per year.  𝑃(𝑆|𝐻) is the spatial probability of the landslide reaching the element at risk.    𝑃(𝑇|𝑆) is the probability the element at risk will be present if the landslide reaches its location.    𝑉 is the vulnerability of the element at risk to the event.  This term captures the degree of damage that occurs given that it is in the area affected by the hazard.    𝐸is the value of the element at risk, or in the case of life loss risk, it is the number of people at risk. Performing landslide risk analysis requires the evaluation of all the terms in Equation [ 1.1 ]; However, the tools and techniques used to evaluate each term are different.  Evaluation of 𝑃(𝐻) requires probabilistic analysis of slope stability that accounts for temporal factors (such as seasonal precipitation) and/or a determination of event frequency.  To quantify 𝑃(𝑇|𝑆), a study of the behavioural habits of the elements at risk is performed.  To quantify 𝑃(𝑆|𝐻) and 𝑉, a runout analysis is performed.  This research focuses on the development of tools that can perform runout analyses.  Chapter 2 provides an overview of tools and techniques that can be used in this context. 12  Chapter 2: Overview of Dynamic Model and Rheologies 2.1 Introduction The purpose of this chapter is to present an overview of numerical models used to perform runout analysis, as well as an overview of DanW and Dan3D, the numerical models used in the present work.  Firstly, an overview of empirical and analytical runout models is presented.  Then, the governing equations of DanW and Dan3D are presented and discussed.  Finally, an overview of common rheologies used to calculate basal resistance to motion is provided, and a new rheology appropriate for liquefied material is introduced. 2.2 Overview of Empirical Methods When performing landslide risk assessment of very rapid to extremely rapid, flow-like landslides, there are many different tools that can be used to determine 𝑃(𝑆|𝐻) and 𝑉 (Equation [ 1.1 ]).  These tools can be classified on a spectrum that ranges from empirical to analytical.  Empirical runout estimation techniques are often based on landslide geometry and include correlations based on angle of reach (Hsu, 1975; Li, 1983; Hungr, 1990; Corominas, 1996; Davidson, 2011; Whittall et al., 2017) as well as correlations based on source volume-deposit area relationships (Iverson et al., 1998; Griswold & Iverson, 2008). 2.3 Overview of Analytical Methods Analytical techniques to perform landslide runout analysis are based on mechanics, as opposed to correlations in historical data.  These techniques allow for site-specific geometry, materials and initial conditions to be accounted for, making them appropriate for site-scale risk analyses.  13  Most of these models solve equations of motion derived from solid or fluid mechanics.  The following is an excerpt from Aaron & Hungr (2016b). A large number of analytical runout models have been introduced in the literature over the last two decades.  A summary was assembled during the landslide runout benchmarking exercise run in 2007 by the Geotechnical Engineering Office in Hong Kong (see Hungr et al. 2007), dividing models into the following groups, based on several criteria:  Dimensions: a) Two-dimensional models assume zero deformation in the y-direction (perpendicular to the direction of motion), similar to plane strain conditions such as flow in a prismatic channel with zero side friction. b) Pseudo-3D models take into consideration user-specified flow width variations in the volume-conservation equation, while neglecting momentum flux in the y-direction (Hungr, 1995). c) Three-dimensional models track both volume continuity and momentum in three dimensions. Hydraulics literature often labels dimensionality in mathematical terms, based on the governing equations.  In such convention, depth integrated 3D models are labelled two-dimensional. This convention will not be used in this thesis.   14  Basic approach:   1) Differential models are based on a complete solution of the partial differential equations of fluid continuum motion.  In this category belong advanced fluid dynamic models such as LS-Dyna (Kwan et al., 2015 ), OpenFOAM (Boetticher et al., 2011) and TOCHNOG (Crosta et al., 2003).  These programs have extensive capabilities to deal with complex geometries, but are highly demanding on computational capacity and input data. 2) Depth-integrated models derive from methods of unsteady flow routing, routinely used in shallow-flow hydraulics (e.g. Chow, 1959).  Some are simply extensions of the Navier Stokes solutions for non-Newtonian fluid flows (e.g. FLO-2D, O'Brien et al., 1993).  Savage & Hutter (1989) introduced non-hydrostatic stress distribution within the stream column, whereby the interior of the flowing mass is considered to be a frictional medium in a state of shear flow. The internal deformation is controlled by an internal friction angle, while basal shear occurs with a different angle of friction (the “SH model”).  Hungr (1995) extended the SH model to consider the possibility that basal shearing is non-frictional and depends on a selectable rheological kernel.  Several three-dimensional models based on the SH theory appeared subsequently (e.g. Chen & Lee, 2002; Pirulli, 2005; McDougall, 2006).  Hungr (2008) proposed a stress correction for cases of deeper flows that violate the assumption of shallow flow that is implicit in the SH model.  As summarized in the following sections, DanW and Dan3D are depth-integrated models. 3) Discrete particle models represent the failed material as an assemblage of spheres or other objects (Campbell & Brennen, 1985; Poisel et al., 2008; Imre et al., 2010).  15  Assumed processes controlling mechanical interaction between the particles are very important. Some dynamic models attempt to incorporate a full mathematical description of a constitutive relationship, derived from micro-mechanics of an assumed granular material or mixture (e.g. Iverson & George, 2014).  Most, however, take a semi-empirical approach, termed “equivalent fluid model” by Hungr (1995).  Equivalent fluid models are described below in more detail.   Most recently, bi-phase flow models have been introduced, which allow for the momentum of the solid and fluid phases to be followed simultaneously (Iverson & Denlinger, 2001; Iverson & George, 2014).  These models require algorithms for mechanical linking of the motion of the two phases and tend to be both complex and demanding of data and constitutive relationships. 2.4 Equivalent Fluid Framework In the semi-empirical equivalent fluid framework, the complex and heterogeneous sliding mass is replaced by an “equivalent fluid” whose behaviour is governed by simple internal and basal rheologies.  The parameters that govern these rheologies are not material properties and can only be determined by back-analysis.  Most models include fixed basal resistance rheologies: frictional (Savage & Hutter, 1989), Bingham (Jeyapalan, 1981; Dent & Lang, 1983; Voight & Sousa, 1994; Huang et al., 2012; Dai et al., 2014),  Herschel-Bulkley (Coussot & Proust, 1996; Imran et al., 2001), or the Voellmy rheology (Christen et al., 2010).  Some models allow a selection of basal rheologies (e.g. Hungr, 1995 ; Bouchut et al., 2003;Pitman et al., 2003;Pirulli, 2005; McDougall, 2006; Pastor et al., 2009).  The various characteristics of some of these basal rheologies are discussed in Section 2.5.2.   16  As will be discussed in Chapter 4, back-analysis is used to determine the values of the parameters that govern the internal and basal rheologies.  The back-analysis process involves matching bulk features of the landslide to simulation results.  These features include the landslide trimline, deposit distribution, landslide velocities and event duration (Hungr & McDougall, 2009). The remainder of this chapter is devoted to providing an overview of the models DanW and Dan3D, which are used extensively throughout this work to address the thesis objectives detailed in Section 1.1. 2.5 DanW and Dan3D Overview DanW, and its three-dimensional extension Dan3D, are depth-averaged Lagrangian implementations of the equivalent fluid approach (Hungr, 1995; McDougall & Hungr, 2004).  The equations that govern these models are similar to the shallow water equations, however, the formulation allows for an anisotropic internal stress distribution using the Savage & Hutter (1989) principles.   The following discussion is an excerpt from Aaron et al. (2017b). The governing equations solved by Dan3D are summarized in Equations [2.1] and [2.2] (McDougall, 2006).  In DanW, only Equation [2.1] is solved.  Only the final form of the equations used in the model are presented; a detailed derivation is presented by McDougall (2006) and Hungr & McDougall (2009).  These equations are depth-averaged and derived in a Lagrangian coordinate system, with the x-coordinate aligned with the local direction of motion and the z-coordinate oriented in the bed normal direction.   17  𝜌ℎ𝐷𝑣𝑥𝐷𝑡= 𝜌ℎ𝑔𝑥 − 𝑘𝑥𝜎𝑧𝜕ℎ𝜕𝑥+ 𝜏𝑧𝑥 − 𝜌𝑣𝑥𝐸  [2.1]  𝜌ℎ𝐷𝑣𝑦𝐷𝑡= 𝜌ℎ𝑔𝑦 − 𝑘𝑦𝜎𝑧𝜕ℎ𝜕𝑦  [2.2]  Where 𝜌 is the density, 𝑣𝑥,𝑦 are the depth-averaged x and y velocities, ℎ is the flow depth, 𝑔𝑥,𝑦 are the x and y components of gravity, 𝑘𝑥,𝑦 are the x and y horizontal stress ratios (ratio of lateral stress to bed normal stress) calculated based on the Savage-Hutter method (Savage & Hutter, 1989), 𝜎𝑧 is the bed normal stress, 𝜏𝑧𝑥 is the basal resistance, and 𝐸 is the entrainment rate.  An overview of the Savage-Hutter method is provided in Section 2.5.1.  A free body diagram that shows the forces acting on a slice of material oriented in the direction of motion is displayed in Figure 2-1.  The first term on the right hand side of Equations [2.1] and [2.2] represents the gravitational stress (the downslope component of the W force in Figure 2-1) , while the second term represents the longitudinal pressure gradient (P force in Figure 2-1).  The basal resistance stress (T force in Figure 2-1) and momentum loss due to entrainment (E force in Figure 2-1) only occur in the x-direction due to the fact that the x-coordinate is aligned with the local direction of motion.  The entrainment rate (𝐸) and density, as well as the parameters that govern 𝑘𝑥,𝑦 and 𝜏𝑧𝑥, are user-specified.   18   Figure 2-1: Conceptual free body diagram of a slice of material oriented in the direction of motion in Dan3D.  W is the weight, T is the basal resistance, P is the internal force due to free surface gradients, and E is the inertial resistance due to entrainment. The solution method used by DanW discretizes the mass into a set of trapezoidal mass blocks.  The equations of motion are solved at the interface between these blocks (Hungr, 1995). The solution method used by Dan3D is smooth particle hydrodynamics (SPH) (Monaghan, 1992).  SPH is a mesh-free method, which allows for bifurcations and large strains to be simulated without mesh distortion problems.   When performing a back-analysis with DanW and Dan3D, the parameters that are commonly calibrated are the internal friction angle (used to calculate 𝑘𝑥,𝑦) and parameters associated with the user-specified basal rheology (used to calculate 𝜏𝑧𝑥).  The entrainment rate is sometimes a calibrated parameter, although it is common to evaluate this parameter based on known estimates of initial and final volumes (McDougall & Hungr, 2005).  The parameterization of 𝑘𝑥,𝑦 will be discussed in Section 2.5.1, and the parameterization of  𝜏𝑧𝑥 will be discussed in Section 2.5.2. 19  2.5.1 Savage-Hutter Method Savage & Hutter (1989) derived an expression for the horizontal stress ratio in the direction of motion (kx) for a depth-averaged 2D model based on the assumption that the material is simultaneously failing internally and sliding along the base (this expression is shown in Equation [ 2.3 ]).  Their derivation was based on the Mohr’s circle shown in Figure 2-2.   𝑘𝑥(𝑚𝑖𝑛/𝑚𝑎𝑥) = 2 (1 ± √1 −  𝑐𝑜𝑠2𝜙𝑖(1 + 𝑘𝑧𝑥2 )𝑐𝑜𝑠2𝜙𝑖) − 1   [ 2.3 ]  Where 𝜙𝑖 is the internal friction angle (a user specified parameter) and 𝑘𝑧𝑥2  is the square of the friction slope (ratio of basal shear stress to vertical stress).    Figure 2-2: Mohr circle of stress for a 2D element simultaneously undergoing internal shear failure and basal slip (after Hungr, 2008).   20  As shown in Equation [ 2.3 ] and Figure 2-2, there are two possible values of 𝑘𝑥.  The value of 𝑘𝑥 depends on the sense of internal shearing.  The maximum value of this coefficient corresponds to the passive stress condition, which occurs when material is compressing, and the minimum value corresponds to the active stress condition, which occurs when material is stretching.  Based on the assumption that principle stresses are aligned with the direction of motion, McDougall (2006) presented Equations [ 2.4 ] and [ 2.5 ] to calculate values for the horizontal stress ratio (ky) in the lateral direction.  As above, the minimum and maximum values of ky correspond to the active and passive conditions respectively.      𝑘𝑦(𝑚𝑖𝑛) = (𝑘𝑥 + 12+ √(𝑘𝑥 − 12)2 + (𝑘𝑧𝑥2 )2) (1 − sin (𝜙𝑖)1 + sin (𝜙𝑖))  [ 2.4 ]  𝑘𝑦(𝑚𝑎𝑥) = (𝑘𝑥 + 12− √(𝑘𝑥 − 12)2 + (𝑘𝑧𝑥2 )2) (1 + sin (𝜙𝑖)1 − sin (𝜙𝑖))  [ 2.5 ]  2.5.2 Basal Rheologies The  𝜏𝑧𝑥 term in Equation [2.1] is parameterized through the use of basal rheologies.  DanW and Dan3D both feature open rheologic kernels, so various basal rheologies can be used.  Through the use of the appropriate rheology, these models can be used to reproduce bulk properties of the range of flow-like landslides described in Section 1.2.  To do this, however, the user must specify an appropriate rheology for a given flow-like landslide type.   21  Three commonly used rheologies (the frictional, Voellmy and Bingham rheologies), as well as a new rheology appropriate for liquefied materials, are described below.  The frictional rheology is shown in Equation [ 2.6 ]: 𝜏𝑧𝑥 =  −𝜎𝑧 tan (∅𝑏) [ 2.6 ] where 𝜎𝑧 is the bed-normal effective stress and ∅𝑏 is the calibrated bulk friction angle, which includes pore-pressure effects.   The Voellmy rheology, given in Equation [2.7 ] is similar to the frictional rheology, with an additional velocity-dependent term (e.g. Voellmy, 1955; Koerner, 1976). 𝜏𝑧𝑥 =  −(𝜎𝑧 𝑓 +  𝜌𝑔𝑣𝑥2𝜉) [2.7 ] where 𝑓 is the friction coefficient (equivalent to tan (∅𝑏)) and 𝜉 is the turbulence coefficient (sometimes referred to as the turbulence ‘parameter’).  Both 𝑓 and 𝜉 are calibrated parameters.  The Bingham rheology, given by Equation [2.8 ], does not assume that the basal resistance is proportional to the bed-normal effective stress: 𝜏𝑧𝑥3 + 3 (𝜏𝑦𝑖𝑒𝑙𝑑2+𝜇𝐵𝑖𝑛𝑔ℎ𝑎𝑚𝑣𝑥ℎ) 𝜏𝑧𝑥2 −𝜏𝑦𝑖𝑒𝑙𝑑32= 0 [2.8 ] where 𝜏𝑦𝑖𝑒𝑙𝑑 is the yield stress and 𝜇𝐵𝑖𝑛𝑔ℎ𝑎𝑚 is the viscosity; both of these parameters are calibrated.  It should be noted that flow resistance calculated by the Bingham rheology is independent of normal stress.      22  2.5.3 New Liquefied Rheology Some researchers (e.g. Hungr & Evans, 2004; Hungr & McDougall, 2009) have noted that the use of the frictional and Voellmy rheologies implies very specific drainage conditions in the basal shear layer.  The friction coefficient used in the frictional and Voellmy rheologies can be expressed as the tangent of a “bulk” friction angle, defined in Equation [2.9].  f =  tan (∅𝑏) = (1 − ru) ∗  tan(∅)          [2.9]  where ∅𝑏 is the bulk friction angle, ru is the pore pressure ratio (ratio of pore pressure to bed-normal stress) and ∅ is the friction angle between the sliding mass and the rupture surface.  The use of a bulk friction angle assumes that ru is constant, which implies that, whenever the bed-normal stress changes (i.e. with a change in depth), the pore pressure changes in a linear proportion, so that the pore pressure ratio remains constant.  For the case of undrained failure, this assumption is inappropriate because it implies changes in effective stress resulting from changes in total stress.  As summarized in Section 1.2, some landslides that are analyzed in this thesis travelled in an undrained condition.   This has important implications when analyzing flow-like landslides that experience significant centripetal acceleration. In a drained, frictional material, centripetal accelerations would increase flow resistance.  However, if the mass were in an undrained condition, the basal resistance stress would not increase due to the added bed-normal stress, and the flow would experience less resistance compared to a frictional material.   Due to these considerations, a new rheology, appropriate for liquefied material, was implemented in both DanW and Dan3D.  This rheology, termed the ‘liquefied Voellmy 23  rheology’ replaces the frictional term in the Voellmy rheology (the first term on the right of Equation [2.7 ]) with a constant liquefied strength that is proportional to the prefailure bed-normal effective stress (Olson & Stark, 2002).  No changes were made to the turbulence term.  The equation for the liquefied Voellmy rheology is given in Equation [2.10].    𝜏𝑧𝑥 = −(𝑠𝑙 +  𝜌𝑔𝑣𝑥2𝜉)       [2.10] where 𝑠𝑙 is the constant liquefied strength calculated using Equation [2.11]: 𝑠𝑙 = 𝑅𝑙𝑖𝑞 ∗ (𝜎𝑛,𝑖 − 𝑢𝑖)                [2.11] where 𝑅𝑙𝑖𝑞 is the liquefied strength ratio,  𝜎𝑛,𝑖 is the initial bed-normal effective stress acting on element i, and 𝑢𝑖 is the initial pore pressure acting on element i.  Olson & Stark (2002) back-analysed 33 liquefaction flow failures in loose, granular material and found that the 𝑅𝑙𝑖𝑞  values range from about 0.02 to 0.1, with most back-analysed values clustered around 0.07.  To implement the liquefied Voellmy rheology into the two dynamic models (DanW and Dan3D), the following procedure is used.  During the first time step, the pre-failure bed-normal effective stress is calculated based on an input ru value.  Then, Equation [2.11] is used to calculate a value of liquefied strength for each computational element.  This liquefied strength is then held constant for the entire simulation.  It should be noted that using constant liquefied strength for the entire duration of the simulation assumes that no drainage occurs during the runout process.  However, the use of a turbulent resisting stress may mimic the effect of pore pressure dissipation due to shear-induced dilation (Hungr & Evans, 2004).  24  As summarized in Aaron et al. (2017a), when the liquefied rheology is used, the basal shear strength is independent of total stress changes, which differentiates it from the frictional and Voellmy rheologies commonly used to analyse extremely rapid flow-like landslides (e.g. Hungr & McDougall, 2009).  The Bingham rheology (e.g. Jeyapalan, 1981) assumes a constant yield strength, however, unlike the liquefied rheology, this strength is constant and independent of pre-failure vertical effective stress. When the liquefied rheology is used, material deposits when the driving gravitational stress (which depends on flow thickness) falls below the value of the constant liquefied strength.  Due to this, material is simulated to deposit in a long, thin sheet covering the length of the path.  This is also the case for the Bingham rheology, however, it is different from the frictional and Voellmy rheologies, where material is simulated to deposit when the local slope angle is less than the bulk basal friction angle.  The difference in the predicted shape of the deposit can be used to justify the use of the liquefied (and Bingham) rheologies in a back-analysis context if field observations are consistent with this deposit morphology.  This will be done in Chapters 5 and 6. 2.5.4 Behaviour of DanW and Dan3D In the analysis that follows, the equations are simplified by ignoring centripetal acceleration and entrainment terms and using the frictional rheology to calculate the basal resistance stress.  This allows for the derivation of simplified equations that demonstrate the behaviour of DanW and Dan3D.  Only the x-direction equation of motion is considered for this analysis.  By making these assumptions, the equation of motion reduces to: 25  𝐷𝑣𝑥𝐷𝑡= 𝑔 sin(𝛼) + 𝑔𝑘𝜕ℎ𝜕𝑥− 𝑔 tan(∅𝑏) cos (𝛼) [2.12] where 𝛼 is the slope angle. Through algebraic rearrangement, this equation can be put in the following form:  𝐷𝑣𝑥𝐷𝑡=𝑔cos(𝛼)(tan(𝛼) − tan(∅𝑏)) + 𝑔𝑘𝜕ℎ𝜕𝑥 [2.13 ] The first term on the right hand side captures the gravitational acceleration and basal resistance to movement, similar to a block sliding down an inclined plane.  The second term on the right hand side expresses the acceleration due to internal pressure gradients.  It is this term that differentiates equivalent fluid models from rigid body models, such as lumped mass models (Heim, 1932).  From Equation [2.13 ], it can be seen that Dan3D simulates two mechanisms that drive landslide motion.  The mass will accelerate when the slope angle is greater than the friction angle, as in the initial path of many rock avalanches, or when there is a strong enough free surface gradient ( 𝜕ℎ𝜕𝑥 in Equation 10 and P force on Figure 4), as in many flowslides.  Equation [2.13 ] also demonstrates that, when a frictional rheology is used and the free surface gradient is small, the mass will only decelerate when the slope angle is less than the friction angle. 2.6 Summary This chapter provided an overview of DanW and Dan3D, the equivalent fluid runout models used in this thesis.  The depth averaged governing equations solved by Dan3D are similar to the shallow water equations, although they feature a number of landslide-specific features.  These 26  features include the simulation of entrainment, anisotropic internal stress distributions and an open rheological kernel. Through the use of different rheologies, the bulk properties of a variety of extremely rapid, flow-like landslides can be reproduced.  Three commonly used rheologies, the frictional, Voellmy and Bingham rheologies, as well as a new rheology appropriate for liquefied materials, were introduced in this chapter.  The parameters that govern these rheologies are empirical, and can only be determined through back-analysis. These rheologies will be used throughout Chapters 3 to 7.    27  Chapter 3: Flexible Block Model This chapter is a lightly edited excerpt from Aaron & Hungr (2016b), reprinted with permission. All models reviewed in Chapter 2 are based on the principles of fluid mechanics, i.e. they do not consider elastic rigidity of the moving material.  However, landslides develop in material which is initially solid and only transitions into a fluid state during movement, as described in the case histories presented later in this chapter. In some landslide types this transition occurs rapidly, e.g. in flowslides (Hungr et al., 2014).  In others, it may be more gradual, as the landslide commences sliding movement as a rigid, or semi-rigid block, then gradually disintegrates into a fluid.  This occurs commonly in rock avalanches.  Fluid mechanics models, applied to such cases, commonly predict excessive lateral and longitudinal spreading of the sliding mass in the source area.  The extent of the path as well as the runout can be substantially distorted by this.  The model described in this paper provides an extension to an existing SH type model that allows for the simulation of this type of extremely rapid landslide. 3.1 Initially Coherent Rock Avalanches It is uncommon for rock avalanches to be in a state of fully developed internal deformation at the onset of their motion.  Instead, many rock avalanches initiate as translational slides which only gradually fragment and turn flow-like.  The physics that govern these two phases of motion are different.  The initial coherent stage is best described by solid mechanics, whereas the flow-like portion is well described with a fluid mechanics solution in the context of an SH model as discussed earlier.  An example of this behaviour can be seen in Figure 3-1 which shows a photo taken during a flyover of the North Nahanni Slide (Wetmiller et al., 1987).  The major part of the 28  deposit features a large intact block covered by vegetation, while the distal part is fluid-like.  The dynamics of the block would be poorly described by fluid mechanics.   Many researchers have noted excessive lateral spreading in the source zone when applying Dan3D to initially coherent rock avalanches (Chalindar, 2005; McDougall, 2006; Fitze, 2010).  Two rock avalanche cases have been selected to demonstrate this problem: Goldau in Switzerland and Mystery Creek in Canada.  It is not the intention of this work to provide a detailed description of these cases.  They are presented in order to demonstrate the results produced by equivalent fluid runout models as a consequence of the implicit assumption of instant fluidization.  References are provided to papers that discuss these cases in more detail.  Figure 3-1: North Nahanni Rock Avalanche.  A large coherent block can be seen on the rupture surface near the centre of the photo, indicating a phase of coherent motion during the event. Photo: O. Hungr. 29  3.1.1 Goldau Rock Avalanche The Goldau Rock Avalanche occurred in Central Switzerland in 1806.  This tragic event claimed 457 lives, destroyed 111 houses and triggered a 20 m high wave in nearby Lake Luarez.  The failure involved the detachement of 35-40 x 106 m3 of material along a planar rupture surface of a dip slope in marlstone and conglomerate (Fitze, 2010).  A detailed investigation of the geology and future hazard potential is given by (Berner, 2004).  A contemporary photo of the source area is shown in Figure 3-2.  Numerical modelling of the failure mechanism is described by Thuro et al. (2006).  It was found that the rupture surface exploits highly weathered marlstones near the main scarp, and then cuts through conglomerate layers lower down.  Friction angles of about 23° were found within the weathered marlstone and evidence of brittle fracture was found within the conglomerates.  Based on this, it was suggested that the slope was near limit equilibrium conditions and progressive failure through the conglomerates initiated brittle failure of the whole sliding mass along a rupture surface parallel to the bedding planes.  The simulations presented here build on a previous Dan3D analysis performed by Fitze (2010).  The topography files and simulation constraints are the same as those used in that analysis.   30   Figure 3-2: Photo of the Goldau source area.  Photo: O Hungr. The model parameters used in the fully fluid simulations are shown in Table 3-1, and the spatial distribution of the parameters is shown in Figure 3-3a.  The frictional rheology is used in the source zone, and the Voellmy rheology is used where the landslide overrode and entrained loose saturated sediments, consistent with previous analyses of similar rock avalanches (Hungr & Evans, 2004; McDougall, 2006).  An overview of the basal rheologies is provided by Hungr & McDougall (2009).  The results of the fully fluid simulation are shown in Figure 3-3b and compared with the reported trimline of the actual landslide.  The fully fluid simulations predict excessive spreading of the sliding material during the early stages of the event, mainly along the unconstrained right flank of the source area, as indicated by the velocity vectors on Figure 3-3b.  31  The fluidized sliding mass is acted upon by gravitational forces which move the slide downslope, but also by lateral forces due to fluid pressure.  These forces cause the sliding mass to spread in the direction perpendicular to the downslope direction.  Comparing the width of the source volume to the width of the trimline during the initial stages shows that the actual sliding mass did not spread laterally.  Presumably, the source block began sliding along a weak bedding plane without much internal deformation.  Internal breakdown only occurred once the block accelerated to high velocity and was shaken by the irregularity of the natural path downslope from the toe of the rupture surface. Table 3-1:  Basal resistance parameters used in the simulation of the two rock avalanche case histories.  Both rock avalanches were simulated using the frictional rheology in the source zone and the Voellmy rheology along the path.  For the spatial locations of the basal rheology switch see Figure 3-3 and Figure 3-4. Case history Bulk Friction Angle (degrees) Voellmy Parameters Goldau 12 f = 0.07 𝜉 = 500 𝑚𝑠2 Mystery Creek 23 f = 0.16 𝜉 = 500 𝑚𝑠2  32   Figure 3-3: A) Initial conditions used to model the Goldau Rock Avalanche.  The frictional Rheology is used to calculate basal resistance north of the red line, and a Voellmy rheology is used south of the red line.  The parameters used in each of these rheologies are shown in Table 3-1.  B)  Final deposit shape and trimline of fully fluid simulation.  Dashed line shows the true trimline (after Fitze, 2010) and the grey area is the predicted trimline.  A cutoff of 0.3 m is necessary due to the solution method used by Dan3D.  The red vectors indicate instantaneous velocity vectors of the particles after 1 second.  The particles are moving laterally as well as downslope, contrary to the observed trimline. It can be argued that the poor fit of the simulation to the landslide trimline could be caused by poorly chosen parameters.  Fitze (2010) performed a back-analysis of the Goldau Rock avalanche using a wide range of input paramters.  No simulations were able to accurately reproduce the characteristics of this rock avalanche.  Preuth et al. (2010) also back-analysed the Goldau rock avalanche using a numerical runout model that simulates a reduction in basal shear resistance due to random kinetic energy.  A similar excessive lateral spreading problem was encountered in these simulations, and attributed to the neglect of fragmentation in their model.  This suggests that the reason for the poor fit is due to the incorrect assumption of instant fluidization.     33  3.1.2 Mystery Creek Rock Avalanche The Mystery Creek rock avalanche is located approximately 20 km north of Whistler, BC.  Carbon dating of charcoal samples found in the debris indicate a minimum age of 800 ± 100 BP (Evans & Savigny, 1994).  There are varying estimates of the volume of the detachment. Eisbacher (1983) assessed the volume of the debris deposit to be 40 x 106 m3, whereas Chalindar (2005) estimated a source volume of 20 x 106 m3 based on a field visit and air photo analyses.  For this analysis, a source volume of 20 x 106 m3 was used.  Since this is a pre-historic rock avalanche there is significant uncertainty as to the pre-slide topography, source mass geometry and impact area.  A description of the geology of the site as well as the failure mechanism is detailed in Nichol et al. (2002).   The initial conditions for both the fully fluid simulation are shown in Table 3-1 and Figure 3-4a.  Similar to the Goldau simulations, the frictional rheology is used in the source zone and the Voellmy rheology is used along the path.  The results of the fully fluid simulation are summarized in Figure 3-4b.  The simulation results in spreading far in excess of the observed trimline.  A detailed look at the topography shows that there are no topographic constraints on the motion of the relatively deep landslide along the right flank.  The fluid model is again influenced by this and responds by over-predicting the initial lateral spreading of the path. 34   Figure 3-4: A) Model setup for the simulation of the Mystery Creek Rock Avalanche.  A frictional rheology is used to calculate basal resistance in the source zone (zone 1) and a Voellmy rheology is used to calculate basal resistance outside the source zone (zone 2).  The parameters used in each of these rheologies are shown in Table 3-1.  B)  Final deposit shape predicted by fully fluid solution for the Mystery Creek Rock Avalanche.  The dashed line shows the true trimline (after Nichol et al., 2002).  Excessive lateral spreading is predicted when the mass is fluidized instantly.  3.2 Model Objectives  The objective of this work is to introduce a solid mechanics solution into DAN3D.  This will allow for the simulation of the initial phase of motion during rock avalanches while the failed 35  mass moves as a slide, before it fragments and turns flow- like.  The requirements of the model are to: 1. Simulate the translation and rotation of an arbitrarily shaped sliding body as it moves across complex 3D terrain 2. Simulate landslides that are initially coherent and then turn flow-like 3. Facilitate the back-analysis process through simplicity of use and computational efficiency Rotation as well as translation is important in the motion of coherent slides, as demonstrated in the Vaiont compound slide (Superchi, 2012), the Mt. Granier rock avalanche (Hungr et al., 2014) and the Mystery Creek rock avalanche (Nichol et al., 2002).  The model must be able to determine slide characteristics based on solid mechanics, and then allow for transition to a fluid mechanics solution during the simulation.   The length of travel which is required for the intact block to completely disintegrate and become flow-like presumably depends on the rock mass quality of the source block and detailed geometry of the path.  Unfortunately, there is no quantitative means at present to simulate the process. This parameter, therefore, will for the time being have to be determined a priori by the user and confirmed by calibration back-analysis.  Some further discussion appears below. Simplicity of use and computational efficiency are necessary due to the semi-empirical framework to which the DAN models subscribe.  The model parameters are determined through back-analysis, and making forward predictions with the model can only be done once a reasonably large dataset of back-analysed parameters exists.  It is important that the model facilitates this extensive back-analysis process. 36  3.3 Model Derivation The model is derived using an approach similar to that of limit equilibrium solutions.  Miao et al. (2001) used a similar methodology to derive a 2-D runout model.  Instead of solving for a factor of safety, the solution algorithm determines the unbalanced force and moment acting on the sliding body, which is then used to determine translational and rotational accelerations.  The accelerations are then integrated to determine translational and rotational velocities and displacements.   The original version of the model was developed specifically for the analysis of the Vaiont Slide and its use for that case is described in Hungr & Aaron (2013).  A preliminary version of the model was also presented in Aaron & Hungr (2014). 3.3.1 Model Assumptions   In the following derivation, the sliding mass is discretized into an assembly of connected columns.  The columns are rectangular and all have the same basal area.  A Cartesian coordinate system is used, and the Z axis is the vertical axis.  The model makes the following assumptions about the sliding mass, justified on the assumption that the thickness of the sliding body is much less than its lateral extent and that the strength of the geomaterials is greater in compression than in shear: 1) The planar bases of the columns are in a state of fully-developed shear failure so that normal and shear stresses on the surface can be related through a user-specified basal rheology. 2) The sliding mass behaves as a flexible block, able to shear in the Z direction but prevented from deforming in the X and Y directions.  In other words, differential 37  movement of columns is permitted in the vertical direction but not in the horizontal plane, where it is prevented from doing so by internal rigidity of the block.  The vertical differential movement is required to ensure that the sliding mass remains in contact with the sliding surface. 3) The sliding mass is assumed to have small height, compared to its length.  This assumption is analogous to the shallow flow assumption used to derive many runout models based on fluid mechanics.  The implication of this assumption is that moments about the horizontal (X and Y) axis can be ignored. 4) For the current formulation, it is assumed that inter-column vertical shear forces can be neglected.  This is the same assumption as is made in simplified limit equilibrium formulations. 5) It is assumed that the weight vector of each column is applied at the centre of the column.   6) It is assumed that the basal shear resistance vector of each column is applied where the vertical axis of the column intersects the sliding surface. 7) It is assumed that the sliding mass moves only through translation and rotation around a vertical axis.  As such, the model is not applicable to sliding events, such as rockfalls, that have some components of free fall, rolling and toppling. 3.3.2 Governing Equations The derivation of the governing equations of the model begins with the equations of linear and angular acceleration of a rigid body translating in 3D space and rotating about an arbitrary axis: 38  𝑚𝑏𝑜𝑑𝑦 ∗ (?̇?𝑥, ?̇?𝑦, ?̇?𝑧)  = (𝐹𝑥, 𝐹𝑦, 𝐹𝑧) [ 3.1 ] 𝐼𝑥 ∗ ?̇?𝑥 − (𝐼𝑦 − 𝐼𝑧) ∗ 𝜔𝑦 ∗ 𝜔𝑧 = 𝑇𝑥 [ 3.2 ] 𝐼𝑦 ∗ ?̇?𝑦 − (𝐼𝑧 − 𝐼𝑥) ∗ 𝜔𝑧 ∗ 𝜔𝑥 = 𝑇𝑦 [ 3.3 ] 𝐼𝑧 ∗ ?̇?𝑧̇ − (𝐼𝑥 − 𝐼𝑦) ∗ 𝜔𝑥 ∗ 𝜔𝑦 = 𝑇𝑧 [ 3.4 ] Where mbody is the mass of the sliding body, (?̇?𝑥, ?̇?𝑦, ?̇?𝑧)  are the translational accelerations of the body, (𝐹𝑥, 𝐹𝑦, 𝐹𝑧) are the components of the unbalanced force acting on the body, (𝐼𝑥, 𝐼𝑦, 𝐼𝑧) are the moments of inertia of the body about the three principal axes, (?̇?𝑥, ?̇?𝑦, ?̇?𝑧) are the angular accelerations of the body about the principal axes,(𝑇𝑥 , 𝑇𝑦, 𝑇𝑧)  are the components of the external torque acting on the body and (𝜔𝑥, 𝜔𝑦, 𝜔𝑧) are the components of angular velocity of the rigid body.  Due to the assumption that the sliding mass behaves as a flexible block that only moves through translation and rotation, the only translational accelerations of interest are (?̇?𝑥, ?̇?𝑦) as free fall is ignored.  Neglecting rolling and toppling of the sliding body means that only 𝑇𝑧 and 𝐼𝑧 are quantities of interest as (𝑇𝑥, 𝑇𝑦) are both assumed to be negligible and therefore (?̇?𝑥, ?̇?𝑦) and (𝜔𝑥, 𝜔𝑦) are zero.  Applying these assumptions results in the following system of equations that describe the translation and rotation of a flexible block over three dimensional terrain:  𝑚𝑏𝑜𝑑𝑦 ∗  ?̇?𝑥  = 𝐹𝑥 [ 3.5 ]  𝑚𝑏𝑜𝑑𝑦 ∗ ?̇?𝑦  = 𝐹𝑦 [ 3.6 ] 39  𝐼𝑧 ∗ ?̇?𝑧 = 𝑇𝑧 [ 3.7 ] The system of equations derived above is solved using the method of columns.  The following derivation will show how ?̇?𝑥, ?̇?𝑦 and ?̇?𝑧 are determined. For a rigid body represented by a system of columns: 𝐹𝑏𝑜𝑑𝑦 =  ∑ 𝐹𝑖𝑁𝑖=1=  ∑ (𝐹𝑥,𝑖, 𝐹𝑦,𝑖)𝑁𝑖=1 [ 3.8 ] 𝑇𝑧 =  ∑ 𝑇𝑖𝑁𝑖=1=  ∑ 𝑇𝑧,𝑖𝑁𝑖=1 [ 3.9 ] Where the subscript i represents an individual column, and N is the number of columns. Since internal forces are balanced within the moving body, the individual net forces (𝐹𝑥,𝑖, 𝐹𝑦,𝑖) acting on each column which contribute to the unbalanced force of the sliding mass are (Figure 3-5): 𝐹𝑥,𝑖 =  𝐺𝑥,𝑖 − 𝐹𝑥𝑏𝑎𝑠𝑎𝑙,𝑖  [ 3.10 ]  𝐹𝑦,𝑖 =  𝐺𝑦,𝑖 − 𝐹𝑦𝑏𝑎𝑠𝑎𝑙,𝑖 [ 3.11 ] Where Fi is the total force acting on the ith column, 𝐺 is the downslope component of gravity and Fbasal is the basal resistance force which acts in the direction of motion. The downslope component of the gravitational force on an individual column can be expressed as: 40  𝐺𝑥,𝑖 =  ℎ𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ sin(𝛼𝑖) ∗ 𝜌 ∗ 9.81 ∗ 𝑔𝑥,𝑖 [ 3.12 ]  𝐺𝑦,𝑖 =  ℎ𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ sin(𝛼𝑖) ∗ 𝜌 ∗ 9.81 ∗ 𝑔𝑦,𝑖 [ 3.13 ] Where hcolumn is the height of the column, 𝛼 the true dip of the local slope, dx and dy are the length and width of the columns, ρ is the density of the sliding material and (𝑔𝑥, 𝑔𝑦) are the x and y components of the unit vector that points in the true dip direction of the basal plane of the column. The basal force, which acts in the direction of motion, depends on the user-specified relationship between normal and shear stress along the sliding surface, which in this derivation is assumed to be frictional.  This does not exclude the possibility of implementing different basal relationships, such as a frictional heating relationship as suggested by Hendron & Patton (1985).  The parameters that govern this relationship, as well as the relationship itself can be varied spatially to account for different path materials.        41   Figure 3-5: Forces acting on a column when internal strength is neglected. G denotes the downslope component of gravity, Fn the component of gravity normal to the slope and F_basal is the resistance force.  Note Fn (blue) is used to calculate F_basal.  F_basal is not collinear with G as it acts in the direction of motion. The magnitude of the basal force, assuming frictional strength, can be expressed by: 𝐹𝑥𝑏𝑎𝑠𝑎𝑙,𝑖 = ( ( ℎ𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ 𝜌 ∗ cos(𝛼𝑖)  ) − (𝜇𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝐴𝑖 ∗ 𝛾𝑤𝑎𝑡𝑒𝑟)) ∗ tan(𝜑𝑖)  ∗ 𝑟𝑥,𝑖 [ 3.14 ]  𝐹𝑦𝑏𝑎𝑠𝑎𝑙,𝑖 = (( ℎ𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ 𝜌 ∗ cos(𝛼𝑖)  ) − (𝜇𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝐴𝑖 ∗ 𝛾𝑤𝑎𝑡𝑒𝑟)) ∗ tan(𝜑𝑖) ∗  𝑟𝑦, 𝑖 [ 3.15 ]  hcolumn dy dx 42  Where μcolumn is the height of the piezometric surface above the base of the column, γwater is the unit weight of water, φ is the friction angle, (𝑟𝑥, 𝑟𝑦) are the x and y components of the resistance force unit vector (which points in the direction of motion) and Ai is the true basal area of the column given by the formula (Hungr & Amann, 2011): 𝐴𝑖 = 𝑑𝑥 ∗ 𝑑𝑦 ∗√(1 − sin(𝛼𝑥,𝑖)2∗ sin(𝛼𝑦,𝑖)2)cos(𝛼𝑥,𝑖) ∗ cos(𝛼𝑦,𝑖) [ 3.16 ]  Where αx and αy are the apparent dip along the x and y axis respectively. If a pore pressure coefficient (𝑟𝑢) is specified instead of a piezometric surface the equations above become:  𝐹𝑥𝑏𝑎𝑠𝑎𝑙,𝑖 = ( ℎ𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ 𝜌 ∗ cos(𝛼𝑖)  ) ∗ (1 − 𝑟𝑢,𝑖) ∗ tan(𝜑𝑖) ∗ 𝑟𝑥,𝑖 [ 3.17 ]  𝐹𝑦𝑏𝑎𝑠𝑎𝑙,𝑖 = ( ℎ𝑐𝑜𝑙𝑢𝑚𝑛,𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ 𝜌 ∗ cos(𝛼𝑖)  ) ∗ (1 − 𝑟𝑢,𝑖) ∗ tan(𝜑𝑖) ∗ 𝑟𝑦,𝑖 [ 3.18 ] The resistance force unit vector is determined by subtracting the position of the column in the previous time step from the columns current position, and then normalizing the resultant vector: 𝑟𝑒𝑠𝑖 = 𝑛𝑜𝑟𝑚(𝑥𝑖𝑡−1 − 𝑥𝑖𝑡, 𝑦𝑖𝑡−1 − 𝑦𝑖𝑡, 𝑧𝑖𝑡−1 − 𝑥𝑖𝑡) [ 3.19 ]  Where the superscript t denotes the current time step, superscript t-1 denotes the previous time step, (x,y,z) are the column positions. The torque exerted by an individual column about a vertical axis passing through the centre of mass of the body can be expressed as: 43  𝑇𝑧,𝑖 = (𝐺𝑦,𝑖 − 𝐹𝑦𝑏𝑎𝑠𝑎𝑙,𝑖) ∗ 𝑥𝑖′ − (𝐺𝑥,𝑖 − 𝐹𝑥𝑏𝑎𝑠𝑎𝑙,𝑖) ∗ 𝑦𝑖′  [ 3.20 ]  Where x’ and y’ are the x and y coordinates of the column relative to the centre of mass of the moving body and the subscript i denotes an individual column.   The mass of the landslide can be expressed as the sum of the mass of the individual columns: 𝑚𝑏𝑜𝑑𝑦 =  ∑ ℎ𝑐𝑜𝑙𝑢𝑚𝑛𝑖 ∗ 𝑑𝑥 ∗ 𝑑𝑦 ∗ 𝜌𝑁𝑖=1 [ 3.21 ]  The moment of inertia about a vertical axis passing through the centre of mass is expressed as: 𝐼𝑧 =  ∑ 𝑚𝑖 ∗ 𝑅𝑖𝑁𝑖=1 [ 3.22 ]  Where mi is the mass of the column Ri is the distance in the x-y plane from the column to the vertical axis. Based on these equations the linear and angular accelerations of the entire sliding body can be determined: ?̇?𝑥 =𝐹𝑏𝑜𝑑𝑦𝑚𝑏𝑜𝑑𝑦=1𝑚𝑏𝑜𝑑𝑦∗ 𝐹𝑥 [ 3.23 ]  ?̇?𝑦 =𝐹𝑏𝑜𝑑𝑦𝑚𝑏𝑜𝑑𝑦=1𝑚𝑏𝑜𝑑𝑦∗ 𝐹𝑦 [ 3.24 ] ?̇?𝑧 =𝑇𝑏𝑜𝑑𝑦𝐼𝑏𝑜𝑑𝑦=𝑇𝑧𝐼𝑧 [ 3.25 ] 44  These accelerations are then numerically integrated in order to determine linear and angular velocities and displacements.  These displacements are then applied to each column to advance the solution through time. Figure 3-6 demonstrates the new solution algorithm using a simple example of a block translating and rotating down an inclined plane.  In this example it is assumed that the sliding mass experiences a net torque in the clockwise direction.  For simplicity, the block is discretized into only four columns, each of which contributes to the net force and torque that acts on the column assembly.  The net force and torque on the column assembly is used to calculate translational and rotational displacements of the sliding mass.  Each column in the assembly experiences the same rotation and translation, so the relative geometry of the columns in the horizontal directions is maintained and columns do not interpenetrate.  In the vertical direction, relative shear displacement between columns is permitted to ensure that the sliding mass remains on the sliding surface (Figure 3-7).  It is assumed that the sliding mass does not experience any internal resistance to this displacement, so the new model is not suitable to simulate motion over strongly compound rupture surfaces.   The implicit assumption behind the described mechanism is that the sliding mass remains coherent in the horizontal direction, but is somewhat flexible when subjected to shear stresses on vertical planes, somewhat like a stiff sliding carpet.  The assumption is fully justified in the case of sliding over planar surfaces.  As the surfaces become curved or compound, demanding strong internal deformation of the sliding body, the method will lose some of its accuracy.  This is not a serious concern, because the proposed method is intended for cases where translational movement free of excessive internal deformation can be assumed in the initial stages of a landslide. 45   Figure 3-6:  Example of a block, discretized as a set of interconnected columns, is simulated to move down an inclined plane.  In this example there is a net torque in the clockwise direction.  The net force causes the centre of mass of the column assembly to translate down the inclined plane, and the net torque rotates the column assemblies.  All the columns experience the same displacement and rotation so the column geometry is maintained and columns do not interpenetrate.  The forces and torques acting on each individual column contribute to the net force and torque experienced by the column assembly.  Figure 3-5 shows the forces acting on each of the individual columns.   46   Figure 3-7: Two dimensional section showing column distortion in the vertical direction for a column assembly moving over a bilinear rupture surface.  Free vertical motion of the columns is permitted in order to ensure the failed mass remains on the sliding surface.  Since it is assumed that inter-column forces are negligible the new model is not suitable to simulate failures over strongly compound rupture surfaces.  3.4 Model Implementation The equations derived in the previous section describe the movement of a flexible block through time over three-dimensional terrain.  The model outputs positions and velocities of the reference columns in a series of time steps.  At a user-determined point in time, these parameters are transferred into the existing DAN3D algorithm and become initial conditions for flow analysis.  The fluidization of the sliding mass uses the following assumptions: 1) The physics of the landslide change from those that describe a flexible block to those that describe the flow of granular material in a single time step. 47  2) The velocities of the particles that describe the flow-like movement of the landslide are initially those of the nearest column. The column velocity used is an aggregate of the translational and rotational velocities, determined based on the displacement of the column during a known timeframe. 3.4.1 Fluidization of the Sliding Mass If the user-specified criterion for fluidization of the sliding mass is satisfied, the fluidization routine is called.  This routine initializes particle positions and velocities.  A grid of current column heights is created based on the nearest neighbor interpolation.  The nodes of this grid are chosen to correspond with the nodes of the path topography file.  This grid is then used as the input grid into the Dan3D initialization routine, which creates the initial fluid particle assembly in a bed-normal framework.    A similar procedure is then used to impart an initial velocity to the particles.  A grid of column velocities whose nodes correspond to the path grid file is created using nearest neighbor interpolation.  The velocities used to create this file are the aggregate rotational/translational velocities of the columns.  This velocity is determined by comparing the column position in the previous time step to the current column position.  Once this grid is created the fluid particles are assigned velocities based on the nearest neighbor column velocity grid node.  The Dan3D solution algorithm then takes over and the solution proceeds based on the fluid mechanics solution.  The integrated model is referred to as Dan3D Flex. 48  3.5 Model Verification In order to verify the implementation of the model, two analyses were conducted.  These verify both the translational and rotational components of the motion, and were completed by comparing model predictions to analytical solutions.  The selected problems are a block sliding down an inclined plane and a rod rotating about a fixed end.  The simple planar sliding model compared perfectly with a closed-form solution.  The configuration of the rod rotating about a fixed end is shown in Figure 3-8.  The code was modified so that rotation occurs about the fixed end of the rod (as opposed to the centre of mass), and a constant force in the positive y direction was applied to the column with (initially) the largest x coordinate.  The rod then rotates about the fixed end in a manner similar to a pendulum as the force remains constant in both magnitude and direction.  Good agreement between model predictions and analytical results were found for both of the tests.  The results of the rod rotating about a fixed end is shown in Figure 3-9.  There is good agreement between the simulated and analytical predictions for this problem.  The slight differences between the simulated and analytical results are likely due to discretization errors.  This case shows that the model is able to accurately solve the governing equations, and confirm that all quantities (such as moment of inertia) are calculated accurately.   49   Figure 3-8: Test configuration for the verification of the rotation algorithm.  The end is pinned and a constant force is applied in the y direction.  The rod rotates about the pinned end in a manner similar to a pendulum.  Figure 3-9: Comparison of centre of mass location predicted by the analytical solution and Dan3D flex at various times 3.6 Back Analyses The two rock avalanches described in the introduction were back-analysed with the new model, the results of which are summarised in the following sections.  These analyses demonstrate that the new model is able to solve the problem of excessive lateral spreading in the source area.  De Blasio (2011) listed possible factors that promote fragmentation in initially coherent rock 50  avalanches.  These include impact with topographic obstacles, free fall, and the influence of rugged topography.  For both the cases simulated here, rugged topography is the only factor that can be used to explain the disintegration of the debris.      3.6.1 Goldau Rock Avalanche 3.6.1.1 Initially Rigid Simulations The results of the simulations where the sliding mass is assumed to initially behave as a flexible block are shown in Figure 3-10.  The trimline and deposit distribution are well simulated.  A plot of the maximum velocities at each node is shown in Figure 3-11.  High velocities are simulated in the area where the model switches from a rigid body simulation to a fluid mechanics simulation.  A high depth gradient is caused by the source geometry being maintained until the fluid mechanics solution takes over, which results in strong fluid pressures within the sliding mass, leading to high velocities.  The sliding mass was fluidized after travelling about 1 kilometer as a flexible block (Figure 3-10).  Examination of the pre-slide topography shows that after travelling this distance the sliding mass has vacated the planar rupture surface and is overriding rugged topography.  This promotes progressive fragmentation of the sliding mass.   Analyses were conducted in order to investigate the sensitivity of model results to the user -specified rigid distance.  The results of the analyses are summarised in Figure 3-12.  Model results are relatively insensitive to the choice of rigid motion distance.  When a small rigid motion distance is used more lateral spreading of the deposit is simulated, and less material is thrust forward into the distal deposit zone.  Conversely, when a long rigid motion distance is used lateral spreading is limited and more material is thrust into the distal zone.   51   Figure 3-10:   final deposit shape predicted by Dan3D Flex for the Goldau Rock Avalanche.  The red outline shows the location where the user specified fluidisation criteria was met.  Figure 3-11: Maximum nodal velocities predicted by Dan3D flex for the Goldau Rock Avalanche. 52   Figure 3-12: Sensitivity of the predicted trimline to rigid motion distance for the Goldau Rock Avalanche.  The blue trimline represents model results of the mass is fluidized at the proximal end of the green zone, and the red trimline shows the result if the mass is fluidized at the distal end of the green zone.  For rigid motion distances in between these two extremes the predicted trimline would be in between the two results. 3.6.2 Mystery Creek Rock Avalanche The results of the initially rigid simulation of the Mystery Creek Rock Avalanche are shown in Figure 3-13.  The sliding mass was fluidized after most of the material had vacated the source zone (Figure 3-13) corresponding to a rigid motion distance of about 1 kilometer.  The pre-slide topography shows that the failed mass moved over a steep, planar surface before interacting with rugged topography.  It is likely that dynamic interaction with rugged topography caused 53  progressive fragmentation in this case, and the rigid motion distance was selected to correspond to this mechanism.    The observed trimline is well simulated, and the runup against the opposing slope at the toe of the deposit is similar to that observed, indicating that the simulated velocities are reasonable.  There are some areas where the model over predicts deposit spreading. This is likely caused by a loss of topographic resolution due to the removal of the deposit from the present day digital elevation model.  The maximum nodal velocities are shown in Figure 3-14.  Due to high depth gradients when the mass is fluidized, high velocities are simulated when the mass fragments.  The sensitivity of the model to the rigid motion distance is summarized in Figure 3-15.  A similar trend to that noted for Goldau can be observed, although there is somewhat greater sensitivity to the rigid motion distance in this case, where the source volume is thicker. 54   Figure 3-13: Mystery Creek final deposit predicted by Dan3D Flex.  The red outline shows where the mass is fluidized. 55   Figure 3-14: Mystery Creek maximum nodal velocities predicted by Dan3D Flex  Figure 3-15: Sensitivity of simulated trimline to rigid motion distance for Mystery Creek.  The blue trimline represents the simulated trimline if the mass is simulated to fluidize at the proximal end of the green zone, and the red trimline shows the result of the mass is simulated to fluidize at the distal end of the green zone.  Intermediate rigid motion distance would plot between these two trimlines.   56  3.7 Selection of Rigid Motion Distance The analyses presented here indicates that the model is relatively insensitive to the choice of rigid motion distance as long as the mass is fluidized within a reasonable distance of leaving the source zone.  Selection of this parameter can be based on examination of the pre-slide topography.  If interaction with a distinct topographic feature would cause the mass to fragment (such as an impact with a topographic obstacle), the rigid motion distance can be selected to correspond with the position of this feature.  If the failure is controlled by a long structural plane, such as a bedding plane, joint or fault, the rigid motion distance can be selected to correspond to the area where the sliding mass has vacated the planar feature.  If the source is controlled by an irregular surface, the flexible block model should not be used as fragmentation is likely to occur immediately following failure.   3.8 Conclusions Numerical runout models, even those that are most advanced, assume that the landslide behaves as a fluid, and can therefore be described by fluid mechanics, albeit modified for granular flow using the SH Method.  Rock avalanches that remain coherent for some or all of their motion do not fit this assumption.  The use of a fluid mechanics- based model to simulate coherent rock avalanches can result in excessive spreading of the moving mass during early motion phases, resulting in erroneous predictions of the landslide impact area. The model presented here introduces a coupling of a solid mechanics solution with an existing semi-empirical equivalent fluid based runout model.  The new model can simulate both translation and rotation of a flexible block as it moves over three-dimensional terrain.  This 57  allows for the simulation of rock avalanches that are semi-coherent, as well as those that are initially coherent before turning flow-like.  This new model only introduces one additional parameter, and is computationally efficient.  Both these features make the new model conducive to inverse analyses.  The additional input parameter is the distance of motion of a semi-solid (flexible) block, before disintegration changes the landslide mass into a granular flow.  The assumption of instant fluidization still represents a major simplification of the real process.  However, it has been shown to greatly improve the results of back-analyses of certain rock avalanches.  The determination of the block motion distance, as a user-controlled input parameter, is guided by an assessment of the slope topography.  The results of analyses are not strongly sensitive to this parameter provided it is chosen to correspond to the area where the failed mass has vacated the source zone, with consideration of geometric features.   The flexible block model has been verified by comparing model outputs to two analytical solutions.  These solutions test both the translation and rotation algorithms.  Model outputs compare favourably to analytical solutions in both cases.  Two full scale rock avalanches were also simulated with the new model.  Once calibrated, the model was able to predict the impact area of these rock avalanches.  The fit between model predictions and field observation would be impossible to achieve using a common fluid mechanics- based runout model.  58  Chapter 4: Calibration of a Runout Model As mentioned in Chapter 2, the parameters that govern equivalent fluid models must be calibrated.  Prior to the present work, calibration was generally performed using subjective, trial-and-error calibration.  This method suffers from a number of weaknesses (summarized below) and it is desirable to have a more objective, systematic methodology to calibrate equivalent fluid models.  The purpose of this chapter is to use techniques from inverse modelling and statistics to develop such a methodology.    Equivalent fluid models have been used to reproduce the velocity, deposit distribution and impact area of a wide variety of extremely rapid, flow-like landslides (e.g. Ayotte et al., 1999; McDougall, 2006; Hungr & McDougall, 2009; Deline et al., 2011). These cases have demonstrated that the models are highly sensitive to the user-specified basal resistance parameters, and the selection of basal resistance parameters is one obstacle to the routine use of equivalent fluid models in forward-analysis. Some researchers have suggested methods to parameterize these models for use in forward-analysis; however, these studies are limited to a small subset of extremely rapid, flow-like landslides, were only performed using two-dimensional terrain data or are based on compilations of calibrated values derived from different models that may not produce similar calibration results (e.g. Hungr & Evans, 1996; Revellino et al., 2004; Sosio et al., 2011, 2012; Quan Luna, 2012).  Back-analysis of case histories with equivalent fluid models is generally performed through trial-and-error calibration (Hungr, 1995). This approach involves manually adjusting the input basal resistance parameters until a satisfactory reproduction of the simulation constraints is obtained. 59  The suitability of a particular simulation is then subjectively assessed. This method of model calibration suffers from four weaknesses: 1. It is very demanding of the user’s time. 2. Model results are subjectively interpreted, meaning that different users could determine different parameters for the same case. 3. This method does not explore the entire parameter space, so there is no guarantee that the best-fit parameters have been determined. An additional difficulty that comes from this weakness is that, if the model conceptualization is wrong (for example, one material is used instead of two), the user will waste a lot of time varying parameters in search of a good fit, as opposed to changing the simulation configuration. 4. This method ignores parameter non-uniqueness. It is often impossible to determine a unique set of best-fit parameters due to the fact that multiple parameter sets produce the same model outputs.   These four weaknesses will be demonstrated using the calibration example presented below.  4.1  Dan3D Calibration Example Before discussing the use of inverse models in the context of Dan3D, it is useful to examine an example calibration case history in order to introduce the Dan3D calibration process, as well as demonstrate the four weaknesses noted above.  For this purpose, a back-analysis of the Mt. Meager case history, performed by McDougall (2016), will be presented. 4.1.1 Mt. Meager Rock Avalanche The Mount Meager rock avalanche occurred on August 6th, 2010, approximately 65 km northeast of Whistler B.C (Guthrie et al., 2012).  Guthrie et al. (2012) estimated the volume of this event to 60  be 48.5 Mm3.  This event generated a seismic signal during its motion, and based on an inversion of this signal, Moretti et al. (2015) suggest that it occurred in three stages.  For the present analysis, the event will be treated as a single failure, as Moretti et al. (2015) show that the total impact area and velocities are similar regardless of failure sequence.  More details of this event are provided in Section A.5. An overview of this event is shown in Figure 4-1.  Initially, the Mt. Meager rock avalanche failed into Capricorn Creek, and travelled approximately 6 kilometers before impacting an opposing valley wall and spreading out down Meager Creek.  While travelling down Capricorn Creek, the rock avalanche superelevated twice, providing the opportunity to estimate flow velocities at two points.  61   Figure 4-1: Overview of the Mt. Meager rock avalanche.  The area of the initial rock slope failure (labelled ‘source zone’), impact area, as well as the locations of two superelevation measurements are shown.  Image: Google Earth 2017: Digital Globe. As input, Dan3D requires a topographic grid file of the source mass and path, as well as the locations of material changes.  In addition to this, the parameters governing the internal and basal rheologies must also be input.  For the present discussion, the calibration results from McDougall (2016) will be used. The objective of the back-analysis is to determine the basal resistance parameters that result in simulations that best reproduce the mapped features of the Mt. Meager rock avalanche.  For this example back-analysis, the Voellmy rheology was used (described in Section 2.5.2).  In this case, the mapped features that were simulated are the impact area and two velocity estimates.  The results of 16 simulations are shown in Figure 4-2.  As can be seen in Figure 4-2, multiple 62  different sets of parameters result in the same simulated impact area.  This is the problem of parameter non-uniqueness mentioned above.  For this particular case, there are velocity estimates available to further refine the parameter estimates (this is not the case for many of the case histories analyzed in this thesis).   Figure 4-2 also shows the maximum simulated velocities at each spatial point in the simulated impact area.  The various parameter combinations result in different simulated velocities.  Only one of the parameter combinations tested (friction coefficient = 0.05, turbulence coefficient = 500 m/s2) reproduced both the observed impact area and velocity estimates, and therefore this is the best-fit parameter combination.       The calibration procedure shown above appears to work well for this case; however, it demonstrates the four weaknesses of subjective calibration procedures listed above:     1.  This calibration process is time consuming.  Each model run must be manually set up, and the results manually plotted, compared and interpreted.  There is potential to automate this repetitive process.   2. The model results are subjectively interpreted.  At least three different researchers have back-analysed this case (Manfredi, 2012; Moretti et al., 2015; McDougall, 2016) and each has interpreted a different location for the distal end of the debris.  Due to this, each researcher has derived different best fit parameter combinations.  Based on Figure 4-2, Manfredi (2012) would determine a best fit friction coefficient of 0.025, McDougall (2016) a friction coefficient of 0.05 and Moretti et al. (2015) a friction coefficient close to 0.075.  This disparity in the interpretation of the results is because the Mt. Meager rock 63  avalanche does not have a clear distal end, due to the fact that the deposits were reworked by post-landslide flooding.  This source of measurement error in the constraints must be accounted for in the back-analysis.  Another source of measurement error is that the velocity estimates are derived from superelevation measurements, which can have large errors associated with them (Prochaska et al., 2008).  This must also be accounted for in the back-analysis.  3. The calibration procedure used above does explore the entire parameter space, however only at a rough resolution (friction coefficients in steps of 0.025, and ξ in steps of 250 m/s2).  Exploring the parameter space at a finer discretization would require even more user time to set up and interpret model results.   4. In the Mt. Meager case, velocity estimates were available to further refine the parameter estimates; however, this is not often the case.  In cases that do not have any velocity estimates, a broad range of parameters provides the same fitness to the constraints.  No current calibration methodology addresses and quantifies this issue.  64   Figure 4-2: McDougall (2016) calibration of the Mt. Meager rock avalanche.  Based on the assumption that much of the apparent impact area of the event is due to post- rock avalanche flooding, McDougall (2016) determined best fit parameters of f = 0.05, turbulence = 500 m/s2.  Note that the ‘turbulence parameter’ labelled in the above Figure is referred to as the turbulence coefficient throughout this thesis.  Figure from McDougall (2016), © 2008 Canadian Science Publishing or its licensors. Reproduced with permission.    65  These four weaknesses can limit the use of back-analyzed parameters in forward-analysis. The fact that trial-and-error back-analysis is subjective has been addressed by McDougall (2006), Galas et al. (2007) and Cepeda et al. (2010). The matrix method, proposed by McDougall (2006) and presented in Figure 4-2, addresses parameter non-uniqueness; however, its utility is limited to model conceptualizations that only include two parameters. The receiver operating characteristic (ROC) method, proposed by Cepeda et al. (2010), can handle more than two parameters and provides an objective method to compare different parameter combinations; however, it is not guaranteed to explore the entire parameter space and is very demanding of the user’s time.   Inverse models, which make use of statistics and optimization theory to find the parameters that best minimize the difference between model results and field observations, can be used to address the four weaknesses listed above. This chapter describes the application of inverse analysis to improve the calibration of equivalent fluid models.                                                                                                            Two methods to calibrate Dan3D, one based on a brute force sensitivity analysis and the other based on optimization theory, are presented.  An example back-analysis to demonstrate the newly-developed methods is also provided. 4.2 Theoretical Background 4.2.1 Mathematical Framework As demonstrated in Section 4.1.1, field investigations of a given landslide case history result in multiple features (or ‘back-analysis constraints’) that characterize an event (e.g. Evans et al., 1994, 2001, 2007, 2009a; Fletcher et al., 2002; Lipovsky et al., 2008; Guthrie et al., 2012; 66  Delaney & Evans, 2014).  In Section 4.1.1, these features included the impact area of the rock avalanche, as well as two point estimates of velocity.  In addition to these two features, in some cases estimates can be made of the volume of material deposited in certain spatial areas, as well as point estimates of deposit thickness.  The role of these features in the mathematical framework is formalized as follows.  This discussion is based on Hsieh (2009) and Gregory (2010). Let 𝑦𝐹 = (𝑦𝐹,1, … , 𝑦𝐹,𝐾)𝑇 denote a vector of K features recorded based on the field investigation of a given landslide case history.  As the impact area of a landslide is a high-dimensional object, we will assume that it has been transformed into a real-valued summary measure.  Therefore, the vector of features for the Mt. Meager case summarized in Section 4.1.1 would have a dimension (K) of three: 𝑦𝐹 = (𝑦𝐹,1, 𝑦𝐹,2, 𝑦𝐹,3)𝑇 where 𝑦𝐹,1 is the real-valued summary measure of the impact area (this measure is discussed in Section 4.3.1), 𝑦𝐹,2 is the velocity estimated at the first superelevation bend (Figure 4-1) and 𝑦𝐹,3 is the velocity estimated at the second superelevation bend (Figure 4-1).  For a different case history, a field investigation may have measured the landslide impact area, the volume of material deposited in two different zones, as well as three point estimates of debris thickness.  In this case, K = 6.  Paulo et al. (2012) emphasizes the importance of treating multiple features simultaneously, an observation also made by Köerner (1976) in the context of calibrating landslide runout models. Dan3D aims to reproduce the runout process of extremely rapid, flow-like landslides, based on a vector of input parameters.  For the following discussion, we will denote this vector of input parameters as b.  As detailed in Section 4.1.1 and in McDougall (2006), this vector includes parameters such as the input topography files, numerical parameters that control the SPH solution algorithm (Section 2.5), as well as material parameters such as the parameters that 67  govern the internal and basal rheologies (Section 2.5.2).  In the discussion that follows, we will only consider the case where the parameters governing the basal rheology are calibrated.  All other parameters will be fixed.   Given the value of the parameter vector b, a Dan3D simulation can be performed.  The simulation outputs the K features of a given case, which can be represented by the vector: 𝑦𝑀(𝒃) = (𝑦𝑀,1(𝒃), … , 𝑦𝑀,𝐾(𝒃))𝑇 Where 𝑦𝑀,𝑖(𝒃)(𝑖 = 1, … , 𝐾) indicates the simulated value of a given landslide feature, for example the velocity at a point.  The goal of model calibration is to determine the parameter vector b that minimizes the difference between 𝑦𝐹 and 𝑦𝑀(𝒃).   For real-valued features, such as point estimates of velocity and deposit depth, the difference between 𝑦𝐹 and 𝑦𝑀(𝒃) can be evaluated by subtracting the simulated and observed value of the feature.  However, when comparing simulated and observed impact area, a more complex ‘difference’ metric must be computed.  This metric will be discussed in detail in Section 4.3.1; however, the outcome is a single number that measures the similarity of the simulated and observed impact area.  Therefore, these differences (sometimes referred to as ‘model errors’ or ‘residuals’), between 𝑦𝐹 and 𝑦𝑀(𝒃) can be collected into a vector 𝒓(𝒃) = (𝑟1(𝒃), … , 𝑟𝐾(𝒃))𝑇 where, for example,  𝑟𝟏(𝒃) is the difference between 𝑦𝑀,1(𝒃) and 𝑦𝐹,1.   To illustrate the notation introduced above, consider the example of the Mt. Meager case run with a friction coefficient = 0.05 and turbulence coefficient = 500 m/s2.  The model output for this set of parameters is shown in Figure 4-2.  For clarity, we will ignore the impact area feature 68  in this example.  Therefore, the vector of features based on the field investigation comprises the values of the two point estimates of velocity: 𝑦𝐹 = (62𝑚𝑠, 73𝑚𝑠 )𝑇.  For this case, the parameter vector is 𝒃 = (𝑓 = 0.05, 𝜉 = 500𝑚𝑠2)𝑇 , where model parameters that are not being calibrated have been omitted.  We then run Dan3D using these parameters, and based on the model output (Figure 4-2), we can create the vector 𝑦𝑀(𝒃) = (70𝑚𝑠, 75𝑚𝑠)𝑇.  By differencing 𝑦𝑀(𝒃) and 𝑦𝐹, we can create the error vector 𝒓(𝒃) = (8𝑚𝑠, −2𝑚𝑠)𝑇.   As mentioned in Section 4.1.1, there is some error in the measurement of the velocity values at the two superelevation locations.  Therefore, even though the model residuals are not zero, ‘𝒃’ may be the ‘true’ parameter vector for this case, with the model misfit due to statistical noise.  The following section quantifies this intuition by assuming a stochastic model for the error vector 𝒓(𝒃).  A procedure is developed to answer the question: “If a given set of parameters results in model errors of a given magnitude, what is the probability that these errors are due to statistical noise?”    4.2.2 Stochastic Model for Error Vector A natural assumption regarding the error vector is that the model residuals follow a multivariate Gaussian (or normal) distribution with zero mean, and covariance matrix ∑.  Under the normality assumption, the density function has the form: 𝑓𝑒𝑟𝑟𝑜𝑟(𝒓|𝒃) =  1√2πK |∑|exp (−12𝒓𝑇∑−1𝒓) [4.1] 69  where 𝒓 = 𝒓(𝒃) is a K-dimensional vector (defined above) and |∑| is the determinant of the covariance matrix ∑.  The expression above demonstrates the dependence of the error vector 𝒓 on the parameter vector b (which controls the numerical model output).  From Equation [4.1], it can be seen that the probability is maximized when all the components of 𝒓 are equal to zero, i.e. where the model output exactly matches the field observations.  Therefore, the goal of model calibration is to find the parameter vector b such that Equation [4.1] is maximized.  This also has a probabilistic interpretation, as we want the probability of the error associated with the observed field measurement to be maximized.  As maximization occurs with respect to the parameter vector b, it is common to view Equation [4.1] as a function of the parameters (as opposed to the error vector).  This function is called the likelihood function and in our case it is given by: 𝐿(𝒃| 𝑦𝐹 , 𝑦𝑀(𝒃)) =  1√2πK |∑|exp (−12𝒓(𝒃)𝑇∑−1𝒓(𝒃)) [4.2] Here, the likelihood of b is conditional on the field measurements  𝑦𝐹 and the numerical model output 𝑦𝑀(𝒃).  Additionally, we emphasize the dependence of the model residuals on b.  Maximizing the likelihood function is equivalent to minimizing the negative log-likelihood function: 𝑙(𝒃) = −𝑙𝑜𝑔𝐿(𝒃| 𝑦𝐹 , 𝑦𝑀(𝒃)) = log (√2πK |∑|) +12𝒓(𝒃)𝑇∑−1𝒓(𝒃) [4.3] If the covariance matrix does not depend on the parameters to be calibrated, then minimizing Equation [4.3] is equivalent to minimizing: 70  Φ(𝒃) =12𝒓(𝒃)𝑇∑−1𝒓(𝒃) [4.4] Note that if the components of the model residual vector are assumed independent, ∑ is a diagonal matrix: ∑ = (𝜎120⋮0 0𝜎22⋮0 …⋯⋱… 00⋮𝜎𝐾2) where 𝜎𝑘’s (k = 1,…,K) are variances of the error components, which quantify the degree of variability in the measurement error associated with the value of a simulation constraint.  For example, if there is a large error in the point estimate of velocity, then the corresponding 𝜎𝑘 will be large.  For the case of independent model residuals, the objective function (Equation [4.4]) becomes: Φ(𝒃) = ∑1𝜎𝑘2𝐾𝑘=1(𝑟𝑘(𝒃))2 [4.5] Equation [4.5] is a weighted sum of the squared error components, with the weights equal to the inverse of the variance of the corresponding feature.  Therefore, if the variance (or error) in the value of a feature is large, then smaller weight will be placed on that component when evaluating the objective function.       If we consider the example of the Mt. Meager rock avalanche given above, and again only consider the two velocity estimates as simulation constraints, the only additional data needed to 71  evaluate Equations [4.2] and [4.4] is the covariance matrix ∑.  This will be discussed in more detail in Section 4.3.5, but for now let us assume that the covariance matrix is given by: ∑ = (5𝑚𝑠00 5 𝑚𝑠) Given the normality assumption, the interpretation of this covariance matrix for the first super elevation bend is as follows. We believe the best estimate of the velocity is 63 m/s, however, there is some measurement error associated with this estimate.  We are 70% sure that the velocity is between 58 and 68 m/s, and 95% sure that the velocity is between 53 and 73 m/s.  This measurement error is quantified by using a standard deviation of 5 m/s.  In this way, error in the measured value of the velocity can be quantified.  This is valuable, as we can rarely estimate features such as velocity and deposit volume precisely, however we can often put bounds on these values.  The results of evaluating Equations [4.2] and [4.4]  for four different parameter sets are presented in Table 4-1.        72  Table 4-1: Value of the objective function and likelihood function for four different parameter combinations.  The parameter combination of 𝒇 = 𝟎. 𝟎𝟓, 𝛏 = 𝟓𝟎𝟎 𝐦𝐬𝟐 (shown in bold) results in the lowest value of the objective function, and therefore the highest value of the likelihood function, indicating that this parameter combination is the best fit of the four combinations tested. Parameter Combination Objective Function Likelihood Function 𝑓 = 0.05, 𝜉 = 200 𝑚𝑠2 19 2.0e-6 𝒇 = 𝟎. 𝟎𝟓, 𝝃 = 𝟓𝟎𝟎 𝒎𝒔𝟐 2.5 8.72e-3 𝑓 = 0.05, 𝜉 = 700 𝑚𝑠2 11 1.2e-4 𝑓 = 0.05, 𝜉 = 1000 𝑚𝑠2 25.5 8.7e-8  As can be seen in Table 4-1, the parameter set that results in the closest match between modelled and field estimated velocities has the highest value of the likelihood function (meaning that it is relatively more probable than the other models) and the lowest value of the objective function, which indicates that it is the best model of the four parameter sets tested.  This agrees with the results presented in Figure 4-2. One key feature to note about the use of inverse models is that they do not require any additional input data when compared to trial-and-error calibration.  It can be argued that the covariance matrix is an additional data requirement of this type of calibration.  However, in trial-and-error 73  calibration the constraints are inherently weighted by the user who subjectively interprets model results.  The use of a covariance matrix (∑ in the equations above) provides a transparent way to express errors in measured features.  By selecting a standard deviation of a measurement, the user is expressing the belief that there is a 70% probability that the measurement is +/- 1 standard deviation from the best estimate value, and a 95% probability that the measurement is +/- 2 standard deviations from the best estimated value.  In this way, error bounds on the values of estimated features can be explicitly included in the calibration process.   4.3 Quantification of Simulation Constraints In order to evaluate Equation [4.2] and [4.5], it is first necessary to define metrics that can be output by Dan3D and compared to field observations of extremely rapid, flow-like landslides.  Four quantities can be used to compare a Dan3D simulation to field data.  These are:  Impact area  Estimates of volumes deposited in spatial areas  Point estimates of deposit depth  Point estimates of flow velocity A procedure has been defined to automatically assess how well Dan3D reproduces these observations.  A post-processor has been created to interpret Dan3D outputs and create a model output vector (𝑦𝑀(𝒃)) at the conclusion of each simulation.  It quantifies model outputs using the procedure detailed in the following sections.  74  4.3.1 Landslide Impact Area When calibrating a 2D runout model, an intuitive and widely used metric is the rock avalanche runout distance.  When performing model calibration in 3D, however, the runout distance is difficult to define because the flow may bifurcate and deposit in multiple lobes (an example of this behaviour is shown in Figure 4-1).  Instead of using runout distance as a metric, in 3D it is necessary to define a metric that quantifies the similarity between simulated and observed impact area. Both Galas et al. (2007) and Cepeda et al. (2010) have proposed such metrics, and the following procedure is based on these two contributions.   Field or remote-sensing investigations of long-runout landslides commonly produce maps of the landslide impact area. These maps can be used to create a grid file (hereafter referred to as the trimline grid file) that has a value of one for areas that are within the impact area and a value of 0 for areas that are outside the impact area. An example of such a grid file is shown in Figure 4-3. One of the outputs of Dan3D is a grid file that shows the maximum flow depth recorded at each node of the inputted sliding surface grid file during the simulation. An example of such a file is shown in Figure 4-4. As can be seen in this figure, the landslide is thickest in the source area, due to the fact that the sliding mass is concentrated in a small area. As the simulation is advanced in time, the sliding mass spreads out and, as a result, the flow depth thins as the sliding mass moves downslope.  75   Figure 4-3: An example of a trimline grid file created based on a field investigation. This grid has a value of one in locations where the field investigation indicated landslide impact and a value of zero where the landslide did not impact. Values of one are coloured dark grey and zeros are coloured light grey.  Figure 4-4: Example of a maximum thickness gridfile output by Dan3D.  Thickness values are reported in meters. In order to quantify how well a given simulation matches the landslide impact area, the maximum thickness file is compared to the trimline grid file. These two grid files are overlapped 76  to create a new grid (hereafter referred to as the trimline fitness grid) that has a value of zero where the trimline file contains a value of one and the maximum thickness file contains a thickness greater than a user specified cutoff value (as summarized in McDougall (2006), a cutoff is necessary due to the numerical solution method used by Dan3D). This indicates agreement between Dan3D and the field investigation that an impact has occurred at that node of the grid. This grid also contains a value of zero where the trimline file has a value of zero and the maximum thickness is less than the user specified cutoff value. This indicates agreement between Dan3D and the field investigation that no impact occurred at that node of the grid. In areas where these conditions are not true, the grid has a value of one. This indicates disagreement between Dan3D and the field investigation. The values of the trimline fitness grid are then summed to give a single fitness number. This number is zero if Dan3D perfectly simulates the trimline grid file, and gets bigger as the fitness gets worse.  4.3.2 Landslide Deposit Distribution An important simulation constraint for some landslides is the deposit distribution. Investigations of long runout landslides often provide estimates of the volume of material deposited in different areas of the landslide path. In order to quantify this simulation constraint, the following routine has been implemented. A grid file of deposit zones is created by the user and input into the post-processor based on the field investigation. This grid file defines zones within the landslide path for which deposit volume estimates are available. At the end of each simulation, the volume deposited in each of the zones is calculated and output. These volumes can then be compared to known volume estimates.  77  4.3.3 Landslide Deposit Depth Estimates In addition to inputting deposit zones, the user can input the spatial coordinates of points where the deposit thickness is known. The post-processor will then output the deposit thickness at these points at the conclusion of a simulation.  The deposit thicknesses can then be compared to field estimates of deposit depth. 4.3.4 Landslide Velocities Estimates of landslide velocities are generally made at specific spatial points. These estimates are commonly of the maximum landslide velocity at a given point in space, often based on superelevation measurements (e.g.  Hungr et al., 1984; McClung, 2001; Prochaska et al., 2008; Guthrie et al., 2012; Scheidl et al., 2014), seismic records (e.g. Allstadt, 2013; Coe et al., 2016) or video evidence (Sosio et al., 2008; Yune et al., 2013).  To quantify velocity constraints, the spatial coordinates of points where velocity estimates are available can be input into the post-processor. Dan3D tracks the maximum velocities simulated at each node during the simulation. At the end of each simulation, the maximum velocity at each of the user-specified points is output.  These simulated velocities can then be compared to field estimates of velocity.  4.3.5 Selection of Standard Deviation of a Measurement The standard deviation of a simulation constraint is a statistic that quantifies the variability in the measurement error in the value of the constraint (see Section 4.2.1).  For the simulation constraints summarized in Sections 4.3.1, 4.3.2 and 4.3.4, selection of this statistic will require subjective judgment.  However, some guidance can be given by exploring the source of errors.  The two main sources are: 78  1. The portion of the error that is a result of uncertainties in a measurement 2. The portion of the error that results from uncertainties in input data as well as the model being an imperfect simulation of reality To understand the first source of error, consider the case of estimating the velocity of a landslide from a superelevation measurement.  Prochaska et al. (2008) show that velocity estimates derived from superelevations are subject to many sources of error.  These sources include estimating the radius of curvature as well as the difficulty of distinguishing splash marks from true superelevations.  For some cases presented by Prochaska et al. (2008), velocity ranges of 5 m/s to 25 m/s appeared defensible.  Therefore, the standard deviation for this constraint should be selected to account for this source of error.  For some superelevation observations, such as the Zymoetz River Rock Avalanche (McDougall et al., 2006), the radius of curvature and banking angle appear well-constrained and the error associated with velocity estimates can be dramatically reduced. An example of the second source of error is demonstrated in the simulation of the trimline of the Zymoetz River Rock Avalanche (Figure 4-3 and Figure 4-4).  By comparing Figure 4-3 and Figure 4-4, it can be seen that Dan3D does not perfectly reproduce the observed trimline (the value of the trimline fitness is > 0).  The observed trimline is derived from accurate remote sensing data, and it is unlikely that this error is a result of measurement error.  Instead, this model residual results from a host of sources that are difficult to rigorously quantify, including uncertainties in pre-failure topography, volume and shape of the initial failure, distribution and character of path materials, as well as the assumptions used to derive the governing equations of Dan3D.  All of these sources of error contribute to the second source of error detailed above.  For 79  features where this term dominates, such as the trimline fitness metric for most rock avalanche cases, good results have been found in this thesis by using one half of the lowest simulated residual as the standard deviation.  This is consistent with the methodologies suggested by Beven  (2005) and Gregory (2010) for calibrating models in other contexts.  Errors in deposit distribution measurements can have a significant contribution from both sources of error.  Selecting a standard deviation for these measurements is difficult, and will likely require subjective judgement tailored to a given case.       4.4 Sensitivity Analyses  One method to find the parameter vector ‘b’ that minimizes the objective function (Equation [4.5]) is to perform a brute-force sensitivity analysis.  This is done by discretizing the parameter space, and evaluating the objective function at each of these parameter combinations.  The parameter vector corresponding to the lowest objective function is the ‘best-fit’ parameter for a given set of constraints.   In the context of equivalent fluid models, a sensitivity analysis can be run to determine the values of the simulation constraints as well as the fitness function for each set of parameter values in the parameter space. The process for running the sensitivity analysis is summarized in Figure 4-5. A given parameter set is input into Dan3D, and available simulation constraints (trimline grid file, deposit zone grid file and superelevation location) are input into the post-processor. At the conclusion of the simulation, the Dan3D outputs are input into the post-processor and the individual fitness metrics are calculated and saved. The fitness function is then 80  calculated by combining these individual fitness metrics using Equation [4.5].  Section 4.7 provides an in-depth example of this process.   Figure 4-5: Steps to run a sensitivity analysis. All parameter combinations are run in Dan3D, which then passes the simulation output to the post-processor. The post-processor accepts the simulation constraints and the Dan3D output as input in order to calculate fitness metrics. The fitness metrics are then combined using the least squares error function. The resulting value of the fitness function is then plotted. Once the dimensionless fitness values for all parameter combinations have been calculated the results are contoured. 4.4.1 Posterior Analysis of Calibrated Parameters This section provides a statistical interpretation of the sensitivity analysis procedure detailed above that is useful for making probabilistic predictions with Dan3D.  When performing model calibration, we may have a prior belief about the possible values of the parameter vector b.  For example, based on previously successful back-analyses, we can often restrict the range of plausible friction and turbulence coefficients when using the Voellmy rheology (Section 2.5.2).  81  We can express this prior belief in the form of a prior distribution, and then update this distribution to include the information gained from evaluating the likelihood function (Equation [4.2]).  The resulting distribution of the calibrated parameters is known as the posterior distribution.  This sort of analysis, referred to as posterior analysis, has been conducted based on the sensitivity analysis results detailed above.  As will be shown in Chapter 4, this is useful when combining multiple back-analysis results for use in a probabilistic forward analysis.   The posterior density of the calibrated parameters can be written using Bayes’ law.  Let π𝑝𝑟𝑖𝑜𝑟 denote the prior density of b, and π𝑝𝑜𝑠𝑡 the posterior density after accounting for the data available about the case history.  These data are comprised of the error vector derived from field measurements 𝑦𝐹 and computer model outputs 𝑦𝑀(𝒃). We then have: π𝑝𝑜𝑠𝑡(𝒃|𝒓) =𝑓𝑒𝑟𝑟𝑜𝑟(𝒓|𝐛) π𝑝𝑟𝑖𝑜𝑟(𝒃)∫ 𝑓𝑒𝑟𝑟𝑜𝑟(𝒓|𝐛∗) π𝑝𝑟𝑖𝑜𝑟(𝒃∗)𝑑𝒃∗ [4.6] Where the denominator is a normalizing constant so that the posterior density integrates to one. In our implementation, we assume a uniform prior on the calibrated parameter vector over a rectangular region B = 𝐼𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑥 𝐼𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑐𝑒, where 𝐼𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 and 𝐼𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑐𝑒 are intervals of feasible ranges of the friction and turbulence coefficients, respectively.  With normally distributed errors, the posterior density has the form:    π𝑝𝑜𝑠𝑡(𝒃|𝒓) =1√2πK |∑|exp (−12 𝒓(𝒃)𝑇∑−1𝒓(𝒃)  ∗ (1/|𝐵|) ∬1√2πK |∑|exp (−12 𝒓(𝒃∗)𝑇∑−1𝒓(𝒃∗) ∗ (1/|𝐵|)𝑑𝒃∗𝐵 [4.7] 82  π𝑝𝑜𝑠𝑡(𝒃|𝒓) =exp (−12 𝒓(𝒃)𝑇∑−1𝒓(𝒃))  ∬ exp (−12 𝒓(𝒃∗)𝑇∑−1𝒓(𝒃∗)) 𝑑𝒃∗𝐵 [4.8] π𝑝𝑜𝑠𝑡(𝒃|𝒓) =exp (−12 Φ(𝒃))  ∬ exp (−12 Φ(𝒃∗)) 𝑑𝒃∗𝐵 [4.9] where |𝐵| denotes the area of 𝐵, and Φ(𝒃) is defined in Equation [4.4]. Equation [4.9] shows that, with the assumption of Gaussian errors and a uniform prior, the posterior probability of the parameters is the likelihood function normalized so that ∫ π𝑝𝑜𝑠𝑡(𝒃|𝒓) 𝑑𝒃 = 1𝐵 , where the parameter space B is defined based on the prior distribution π𝑝𝑟𝑖𝑜𝑟.  Parameter combinations that, when used in Dan3D, well reproduce the field observations will be given high probabilities, and those that poorly reproduce the field observations will be given low probabilities.   4.4.2 Limitations of the Sensitivity Analysis Approach The biggest limitation of the sensitivity analysis approach is the number of model runs it requires.  When two parameters are being calibrated, it is feasible to run a sensitivity analysis.  Chapter 4 makes extensive use of this approach to calibrate Dan3D to 24 rock avalanche case histories.  When parallelized on a desktop computer with 12 processors, run times for these sensitivity analyses ranged from 1 to 5 days. The number of simulations required to perform a brute-force sensitivity analysis increase exponentially with the number of parameters being calibrated.  When more than two parameters 83  are being calibrated, the sensitivity analysis approach may no longer be feasible due to long runtimes.  In such cases, an approach based on optimization theory can be used to calibrate Dan3D.  This approach, described below, can dramatically cut down on the number of model runs required to determine the ‘best-fit’ parameters.  This increased efficiency comes at the expense of transparency in the calibration process.  This approach was used to calibrate the model to the coal mine waste flowslide described in Section 6.2.   4.5  Optimization Approach - The Gauss-Marquart-Levenberg Algorithm There are many different algorithms that can be used to minimize the objective function (Mikosch et al., 2006).  The algorithm used in the present work is the Gauss-Marquart-Levenberg (GML) algorithm (Levenberg, 1944; Marquardt, 1963).  The following discussion is adapted from Doherty (2010).  To understand the GML algorithm, it is useful to first look at the problem of calibrating a linear model using the objective function defined in Equation [4.5].  In the optimization literature, this objective function is referred to as the least squares objective function (e.g. Mikosch et al., 2006).  These principles will then be applied to fitting parameters to a non-linear function (such as Dan3D) using the least squares objective function.  A linear model can be written in the following form: 𝑐1 =  𝑏1 + 𝑏2𝑋11 + 𝑏3𝑋12+. . . +𝑏𝑛𝑋1𝑛 𝑐2 =  𝑏1 + 𝑏2𝑋21 + 𝑏3𝑋22+. . . +𝑏𝑛𝑋2𝑛 𝑐𝑖 =  𝑏1 + 𝑏2𝑋𝑖1 + 𝑏3𝑋𝑖2+. . . +𝑏𝑛𝑋𝑖𝑛 84  Where c is a model output that can be compared to a field observation, i is the number of observations, n is the number of parameters and X are known values of independent variables, such as distance or time.  In matrix notation this can be expressed as: 𝑿𝒃 = 𝒄 [4.10] Where X is a matrix that has a number of columns equal to the number of parameters, and a number of rows equal to the number of observations.  In the following discussion, the observation weights (𝜎𝑘’s in Equation [4.5]) will be omitted.  Although important when implementing the Gauss-Marquart-Levenberg algorithm, the weights are not needed to understand its theoretical background.  Using matrix notation, the least squares objective function (Equation [4.5]) can be re-written: Φ =  𝒓(𝒃)𝑇𝒓(𝒃)  [4.11] The parameter vector b that minimizes equation [4.11] can be calculated by (e.g. Mikosch et al., 2006): 𝑑𝑑𝑏Φ =  𝑑𝑑𝑏𝒓(𝒃)𝑇𝒓(𝒃)= 0 [4.12] Solving equation [4.12] for b gives:  𝒃 = [𝑿𝑇𝑿]−1𝑿𝑇𝑦𝐹 [4.13] With the inclusion of parameter weights, equation [4.13] becomes (e.g. Doherty, 2010):  𝒃 = [𝑿𝑡∑𝑿]−1𝑿𝑡∑𝑦𝐹 [4.14] 85  where ∑ is the covariance matrix discussed in Section 4.2.2.  When the simulation constraints (or features) are uncorrelated, as assumed in this chapter, ∑ contains the parameter weights along the main diagonal, and zeros elsewhere (see Section 4.2.2). Therefore, for a linear model, evaluating Equation [4.14] will provide the parameter vector that minimizes the least squares objective function (Equation [4.5]).  For non-linear models, additional steps are needed in order to apply the theory detailed above, however, the core principle is the same.   When calibrating a non-linear model, such as Dan3D, using least squares optimization theory, it is necessary to iteratively solve the problem using Equation [4.14] and a linearized model.  The model is linearized using a Taylor expansion.  If we neglect the terms with derivatives of higher order than 1, we can approximate a non-linear function using a linear function: 𝑦𝑀(𝒃) ≈ 𝑦𝑀(𝒃0) + 𝐽(𝑏 − 𝑏0) [4.15] Where 𝑦𝑀(𝒃0) is the simulated value of the observations at the parameter value 𝑏0, and ‘J’ is the Jacobian matrix.  The Jacobian matrix has a number of columns equal to the number of parameters, and a number of rows equal to the number of simulation constraints (such as impact area or velocity at a point).  The elements of the Jacobian contain the derivatives of the simulated value of the mapped features with respect to the parameters.  For example, if the simulated velocity at a point is highly sensitive to the input turbulence coefficient, the derivative of the simulated velocity with respect to the turbulence coefficient will be large.   This linearized model can then be used in Equation [4.13].  As this linearized model is not a perfect representation of the non-linear model (the difference being the neglect of the higher 86  order derivatives), the minimum parameter combination for the linearized model will not necessarily correspond to the non-linear model.  Therefore, the linearization process must be repeated multiple times until the best-fit parameter combination corresponding to the non-linear model is found.  When the algorithm is far from the best-fit parameter combination, it has been found that incorporating some information about the direction of steepest descent can improve the efficiency of the algorithm.  This is done through the use of Equation [4.16].  This algorithm is referred to as the Gauss-Marquart-Levenberg (GML) algorithm. Further details of this algorithm can be found in Mikosch et al. (2006) and Doherty (2010).  𝒃 = 𝒃0 + [𝐽𝑡∑𝐽 + 𝛼I]−1𝐽𝑡∑𝒓(𝒃0) [4.16] where the new parameter 𝛼 is a hyperparameter that controls how much the next iteration will follow the direction of steepest descent, and I is the identity matrix.   To use this algorithm to find the best-fit basal resistance parameters, the user specifies 𝑦𝐹and ∑, as well as an initial guess of the best-fit parameter values (used to calculate 𝑦𝑀(𝒃0) and J in Equation [4.15]). The algorithm then runs the model multiple times to determine the residuals, as well as the elements of the Jacobian matrix (‘J’ in Equation [4.15]). Based on the residuals and the Jacobian matrix, the algorithm then selects a new set of model parameters that reduces the value of the fitness function (Equation [4.5]). This process is repeated until the residuals are reduced to a user-specified value, the parameter change from one iteration to the next is below a user-specified threshold or the simulation is manually terminated.     87  4.6 Implementation The parameter estimation package PEST (Doherty, 2010) has been used for the inverse simulation detailed in this work. PEST is a model-independent parameter estimation package and contains an implementation of the GML algorithm that can handle both overdetermined (number of parameters greater than the number of constraints) and underdetermined (number of constraints greater than the number of parameters) inverse problems. In addition to providing a set of optimized model parameters, the PEST algorithm outputs a parameter correlation matrix, which provides information on the uniqueness of the best-fit parameters. Details of how PEST calculates parameter correlation are found in Doherty (2010).    In the current implementation, the Jacobian matrix (J in Equation [4.16]) is calculated using the forward difference or central difference approximation (Doherty, 2010). These approximations are the simplest method to calculate model derivatives, however they suffer from two major weaknesses: 1. They require a large number of model runs to calculate. 2. They can be inaccurate. The calculation of sensitivities with these methods requires one or two model runs per parameter being estimated. This is not seen as a large impediment when estimating parameters in Dan3D due to the fact that the number of parameters being estimated is small. Inaccurate derivatives can potentially be an issue; however, using PEST to calibrate Dan3D to real case histories as part of this work has shown that the model can converge to the best-fit parameters, indicating that the derivatives are suitably accurate.  88  4.7 Example Case History- Zymoetz River Rock Avalanche The purpose of the analysis described below is to demonstrate both the sensitivity analysis approach to derive a parameter probability density function, as well as the GML algorithm. The selection of standard deviations for the field constraints is also explained.  4.7.1.1 Description of the Event The Zymoetz River Rock Avalanche (ZRRA) occurred in British Columbia, Canada about 18 km east of the city of Terrace, B.C. An aerial photograph of this landslide is shown in Figure 4-6. The landslide initiated as a slide in volcanic bedrock, with an initial volume of about 900,000 m3. The landslide travelled about 4 km down a sinuous channel, entraining an additional 500,000 m3 of material (McDougall et al., 2006). The volume of material entrained during this event is greater than 25% of the final volume, so this event is classified as a “rockslide-debris avalanche” in the terminology of Hungr & Evans (2004). This highly mobile landslide severed a gas pipeline, resulting in an estimated indirect cost of $30 million (Schwab et al., 2003). 89   Figure 4-6: Overview of the Zymoetz River rock avalanche.  The simulation constraints used are the volume of the deposit near the 'C' on the figure (100,000 m3), the velocity of the flow at the cross (17 m/s) and the impact area (extending from 'A' to 'D').  The rheology change was implemented in the channel, downstream of the location labelled ‘B’. Image modified from McDougall et al. (2006).  Image: Province of British Columbia, Copyright © Province of British Columbia. This event has been described in Schwab et al. (2003), Boultbee (2005) and Boultbee et al. (2006). A previous Dan3D back-analysis of this event is provided in McDougall et al. (2006). It was found that the event could be well-simulated using a frictional rheology in the source zone and then switching to the Voellmy rheology when entrainment occurred (the location of the rheology change is shown as ‘B’ on Figure 4-6). The friction angle in the source zone was set at 30 degrees, close to the ‘expected’ friction angle for dry fragmented rock (Hsu, 1975). Due to this, the only calibrated parameters are the two parameters that control the Voellmy rheology. McDougall et al. (2006) found that a friction coefficient of 0.1 and turbulence coefficient of 1000 m/s2 resulted in an acceptable simulation of the bulk landslide behaviour based on a visual assessment of trial-and-error simulation results (using the method demonstrated in Figure 5). 90  4.7.1.2 Simulation Constraints Three simulation constraints were used for this inverse analysis. These constraints include the impact area, deposit distribution and a velocity estimate. The impact area of the landslide was determined from post-event aerial imagery. A trimline grid file was created based on this information, and is shown in Figure 4-3.  It was estimated that a volume of 600,000 m3 deposited in the upper cirque basin, 200,000 m3 in the channel (about 100,000 m3 deposited downstream of ‘C’ on Figure 4-6) and 600,000 m3 in the river at the bottom of the channel (See Figure 4-6 for the location of these deposit zones).  Figure 4-6 shows the location where the flow rounded a bend and superelevated. Based on the forced vortex equation, a velocity of 16 to 18 m/s was estimated (Boultbee et al., 2006).  4.7.1.3 ZRRA - Sensitivity Analysis A sensitivity analysis was run using friction coefficients of 0.04 to 0.19, varied by steps of 0.01, and turbulence coefficient values of 100 to 2000, varied by steps of 100. The sensitivity analysis therefore comprises 285 model runs. The results of the sensitivity analysis are succinctly summarized in Figure 4-7 to Figure 4-9. These types of figures show contour plots of individual fitness metrics, as well as contours of the fitness function for each parameter set tested.  Figure 4-7 shows contours of trimline fitness for each parameter set. It is interesting to note that a wide range of parameters lead to the same fitness value. Parameter non-uniqueness associated with the two-parameter Voellmy rheology was first pointed out by Koerner (1976) and is displayed here. When only the trimline is used as a constraint, a wide range of parameters give a similar fit. This is particularly true of this case due to the fact that the landslide ran out into a river at its distal 91  end. This means that any set of resistance parameters that lead to the landslide staying in the channel and reaching the river will have approximately the same trimline fitness. This is a unique feature of the ZRRA, as most rock avalanches have a well-defined impact area.   The fitness values have been normalized to have a mean of 0 and standard deviation of 1.  This was done by subtracting the mean of residuals (0) and dividing by the standard deviation (Hsieh, 2009).  The standard deviations used for each of the residuals are summarized in  Table 4-2.  Table 4-2: Standard deviation of measurements used to normalize the residual values. Fitness Metric Standard Deviation Trimline 1150 Velocity 1 m/s Deposit Distribution 15,000 m3  In  Table 4-2, and following the discussion in Section 4.3.5, the standard deviation of the trimline was selected based on half the lowest trimline residual in the sensitivity analysis.  The standard deviation of the velocity measurement was selected based on the error estimate of +/- 1 m/s made by Boultbee et al. (2006), which indicates velocities of 16-18 m/s.  The standard deviation of the deposit distribution was assigned based on a subjective confidence interval of 85,000 m3 to 115,000 m3.   92  When subjectively interpreting model results, there is no need to explicitly state the error associated with a field measurement.  Instead, the modeller subjectively determines that a given set of model outputs are ‘acceptable’.  The use of standard deviations in the calibration process explicitly defines when model results are ‘acceptable’.  In this way, the new calibration methodology forces the modeller to be transparent about why they have accepted a given model as well calibrated.     Figure 4-7: Normalized trimline fitness residual for all parameter combinations in the parameter space. The contours of deposit depth (Figure 4-8) show that this constraint is insensitive to the turbulence coefficient. This is due to the fact that, when the Voellmy rheology is used, the friction coefficient determines the slope angle where material deposits. The contours of velocity (Figure 4-9) show that this constraint is sensitive to both the turbulence and friction coefficient, with the highest velocity corresponding to the lowest resistance parameters.  93   Figure 4-8: Normalized residual of volume deposited in the channel.  One standard deviation around the mean is highlighted in red.  Figure 4-9: Normalized residual of velocity simulated at the bend in the channel.  One standard deviation around the mean is highlighted in red. The value of the objective function (Equation [4.5]) calculated based on these residuals is shown in Figure 4-10.  A narrow parameter zone, with friction coefficients from 0.09 to 0.105, and turbulence coefficients from 500 to 2000 m/s2 defines the best-fit parameter space.  The posterior PDF, obtained by evaluating Equation [4.9], is shown in Figure 4-11. The posterior PDF defines a narrow parameter range, with 3 peaks.  These peaks are due to the non-linearity in the velocity 94  residual (Figure 4-9).  If further simulation constraints were available for this case the zone of best fit parameters would likely shrink.  Figure 4-10: Contours of the objective function (Equation [4.5]) for the Zymoetz River Rock Avalanche, calculated based on normalized simulation constraints.  Figure 4-11: Zymoetz River Rock Avalanche posterior PDF.  High probabilities are assigned to parameter combinations that well reproduce the field constraints. 95  4.7.2 Application of the GML Algorithm The results of the sensitivity analysis described above indicate a narrow zone of best-fit parameter combinations (Figure 4-10 and Figure 4-11). The results of the sensitivity analysis will be used to understand the behaviour of the GML algorithm, as the direction of steepest descent used by the GML algorithm, as well as the optimization steps, can be visualized on Figure 4-12.  Figure 4-12: Optimization steps taken by the GML algorithm for the Zymoetz River Rock Avalanche, overlaid on the contours of the objective function.  The red crosses show the steps taken by the GML algorithm, obtained by iteratively solving Equation [4.16]. PEST provides the option to calculate the sensitivity of model outputs to input parameters based on scaled values of the input parameters. Making use of this feature is crucial when estimating the parameters that control the Voellmy rheology. Since the friction coefficient typically ranges between 0.01 and 0.2 and the turbulence coefficient ranges from about 100 to 2000 m/s2, the outputs are much more sensitive to small changes in the friction coefficient than in the turbulence coefficient. If unscaled parameters are used, the parameter estimation process will be unable to determine optimal values of the turbulence coefficient due to the fact that its sensitivity 96  will be low. In order to avoid this issue, the friction coefficients were multiplied by a scale factor such that a value of 0.01 scales to 100 and a value of 0.2 scales to 2000.  This is a universally applicable scale factor for all PEST analyses used to calibrate the Voellmy model.  The optimization steps are shown on Figure 4-12. The GML algorithm takes 42 model runs in order to converge to the minimum value of the fitness function.  The number of model runs is greater than the number of optimization steps because calculating the Jacobian matrix, which must be done at each step, requires multiple model runs. The initial guess was purposely chosen to be far from the optimal parameter set. A poor initial guess increases the number of iterations it takes to converge to the optimal parameter set; however, it does not prevent model convergence. The parameter correlation matrix is presented in Table 4-3. This matrix indicates that parameter correlation is moderate. Figure 8 shows that a range of parameter values from friction coefficients of 0.095 to 0.11 and turbulence coefficients from 1300 to 1900 m/s2 all give similar fitness values. The moderate parameter correlation is a reflection of this.  Table 4-3: Parameter correlation matrix calculated at the conclusion of the inverse model. The moderate correlation between the two parameters reflects the fact that multiple parameter combinations can give similar fitness results.  Friction Coefficient Turbulence Coefficient Friction Coefficient 1 -0.59 Turbulence Coefficient -0.59 1  97  4.8 Discussion and Conclusions Both the GML algorithm and the sensitivity analysis approach can be used to effectively calibrate Dan3D for a given case.  The GML algorithm can determine a set of best-fit parameters using relatively few model runs (as compared to the sensitivity analysis approach). The calibration procedure is performed automatically, and only needs minimal user intervention. Compared to trial-and-error calibration, the user is required to specify a covariance matrix.  This matrix explicitly quantifies error in measured values, allowing for a user to transparently define when a given model residual is acceptable (a process implicit in subjectively interpreting model results).  At present, subjective judgement is required to select this matrix.  Future work could focus on more objective methodologies to define the error in the measured values of the various types of back-analysis constraints. The new calibration methodology allows the user to focus on testing different problem configurations, and provides some assurance that the best set of parameters for a given configuration has been found.  The drawback to using the optimization approach is that parameter uniqueness can only be assessed using the parameter correlation matrix, which is more difficult to interpret than the results of sensitivity analyses.  Therefore, the use of the GML algorithm is only recommended when more than two parameters are being calibrated. For simulations where one or two parameters are calibrated, sensitivity analyses such as that detailed in Figure 8, as well as the methods proposed by McDougall (2006) and Cepeda et al. (2010), can identify parameter non-uniqueness and, if performed diligently, will explore the entire parameter space.  98  The error associated with back-analyzed parameters can be quantified using the techniques presented in this chapter. Figure 4-10 shows that, even for the well-constrained case provided, a zone of parameter values all result in similar values of the fitness function. Based on the assumption of uncorrelated, Gaussian model errors, Equation [4.9] can be used to derive a posterior distribution for the input parameters.  This posterior distribution is useful when combining the results of multiple back-analyses to conduct a probabilistic forward analysis with Dan3D.  This application will be expanded upon in Chapter 5, where the posterior probability density functions from multiple rock avalanche case histories will be combined.   99  Chapter 5: Rock Avalanches 5.1 Introduction Rock avalanches are extremely rapid flows of fragmented rock that can impact people and property far from their source.  Predicting the potential impact area of a rock avalanche before it initiates is a crucial step in the risk analysis of these flows.  When performing these sorts of forecasts with runout models, a modeller must be able to select a model parameterization before the event occurs.  When using an equivalent fluid model such as Dan3D, parameter selection requires both an understanding of rock avalanche movement mechanisms, as well as a database of successful rock avalanche back-analyses to guide forward analysis.   A database of thirty rock avalanche case histories has been assembled.  Of these cases, twenty-four have been back-analysed using the methodology detailed in Chapter 4.  Six cases were not analysed either because they have been well calibrated by other researchers, or they were reserved for validation of the proposed probabilistic prediction framework.  The results of this calibration effort are interpreted to infer information about rock avalanche movement mechanisms.  A methodology for probabilistic prediction of rock avalanche motion is also presented. This chapter is laid out as follows.  Firstly, a review of rock avalanche movement mechanisms is provided.  Secondly, a database of rock avalanche case histories, assembled in this research, is summarized.  Thirdly, the systematic back-analyses of these case histories are compared to infer information about the mechanisms that govern rock avalanche mobility.  Finally, a framework 100  for probabilistic runout predictions is developed and example forecasts based on this framework are given.   5.2 Background       Rock avalanches typically initiate as large rock slope failures on steep mountain slopes.  After initiation, the failed rock mass can travel some distance as a coherent block, before it disintegrates and turns flow-like (more detail on this process was provided in Chapter 3).   Beginning with the work of Heim (1932), many researchers have noted an apparent increase in rock avalanche mobility with increasing volume (e.g. Heim, 1932; Scheidegger, 1973; Hsu, 1975; Li, 1983; Legros, 2006; Davidson, 2011; Whittall et al., 2017).  This observation is based on plots of volume vs. angle of reach (defined as the inclination from horizontal of the line connecting the highest point on the failure scarp to the distal end of the deposit), suggesting that higher volume events have greater mobility.  The H/L (or ‘Fahrböshung’ in the terminology of Heim (1932) ) ratio is explained in Figure 5-1.   Heim (1932) showed that, for a frictional material, the inclination of the line connecting the centers of mass of the source and deposit (dashed red line on Figure 5-1) is equivalent to the average apparent friction angle of the flow.  The H/L ratio can be seen as an approximation of the inclination of the line connecting the centers of mass, and is therefore an approximation of the average apparent friction coefficient.  The friction coefficient for dry, fragmented rocks is expected to be approximately 0.6 (Hsu, 1975).  As shown on Figure 5-2, the average friction coefficient of many rock avalanches is lower than this value.         101   Figure 5-1: Explanation of the H/L ratio.  The dashed grey line represents the line connecting the top of the back-scarp to the distal end of the deposit.  The inclination of this line from horizontal, defined above as the angle of reach, is tan-1(H/L).  The dashed red line shows the line connecting the center of mass of the source to that of the deposit.  For a frictional material, the inclination of this line from horizontal is the average friction coefficient experienced by the moving mass (Heim, 1932). As shown in Figure 5-2, the motion of rock avalanches cannot be explained using the frictional mechanics of flowing fragments of rock.  The mechanism(s) that contribute to this apparent volume-mobility trend remain debated (e.g. Hungr & Evans, 2004).  It should be noted that the disintegration of a rock mass results in an increase in volume of approximately 20%, which removes the possibility of pre-failure pore-pressures within the source mass having a significant role in rock avalanche runout (Hungr & Evans, 2004).      102   Figure 5-2: Volume vs. H/L of cases in the database of rock avalanche case histories.  For comparison, H/L data collected by Scheidegger (1973), Li (1983) and an unpublished database of Canadian rock avalanches (Brideau, M.A, BGC Engineering, unpublished data) are shown.  The cases analysed in this chapter appear to be more mobile than this background data. Cases are sorted by path material.  Descriptions and back-analyses of the majority of these case histories are summarized in Appendix A.  For cases not back-analyzed in this thesis, references to DanW and/or Dan3D analyses are provided.  Labels: 1. Zymoetz, 2. Crammont, 3. Six des Eaux Froides, 4. Huascaran, 5. Kolka (Huggel et al., 2005; McDougall, 2006; Evans et al., 2009b), 6. Mt. Meager, 7. Mt. Steele, 8. Nomash River, 9. Sherman Glacier (Sosio et al., 2012), 10. Thurweiser, 11. McAuley Creek, 12. Val Pola, 13. Avalanche Lake, 14. Goldau, 15. Mystery Creek, 16. Turnoff Creek, 17. Madison Canyon, 18. Chisca, 19. Hope, 20. Pandemonium Creek (Evans et al., 1989), 21. West Salt Creek, 22. Frank, 23. Guinsaugon, 24. Bingham Canyon, 25. Sentinel, 26. Daubensee, 27. Rinderhorn, 28. Rautispitz, 29. Platten, 30. Chehalis. 103  A number of theories have been proposed to explain the apparent correlation between mobility and volume of rock avalanches.  Many of these theories are reviewed by Legros (2002) and Hungr & Evans (2004).  The theories can be broadly grouped into mechanisms that are due to intrinsic characteristics of the rock avalanche material, and those that rely on extrinsic factors such as path material. The following recently-published theories demonstrate that the debate surrounding rock avalanche movement mechanisms is still ongoing.  These include:    Johnson et al. (2016) showed results of discrete particle models that predict an increase of mobility with increasing volume.  They propose that this phenomenon arises from acoustic waves propagating through the particle assembly that reduce intergranular stresses, consistent with the theory of acoustic fluidization (e.g. Melosh 1979).    Manzanal et al. (2016) proposed that rock avalanches dilate upon failure, however, as fragmentation proceeds, the reduction in grain size results in a switch from dilative to contractive behaviour, resulting in generation of pore-air pressures.  High pore air pressures had originally been proposed by Shreve (1968) to explain the anomalous mobility of rock avalanches, although Shreve (1968) proposed that these are generated by the rock avalanche riding on a cushion of trapped air.    Lucas et al. (2014) proposed a velocity weakening rheology, and showed that a consistent set of parameters could reproduce field observations from three rock avalanche case histories.  They noted that their rheology is consistent with the mechanism of flash heating.  Erismann (1979) originally proposed the theory of frictional heating as a rock avalanche mobility mechanism.   104   Bowman et al. (2012) presented geotechnical centrifuge experiments that suggest a positive correlation between degree of fragmentation and runout length.  This is the mobility theory advanced by Davies et al. (1999).  Their experiments used a bilinear path with a high angle sloped portion (70°).  Blasio & Crosta, (2015) demonstrated that a steep path combined with isotropic fragmentation can increase centre of mass displacement, however, the effect disappears for slope angles typical of rock avalanche paths.    One of the oldest mobility theories is that rock avalanche motion is governed by the character of the path materials (Buss & Heim, 1881; Sassa, 1985; Voight & Sousa, 1994; Hungr & Evans, 2004).  Coe et al., (2016) and Aaron & Hungr (2016a) both invoked low basal friction due to entrainment and overriding of saturated soil to explain the dynamics of the West Salt Creek Rock Avalanche and the Avalanche Lake Rock Avalanche, respectively.  In both cases, this hypothesis was supported by field evidence of entrained path material at the base of the deposit.           Sections 5.3 to 5.6 detail an analysis that tests whether the shear resistance experienced by rock avalanches can be explained based on the character of the path material.  As this analysis uses a database of case histories that overran glacial ice, snow, sediment and bedrock, the expected shear characteristics of these materials is summarized below:  Glacier Ice:  The most widely accepted path material that has been demonstrated to increase rock avalanche mobility is glacial ice (e.g. Evans et al., 1989, 1996, 2009a; Delaney & Evans, 2014).  Sosio et al. (2012) shows that Fahrboshung plots of case histories that overran glaciers tend to be more mobile than those that did not.  Sosio et al. (2008) shows that, for a Dan3D simulation to match video evidence of the velocity of the 105  Thurweiser rock avalanche, a low strength along the Thurweisser glacier had to be used, and a high strength along the sections where the rock avalanche overran bedrock.  Schneider et al. (2010) also show that a low strength along the glacier is necessary for simulations of the Thurweisser rock avalanche to accurately reproduce the seismic signal generated by this landslide.  Therefore, it can be expected that cases that overrun glacier ice will experience relatively low resistance to motion.   Snow:  Snow cover increases the likelihood that the underlying soil is saturated, and if debris overrides the snow it can act as a low strength basal layer.  Boultbee et al. (2006) describes field evidence at the Zymoetz River Rock Avalanche that indicates that an initial small volume failure overrode snow and attained a high velocity, indicated by streamlines in the snow and a superelevation that provides velocity estimates of 26 m/s.  The Crammont rock avalanche projected a small volume of debris that moved on a layer of snow and travelled over 3 km (Deline et al., 2011).  It appears as though direct lubrication of rock avalanche debris by snow can only enhance the mobility of small volume failures, as large volume rock avalanches tend to plow the snow layer.  Sediments:  Hungr & Evans (2004) hypothesized that rock avalanche interaction with loose saturated sediments can enhance the mobility of rock avalanches.  They suggested that this is caused by rapid undrained loading of the saturated sediments (e.g. Sassa & Wang, 2005).  As summarized in Section 6.2 in the context of debris avalanches, this mechanism can result in a rock avalanche experiencing extremely low basal resistance. If the path material is unsaturated, then it is expected that basal resistance would be higher than if the path material were saturated.    106   Bedrock:  For cases that overrun bedrock, it is expected that shear resistance will be governed by frictional mechanics, with a friction angle close that expected for dry fragmented rocks (~30°).  Whittall et al. (2017) presented a database of H/L ratios for rock avalanches that occurred in open pits, and showed that that failure in ‘fresh, strong rocks’ behave as dry granular flows with limited mobility.  However, some of these failures in ‘fresh, strong rocks’, such as the Bingham Canyon rock avalanche, displayed high mobility.  As will be argued in Section 5.6, this can likely be explained by a mechanism that reduces basal resistance in the source zone but would not be acting along the path.         5.3 Back-Analysis Methodology Sections 5.5 and 5.6 will use back-analysis results to infer rock avalanche movement mechanisms.  To avoid conclusions based on using a subjective calibration methodology, all the cases in the database have been back-analysed using the techniques developed in Chapter 4.  An example back-analysis is provided in Section 4.6.  Unless otherwise noted, all back-analyses assumed that failure occurred in a single, dominant phase.  This is consistent with previous investigations of the analysed case histories.     For all cases, multiple model parameterizations (including multiple rheologies) were tested in order to determine the best-fit rheology and parameters.  For cases that failed along a continuous structural feature, such as a bedding plane, the flexible block model was used.  For most cases, the frictional rheology was used in the source zone, and, depending on the path materials, the rheology was changed to the Voellmy or Bingham rheology.  The friction angle in the source zone was constrained primarily with observations of the deposit distribution.  As shown on 107  Figure 5-3, some cases had thick deposits at the toe of the source slope.  To reproduce this observation, a high friction angle in the source zone had to be used.    Figure 5-3: Photo of the McAuley Creek rock avalanche.  Thick deposits at the toe of the source slope can be observed.  Image: Google Earth, Province of British Columbia. As will be shown when discussing the Bingham Canyon rock avalanche (Section 5.6), for other cases there are no deposits left in the source zone, despite failure on moderate slopes.  To reproduce this observation a lower friction angle had to be used in the source zone.    Rheology changes were implemented at locations where there is a likely change in the character of the path material, often at the toe of the steep slope that the rock avalanche descended.  For both the Voellmy and Bingham rheology, a range of input parameters were tested so that the zone of best possible fit to available simulation constraints could be determined using a brute force sensitivity analysis (Section 4.4).  Conducting systematic back-analyses in this way allows 108  for the fitness of various model parameterizations to be objectively compared.  This can be used to infer movement mechanisms based on the following three axioms: 1. If two cases are analyzed with the same rheology, and one case has parameters that correspond to lower basal resistance, then the mechanism that govern basal resistance leads to lower resistance for this case. 2. The spatial distribution of the back-analyzed rheological parameters is correlated with the spatial distribution of basal resistance that results from the mechanism that governs resistance to motion.  For example, consider a case history where high resistance along one part of the path is back-analyzed to reproduce a thick deposit, and low resistance along a different part of the path is required to produce a long, thin deposit.  This axiom states that this information can be interpreted to mean that the flowing mass experienced high resistance in one area of the path and low resistance in another area.  3. If one rheology provides better reproduction of simulation constraints than an alternative rheology, then the mechanism governing resistance to motion produces a basal resistance more consistent with the assumptions of the best-fit rheology.  For example, if the deposit distribution of a case history is better reproduced with the Bingham rheology as opposed to the Voellmy rheology (an overview of these rheologies is provided in Section 2.5.2), then the mechanism governing basal resistance result in a constant yield stress. Using the three axioms detailed above, the results of the back-analysis of the cases in the rock avalanche database can be combined with field observations to infer movement mechanisms.  Using these axioms, the back-analysed rheologies and parameters provide a useful means of 109  interpreting field observations.  This is analogous to the use of a telescope to study the night sky.  When observing the night sky with a telescope, as opposed to the naked eye, new observations can be made to test existing theories.  Similarly, when back-analysis results are interpreted with the three axioms detailed above, the results provide new ways of interpreting field data by allowing for field observations to be interpreted based on a macroscopic force balance that accounts for 3D path topography, and is constrained by observed 3D impact area, deposit distribution and velocity estimates (where available). The following three rock-avalanche mobility hypotheses will be tested:   1. Rock avalanche mobility is governed by a universal, volume-dependent mechanism.  If this is the case, then all rock avalanches of a given volume should share the same rheology and similar basal resistance parameters. 2. Rock avalanche mobility is governed by site-specific mechanisms.  If this is the case, then for two cases of a given volume, site specific factors can lead to dramatically different runout behaviour. 3. Both universal and site-specific mechanisms act together to govern rock avalanche mobility.   These hypotheses are tested by comparing the back-analysed shear strength parameters to those that would be predicted based on the hypothesis that shear resistance is governed by the character of the path materials.  As was discussed in Section 5.2, we can hypothesize that certain types of path materials will lead to different basal resistance.  If the back-analysed strengths are consistent with those that would be hypothesized based on the path material, then the results of the present work would support hypothesis 2 (mobility being governed by site-specific 110  mechanisms).  This would not rule out hypothesis 3; however, it would provide evidence that site-specific mechanisms must be considered when analyzing rock avalanche runout.  If the back-analysed strengths are inconsistent with those hypothesized based on the character of the path material, then the results of the present work would support hypothesis 1 (mobility governed by a universal, volume-dependent mechanism).  Overall, if hypothesis 2 is more likely, it is expected that cases that overrun bedrock will experience higher resistance to motion than cases that overrun saturated substrate, glacier ice and/or snow.  It is expected that cases that overrun unsaturated substrate will experience basal resistance that is in between saturated substrate and bedrock.  It is also hypothesized that back-analysed strength parameters of the path material should be somewhat independent of volume, as mechanisms such as rapid undrained loading (Section 5.2 and 6.2) can enhance the mobility of both small and large volume rock avalanches.  5.4 Case Histories Twenty-four rock avalanche case histories have been back-analyzed using the calibration methodology detailed in Chapter 4.  With the exception of the Bingham Canyon rock avalanche, all cases in the database are natural rock avalanches.  Detailed descriptions of the case histories can be found in Appendix A.  The volume of the cases in the database spans three orders of magnitude from 1.0*105 m3 to 4.0*108 m3.  Most cases are between 1.0*107 m3 to 1.0*108 m3.  Figure 5-2 shows the volume vs. H/L relationship of the cases in the rock avalanche database.  The cases span a wide range of mobility, from the ‘expected’ value for dry fragmented debris of 0.6 (Hsu, 1975) to the extraordinarily mobile Sherman Glacier rock avalanche (H/L of 0.1).  Figure 5-2 also compares the H/L values measured for the cases in the database to those 111  measured by Scheidegger (1973), Li (1983) and Canadian rock avalanches (Brideau, M.A, BGC Engineering, unpublished data).  Compared to this data, on average the cases analysed in this chapter appear more mobile than those compiled by Scheidegger (1973) and Li (1983), and similar to the Canadian cases.  Therefore, it can be expected that the cases analysed in the present work are biased towards highly mobile rock avalanches.  As will be discussed in Section 5.8, this will have important implications when using these cases to guide forward analysis.   As shown on Figure 5-2, most cases in the database likely overran saturated substrate, however cases that overran bedrock, glaciers and likely unsaturated sediments were also back-analyzed.  Table 5-1 summarizes the back-analysis parameterization, available constraints and path materials of the 24 cases back-analysed in the present work.  Two or three rheology simulations were conducted for 18 of the case histories, as these cases encountered multiple types of path material.  To back-analyse four cases (Mt. Meager, West Salt Creek, Sentinel and Huascaran), only one rheology was used, despite the fact that these cases encountered multiple path materials.  This was done because the length of the source zone relative to the length of the path was so small that changing the parameters in the source zone had no discernable effect on the simulation results.  The Chehalis and Daubensee cases both only overran one type of path material, so only one rheology was used for these simulations.   For all cases, the impact area (which, as discussed below, excludes the ‘splash zone’ surrounding many rock avalanche case histories) was available as a constraint. Additionally, estimates of the deposit distribution were available for 14 of the cases, and velocity estimates were available for five of the case histories.  112  Table 5-1: Summary of the 24 back-analysed rock avalanche case histories. Name Setup Constraints Path Materials* Crammont  Two rheologies used with the transition selected to account for plowing of snow cover  Frictional in source area  Voellmy along path  Impact area  Deposit distribution  Snow Huascaran  Entrainment used   Same rheology used in source zone and path  Impact area  Deposit distribution  Velocity  Glacier ice  Loose, saturated substrate Madison Canyon  Two rheologies used with the transition selected when the mass moves from bedrock to path sediments  frictional in the source zone  Voellmy on the valley floor  Used flexible block model  Impact area  Deposit distribution  (likely) unsaturated, coarse grained sediments McAuley  Two rheologies used with the transition selected downstream of the source zone  Frictional rheology in the source zone  Voellmy rheology along the path  Impact area  Deposit distribution  Saturated sediments Mt. Meager  One rheology used for source zone and path  Impact area  Velocity  Deposit distribution  Loose, saturated sediments Mt. Steele  One rheology used for source zone and path  Impact area  Glacier ice Nomash  Entrainment used  Two rheologies used, with the transition selected to correspond with zone where entrainment begins  Frictional rheology used in the source zone  Voellmy rheology used at the onset of entrainment  Impact area  Entrained volume  Loose, saturated sediments Six des Eaux Froides  Two rheologies used, with the transition selected at the toe of the slope  Frictional rheology used in the source zone  Voellmy rheology used at toe of the slope  Used flexible block model  Impact area  Saturated sediments  113  Name Setup Constraints Likely Path Materials* Val Pola  Two rheologies used, with the transition implemented at the toe of the slope  Frictional rheology in the source zone  Voellmy rheology along the path  Impact area  Saturated sediments Thurweisser  Two frictional rheologies used, one for bedrock and the other for the glacier  Impact area  Velocity  Deposit distribution  Bedrock  Glacier Ice West Salt Creek  Single Bingham rheology used for entire runout path  Impact area  Deposit distribution  Velocity  Saturated fine grained sediments Sentinel  Single Voellmy rheology used for both source and path  Used flexible block model  Impact area  Unsaturated sediment Daubensee  Single frictional rheology used for both source and path  Used flexible block model  Impact area  Bedrock Rinderhorn  Two rheologies used with transition selected in area where the mass overrode valley sediments  Frictional rheology in the source zone  Voellmy rheology along the path  Used flexible block model  Impact area  Saturated sediments Bingham Canyon  Two frictional rheologies used, with the transition slelected where the mass vacates the source zone  Used flexible block model  Impact area  Velocity  Deposit distribution  Bedrock Zymoetz  Two rheologies used with transition selected to account for plowing of snow  Frictional rheology in the source zone  Voellmy rheology along the path  Impact area  Velocity   Deposit distribution   Snow  Saturated sediments Goldau  Two rheologies used with the transition selected at the toe of the slope  Frictional rheology in the source zone  Voellmy rheology along the path  Used flexible block model  Impact area  Deposit distribution  Bedrock  Saturated sediments  114  Name Setup Constraints Path Materials* Avalanche Lake  Two rheologies used with transition selected to correspond where the mass overrode valley fill sediments  Frictional rheology in the source zone  Voellmy rheology along the path  Impact area  Deposit distribution  Saturated sediments Mystery Creek  Two rheologies used with the transition selected at the toe of the slope   Frictional rheology in the source zone  Voellmy rheology in the path  Used flexible block model  Impact area  Saturated sediments Frank  Two rheologies used with the transition selected at the toe of the slope  Impact area  Deposit distribution  Saturated sediments Rautispitz  Two rheologies used with the transition selected at the toe of the slope  Used flexible block model  Impact area  Saturated sediment Platten  Two rheologies used with the transition selected at the toe of the slope  Used flexible block model  Impact area   Saturated sediment Chehalis  Single frictional rheology used  Impact area  Deposit distribution  Bedrock Guinsaugon  Three rheologies used with the transition selected to correspond to where the mass interacted with various path materials  Frictional rheology in the source zone  Two Voellmy rheologies along the path  Used flexible block model  Impact area  Deposit distribution  Loose saturated sediments  Flooded paddy field  *Justification of the likely path materials is provided in Appendix A . 115  5.5 Back-Analysis Results A plot of volume vs. source zone friction angle is shown in Figure 5-4.  As summarized in Section 5.3, the source zone friction angle was constrained through observations of the deposit distribution.  If all the material was observed to vacate the source zone then a low friction angle was back-analysed, and if some material deposited in the source zone then a higher friction angle was back-analysed.  Figure 5-5 shows the best fit zones of the path material for cases that overran sediments colored by event volume.  These cutoff zones represent 70% ‘credible regions’ based on the posterior probability distributions derived for each of the case histories (Gregory, 2010).  These  represent the regions of the parameter space with the highest probability of containing the best-fit parameter values (Gregory, 2010).  A 70% credibility region was selected for interpretability, however wider credibility regions (for example a 95% credible region) will increase the area of the best fit zones, but do not change the overall interpretation of the plot (in terms of the relative basal resistance experienced by each of the case histories).  Due to parameter non-uniqueness, discussed in Chapter 4, only a best fit zone could be determined for many of these cases.  Figure 5-6 shows the best fit parameter zones for the cases that encountered sediments and/or snow along the path.  The West Salt Creek rock avalanche encountered saturated sediment along the path, however, the impact areas and velocities of this event were best reproduced with the Bingham rheology.  Figure 5-7 shows the best fit friction angles for cases that overran bedrock as the path material.   116   Figure 5-4: Back-analysed friction angles in the source zone.  As mentioned in Section 5.3, these values are constrained by observations of the deposit distribution.  There is a reduction in source zone friction angle with increasing volume possibly due to a reduction in frictional strength due to shearing under high normal stresses.  See Figure 5-2 for case names that correspond to the case numbers.    Figure 5-5: Best fit path material parameter zones for each of the cases analyzed with the Voellmy rheology.  Each polygon represents the 70% credibility region for one case, the case names corresponding to the case numbers can be found in the caption of Figure 5-2.  The cases are coloured based on volume.  The strength in the path materials appears to be independent of volume.  For all cases except 4, 8 and 25 (Huascaran, Mt. Meager and Sentinel, respectively) the frictional rheology was used in the source zone. 117   Figure 5-6: Best fit Voellmy parameters colored by path material.  Each polygon represents one case, case names can be found in the caption of Figure 5-2.  For all cases except Mt Meager (8), Huascaran (4) and Sentinel (25) the frictional rheology was used in the source zone, with a friction angle shown on Figure 5-4.       Figure 5-7: Best fit friction angles for cases that overran bedrock.  Each point represents one case.  Case names can be found in the caption of Figure 5-2.    118  5.6 Discussion For the majority of cases, good results were obtained using a frictional rheology in the source zone and then a change in resistance parameters and/or rheology for the path materials, over-ridden downslope of the source area.  Figure 5-4 appears to show that the source zone friction angle is volume-dependent.  This suggests that some mechanism is required to significantly reduce the basal resistance along the rupture surface.  Many of the rupture surfaces of the large volume rock avalanches develop along continuous planar features.  One likely factor which can significantly reduce the strength along a continuous planar feature is extreme polishing and removal of asperities, which leads to a reduction of the friction angle to the “ultimate value”, as proposed by Cruden & Krahn (1978).  This mechanism would be volume-dependent, as higher volume cases would have higher normal stresses along the planar feature, which would increase polishing and removal of asperities.   As shown in Figure 5-5, the volume-dependence of basal shear strength appears to disappear once the rock avalanche has vacated the rupture surface and is overriding path material.  Figure 5-6 and Figure 5-7 show that the character of the path material is a plausible explanation for the variance in the shear resistance experienced by the moving rock avalanche once it has vacated the source zone.  Figure 5-6 shows that, independent of volume, the cases that overran saturated substrate experienced relatively low resistance to motion, with best fit friction coefficients as low as 0.05.  These cases span a range of volumes from 0.2 Mm3 to 200 Mm3.   Two of the least mobile case histories in the database, Sentinel and Madison Canyon, overran sediments; however, it is likely that these sediments were unsaturated.  Figure 5-6 shows that the basal resistance parameters for these cases are relatively high.  Sentinel and Madison Canyon are both 119  located in arid regions (Hadley, 1978; Castleton et al., 2016; Wolter et al., 2016).  Additionally, Hadley (1978) showed that there was no precipitation in the six weeks leading up to the Madison Canyon event.  Thus, it is likely that these cases were relatively immobile because they overran unsaturated substrate materials.   The velocity dependent term in the Voellmy rheology has the potential to introduce some volume dependence in the basal resistance (the equation used to calculate basal resistance based on the Voellmy rheology is shown in Section 2.5.2).  This is because larger volume cases tend to have greater flow depth then smaller volume cases, however for a given velocity the velocity dependent term will be constant.  Therefore thicker flows will have a greater difference between driving and resisting stress.  Figure 5-5 shows that, for a given volume class, basal resistance appears independent of volume.   Therefore it is not expected that the volume dependence implicit in the Voellmy rheology is masking a volume dependent effect in the back-analysed basal resistance parameters. One case that appears to disagree with this overall trend is Val Pola (case 12 on Figure 5-6).  The best fit basal resistance parameters back-analyzed for the path material in this case is closer to those analyzed for the two cases that likely overran unsaturated substrate.  It failed after a period of extremely heavy rainfall, so it is likely the ground was saturated.  Further research is required to better understand why this case history was not more mobile.       Figure 5-7 shows that the cases that overran bedrock all experienced relatively high resistance to motion, regardless of volume (friction coefficients ranging from 0.38 to 0.53).    Based on this, the back-analysed strengths along the path appear to be poorly explained by a volume-dependent 120  movement mechanism, however, they are consistent with those that would be hypothesized based on the character of the path material.  The contrast of shear strength in the source zone and path is clearly demonstrated by comparing the distribution of shear strength derived for the Bingham Canyon rock avalanches with that derived for the Nomash River rock avalanche.  Overview images of these two case histories are shown in Figure 5-8.      121   Figure 5-8: Comparison of the Bingham Canyon and Nomash River Rock Avalanches.  For Bingham Canyon, low strength in the source zone lead to no material depositing on the basal fault, which is inclined at 23°.  A lack of runup at the toe of the landslide indicates high basal resistance along the path.  Conversely, at Nomash River, low strength along the path lead to runup features, as well as long runout on a shallow slope.  This distribution of shear strengths is supported by the back-analyses of these two case histories.  Bingham canyon image is from Aaron et al. (2017b), who modified an image by Pankow et al. (2014) , Nomash River image was taken by Dana Ayotte, and published in McDougall & Hungr (2005), © 2008 Canadian Science Publishing or its licensors. Reproduced here with permission. The 30 Mm3 first phase of the Bingham Canyon rock avalanche initiated on a shallowly- dipping fault plane, and only a small volume of material was deposited in the source zone.  This observation places a strong constraint on the friction angle in the source zone, which was back- analyzed to be 10°.  This low value was required in order to have the entire failed volume vacate 122  the rupture surface.  The rock avalanche then flowed across the pit floor and did not run up the opposing pit wall.  To reproduce this observation, a friction angle of 26° had to be used outside the source zone.  This high angle is close to that expected for dry fragmented rock, and no further mechanisms of strength reduction are needed to explain the runout.   The Nomash River case history is a 0.375 Mm3 rock avalanche that failed along an irregular rupture surface (Hungr & Evans, 2004; McDougall & Hungr, 2005).  The back-analysed friction angle in the source zone is 30°, which is close to the limit equilibrium friction angle.  This high strength in the source zone is likely due to the relatively small volume of the rock avalanche, combined with the lack of a continuous structural feature controlling the rupture surface.  After failure, this small volume rock avalanche (well below the typical cutoff of 106 m3 usually quoted for excessive mobility (e.g. Scheidegger, 1973)) overrode and entrained an additional 0.36 Mm3 of saturated sediments (Hungr & Evans, 2004; McDougall & Hungr, 2005).  In order to reproduce the observed runup features, as well as the long runout on a shallow slope (Figure 5-8), the Voellmy rheology was used, with back-analysed parameters corresponding to relatively low basal resistance.  The coincidence of these two factors (entrainment of material corresponding to low back-analysed strength) is interpreted to indicate that interaction with path materials dramatically increased the mobility of the Nomash River rock avalanche.  This interpretation was first suggested by Hungr & Evans (2004).  Therefore, the back-analyses of these two case histories appear to support the hypothesis of shear resistance being dependent on the character of the path material.  In the case of Bingham Canyon, the path material being bedrock likely lead to high basal resistance, whereas for Nomash River, entrainment of saturated sediments likely lead to low basal resistance.    123  5.6.1 Comparison of Two Extremely Large Volume Rock Avalanches Another example of the site-dependence of rock avalanche mobility can be seen by comparing the 286 Mm3 Sentinel Rock Avalanche (Sentinel) with the 200 Mm3 Avalanche Lake Rock Avalanche (Avalanche Lake).  These two rock avalanches are shown in Figure 5-9.   124   Figure 5-9: Comparison of the Avalanche Lake and Sentinel rock avalanches.  At Avalanche Lake, the failed mass rapidly descended the source slope, and ran up a 640 m high adverse slope, creating the largest rock avalanche runup feature ever documented.  The spectacularly energetic mass also spread out along the valley floor, creating the main deposit and south lobe.  The Sentinel rock avalanche, which is similar in volume to Avalanche Lake, also impacted an adverse slope.  However, it only created a moderate runup against this slope, and only displayed limited spreading along the valley floor.  Avalanche Lake photo: Oldrich Hungr, Zion Canyon photo: Google Earth, USDA Farm Service Agency. 125  The shear strength parameters back-analysed for Sentinel show that it experienced much greater basal resistance than Avalanche Lake.  Both these rock avalanches have similar path geometry, as they both impacted an adverse slope.  Sentinel was stopped by the adverse slope, and spread out moderately in the cross-slope direction. Avalanche Lake overtopped a steep adverse slope, creating the highest documented runup of any rock avalanche ever recorded.  The volumes of these two rock avalanches are of the same order of magnitude, so any universal, volume-dependent mobility mechanism cannot explain these contrasting runout behaviours.  Evans et al., (1994) provides field evidence that indicates that Avalanche Lake overran and entrained saturated sediment after it had vacated the source zone.  Conversely, Castleton et al. (2016) suggest that Sentinel failed in a dry environment, where the path materials were unlikely to be saturated.  The contrast of back-analysed strength, combined with the contrast of the character of the path material, is interpreted to indicate that the basal resistance of these two rock avalanches is likely governed by the character of the path material.  Another difference between these two events is that Avalanche Lake failed along a planar rupture surface, whereas Sentinel likely did not fail along a continuous structural feature.  Thus the source zone strength experienced by Avalanche Lake (back-analysed value of 15°) was likely less than that for Sentinel (back-analysed value of 23°) due to a reduction of strength in the source zone due to shearing along a planar surface under high normal stress.   5.6.2 Two Rock Avalanches in the Bernese Alps Another demonstration of the site-dependence of rock avalanche mobility can be seen by contrasting the runout behaviour of the Rinderhorn and Daubensee rock avalanches, two rock 126  avalanches of roughly the same age that failed within 2 km of each other.  These two rock avalanches are shown on Figure 5-10.    Figure 5-10: Comparison of the Daubensee and Rinderhorn rock avalanches.  The Daubensee rock avalanche overran glacially-abraded bedrock, which limited its mobility.  The Daubensee rock avalanche initially overran bedrock, but after descending its source slope it overrode fluvial sediments, which greatly enhanced its mobility.  After Grämiger et al. (2016), reprinted with permission. 127  Grämiger et al. (2016) presents cosmogenic nucleid exposure dates that indicate these two rock avalanches are coincident in age, and field evidence suggests a seismic trigger.  Grämiger et al. (2016) shows that the Daubensee rock avalanche overran glacially-abraded bedrock, and has correspondingly high back-analysed strength parameters.  Grämiger et al. (2016) also show that the Rinderhorn rock avalanche initially overran bedrock, however, it encountered valley sediments after descending the source slope.  Back-analysis of the Daubensee rock avalanche indicates that it is best reproduced using a single frictional rheology, with a friction angle of 22°.  Back-analysis of the Rinderhorn rock avalanche indicates that, in order to reproduce the observed long runout and deposit distribution, the flow resistance must be dramatically reduced when the rock avalanche overruns the valley sediments.  This contrast of shear resistance is interpreted to mean that interaction with valley sediments reduced the basal resistance experienced by the Rinderhorn rock avalanche, enhancing its mobility. 5.6.3 The West Salt Creek Rock Avalanche Contrasting the six case histories presented above shows that basal resistance must be selected to correspond to changing path materials, and that the volume of the rock avalanche has little influence on the back-analysed shear resistance once the flow has vacated the rupture surface.  This section will present a case history whose runout characteristics cannot be explained using frictional mechanics.  It will be shown that this case likely overran fine-grained sediments, and that the back-analysed shear resistance is consistent with the shear behaviour of this type of path material. The following is paraphrased from Aaron et al. (2017b). 128  The West Salt Creek rock avalanche occurred on May 25th, 2014, and claimed the lives of three people.  This landslide released from the northern flank of Grand Mesa, in western Colorado.  The event had a complex, two-stage failure mechanism.  The first stage included reactivation of an ancient slump block in a unit consisting of shales and marlstones, with an estimated total volume of rock displaced by the slump of 54 Mm3 (White et al., 2015; Coe et al., 2016).  It is thought that this reactivation was triggered by a rain-on-snow event (White et al., 2015).  The second phase of the failure consisted of rapid evacuation of a rock avalanche from the toe of the displaced slump block (Coe et al., 2016).  The rock avalanche had a source volume of approximately 12 Mm3.  The hypothesized initiation mechanism of the rock avalanche, based on Coe et al. (2016), is shown in Figure 5-11.    White et al. (2015) identified a prehistoric landslide that travelled part of the way down West Salt Creek.  White et al. (2015) also noted that the debris of the 2014 event underwent rapid slaking (in the months following the event), which transformed the shales and marlstones into disaggregated, loose clasts of fine-grained debris.  Additionally, Coe et al. (2016) hypothesized that the surficial sediments present in the West Salt Creek channel likely consisted of a mixture of alluvium and landslide deposits.  Based on these observations and hypotheses, it is likely that, after failure, the rock avalanche overran loose fine-grained sediments that had a high degree of saturation from the high precipitation and snowmelt that preceded the rock avalanche.  As can be seen in Figure 5-12, the rock avalanche overtopped a 40 m high ridge, and superelevated through three bends along the runout path.  Based on these superelevations, White et al. (2015) estimated runout velocities of 37 m/s, 25 m/s and 9 m/s at the three bends (from upstream to downstream, respectively) using the forced vortex equation (e.g. Chow, 1959).  Coe 129  et al. (2016) also provided dynamic constraints on the motion of the rock avalanche through interpretation of radiated seismic signals; they estimated that the slide was travelling at an average velocity of 21 m/s.     Figure 5-11: Failure sequence for the West Salt Creek Rock Avalanche hypothesized by Coe et al. (2016).  The geometry after the slump (green line on panel b and c) was derived from a Dan3D-Flex analysis.  The section line is shown on Figure 5-13. LiDAR data were collected after the event to constrain the post-slide geometry.  Pre-event topographic data are available on a 10-m spaced grid.  Based on these data, an accumulation/depletion map was derived (Figure 5-13).  Immediately down slope of the slump block, there is little change in the topography, indicating that either there was no deposition in this zone, or there was erosion of path materials that were later replaced by deposition of rock avalanche debris.  Significant deposition begins towards the distal end of the channelized portion 130  of the path.  We estimate the volume of material that overtopped the ridge is between 100,000 m3 and 150,000 m3 (Figure 5-13).   Figure 5-12: Overview of the West Salt Creek Rock Avalanche (Photo: J Coe). A back-analysis of the West Salt Creek Rock Avalanche was conducted based on the failure mechanism described by Coe et al. (2016).  The 3D rupture surface of the slump block was input into Dan3D-Flex, and a frictional rheology was used to simulate initial rotational failure. We used 7015 columns to represent the failed mass, and the mass was kept rigid throughout the entire simulation using the flexible block model.  The friction angle was adjusted by trial-and-131  error until the back-tilted portion at the top of the slope best matched the post-slide LiDAR surface.  The best fit bulk friction angle was determined to be 11°.  The results of this back-analysis are shown in Figure 5-11.  The back scarp of the rock avalanche is visible on the post-slide LiDAR surface, and this was combined with the final Dan3D-Flex geometry to create a 3D rupture surface for the second phase of motion (i.e. initiation of the rock avalanche).  This process is shown schematically in Figure 5-11.  Our reconstruction resulted in a modelled rock avalanche source volume of 12 Mm3, very close to that estimated from the accumulation/depletion map.  Figure 5-13: West Salt Creek Rock Avalanche accumulation and depletion map. Coe et al. (2016) noted that the estimated vertical error of the digital elevation data is ± 4.72 m.  The section line refers to Figure 9. Initially, we used the Voellmy rheology to parameterize the basal resistance force.  The best fit results, obtained by testing 400 different parameter combinations, are shown in Figure 5-14.  As can be seen in Figure 5-14, no combination of friction and turbulence coefficients can simultaneously reproduce the overtopping of the ridge and the distal runout extent.  Additionally, when material is predicted to overtop the ridge, it does not deposit in the correct location.   132  To overcome these shortcomings, we next tested the Bingham rheology.  This rheology is appropriate to simulate rapid shearing of fine-grained material, which in the present case represents the saturated, fine-grained material along the valley floor that was overridden by the rock avalanche.  The material within the body of the rock avalanche had high frictional strength (Coe et al., 2016). The results of a sensitivity analysis using the Bingham rheology are shown in Figure 5-15.  As can be seen in Figure 5-15, τyield =32 KPa and μbingham = 7 KPa*s provide the best compromise between simulating both the impact area and deposition on the 40-m high ridge (based on the accumulation depletion map, we expect that the volume deposited on the ridge is between 100,000 m3 and 150,000 m3).  The simulation results for this best-fit combination are shown in Figure 5-16; we obtained good agreement between field observations and model results, both in terms of impact area and deposit thickness distribution.  Runout velocities predicted by the model are approximately 30% higher than the maximum velocities estimated by White et al. (2015); however, they broadly agree with the average velocity estimated by Coe et al. (2016).       Figure 5-14: Final deposit depth and predicted impact area when basal resistance is parameterized with the Voellmy rheology.  The red outline shows the observed impact area.  A minimum deposit depth value of 0.3 m is necessary due to the solution method used by Dan3D. 133    Figure 5-15: Results of the sensitivity analysis used to determine the best-fit Bingham parameters for the West Salt Creek rock avalanche.  Impact area fitness is calculated using a dimensionless number that measures the misfit between a user specified impact area and the simulated impact area (Section 4.3.1).  Lower numbers indicate better fitness (a value of zero indicates perfect agreement between observed and simulated impact area).  A good compromise between simulating the observed impact area and deposit distribution is found for τyield =32 KPa and μbingham = 7 KPa*s.  Volume is in m3.   134   Figure 5-16: Predicted impact area and simulated deposit depths when basal resistance is parameterized with the best fit Bingham rheology.  The red outline shows the observed impact area.  A minimum deposit depth value of 0.3 m is necessary due to the solution method used by Dan3D. There are two reasons for the improved results when basal resistance is parameterized with the Bingham rheology.  The first is that centripetal accelerations do not increase basal resistance (as in the Voellmy rheology); so the flowing mass expends less momentum overtopping the ridge.  The second is that deposition is now controlled by both flow depth and slope angle, as opposed to the frictional and Voellmy rheologies, where deposition is less strongly controlled by flow depth, and is strongly influenced by slope angle.  This allows the mass to deposit both on the steep overtopped ridge, as well as at the distal toe.  These two factors provide strong justification for the use of the Bingham rheology to simulate the West Salt Creek rock avalanche.  5.6.4 Conclusions about Mobility Mechanisms The stark contrast of the mobility of these case histories makes it clear that rock avalanche mobility is not solely governed by a mechanism that is universal and volume-dependent.  The 135  scatter on plots of H/L (e.g. Figure 5-2) indicates this; however, the present study shows that the character of the path material is a possible explanation for this scatter.  The present work cannot definitively rule out the possibility that multiple mechanisms (including those summarized in Section 5.2) are simultaneously acting to reduce basal resistance; however, Section 5.6 shows that no other explanations in addition to the shear characteristics of the path material are needed to explain the bulk characteristics of the analysed rock avalanches.  The dependence of runout behaviour on path materials is repeated throughout the calibrated cases, with the 286 Mm3 Madison Canyon Rock Avalanche exhibiting moderate mobility, while the .9 Mm3 Zymoetz rock avalanche exhibits high mobility.  Additionally, back-analysis of the West Salt Creek rock avalanche (Section 5.6.3) indicates that the character of the path material can lead to rock avalanche runout behaviour that cannot be explained with frictional mechanics.  Taken together, these results show that consideration of path material is crucial in understanding and predicting rock avalanche motion. However, the back-analysis of the Val Pola case history, as well as the scatter on Figure 5-6, indicates that further research is needed to better understand rock avalanche interaction with sediments.   The only universal, volume-dependent mechanism needed to explain the shear strength distribution back-analysed for the twenty-four case histories presented here is extreme polishing along planar features in the source zone.  Hungr & Evans (2004) hypothesized that larger volume rock avalanches cover more spatial area, so are more likely to encounter weak substrate materials.  This mobility mechanism, modified to account for possible volume-dependent resistance in the source zone, appears to be a plausible explanation for rock avalanche mobility. 136  5.7 Disintegration Process The new flexible block model (described in Chapter 3) demonstrates the need to consider failure kinematics when performing runout analysis of rock avalanches.  For case histories where the rock avalanche initiated on a planar feature that did not require internal distortion for movement to be kinematically admissible, the landslide mass can move substantial distances as a flexible block.  This conclusion is based on a number of case histories where the unmodified version of Dan3D predicted excessive lateral spreading in the source area.  This problem is most pronounced in the back-analyses of Mystery Creek, Goldau, Bingham Canyon, Zion Canyon, Rinderhorn, Daubensee, Oberarth and Guinsaugon.  In all cases analyzed in the present work, the user-specified location where the flexible block is fluidized corresponds to changes in slope in the runout path, a process described mechanistically by De Blasio & Crosta (2013). As demonstrated in Sections 3.6.1 and 3.6.2, the predicted impact area is relatively insensitive to the distance travelled as a flexible block, so long as the mass is fluidized within a reasonable distance of the source zone.     5.8 Proposed Probabilistic Runout Analysis Framework Based on the analysis of the controls of rock avalanche motion detailed in Sections 5.5, 5.6 and 5.7, a probabilistic runout analysis framework has been developed.  For a given potential failure volume, the following three factors must be parameterized in order to forecast rock avalanche runout:  The distance travelled as a flexible block.  This parameter depends on the disintegration process and can be estimated based on the pre-failure topography. 137   The strength in the source zone.  This parameter is volume-dependent if shearing is localized to a discrete plane.  The strength outside of the source zone.  The parameters that govern strength outside the source zone depend on path materials. A decision tree summarizing a methodology to parameterize these factors is shown in Figure 5-17.  The first node on the decision tree represents the possibility of catastrophic failure, defined as failure of the entire unstable volume.  Catastrophic failure is contrasted with piecemeal failure, where the unstable volume unravels.  All of the cases analyzed in the database failed catastrophically, although the Crammont case history had a secondary failure that was piecemeal.  For cases that fail in a piecemeal manner, it is expected that runout will be much shorter than if catastrophic failure occurs.  An example of this is the Randa rockfall, where 30 Mm3 of rock unravelled in a series of small volume failures, creating a gigantic talus cone at the base of the slope (Eberhardt et al., 2004).  Had this mass failed catastrophically, the runout would likely have been significantly longer.  In a forward analysis, there will likely be significant uncertainty as to whether a failure mechanism that can lead to catastrophic failure is possible, and probabilities should be assigned based on a susceptibility analysis.  The decision framework presented in Glastonbury & Fell (2008) can be used to guide the assessment of the likelihood of catastrophic failure.  All of the cases analyzed in the present work failed catastrophically, so no guidance can be given about assessing the runout of a piecemeal failure mechanism on the basis of these cases.  The rest of this chapter is devoted to forecasting the runout of rock avalanches that undergo catastrophic failure.   138  The next node in Figure 5-17 should be evaluated based on an analysis of the potential rupture surface.  If failure is expected to occur along a continuous planar feature then the flexible block model should be used.  As summarized in Section 3.7, the rigid motion distance can be selected based on an examination of the pre-failure topography.  All rock avalanche case histories analyzed in the database were back-analysed with this methodology, and it was found unnecessary to extensively calibrate the rigid motion distance parameter. Additionally, an analysis was performed to test the sensitivity of results to the rigid motion distance.  This sensitivity analysis was performed for the Goldau and Mystery Creek case histories (Section 3.6.1 and 3.6.2), and showed that the final predicted impact area is insensitive to the choice of rigid motion distance.  Therefore, the rigid motion distance can be selected a-priori and it is not recommended that this parameter be varied in a probabilistic context.  If failure is along an irregular surface then the flexible block model should not be used, as fragmentation is likely to occur close to the onset of failure.   139   Figure 5-17: Decision tree to guide the selection of parameters for the prediction of rock avalanche runout using Dan3D 140   When failure occurs along a continuous structural feature, the strength in the source zone can be assigned based on the failure volume.  The cases analyzed in the database appear to show that frictional strength in the source zone depends on source volume (Figure 5-4).  For cases with a volume less than 5 Mm3, the back-analysed frictional strength corresponds to a friction angle a few degrees less than the limit equilibrium value.  Of the analyzed cases below this volume threshold, the friction angles ranged from 25° to 30°.  For cases with volumes greater than 5 Mm3, the friction angle in the source zone can be significantly reduced from the limit equilibrium value due to extreme polishing (Cruden & Krahn, 1978).  For these cases, it is recommended that the source zone friction angle be selected based on Figure 5-4.   The next node of the decision tree concerns path materials.  If the path material is bedrock then the frictional rheology should be used to parameterize basal resistance.  Four cases in the database overran bedrock (Thurweiser, Bingham Canyon, Daubensee, and Chehalis).  As shown in Figure 5-7, these cases were all well reproduced with friction angles ranging from 21 to 28°.  For conservative deterministic predictions, a friction angle of 20° can be used, however, for probabilistic predictions, a range of friction angles can be tested and the associated exceedance probabilities assigned to each of them.  Due to the limited number of case histories in the database that overran bedrock, it is difficult to give guidance on the exceedance probabilities to assign to these values of the friction angle.   For cases that overrun sediments, the selection of strength parameters along the path is a highly uncertain part of a forward runout prediction.  As shown in Figure 5-5 and Figure 5-6, these 141  parameters can vary widely, and do not appear to be volume-dependent.  As will be shown in Section 5.8.2.1, using lower bound basal resistance can result in an extreme over-prediction of the impact area.  Forward predictions must account for uncertainty in these parameters, while not making over-conservative forecasts.  The procedure suggested for doing this is to parameterize the path materials in a probabilistic manner, using the Bayesian methodology detailed below.   5.8.1 Bayesian Parameter Estimation Framework With the exception of the Guinsaugon case history (Section A.21), a brute-force sensitivity analysis has been conducted for each of the cases that overran sedimentary substrate material, using the methodology detailed in Section 4.4.  An example back-analysis using this methodology is presented in Section 4.7.  The posterior density functions derived for each of the case histories is presented in Appendix A.  The individual posterior density functions can be combined to produce a probability density function for the path parameters, which can in turn be used in a parametric or Monte Carlo type analysis to derive runout exceedance probabilities.  This is done using the procedure described below.   The conventional approach to deriving a probability density function for the input parameters, used by McKinnon (2010) and Quan Luna (2012), for example, is to derive a single set of best-fit basal resistance parameters for each case history.  This process can be repeated for many case histories, and the best-fit parameter values can then be used to construct a histogram.  This histogram can then be fit with a probability distribution. As shown in Chapter 4, there is no single set of best-fit basal resistance parameters for a given case history.  In fact, even for the well-constrained Zymoetz River Rock Avalanche, Figure 4-11 142  shows that a range of parameter combinations is supported by the field data.  For this reason, the standard approach of constructing a histogram based on a single set of best fit parameters is inappropriate to this problem.  Instead, a kernel density estimator was used to construct a probability density function for the parameters (e.g. Rosenblatt, 1956).   As shown in Figure 5-18, kernel density estimation is similar to constructing a histogram.  In the example shown in Figure 5-18, 6 data points are used to construct a histogram, as well as a kernel density estimator.  In the case of the histogram, each data point increases the height of its corresponding bin.  There is no way to capture variance in the measurement of the value of the data point using this method (i.e. in some problems, such as calibrating Dan3D parameters, there is a non-zero probability that the measured data point could be in two different histogram bins).  The use of the kernel density estimator allows for variance in the measured data to be captured, as each data point has a probability density function associated with it.  The superposition of the probability density functions gives an estimate of the probability density function associated with the random variable of interest.   143   Figure 5-18: Comparison of histogram (left) and kernel density estimator (right) for the same data.  The kernel density estimator allows for variance in the measured values of ‘x’ to be accounted for.  In most Dan3D back-analyses, a zone of best fit parameters is supported by the data.  Selecting a single best fit parameter set, as is required to construct a histogram, cannot capture this source of variance.  A kernel density estimator, through the use of a probability density function associated with each data point, captures variance in the parameter values.  Image: Drleft at English Wikipedia, via Wikimedia Commons. An expression for the kernel density estimator is shown in Equation [5.1].     𝜋𝑓(𝒃𝑓|𝒃1 … 𝒃𝑛) =∑ π𝑝𝑜𝑠𝑡(𝒃|𝒓)𝑛𝑛=𝑛𝑐𝑎𝑠𝑒𝑠𝑛=1𝑛𝑐𝑎𝑠𝑒𝑠 [5.1] where π𝑝𝑜𝑠𝑡(𝒃|𝒓) is defined in Section 4.4.1, and 𝑛𝑐𝑎𝑠𝑒𝑠 is the number of case histories being combined to construct the kernel density estimator.  The subscript f refers to the forward analysis case of interest, and the subscripts 1 to n refer to previously calibrated case histories.        144  The standard kernel density estimator (Equation [5.1]) must be modified in order to appropriately represent a prediction made within the equivalent fluid framework.  An additional term must be added in order to assess the probability that a past case history will be similar to the case of interest. 𝜋𝑓(𝒃𝑓|𝒃1 … 𝒃𝑛) =∑ π𝑝𝑜𝑠𝑡(𝒃|𝒓)𝑛 ∗ 𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 (π𝑝𝑜𝑠𝑡𝑛|π𝑝𝑜𝑠𝑡𝑓)𝑛=𝑛𝑐𝑎𝑠𝑒𝑠𝑛=1𝑛𝑐𝑎𝑠𝑒𝑠 ∗ ∑ 𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 (π𝑝𝑜𝑠𝑡𝑛|π𝑝𝑜𝑠𝑡𝑓)𝑛𝑛=𝑛𝑐𝑎𝑠𝑒𝑠𝑛=1 [5.2] Where the additional term,𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 (π𝑝𝑜𝑠𝑡𝑛|π𝑝𝑜𝑠𝑡𝑓), is the probability that the case of interest is similar to a given back-analyses.  At present, this probability is difficult to accurately assess, however, some guidance can be taken from the information presented in Sections 5.5 and 5.6.  More detail about selecting the probability of a case being similar to the case of interest will be provided in Section 5.10.2. The numerator on the right side of equation [5.2] represents a weighted combination of the parameter values of the back-analysed cases.  The resulting PDF for the case of interest will therefore be the combined back-analysed parameter values, weighted by the similarity between the back-analysed cases and the case of interest.  The denominator on the right side of Equation [5.2] is a normalizing constant that ensures the volume under the PDF is equal to 1.   The result of combining all of the case histories into one probability density function and assuming that all cases are equally likely (assuming 𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 is given by a uniform distribution) is shown in Figure 5-19.  In Figure 5-19, the high probability zones correspond to 145  places where the best fit parameters of many case histories overlap (this can be seen by comparing Figure 5-19 and Figure 5-6).  Figure 5-19: Probability density function derived from combining the best fit parameters from all the case histories.  High probability zones correspond to parameter ranges that fit a large number of case histories (compare to Figure 5-6). As argued in Section 5.6, the mechanism that likely governs rock avalanche motion is the character of the path materials.  If it is expected that the rock avalanche will encounter loose, saturated substrate then a higher probability should be assigned to parameter combinations that correspond to those types of cases in the database.  The cases analyzed in the database tended to be extremely mobile, so Figure 5-19 provides a reasonable posterior distribution for cases that are expected to overrun loose, saturated sediments.  If unsaturated coarse grained materials are 146  expected then the cases that overran such path materials should be weighted higher.  This can be done through the use of a subjective prior distribution (𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 in Equation [5.2]).   5.8.2 Example Runout Forecast Posterior parameter distributions, such as that shown in Figure 5-19, can be used to derive probabilistic estimates of any intensity parameter of interest.  In the present work, the only parameter that will be derived is spatial impact area.  To do this, a set of parameters is chosen that represents the parameter space.  For the case detailed below, values of friction coefficient were selected between f = 0.05 to 0.25 with steps of 0.05 and turbulence coefficients were selected with irregular spacing between 100 m/s2 to 1400 m/s2.  Each of these parameter combinations were then input into Dan3D, resulting in an estimate of impact area.  The impact area estimates for the individual simulations are then combined using a weighted sum, with the weights determined by the posterior distribution (derived in using in Equation [5.2]) and parameter sampling interval.  The methodology detailed above has been applied to the Turnoff Creek rock avalanche, a case history that was not included in the calibration database.    5.8.2.1 Turnoff Creek The Turnoff Creek rock avalanche occurred in 1992, and involved an estimated 4 Mm3.  It failed along a continuous bedding plane (Beguería et al., 2009).  An overview of this rock avalanche is shown in Figure 5-20.  Following the methodology presented in Figure 5-17, a friction angle of 21°, a few degrees below the limit equilibrium angle, was selected in the source zone.  A rigid motion distance was selected to correspond with the landslide fragmenting when most of the failed material had left the source zone.     147   Figure 5-20: Overview of Turnoff Creek.  Image: Google Earth, Digital Globe. A probabilistic analysis was conducted to parameterize the path materials.  To do so, a probability density function was derived using Equation [5.2].  All cases were equality weighted except for Mt. Meager and Nomash River. Both Mt. Meager and Nomash River entered channels and entrained large quantities of loose, saturated sediment, which is thought to have greatly enhanced their mobility (See Sections A.5 and A.7 respectivley).  Beguería et al. (2009) did not observe evidence of entrainment at Turnoff Creek, and reported that ‘little soil occurs in the displaced material’.  If the shear characteristics of the path materials are the mechanism governing the mobility of rock avalanches, as argued in Section 5.6, than assigning these case histories a low similarity probability appears justified.  However, the key question is whether these case histories could be justifiably excluded based on 148  a field investigation prior to the event.  Such an investigation would have to focus on the spatial distribution and thickness of entrainable substrate.  Based on Beguería et al. (2009) , such an investigation would likely have shown limited availability of entrainable substrate.  Based on this, the Mt. Meager case was assigned a 𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 of 0, and Nomash River was assigned a 𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 of 1/10 that of the rest of the cases.  Although subjective, these 𝜋𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 values reflect the fact that Mt. Meager is not a good precedent to use when predicting the motion of the Turnoff Creek rock avalanche, however due to similar scale and morphology it is difficult to exclude Nomash River entirely.  The resulting 𝜋𝑓 is shown in Figure 5-21 below, and the resulting runout exceedance probabilities are shown in Figure 5-22.   149   Figure 5-21: Probability density function for the Turnoff Creek path material parameters derived based on evaluating Equation [5.2].  Since Mt. Meager was excluded, and Nomash River assigned a low similarity probability, the probability of the parameters having a friction coefficient less than 0.08 are lower than in Figure 5-19. As can be seen in Figure 5-22, using a range of parameters corresponding to friction coefficients of 0.1 to 0.25 provides a reasonable forecast of the runout extent (the 0.05 runout exceedance probability corresponds to simulations using friction coefficient = 0.1, turbulence coefficient = 1400 m/s2).  When a friction coefficient of 0.05 is used the resulting simulation predicts an impact area far in excess of that observed.  This is due to the inclusion of Nomash River in the combined PDF that is used to derive the exceedance probabilities.  Since Nomash River was subjectively assigned a low similarity probability, the predicted exceedance probability is also low.   150   Figure 5-22: Results of a parametric analysis using 25 different parameter combinations drawn from the low strength PDF.  The contours on the image are exceedance probabilities.  The extremely long runout represented by the 0.05 exceedance probability is due to the inclusion of Nomash River when evaluating Equation [5.2]. 5.9 Deficiencies in the Database A number of deficiencies in the database of calibrated case histories have been identified that should be noted when making predictions based on the methodology described in Sections 5.8.  These deficiencies include: 1) a lack of cases with glacial ice as the dominant path material, 2) the neglect of the ‘splash zone’ that surrounds many rock avalanches (defined and discussed below) and 3) a lack of detailed information about the character of the path material.   151  The Mt. Steele case history is the only case in the database where glacier ice is the dominant path material.  Therefore, no strong guidance can be given as to the selection of basal resistance parameters for cases that overrun glaciers.  Sosio et al. (2012) provide the results of the back-analysis of 18 case histories that overran glaciers, and this work can be used to guide parameterization of these case histories. When back-analyzing the case histories in the database, no attempts were made to reproduce the splash zone observed around many of the rock avalanches.  However, consideration of the splash zone is crucial when performing risk analysis of rock avalanches.  This feature, termed ‘Spritzone’ by Heim (1932) is formed by the projection of liquefied path materials in front of the flowing rock avalanche.  Splash zones have been observed surrounding the main deposit of many rock avalanches (e.g. Heim, 1932; Mathews & McTaggart, 1978; Hungr & Evans, 2004; McDougall & Hungr, 2005).  At Goldau (Sections 3.6.1 and A.14), a mud wave radiated out around the main deposit and triggered a tsunami in Lake Lauerz (Bussmann & Anselmetti, 2010).  A large splash zone was observed around the deposits of the Frank Slide (Section A.18), and much of the destruction associated with this event was caused by this splash zone (Cruden & Hungr, 1986).  A splash zone around the Mt. Meager event (Section 4.1.1 and A.5) dramatically increased its impact area.   At present, the mechanics of splash zones are poorly understood and any attempt to reproduce this phenomenon with equivalent fluid models is difficult to justify.  It is likely that the highly fluid material that creates the splash zone is pushed ahead of the main rock avalanche mass, and continues to spread after the fragmented rock phase has come to rest.  Therefore, this is a two phase process that will be difficult to capture with a one phase model.  It is possible that a 152  rheology change at the distal end of the runout path could approximate this process, however, at present it is difficult to anticipate where such a change would occur. The final deficiency in the database concerns the path materials that the rock avalanches overran.  There is still significant variance in the back-analysed strengths for cases that overran sediments.  Figure 5-22 shows that this variance leads to a range of predicted runout extents.  This variance can be reduced by further understanding the role of path material in enhancing runout.  Future work should focus on better characterizing path materials that past rock avalanches have overran, and using this information to classify runout behaviour. 5.10 Complications with Applying Proposed Prediction Framework 5.10.1 Location of Rheology Change The forward analysis procedure summarized in Figure 5-17 recommends using different shear strength parameters in the source zone and along the path.  This is because the mechanism controlling movement is likely different in the source zone, where shearing is controlled by the strength on the rupture surface, and along the path, where path materials likely alter the dynamics of the flow.  The disadvantage of using differing shear strengths is that the user must select the location of the change in rheology.       For many cases, such as Huascaran, Thurweiser, Bingham Canyon, Nomash River, Madison Canyon, Mt Meager, Avalanche Lake, Guinsaugon, Rinderhorn, Rautispitz and Platten, the location of the rheology change is easy to predict as there is a clear distinction between the source and path materials.  Eaux Six, Val Pola, Mystery Creek, Goldau and Frank all descended steep source slopes before spreading out on a valley floor.  For these cases, good results were 153  found by using the source zone strength parameters until the rock avalanches encounter valley floor sediments.  For Daubensee and Chehalis, the path material was bedrock, so no change in rheology was implemented between the source zone and path.   The only case that did not fit well into this framework was McAuley Creek. The McAuley Creek rock avalanche failed into a narrow valley, and deposited a significant quantity of debris at the toe of the source slope (Figure 5-3).  A small tongue of highly mobile debris projects from this low mobility proximal deposit.  As shown in Section A.4, McDougall (2006) and Brideau et al. (2012a), in order to reproduce these two observations with Dan3D, a two rheology simulation was necessary.  The rheology change was implemented at the interface between the low and high mobility deposits, however, it likely does not correspond to any feature that could have been mapped prior to the failure (McDougall, 2006). There are two possibilities that can explain this deposit morphology.  The first is that the mass failed all at once, and the majority of the failed volume plowed the loose valley sediments, as opposed to overriding them.  In this scenario, a portion of the failure, located on the downstream margin, would have overrode loose saturated sediments and created the high-mobility distal tongue.  The second possibility is that the failure occurred in multiple phases, with the first failure a moderate volume failure that spread down the valley, and then a less mobile, secondary failure resulted in the thick deposits at the toe of the slope.  Both scenarios appear to be plausible, and with present information it is difficult to determine which is more likely.  The proposed risk analysis methodology, summarized in Figure 5-17, would likely provide conservative runout estimates for this case, as the entire valley floor would be parameterized with low strength.  154  5.10.2 Selection of Case Weighting In Section 5.8.1 it was suggested that, when deriving a posterior probability density function based on successful back-analyses, the cases be weighted based on a subjective assessment of how similar they are to the future case of interest.  In practice, this will be difficult to do, especially when making a decision to exclude the more mobile cases in the database (due to the fact that the resulting predictions will be less conservative than if these cases are included).  Further research is required to better understand how to select case weightings; however, the following guidance can be taken from the database of calibrated case histories. The cases with the lowest best-fit friction coefficients for the path material are Mt. Meager, Nomash River and Huascaran.  The debris involved in the Mt. Meager and Nomash River cases became channelized, and in both cases the moving mass entrained significant quantities of surface water and loose saturated sediments.  For forward analysis where the rock avalanche will not become channelized, these case histories can be assigned a low weight when deriving a PDF for the parameters with Equation [5.2].  Similarly, the Huascaran rock avalanche became channelized, but also had its mobility enhanced by glacial ice in the source zone.  It can be assigned a low weight when evaluating Equation [5.2] for forward analyses of cases that do not have these characteristics.   Besides these cases, at present it is difficult to provide guidance as to how to weight the other cases.  Until more research is done to link specific substrate conditions to mobility, it is recommended that all cases be assigned equal weight when using Equation [5.2] to derive a PDF for the parameters.  This will likely provide relatively conservative predictions, as the cases in the database tend to be very mobile (Figure 5-2). 155  5.11 Conclusions A database of thirty rock avalanche case histories, all of which have been analysed with Dan3D, has been compiled.  Twenty-four of these case histories have been back-analysed using the methodology detailed in Chapter 4.  The back-analysis results show that a site-specific mechanism likely governs the mobility of these case histories.  For most cases, the strength in the source zone was best simulated with the frictional rheology using a volume-dependent friction angle.  This is consistent with the mechanism of polishing of the rupture surface due to shearing under high normal stresses.  The shear strength distributions back-analysed along the path are consistent with the hypothesis that rock avalanche mobility is controlled by path materials. A probabilistic rock avalanche runout methodology has been developed based on the database of calibrated case histories.  To forecast rock avalanche runout, a modeller must select the distance the mass will travel as a flexible block, the strength in the source zone and the strength along the path.  The most uncertain part of this analysis is selecting the shear strength along the path when the path material is sediment.  For these cases, it is recommended that a parametric or Monte-Carlo analysis, based on a probability density function derived by combining the results of multiple back-analyses, is used to forecast runout.  156  Chapter 6: Analysis of an Unusual Debris Avalanche 6.1 Introduction Debris avalanches (see Table 1-2 for the definition from Hungr et al., 2014) can be devastating events.  The 2011 Umyeonsan debris avalanches in Seoul, South Korea reached velocities of 25 m/s (based on video evidence) and devastated residential areas located at the toe of the slope (Yune et al., 2013).  These events often occur as swarms, an example of which is the Vargas State disaster that occurred in northern Venezuela in 1999.  In this event, thousands of debris avalanches moved down steep slopes into heavily inhabited areas, causing an estimated 15,000 deaths (Larsen & Wieczorek, 2006). Quantifying the risk posed by debris avalanches requires consideration of their runout characteristics.  As will be summarized in Section 6.2, previous work has found good results by modelling debris avalanches with the frictional or Voellmy rheology (described in Section 2.5.2).  However, the assumption that debris avalanches experience frictional resistance may not be suitable in all cases.  The purpose of this chapter is to present a back-analysis of the Johnsons Landing debris avalanche, and discuss its implications to the dynamic analysis of debris avalanches in general.  As will be shown, this case history is unique from most previously-analysed debris avalanches, as it appears to have travelled in an undrained condition. 6.2 Background Debris avalanches often initiate as shallow debris slides on steep slopes, defined by Hungr et al. (2014) as “Sliding of a mass of granular material on a shallow, planar surface parallel to the 157  ground.”  The substrate of these steep slopes is often formed by a strong material, such as bedrock, and the granular soil veneer is only stable due to ephemeral effects such as root cohesion and/or negative pore pressures (Hungr et al., 2014).  Initiation of these events is often due to loss of these transient sources of cohesion.  It is important to note that this mechanism is different from that of flowslides.  As summarized in Chapter 7, in flowslides, strength loss and failure are due to liquefaction of a portion of the source volume in the source zone.  Although rare, liquefaction-prone materials can sometimes be found on steep slopes (one likely example of this will be presented in this chapter).  Such events are transitional between flowslides and debris avalanches, and this chapter will show that this has implications for the dynamic analysis of their motion.             Once initiated by a shallow debris slide, the mechanism of strength loss in debris avalanches is complex, as it involves dynamic pore-pressure effects (Iverson, 1997).  These events often entrain significant quantities of debris from the path, a process that can increase the volume by orders of magnitude.  As shown in Figure 6-1, the pore-fluid effects during this entrainment process are one process that can lead to low basal strength, high velocities and long runout.  Using results from an undrained ring shear device, Sassa & Wang (2005) demonstrated the process of rapid undrained loading leading to excess porewater pressure.  When a debris avalanche overrides saturated path material, it increases both the total shear and normal stresses acting on the substrate.  If drainage is restricted, the effective normal stress on the path material will not increase.  Thus, if the soil is not contractive, the shear resistance will be governed by the effective stress acting on the path material prior to being overridden.  If the path material is contractive, then it will liquefy, leading to very low basal resistance (see Section 7.2).  A key 158  feature to note about this process is that the basal resistance will be governed by a constant yield stress, proportional to the internal friction angle, density and thickness of the entrained material.  This assumes that the timescale for drainage is much greater than the emplacement time of the debris.  This phenomenon has been shown experimentally by Iverson et al. (2010).  Figure 6-1: Schematic of the undrained loading process described by Sassa & Wang (2005).  Panel C and D show the shear and normal stress acting on the soil element highlighted in panels A and B.  Before being overridden, panel C shows that the soil element (red dot in C and D) is stable.  When the soil element is overridden (B and D), both the shear and normal stress acting on the element increase.  If drainage is restricted but no contraction occurs, shear stress increases until the column fails, while effective normal stress remains constant (panel (B) point A).  If the overridden mass is loose and liquefiable, then shear stress will decrease to a very low value.  ϕi is the internal friction angle of the path material.   159  In debris avalanches, frictional resistance can develop if the flow develops a frictional front, or if the source/entrained material has a significant coarse fraction.  A debris avalanche can develop a frictional front through entrainment of coarse grained material or timber.  Another mechanism that could lead to development of a frictional front is longitudinal sorting, a process that is fundamental to debris flows (e.g. Iverson, 1997; Hungr, 2000; Iverson & George, 2014).  This mechanism may occur in some debris avalanches that become channelized.     As described below, runout modelling of debris avalanches using equivalent fluid models has primarily used either the frictional or Voellmy rheology.  The use of these two rheologies assumes at least some frictional resistance to motion, which is in apparent contradiction to the rapid undrained loading mechanism that governs basal resistance of some debris avalanches.  However, as described above, frictional resistance may develop in debris avalanches, which would justify the use of the frictional and Voellmy rheologies (described in Section 2.5.2).      Revellino et al. (2004) back-analysed 17 debris flows/avalanches that occurred between 1997 and 2001 in the Campania region of southern Italy.  Revellino et al. (2004) found that a single set of Voellmy parameters (friction coefficient = 0.07, turbulence coefficient = 200 m/s2) could adequately reproduce field estimates of impact area, velocity and deposit distribution for the 17 cases.  However, when these parameters were applied to the Nocera Inferiore debris avalanche, which occurred in the Campania region in 2005, they provided overly-conservative predictions of runout extent (Revellino et al. 2013).  The Geotechnical Engineering Office (GEO) of Hong Kong has published guidelines suggesting rheologies and parameters to use when assessing debris avalanches in Hong Kong (The GEO uses the terminology “open hillslope failures” and “failures within topographic depressions”).  160  For “open hillslope failures”, they recommend the use of the frictional rheology, and a volume-dependent friction angle.  For failure volumes less than 400 m3, a 25° bulk friction angle is recommended, and for failure volumes greater than 400 m3, a bulk friction angle of 20° is recommended (GEO, 2012).  The database used to constrain these values has a maximum event volume of 4000 m3.  For “failures within topographic depressions”, defined as debris avalanches that become channelized, the Voellmy rheology is recommended with a friction coefficient of 0.32 and turbulence coefficient of 1000 m/s2.  This recommendation is based on a database of 46 debris avalanches, with a maximum volume of approximately 3000 m3 (GEO, 2013).  Lam (2013) back-analysed a database of 158 debris avalanche case histories using both the frictional and Voellmy rheologies.  The volumes of these case histories ranged from 50 m3 to 40,000 m3.  It was found that the Voellmy rheology, using a volume-dependent friction coefficient and a turbulence coefficient of 500 m /s2, provided a reasonable fit to the analysed cases.  The volume-dependent friction coefficients ranged from 0.28 to 0.53.  Based on the above, it appears as though the Voellmy and frictional rheologies can provide good results when simulating debris avalanche motion.  The likely reason for this is that, due to the mechanisms detailed above, debris avalanches will experience some frictional resistance. However, as shown in the following back-analysis of the Johnsons Landing debris avalanche, this modelling approach does not work for all debris avalanches.    6.3 Johnsons Landing Debris Avalanche The Johnsons Landing debris avalanche occurred on July 12th 2012 about 2 km northeast of the small community of Johnsons Landing, located on Kootenay Lake.  An image taken soon after 161  the debris avalanche is shown in Figure 6-2.  This tragic event claimed four lives (Nicol et al., 2013).  Figure 6-2:  Overview of the Johnsons Landing debris avalanche.  Image: Province of British Columbia, Copyright © Province of British Columbia. As shown in Figure 6-2, the debris avalanche was initially confined in a channel (Gar Creek).  Just above the community of Johnsons Landing, this channel has a 70° bend (the bend is labelled ‘avulsion point’ on Figure 6-2).  Debris flows have been occurring repeatedly in the Gar Creek drainage, but they have always followed the curving path of the gully.  A debris fan built up at the mouth of the gully, labelled ‘Gar Creek Fan’ on Figure 6-2.  The 2012 debris avalanche avulsed from the channel at this bend, spread out over the terrace surface and destroyed three homes located on the Johnsons Landing bench (the location of the bench is shown on Figure 6-2.)     During the investigation following this tragic event, test pits were excavated through the debris on the bench.  These test pits did not show any evidence of other landslide deposits on the bench 162  since deglaciation (Nicol et al., 2013).  The 2012 event was therefore the first event to over-run the sharp bend and impact this bench in at least 7,700 years (Nicol et al., 2013).    6.3.1 Event Description An in-depth investigation was conducted by Nicol et al. (2013) immediately following the Johnsons Landing debris avalanche.  Previous terrain maps had identified the source zone, which is composed of glacial sand and silt, as a zone of instability, subject to multiple rotational sliding (“Failing” designation of the terrain polygon).  This previously disturbed material failed following a rain-on-snow event (Nicol et al., 2013). Nicol et al. (2013) provides estimates of source and deposit volumes.  These estimates are based on a reconstruction of the pre-event topography made using photogrammetry, and adjusted to be consistent with test pits excavated in the debris field. It is estimated that the source volume was 320,000 m3 and the deposit volume was 364,000 m3.  This discrepancy in source and deposit volume can be attributed to inaccuracies in the volume estimates, as well as bulking of the debris after failure.  Nicol et al. (2013) divides the deposit into three zones: The bench, the mid channel and the upper channel.  These zones are shown on Figure 6-2.  It was estimated that 169,000 m3 of material deposited on the bench, 55,000 m3 deposited in the mid channel and 140,000 m3 deposited in the upper channel.  Rough velocity estimates based on superelevation data, as well as an eyewitness account, indicate flow velocities of between 25-35 m/s as the landslide travelled down the channel (Nicol et al., 2013).  One key constraint on the dynamic analysis is that only about 5% of the material from the initial failure travelled through the sharp bend and all the way down the channel to the fan during the 163  initial catastrophic debris avalanche.  This observation is based on an investigation performed on the day of the debris avalanche (Nicol et al., 2013).  A subsequent failure on July 13th resulted in a significant amount of debris and timber moving down the established channel and depositing in Kootenay Lake.  A video of this secondary landslide is available online (https://www.youtube.com/watch?v=n1cCs-S5EKc). 6.3.2 Previous Dynamic Analysis There have been two previous attempts to back-analyze the dynamics of this event.  Marinelli et al. (2015) conducted an analysis using DanW, and Nicol et al. (2013) presented an analysis using Dan3D.  It was found that, in order to reproduce the large volume that deposited on the bench, a low basal resistance to flow had to be used in the channel, and a high flow resistance had to be used on the bench.  Both Nicol et al. (2013) and Marinelli et al. (2015) hypothesized that interaction with trees located downstream of the 70° bend was responsible for the increased flow resistance.  In all Dan3D simulations, Nicol et al. (2013) found that too much volume was predicted to deposit in the channel downstream of the bend.  In other words, the analysis underestimated the tendency of the avalanche front to overtop the curving bank of the channel at the bend.  Additionally, a channel obstruction was assumed to exist downstream of the 70° bend to limit the volume carried into the established channel.  Nicol et al. (2013) hypothesized that this obstruction was due to an accumulation of timber, stripped from the upper slopes of the gully and entrained at the flow front, getting jammed due to a narrowing of the channel width.  An overview of this simulation setup is shown in Figure 6-3 and the simulation results are shown in Figure 6-4.  Overall, the results appear to reproduce the observed deposit distribution; however, the assumption of two rheologies and a channel obstruction is necessary to achieve 164  these simulation results.  The appropriateness of these assumptions will be investigated in Sections 6.4 to 6.10.  Figure 6-3: Model basal resistance parameterisation used for simulations that hypothesize that trees dramatically increase basal resistance.  The black area shows areas of low resistance, and the grey area shows areas of high resistance.  Figure 6-4: Johnsons Landing simulation results when a two rheology simulation is used.  The deposit on the debris field matches the observed deposit, however, too much material is simulated to deposit in the mid-channel, and not enough in the upper channel.  A channel obstruction was assumed to limit the amount of debris that deposited in the section labelled “lower channel”. 165  6.4 Methodology This analysis will test two hypotheses regarding the behaviour of the Johnsons Landing debris avalanche:  It is incorrect to assume frictional resistance to movement  Interaction with trees can lead to a substantial increase in flow resistance These hypotheses are tested using DanW and Dan3D (described in Chapter 2).  Both models were modified to include a rheology that is appropriate for liquefied materials.  The liquefied rheology is described in Section 2.5.3.  This new rheology was then used to back-analyse the Johnsons Landing debris avalanche to determine if better simulation results can be achieved when compared to the analyses detailed in Section 6.3.2.   As described in Section 6.6, DanW has been modified to explicitly include the dynamic interaction of the debris avalanche with forest.  This modified model was then parameterised based on estimates of tree density and geometry provided in forestry reports of the Johnsons Landing area.  The simulation results provided by the modified model give order of magnitude estimates of the increase in flow resistance that can be expected due to interaction of the flowing debris with the forest.   6.5 Model Rheology As mentioned in Section 6.4, to test the hypothesis that the source volume involved in the Johnsons Landing debris avalanche liquefied, flow resistance will be calculated with the liquefied Voellmy rheology (Section 2.5.3).  As discussed in Section 2.5.3, the use of a constant 166  liquefied strength instead of a frictional strength component implies that increased centripetal accelerations do not increase flow resistance.   This has important implications when analyzing the dynamics of the Johnsons Landing debris avalanche.  When the mass of failed debris avulsed from the channel, it overtopped a 10 m high ridge.  The failed mass would have experienced significant centripetal acceleration in passing through the toe of the adverse slope, which would locally lead to a strong increase in the total bed-normal stress in a drained, frictional material and would substantially increase flow resistance of the part of the flow, passing through the vertical curve.  However, if the mass were in an undrained condition, the basal resistance stress would not increase due to the added bed-normal stress, and the debris avalanche would lose less momentum as it avulsed from the channel.   The added flow resistance due to centripetal acceleration under assumed drained conditions could explain why previous back-analyses of this debris avalanche using the Voellmy rheology had to rely on ad-hoc assumptions regarding the flow resistance.  It is conceivable that, in order to overcome the added flow resistance due to centripetal accelerations, the assumed basal resistance had to be low; however, such a low resistance does not lead to any deposition on the bench (due to the moderate slope of the bench).  Thus, the resistance had to be increased on the bench. 6.6 Quantifying the Role of Trees on the Bench In addition to the previous dynamic analyses performed with DanW and Dan3D, other studies have concluded that interaction with timber can have an important role in dissipating energy 167  during debris avalanche runout (e.g. Mizuyama & Narita, 1988; Ishikawa & Kawakami, 2003).  However, none of the previous studies have considered large volume (>100,000 m3) debris avalanches, with most studies focused on debris avalanches with volumes around 1,000 m3 – 10,000 m3.  Snow avalanche researchers have also looked at the effect of forests on snow avalanche dynamics (e.g. Bartelt & Stöckli, 2001; Feistl et al. 2015).  These researchers have found that the dynamics of small volume snow avalanches can be significantly altered by interaction with trees; however, this effect is minimal for large volume snow avalanches.     In order to test the assumption of whether interaction with forest can significantly increase flow resistance, a new module was implemented into DanW.  Bartelt & Stöckli (2001) identified two mechanisms through which trees can dissipate momentum: stem breakage and overturning.  Additionally, a momentum transfer occurs when accelerating stationary trees to the velocity of the moving Lagrangian reference frame.  Post-event photos of the Johnsons Landing debris avalanche show that the majority of trees incorporated into the flow failed through stem breakage, so overturning is not considered in this analysis.   Based on the assumption that trees can be idealized as circular beams, Equation [7] can be used to estimate the critical moment required to break a tree trunk (e.g Bartelt & Stöckli, 2001; Dorren & Berger, 2006).   𝑚𝑐𝑟𝑖𝑡 =  𝜎𝑚𝑜𝑟∗ 𝜋 ∗𝑑332                                                      [7] where 𝑚𝑐𝑟𝑖𝑡 is the critical moment required to induce trunk breakage,  𝜎𝑚𝑜𝑟 is the modulus of rigidity of the tree (a material property) and 𝑑 is the trunk diameter.  As shown in Figure 6-5, 168  assuming that the flow front has a rectangular cross section, the critical force required to break a tree trunk is given in Equation [8]. 𝑓𝑐𝑟𝑖𝑡 =  2 𝑚𝑐𝑟𝑖𝑡ℎ                                                             [8] where 𝑓𝑐𝑟𝑖𝑡 is the critical force and ℎ is the flow depth.  A force equal to 𝑓𝑐𝑟𝑖𝑡 will be exerted on the debris avalanche flow front when it breaks a tree.        Figure 6-5: Derivation of resultant force applied to the debris flow front due to tree breakage. In order to incorporate this force into the equations of motion, the distance over which this force acts must be known.  Based on full-scale rockfall experiments, Dorren & Berger (2006) estimate that the breaking of tree stems of various tree species with similar trunk diameters as those found on the Johnsons Landing Bench dissipates about 400 kJ of kinetic energy.  Based on the work-kinetic energy theorem, the distance over which 𝑓𝑐𝑟𝑖𝑡 acts can be calculated as: 𝑓𝑐𝑟𝑖𝑡 ∗ 𝑠 = 𝑊 = ∆𝐾𝐸                                                    [9]   where 𝑊 is work, ∆𝐾𝐸 is the change in kinetic energy and 𝑠 is the distance over which the force acts.   169  In order to implement Equations 8 and 9 in DanW, estimates of the modulus of rigidity, tree spacing and tree diameter are required.  Based on pre-event aerial photographs and forestry reports in the Johnsons Landing area, the average tree spacing is about 5 metres and the primary species are Douglas Fir and Western Hemlock (British Columbia Ministry of Forests, 2011).  The modulus of rigidity of these tree species ranges from 45 MPa to 85 MPa (Green et al. 1999).  In the analyses that follow, a representative average value of 65 MPa was used.  Bartelt & Stöckli (2001) derived an expression for the number of trees entrained per unit time, and found that tree entrainment results in a velocity-dependent resistance due to momentum transfer effects.  Based on this result, they suggested that the turbulence coefficient in the Voellmy rheology could be reduced to account for the effect of tree entrainment.  In the present work, a more direct approach is used where the momentum expended to accelerate broken trees to the local flow velocity is explicitly accounted for. To account for entrainment of trees, the DanW entrainment algorithm was modified.  Based on the tree spacing and flow width, the number of trees entrained after stem breakage occurs can be estimated.  Then, based on an assumed weight per tree (the present analysis uses 500 kg (British Columbia Ministry of Forests, 2011)), the momentum equation was modified to include the momentum transfer that occurs when accelerating the tree mass from rest to the velocity of the flow.  In Equation [1], this means changing the final term (𝜌𝐸𝑣𝑥) to 𝑚𝑡𝑣𝑥, where 𝑚𝑡 is the mass of trees entrained per time step.   This procedure to account for the interaction of the flowing debris with forest makes a number of key assumptions.  The force applied to the front of the landslide due to stem breakage is 170  controlled by the energy dissipated when the debris avalanche breaks a tree trunk, a parameter well-constrained by the experiments of Dorren & Berger (2006).  The other assumptions regarding the trees behaving as circular beams and regarding the modulus of elasticity are less important, as these serve to determine the distance over which the energy is dissipated.  Similarly, the momentum expended entraining trees is dependent on the mass of the trees, a parameter well-constrained by site-specific forestry reports.  Therefore, this procedure is expected to provide reasonable estimates of the influence of forests on debris avalanche dynamics.  It is possible that the accumulation of logs at the flow front could act similarly to the boulder front of a debris flow, and add a component of frictional resistance to the flow front.  This phenomenon is not accounted for explicitly in the analysis presented above.  By neglecting to explicitly account for this effect, the added frictional resistance due to trees will be implicitly accounted for in the back-analysed basal strength parameters.  The potential for this timber front to jam and create a flow obstruction will be explored in Section 6.9.        6.7 Model Input For the analyses that follow, it is assumed that the observed upper channel deposit is the result of secondary failures after the main failure.  For this reason, the dynamic analyses used a reduced source mass, which corresponds to the volume deposited in the mid channel and on the bench.   DanW analyzes landslide motion along a representative cross section.  A close up of the avulsion location is shown in Figure 6-6.  The vertical radius of curvature at this location is approximately 160 m.  Nicol et al. (2013) estimated velocities of approximately 33 m/s in the channel, and modelled velocities (presented below) are approximately 40 m/s at this location.  This indicates 171  centripetal accelerations of 7 m/s2 to 10 m/s2.  Thus, the influence of centripetal accelerations would approximately double the normal stress experienced by the flowing material.  Figure 6-6: Close up of the location where the material avulsed from the channel.  For this section the topography has not been smoothed, however in the dynamic analyses that follow smoothed topography was used.  The radius of curvature at this location is approximately 160 m, indicating centripetal accelerations of 7 m/s2 (based on velocity estimates from Nicol et al. (2013) to 10 m/s2 (based on modelled velocities presented below).  This indicates that centripetal accelerations approximately double the normal stress. The section line that was used for the DanW analysis is shown in Figure 6-2.  The cross section corresponding to this section line is shown in Figure 6-7.  The width in the source zone was systematically adjusted until the appropriate volume was obtained (210,000 m3).  For the Dan3D simulations, a 3D rupture surface and 3D source geometry must be input.  The rupture surface was created by Nicol et al. (2013) as part of the risk analysis immediately following the debris avalanche.  The source volume that was used in the analysis was 206,000 m3.  Nicol et al. (2013), based on an extensive field investigation, concluded that entrainment of path material (aside from trees) was not important for this event, so entrainment was neglected in the dynamic analysis below.  The volume estimates used as input into the model are based on the deposit 172  volume, which is bulked compared to the source volume.  By using the bulked deposit volume, it is assumed that bulking occurs instantly following failure.   6.8   Dynamic Analysis Results 6.8.1 DanW The DanW results that best reproduce the observed runout distance and overtopping volume are shown in Figure 6-7.  These results used a liquefied strength ratio of 0.08 and a turbulence coefficient of 1,400 m/s2.  The runout distance is slightly overestimated and the velocities in the channel are higher than those estimated by Nicol et al. (2013).  Figure 6-7 also shows the effects of explicitly accounting for debris avalanche interaction with forest.  The simulated runout is approximately 45 m shorter when the interaction with trees is accounted for.  While not negligible, this difference cannot justify the dramatic increase in basal resistance described in Section 6.3.2.   173   Figure 6-7: Final deposit depths and velocities predicted by DanW using the liquefied Voellmy rheology.  When interaction with trees is explicitly accounted for, the simulated runout distance is reduced by 45 m.  6.8.2 Dan3D A sensitivity analysis, using the methodology described in Section 4.4, was conducted to determine the best-fit parameters for both the frictional Voellmy and undrained Voellmy rheologies (Figure 6-8).  The Dan3D results that best reproduce the observed impact area and deposit distribution for the liquefied Voellmy and frictional Voellmy models are shown in Figure 6-9.  In the liquefied Voellmy simulations, the best-fit parameters were determined to be a liquefied strength ratio of 0.08 and turbulence coefficient of 1,400 m/s2, comparable to the 2D 174  analysis.  In the frictional Voellmy simulations, the best simulation results were attained with a friction coefficient of 0.15 and turbulence coefficient of 1,800 m/s2.  The volume of material simulated to deposit on the bench is shown in Table 6-1.  The Dan3D results under-predict the volume deposited on the bench by about 50%, and over-predict the volume that went down the channel after the 70° bend.  The simulated velocities (30-45 m/s) are somewhat higher than those determined from field evidence (Nicol et al., 2013) provided velocity estimates of 25 to 35 m/s).    Figure 6-8: Sensitivity analysis to compare the undrained Voellmy rheology vs. the frictional Voellmy rheology.  Trimline fit is assessed with the dimensionless fitness number described in Section 4.3.1.  Bench volume is in cubic metres.  The white dot on each image shows the best-fit parameter combination.  175     Figure 6-9: Comparison of simulated deposit depths (top two panels) and velocities (bottom two panels) when the liquefied Voellmy rheology is used (left) and when the frictional Voellmy rheology is used (right).  The inset into the top left panel shows the deposit derived from LiDAR. Table 6-1: Landslide debris volume simulated in the deposit zones  Bench Deposit Volume (m3) Dan3D- Liquefied Voellmy 62,000 Dan3D- Frictional Voellmy 45,000 Dan3D- Channel Obstruction 72,000  176  6.9 Dynamic Analysis Discussion Figure 6-9 and Table 6-1 show that the use of the liquefied Voellmy rheology improved simulation results when compared to the frictional Voellmy rheology.  There are two reasons for this.  Firstly, as mentioned previously, centripetal accelerations experienced by the sliding mass do not result in increased resistance in the liquefied model, so the mass experiences less resistance to overtopping the bend.  Secondly, as the mass spreads out on the Johnsons Landing bench, the flow depth gets thinner.  When a frictional rheology is used, both the driving and resisting force are reduced when the flow depth reduces, and it is the slope angle that controls where deposition occurs.  When a constant yield stress is used, a reduction in flow depth only reduces the driving force, not the resisting force.  This results in deposition being controlled by flow depth as well as the slope angle, as is common with rheologies containing a plastic term.  The slope angles in the lower part of the channel and on the bench are nearly identical; however, the moving mass attained high velocities in the channel and deposited on the bench.  This implies that deposition is controlled by factors other than the slope angle, and may be indirect evidence that the liquefied Voellmy rheology is the most appropriate to model this type of event. The use of the liquefied Voellmy rheology obviates the need to use multiple rheologies in order to achieve satisfactory simulation results.  Thus, the ad hoc assumptions regarding increased flow resistance due to interaction with forests are unnecessary.  However, the simulations still predict too much volume deposited in the channel downstream of the 70° bend.  Therefore, it appears that the assumption regarding a channel obstruction is still needed.  Figure 6-11 shows simulation results that assume that a channel obstruction was present where the channel narrows.  177  This channel obstruction is thought to have been created during the main event and was composed of entrained trees (Nicol et al., 2013).  There is some field evidence that such an obstruction developed during the movement of the debris avalanche.  Downstream of the 70 degree bend the channel narrows, which would constrict flow and promote deposition (Figure 6-2).  Additionally, on the day following the main event, a flow surge, composed mainly of timber, was filmed.  Figure 6-10 shows a still from this video, which shows the composition of the flow.  Figure 6-10: Still from video of surge of debris on the day following the main debris avalanche at Johnsons Landing.  The high proportion of timber within this surge provides indirect evidence of a channel obstruction composed of timber that may have failed, leading to this surge.  Video: Global News: https://www.youtube.com/watch?v=n1cCs-S5EKc. This simulation used the same parameters as the liquefied Voellmy simulation described in Section 6.8.2.  The volume deposited on the bench when a channel obstruction is present is shown in Table 6-1.  Both the simulated impact area and overtopping volume are closer to that observed in the field when a channel obstruction is assumed to be present.  178  As can be seen in Figure 6-7, theoretically, interaction with forest has a small influence on the simulation results.  The simulated flow front is stopped sooner than when the effect of trees is ignored; however, this does not produce a substantial increase in flow resistance.  The interaction with forest only becomes important when the flow depth is thin, as this increases the force required to achieve the critical moment (Equation [7] and [8]).  In the present analysis, the effect of trees became pronounced when the flow depth dropped below about 0.5 m.  Therefore, it is reasonable to conclude that smaller volume debris avalanches may be significantly influenced by interaction with forests, however, the effect on large volume debris avalanches is small.  As discussed above, this analysis does not consider the increased frictional resistance that may result from entraining timber at the flow front.    Figure 6-11: Final deposit depths predicted by Dan3D when a channel obstruction is assumed to be present in the channel downstream of the 70° bend. 6.10 Summary – Johnsons Landing The Johnsons Landing debris avalanche was re-analyzed using a new rheology that is more appropriate than typical frictional rheologies for slide debris moving in an undrained condition.  This rheology will likely provide better results for cases where the failed volume liquefies and 179  moves in an undrained condition.  Compared to previous analyses, the use of the new rheology gives reasonable results without requiring ad hoc assumptions regarding the interaction of the debris avalanche with a forest.  In order to closely reproduce the field observation that very little volume was deposited downstream of the 70° bend, a channel obstruction, hypothesized to be composed of timber, must be included in the analysis.  The likelihood of occurrence of this channel obstruction would be very difficult to predict a-priori. This case highlights the necessity of introducing forced avulsions into the model when performing quantitative risk analysis of these types of events. 6.11 Appropriate Rheology for Debris Avalanches The Johnsons Landing case shows that not all debris avalanches can be accurately analyzed using a frictional or Voellmy rheology.  As mentioned in Section 6.2, other researchers and practitioners have found good results using the Voellmy rheology to analyze debris avalanches. The difference between those results and the Johnsons Landing results is likely the material that makes up the flow.  Johnsons Landing was composed of primarily fine grained debris, and likely did not develop a frictional front.  For debris avalanches that contain a significant portion of coarse grained debris, such as those analysed by the GEO in Hong Kong, the flow can experience significant frictional resistance, and the frictional or Voellmy rheology is likely appropriate.  Therefore the selection of the appropriate rheology to model debris avalanches should be made based on the expected material that will be mobilized during failure and runout (including entrained materials).  Compared to debris avalanches that contain coarse materials, it is expected that fine grained flows will travel faster, and experience less resistance to channel avulsion and overtopping. 180  The liquefied Voellmy rheology, as implemented in Section 6.5, is appropriate for debris avalanches where the majority of the source volume liquefies, and entrainment is limited.  This is due to the fact that the liquefied strength is normalized to pre-failure vertical effective stress.  These events are transitional between debris avalanches and flowslides (Chapter 7).  Dahl et al. (2012) describes three debris avalanches on the Faroe Islands that are likely of this type.   The liquefied Voellmy rheology is inappropriate for events that are governed by the mechanism of rapid undrained loading, and that do not contain a significant quantity of coarse grained material. In such cases, the basal shear strength will not be proportional to pre-failure vertical effective stress of the source material.  Sassa & Wang (2005) shows that the shear strength of material entrained through the mechanism of rapid undrained loading is proportional to the thickness, density and internal friction angle of the entrained layer (and independent of the added normal stress of the overriding debris avalanche).  If these characteristics of the entrainable material can be assumed to be relatively constant, then a rheology that assumes a constant yield stress, such as the Bingham rheology (described in Section 2.5.2), may be appropriate.  Further work on this type of debris avalanche is needed to prove this.     181  Chapter 7: Flowslides  7.1 Introduction Flowslides, which involve the extremely rapid motion of liquefied soil, are extremely mobile events that have tremendous destructive potential (Hungr et al., 2014).  The 1920 Haiyuan earthquake triggered a large number of flowslides in loess, which resulted in an estimated 100,000 fatalities (Zhang & Wang, 2007).  As summarized in Chapter 1, quantitative risk analysis of flowslides requires prediction of their dynamic characteristics.  This requires both an understanding of the types of material that are susceptible to liquefaction, as well as appropriate rheologies for simulation of flows moving in an undrained condition.      This chapter details the application of an equivalent fluid runout model to simulate the motion of flowslides.  Firstly, an overview of flowslides will be provided, as well as an example back-analysis of a flowslide case history. Then, the back-analysis of two unusual flowslides, both composed of fine grained colluvium, will be presented.  It will be argued that the list of liquefaction-prone materials should be expanded to include this material type.   7.2 Background The key material characteristic that defines flowslides is that they are triggered by liquefaction of a portion of the failed volume within the source zone (Hungr et al., 2014).  Due to this characteristic, flowslides can only occur in a small subset of liquefaction-prone soils.  These soils have been widely recognized to include loose, saturated granular material and quick clay (e.g. 182  Fell et al., 2007; Hungr et al., 2014).  An overview of these two liquefaction mechanisms is provided below. 7.2.1 Liquefaction of Granular Materials Numerous critical reviews of the mechanism of liquefaction of granular materials exist (McRoberts & Sladen, 1992; Olson, 2001; Wang, 2008).  Figure 7-1 shows a typical stress path for a loose, saturated granular material subject to a static overstress.  The soil reaches a peak shear strength (point B on Figure 7-1) before undergoing significant strength loss.  As summarized by Olson (2001), the phenomenon of liquefaction of granular materials can be explained based on critical state soil mechanics.  Casagrande (1936) showed that for a given confining pressure, and regardless of initial density, a soil subject to sustained shearing will reach a steady state density.  Therefore, if the initial density is less than this steady state density, the soil will tend to contract, and the soil will dilate when denser than critical.  The phenomenon exhibited in Figure 7-1 occurs for soils whose density is loose of critical, and where loading occurs faster than drainage can occur.    183   Figure 7-1: Schematic showing the undrained response of a loose, granular material subject to static overstressing (after Olson (2001) and Wang (2008)).  Prior to loading the stress state of the material is at point ‘A’.  As the material is loaded, the stress state reaches point ‘B’, the peak shear strength.  Loading beyond the peak shear strength results in soil structure collapse, and liquefaction.  The strength reduces to point ‘C’, which corresponds to the liquefied strength. Figure 7-2 shows a schematic of the response of a loose granular material to both drained and undrained shearing.  When a loose granular material is sheared in a drained condition, the particles re-arrange into a denser packing, and porewater pressures do not change.  When the same material is sheared in an undrained condition, no particle re-arrangement can occur because the pore-fluid cannot drain.  This leads to generation of excess pore-water pressures due to load being transferred from the particles to the pore fluid.  This is shown schematically by the grey arrows on Figure 7-2. 184   Figure 7-2: Top: drained shearing of a loose granular material.  The particles rearrange into a denser packing, and no excess pore pressures develop.  Contraction occurs and porewater is drained from the soil skeleton.  Bottom: Undrained shearing of a loose granular material.  Because drainage is restricted, the particles cannot densify.  This results in a transfer of load from the particles to the pore fluid (grey arrows), and an excess porewater pressure develops. As shown in Figure 7-2, liquefaction of granular materials occurs due to contractive shearing under undrained conditions.  Castro (1969) demonstrated experimentally that a difference in initial void ratio (volume of voids to volume of solids) of 0.008 can lead to a dramatically different undrained stress-strain response, if this difference changes the density from loose of critical to dense of critical.  This change in density was enough to change soil behaviour from contractive to dilative, which changes the undrained stress response from strain softening to strain hardening.  This likely explains the bifurcation of simulation results observed by Iverson 185  & George  (2016) when simulating the Oso flowslide as though it were a loose granular material.  In their simulation, a small change in initial density resulted in a dramatic difference in observed runout. Fell et al. (2007) used an analysis of 350 case studies to determine approximate bounds on the particle size distribution of soils susceptible to static liquefaction.  Due to permeability constraints, they suggest that sandy gravels with trace silt, observed at coal mine waste dumps, represent the coarsest statically-liquefiable material.  For fine grained soils, Fell et al. (2007), based on the work of Seed et al. (1982), suggested that clayey soils can liquefy provided they have a high water content (0.9 times the liquid limit, so long as the liquid limit is less than 35%) and clay content less than 15% by weight.               Sassa et al. (2005) showed that grain crushing can also lead to the generation of excess pore-pressures, a phenomenon they termed ‘sliding surface liquefaction’.  This process is somewhat analogous to the one presented in Figure 7-2, because when grain crushing occurs, particles are able to pack more densely.  If drainage is impeded, then the particles will not be able to re-arrange, and load will be transferred to the pore-fluid.  A key feature of sliding surface liquefaction is that it will occur after large displacements, on the order of metres.  These displacements must be triggered by some other mechanism.   7.2.2 Liquefaction of Quick Clay Flowslides can also occur in quick clays (e.g. Crawford, 1968; Tavenas et al., 1971; Gregersen, 1981; Quinn et al., 2011).  An example of a quick clay flowslide is shown in Figure 7-3.  Quick clays occur due to very specific deposition conditions.  Due to chemical effects, clays can form a 186  ‘cardhouse’ structure with large pore spaces when deposited in high salinity environments (Figure 7-4).  Subsequent leaching of salt from these clays can cause this structure to be metastable (if this structure is disturbed it will not reform).  When disturbed, the quick clay structure collapses, and the clay particles disperse into the pore fluid.  This results in a change in state of the material from a solid into a viscous liquid capable of rapid flow.  When this change of state occurs, the shear strength can be reduced by 1 to 3 orders of magnitude (Rankka et al., 2004).  Figure 7-3: Overview image of the Lemieux quick clay flowslide.  Photo: S.G Evans, with permission. 187   Figure 7-4: Schematic of a quick clay failure.  When the pore fluid has a high salt content, the cardhouse structure is favoured, and when the pore fluid has a low salt content, the dispersed structure is favoured.  If the clay is formed in a high-saline (e.g. marine) environment, a cardhouse structure will form.  Leaching of the pore fluid will result in this structure being metastable.  A disturbance of the clay will collapse the cardhouse structure and disperse the clay particles into the pore fluid, resulting in a change of state from a solid to a viscous fluid.   7.2.3 Summary of Liquefaction Mechanisms Based on Section 7.2.1 and 7.2.2, it appears as though only very specific conditions can lead to the formation of liquefaction (and therefore flowslide) prone materials.  As described above, these conditions include deposition of granular materials in a loose condition, such as when uncompacted fill is placed for construction.  Deposition of clays in marine environments can also lead to the development of flowslide-prone material if this clay is subsequently leached by fresh water.  Based on the above, we would expect that materials prone to flowsliding should be identifiable based on conventional geotechnical testing, such as SPT/CPT tests to measure in-situ density of granular soils (e.g. Olson & Stark, 2002).  Section 7.3 will present the back-analysis of a flowslide in granular material to demonstrate the application of liquefaction theory to the analysis of a real case history.     188  7.3 Coal Mine Waste Dump Failure The purpose of this section is to present a back-analysis of a flowslide that originated on a coal mine waste dump.  As summarized by Kent et al. (1995) , Dawson et al. (1998), Hungr et al. (2002) and Hungr (2017), there have been numerous long runout flowslides originating on coal mine waste dumps.  For example, Kent et al. (1995) documented 41 such failures.  Dawson et al. (1998) showed that these flowslides can travel distances up to 3.5 km.   Dawson et al. (1998) performed triaxial tests on samples of coal mine waste from four different sites, and demonstrated that this material is liquefiable.  Dawson et al. (1998) also performed limit equilibrium back-analyses of three coal mine waste flowslides.  They showed that when strengths corresponding to the failure envelope (Figure 7-1) are used, the dumps are stable with a factor of safety of approximately 1.2.  When strengths corresponding to the yield strength envelope are used, corresponding to the peak strength mobilized before liquefaction (Figure 7-1), the three failures analysed have a factor of safety of approximately 1.  Therefore, it appears as though liquefaction of the source material triggered these flowslides.  Hungr (2017) also suggests that sliding surface liquefaction can trigger flowslides from coal mine waste dumps. Hungr et al. (2002) back-analysed the runout characteristics of 44 coal mine waste dump flowslides using Dan-W.  They found that the majority of cases could be simulated using a back-analysed friction angle of 21°.  This value of bulk friction angle was explained as an average resistance composed of both liquefied and non-liquefied portions of the source mass.  About 30% of the cases analysed by Hungr et al. (2002) required a two-rheology simulation in order to accurately reproduce field observations.  A frictional rheology was used in the source zone, and the Voellmy rheology was used along the path.  The use of the Voellmy rheology was justified 189  because the failed material overran loose saturated substrate, similar to many rock and debris avalanches (Chapters 5 and 6).   A well-documented coal mine waste dump flowslide has been back-analysed using Dan3D.  An overview of this event is shown in Figure 7-5 and Figure 7-6.  The topography for this flowslide was digitized from a pre-event topographic map.  The initial volume of the failed material was 700,000 m3.  An estimated 500,000 m3 deposited near the source zone on Figure 7-6, while a further 200,000 m3 of material became channelized, and travelled approximately 1 km further.  Velocity estimates of 17 m/s were made based on a superelevation measurement at the point shown on Figure 7-5 and Figure 7-6. 190   Figure 7-5: Three images of the coal mine waste dump failure.  ‘A’ shows an image of the source zone, ‘B’ shows an image of the superelevation feature used to estimate flow velocity, and ‘C’ shows an image of the highly mobile, channelized portion of the flow.  Photos: O. Hungr.   191   Figure 7-6: Coal mine waste flowslide simulation constraints. The parameter estimation program Pest, described in Section 4.5, was used to calibrate the model.  For this back-analyses, a two rheology simulation was used, with the boundary between the source slope and the channel (shown on Figure 7-6).  This boundary corresponds to the location where the flowslide overran loose, organic-rich path material that was saturated by snowfall (Hungr, 2017).  A frictional rheology was used in the source zone, and the Voellmy rheology was used when the material became channelized.  The best fit parameters were found to be 21° for the source material, and a friction coefficient of 0.06 and turbulence coefficient of 680 m/s2 for the channelized portion of the runout path.  The best fit results are shown in Figure 7-7. 192   Figure 7-7: Top: Deposit depths and impact area simulated by the model.  The deposit in the source zone, as well as the distal runout distance, is well reproduced.  Bottom: Maximum velocities simulated by the model.  The velocity at the superelevation is reproduced. 193  The best fit rheological parameters agree well with those analyzed by Hungr et al. (2002).  Therefore, it appears as though the mechanism of this flowslide can be explained by two runout mechanisms.  In the source zone, partial liquefaction of the source material resulted in brittle failure.  As previously mentioned, the friction angle of 21° is likely an average value between liquefied and non-liquefied portions of the rupture surface.  After vacating the source zone, a portion of the initial failure encountered loose, saturated substrate.  This substrate was overridden, leading to the extremely long distal runout. This case shows the complexity of analyzing flowslide motion for full scale case histories.  Full or partial liquefaction in the source zone is required for the mass to move initially.  Following initial liquefaction, the mass can interact with path materials to further runout.  The source material must have special characteristics to liquefy, which in this case is due to end dumping of mine waste.  This method of emplacement can lead to material being deposited in a loose condition, and the angularity of the waste fragments can leave them susceptible to grain crushing, which can result in sliding surface liquefaction.   7.4 Flowslides in Overconsolidated Silt and Clay A class of flowslides is emerging that is difficult to explain using the mechanisms detailed in Sections 7.2.1 and 7.2.2.  This class includes the 1973 Attachie slide (Fletcher et al., 2002) and the 2014 Oso flowslide (Keaton et al., 2014; Hibert et al., 2015; Iverson et al., 2015; Stark et al., 2017).  In both cases, the failed material consists of overconsolidated silt and clay. As detailed below, extensive surface and subsurface investigations have not discovered liquefaction-prone material at either of the sites (Fletcher et al., 2002; Stark et al., 2017).  In fact, prior to catastrophic failure, both sites experienced multiple episodes of slow, compound sliding leading 194  to displacements of tens of metres.  If a liquefaction-prone material were initially present at the sites, then these past movement episodes should have triggered catastrophic failure.  Therefore, a mechanism must be acting at these sites that transforms the material undergoing slow landsliding into a liquefaction-prone material.   A mechanism to describe this behaviour was provided by Fletcher et al. (2002), and termed ‘macroscopic brittleness’.  This mechanism is similar to the liquefaction of granular material described in Figure 7-1 and Figure 7-2.  Fletcher et al. (2002) proposed that previous failures result in deposits of blocks (on the order of m3) of overconsolidated silt and clay that are separated by cracks and cavities.  These loosened blocks would undergo a process called ‘softening’, whereby the blocks would degrade, resulting in the cracks and cavities between the blocks being filled by loose, fine grained material.  Due to softening, the permeability of the joints between the blocks would be low, and the infill material would contract when sheared.  If such a loosened and softened mass were saturated and loaded, it is conceivable that it could liquefy.  The rest of this chapter will test the hypothesis that fine-grained colluvium has the potential to liquefy through the back-analysis of the Attachie and Oso case histories. 7.5 Methodology To back-analyse the Attachie and Oso case histories, the new liquefied rheology (summarized in Section 2.5.3) was used.  As shown by Fletcher et al. (2002) and Aaron et al. (2017a), both case histories have a distinct morphology, with fluid distal deposits, and intact proximal deposits.  Both cases had thick deposits of loosened and softened colluvium within the source mass (Fletcher et al., 2002; Aaron et al., 2017a).  In the dynamic analyses, this colluvium is parameterized using the new liquefied rheology, whereas the intact material is parameterized 195  with the frictional rheology and a moderate pore pressure.  If back-analyzed coefficients comparable to other liquefaction case histories can be found that reproduce the observed runout extent, this would provide strong evidence that the colluvium liquefied. 7.6 Model Rheology As mentioned in Section 6.1, flowslides are defined by liquefaction of material in the source zone.  The frictional and Voellmy rheologies are unsuitable to simulate this class of flow-like landslide due to the fact that these rheologies assume drainage during flow (Section 2.5.3).  To simulate flowslides, a constant liquefied strength should be used (Olson & Stark, 2002).      In the analyses that follow, the basal resistance will be parameterized with the new liquefied rheology, introduced in Section 2.5.3. Additionally, the velocity-dependent term will be neglected.  Future work could consider adding rate-dependent resistance, either by incorporating the liquefied shear strength into the Herschel-Bulkley rheology, in a manner similar to De Blasio et al. (2011) or through using the liquefied Voellmy rheology (described in Section 2.5.3).  However, the velocities obtained in the analysis are not unrealistically high and introduction of rate dependence does not seem to be essential.  For the analyses that follow, it is assumed that the flow experienced only negligible viscous or turbulent resistance.     7.7 Attachie The Attachie flowslide occurred on May 26, 1973 near Fort St. John, B.C.  Van Esch (2012) estimated that this landslide had a total volume of 14.8 Mm3.  This flowslide travelled approximately 1 km across the flat floodplain of the Peace River.  Based on air photos taken in 1952, prior to catastrophic flowsliding, the Attachie slide exhibited behaviour typical of 196  compound slides in overconsolidated clay: slow to rapid displacements, likely controlled by pore-pressure variations (Fletcher et al., 2002).  Similar to the Oso flowslide, no sensitive materials have been identified at this site in the extensive investigations performed following the highly mobile failure (Fletcher et al., 2002; BGC, 2012). Pre and post event images of this event are shown in Figure 7-8 and Figure 7-9.  The debris of the highly mobile portion of the Attachie flowslide appears to consist primarily of blocks of glaciolacustrine silt and clay supported by a remoulded matrix (Fletcher et al., 2002).  This deposit is estimated to have an average thickness of 7.6 m (Evans et al. 1996).  The highly mobile debris temporarily dammed the Peace River, and projected a wave that ran up the opposite bank.    Figure 7-8: Pre-event image of the Attachie flowslide. (Airphoto credit: Province of British Columbia.  Airphoto BC7279-70, 1970, Copyright © Province of British Columbia.  Reprinted with permission.) 197   Figure 7-9: Post event image of the Attachie flowslide . (Airphoto credit: Province of British Columbia.  Airphoto BC5529-75, 1973, Copyright © Province of British Columbia.  Reprinted with permission) Fletcher et al. (2002) reconstructed the 3D rupture surface of the Attachie flowslide based on numerous surface and subsurface observations, including borehole logs and inclinometers.  The rupture surface is interpreted to have two backscarps separated by a step (Figure 7-10).  Fletcher et al. (2002) estimate that 6 Mm3 of material deposited in the source zone (the source zone is shown on Figure 7-11).  Similar to the Oso flowslide, this debris in the upper portion of the source zone is intact but highly disturbed (Fletcher et al., 2002).     198   Figure 7-10: Cross section through the Attachie rupture surface showing the stepped rupture surface and the material boundaries used in the dynamic analysis. An overview of the constraints used for the dynamic analysis is shown in Figure 7-11.  For the dynamic analysis of the Attachie slide, the colluvium was parameterized with the liquefied rheology, and the intact material was parameterized with the frictional rheology.  The hypothesized boundary between these two material types is shown on Figure 7-10.  This boundary was selected based on Fletcher et al. (2002) who suggested that the material on the lower portion of the rupture surface liquefied, whereas the material in the upper portion only had limited mobility.   For these simulations, Dan3D was modified to link material properties to particles as opposed to spatial zones.  Therefore, the basal resistance for the intact material was calculated based on the frictional rheology throughout the entire simulation.  Since Dan3D is a depth-averaged model, a vertical boundary between the two material types had to be used.  This represents a necessary simplification of reality, as the transition between the two material types is likely gradual.   199   Figure 7-11:  Constraints used to calibrate the Attachie runout analysis.  The constraints include the deposit thickness, deposit distribution and impact area. Fletcher et al. (2002) back calculated a pore pressure ratio of 0.26 based on 2-D limit equilibrium analyses and Spencer’s method of slices.  This value was used in the back-analysis of the Attachie flowslide, and the liquefied strength ratio was calibrated based on trial-and-error analysis.  The best fit liquefied strength ratio was found to be 0.05, with an internal friction angle of 15°. As mentioned in Section 2.5.4, flowslide motion can be controlled by free surface gradients (P force on Figure 2-1), so the internal friction angle must be considered when analyzing their motion.  A plan view of the best fit results is shown in Figure 7-12, and a section through the results is shown in Figure 7-13.  200   Figure 7-12: Back-analysis results of the Attachie flowslide.  The red dashed line is the observed impact area.  The thick deposits in the source zone are well simulated, as is the average thickness of debris along the valley floor.  The leading edge of the debris is not simulated to deflect downstream.  This deflection can be attributed to factors not simulated by the model (see text).  Figure 7-13: Comparison of model results to post event LiDAR.  The simulated deposit thicknesses are similar to those derived from the LiDAR data.  The section line is shown on Figure 7-11. 201  Figure 7-12 and Figure 7-13 shows that the distal extent and thickness of the debris is well reproduced, however the location of the impact area is poorly simulated.  It appears as though the leading edge of the debris was deflected downstream, an observation not reproduced by the runout model.  The deflecting of the debris can be attributed to a number of factors not included in the model, including the momentum of the river pushing the debris, detailed topographic features of the pre-event topography obscured by the debris, varying water depths within the channel and reworking of the debris during the gradual breach of the landslide dam.  Given this, the simulation results reasonably reproduce the simulation constraints, and support the hypothesis that the previously disturbed material liquefied. 7.8 Oso  This section is an excerpt from Aaron et al. (2017a) reprinted with permission from ASCE. On March 22, 2014 a terrace slope located north-east of Oso in the state of Washington, USA, failed catastrophically.  The resulting flowslide killed forty-three (43) people in the small community of Steelhead Haven and buried Washington Highway SR530.  The slope on which the landslide originated had failed multiple times in the past at lower elevations, most recently in 2006, and had a thick accumulation of landslide debris on its surface and at its toe.  In comparison to the previous failures, the March 2014 event initiated at a higher elevation, was extremely mobile, and projected debris across the width of the valley floor, causing the noted death and damage.  Understanding the causes of this rapid movement is an important step in recognizing and preventing future disasters in similar settings.   202  The pre-2014 landslides that occurred at this and other locations in the Stillaguamish Valley have been studied, however, most of this work focused on landslide-induced turbidity in the Stillaguamish River and its impact on fish populations (Miller, 1999; Shannon & Wilson, 1952).  These investigations recognized two main types of landslides that occurred at the site.  The first is compound sliding and motion of intact blocks, an example of which is the 1967 landslide (Miller, 1999).  The second type is small-volume flow-like landslides with moderate velocities, similar to earthflows.  Both Miller (1999) and Shannon & Wilson (1952) attribute these flows to the disintegration of failed glaciolacustrine blocks.  The landslides detailed in these reports are all small when compared to the volume and impact area of the 2014 landslide and much less mobile.  The most mobile of these previous failures occurred in 1967 and resulted in a temporary damming of the river channel (Miller, 1999).   The 2014 flowslide represents a distinct mechanistic change in the failure behaviour of this terrace slope (the terrace is referred to herein as the Whitman Bench).  The failure travelled over 1.4 km on a nearly horizontal runout surface and exhibited behaviour typical of flowslides in liquefiable granular material or sensitive clay.  Based on site investigations performed before and after the flowslide, such materials have not been identified at this site (Stark et al. 2017).  After the 2014 landslide, it was unclear how a landslide that took place primarily in overconsolidated, insensitive glaciolacustrine silt and clay could transform into a flowslide as evidenced by the greatly varying failure mechanisms proposed by others (Keaton et al., 2014; Iverson et al., 2015; Iverson & George, 2016; Wartman et al., 2016).   Nevertheless, the occurrence of extreme undrained strength loss/liquefaction is clearly implied by the observed dynamic behaviour of the event. 203  The discussion of the Oso flowslide is presented as follows.  First, the morphology of the debris field will be described, as this information is used in all subsequent analyses.  Next, the rupture surface used in the back-analyses is described.  Much of the rupture surface is obscured with slide debris, so the geometry had to be inferred from the morphology of the debris, as well as borehole information.  Finally, the results of the back-analysis using the new liquefied rheology are presented.  7.8.1 Data Sources The Oso flowslide was well documented through the authors’ site visits, multiple field surveys, airphoto imagery, and high quality topographic data.  LiDAR surveys of the valley were conducted in 2003, 2013, and immediately following the 2014 event.  Airphoto imagery of the slope is available dating back as far as the 1920’s, and many pre- and post-event oblique images are available.  Both Keaton et al. (2014) and Iverson et al. (2015) document many features of the debris field and source area, providing valuable information that aided in the present reconstruction of the event dynamics.  A detailed site investigation including multiple deep exploration drill holes, some of which intercepted the rupture surface of the 2014 slide, became available to the authors in 2016 (Badger, 2015).  All of these data sources, as well as additional data gathered during a 3 day field visit by the authors in 2014, were used to guide the present analysis. 7.8.2 Morphology of the Debris Field To reconstruct the trajectories of various parts of the deposit, the trees preserved on the slide surface were correlated with the original forest on and below the Whitman Bench.  For this 204  purpose, the debris field is discussed in three separate zones: the source zone, the valley floor, and the “splash” zone (see Figure 7-14).   Figure 7-14: Landslide zones for dynamic analyses: the dashed orange line outlines the splash zone, the dash-dot red line outlines the valley floor deposit, and the solid black line outlines the source zone.  The locations of the boreholes used to constrain the rupture surface are shown (Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation). The source area is understood here as the planimetric footprint of the source volume.  Figure 7-15 shows that the proximal source area features thick deposits, consisting largely of a relatively intact but heavily internally sheared block, covered by trees (Polygons 1 and 2 in 205  Figure 7-15).  Apart from a series of scarps formed by internal shears, the forest floor is mostly intact on this block.  This suggests that the proximal block remained relatively coherent during its movement, although the abundance of internal shear surfaces exposed by normal offsets testifies to strong internal shear deformation and suggests non-rotational shape of the sliding surface.  Allowing for some spreading, the area of the forest floor that remained intact on this block was measured to be 59,000  m2.  By overlaying the impact area of the 2014 landslide on a pre-slide orthophotograph taken in 2013, it is possible to determine where the trees present on this block must have originated.  Figure 7-16 shows that the intact block of trees derived from the surface of Whitman Bench located at the head of the slide and from a part of the prehistoric scarp (Polygon 16, which also has an area of 59,000 m2).  The heavily forested area lying originally downslope of the Whitman Bench crest (Polygon 17) consists of an ancient slide block, likely made up of previously displaced material (Stark et al. 2016).   206   Figure 7-15: Polygons of intact forest blocks in 2014 slide mass used to constrain the origin of different features of the debris.  Polygons 1 and 2 have a combined area of 59,000 m2, corresponding to the pre-failure area of trees on the Whitman Bench (see Figure 7-16).  Polygons 3 through 15 have a combined area of 50,000 m2, corresponding to the area of the trees in front of the Whitman Bench (see Figure 7-16).  Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation. 207   Figure 7-16: Measurement of tree areas on pre 2014 ortho-photograph where Polygon 16 and 17 have a measured area of 59,000 m2 and 54,000 m2, respectively (Background image from National Agriculture Imagery Program, data available from the U.S Geological Survey). To the southeast of this proximal block, the landslide deposits exhibit blocks of coherent material that appear to have dropped down while moving over a sloping surface (see Figure 7-17).  This indicates that the material forming the proximal debris deposits underwent sliding failure on a compound rupture surface, which necessitated disruption on multiple internal shear surfaces.  However, in contrast to the distal deposits, there is no evidence of liquefaction in this part of the debris.    208   Figure 7-17: Post event image of 2014 Oso flowslide source zone with intact blocks that appear to have undergone extension while traveling over an inclined surface (see square on left photo).  The photo on the right shows a side view of the dropped down blocks (Photos: J. Aaron). The 2014 debris that deposited on the valley floor has a much more disturbed morphology.  These deposits are disaggregated and mostly consist of glaciolacustrine deposits overlain by discontinuous ridges and hummocks of sand, some of which are topped with mature trees in varying degrees of tilt.  Most of this sand represents blocks from below the crest of the Whitman Bench, although some of it probably originated from sand blocks moved by previous landslides and resting originally on the upper slopes.  Small rafts of intact forest floor can be found within the valley floor debris.  Measuring the combined area of these relatively intact blocks (Polygons 6 through 15 on Figure 7-15) and correcting for spreading (by making the assumption that these blocks were stretched by 15%) yields a total forest area of 50,000 m2.  The forest originating on the Whitman Bench surface is already accounted for by the source zone deposit, so these trees must have originated from the area downslope of the Whitman bench, including the surface of the ancient slide block and the surrounding upper slopes.    209  The splash zone is located at the distal margins of the 2014 slide deposit (see Figure 7-14).  Based on the accumulation/depletion map (see Figure 7-18) it can be seen that the deposits in this zone are extremely thin.  Comparing Figure 7-15 and Figure 7-16 shows that some trees that originally grew on the floodplain in this zone remained standing after the 2014 slide, indicating that they resisted the impact of the fluid debris.  A common feature of large landslides is a splash zone surrounding the distal end of deposits (termed a “Spritzone” by Heim, 1932).  It is likely that this area was over-run by water and highly liquid colluvium pushed in front of the main debris mass.  Some of the debris in the splash zone also originated from channel and floodplain deposits entrained or pushed together with surface water from the path of the rapidly-moving flowslide front.  Figure 7-18: Accumulation and depletion zones of 2014 Oso landslide that shows deposits in the splash zone are thin.  The thin deposits extending to the north-east are due to post landslide flooding (Keaton et al. 2014). 210  Based on the analysis described above, the morphology of the deposits can be summarized as follows:    Source zone, which contains a nearly intact, though heavily deformed block at the head, bordered by highly sheared but massive blocks that appear to have travelled for a limited distance down a sloping, step-like surface from a higher elevation.    Valley floor, which consists of widely-spread fluid deposits bearing rafts of intact sand and clay, and  Distal splash zone, which consists of only fluidized material and organic debris. 7.8.3 Rupture Surface Reconstruction Reconstructing the rupture surface is a difficult step in the dynamic analysis of the Oso landslide.  A portion of this surface is visible in the main scarp, however, the rest of the failure surface is obscured by slide debris.  Therefore, a large part of the surface on which the failed mass moved must be inferred from field evidence.  The primary source of data used to reconstruct the failure surface is the morphology of the deposits, as observed on the pre and post event LiDAR topography data, as well as four boreholes drilled by the Washington Department of Transportation (WSDOT) (Badger, 2016).   The geometry of the Phase A failure surface was chosen to correspond with the mechanism described in Stark et al. (2017), where a relatively small initial failure in the upper part of the slope induces a large strength loss in the colluvium accumulated at the toe of the slope.  Multiple pieces of evidence support this interpretation.  As discussed above, the measured area of intact blocks of forest floor rafted onto the valley indicates that the valley floor deposits are derived 211  primarily from material originally located to the southeast of the Whitman Bench.  In its pre-failure position, this material was traversed by two scarps: the scarp associated with the ancient landslide and the scarp associated with the 2006 landslide.  The locations of these scarps correspond to the dropped down-slabs of material identified in the 2014 debris field (see Figure 7-17), indicating that the material originally located between the scarps and the 2013 ground surface was displaced during the flowslide.  Therefore, both the ancient and 2006 scarps were used as the back scarp of the Phase A failure surface (see Figure 7-19).  The 2013 LiDAR shows a thick accumulation of colluvium derived from the 2006 and other landslides that span from the toe of the slope to the river.  This colluvium is assumed to have undergone a substantial amount of strength loss during the Phase A failure, so the rupture surface was chosen to correspond to the 2003 ground surface in this section.       212   Figure 7-19: Simplified stratigraphic section through the 2014 Oso landslide showing the borehole results presented by Badger (2016).  The materials comprising phases ‘A’ and ‘B’ are indicated. The dashed red line shows the proposed rupture surface, which agrees with available borehole data and surface observations.  The locations of the two intermediate scarps correspond to observed pre-failure scarps, and the depth of the rupture surface was determined from borehole data.  The location of the boreholes and section line are shown on Figure 7-14. The Phase B failure surface is interpreted as a compound surface with a near horizontal plane located in the Glacio-Lacustrine unit (see Figure 7-19).  There is a large intact block that came to rest in the source zone of Phase A, though deformed along multiple inclined shears.  The style of this deformation is indicative of a near horizontal plane located in the Glacio-Lacustrine unit, typical of a bi-linear compound slide (Hutchinson, 1988).  In front of this intact block, slabs of material “scalloped” or dropped down, exhibiting marked extension while moving over a steeply-inclined, step-like surface segment (see Figure 7-17). The elevation of the near horizontal planes in the Upper Glacio-Lacustrine unit can be inferred from the four WSDOT boreholes (Figure 7-14).  In EB-04si-15 the elevation of the rupture 213  surface can be determined by the contact between a sandy unit (likely glaciofluvial) and a silt/clay unit (likely glaciolacustrine).  In H-12si-15 the borehole log notes variably inclined partings at an elevation of approximately 105 m.  In EB-07si-15 and EB-09si-15 borehole logs indicate slickensides/disruption at elevations of 105 m.   Because the proposed failure surface is based partly on indirect evidence, there is some uncertainty associated with it.  However, compared to the failure surfaces proposed by other researchers (Keaton et al., 2014; Iverson et al., 2015; Wartman et al., 2016), it appears to be the most consistent with field observations and site conditions.  For example, varved silt and clay is expected to have anisotropic strength properties.  This, combined with evidence of strong internal distortion of the source volume, as well as the observation that there is little elevation change between the 2013 and 2014 LiDAR southeast of the large intact block in the source zone (see Figure 6), negates the possibility that the failure surface is circular (Keaton et al., 2014; Wartman et al., 2016) or a logarithmic spiral (Iverson et al., 2015).  The proposed compound surface agrees with field observations as well as the expected failure behaviour of the overconsolidated Glacio-Lacustrine unit, while corresponding to the positions of shear surfaces identified in the 2015 drillholes (Badger, 2016).         7.8.4 Numerical Simulation Results The rupture surface described above has been used in the dynamic models DanW and Dan3D to back-analyse the runout of the Oso flowslide.  As mentioned previously, in both simulations the colluvium (‘A’ on Figure 7-19) was parameterized with the liquefied rheology, and the intact material (‘B’ on Figure 7-19) was parameterized with the frictional rheology. 214  7.8.5 2D Simulations The results of the 2-D simulations, where the two phases are assumed to be separated by a time gap are summarized in Figure 7-20.  In particular, Figure 7-20 shows a comparison between the final surface predicted by DanW and that derived from the 2014 LiDAR.  Comparison of these two surfaces shows that DanW is able to reproduce the runout distance and deposit distribution.  The blocky nature of the debris is not reproduced, because DanW models the sliding mass as a homogeneous material.  The distal end of the deposit is also thinner than indicated on the 2014 LiDAR, likely due to the neglect of rate-dependent resistance and entrainment of debris and vegetation by the flow front.  A bulk friction angle of 12 degrees, consistent with residual friction angles measured by Stark et al. (2017), was used to simulate the Phase B material.   The results of 2-D simulations, where no time gap is simulated between the two phases, are shown in Figure 7-21.  The results, in terms of runout distance and deposit distribution, are nearly identical to those determined for the simulations where the phases moved separately.  The similarity between the two sets of simulations demonstrates that the observed deposit shape is not controlled by the temporal sequence of failure but instead by the distribution of shear strength along the failure surface.  The large intact block deposited in the source zone is well reproduced, because this material has a higher strength and remained frictional.   215   Figure 7-20: Final deposit depths predicted by DanW when the two phases are simulated to occur at different times.  The final deposit surface agrees well with the 2014 LiDAR.  Figure 7-21: Final deposit predicted by DanW when the two phases occur at the same time.  The results are similar to those presented on Figure 7-20. 7.8.6 3D Simulations The final deposit depths predicted by the 3D simulations for Phase A and Phase B are shown in Figure 7-22 and Figure 7-23, respectively.  These results cannot be directly compared to Figure 216  7-18 because these deposit depths are calculated based on the difference in elevation between the top of the deposit and the failure surface, whereas accumulation and depletion maps are calculated based on the difference between the pre-event ground surface and top of the slide deposit.  In areas where material has been removed and then partially replaced, such as the source zone, deposit depths will be positive but the map will show depletion.  To better facilitate comparisons between model results and field observations, a map showing the zones of accumulation and depletion predicted by Dan3D was created by adding the calculated deposit depths to the map of the failure surface and sliding/flow path (see Figure 7-24).  These simulations treat the failure as two discrete events, using the same parameters as used in the 2D analyses.  The model adequately reproduces the impact area and deposit distribution.  Similar to the 2D analyses, the distal end of the predicted deposit is thinner than that observed in the field, for reasons stated earlier.  The model does not reproduce the deposits in the south-west corner (the “splash” area) because these deposits represent a small volume of material entrained in the valley that mixes with surface water and is projected ahead of the main debris mass. This process is not simulated with the present dynamic models.   217   Figure 7-22: Results of Dan3D simulation of first phase of movement with impact area and deposit distribution in agreement with field observations. No effort was made to reproduce the splash zone (Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation) 218   Figure 7-23: Results of Dan3D simulations of Phase B overlain on post 2014 event ortho-photograph (Background Image Courtesy of Esri Disaster Response and Washington State Department of Transportation. 219   Figure 7-24: Accumulation and depletion zones predicted by Dan3D.  The splash zone was not modelled, so the impact area in the western section is less than that observed. To further verify the 3D simulations, the 3D model was used to simulate the structural damage caused by the 2014 landslide.  For this purpose, the model results were used to derive the “debris flow intensity index” proposed by Jakob et al. (2012).  This empirical index has been correlated to structural damage for a wide variety of case histories of damaging debris flows.  It is calculated with equation [2] where 𝑙𝑖𝑑𝑓 is the debris flow intensity index, ℎ is the flow depth, and 𝑣 is the velocity.    220  𝑙𝑖𝑑𝑓 = ℎ ∗ 𝑣2 [4] Values of the intensity index greater than 100 usually result in complete destruction of weak structures, while values between 10 and 100 tend to result in major structural damage (Jakob et al. 2012).  Figure 7-25 shows the maximum value of the intensity index calculated by the DAN3D model, compared to the location of houses in the deposit zone.  Model estimates of the intensity index correlate well with those inferred from field evidence, suggesting that the spatial distribution of the simulated velocities and depths is realistic.  Figure 7-25: Model simulated intensity index compared to building damage.  The units of intensity index are m3s-2.  According to Jakob et al. 2011 values of the intensity index between 1 m3s-2 and 100 m3s-2 correspond to some/major structural damage, from 100 m3s-2 to 1000 m3s-2 correspond to major structural damage/complete destruction and greater than 1000 m3s-2 indicates complete destruction (Background image from National Agriculture Imagery Program, data available from the U.S Geological Survey).   221  A representative cross-section through the 3D model results is shown in Figure 7-26.  The model is able to reproduce the deposit distribution derived from the 2014 LiDAR.  Comparing Figure 7-26 with Figure 7-20 shows the DanW and Dan3D results are nearly identical.  Cross sections through the 3D model results at different times during the motion are shown in Figure 7-27.    Figure 7-26: Representative cross section through the Dan3D results showing the simulated deposit depths.  The results are in good agreement with the deposit surface derived from the 2014 LiDAR.  Figure 7-27: Sections through the Dan3D Phase A simulation at ten second intervals showing slide debris piling up at the slope toe before spreading over valley floor. 7.9 Discussion and Conclusions Back-analysis of the Oso and Attachie case histories support the hypothesis that the highly mobile phase of each event consisted of liquefied colluvium.  This agrees with the findings of 222  Fletcher et al. (2002) regarding the mobility of the Attachie slide.  Overconsolidated glaciolacustrine silt and clay is not a material traditionally recognized as liquefiable.  The analyses presented in this chapter show that this material can liquefy, and the resulting flowslides can have catastrophic consequences.  Risk analysis in similar settings must consider flowslides as a potential hazard.  Although the analyses demonstrate that the colluvium likely liquefied, since they are inverse analyses the mechanism of liquefaction still remains unclear.  Excluding the possibility that either of these sites contained sensitive material (none have been identified despite multiple in depth investigations) the macroscopic brittleness mechanism proposed by Fletcher et al. (2002) appears to be a possible explanation for the runout behaviour of these flowslides.   This mechanism is difficult to test experimentally, as it involves the interaction of blocks with volumes on the order of cubic meters.  This phenomenon could potentially be modelled using discontinuum numerical models; however, the complex pore-pressure feedbacks may lead to extremely long model run times. Regardless of the underlying mechanism that leads to liquefaction of fine grained colluvium, the process appears to be relatively rare.  Of the 1610 landslides documented along the Peace River, the Attachie slide appears to be the only flowslide in overconsolidated glaciolacustrine silt and clay (Severin, 2004).  BGC (2012) mapped 4,010 individual landslides along the Peace River and its tributaries, and found that 52% of these were flowslides in normally consolidated glaciolacustrine deposits.  BGC (2012) suggested that these were triggered by the collapse of a metastable structure, a different mechanism from that observed at Oso and Attachie.  There are two apparently other long runout, flow-like landslides near Oso (LaHusen et al., 2016).  The La 223  Conchita slide, which occurred in 2005, appears to be of this type (Jibson, 2006), although to date no detailed dynamic analysis has been performed.  With the caveats that river activity may obscure evidence of long runout, and that LiDAR technology may lead to many more of these types of flowslides being identified, there appears to be relatively few examples of colluvium liquefying in comparison to the large number of cases of slow moving landslide complexes in overconsolidated silt and clay.   The dynamic analyses presented here can be used as a precedent to perform forward analysis of potentially liquefiable colluvium.  In a forward analysis, the colluvial accumulation should be parameterized with the liquefied rheology, using the liquefied strengths determined above (0.07 for Oso and 0.05 for Attachie).  Brittle failure of intact material (for example ‘Phase B’ at Oso) can potentially have implications for a risk analysis, and this phase of motion can be modelled by parameterizing resistance with the frictional rheology with a friction angle typical of the residual strength of the material and moderate pore pressures.   Since the impact area and vulnerability parameters can be estimated using analyses similar to those presented in this chapter, the most uncertain part of a risk analysis is likely to be the probability of catastrophic failure (PH in Equation [ 1.1 ]).  At present, the mechanism of liquefaction of fine grained colluvium is not very well understood.  Fletcher et al. (2002) suggests that the presence of non-plastic (or low-plasticity) units is a necessary condition for catastrophic failure, however, beyond this, little is known about how to distinguish liquefiable from non-liquefiable fine grained colluvium.  There is potential to assign probabilities of catastrophic failure empirically, based on the inventories detailed above, however more research 224  is needed before this can be routinely done.  Further research into the mechanism of colluvium liquefaction is required to defensibly choose this probability.           225  Chapter 8: Conclusions 8.1 Introduction Extremely rapid, flow-like landslides, such as rock avalanches, debris avalanches and flowslides, are a significant hazard worldwide.  Managing the risk associated with these landslides requires probabilistic predictions of the flow depth, velocity and impact area of an event before it occurs.  Numerical runout models, based on fluid mechanics, are one tool that can be used to make these predictions.  However, the use of these tools in this context is not routine.  This is due to both a lack of understanding of the mechanisms that govern the motion these types of landslides, as well as a lack of a database of case histories that can be used to guide predictions. The goals of this thesis were to both improve our understanding of the mechanisms governing the motion of extremely rapid, flow-like landslides, and to develop practical tools that can be used to predict their runout characteristics.  To do so, a database of rock avalanche, debris avalanche and flowslide case histories has been assembled.  New tools and techniques were developed to back-analyse this database, and the results have been used to infer movement mechanisms.  Additionally, the implications of these back-analysed results to the prediction of flow-like landslide motion were detailed.  This chapter summarizes the main conclusions that have resulted from this work. 8.2 Background Information A detailed literature review of existing numerical runout models, as well as existing approaches to model the motion of rock avalanches, debris avalanches and flowslides was presented.  This 226  review is summarized in Chapters 1 and 2.  Several knowledge gaps in the analysis of extremely rapid, flow-like landslides were identified.  Based on this, the following specific research objectives were defined:  1. Develop a new rheology appropriate for the simulation of liquefied material.   2. Develop a new numerical model to simulate the motion of initially coherent landslides.   3. Develop a new calibration methodology that removes some of the subjectivity associated with trial-and-error calibration.   4. Compile and back-analyse a database of rock avalanche case histories in order to explore rock avalanche movement mechanisms.    5. Develop a new methodology that uses the rock avalanche database to perform probabilistic forward analysis.   6. Back-analyse the Johnsons Landing debris avalanche to infer its movement mechanisms, and comment on the implication of this case to the analysis of debris avalanche motion.   7. Back-analyse two flowslides that occurred in overconsolidated glaciolacustrine colluvium to infer their movement mechanisms. 8.3 New Techniques to Analyse Extremely Rapid, Flow-like Landslides New techniques to analyse the motion of extremely rapid, flow-like landslides were presented in Chapters 2, 3 and 4.  These include the development of a new rheology to simulate the motion of liquefied material (Chapter 2), a new dynamic model to simulate the motion of initially coherent rock and debris avalanches (Chapter 3), as well as a new methodology to calibrate equivalent fluid runout models (Chapter 4).  The main findings from these chapters are:     1. A new rheology, appropriate for the simulation of extremely rapid, flow-like landslides moving in an undrained condition, was devised and implemented.  This rheology 227  assumes basal resistance is governed by a constant yield stress, which is proportional to the pre-failure bed-normal effective stress.  Since this new rheology assumes that basal resistance has no frictional component, centripetal accelerations do not increase basal resistance. 2. Existing runout models are based on fluid mechanics, and applying them to some rock avalanche case histories results in predictions of excessive lateral spreading.  A new numerical model that accounts for the initially rigid phase of motion was derived, implemented and tested. 3. The new model, Dan3D-Flex, treats the landslide as a flexible block, which translates and rotates over three-dimensional topography, remaining coherent in plan, but flexing vertically to maintain contact with the irregular terrain.  At a user-specified distance, the solution method switches to the original Dan3D algorithm, which simulates the disintegration and flow-like motion of the landslide. 4. It was found that including an initially rigid phase of motion was necessary to accurately reproduce the impact area of certain rock avalanches.  It was suggested that the distance travelled as a flexible block can be selected based on an examination of the pre-failure topography.    5. A review of existing methodologies to calibrate equivalent fluid models was presented.  This review revealed that current methodologies can be subjective; they often do not account for parameter non-uniqueness and can be very demanding of the users’ time.  Two new calibration methodologies, based on optimization theory and statistics, have been developed and implemented to address these deficiencies. 228  6. The new calibration algorithm greatly increases the efficiency of calibration and accounts for some sources of error.  It also provides a way to use previously successful back-analyses to make forward predictions. 8.4 Rock Avalanche Movement Mechanisms and Prediction As summarized in Chapter 5, a review of rock avalanche literature revealed that there is still controversy surrounding the mechanisms governing their motion.  A database of 30 back-analysed rock avalanche case histories was assembled in order to infer movement mechanisms, and develop a methodology for probabilistic prediction of rock avalanche motion.  The main findings of Chapter 5 include: 1. The results of the back-analysis of the database of rock avalanche case histories revealed that the dominant mechanism of natural rock avalanche motion is likely the shear behaviour of the path materials controlled by pore-pressure effects, although other proposed mechanisms could not be conclusively disproven. 2. By combining the back-analysis results, it was found that rock avalanches can be classified based on detachment process and path materials. 3. Selection of rock avalanche parameters for forward analysis can be guided by the probabilistic risk analysis framework proposed here.  If the rupture surface is formed by a continuous structural feature then the flexible block model and a volume-dependent friction angle, should be used in the source zone.  The rheology and parameters used along the path should be selected based on the path materials.  4. An example forecast of the runout of a rock avalanche that likely overran sedimentary substrate materials was provided.  It was found that the predicted exceedance 229  probabilities well describe the observed runout, however, the exceedance probability contours below 0.05 corresponded to extremely conservative forecasts.      8.5 Analysis of the Johnsons Landing Debris Avalanche A review of the application of equivalent fluid models to simulate debris avalanches revealed that most researchers and practitioners have used either the frictional or the Voellmy rheology to simulate debris avalanche motion.  These rheologies assume some component of frictional resistance to motion, and would be inappropriate for a mass moving in an undrained condition.  In Chapter 6, a back-analysis of an unusual debris avalanche that occurred in British Columbia in 2012 was presented.  This analysis showed that not all debris avalanches can be accurately modelled with the frictional or Voellmy rheologies.  The main findings from Chapter 6 are: 1. The Johnsons Landing debris avalanche can be simulated using the Voellmy rheology, however, good results can only be attained if interaction with forest dramatically increased the resistance experienced by the flow. 2. A theoretical analysis of debris flow interaction with forest, which accounts for tree breakage and entrainment, revealed that these two mechanisms can influence flow dynamics, but not substantially enough to justify a dramatic increase in flow resistance. 3. The new liquefied rheology was able to well reproduce the observed deposit distribution of the Johnsons Landing event without the need for ad-hoc assumptions regarding the interaction of forest with the debris avalanche.  This is likely due to the fact that the liquefied rheology does not predict increased basal resistance due to centripetal accelerations, and that the constant yield stress enables deposition on moderate slopes if the flow depth is thin. 230  4. It was suggested that the Johnsons Landing debris avalanche is a transitional event between debris avalanches and flowslides.  Not all debris avalanches will be accurately simulated using the liquefied rheology, and the appropriate choice of rheology likely depends on the material composing the debris avalanche. 8.6 Analysis of Two Flowslides in Fine Grained Colluvium As summarized in Chapter 7, flowslides are defined by liquefaction of a portion of the source volume in the source area.  A review of liquefaction literature showed that two material types are commonly recognized as liquefiable.  These are loose granular soils and sensitive clay.  A back-analysis of a coal mine waste dump flowslide was presented to demonstrate a flowslide case history. Two flowslides, neither of which contained sensitive clay or loose granular soil, were then presented.  These flowslides both contained a significant portion of colluvium derived from overconsolidated glaciolacustrine silt and clay.  It was found that parameterizing this colluvium using the new liquefied rheology well reproduced their observed runout.  This result was used to infer that the fine- grained colluvium likely liquefied.  This is consistent with the findings of  Fletcher et al. (2002), who hypothesized that reworked material infilling the space between disaggregated, intact blocks of colluvium can liquefy, leading to brittle failure of the entire colluvial accumulation. This material is not conventionally recognized as liquefiable. 8.7 Recommendations for Future Work Several limitations and opportunities for future work have been identified based on the results presented in this thesis.  These include: 231  1. Further research into the rock avalanche disintegration process could be conducted.  This could result in an automatic criterion to model rock avalanche disintegration, as opposed to the current implementation where disintegration occurs instantaneously at a user-specified distance. 2. Although some guidance has been given, a more objective criterion to select standard deviations for model calibration could be developed.  The quantification of errors that result from the model being an imperfect simulation of reality, as well as possible correlations between the errors associated with different features, could be further explored.    3. New calibration constraints could be incorporated into the calibration framework presented here.  In particular, landslide force histories derived from seismic signals could be used as an additional calibration constraint (Favreau et al., 2010; Schneider et al., 2010; Moretti et al., 2012, 2015; Allstadt, 2013; Hibert et al., 2015; Iverson et al., 2015). 4. Further research into the role of path materials in rock avalanche motion should be conducted.  The results of this work show that there is significant variation in the basal resistance experienced by rock avalanches when they override sediments.  It is hypothesized that better characterization of the path materials will lead to an explanation of this variance.     5. The conclusions regarding the mobility of rock avalanches that overrun bedrock can be strengthened by using the database collected by Whittall et al. (2017).  Dynamic analysis of the cases classified as ‘fresh, strong rock’ can be used to test the hypothesis that the strength in the source zone can be reduced by a volume-dependent mechanism; however, strength along the path is close to that expected for dry, fragmented rock. 232  6. Further research is needed to better understand how to weight cases when they are being combined for use in forward analysis.  Research into the role of path materials will likely lead to a better understanding of how to select case weightings. 7. The rock avalanche runout predictions summarized in this thesis use a simple method to sample the path parameter probability distribution.  A rigorous Monte-Carlo approach could be implemented, and the results compared to the simple approach used here.  Future work in this area should focus on developing simple methods that can be used by practitioners.  8. Future work could explore the sensitivity of model results to the assumed failure sequence.  In particular, future work could explore whether the calibrated rock avalanche parameters are sensitive to the assumption that failure occurs in one dominant phase.  9. Future work could develop methodologies to simulate the splash zones that surround many extremely rapid, flow-like landslides, which are often associated with substantial damage. 10. The application of constant yield stress rheologies, such as the Bingham rheology, to debris avalanches should be explored.  It is expected that these rheologies will be appropriate for cases that do not have a significant portion of coarse grained material, and that have their mobility enhanced through rapid undrained loading of path material.   11. The results presented in this thesis support the hypothesis that some fine grained colluvium can liquefy.  Further research into the mechanics of this process is needed to develop criteria to predict whether a colluvial mass can liquefy.   12. The calibration and probabilistic runout procedures detailed in this work for rock avalanches could be applied to other classes of extremely rapid, flow-like landslides, such 233  as debris flows.  Different fitness metrics than the ones described here may prove more useful when back-analyzing different event types. 8.8 Conclusion This work was undertaken to better understand and predict the motion of rock avalanches, debris avalanches and flowslides.  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(2011) to be 370,000 m3, well below the volume threshold proposed by many researchers for long runout rock avalanches (Scheidegger, 1973; Johnson et al., 2016).    Figure A-1: Overview of the Crammont Rock Avalanche.  Figure from Deline et al. (2011), distributed under Creative Commons Attribution 3.0 License. 261  The Crammont rock avalanche initiated as a planar failure of a rock spur located on the north face of Mont Crammont Alps (Deline et al., 2011).  Mont Crammont is composed of two main formations; a conglomerate with schistose interbedding and a unit composed of alternating beds of limestone, calcareous and phyllitic schists and mollasic sandstones (Deline et al., 2011).  Both pre and post event LiDAR is available for this event.  The rupture surface was clear of slide debris following the event, allowing for an accurate estimate of detachment volume based on differencing of the pre and post event LiDAR.  Deline et al. (2011) estimate a total detachment volume of 535,000 m3.  As the post event LiDAR was collected in 2009, the volume estimate includes both the initial detachment as well as a significant amount of material that was destabilised and unravelled after the initial rock avalanche Alps (Deline et al., 2011).  This rock avalanche overrode and entrained a significant amount of snow, however, the snow had melted out of the rock avalanche deposit at the time the post-event LiDAR was collected. The deposit consists of two main zones shown in Figure A-1.  The first zone contains a large deposit located at the toe of the slope and consists of deposits derived from the rock avalanche as well as a significant volume of material that unravelled after the main detachment (Deline et al., 2011).  The bulk of the rock avalanche material deposited in this zone (Deline et al., 2011).  The second zone consists of the deposit in the torrent gullies.  The deposit in this zone is thin, and represents a small percentage of the total volume of failed material (Deline et al., 2011).   Deline et al. (2011) provides a Dan3D analysis of the Crammont rock avalanche.  This analysis uses two different rheologies; a frictional rheology in the upper part of the path and a Voellmy rheology in the lower part of the path, corresponding to the rock avalanche overrunning and entraining significant quantities of snow.  Deline et al. (2011) found that, if all of the detachment 262  volume is released at once, the deposit at the toe of the slope is poorly simulated.  The simulation results were improved by initially releasing 370,000 m3 of material and using a 30 degree bulk friction angle in the upper part of the path and a friction coefficient of 0.17 and turbulence coefficientof 1000 m/s2 in the lower part of the path.  Following the simulation of the initial release, the rest of the volume was released using a bulk friction angle of 37 degrees in order to simulate unravelling of the rock mass destabilized by the initial rock avalanche.  Deline et al. (2011) note that simulating the unraveling of the rock mass as a single event is a major simplification of reality as it is likely that this occurred during many small failures.   The Crammont rock avalanche was reanalysed for two reasons.  The first is that it appears as though the topography used by Deline et al. (2011) was not smoothed.  In order for the parameters derived from this analysis to be comparable to the other cases analysed in this work, the topography must be smoothed.  The second reason for reanalyzing this case is that Deline et al. (2011) did not report any results about uncertainties associated with the best fit basal resistance parameters.  By using the calibration techniques described in Section 4.4 to analyse this case, the variance associated with the best fit basal resistance parameters can be explored. For the analysis presented here, the same parameterization as Deline et al. (2011) is used.  The friction angle in the upper zone was kept constant at 30 degrees, as this friction angle adequately reproduces the deposit in the upper zone.  A sensitivity analysis was run in order to explore the variance surrounding the two parameters that govern the Voellmy rheology used in the lower zone.  The process used for running the sensitivity analysis is explained in Section 4.4.  Both the trimline and deposit distribution were used as constraints for the back-analysis.  No velocity estimates are available for this event, so a velocity constraint was not used.     263  The results of the sensitivity analysis are summarized in Figure A-2 and Figure A-3.  The best fit parameters are similar to those found by Deline et al. (2011), although the turbulence coefficient is lower.  This lower turbulence coefficient is likely due to smoothing the topography.  The analysis presented here has supported the important conclusion of Deline et al. (2011) that snow can enhance the mobility of small volume rock avalanches.  It is likely that, without the presence of snow, the rock avalanche would not have travelled down the two torrents.  Only a small portion of the total volume overrode the snow cover and demonstrated high mobility; the majority of the failed volume deposited in the proximal part of the path.  A-priori prediction of the location of the material change is challenging, and assuming that the entire failed volume would have its mobility enhanced by snow is over conservative.           Figure A-2: Crammont fitness number plot; the best fit parameters are friction coefficient of 0.17 and turbulence parameter of 300 m/s2.     264   Figure A-3: Best fit Crammont results. The standard deviation of the trimline fitness metric used to derive a PDF for this case is shown in Table A-1, and corresponds to half the lowest simulated model residual (Section 4.3.5).  The resulting PDF is shown in Figure A-4.   Table A-1: Standard deviation for the trimline fitness metric used in the Crammont case history. Fitness Metric Standard Deviation Trimline 1250 265    Figure A-4: Posterior probability density for the Crammont case history A.2 Huascaran The tragic Nevados Huascaran rock slide- debris avalanche (using the