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Statistical estimation and prediction of avalanche activity from meteorological data for the Rogers Pass… Salway, Anthony Austen 1976

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STATISTICAL ESTIMATION AND PREDICTION OF AVALANCHE ACTIVITY FROM METEOROLOGICAL DATA ' for the Rogers Pass area of Br i t i s h Columbia by ANTHONY AUSTEN SALWAY M.Sc., University of Birmingham, England, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGRES OF DOCTOR OF PHILOSOPHY in The Faculty of Forestry (Interdisciplinary) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Apri l 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Co lumbia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and study . I f u r t h e r agree t h a t p e r m i s s i o n fo r e x t e n s i v e copy ing o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copy ing or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of F o r e s t r y The U n i v e r s i t y of B r i t i s h Columbia 20 75 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date April 2 6 , 1976 i l ABSTRACT The p r e d i c t i o n o f avalanche a c t i v i t y , by o b s e r v e r s i n the f i e l d , i s l a r g e l y a c h i e v e d a l o n g c a u s a l - i n t u i t i v e l i n e s , depending f o r i t s s u c c e s s upon the e x p e r i e n c e o f the o b s e r v e r i n h i s own p a r t i c u l a r a r e a . V a r i o u s attempts have been made i n the p a s t t o q u a n t i f y such p r o c e d u r e s u s i n g p r e d i c t i v e models based upon m e t e o r o l o g i c a l measurements. M o d i f i e d f o r m s ' o f a m u l t i v a r i a t e s t a t i s t i c a l t e c h n i q u e known as l i n e a r d i s c r i m i n a n t a n a l y s i s , have been t r i e d (Judson and E r i c k s o n (1973). B o i s e t a l . (1974) and B o v i s (1974)) w i t h o n l y p a r t i a l s u c c e s s . The n o n - i n c l u s i o n o f time l a g decay terms, a u t o c o r r e l a t i o n s i n the d a t a , i n s u f f i c i e n t v a r i a t i o n i n the dependent v a r i a b l e and sampling d i f f i c u l t i e s , combine to weaken the d i s c r i m i n a n t a p p r o a c h . These problems and the n a t u r e o f the phenomenon suggest t h a t a time s e r i e s approach i s r e q u i r e d . A c o m p l e t e l y f l e x i b l e system o f d a t a s t o r a g e , r e t r i e v a l and computer a n a l y s i s has been d e s i g n e d t o f a c i l i t a t e the development o f time s e r i e s models f o r p r e d i c t i n g avalanche a c t i v i t y from m e t e o r o l o g i c a l o b s e r v a t i o n s f o r the Rogers Pass a r e a o f B r i t i s h C o l u m b i a . These methods i n v o l v e a u t o r e g r e s s i v e i n t e g r a t e d moving average (ARIMA) s t o c h a s t i c p r o c e s s d e s -c r i p t i o n t e c h n i q u e s , as w e l l as t r a n s f e r f u n c t i o n and s t o c h a s t i c n o i s e i d e n t i f i c a t i o n and e s t i m a t i o n p r o c e d u r e s . Such methods not o n l y o p t i m i z e the s e l e c t i o n o f the most a p p r o p r i a t e i n t e r c o r r e l a t e d independent v a r i a b l e s f o r model development, b u t a c t u a l l y e x p l o i t these i n t e r c o r r e l a t i o n s to c o n s i d e r a b l e a d v a n t a g e ; A n u m e r i c a l w e i g h t i n g scheme was d e v i s e d f o r the r e p r e s e n t a t i o n o f avalanche a c t i v i t y i n t e r m s . o f t e r m i n u s , s i z e and m o i s t u r e c o n t e n t codes i i i f o r each event. Various types of c o r r e l a t i o n analysis were performed on the data f o r the period, 1965-73» * n which the r e l a t i o n s h i p between avalanche a c t i v i t y and a comprehensive set of simple and complex meteoro-l o g i c a l variables was examined. Models were then developed f o r i n d i -vidual years and the entire period, using the three best weighting schemes f o r avalanche a c t i v i t y representation, and the most promising meteorologi-c a l variables, as indicated by the r e s u l t s of the c o r r e l a t i o n analyses. Multiple c o r r e l a t i o n c o e f f i c i e n t s as high as 0.87, using a simple two-term model, based on a composite s e r i e s , involving snowpack depth, water equivalent of new snow and humidity, have been obtained f o r i n d i v i d u a l years, and as high as 0.81, using a single six-term model consisting of only two composite meteorological ser i e s , f o r the entire period. Predic-t i o n p r o f i l e s , plotted from these models, indicate that a high l e v e l of forecasting accuracy could be possible i f such models are f i t t e d to future years. A simulated forecast was performed on data f o r the period, 1969-73* using a model developed f o r the period, I965-69, with a multiple corre-l a t i o n c o e f f i c i e n t of O.83. A value of O.76 was r e a l i z e d f o r the simulated forecast i n d i c a t i n g a high degree of precision. During t h i s study, great emphasis was placed on keeping the procedures general, rather than s p e c i f i c , so that, besides producing an accurate evaluation of the avalanche hazard at Rogers Pass, i t would also be possible to successfully apply such methods to other areas which have an avalanche problem. i v TABLE OF CONTENTS Page LIST OF TABLES v i i LIST OF FIGURES . . . v i i i ACKNOWLEDGMENTS ix Chapter I INTRODUCTION 1 Geographical Considerations 1 Avalanche Hazard Evaluation and Control . . . . . . 2 Development of Models 7 II STATISTICAL EVALUATION OF AVALANCHE HAZARDs A REVIEW 9 Delineation of the Most Significant Meteorological Factors 9 Discriminant Analysis 11 Summary and Conclusions . 19 III AVALANCHE ACTIVITY AS A DEPENDENT VARIABLE 20 Field Observations of Avalanche Occurrences . . . . 20 The Avalanche Data F i l e 22 The Avalanche Activity Index 22 IV METEOROLOGICAL FACTORS AS INDEPENDENT VARIABLES 27 The Meteorological F i l e 27 Depth of Snowpack . 27 New Snowfall 28 Precipitation 28 Air Temperatures 31 V Chapter P aS e Wind 32 Cloud Cover 33 Humidity 33 The Shear Test 34 Settlement 36 Wind Terms 36 Other Terms . 36 Air Temperature Gradient 37 Temperature Gradient within Snowpack 37 Density 38 Further Terms . 38 V CORRELATION ANALYSES 40 Identification of Dates 41 Independent Variables . . . . 43 East-West Division of Data 5* Selection of Best Weights for Avalanche Activity 52 Revision of Dates 57 A r t i f i c i a l and Natural Avalanches 59 Reduction of Sites 60 Storm Periods 62 Twice Daily Data 63 Individual Years 63 VI TIME SERIES ANALYSIS . . . 6 9 Stochastic Processes . . . . . . . . . 69 Transfer Function Representation . . . . 75 v i Chapter * Page Computational Procedures . . . . . . . . . . 79 VII THE TIME SERIES MODELS 82 SWH Model for 1967-68 82 Models for Individual Years 84 SWH,W.E*TMI Model for 1965-73 85 Confidence Limits 99 Simulated Forecast . . . . . 104 Decomposition of Avalanche Activity 106 Domain Analysis . 107 VIII CONCLUSIONS 108 BIBLIOGRAPHY 110 APPENDIX A AVALANCHE SITES ON THE TRANS CANADA HIGHWAY AT ROGERS PASS 114 APPENDIX B THE AVALANCHE SITES 115 APPENDIX C STORM PERIODS . . . . . 118 v i i LIST OF TABLES Table Page I Avalanche Activity Index Weighting Schemes . . . . . . . 21 II Variables Treated as Independent . . . . . 29 III Starting and Finishing Dates kZ IV Correlation Analysis, Rogers Pass Meteorological Data, Period 1965-73, Daily Observations, (12,4,2) Weights, SML, AN 45 V Total Annual Avalanche Activity, (12,4,2) Weights, SML AN, Versus Total Annual Snowfall and Water Equivalent Using Rogers Pass Meteorological Data 47 VI Avalanche Activity Index Weighting Schemes Used i n the Analysis 53 VII Reduced Set of Correlations for Various Weighting Schemes, Rogers Pass Meteorological Data, Period 1965-73, Daily Observations, A l l Sites, AN . 55 VIII Revised Starting and Finishing Dates 58 IX .Reduced Set of Correlations for Various Subsets of Avalanche Activity, Rogers Pass Meteorological Data, Period 1965-73, F i r s t Parts, Daily Observations, (1,1,1) Weights, ML 60 X Total Annual A r t i f i c i a l and Natural Avalanche Activity . . 6l XI Reduced Set of Correlations for Individual Years, Rogers Pass Meteorological Data, F i r s t Parts, Daily Observations, A l l Sites, AN 65 XII Time Series Models for Avalanche Activity, Daily Observations, Rogers Pass Meteorological Data, F i r s t Parts, A l l Sites, A r t i f i c i a l and Natural Avalanches . . . . . . . . . . . 86 v i i i LIST OF FIGURES Figure P a g e 1. Avalanche Hazard Forecast , . 8 2. Avalanche Activity Prediction Profile for 1967-68 based on the Model AVAL = 13.08 SWH + 4.010 S'wHl 94 3. Avalanche Activity Prediction Profiles for I965-73 Based on the Single Model, AVAL = .0947 AVALl + .0589 AVAL2 + 6.85 SWH - .937 SWHl + .131 W.E*TMI - .035 W.E*TMI2 100 ACKNOWLEDGMENTS The avalanche a c t i v i t y and meteorological data, which formed the basis of these studies, were made available by kind permission of Parks Canada. Credit for the acquisition of these data, which are of the highest quality, goes to the avalanche analysts, V.G. and W.E. Schleiss, who have been in charge of the Snow Research and Avalanche Warning Section at Rogers Pass since 19&5. For their willing co-operation and frequently useful and constructive advice, the author wishes to express his sincere gratitude. The author Is indebted to his wife, Judy Moyse, who not only typed the thesis, but also coded most of the data for the computer, wrote the majority of the data handling and retrieval, programs and assisted with a great deal of the computer analysis. Special thanks are due to P.A. Schaerer, Senior S c i e n t i f i c Research Officer, NRC, for his many valuable criticisms and suggestions throughout the research program and the f i n a l thesis preparation. Further acknowledgment and gratitude i s extended to Dr. J.P. Demaerschalk, Dr. M.C. Quick, Dr. H.O. Slaymaker and Dr. R.P. Willington, for their assistance during these studies and careful reading of the manuscript. The author i s also indebted to Dr. R.I. Perla, with whom many stimulating and controversial arguments were pursued, in Calgary during the summer of 1975. Financial support throughout this program of research was provided via a teaching assistantship from the Faculty of Forestry, UBC, a project grant from the Centre for Transportation Studies, UBC, and the f i r s t X installment of a Transportation Development Agency Fellowship Award. Further necessary support during a particularly lean period was provided by Parks Ganada, for which the author Is most grateful. The map of "Avalanche Sites of the Trans Canada Highway at Rogers Pass" due to P.A. Schaerer, was included as Appendix A with his generous approval. 1 Chapter I INTRODUCTION Geographical Considerations The Rogers Pass, at an elevation of.approximately 4350 f t . , provides an Important east-west route through the Selkirk Mountains of E r i t i s h Columbia, via the Trans Canada Highway and the Canadian Pacific Railway. It i s also one of the most active avalanche areas in Western Canada. A combination of steep-sided mountains, a characteristic of the Selkirk Range, and heavy winter snowfalls, cause more than ninety major sites to affect the highway along a t h i r t y mile length, from the east gate of Glacier National Park to just beyond the west boundary. The greatest concentration of these sites exists between two narrow defiles formed by Mts. Tupper and MacDonald, just east of the Pass, and Mts. F i d e l i t y and Fortitude i n the western section (see Appendix A). The terrain and climate of the area have been described by Schaerer (1962:2-5), who categorizes the Selkirks as the northern extension of the middle alpine zone after Roch, "characterized by heavy snowfalls of moist to dry snow, medium temperatures only occasionally below zero degrees Fahrenheit and strong wind action on the mountains." Schleiss (1970:115) recognizes three different climate sub-zones for the area, stating that, "the west side of the park i s influenced by the Pacifi c weather systems, the east side by Arctic weather fronts and the clashing of both systems influences the weather in the central section." This rather complex meteorological situation necessitated the establishment of two major observatories, one at the Rogers Pass headquarters to monitor weather conditions for the 2 eastern section, and the other on Mt. Fi d e l i t y , at an elevation of 625O f t . to monitor the western section. These two observatories provide infor-mation on snowpack conditions on a continuous basis throughout the avalanche season, as an aid in the forecasting and control program. This information i s supplemented by a i r temperature, wind velocity and direction, and humidity data, which i s telemetered from two remote observatories, MacDonald West Shoulder (elevation 65OO f t , ) , located above the Rogers Pass, and Roundhill Station (elevation 6900 f t . ) , at Mt. Fi d e l i t y . Avalanche Hazard Evaluation and Control The Snow Research and Avalanche Warning Section (SRAWS), under the jurisdiction of Parks Canada and the leadership of the snow and avalanche analysts, V.G. and W.E. Schleiss, conducts an ongoing program of avalanche hazard evaluation and control for the Rogers Pass area. The operational objectives involve the maintenance of an optimum balance between minimum highway closure times and the safety of the public and parks personnel. This balance can only be achieved by the accurate evaluation of avalanche hazard, backed up by prompt action in the form of a r t i l l e r y control. Potential for Avalanchlng. The avalanche hazard evaluation i s based on an evaluation of the s t a b i l i t y of the upper, often new snow layers and the lower layers within the snowpack, combined with an assessment of the amount of available snow for avalanching at each s i t e . Ideally, s t a b i l i t y measurements should be made in the starting zone and avalanche track, but l o g i s t i c a l d i f f i c u l t i e s , inaccessibility and danger to the observer prevent th i s . Ski-tests are performed, however, whenever possible, on short slopes at high elevations, which are repre-sentative of conditions i n the slide paths. Such tests often reveal i n s t a b i l i t y i n the upper layers, when they fracture and move under the skier's weight, and hence provide a direct indication of i n s t a b i l i t y . More usually however, i t i s necessary to rely upon less direct structural measurements made at the study plot and indirect indicators i n the form of meteorological observations. The presence of a weak layer i n the new or p a r t i a l l y settled snow may be detected and some form of strength test applied. The amount of new snowfall, a i r temperature and wind w i l l also provide an indication of the s t a b i l i t y of the upper layers. The s t a b i l i t y of the lower layers within the pack can be inter-preted from current snow p i t data, or interpolated from past data. The analyst w i l l be aware of any deep-seated i n s t a b i l i t i e s within the pack, for example, a persistent surface hoar layer which has been responsible for several avalanche cycles so far that winter. Finally, to complete the evaluation, the analyst refers to his past records of avalanche a c t i v i t y to determine the a v a i l a b i l i t y of avalanchable snow for each particular s i t e on an individual basis. As LaChapelle (1970:108) has observed, "The hazard evaluation i s amenable to numerous refinements. For large avalanches f a l l i n g over long paths, the volume of snow apt to reach the valley floor can be estimated by taking into account the amount of unstable snow i n the middle and lower reaches of the path." For example, avalanches may recently have occurred at some sites resulting i n the removal of the upper unstable layers and perhaps also the lower layers, i f they were su f f i c i e n t l y unstable. Furthermore, at other sites, the lower layers may no longer be present, as a result of previous avalanching that occurred some time i n the past. Therefore, a complete h i s t o r i c a l record of avalanche a c t i v i t y at each s i t e , since the "beginning of the season, i s a necessary requirement for the determination of the amount of avalanchable snow l i k e l y to be available. Hence, an accurate evaluation of the potential for avalanching r e l i e s upon three factors, as depicted in Figure 1, the s t a b i l i t y of the upper layers, which w i l l often be trigger snow, the s t a b i l i t y of the lower layers, which may constitute the main mass of the avalanche released or set i n motion by the trigger snow and the a v a i l a b i l i t y of snow for avalanching at each particular s i t e . Avalanche Hazard Evaluation. The evaluation of avalanche hazard r e l i e s heavily, but not exclusively, on the evaluation of the potential for avalanching, as defined above. Consideration must also be given to the possible effect of such avalanching on human l i f e and property, which, in the case of the Rogers Pass, can be identified with the Trans Canada Highway. For example, the potential for avalanching on some sites may be extremely high, but these sites may not affect the highway, therefore the hazard to the highway would be low. In areas other than the Rogers Pass, hazard might perhaps be identified with respect to skiers in relation to ski areas or back country travel, i n which case the hazard evaluation would be different. Avalanche Hazard Forecast. Finally, the avalanche hazard evalu-ation can be combined with the weather forecast to produce an avalanche hazard forecast, which may be either short term or long term, depending on the nature of the weather forecast. Figure 1 summarizes the important steps in the evaluation and forecasting procedures just described. At the Rogers Pass, operational decisions with regard to highway 5 closures are based upon the avalanche hazard evaluation. An avalanche hazard forecast, in the s t r i c t sense, i s seldom, i f ever, attempted due to the u n r e l i a b i l i t y of mountain weather forecasting data. However, a current evaluation i s a l l that i s generally required as a basis for operational decisions. As LaChapelle (1970:107) points out, "The hazard evaluation seeks to ascertain current snow s t a b i l i t y . It i s the basis on which operational decisions (road closures, control measures, etc.,) are most often made. This i s the most common function and the one which i s usually labelled 'avalanche forecasting' i n the loose sense." Avalanche Control. Hence, the avalanche hazard evaluation may lead directly to a decision with regard to the possible closure of the highway, after which a r t i l l e r y control measures may be implemented. From various established gun positions alongside the highway, 105mm howitzer s h e l l f i r e i s directed at predesignated target areas, usually trigger zones which are generally situated above the main avalanche starting zones. These trigger zones often consist of small localized deposits of highly unstable snow, the release of which loads the lower slopes causing them to avalanche. Whenever possible, such 'sta b i l i z a t i o n shoots', as they are ; called, are implemented before large buildups of snow have occurred in the starting zones and slide paths, so that any avalanches which result do not reach unreasonable proportions, (such occurrences are referred to as " a r t i f i c i a l ' avalanches as opposed to 'natural' avalanches which take place without human intervention). Minimizing the occurrence of large avalanches i n this way not only decreases the hazard to the highway, but also reduces the times required for cleanup operations. Ideally however, controlled avalanches should be of a significant size, resulting in a substantial reduction of snow i n the accumulation zones. The major benefit of the avalanche control program l i e s not so much i n the reduction of avalanche size, but in the fact that any a r t i -f i c i a l avalanches which occur as the result of stabilization procedures, do so under re l a t i v e l y safe conditions during periods of highway closure. Furthermore, a rigorous control program, executed during the height of the season, by preventing excessive buildup of snow i n the avalanche paths, effectively cuts down the size and number of the more dangerous and unpredictable wet snow avalanches that take place i n the spring. These spring avalanches are not amenable to a r t i l l e r y control due to the damping effect of wet snow which severely limits the propagation of the explosive energy through the snowpack. Purpose of this Study. Returning to the problem of hazard evaluation, i t i s not always possible to identify periods of i n s t a b i l i t y i n t u i t i v e l y , and even those that are identified may be of short duration, i f , for example, the snow i s settling rapidly after a heavy snowfall. The time interval between the decision to perform a r t i l l e r y control and the f i r i n g of the f i r s t round may be such that the period of i n s t a b i l i t y i s missed. This study addresses i t s e l f to the problem of s t a t i s t i c a l e s t i -mation and prediction of avalanche a c t i v i t y from meteorological data using analytical techniques. Mathematical models have^een developed which describe the phenomenon i n terms of the s t a t i s t i c a l behaviour of past data. It i s hoped that such models w i l l ultimately be used, by the avalanche analyst, as an important aid along with the other somewhat intuitive approaches, to enable him to more accurately evaluate the 7 hazard situation, and identify and predict periods of i n s t a b i l i t y with greater certainty. Besides providing e f f i c i e n t working models, i n the operational sense, the application of s t a t i s t i c a l methods to this complex problem should also eventually help to reveal the physical processes which govern the formation of avalanches. Development of Models A completely f l e x i b l e system of data storage, retrieval and com-puter analysis has been designed to f a c i l i t a t e the development of simple or complex time series models involving auto-regressive integrated moving average (ARIMA) process description techniques, as defined by Box and Jenkins (1970), as well as transfer function and stochastic noise iden-t i f i c a t i o n and estimation procedures. These methods not only f a c i l i t a t e the optimum selection of intercorrelated independent variables, but actually exploit these intercorrelations to considerable advantage.* A suite of computer programs was written i n FORTRAN and thoroughly tested using avalanche and meteorological data for the period, 1965-73. The data were then systematically analysed and the best forecasting models deve-loped, both for individual years and for the entire period. Multiple correlation coefficients as high as 0.87, using a simple two-term model have been obtained for individual years, and as high as 0.81, using a * The usual backwards, forwards, or stepwise selection procedures, employed in normal least squares regression and discriminant analysis, break down i f strong intercorrelations exist among the independent variables (Draper and Smith,1966:163-195). As Judson and Erickson(l973), Bois, Obled and Good(l974), and Bovis et al . ( l 9 7 4 ) have discovered, such conventional approaches can lead to complicated but rela t i v e l y weak models, peculiar to the particular data sets analysed, consisting of large numbers of inter-related unlagged meteorological terms, many of which are only just s i g n i -ficant. 8 Figure 1 AVALANCHE HAZARD FORECAST STABILITY EVALUATION OF UPPER LAYERS AVALANCHING STABILITY EVALUATION EVALUATION OF POTENTIAL FOR AVALANCHING AVALANCHE HAZARD OF LOWER LAYERS — • / EVALUATION / ESTIMATION OF POTENTIAL FOR AVALANCHES TO AFFECT / WEATHER FORECAST AVAILABLE SNOW FOR HUMAN LIFE AND PROPERTY AVALANCHE HAZARD FORECAST single six-term model, consisting of only two meteorological series of a composite nature for the entire period. Prediction profiles using these models have been plotted and a high degree of accuracy can be demonstrated. During this study, great emphasis was placed on keeping the procedures general, rather than specific, so that besides producing an accurate evaluation of the avalanche hazard at Rogers Pass, i t would also be possible to successfully apply such methods to other areas which have an avalanche problem. 9 Chapter II STATISTICAL EVALUATION OF AVALANCHE HAZARD: A REVIEW Delineation of the Host Significant Meteorological Factors Atwater's Precipitation Intensity Term. Atwater (1952), was among the f i r s t to recognize the importance of precipitation intensity, P.I., measured on an hourly basis, as an "excellent indicator of avalanche hazard" (Atwater, 1952:17). Based on studies at three stations: Alta i n Utah, Stevens Pass in Washington and Berthoud Pass in Colorado, he was able to devise the following 'rule of thumb', "P.I. continuously above 0.10 i n . per hour, at wind velocities 15 mph or over and i n the absence of s l u f f cycles equals a high degree of avalanche hazard whenever total precipitation i s one inch" (Atwater, 1952:18). Perla's Contributory Factors in Avalanche Hazard Evaluation. Perla (1970) investigated twenty years of storm and ramsonde pr o f i l e data mea-sured at Alta, Utah for the period 1950-69, considering only large avalanches on south facing slopes. After performing a contributory analysis, he found that, "the probability of an avalanche hazard varies considerably with precipitation and wind direction, only s l i g h t l y with temperature change, and seems to have no definite relationship to wind speed and snow settlement"(Perla, 1970:418). Hence, while there i s a concensus of opinion on the importance of precipitation, the role played by wind speed or direction i s less clearly defined. However," the greater influence of wind direction compared to wind speed may simply be a conse-quence of the uniform orientation of the set of avalanche sites studied by Perla. Judson's Univariate Analysis. Judson and Erickson (1973) con-ducted a univariate analysis similar to Perla's (1970) analysis of con-tributory factors i n avalanche hazard evaluation. This analysis was per-formed on twenty-three avalanche paths, nineteen of which were controlled by explosives, located in the Central Rockies of Colorado's Front Range 2 near Berthoud Pass, the Urad Mine and Loveland Pass. Seven winters of data (1963-70) were used but the analysis was restricted to storm periods only. Simple linear regression analysis was applied using the number of avalanches from the twenty-three paths as the dependent variable and single weather factors or simple combinations of them as independent variables, in an attempt to identify the most significant terms. The four factors so identified were 24-hour water equivalent, 24-hour snow-f a l l , maximum precipitation intensity and maximum precipitation intensity modified for excessive wind. "The factor best correlated with avalanche a c t i v i t y was the sum of the maximum precipitation intensities multiplied by a constant for excessive wind speed" (Judson and Erickson, 1973*2), which was termed the "storm index". Recognition of Need for a Time Series Approach. Although the storm index seemed quite promising, Judson and Erickson (1973:4) saw the need for a time series approach, "The main drawback with the storm index i s that the index i s highest near the end of storms, even though 2 A separate analysis was performed on twenty-three uncontrolled avalanche paths which resulted i n weaker correlations, implying that, "data from uncontrolled paths are d i f f i c u l t to interpret and are less reliable as forecast guides." (Judson and Erickson, 1973:4) 11 hazard may be decreasing because some avalanches have already f a l l e n and the snow i s s t a b i l i z i n g . A way of reducing the index toward the end of the storm (a decay function) i s badly needed and i s now under study." Discriminant Analysis The Linear Discriminant Analytical Procedure. Various attempts have been made, notably by Judson and Erickson (1973)» Bois, Obled and Good (1974), and Bovis (1974), to produce forecasting models for avalanche occurrences using modified forms of a multivariate s t a t i s t i c a l technique known as linear discriminant analysis. This procedure, which i s closely related to linear regression analysis, involves, in i t s simplest form, the assignment of 'cases' into one or other of two groups, using a linear discriminant function. The function consists of a linear combination of independent variables, multiplied by appropriate coefficients, which are least squares estimates, obtained by maximising the ratio of the between groups variance to the within groups variance (Rao, 195 2). After obtaining the function, the mean value of the discriminant for each group may be calculated by substituting the group mean values of each independent variable into the function. The difference between the two mean values of the discriminant i s known as the generalized or Mahalanobis distance, and the average of the two multivariate group means, known as the discriminant index, serves as a cri t e r i o n for the c l a s s i f i -cation process. Significance tests may be performed on each independent variable and the Mahalanobis distance. A 'probability of misclassification* may be obtained by comparing the value of the Mahalanobis distance with a cumulative normal frequency distribution table of the normal deviate. The method i s capable of extension into three or more group classi f i c a t i o n s , 12 i n which case two or more discriminant functions are required to be c a l -culated . Judson's Discriminant Analysis. After i d e n t i f y i n g t h e i r most s i g n i f i c a n t v a r iables, Judson and Erickson next performed a multivariate l i n e a r discriminant analysis using eight controlled s i t e s on an Individual basis, and data fo r the period, 1952-71. Group c l a s s i f i c a t i o n s were based on control r e s u l t s . Days were assigned to group 1 when control e f f o r t s produced a s l i d e , or when a natural avalanche occurred, and to group 2, when control e f f o r t s f a i l e d to i n i t i a t e an avalanche. Discriminant functions were developed f o r each s i t e containing the following terms: (1) a p r e c i p i t a t i o n term made up of the sum of the maximum consecutive 3-hour p r e c i p i t a t i o n i n t e n s i t i e s within each 6-hour period decayed over an i n t e r v a l . "The function i s held a t one for the f i r s t 2 days, reaches 0.5 on the 5th day, and l e v e l s o f f a t 0.2 from the 9th day on." (Judson and Erickson, 1973:10), (2) a temperature term consisting of the sum of the 6-hour negative temperature departures from 20°F, (3) a wind term made up of the sum of the wind speeds greater than or equal to 15 mph resolved to an optimum d i r e c t i o n f o r each path. P r o b a b i l i t i e s of m i s c l a s s i f i c a t i o n ranged from 21 to 30%. Problems with Judson's Discriminant Analysis. Judson and Erickson's models are useful i n that they indicate which of the meteorological factors are most s i g n i f i c a n t . However, they are f a r too weak to be used i n a r e a l s i t u a t i o n f o r avalanche hazard evaluation f o r the following reasons: (1) Lagged Variables. The functions r e l y e x c l u s i v e l y on current weather factors, although an attempt was made to introduce c e r t a i n a r b i -t r a r y decay terms to overcome t h i s d e f i c i e n c y . Perla and Judson (1973) 13 have investigated the po s s i b i l i t y of introducing fading memory terms, without arbitrary factors, into the discriminant analysis procedures. However, discriminant analysis does not readily lend i t s e l f to time series applications. A stochastic transfer function time series approach, on the other hand, i s far superior in that i t automatically involves lagged values of precipitation, temperature and wind terms, the coefficients of which are least squares best estimates determined from the actual data. (2) Intercorrelated Variables and Autocorrelated Data. Strong intercorrelations between the Independent variables, a normal feature of weather data, are not handled well by conventional regression methods l i k e discriminant analysis. Furthermore, the meteorological time series are usually quite strongly autocorrelated, or i n other words, adjacent observations i n time are not independent. Such intercorrelations and the interdependence of observations adjacent i n time are regarded as an undesirable feature of the data, in a conventional regression situation, resulting i n the interference of normal variable selection procedures such as 'forwards selection' and •backwards elimination*. This leads to models which do not necessarily contain the 'best' set of independent variables. Time series analysis procedures, on the other hand are designed to operate on observations which are dependent and, "where the nature of this dependence i s of interest In i t s e l f " (Box and Jenkins, 19?0:vii). The time series approach exploits these intercorrelations to the f u l l e s t advantage, producing much more powerful models, containing an optimum selection of lagged and unlagged meteorological terms. Furthermore, such 14 models, i f developed for separate years, tend to display greater simi-l a r i t y than discriminant functions, which are often uniquely different. Model similarity between years i s , of course, a desirable feature i f the prediction of a c t i v i t y for future years i s contemplated, (3) Variation of the Dependent Variable. The assignment of a l l avalanche days, whether the level of a c t i v i t y i s high or low, into one class i s bound to lead to weak models. It i s far better to treat avalanche a c t i v i t y as an ordinary dependent variable, allowing i t to take . on values corresponding to various levels of a c t i v i t y , thereby more accurately reflecting the changing meteorological conditions which give r i s e to the. phenomenon, (4) Data Imbalance between Avalanche Days and Non-Avalanche Days. A further undesirable feature of discriminant analysis,in i t s application to the avalanche forecasting problem,lies i n the imbalance between avalanche and non-avalanche days. There are usually far more non-avalanche days, which results in discriminant functions which are biased i n the direction of the non-avalanche group. Hence, a greater proportion of the avalanche days are misclassified than non-avalanche days. To overcome this d i f f i c u l t y , Judson and Erickson (1973) use a weighted average of the dis^ criminant means for each group as their discriminant index. This somewhat a r t i f i c i a l and unsatisfactory device causes the probabilities of misclassi-fication for avalanche and non-avalanche days to be approximately equal, but does l i t t l e to improve the overall c l a s s i f i c a t i o n scheme. Bois et a l . (1974), and later, Bovis( l974), try to overcome this 'zero imbalance' by a different device, which involves the selection of a random sample of non-avalanche days equal in number to the avalanche days. 15 However, tests using the Rogers Pass data have shown that ran-domly sampling non-avalanche days, in this fashion, gives r i s e to d i s -criminant functions which are significantly different for the same block of avalanche data. Ten runs were made using data for the period, 1972-?3» and avalanche occurrences at a single avalanche s i t e called 'Portal'. A random sample of non-avalanche days, equal in number to the avalanche days, was selected for each run. After backwards elimination, using the same i n i t i a l set of independent variables for each run, ten unique models were obtained, consisting of a minimum of two and a maximum of eight significant precipitation, temperature and wind terms, with probabilities of misclassification ranging from 7 to 2k%. Thus, the models appeared to be a function of the particular set of non-avalanche days, even though the sets were chosen randomly. Hence, such a procedure must be viewed with a great deal of scepticism. Bois' Discriminant Analysis. Bois, Obled and Good (1974) have analysed avalanche and meteorological data from the Parsenn area of Switzerland, for the period, 1961-70, r e s t r i c t i n g their analysis to natural occurrences only. They use a three-way discriminant analysis approach i n an attempt to distinguish between wet snow avalanche days, dry snow avalanche days and non-avalanche days. A single event, on any s i t e , serves to classify a day as an avalanche day. The ten-year sampling period was analysed on a monthly basis, for example, a l l Januaries in the ten-year sampling period were taken as the total population for that month. This procedure was adopted presumably on the assumption that similar conditions occur during the same month each year on a regular basis. This i s not generally the case since some winters may be more advanced than others on a particular date each year. 16 As previously mentioned, Bois et a l . (1974) select a random sample of non-avalanche days, approximately equal i n number to the avalanche days, in order, not only to eliminate the 'zero-imbalance', but also because, "this eliminates s e r i a l correlation between successive days" (Bois et a l . , 1974:7). It has already been pointed out that meteorological and avalanche observations adjacent in time are generally not independent, that i s to say, the series are autocorrelated. The auto-correlation functions of such series reveal a great deal about the processes involved and should certainly not be eliminated. Serial cor-relations should be exploited by use of proper time series procedures. Bois et a l , (1974) have documented the results of their analysis for March only, which indicate that, (1) height of settled new snow summed over precipitation sequence, (2) temperature at 1:00 P.M. on the previous day, plus 3'C, (3) the number of precipitation sequences (longer than 2 days) since the beginning of the winter, are the three most important variables for dry snow avalanche c l a s s i f i c a t i o n , and, (1) temperature at 1:00 P^ M. on the previous day, (2) the number of avalanche days in the test area per number of precipitation sequences, and, (3) absorbed radiation flux, are the three most important variables for wet snow avalanche c l a s s i f i c a t i o n . Probabilities of misclassification for both wet and dry avalanches for March using these variables were of the order of 19$. Bovls' Discriminant Analysis. Bovis (1974) has performed a s t a t i s t i c a l analysis of avalanche events along station 152 (Highway 550) 17 in the San Juan Mountains of southwestern Colorado for the 1972-73 and 1973-74 seasons, Bovls' approach is similar to that of Bois et a l . , in that he employs a linear discriminant analysis technique in order to discriminate between wet, dry and non-avalanche days. Random selection of a sample of non-avalanche days equal in number to the avalanche days is also used by Bovis. Bovis has however introduced two important refinements. First ly , avalanche events are stratif ied on the basis of magnitude for both the dry and wet seasons. Four magnitude classes of avalanche activity for the area are recognized. This is similar to a regression situation in which the dependent variable is allowed to take on any one of five values, including zero. As discussed previously, such a scheme, by decreasing the restrictions on the effective variation of the dependent variable is bound to result in stronger models. However, stratification of avalanche events in this way does unfortunately result in a reduction in the sample sizes. As Bovis (1974: 71) points out, "stratification on the basis of magnitude provides a variable operational definition of an avalanche day, although i t is constrained by considerations of sample size." If the sample sizes are too small, the discriminant analysis procedure breaks down. At least thirty cases are generally regarded as necessary to provide a good estimate of the group mean and variance. Hence, as Bovis has recognized, his data base is rather too small to produce reliable samples and hence discriminant functions from which any fundamental conclusions may be drawn. Spurious terras appear in his models, for example, "although the 18 Importance of variable 2 In the table 16 comparisons can be related to slope loading, the interpretation of a i r temperature i s less clear", (Bovis, 1974":8l) and, "no physical significance can be attached readily to variable 8 (mean wind speed during preceeding 24 hours) i n the three time integrations in table 17 since Its average value i s lower over the avalanche day group, indicating a higher wind-loading potential for non-avalanche days i n this instance" (Bovis, 1974V85). Of course, i f avalanche a c t i v i t y i s treated as a normal dependent variable and time series methods employed instead of discriminant analy-s i s , no such sampling problem exists. The second significant feature of Bovis' work i s his use of meteorological and snowpack parameters, integrated over two, three or f i v e days, as independent variables. This i s certainly one way to introduce the effects of past conditions into the models, rather similar to Judson's arbitrary decay terms, except that, in Bovis' analysis, each term, inte-grated over the time interval, i s equally weighted. These attempts further serve to i l l u s t r a t e the need, for a time series approach, in which the lagged variables appear as a necessary and elegant consequence of the procedures involved. It i s useful to compare Bovis' most significant variables with those of Bois et a l . and Judson and Erickson, previously quoted. For dry slides and for the 1972-73 winter, Bovis found that, (1) maximum 6-hour precipitation intensity i n the 24-hour period, (2) total precipitation over two, three or five days prior to the event, and, (3) certain temperature terms, were the most important factors 19 for the unstratifled events, and natural slides greater than or equal to magnitude 2. Overall.probabilities of misclassification were of the order of 35%. Sample sizes for wet slides were too small to provide useful indicators of significant variables. Summary and Conclusions In summary, i t i s f e l t that the time series procedures about to be described in this study of avalanche a c t i v i t y as a function of meteoro!-log i c a l parameters, are superior to the discriminant analysis techniques employed in the past, for the following reasons: (1) Lagged variables (decay terms), representing the effects of previous precipitation amounts, temperatures, winds, etc., can be intro-duced into the models conveniently and elegantly, in the most e f f i c i e n t manner. Discriminant analysis does not lend i t s e l f to the introduction of such time series terms. (2) Intercorrelated variables and autocorrelated data, a drawback in normal regression and discriminant analysis, can be exploited i n the time series approach, to produce the 'best' models in an optimum sense. (3) Avalanche a c t i v i t y i s treated as a dependent variable, and allowed to take on values corresponding to various levels of a c t i v i t y , i n unison with the independent variables. This results i n much more powerful models. (4) Problems related to small sample sizes of discriminant groups and the imbalance between avalanche and non-avalanche days are eliminated i f a time series approach i s employed. 20 Chapter III AVALANCHE ACTIVITY AS A DEPENDENT VARIABLE Fie l d Observations of Avalanche Occurrences Approximately one hundred active avalanche sites are recognized by the Snow Research and Avalanche Warning Section, and have been c l a s s i f i e d by name, number and mileage from the east boundary of Glacier National Park (see Appendix B). Natural occurrences are recorded generally on a twice daily basis, often after the event, according to a prescribed format. The site name, date, and i f possible, the time of occurrence i s noted, along with the observer's estimate of size, terminus and moisture content, indicated by the designations i n Table I. The continuous monitoring of such an inter-mittent phenomenon i s often aggravated by limitations on the a v a i l a b i l i t y of man power, high hazard and poor v i s i b i l i t y , particularly during periods of intense a c t i v i t y when observations are most needed. These problems, combined with the necessarily subjective nature of the measurements, set the l i m i t on the overall accuracy of the data and ultimately determine the l e v e l of random noise in the prediction models. A r t i f i c i a l occurrences are noted during the stabilization shoot and can therefore be timed r e l i a b l y when v i s i b i l i t y i s good. Size, terminus and moisture content are also recorded whenever possible. Both for a r t i f i c i a l s and naturals, size i s estimated relative to the actual size of the particular s i t e , either from the visual appearance of the s i t e and size of the deposit, i n the case of naturals after the event, or from a visual impression of mass and energy, i f the avalanche 21 Table I Avalanche Activity Index Weighting Schemes Designation (12,12,6) (12,4,2) (12,2,2) (12,3,1) (12,1,1) (1,1,1) SML SML ML SML SML ML Terminus or 1/3 path 1 . 1 - 1 1 1 1 j path 2 2 2 2 2 1 2/3 or 3/4 path 3 3 . 3 3 3 1 End path, to fan, or gully 4 4 • 4 4 4 1 £ fan 5 5 5 5 5 1 1/3 fan 6 6 6 6 6 1 i fan 7 7 7 7 7 1 2/3 fan 8 8 8 8 8 1 3/4 fan, Old HR, or Bench 9 9 9 9 9 1 Over fan, or Mounds 10 10 10 10 10 1 Edge TCH 11 11 11 11 11 1 Over TCH 12 12 12 12 12 1 Size Small 1 1 0 1 1 0 Medium 6 2 1 2 1 1 Large 12 4 2 3 1 1 Moisture Content Dry 1 1 1 1 1 1 Damp 3 1.5 1.5 1 1 1 Wet 6 2 2 1 1 ' 1 22 Is actually observed, as i s often the case with a r t i f i c i a l s . The terminus c l a s s i f i c a t i o n gives an indication of the farthest point reached by the avalanche, but does not include any information on the actual distance travelled from the starting zones. A low cloud base frequently obscures the starting zones, thereby preventing the point of origin or fracture line from being recorded. However, since Individual sites consistently avalanche from the same rupture area, often at the base of c l i f f s , the terminus does provide a good indication of distance travelled. The Avalanche Data F i l e The greatest overall accuracy that could reasonably be obtained from the records for natural event times was twice daily. Accordingly, therefore, both naturals and a r t i f i c i a l s were coded on a twice daily basis, along with size, terminus and moisture content, for the period, 1965-73. After sorting into two subsets of daily and twice daily observations, by si t e within date, the data was stored on a computer tape f i l e , ready for analysis. The Avalanche Activity Index Definition and Physical Interpretation. The f i r s t problem prior to the application of s t a t i s t i c a l techniques i s to devise a suitable index of avalanche a c t i v i t y which can be used as a dependent variable. In a p i l o t study,' based on data for the winter of l972-73» an "index of mass movement" was defined as the product of terminus, size, and moisture content for a particular event, after assigning arbitrary numerical codes of one to twelve for terminus, one to twelve for size, and one to six for moisture content, as outlined i n Table I, column 1 . The column heading 23 (12,12,6) SML w i l l be explained later. . It i s probable that this index i s a good measure of avalanche ac t i v i t y , since i t not only includes an indication of the size, and there-fore the amount of snow picked up from the lower zones after the i n i t i a l movement, but also an indication of the energy associated with the avalanche i n terms of distance travelled. However, both Schaererand Shimizu have shown that the logarithm of mass may be a more useful measure of avalanche size than mass alone. Shimizu (1967), i n fact, proposes and defines three measures of avalanche magnitude, (1) Mass Magnitude—the logarithm of the mass of avalanched snow, (2) Potential Magnitude—the logarithm of the product of mass and vert i c a l distance moved by the avalanche; a measure of potential energy, (3) Destructive Magnitude—the logarithm of the product of mass and the square of the sine of the slope angle divided by the square of a resistance coefficient; a measure of kinetic energy. However, the somewhat subjective assignment of avalanches i n the Rogers Pass area, into small, medium and large, by the f i e l d observer i s probably already i n t r i n s i c a l l y logarithmic, since, as Schaerer (I97i:2) points out, " i t has also been found that an experienced observer, using visual observations only, would usually assign avalanches to the same class," The index i s therefore similar to the "Potential Magnitude" measure proposed by Shimizu. Since the size of the avalanche i s estimated relative to the s i t e , the index does not provide an absolute estimate of the energy associated 2k with the avalanche. Some researchers regard an estimate of the absolute size and energy of an avalanche as a more meaningful measure of avalanche ac t i v i t y . The schemes of both Schaerer (1971) and Shimizu (1967) are based on absolute sizes and Perla (1976) c l a s s i f i e s avalanches according to their estimated destructive power on a scale of one to f i v e , i r r e s -pective of the size of the s i t e . However, such measurements are no less subjective than the relative size measurement, which may be easier to make. Besides, i f i n s t a b i l i t y i s regarded as more important than absolute size in the assessment of hazard, relative sizes may provide a better measure, since a small s i t e may never experience a large avalanche i n the absolute sense, no matter how unstable the snow. The question arises at this point as to whether the prediction of i n s t a b i l i t y or absolute size of avalanches i s the prime requisite. How-ever, this question can be resolved after consideration of the main purpose of procedures developed from this study, which i s to provide the avalanche control crew with an indication of the optimum time for the stabilization shoot. This surely coincides with the period of greatest i n s t a b i l i t y and therefore, models should be developed to predict insta-b i l i t y , rather than the absolute size of avalanches. Furthermore, i t i s l i k e l y that a measure of i n s t a b i l i t y i s more closely related to meteorological processes, since absolute size depends to a large extent on the topography of the area. Therefore, i t can be expected that greater success w i l l be obtained i n the development of prediction models, i f a measure of i n s t a b i l i t y based on relative avalanche size measurements i s used. Such models should also be more generally appli-cable to other areas possessing different terrain characteristics, but 2 5 similar meteorological conditions. Computation. The avalanche a c t i v i t y index, when computed for individual events, can be summed for a l l sites, or a group of sites, on a daily or twice daily basis, resulting in values of avalanche a c t i v i t y which vary smoothly and continuously, and therefore lend themselves to the successful application of multiple regression and time series techniques. LaChapelle (1970:106) recognizes that the greatest potential of the " s t a t i s t i c a l approach" l i e s in this direction, since he states, " i t Is most useful when dealing with hazard probabilities over large areas, where individual avalanches f a l l effectively at random, but the patterns of their occurrence i n time are related to snow and weather." Effects of random errors caused by individual s i t e peculiarities and the subjective nature of the data are minimized. Not only does this index contain a l l three basic characteristics of the avalanche measured i n the f i e l d , but also the relative contribution of each measurement can be altered by choosing a new set of weights. Numerical Convention for Representation. Since a multitude of weighting schemes have been used i n this study, i t i s necessary, at this point, to introduce a simple convention for their abbreviated represen-tation. In this convention, the weights, as outlined in Table I, column 1, are referred to as (12,12,6) weights, the f i r s t figure indicating the maximum terminus code, the second figure, the maximum size code, and the third, the maximum moisture content. Values of unity are usually assigned to a quarter path, small, and dry, and the weights are evenly distributed, between the other categories. Hence (12,4,2) weights indicate that terminus ranges from one to twelve, size ranges from one to four, and moisture content, from one to two, as shown i n Table I, column 2. If 26 small avalanches are omitted, unity i s often assigned to mediums, as shown in Table I, column 3. The designation, SML, refers to small, medium and large avalanches. Besides the (12,12,6) weights used i n the p i l o t study, (12,4,2) weights were tr i e d with considerable success. Later, i t w i l l be shown that (12,3,1), (12,1,1), and (1,1,1) weights result in the best models. It should be noted that (12,1,1) weights indicate that terminus alone determines the value of the index, and (1,1,1) weights imply equal weighting for a l l classifications, and hence the index i s simply a frequency count of the number bf avalanches. 27 Chapter IV METEOROLOGICAL FACTORS AS INDEPENDENT VARIABLES The Meteorological F i l e , Meteorological data from the Rogers Pass and F i d e l i t y observatories for the period, 1965 - 7 3• consisting of twice daily observations, measured at approximately 0700 and 1600 hours, of snow accumulation, new snow depth, water equivalent, maximum and minimum a i r temperatures, wind speed and direction, cloud cover, shear test data, and humidity, were transcribed from f i e l d books and stored on computer tape. Snow profiles are included with a maximum frequency of two weeks, consisting of density, wetness, crystal size and type, hardness and temperature of each snow layer, as well as a ram penetrometer prof i l e for the snowpack. Table II contains a l i s t of a l l the variables used i n the analysis, together with their computer labels for future reference. The following i s a summary of these variables, and the physical processes associated with them, which are thought to influence the level of avalanche a c t i v i t y . Depth of Snowpack SAC i s the depth of the snowpack, obtained from snow stakes at the study plots, and i s a direct measure of available snow for avalanching. According to Schaerer (1962:17), a certain minimum depth, i n the order of seventy centimeters, i s required for the Rogers Pass area, to cover the rocks and vegetation i n the slide paths before the avalanche season i s established. This figure agrees closely with Mellor's estimate for typical mountain terrain, in his discussion of the ten point system employed by the U.S. Forest Service (Mellor, 1968:148). 28 New Snowfall SNO i s the new snowfall measured from snow stakes at the study plots, and i s the primary and obvious cause of direct action avalanching. According to Mellor (1968:14-9), "the depth of new snow giyes a good measure of the quantity of snow l i k e l y to be released. As the depth of new snow increases above 1 f t . (30 cm.) or so, the probability of wide-spread avalanches of significant size tends to increase," Schaerer (1971) has shown that there i s considerable variation i n precipitation with elevation for the Rogers Pass area. Thirty year maximum water equivalents were computed and they range from 0.9 m. at 1220 m, to 2.0 m. at 2200 m. for the east and 1.1 m. at 1200" m. to 2.4 m. at 2200 m. for the west. Conditions at the Mt. Fi d e l i t y observatory, because of i t s higher elevation, more closely approximate those in the starting zones, but since the Illecillewaet valley usually receives about twenty-five percent more snowfall than the Tupper area, forecasts based on Fide l i t y obser-vations are more applicable to the western section. The extent to which avalanching occurs, depends on the rate of stress build-up in relation to the rate of increase of strength by compac-tive creep, sintering and bond growth. According to Mellor (1968:149), an accumulation rate of one inch per hour or more, sustained for several hours i s l i k e l y to produce major avalanching. Precipitation W.E i s the precipitation or water equivalent, and i s obtained as the product of snowfall and new snow density or from precipitation gauges. However, r a i n f a l l i s also included i n the data. Precipitation i s associated more strongly than snowfall with Table II Variables Treated as Independent Labels Used In Computer Analysis Description Units SAC Depth of snowpack, (snow accumulation) cm SNO Depth of new snow cm W.E Precipitation, (water equivalent for new snow and/or r a i n f a l l ) mm TMA Maximum a i r temperature °F TMI Minimum a i r temperature °F WNO North wind component mph WWE West wind component mph wso South wind component mph WEA East wind component mph CLO Cloud cover 1=25% of sky HUM Humidity (relative) % CD1 C r i t i c a l depth to f i r s t weak layer cm SW1 Shear weight of snow above 1st weak layer, (pressure) gm/cm SSI Shear strength of 1st weak layer (at zero normal stress) / 2 gm/ cm CI1 F i r s t c r i t i c a l index (SWl/SSl) . CD2 C r i t i c a l depth of second weak layer cm SW2 Shear weight of snow above second weak layer / 2 gm/cm SS2 Shear strength of second weak layer / 2 gm/cm CI2 Second c r i t i c a l index (SW2/SS2) SET Settlement (SAC1*+ SNO - SAC) cm WIN**2 Wind speed squared (mph)2 WIN Wind speed mph SNO*WIN Product of new snow depth and wind speed cm.mph TMI*WIN Product of minimum temperature and wind speed "F.mph SNO*TMI Product of new snow depth and minimum a i r temperature cm.°F TGR Minimum a i r temperature gradient (TMI - TMIl) •p TMI/SAC Quotient of minimum a i r temperature and depth of snowpack °F/cm DEN Density of new snow (W.E/SNO) gm/cc DEN*TMI Product of new snow density and minimum temperature (gm/cc).°F Table I I (continued) 30 L a b e l s D e s c r i p t i o n U n i t s •DEN*WIN Product o f new snow d e n s i t y and wind speed (gm/cc) .mpl DEN*HUM P r o d u c t o f new snow d e n s i t y and humidity gm/cc SNO*HUM P r o d u c t o f new snow d e p t h and humidity cm TMI*HUM P r o d u c t o f minimum a i r temperature and humidity ° F WIN*HUM P r o d u c t o f wind speed and humidity mph NVTH P r o d u c t o f new snow d e p t h , wind speed, minimum a i r temperature and h u m i d i t y cm.mph. °F W.E*WIN P r o d u c t o f p r e c i p i t a t i o n and wind speed mm.mph W.E*TMI Product o f p r e c i p i t a t i o n and minimum temperature mm.°F W.E*HUM Product o f p r e c i p i t a t i o n and humidity mm WVTH Product o f p r e c i p i t a t i o n , wind speed, minimum a i r temperature and h u m i d i t y mm.mph.°F SAC*W.E P r o d u c t o f depth o f snowpack and p r e c i p i t a t i o n cm. mm SWH Product o f depth o f snowpack, p r e c i p i t a t i o n , and humidity cm. mm SWHT Product o f depth o f snowpack, p r e c i p i t a t i o n , h u m i d i t y and minimum a i r temperature cm.mm.°F SWHTV Product o f depth o f snowpack, p r e c i p i t a t i o n , h u m i d i t y , minimum temperature and wind speed cm.mm."F.mph Note: AVAL i s the l a b e l used to d e s c r i b e the dependent v a r i a b l e , avalanche a c t i v i t y i n d e x , used i n the a n a l y s i s . SAC1— t h e f i r s t l a g o f SAC. 31 3 avalanche occurrence and notably with the formation of slab avalanches (USFS, 1961:34). This has been borne out by analysis as w i l l be seen :. la t e r . Air Temperatures Maximum and minimum a i r temperatures, TMA and TMI, are read from maximum and minimum thermometers at the study plots. Upper a i r temperatures correlate with the type of snow which f a l l s . Large intricate crystals occur at high temperatures, whereas small elemen-tary crystals are most common at low temperatures. "Thus, a i r temperature i s related to the type and density of the new snow, and hence to the i n i t i a l mechanical properties" (Mellor, 1968:151). The type and rate of metamorphism that occurs after the snow has fal l e n i s also largely determined by a i r temperature. High temperatures 4 induce equl-temperature (or destructive) metamorphism and high rates of settlement, causing the snow to s t a b i l i z e quickly. However, i f tempera-tures r i s e above freezing during snowfall, the snow may turn to rain and melting may occur, creating a serious avalanche hazard. At the Rogers Pass, "rain following a snowfall in the avalanche rupture zones can start avalanches within one or two hours of i t s beginning" (Schaerer, 1962:18). Low temperatures can induce temperature gradient (or constructive) J Two principal types of snow avalanche are widely recognized and referred to by the terms 'slab' or 'loose' (LaChapelle,1970bV8). Slab avalanches are usually characterized by a well defined fracture li n e and involve a mass of snow exhibiting some degree of internal cohesion. Loose avalanches generally start from a point and involve loose cohesionless snow. 4 Transport of water molecules from convexities to concavities in the ice skeleton due to a vapour pressure difference, thereby producing smaller, more rounded crystalline grains and stronger inter-crystalline bonds. 32 5 metamorphism causing increasing i n s t a b i l i t y , and low rates of settlement resulting i n a slow rate of s t a b i l i t y gain. "Cold weather in January and February with a period of no snowfall for two or more weeks may cause considerable metamorphism of the snow at the surface. This snow layer has low cohesion and may fracture under the weight of new snow or during the snow-melt period", at the Rogers Pass (Schaerer, 1962:16), Roch (1966:86-99) has also shown that the tensile strength, and Losev (1966:50), the shear strength, of given types of snow increases as temperature decreases, but as Mellor (1968:151) points out, "the probability of avalanche release tends to increase as temperature decreases over the usual range of sub-freezing temperatures". Hence the overall effects of temperature are extremely complex and d i f f i c u l t to evaluate. Besides being significantly correlated with almost every other meteorological variable associated with avalanche a c t i v i t y , temperature undoubtedly has a non-linear relationship with the le v e l of a c t i v i t y . Wind Wind speed and direction, measured by anemovane, i s telemetered from the MacDonald West Shoulder and Roundhill stations to the Rogers Pass and F i d e l i t y observatories, where i t i s recorded on anemographs. For the purpose of this analysis, i t has been resolved into four rectangular components, WNO, WWE, WSO, and WEA, which can be treated as separate variables. Transport of water molecules from warmer to colder grains due to a vapour pressure difference, thereby producing larger more angular crystals and a weakening of the ice skeleton. 33 Strong winds accompanying snowfalls often lead to a high level of direct action avalanching, by causing d r i f t i n g in areas of low wind stress, such as gullies and lee slopes. For the Rogers Pass, "prolonged wind strengths of 15 mi/hr in the west area and 25 mi/hr for the centre and east area are critical"(Schleiss., 1970:117). The f i r s t figure i s iden-t i c a l to Mellor's (1968:151), who states that, "significant wind transport and wind packing begins when the wind speed exceeds about 15 mph", for any avalanche prone area i n general. The pattern of distribution and redis-tribution of snow i s a complex function of wind speed, direction and topo-graphical characteristics of the terrain. Erosion zones may be more vulnerable to temperature gradient metamorphism iif the pack i s thin, possibly leading to greater avalanche hazard later in the season. Snow that i s transported i n the wind stream by saltation and turbulent suspension i s fragmented and may be deposited i n the form of wind slab^ i f the humidity i s high' enough. Cloud Cover Cloud cover, CLO, i s recorded as the amount of overcast in approximate quarters. It i s an important factor in determining the radiation balance of the snowpack, but i s probably more directly cor-related with storm periods than with avalanche a c t i v i t y . Humidity Relative humidity, HUM, i s measured with hygrographs and psych-rometers at the study plots, and i s expressed as a percentage of Wind slab consists of snow grains held together by inter-granular bonds. A gradation from soft slab to hard slab, depending on the degree of cohesion, i s generally recognized. 34 saturation. According to Schleiss (1970:117), "data indicate that a relative humidity of 80 per cent and over, i n combination with wind speeds of 15 mi/hr causes the formation of slab avalanches", at the Rogers Pass. Soft slab conditions, a characteristic feature of the Middle Alpine Zone (Mellor, 1968:154) are a common occurrence at the Rogers Pass and are undoubtedly responsible for a major portion of the avalanche a c t i v i t y . Seligman (1936:194-95) also recognizes the importance of humidity i n the formation of wind slab, stating that a value of eighty-five per cent or over causes wind packing. This situation i s reflected in the prediction models, in which, as i t w i l l be shown later, humidity appears to play an important role. The Shear Test Shear test data, the most subjective of the study plot measure-ments employed by the Snow Research and Avalanche Warning Section at the Rogers Pass, consists of three structural observations designed to identify and estimate the strength in relation to loading of c r i t i c a l layers, frequently thin and fra g i l e , i n the new or p a r t i a l l y settled snow of the upper section of the pack. Such layering or s t r a t i f i c a t i o n i s a common cause of direct and delayed action avalanching at the Rogers Pass. These layers, the depths and weaknesses of which are a complex function of the antecedent meteoro-lo g i c a l conditions, often originate at the surface in the form of surface hoar, surface layers produced by temperature gradient metamorphism, rain, sun, melt, or wind crust. However, even a l i g h t sprinkling of loose powder snow on the old surface can result i n the poor bonding of a new and heavier snowfall. 35 Shear plane depth, In centimeters, GDI, (and CD2, i n the case of a second layer), i s measured from the top of a sample block (approximately eighteen inches cube) resting on a th i r t y - f i v e degree t i l t table, down to the 'shear plane' after shear has been induced by a sharp 'tap* on the underside of the table. Shear weight, SW1, (SW2 for a second layer), i s the weight of snow above the shear plane in grams per square centimeter, and shear strength, SSI, (SS2 for a second layer), at zero normal stress, 2 i s measured using a Roch 100cm frame just above the shear plane, and reduced to grams per square centimeter. The ratio of shear strength to shear weight called the s t a b i l i t y factor, the reciprocal of which i s defined as the c r i t i c a l index, G i l , (CI2 for a second layer), in this study, obtained from the above measure-ments, i s thought by Schleiss(l970:116) to be fundamentally related to the lev e l of avalanche a c t i v i t y . In fact, a s t a b i l i t y factor of 1.5 or less for the Rogers Pass area i s considered c r i t i c a l . Also, i f the shear plane depth i s greater than twenty centimeters i n combination with other factors, the hazard i s l i k e l y to be high. There i s no doubt a strong relationship between the depth and weakness of c r i t i c a l layers within the snowpack and the level of avalanche ac t i v i t y , but there are a number of problems associated with the inter-pretation of such measurements. F i r s t l y , the shear test i s d i f f i c u l t to perform consistently and reliably, requiring the s k i l l and practice of an experienced man. Secondly, the results of this test made at the study plot may not have a great deal of bearing on conditions in the fracture zone, unless such conditions are widespread and pronounced. As far as the s t a t i s t i c a l analysis used i n this study i s concerned, the measurements are too discontinuous and intermittent to produce reliable correlations. 36 Settlement Settlement, SET, i s calculated by adding the current new snow depth to the previous snow accumulation and subtracting the current value of snow accumulation, but can be calculated from a storm stake with per-haps greater r e l i a b i l i t y . The rate of settlement or densification determines, to a large extent, the rate at which the snow i s gaining strength. In general, the faster the snow settles, the faster i t gains strength. However, the rate depends on temperature and the i n i t i a l density of the snow. "Low density snow has l i t t l e i n i t i a l strength but settles rapidly; high density snow has high i n i t i a l strength but densifies slowly, tending to gain strength more by sintering than by compaction"(Mellor, 1968:151). Hence, settlement rate and avalanche a c t i v i t y have a complicated relationship which i s aggravated by the fact that measurements of s e t t l e -ment are made at temperatures which may be quite different from those i n the fracture zones. Wind Terms Wind speed squared, WINM2, i s direc t l y related to the energy of the wind, which determines i t s carrying capacity for snow transport and i t s a b i l i t y to create stress i n the fracture zones. Wind speed, WIN, represents the scalar effect of wind. Other Terms SNO*WIN may be more highly correlated with slab formation than snowfall alone, besides being a measure of the amount of d r i f t i n g during snowfall. TMI*WIN may provide a useful indication of the thermal con-duction rate for spring avalanching. SNO*TMI i s related to the type of 37 snow crystal which f a l l s and the i n i t i a l structural properties of the snow on the ground. Air Temperature Gradient Minimum a i r temperature gradient, TGR, i s the difference between the previous and current values of minimum temperature. A sudden or large temperature change may trigger avalanches, according to Losev, (1966:48) who states that, "avalanches related to an abrupt temperature drop are formed when the volume of the snow cover undergoes thermal contraction. This produces additional stresses within the snow layer so that avalanches are formed." Temperature Gradient within Snowpack TMl/SAG i s the quotient of minimum a i r temperature and the depth of the snowpack and provides an indication of the average temperature gradient within the pack, since the temperature at the base of the pack i s usually f a i r l y constant and close to freezing point throughout the winter, provided that the pack i s thick enough to supply suf f i c i e n t insulation. Temperature gradient within the pack determines the rate and type of meta-morphism. A gradient in excess of ten degrees Centigrade per meter can cause significant temperature gradient metamorphism and the formation of depth hoar. Losev (1966:73)t who has tr i e d to quantify certain forecasting, procedures i n terms of analytical equations, principally concerned with establishing the time of onset of the avalanche a c t i v i t y period, assumes a direct proportionality between the stresses and temperature gradient within the snowpack, thus suggesting that there i s a direct relationship 38 between temperature gradient in the pack and i n s t a b i l i t y . Density Density of new snow, DEN, expressed as the quotient of water equivalent and snowfall, W.E/SNO, i s closely related to snow strength. Using a specially developed centrifugal or spin tester to measure tensile strength, Martinelli (1971:7-10) has demonstrated that snow strength increases rapidly with density over the range of samples tested. Since the i n i t i a l snow density i s largely determined by crystal type and mode of deposition, this parameter may be a good indicator of s t a b i l i t y . However, the relationship between density and s t a b i l i t y i s probably non-linear, since, " i t has been observed that when new snow density at a particular si t e departs widely from the mean density for that s i t e , avalanches are l i k e l y " (Mellor, 1968:149). Schaerer (1962:19) has also noted that for the Rogers Pass, "new snow with specific gravities lower than 0.07 and higher than 0.10 are more l i k e l y to cause avalanches." Unusually low densities, 'wild snow', may indicate a lack of cohesion, and high densities, the presence of free water i f temperatures are high,, resulting i n i n s t a b i l i t y i n both cases. High densities may also be associated with slab conditions. Further Terms Further terms have been included in the analysis in the hope that higher correlations with avalanche a c t i v i t y would be realized. From Table II, i t can be seen that these terms consist of certain combinations of the primary variables already discussed, which might be more strongly related to i n s t a b i l i t y than the simpler terms. DEN*TMI i s associated with i n i t i a l snowfall structure and free water content. DEN*WIN appertains to 3 9 the type of snow deposit, a high value possibly Indicating slab conditions. Humidity terms, DEN*HUM, SNO*HUM, TMI*HUM, and WIN*HUM may a l l be asso-ciated with slab formation, and NVTH i s perhaps a composite wind slab term, in which snowfall i s modified by wind speed, minimum a i r temperature and humidity. Water equivalent terms, W.E*WIN, W,E*TMI, W.E*HUM, and WVTH have been included, since i n general, precipitation i s more strongly correlated with i n s t a b i l i t y than snowfall, as w i l l be seen later. The remaining variables and their evolution w i l l be discussed i n the chapter on correlation analyses, where i t w i l l be shown that two important composite terms emerge which correlate more highly with avalanche a c t i v i t y than any other previous factors. 40 Chapter V CORRELATION ANALYSES This phase of the study i s concerned with the identification of appropriate starting and finishing dates for avalanche a c t i v i t y periods, suitable independent variables and the best act i v i t y index weighting schemes, as defined in Chapter III, to be used in the subsequent develop-ment of linear time series prediction models. Simple linear correlation 7 coefficients can be used to provide a good i n i t i a l indication of the potential strength of such models. In order to allow complete f l e x i b i l i t y in the application of these procedures, a data selection program was written. This program takes the meteorological information for any ranges of dates requested, computes the appropriate avalanche a c t i v i t y indices for each day (or half day), and writes the entire record, including the index, onto a f i l e ready for input to the analysis programs. Apart from allowing total freedom i n the choice of dates, the program permits any combination of sites (or ranges of sit e s ) , types (natural or a r t i f i c i a l ) sizes and moisture contents, 7 Defined by the familiar equation, I (X - X) (Y - ?) - - 1 ' X y Z N ~ E (X - X T I (Y - Y)* where Y and X are the dependent and independent variables respectively, and N i s the number of observations i n the sample. 41 to be specified as c r i t e r i a for including any one avalanche as part of the ac t i v i t y index. Since the data does not actually consist of equi-spaced twelve hourly observations, but more closely resembles nine and fifteen hourly measurements, i t was reduced to daily values for the purpose of the majority of the analyses. This was achieved by integrating avalanche a c t i v i t y , snowfall and precipitation, selecting minimum and maximum a i r temperatures, and averaging wind speeds, cloud cover, humidity and shear test data. As w i l l be shown later, the use of twice daily observations i s not j u s t i f i e d , due to their high noise l e v e l . Identification of Dates As a f i r s t step in the investigation, starting and finishing dates were defined as the dates of the f i r s t and l a s t avalanche occurrences for the eight avalanche seasons, as indicated i n Table III. Division of each season into a f i r s t part, which i s primarily snowfall dependent, and a second part, which i s primarily temperature dependent, i s important, for, as Schaerer (1962:7) points out, "there are two avalanche periods each year. In the f i r s t period, between early November and late February, avalanches are caused mainly by snowfalls, wind action, and rain i n association with snowfalls. In the second period, between late March and mid-May, ava-^ lanches are caused mainly by warm weather and melting of the snow," Models should then be developed for each part, which best describe the two types of avalanching. Bois et al.(1974:5) and Bovis (1974:71) distinguish be-tween dry and wet avalanches, which leads to a similar but not identical division of the data, since dry avalanches are not confined to the f i r s t nor wet avalanches to the second part of the winter. However, according to Bovis (1974:71) f "dry avalanche and wet avalanche periods are defined "by the transition (usually abrupt i n the San Juan Mountains) from dry to wet slides." a simple procedure, since there i s always a transition period between the two parts, during which both types of avalanching occur. However, i t i s possible, by visual inspection of the data correlations, to identify the interval over which avalanches tend to become more dependent on temperature. This somewhat subjective approach has been considerably improved by the introduction of a technique which can be referred to as 'incremental correlation analysis'. A subroutine which computes the correlation coef-ficients between the dependent and the individual independent variables, after each sequential data record i s read and added to the f i l e , has been written and incorporated into the main analytical program. , The identification of suitable transition dates i s by no means Table III Starting and Finishing Dates Total Season F i r s t Parts Second Parts Date of F i r s t Avalanche Transition Date Date of Last Avalanche 14/H/ 6 5 19/10/66 20/10/67 15/10/68 5/U/69 16/11/70 25/10/71 25/11/72 29/ 1/66 28/ 1/67 1/ 2/68 2/ 2/69 5/ 2/70 1/ 2/71 5/ 2/72 1/ 2/73 30/ 5/66 30/ 5/67 31/ 5/68 13/ 5/69 23/ 5/70 25/ 5/71 31/ 5/72 28/ 5/73 43 Hence, after a starting date has been established, the behaviour of correlations between avalanche a c t i v i t y and snowfall, for example, can be monitored as the winter progresses. For most winters, i t i s observed that such correlation coefficient values r i s e to a peak, near the end of January, or beginning of February, after which they drop off sharply, as snowfall becomes less significant than temperature. This procedure was ; applied to data for each winter using an avalanche a c t i v i t y index based on (12,4,2) weights for a l l sites and a l l avalanche events, with water equivalent, as well as snowfall as independent variables. Optimum dates for the separation of the data into f i r s t parts and second parts, were established for each winter, and are indicated i n Table III. Independent Variables Employing these dates, a complete correlation analysis was per-formed using a l l the independent variables, described i n Chapter IV, and meteorological data from the Rogers Pass observatory. Avalanche a c t i v i t y indices were computed using (12,4,2) weights, a l l sites, and a l l avalanche events, both a r t i f i c i a l and natural (AN) and small, medium and large (SML). The results are indicated i n Table IV, i n which correlation coefficient values have been multiplied by one hundred for convenience in represen-tation. Total Seasons. The values in column (1) were obtained by com-bining total seasons, as defined i n Table III, for the entire period, 1965-73. This sample consists of 1654 daily records and therefore, absolute values of the correlation coefficients i n excess of 0.06 can be regarded as significant at the ninety-nine per cent l e v e l . It i s immediately apparent that a l l the independent variables are significantly 44 correlated with avalanche a c t i v i t y , with the exception of TMl/SAC. Temperature and density terms display only weak correlations, probably because of their non-linear association with avalanche events, as mentioned in Chapter IV; Wind terms are also not as significant as might be expected, possibly because wind observations suffer from a high level of noise, as w i l l be discussed later. Precipitation terms, predictably, have the strongest correlations. I t i s Important to note that W.E, that i s , SNO*DEN, i s more important than SNO alone, as was suggested i n Chapter IV. At this point, a peripheral study was made to determine to what extent the tot a l avalanche a c t i v i t y for each year i s correlated with the tota l amount of snowfall or water equivalent, for as Mellor (1968:157) points out, " i t seems l i k e l y that avalanche a c t i v i t y w i l l correlate close-l y with the amount of winter precipitation, although this i s yet to be demonstrated." Using the (12,4,2) weights, and avalanche seasons defined by f i r s t and la s t occurrences, correlation coefficients of 0.92 and 0.97 were obtained for snowfall and water equivalent respectively, as shown in Table V. This not only indicates an extremely high level of correlation between t o t a l avalanche a c t i v i t y f o r each year and total snowfall, but i t also demonstrates clearly that water equivalent i s more important than new snow depth i n determining avalanche a c t i v i t y . It i s also interesting to compare the average annual snowfall of 1064 cm. (419 in.) for the period, 1965-73, and the maximum of 1530 cm. (602 i n . ) , for the winter of I966-67, measured at the Rogers Pass, with the average of 342 i n . for the period, 1921-51, and the maximum of 680 i n . for the winter of 1953-54, measured at Glacier (Schaerer, 1962:4). Table IV Correlation Analysis Rogers Pass Meteorological Data Period 1965-73 Daily Observations (12,4,2) Weights, SML, AN (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) AT AF AS AT AF • ET EF ES WT WF ws SAC 24 33 21 24 27 23 31 20 22 29 19 SNO 46 54 41 - 4 - 4 45 53 43 33 39 30 W.E 54 62 50 0 0 54 62 54 45 .53 39 TMA 8 22 - 2 16 3 8 22 -3 . 7 17 - 1 TMI 18 22 16 13 1 19 23 17 14 17 13 WNO -6 -6 - 7 - 2 1 -6 -5 - 8 -6 -6 -6 WWE 16 20 11 6 6 16 21 12 12 16 8 WSO 22 26 17 5 8 20 23 17 21 27 14 WEA - 7 - 1 2 - 1 0 - 1 -9 - 1 4 - 4 - 2 - 8 5 CLO 19 23 18 -7 -7 21 24 20 13 16 12 HUM 19 15 24 0 1 19 14 24 16 12 21 CD1 25 31 21 - 2 2 26 31 25 18 23 12 SW1 22 29 17 0 6 23 29 19 16 22 9 SSI 8 8 11 1 4 8 8 13 5 6 7 CI1 19 20 18 - 1 - 4 21 23 20 11 10 12 CD2 13 13 14 -3 -3 15 15 16 6 6 7 SW2 12 13 12 -3 - 2 14 16 14 6 6 6 SS2 8 9 8 -3 - 1 9 10 10 4 4 4 CI2 15 11 17 6 -5 15 14 16 12 4 16 SET 36 11 - 1 6 - 8 36 11 - 2 30 9 2 WIN**2 26 31 22 8 12 24 27 22 25 32 19 WIN 20 24 17 6 9 18 21 16 21 26 16 SNO*WIN 41 45 40 3 4 41 43 42 35 40 30 TMI*WIN 27 33 22 14 9 26 31 21 . 24 29 20 SN0*TMI 44 46 42 3 2 45 48 45 33 36 31 TGR 14 15 8 3 -3 15 16 9 10 11 6 TMI/SAC - 4 -5 -6 -5 - 4 - 4 . -5 -5 _4 - 4 - 6 Table IV continued (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ( U ) P P AT AF AS AT AF ET EF ES WT WF WS DEN 22 21 24 0 - 4 20 19 22 22 22 24 DEN*TMI 25 26 25 6 c-3 22 24- 22 25 26 26 DEN*WIN 29 28 31 9 6 25 24 27 31 32 31 DEN*HUM 24 23 27 1 - 3 22 20 24 24 23 27 SNO*HUM 45 52 42 -2 - 3 46 53 45 34 40 31 TMI*HUM 24 23 28 13 2 25 23 29 19 18 23 WIN*HUM 25 27 25 7 9 23 23 24 25 28 22 NVTH 43 44 44 9 8 43 43 46 35 38 33 W.E*WIN 51 57 47 8 10 48 52 48 46 54 38 W.E*TMI 53 58 50 8 9 53 56 52 45 50 39 W.E*HUM 55 63 52 2 2 55 62 54 46 54 40 WVTH 52 54 54 16 16 50 50 54 48 51 44 SAC*W.E 59 70 51 58 67 52 51 62 41 SWH 61 71 53 60 68 53 52 64 42 SWHT 63 75 57 62 70 58 55 70 46 SWHTV 57 65 53 54 57 52 54 . 67 45 = a l l sites, E = eastern sites, W = western sites, = total seasons, F = f i r s t parts, S = second parts, = par t i a l correlation coefficients. 47 F i r s t Parts. Column (2) of Table IV was obtained by combining f i r s t parts, as defined i n Table III, for the entire period, 1965-73. This sample consists of 739 daily records resulting i n a ninety-nine per cent significance level of 0.10 for simple correlation coefficients. The results suggest that models based on precipitation terms should achieve a high degree of predictive accuracy for the f i r s t part of the season, par-t i c u l a r l y i f individual years are used. Table V Total Annual Avalanche Activity, (12,4,2) Weights, SML AN, Versus Total Annual Snowfall and Water Equivalent Using Rogers Pass Meteorological Data YEAR N E AVAL E SNO (cm.) I W.E (mm.) 1965-66 198 24100 1074 897 1966-67 224 38700 1530 1260 1967-68 225 323OO 982 975 1968-69 211 23030 903 792 1969-70 200 16610 780 656 1970-71 191 22900 1000 903 1971-72 220 37400 1475 1263 1972-73 185 I658O 766 633 R = 0.92 R - 0.97 N = Number of days in season from f i r s t avalanche to l a s t avalanche, as per Table III. Second Parts. Column (3) of Table IV was obtained by combining second parts, as defined in Table III, for the entire period, 1965-73. Consisting of 915 daily records, this sample results i n a ninety-nine per cent level of significance for simple correlation coefficients of 0 . 0 9 . ; As expected, precipitation terms are less important than for f i r s t parts, hut temperature terms, probably because of their non-linear effects, are also poorly correlated with avalanche a c t i v i t y . However, humidity and density terms appear to be s l i g h t l y more important during second parts, but correlation values do not indicate a strong dependence. It seems l i k e l y that non-linear terms w i l l have to be introduced into forecasting models designed s p e c i f i c a l l y for second parts of the avalanche seasons, i f an acceptable degree of accuracy i s to be achieved. Suitable terms are under consideration and w i l l be incorporated into future analyses. Evolution of the Best Independent Variables. Recapitulating, W.E for total seasons and f i r s t parts, i s by far the most important of the simple variables in i t s association with the level of avalanche a c t i v i t y , in terms of the avalanche a c t i v i t y index. However, there i s a suggestion that W.E*HUM may be more significant than W.E alone, and perhaps a more complex composite term may display an even higher correlation. In order to test the v a l i d i t y of this proposition and also to identify any secon-dary variables which might be important after the variation accounted for 8 by W.E has been subtracted out, p a r t i a l correlation coefficients were Defined by the following equation (Freese, 1964:104), „ ry2 " r y l r 2 i ry2.1 where r y g ^ i s the correlation coefficient between y and x^ after x^, r^2 I s "the correlation coefficient between y and x^, r ^ i s the correlation coefficient between y and x^, and, i s the correlation coefficient between x^ and x^. computed f o r t o t a l seasons and f i r s t parts. The r e s u l t s appear i n columns (4) and (5) of Table IV, and c l e a r l y indicate that a f t e r W.E, SAC i s the next most Important term, A model containing water equivalent and snow accumulation should therefore be stronger than one containing water equivalent alone. However, there are good reasons why SAC cannot be 9 introduced as a secondary variable a f t e r W.E, but i f SAC i s used as a facto r modifying W.E no such problem e x i s t s . Hence, SAC*W.E was i n t r o -duced as a new variable i n the analysis. Referring back to columns (1) and (2) of Table IV, i t can be seen that c o r r e l a t i o n c o e f f i c i e n t s f o r SAC*W.E are 59 and 70 f o r t o t a l seasons and f i r s t parts, as opposed to 54 and 62 f o r W.E, i n d i c a t i n g a substantial improvement.*^ This suggests that the amount of avalanche a c t i v i t y f o r a given quantity of p r e c i p i -t a t i o n increases with increasing snowpack depth. This improvement i n the co r r e l a t i o n could not merely be the r e s u l t of the minimum snowpack depth c r i t e r i o n required f o r avalanching to s t a r t , as discussed i n Chapter IV, since t h i s depth has already been established on or near the s t a r t i n g dates which were used i n t h i s a n a l y s i s . Therefore, the e f f e c t i s undoub-tedly ' r e a l ' . Furthermore, a s i m i l a r e f f e c t has been reported i n the l i t e r a t u r e . Losev (1966:75). quotes r e s u l t s obtained by V. Sh. Tsomaya and K. L. Abdushelishvili (1962) f o r a slope i n the High Caucasus i n the ^ Since the snow accumulation se r i e s i s non-stationary and highly autocorrelated, i t cannot be introduced separately into a time series model containing the water equivalent s e r i e s , which i s e s s e n t i a l l y stationary. F i r s t differences of snow accumulation are too highly correlated with water equivalent to r e s u l t i n a s i g n i f i c a n t contribution, a f t e r the e f f e c t of water equivalent has been subtracted. *^ The improvement i s highly s i g n i f i c a n t at the .999 l e v e l , as described by "Hotelling's t-Test". (Freese, 1964:108) 50 region of the Krestov Pass, which clearly demonstrate that the onset of avalanching requires progressively less precipitation as the snowpack increases i n depth. The authors have empirically deduced the following equation: = 55 - 2.8 J~h , where i s the minimum precipitation, in millimeters, required for avalanching and h i s the depth of old snow, i n centimeters. Such a relationship strongly suggests that the amount of avalanche a c t i v i t y for a given quantity of precipitation increases with increasing snowpack depth. The physical interpretation of this result i s that the snowpack, as i t gets deeper, participates more and more i n the avalanche a c t i v i t y , presumably as a consequence of an increase i n the available amount of avalanchable snow. Of course, i t should be pointed out that SAG measured at the study plot certainly does not represent directly the amount of accumulated snow in the avalanche paths, which may have already run several times so far during the winter. However, SAG, like a l l the other meteorological factors measured at the study plots, i s an indicator of conditions i n the slide paths. I t i s probable that the importance of SAC i s indicative of a delayed action effect, which may represent the formation of soft slab conditions, particularly during the f i r s t parts of the seasons. This proposition led to the development of the composite terms, SWH, SWHT, and SWHTV. Referring back to columns (l) and (2) of Table IV, i t can be seen that correlation coefficient values for SWH and SWHT are 61 and 63 for tot a l seasons, and 71 and 75 for f i r s t parts. Thus, humidity and temperature are also important modifying factors** probably associated 1 1 SWH i s sig n i f i c a n t l y better than SAC*W.E at the .999 l e v e l , SWHT i s sig n i f i c a n t l y better than SWH at the .95 l e v e l , as described by "Hotelling's t-Test" (Freese, 1964:108). 51 with slat formation. The values for SWHTV are only 57 and 65, which sug-gests that wind i s not an important modifying factor. However, wind undoubtedly has a strong influence on slab formation, but this may be masked since i t s effect i s d e f i n i t e l y non-linear, diminishing as wind speeds exceed a c r i t i c a l l e v e l . "There also appears to be an upper c r i t i c a l wind l e v e l , not clearly defined, above which, snow tends to form wind pack rather than slab" (USFS, 1961:35). Very high winds may also cause too much erosion since, as Mellor (1968:151) points out, "in some locations very strong ( f u l l gale) winds may be less effective than moderately strong winds in loading up the release zones." East-West Division of Data The remaining columns (6) to (11) of Table IV contain correlation coefficient values for eastern and western avalanche sites, for, "at Rogers Pass there are two major climate areas, and the avalanche hazard for each should be evaluated separately. The two areas are: —The Tupper area on the east side of the Pass, —The Illecillewaet Valley on the west side of the Pass" (Schaerer, 1962:15). For this analysis, the division between eastern and western sites was established at mile 16.53» measured from the east boundary of Glacier National Park. Since the analysis i s based on data from the Rogers Pass observatory, correlations for the eastern sites might be expected to be higher than those using the eastern and western sites combined. However, although the snowfall terms support this premise, water equivalent terms suggest the opposite. In other words precipitation measured at the Rogers Pass i s a good indicator of avalanche ac t i v i t y for the entire area. This argument i s supported by Schaerer (1962:15) who states that, "observations during the two winters between 52 1957 and 1959 showed that the average total snowfall i n the Tupper area was 80 percent of the snowfall measured in the Illecillewaet Valley. As less snowfall i s required to cause avalanches on Mount Tupper, the avalanche hazard i s usually about equal i n both areas." Hence, i t appears that Rogers Pass meteorological data i s truly representative of the entire area, i n terms of precipitation, as i t relates to avalanche act i v i t y , and therefore, an east-west s p l i t may not by worthwhile. Selection of the Best Weights for Avalanche Activity The next phase of the study i s concerned with the selection of the best set of weights to be used i n the determination of the avalanche a c t i v i t y index. Various sets of weights were chosen, as outlined in Table VI, and a complete correlation analysis performed on a l l the independent variables using total seasons, f i r s t parts and second parts, for the period, 1965-73, as defined in Table III, a l l sites, a r t i f i c i a l and natural avalanches, and meteorological data from the Rogers Pass observatory. Table VII contains a summary of the results of this study in terms of a reduced set of correlations. The variables, SNO, W.E, SWH, and SWHT were used as appropriate indicators of performance and the results quoted for totals, f i r s t parts and second parts. Moisture Content. (12,4,1) Sift weights result in generally better correlations than (12,4,2) SML weights, Indicating that equal weights applied to the moisture content c l a s s i f i c a t i o n w i l l lead to stronger models. The i n a b i l i t y of moisture content to improve the index for either f i r s t parts or second parts, has been observed using various other sets of weights for terminus, size and moisture content. Therefore, i t must be concluded that- either moisture content i s not l i k e l y to be an important 53 Table VI Avalanche' Activity Index Weighting Schemes Used in the Analysis Designation (12,4,2) (12,4,1) (12,36,l) . ( 1 2 , 1 2 , l ) (12,12,1) (12,3,1) SML SML SML " SML ' ML SML Terminus £ or 1/3 path 1 1 1 1 1 1 •§- path 2 2 2 2 2 2 2/3 or 3/4 path 3 3 3 3 3 3 End path, to fan, or gully 4 4 4 4 4 4 ij-fan 5 5 5 5 5 5 1/3 fan 6 6 6 6 6 •6 T fan 7 7 7 7 7 7 2/3 fan 8 8 8 8 8 8 3/4 fan, Old RR, or Bench 9 9 9 9 9 9 Over fan, or Mounds 10 10 10 10 10 10 Edge TCH 11 11 11 11 11 11 Over TCH 12 12 12 12 12 12 Size Small 1 1 1 1 0 1 Medium 2 2 18 6 . 6 2 Large 4 4 36 12 12 3 Moisture Content i Dry 1 1 1 1 1 1 Damp 1 1 1 1 1 Wet 2 1 1 1 1 1 Table VI continued 54 Designation (12,1,1) SML (12,1,1) ML (1,4,1) SML (1,3,1) SML (1,2,1) ML (1,1,1) SML (1,1,1) ML Terminus £ or 1/3 path 1 1 1 1 1 1 1 -§- path 2 2 i 1 1 1 1 2/3 or 3/4 path 3 3 1 1 1 1 1 End path, to fan, or gully 4 4 1 1 1 1 1 £ fan 5 5 1 1 1 1 1 1/3 fan 6 6 1 1 1 1 1 •§- fan 7 7 1 1 1 1 1 2/3 fan 8 8 1 1 1 1 1 3/4 fan, Old HR, or Bench 9 9 1 1 1 1 1 Over fan, or Mbunds 10 10 1 1 1 1 1 Edge TCH 11 11 1 1 1 1 1 Over TCH 12 12 1 1 1 1 1 Size Small 1 0 1 1 0 1 0 Medium 1 1 2 2 1.5 1 1 Large 1 1 4 3 2 1 1 Moisture Content Dry 1 1 1 • 1 1 1 • 1 Damp 1 . 1 1 1 1 1 1 Wet 1 1 1 1 1 1 1 55 Table VII Reduced Set of Correlations for Various Weighting Schemes Rogers Pass Meteorological Data, Period 1965-73 . ' ' Daily Observations, A l l Sites, AN SNO W.E SWH SWHT AT AF AS AT AF AS AT AF AS AT AF AS (12,4,2), SML 43 50 41 54 62 50 60 71 53 63 74 57 (12,4,1), SML 57 61 52 61 66 56 67 77 59 64 72 60 (12,36,1), SML 54 59 50 59 63 54 66 77 58 62 71 59 (12,12,1), SML 55 60 51 59 64 55 67 77 58 63 71 59 (12,12,1), ML 54 58 50 58 63 54 66 76 57 62 71 59 (12,3,1), SML 58 63 54 62 67 56 68 78 59 64 73 60 (12,1,1), SML 60 66 55 63 70 56 67 78 59 64 74 60 (12,1,1), ML 56 62 51 60 66 54 67 78 58 62 72 58 (1,4,1), SML 56 61 52 61 66 56 66 76 58 64 73 60 (1.3,1), SML 57 63 53 61 67 56 66 77 58 64 74 60 (1,2,1), ML 54 60 49 58 64 53 65 76 56 62 72 57 (1,1,1), SML 58 64 54 61 68 55 65 74 57 63 72 59 (1,1,1); ML 53 60 49 58 65 53 65 77 55 62 73 57 1(1,1,1), ML 55 62 49 58 65 53 64 75 55 61 71 56 1(1,3,1), SML 57 63 53 61 66 56 66 74 58 63 71 60 A = a l l s i t es, T = to t a l seasons, F = f i r s t parts, S = second parts, I refers to individual site weights. Correlation coefficients are multiplied by 100 for convenience in represen-tation. 56 factor in assessing avalanche a c t i v i t y in terms of the meteorological variables used i n this study, or the observation of moisture content i s too subjective to be useful. Since avalanches may start dry but appear wet at the terminus, as a result of higher temperatures i n the valley, picking up wet snow i n the avalanche track, or pulverization, the former p o s s i b i l i t y i s l i k e l y . Terminus and Size. Progressive gains are realized as the index i s changed from (12,36,1) SML, to (12,12,1) SML, to (12,4,l) SML, to (12,3,1) SML, to (12,1,1) SML. Thus the terminus c l a s s i f i c a t i o n provides a better indication of avalanche a c t i v i t y than size. This i s borne out by a pro-gressive loss as the index i s altered from (1,1,1) SML, to (1,3,1) SML, to (1,4,1) SML. From an observational standpoint, terminus i s certainly a less subjective and more precise estimate than size, but perhaps there i s a more fundamental reason why terminus seems to be a better measure of avalanche a c t i v i t y . Distance travelled may be more indicative of insta-b i l i t y i n terms of meteorological factors than the size of the avalanche. In any case, a weighting scheme based on terminus alone, that i s , (12,1,1) SML, gives the best results. Small Avalanches. Dropping the small avalanches, as i n (12,1,1) ML, weakens the index. Hence, i t seems important to include the smalls, but their influence on the index i s no doubt minimized by their generally small terminus codes. Site Weightings. Individual site weightings, (see Appendix B) based on estimated s i t e sizes, taken from highway and a e r i a l photographs, were incorporated into the (1,1,1) ML and (1,3,1) SML indices. The f i r s t scheme effectively converts the data into an absolute one biased towards the sizes of the sites and the second into a more absolute measure of 57 avalanche sizes. In both cases, the indices are weakened by the conversion. The Best Weights. Besides the (12,1,1) SML scheme identified pre-viously as the best, the (12,3,1) SML weights are of interest, since here, sizes are incorporated with the simple weights, 1, 2, and 3 for small, medium and large. Both the (12,1,1) SML and the (12,3,1) SML schemes are powerful, and i n both, the small avalanches carry l i t t l e weight. The (1, 1,1) ML scheme i s also of considerable interest, since this i s merely a frequency count of the number of medium and large avalanches per day. I f the smalls are included, as in the (1,1,1) SML-scheme, the index i s weak-ened. Therefore, smalls should be excluded i f the index i s based on 12 frequency alone, as suggested by Schaerer. Hence, the three best weighting schemes selected for further analysis are the (12,1,1) SML, (12,3,1) SML and (1,1,1) ML schemes. Revision of Dates Transition dates for the f i r s t and second parts were revised using these weights and SNO, W.E, SWH and SWHT as independent variables, i n a repeat of the incremental correlation analysis procedure, previously described. These dates are recorded i n Table VIII, along with a new set of finishing dates for each of the avalanche seasons, also determined from the incremental correlation procedure. After r i s i n g to peak values which establish the transition dates, the correlations between avalanche a c t i -v i t y and precipitation terms gradually taper off u n t i l the new finishing dates are reached, after which, even correlations for temperature terms suddenly plummet, indicating that the season i s effectively over, although Studies at Rogers Pass, unpublished. 58 a few late spring avalanches have yet to take place. It i s unwise to define the end of the season as the date of the l a s t avalanche, for, as Schaerer points out, a large interval of negligible a c t i v i t y may separate 13 the effective end of the season from this f i n a l event. • Table VIII Revised Starting and Finishing Dates Total Season F i r s t Parts Second Parts Date of F i r s t Transition Revised Dates for Avalanche Date End of Season 14/H/65 29/ 1/66 7/ 5/66 19/10/66 28/ 1/67 8/ 5/67 20/10/67 22/ 1/68* 30/ 4/68 15/10/68 2/ 2/69 24/ 4/69 5/11/69 5/ 2/70 5/ 5/70 16/11/70 11/ 2/71* 24/ 4/71 25/10/71 27/ 2/72* 23/ 5/72 25/11/72 1/ 2/73 14/ 5/73 Indicates revised transition dates The complete correlation analysis was repeated using these revised dates and the three best weighting schemes, for a l l sites, a r t i -f i c i a l and natural avalanches, and total seasons, f i r s t parts and second parts, for the entire period, 1965-73. Correlation coefficients of 67, 65 and 63 were realized for SNO, 71, 68 and 66 for W.E, 78, 79 and 77 for SWH, and 75, 74 and 74 for SWHT for (12,1,1) SML, (12,3,1) SML, and (1,1,1) ML, respectively, for f i r s t parts, Thus SWH i s the strongest variable. Later, i t w i l l be shown that such high correlations lead to powerful models both 13 Personal communication 59 for individual years, and for the eight year period.. A single model i s developed for the eight f i r s t parts of the entire period, 1965-73, with a multiple correlation coefficient of 0.81, A r t i f i c i a l and Natural Avalanches A complete correlation analysis was performed using natural avalanches only, for a l l sites, (1,1,1) ML weights, f i r s t parts, for the period, 1965-73, and Rogers Pass meteorological data. Correlations for SNO, W.E, SWH, and SWHT are recorded in Table IX. The results indicate that naturals alone are not as good as naturals and a r t i f i c i a l s combined, probably because a r t i f i c i a l s account for a high percentage of tot a l avalanche a c t i v i t y for the Rogers Pass area, and therefore should not be excluded. Table X indicates the percentage contributions of a r t i f i c i a l to total avalanche a c t i v i t y for a l l sites, i n terms of (1,1,1) SML and (1,1,1) ML weights, for each individual year. Values range from 23 to 41 for (1,1,1) SML and 23 to 41 for (1,1,1) ML, and display a generally increasing trend from 1966-73, indicating that the control program has improved over the years. However, the percentages are anomalous for the year, 1965-66. It should be noted that the exclusion of the small avalanches does not a l t e r these percentages significantly. The stronger correlations between total avalanche a c t i v i t y and meteorological factors, for a r t i f i c i a l s and naturals combined, as opposed to naturals only,- implies that the decision to shoot i s usually made during optimum conditions for natural avalanching, which i s an obvious conse-quence of the operational procedures involved. The control program may be improved i f procedures can be developed to forecast i n s t a b i l i t y prior to the onset of natural avalanche cycles. It i s hoped that this study w i l l 60 ultimately lead to such procedures. Table IX Reduced Set of Correlations for Various Subsets of Avalanche Activity Rogers Pass Meteorological Data, Period 1965-73» F i r s t Parts, Daily Observations, (1,1,1) Weights, ML SNO W.E SWH SWHT A l l Sites, AN 63 66 77 74 A l l Sites, N 57 62 68 71 * Tapper Gullies, AN 62 63 74 68 MacDonald Gullies, AN 55 58 63 64 Lens, AN 50 53 58 53 Crossover, AN 46 48 51 52 Ross Peak, AN 13 15 13 14 A l l Sites, Storm Periods, AN 57 61 63 61 Sites designated Tupper and MacDonald Gullies are l i s t e d in Appendix B. Reduction of Sites Two groups of sites, the Tupper gullies and the MacDonald gu l l i e s , (Appendix B), were examined using Rogers Pass data, (1,1,1) ML weights, a r t i f i c i a l s and naturals, for f i r s t parts and the period, 1965-73. Cor-relations for SNO, W.E, SWH and SWHT are indicated in Table IX. The Tupper gulli e s appear to be representative of the entire area in terms of these coefficients and powerful forecasting models could be based on these sites alone. The MacDonald gu l l i e s , however, display somewhat weaker correlation. Correlations for the individual sites known as Lens, Crossover and Ross Peak are also indicated i n Table IX. While Lens gives r i s e to moderately 61 Table X Total Annual A r t i f i c i a l and Natural Avalanche Activity (1,1,1) Weights Year I AN £ A £A/£AN £ AN £ A IA/£AN SML • SML SML ML ML ML Percent Percent 1965-66 :':3?2 131 35 238 96 40 1966-67 2953 715 24 1668 435 26 1967-68 1508 403 27 918 238 26 1968-69 1141 308 27 723 164 23 1969-70 936 214 23 567 130 23 1970-71 1159 410 35 638 240 38 1971-72 1815 7^ 9 41 1011 410 41 1972-73 1193 393 33 839 314 37 £ AN i s the total number of a r t i f i c i a l and natural avalanches £ A i s the total number of a r t i f i c i a l avalanches SML indicates that small,- medium and large avalanches are included ML indicates medium and large avalanches only. 62 high correlations, Crossover i s less strong and Ross Peak i s quite weak. Restricting the variation of the avalanche a c t i v i t y index in this way must inevitably result i n such weak correlations, since there are many more non-avalanche days, during some of which, meteorological conditions may be 14 quite favorable for avalanching at other site s . There are i n s u f f i c i e n t occurrences at individual s i t e s , even sites such as Lens, which i s most active, to produce individual forecasting models of high precision. The development of models using a l l the sites i s a much better procedure, leading to considerably more powerful models. As w i l l be described later, the level of avalanche a c t i v i t y for the entire area can then be predicted, using these models, and the predictions decomposed into the most probable distribution of individual s i t e a c t i v i t i e s . Storm Periods A peripheral analysis was conducted for individual storm periods, which were identified after careful scrutiny of the precipitation patterns for the eight avalanche seasons. Some winters consist of very clearly defined storm periods, whereas others may be made up of longer periods of intermittent snowfall. In such cases, the identification of specific storm periods i s rather subjective. Nevertheless, forty-five storms (see Appendix C), were combined and correlation coefficients.calculated. How-ever, treatment of the data in this way i s not only d i f f i c u l t to interpret, but also results in correlations which are in fact weaker than those for f i r s t parts as shown in Table IX. The discriminant analysis techniques employed by Judson and Erickson(l973), Bois et a l . (1974) and Bovis (1974) suffer from this major fault. Any attempt to assign a l l avalanche a c t i v i t y into a single class, or portions into a restricted number ,of sub-classes, results i n high probabilities of misclassification. 63 Twice Daily Data Finally, a complete correlation analysis was performed using twice daily data from the Rogers Pass observatory, i n order to determine whether greater accuracy could be achieved. Avalanche a c t i v i t y indices were com-puted for a l l sites, a l l avalanches, and (12,1,1) weights, using data for the entire period, 1965-73, f i r s t parts only. Correlation coefficients of 62, 62, 68 and 67, were obtained for SNO, W.E, SWH and SWHT, as opposed to 67, 71, 78 and 75 for daily data. Hence, models based on twice daily data would be weaker than daily models, probably as a result of the relatively high l e v e l of noise i n the twice daily data. As indicated previously, avalanche times of occurrence are not relia b l e enough to ju s t i f y the use of twice daily observations, besides which, twice daily meteorological observations are not equispaced. Individual Years Using the three best sets of weights for avalanche a c t i v i t y , a complete correlation analysis was performed for the f i r s t parts of i n d i v i -dual years as defined in Table VIII. The results are summarized in Table XI i n terms of a reduced set of correlation coefficients for SNO, W.E, SAC*W.E, SWH, SWHT and SWHTV. As for the t o t a l f i r s t parts for the period, 1965-73, correlations for each individual year indicate that i n general, (12,1,1) SML weights are better than (12,3,1) SML weights, which are i n turn better than (1,1,1) ML weights, W.E i s more highly correlated with avalanche a c t i v i t y than SNO for the years, 1966-67, 1967-68, 1968-69, 1970-71, and 1972-73 but, for the remaining years, i t i s somewhat weaker. As discussed previously, W.E, that i s , SN0*DEN, seems to be more important than SNO, in i t s association with avalanche a c t i v i t y , perhaps 64 for the following reasons. F i r s t l y , SNO*DEN i s a more direct measure of 'shear weight* than SNO alone. Secondly, DEN contains the temperature effect, a high value of DEN possibly indicating high temperatures and perhaps free water. It i s significant to note that, for the three years for which SNO correlations were stronger than W.E correlations, SWHT was also weaker than SWH, suggesting that temperature was not important as a modifying factor. Finally, DEN may also be a strong indicator of slab conditions. This i s supported to some extent by the fact that, the years displaying an increase i n correlation values for W.E compared to SNO, also tend to have a higher proportion of recorded slab avalanches. The importance of SAG as a modifying factor for W.E i s clearly shown in Table XI, i n which correlation coefficient values for SAC*W.E indicate a very substantial improvement over values for W.E, for every individual year. This effect has already been discussed, and i t was f e l t that SAC represents a 'delayed, action effect', which could perhaps be associated with soft slab build-up. It i s also worth noting that high values of SAC imply the participation of more snow layers i n the avalanching and the a v a i l a b i l i t y of a greater number of potential s l i d i n g surfaces. Furthermore, SAC may contain a settlement effect, a high value of SAC indicating less settlement, and hence, greater i n s t a b i l i t y . Except for 1971-72 and 1972-73, correlation coefficient values for SWH are significantly better than those for SAC*W.E. Values for SWHT, on the other hand, are generally lower than those for SWH, except for the years;, 1967-68 and 1970-71. The possible influence of humidity on the build-up of slab conditions has been f u l l y described by Seligman (1936:195), who refers to the mechanism of slab formation as a condensation of water 65 Table XI Reduced Set of Correlations for Individual Years Rogers Pass Meteorological Data, F i r s t Parts, Daily. Observations, A l l Sites, AN - SNO W.E SAC*W.E SWH SWHT SWHTV 1965-66 (12,1,1) SML 73 71 82 83 62 46 (12,3,1) SML 74 70 83 83 58 43 (1,1,1) ML 72 66 80 80 52 38 1966-67 (12,1,1) SML 69 73 84 84 80 69 (12,3,1) SML 68 72 84 84 80 69 (1,1,1) ML 68 73 84 85 81 72 1967-68 (12,1,1) SML 58 59 81 84 85 80 (12,3,1) SML 51 53 77 80 84 79 (1,1,1) ML 50 52 76 79 83 81 1968-69 (12,1,1) SML 67 72 78 79 71 70 (12,3,1) SML 64. 70 79 81 73 71 (1,1,1) ML 62 66 78 80 71 69 1969-70 (12,1,1) SML 71 65 83 84 76 71 (12,3,1) SML 66 61 80 80 71 67 (1,1,1) ML 59 55 75 75 63 60 1970-71 (12,1,1) SML 50 68 76 78 79 77 (12,3,1) SML 48 6 ? •79 • 80 82 80 (1,1,1) ML 48 65' 79 81 83 79 1971-72 (12,1,1) SML 78 76 82 81 67 63 (12,3,1) SML 77 76 84 83 65 62 (1,1,1) ML 74 73 81 80 65 62 66 Table XI continued SNO W.E SAC*W.E SWH SWHT SWHTV 1972-73 (12,1,1) SML 76 78 83 82 79 75 (12,3,1) SML 73 76 82 81 81 71 (1,1,1) ML 66 70 74 73 80 64 vapor onto crystals or crystal fragments, brought together by moderate winds, and their subsequent cementing together. Avalanching often occurs at the Rogers Pass, along with humidities of Q0% or over. Substantial levels of ac t i v i t y have also been observed i n the data, for the f i r s t parts of the seasons, in association with humidities of over 85%, temperatures just above freezing, and some r a i n f a l l , measured at the Rogers Pass observatory. There are three such days i n the f i r s t part of I967-68, and another three for 1970-71, which undoubtedly give r i s e to the s l i g h t l y higher correlation values for SWHT, as opposed to SWH. However, i t i s unlikely that a great deal of precipitation f e l l i n the form of rain on the upper slopes, since temperatures were so close to freezing for these periods. Hence, although a portion of the humidity effect may be attributable to r a i n f a l l , indeed humidity may be regarded as a good indicator of r a i n f a l l , the major influence of humidity i s probably related, to slab formation, particularly since SWHT correlation values are lower than SWH values for the f i r s t parts of most of the winters. The non-linear effect of wind has already been mentioned. However, even though the formation of slab conditions requires only moderate winds, i t might be expected that, for some years at least, wind should play a 6 ? more important role than the results of this analysis indicate. Wind terms may be more important than the data suggests, for the following reasons. There are several periods of missing observations, during which the anemovane was not operational, often because of icing problems. This frequently occurs during the height of snow storms, just when the measure-ments would be most significant. Since wind speed and direction varies radically from day to day, and indeed, even from hour to hour, i t seemed only reasonable to substitute eight year averages for these missing values, rather than try to interpolate between measured values, particularly since the periods of missing wind speeds may be up to two weeks i n length. Daily mean wind speed and directions were used in the analysis, but as Schaerer suggests,*^ maximum values should be more significant, since gusting speeds may be a better indicator of wind effects i n the upper zones. Indeed, i t may be necessary to use more frequent wind observations,, per-haps three or six hourly, since this parameter, above a l l others, varies most radically. Although the MacDonald West Shoulder wind station i s ideally located, as far as point measurements are concerned, and may be entirely representative of the area as a whole, a network of stations, or even one other station, perhaps in the Hermit Meadows, would be quite advantageous in providing better control of these observations. It i s hard to believe that one station can accurately describe the wind conditions on both sides of the Pass. In conclusion, the results of this phase of the analysis Indicate that, for the f i r s t parts of individual years, models based on SWH terms Personal communication 68 should achieve a high level of precision. After presenting a b r i e f out-li n e of the theory of time series procedures i n the next chapter, models w i l l be developed i n Chapter VII for the f i r s t parts of individual years, and for total f i r s t parts of the entire period, 1965-73, using (12,1,1) SML, (12,3,1) SML and (1,1,1) ML weights for the avalanche a c t i v i t y index, and SNO, W.E, SWH and SWHT as independent variables. Of course, models could be developed using any of the significant meteorological factors, such as are indicated in Table IV. For example, models based on wind terms or temperature terms could be designed. However, because a l l the meteoro-logi c a l factors are intercorrelated and since i t i s desirable to produce the strongest possible models, i t i s best to concentrate on those indepen-dent variables given above, which are most highly correlated with avalanche a c t i v i t y . 69 Chapter VI TIME SERIES ANALYSIS In this chapter, procedures used i n the development of multi-linear time series models, which best describe the processes governing the association between avalanche a c t i v i t y and the various meteorological factors, are discussed. Such procedures involve the determination of a suitable transfer function, or dynamic input-output relationship for the system, based on discrete observations, which are equispaced i n time, after which the stochastic*^ noise component i s identified and estimated in terms of an autoregressive, moving average or mixed process, as defined by Box and Jenkins (1970). "The stochastic models we employ are based on the idea, (Yule,1927), that a time series i n which successive values are highly dependent can be usefully regarded as generated from a series of independent 'shocks', a^. These shocks are random drawings from a fixed distribution, usually assumed Normal and having mean zero and variance (Box and Jenkins, 1970:8). Some of the concepts involved in this approach w i l l now be i l l u s -trated together with an outline of the specific procedures employed i n this study. For a more complete and thorough exposition of the theory of linear time series processes, the reader i s advised to consult Box and Jenkins. Stochastic Processes F i r s t Order Autoregressive Process. The f i r s t order autoregressive (Markov) process, AR(l), (Box and Jenkins, 1970:56) i s of considerable * A s t a t i s t i c a l phenomenon that evolves i n time according to probabilistic laws i s called a stochastic process. 70 practical importance and can be represented by the following equation, Z t = ^1 Z t - 1 + a t ' ^) where z^ and ^ are values of the series (usually deviations from the mean), at times t and t-1 respectively, <fc>^ i s the f i r s t order autoreg-ressive coefficient and a^ i s the residual error or random 'white noise' term at time t. <p ^  must satisfy the condition, -1 < <p^ < 1, for the 17 process to be stationary. 1 Rearranging equation ( l ) , (1 - cf>t B) z t = a t, (2) where B i s the backward s h i f t operator defined by, B z t = Z t - l « Hence, dividing throughout in equation (2) by (1 - <£^B) gives, z t = (1 + ^ B + 4>\ B 2 + ...) a t . (3) That i s , z^ . can be expressed i n the form of an i n f i n i t e moving average process (Box and Jenkins, 1970:10). The autocorrelation function, Y k f z z />, = ~ = t^-a6 z t Z t - k > k = 1, 2 oo, (4) K y0 oo 2 + ** t=-oo where y k i s the covariance and YQ i s the variance, i s a powerful tool used in the identification and estimation procedures (Box and Jenkins, 1970:28). Multiplying throughout in equation (1) by z^ ^ results i n , Z t - k Z t - * 1 Z t - k Z t - 1 + Z t - k a t ' <5> A stationary process i s said to be s t r i c t l y stationary i f i t s properties are unaffected by a change of time origin. 71 and taking expectations, Y k - 4>t Y k - 1 » k > 0. (6) Note that the expectation E [z^ ^  a.J vanishes when k > 0, since ^ can only involve the shocks a. up to time t-k, which are uncorrelated 3 with a^. Dividing throughout i n (6) by YQ» ^ k = Fk-V k > °' (?) which i s the Yule-Walker equation for the f i r s t order autoregressive process. Setting /> = 1, equation (7) has the solution, />k-4>J, k > 0 . (8) Hence, the autocorrelation function decays exponentially to zero, when <p ^  i s positive, but decays exponentially to zero and oscillates in sign when 0 ^ i s negative. In particular, i t should be noted that, ?l - <PV . (9) In general, f i n i t e autoregressive processes of any order p, that i s , AR(p) processes, have unique autocorrelation functions, and therefore, in principle, the characteristic features of these functions can be used to identify the processes from which they are generated. For f i n i t e time series, the autocorrelation function can be estimated from, r k = c k / c 0 , (10) i N" k °k ~ N + ? : Z t Z t - k ' k - °. 1. 2 ' K -ck and C Q are the sample covariance and variance respectively, N i s the number of observations in the sequence, r ^ i s called the sample autocor-72 relation function. "In practice, to obtain a useful estimate of the auto-correlation function, we would need at least f i f t y observations and the estimated autocorrelations r ^ would be calculated for k =0, 1, 2, K, where K was not larger than say N/4" (Box and Jenkins, 1970:33). General Autoregresslve Process. The general autoregressive process of order p, that i s , the AR(p) process can be written, Z t " 0 1 Z t - 1 + 0 2 Z t - 2 • » . • * p V p + a f (") Multiplying throughout i n ( l l ) by z. T and taking expectations, gives, X»"~rv ^ = *1 ^k - 1 + * 2 ^k - 2 + + ^p/'k-p' k > 0 > ( 1 2) which i s analogous to the difference equation sati s f i e d by the process i t -self. Substituting, k = 1, 2, ..., p in (12), the following set of linear equations for <p^,4>^, ..., i n terms of f>^, ..., i s obtained, A = 0 i + ^2 /°i + ••• + *p />p-l» ^2 * 0 1/' 1 + 4>2 + ... + <pp . . (13) • • • • • • • • />p " ^ 1 /'p-l + * 2 / p - 2 + + V These are usually called the Yule-Walker equations from which estimates of the parameters can be obtained by replacing the theoretical autocorrelations, f^i by the estimated autocorrelations r f c (Box and Jenkins, 1970:54-56). The equations are identical to the reduced 'normal' equations of multi-linear regression analysis, which lead to the familiar least squares e s t i -mates . Another useful tool i n the identification process i s the pa r t i a l autocorrelation function, which i s defined as the l a s t autoregression 73 coefficient obtained after successively f i t t i n g increasing orders of autoregressive process to the data. "For an autoregressive process of order p, the pa r t i a l autocorrelation function, 0 ^ w i l l be nonzero for k less than or equal to p and zero for k greater than p" (Box and Jenkins, 1970:65). As a useful general rule i n f i t t i n g autoregressive models of order p, the autocorrelation function of a stationary autoregressive pro-cess i s i n f i n i t e in extent and consists of a mixture of damped exponentials and damped sine waves, whereas the pa r t i a l autocorrelation function i s f i n i t e with a cutoff after lag p. F i r s t Order Moving Average Process. The f i r s t order moving average process, MA(1), (Box and Jenkins, 1970:69) i s represented by the following equation, Z t " a t " 61 a t - l ' which has the following alternative forms, Z t = ^ " 61 B) a t ' a n d ' ( 1 5 y a t = (1 + 6jB + 0 2 B 2 + ...) z t . (16) Hence, a^ can be expressed i n the form of an i n f i n i t e autoregressive process. Multiplying throughout in equation (14) by z^ results i n , V k z t • ( a t - k - e i at-k-i> K - e i a t - i > <17> and on taking expectations, ~ 61 • , k = 1, I i + G " (18> K 1 0, k > 2. Thus, in contrast to the AR(p) process, the autocorrelation function for the MA(q) process has a cutoff after lag q and the partial autocorrelation 74 function t a i l s off, and i s dominated by a mixture of damped exponentials and damped sine waves. Mixed Process. "To achieve greater f l e x i b i l i t y in f i t t i n g of actual time series, i t i s sometimes advantageous to include both auto-regressive and moving average terms in the model" (Box and Jenkins, 1970: 11). The f i r s t order mixed autoregressive-moving average process, ARMA(l,l) (Box and Jenkins, 1970 :?6) i s represented by the following equation, Z t " *1 Z t - 1 * a t " 61 at-l« <19> that i s , Therefore, (1 - 4>1 B) z t = (1 - 6 l B) a t . (20) z t = (1 - 61 B ) ( l - 0 1 B)" 1 a t, (21) a t = (1 - 0 1 B ) ( l - et B ) " 1 z t . (22) Hence, both the autocorrelation and the p a r t i a l autocorrelation functions are i n f i n i t e in extent. If the stochastic time series exhibits non-stationary behaviour, usually indicated by a slow and linear tapering of the autocorrelation function, i t may be necessary to apply some degree of differencing to the data (Box and Jenkins, 1970:85-119). Of particular interest i n this respect i s the f i r s t order auto-regressive integrated moving average process, ARIMA(1,1,l), represented by the following equation, W t " *1 w t - l = a t " e l at-l» (23) where, wt = v z t, (24) and V i s the difference operator defined by, 75 V = 1 - B (25) that i s , v z t = z t " z t - r <26) Transfer Function Representation Let, y t = v(B) x t, (27) be a linear representation of a deterministic process, known as a linear f i l t e r , where x^ and y^ are the independent variable (input series) and dependent variable (output series), respectively, and, x t m ( X t - X), and y t - < yt - Y>-The function, v(B) - (v Q + v1 B + v 2 B 2 + . . . ) , (28) is called the transfer function of the process. The weights, v^, v^, v^, are called the impulse response function (Box and Jenkins, 1970:14), The linear f i l t e r i s stable i f the transfer function converges, that i s , i f the series i s f i n i t e or i n f i n i t e and convergent. For 'real' data, y t = v(B) x t + n t, (29) where v(B) i s a deterministic transfer function and n^ i s stochastic noise, with x^ and n^ assumed independent, x^ i s also assumed to be a stochastic noise process (Box and Jenkins, 1970:371). n t - y(B) a t, (30) where y(B) i s a stochastic transfer function, that i s , n, i s the output 76 of a linear f i l t e r whose input i s the 'white* noise process, a^ .. A l l ARIMA processes can be represented in this way. Alternatively, (29) may take the form, y t = 6 " 1 (B) w(B) x t + cp"1 ( B) e(B) a t . (31) The cross correlation function, as defined by the following equation, Y Y V(k) "xy xy % ' c o j x wy OO . z x t - k v t t=-oo + * x t - k + 2 y t k - +1, +2 ± oo, (32) where v (k) i s the covariance, and a , c are the standard deviations, i s a powerful tool used in the identification and estimation procedures. For f i n i t e time series, the cross correlation function can be estimated from, c (k) r v v ( k ) - - S L . . (33) where, xy^ ' s s J x y . N-k c x y 0 O = | I x t_ R y t, k = 0, 1, 2 K. c (k) i s the covariance and s and s are sample standard deviations for xy v ' x y x and y. N i s the number of observations and r (k) i s called the sample xy 18 cross correlation coefficient. "In practice, we would need at least 50 pairs of observations to obtain a useful estimate of the cross correlation 18 Unlike the autocorrelation function, the cross correlation function i s generally asymmetrical. One half i s 'physically realizable' and i s known as the 'memory function', while the other half i s 'not physically realizable' and i s called the 'anticipation function'. Only the memory function i s here defined. 77 function" (Box and Jenkins, 1970:374). Suppose that, y t " V 0 X t + > 1 X t - 1 + v 2 X t - 2 + + V W i s a linear transfer function model, where and are the output and i n -put series respectively, suitably differenced to induce stationarity, and VQ, VJ^ » ...» etc. i s the impulse response function. "We assume that a degree of differencing, d, necessary to induce stationarity has been achieved when the estimated auto- and cross correlations r (k), r (k) xx v " yy v ' and r xy(k) of x^ = V d X^, and y^ = V d Y^ damp out quickly. In practice, d i s usually 0, 1, or 2" (Box and Jenkins, 1970:378). Multiplying through-out in equation (34) by x^ ^, gives, x t - k y t - v o x t - k x t + v i x t - k x t - i + ••• + x t - k V ( 3 5 ) Assuming that x^ ^ and n^ are uncorrelated for a l l k > 0, then, taking expectations, Y x y ( k ) = v 0 Y x x ( k ) + v l Y x x ( k _ 1 ) + «... k = 0. 1. 2 (36) or, /> x y( k) - f x [ v 0 / x x ( k ) + v l/ ? xx ( k " 1 ) » k = 0 ' 1 » 2 " . . » ( 3 7 ) y which are similar to the Yule-Walker equations. However, these equations, analogous to the reduced 'normal' equations of regression analysis, do not in general, provide e f f i c i e n t estimates of the transfer function coef-f i c i e n t s . Considerable simplification i n the identification procedure i s achieved and more e f f i c i e n t estimates of the parameters obtained i f the i n -put and output series are 'prewhitened' prior to analysis (Box and Jenkins, 1970:379). The procedure i s as follows. Given that, y t = v(B) x t + n t, (38) 78 the series x^ i s represented by the ARIMA model, 0 X(B) e^(B) x t - a t, (39) which, to a close approximation, transforms the correlated input series, x^, into the uncorrelated white noise series a^. The same transformation i s applied to y^ to obtain, B t = 0 x(B) e^(B) y t. (40) Hence, equation (38) can be written, P t - v(B) a t + € t , (41) where € ^ i s the transformed stochastic noise series, £ t = 0 x(B) 6 X X ( B ) n t. (42) Multiplying throughout in equation (41) by k and taking expectations, gives, V k ) = v k V ( 0 ) v <*3> since ct^ . i s a white noise process. Thus, 2 0\ v k = -^V-' (44) a or in terms of cross correlations, v k aP g P ' k = 0, 1, 2 (45) a Hence, i n i t i a l estimates for the transfer function coefficients v. t roay be K. obtained dir e c t l y from the sample cross correlation function, using equation (45) rewritten i n terms of sample estimates, that i s , r * f i ( k ) Sft k = 0, 1, 2 (46) K s a 79 In practice, least squares estimates are obtained after prewhitening the input and output series, using an a l l combination approach, since pre-whitening i s always imperfect. The stochastic noise process may now be identified and estimated, since, n t = y t - 0(B) x t. • (47) An ARIMA model i s f i t t e d to the estimated noise process of the form, n t = 0^(B) e*(B) a t, (48) giving the total model, y t = v(B) x t + 0" 1(B) e A(B) a t . (49) Since the transfer function and stochastic noise components are identified separately, estimates of the parameters are necessarily inef-f i c i e n t (Box and Jenkins, 1970:386). To obtain more e f f i c i e n t estimates after the identification procedures, the transfer function and noise models may be combined in a single least squares estimation. If more than one input series i s used, the prewhitening and trans-fer function estimation procedures are repeated for the second input series and an output series, formed by subtracting the transfer function for the f i r s t series from the original output series. Symbolically, y + - V ( B ) X l t = U^ B) X 2 t + n t ' (50) replaces equation (38), where x ^ and x ^ are the two input series. This procedure can be repeated for any number of input series. Computational Procedures An extremely flexible operational system was devised for the development of optimum transfer function/stochastic noise models for 80 avalanche forecasting. The main instrument i n these procedures i s a computer program, which was written s p e c i f i c a l l y for this study, incor-porating a l l the features described in the foregoing theory. Using the program, the following steps are performed. 1) Autocorrelation functions for the input series, together with cross correlation functions between input and output are computed up to twenty lags, and examined for stationarity. In the case of avalanche act i v i t y , and most of the composite meteorological series, the correlation functions decayed s u f f i c i e n t l y rapidly such that differencing was unnecessary. Tests have been carried out using f i r s t and second differences to see whether such treatment would lead to more powerful models. The contrary 2 19 seems to be the case, since although R - values y were increased, this was offset by an increase i n the dispersion of the data, such that the residual errors were just as high as i n the undifferenced data. Differen-cing also resulted in an increase in the stochastic noise component re-sulting in more complex models, an undesireable feature of the process. 2) Partial autocorrelation functions are computed for the input series up to twenty lags and suitable ARIMA models describing these processes iden-t i f i e d and estimated. 3) The ARIMA model describing the f i r s t (primary) input series i s used to transform this series into an approximate white noise process, and the same transformation applied to the output series. 4) The transfer function i s then identified and least squares estimates obtained for the transformed input and output series using ah a l l com-19 Multiple correlation coefficient squared 81 bination approach up to five lags. 20 5) I f a secondary input series i s contemplated, the transfer function obtained in step 4 i s subtracted from the output series and steps 3 and 4 repeated using this secondary series, 6) Finally, the complete transfer function i s subtracted from the output series to obtain the stochastic noise process. 7) The autocorrelation and pa r t i a l autocorrelation functions for the stochastic noise process are computed up to twenty lags and a suitable ARIMA model describing the process identified and estimated. 8) The transfer function and noise models are then combined and e f f i c i e n t least squares estimates of the parameters obtained. Any insignificant terras are eliminated and the f i n a l , complete model re-estimated. 9) In this l a s t step, tests of model adequacy are performed as described by Box and Jenkins (1970:392-5). Among other things, insignificant autocorrelation i n the residuals i s confirmed. The selection of the secondary series i s achieved by calcula-ting p a r t i a l correlation coefficients, as described i n Chapter V, after the effect of the primary independent (meteorological) variable has been subtracted from the dependent (avalanche activity) variable. 82 Chapter VII THE TIME SERIES MODELS Employing the procedures outlined in Chapter VI, transfer function/ stochastic noise functions were developed using the three best sets of weights for avalanche ac t i v i t y , as defined in Chapter V. These models are l i s t e d in Table XII, along with appropriate s t a t i s t i c s indicating their strength. Models for individual years, besides the entire period, 1965-73, are quoted for f i r s t parts only, as defined i n Table VIII. SNO, W.E, SWH, and SWHT, obtained from the Rogers Pass meteorological measurements, were used as input series and avalanche a c t i v i t y indices were computed for a l l sites, a r t i f i c i a l and natural avalanches combined and (12,1,1) SML, (12,3, 1) SML, and (1,1,1) ML weights. The models have been l e f t in their trans-fer function/stochastic noise form so that their basic structure can be better i l l u s t r a t e d . A f i n a l least squares estimation would be performed prior to their implementation as prediction models. As anticipated from the results of the correlation analysis, given in Table XI, the highest R - values were realized with SWH models for individual years, with the exception of 1970-71, which has somewhat stronger SWHT models. To il l u s t r a t e the procedures involved, by way of an example, the development of the SWH model for 1967-68 using (12,1,1) SML weights w i l l now be shown. It should be noted that each series i s reduced to dev-iations from i t s mean, prior to analysis. SWH Model for 1967-68 l ) Correlation functions decrease to insignificance at or before the third lag. Therefore, stationarity i s assumed and no differencing i s 83 applied. The number of observations, N = 95. For the A V A L series, the corrected sum of squares, S S T O T ( A V A L ) - 1353670, hence, the standard deviation, S D ( A V A L ) = 1 2 0 . 0 . For the SWH series,, the corrected sum of squares, SS T 0 T(SWH) = 4593.87, and, SD(SWH) = 6 . 9 9 1 . 2) Inspection of the autocorrelation and part i a l autocorrelation functions suggests a (1 ,0,0) model for the SWH series. Hence, 0 1 = 0.3362, that i s , SWH = 0.3362 SWH1 + a* rearranging, (1 - 0.3362 B) SWH = a. 3) Thus, the prewhitening operator i s (1 - O.3362 B) a = ( 1 - 0.3362 B) SWH, !; 6 = (1 - 0.3362 B) A V A L . 4) The transfer function obtained by least squares i s , 0 = 13.02 a + 3.683 a 1 . 5) Hence, NSE = A V A L - 13.02 SWH - 3.683 SWH1, i s the noise series. For the NSE series, the corrected sum of squares, For simplicity, the 't' subscript notation has been dropped. S W H t - l h a s t h e a b t , r e v i a " t e d f o r m SWH1, etc., and a t ^ has the abbreviated form etc. 84 S S T Q T ( N S E ) = 329660. 6) Inspection o f the a u t o c o r r e l a t i o n and p a r t i a l a u t o c o r r e l a t i o n functions i n d i c a t e s t h a t the noise s e r i e s i s e s s e n t i a l l y random, tha t i s , there i s no s t o c h a s t i c noise component f o r t h i s model. 7) Hence, the complete model may be written, AVAL - 13.02 SWH + 3.683 SWH1, and re-estimated more e f f i c i e n t l y using l e a s t squares. The re-estimated model i s , AVAL - 13.08 SWH + 4.010 SWH1, 2 which has an R of O.756. Figure 2 shows p r e d i c t e d values obtained with t h i s model, p l o t t e d together with a c t u a l values of the avalanche a c t i v i t y index. Thus, a very close agreement has been achieved by use of t h i s simple two-term model based on the independent v a r i a b l e SWH. Besides the unlagged term, SWH, the f i r s t l a g term i s h i g h l y s i g n i f i c a n t and would have made a strong cont-r i b u t i o n to r e a l - t i m e f o r e c a s t s of avalanche a c t i v i t y f o r the period. Since SWHl would be p r e c i s e l y known a t the time the f o r e c a s t i s made, the model does not r e l y e x c l u s i v e l y on the weather f o r e c a s t . S i m i l a r accuracies could be achieved f o r the other years, using the SWH models described i n Table XII, Models f o r I n d i v i d u a l Years A number of conclusions may be drawn a f t e r c l o s e examination of a l l 2 the models depicted i n Table XII, R - values f o r the models, c o n s i s t e n t l y d i s p l a y a progressive increase f o r (1,1,1) ML weights, to (12,3,1) SML weights, up to (12,1,1) SML weights. SWH models are g e n e r a l l y b e t t e r than SWHT models, which are always b e t t e r than W.E models. W.E models are more 85 powerful than SNO models, except for the years, 1965-66, 1969-70, and 1971-72. Models for the entire period, 1965-73, have more significant lagged 2 terms than those for Individual years, but R - values are lower as a result of the larger sample size. No models have terms which are higher than the thi r d lag, a significant result. Models for each individual year appear to be structurally quite unique, but some si m i l a r i t i e s do exist. SNO models for 1966-67 and I968-69 both consist of only one term, the unlagged SNO term, for which the coefficients are i n close agreement. W.E models for these years are also similar. However, the SWH models d i f f e r s l i g h t l y in structure, but the unlagged SWH coefficient values are almost identical for these two years, the same argument applying to the SWHT models. The SNO model for the 1972-73 year also consists of only one term, the unlagged SNO term, but the coefficient i s higher than those for the two years just mentioned, indicating that equal amounts of precipitation produced more avalanching i n 1972-73, than i n I966-67 or I968-69, possibly a temperature effect. It i s worthwhile to examine the models depicted i n Table XII quite closely, and attempts can be made to group years together according to the class of model describing their avalanche a c t i v i t y . SWH.tt.E*TMI Model for 1965-73 However, for the practical forecasting of avalanches, i t i s neces-sary to have, at one's disposal, a single general model, which can be applied without having to assign the winter to a particular class. To this end, the SWH model for the entire period, 1965-73, using (12,1,1) SML weights, was developed and a secondary series, the W,E*TMI series, incorporated into the model in order to improve i t s forecasting accuracy. 86 Table XII - Time Series Models f o r Avalanche A c t i v i t y , Daily Observations, Rogers Pass Meteorological Data, F i r s t Parts, A l l S i t e s , A r t i f i c i a l and Natural Avalanches Transfer Function Stochastic Noise Overall SD Year SNO SNOl' SN02 SN03 NSE1 NSE2 NSE3 R 2 SE 1. SNO Models (12,1,1) SML 1965-73 6.740 I.252 .6009 .1490 .0695 . .0926 .494 103.3 73.6 1965-66 4.275 2.005 .2423 .636 79.3 48.2 1966-67 7.367 .485 126.1 90.5 1967-68 6.941 4.771 .2438 .2803 .508 120.0 85.2 1968-69 6.347 - ..•451 78.4 58.1 1969-70 8.564 .2554 .527 80.0 55.3 1970-71 4.966 2.578 .2263 .289 114.2 96.8 1971-72 7.147 1.767 .620 117.5 72.5 1972-73 9.737 .566 114.3 75.3 (12,3,1) SML 1965-73 12.87 2.365 1.347 .1843 .09096 .466 205.7 150.6 1965-66. . 8.546 3.377 .600 154.1 97.4 1966-67 14.47 .467 250.2 182.7 1967-68 13.27 10.41 .2721 .2913 .4?8 263.3 192.4 87 Table XII continued Transfer Function Stochastic Noise Overall SD Year SNO SN01 SN02 SN03 NSE1 NSE2 NSE3 R2 SE 1968-69 12.98 .423 1 6 5 . 5 125.7 1969-70 14.82 .2684 .468 146.5 107.5 1970-71 8.922 4.947 .3095 .299 224.3 188.8 1971-72 14.20 3.819 .616 240.1 148.7 1972-73 15.42 .522 191.0 I32.I (1,1,1) ML 1965-73 . 5242 .0815 .0516 .0579 .1983 .438 8.60 6.45 1965-66 .3525 .1048 .550 6.34 4.25 1966-67 .6258 .462 10.80 7.93 1967-68 .5322 .4259 .2261 .2650 .441 10.60 7.99 1968-69 .5234 .398 6.88 5.34 1969-70 .6416 .2564 .386 6.84 5.39 1970-71 .3748 .4423 .321 9.72 8.06 1971-72 .5726 .1427 .566 10.10 6.63 1972-73 .4535 .1711 .499 6.61 4.68 8 8 Table XI I 'cont inued Transfer Function Stochast ic Noise Overal l SD Year W.E W.El. W.E2 W.E3 NSE1 NSE2 NSE3 . R 2 SE 2 . W.E Models ( 1 2 , 1 , 1 ) SML 1 9 6 5 - 7 3 9 . 5 0 4 1 . 1 3 4 .1674 . 0 8 1 3 . 5 3 3 1 0 3 . 3 7 0 . 7 1 9 6 5 - 6 6 6 . 3 6 3 2 . 0 8 6 . 3 6 5 3 . 6 0 1 7 9 . 3 5 0 . 4 1 9 6 6 - 6 7 11.08 .540 1 2 6 . 1 8 5 . 5 1 9 6 7 - 6 8 8 .194 3 . 0 9 7 . 2 7 2 8 . 2 6 7 2 .512 1 2 0 . 0 8 4 . 8 I 9 6 8 - 6 9 8 . 7 6 9 . 5 1 2 7 8 . 4 5 4 . 8 I 9 6 9 - 7 O 9.880 . 2 2 9 4 . 4 5 3 8 0 . 0 5 9 . 5 1 9 7 0 - 7 1 9.505 . 4 5 6 114 .2 84.3 1 9 7 1 - 7 2 9 . 8 2 2 . 5 7 7 1 1 7 . 5 7 6 . 4 1 9 7 2 - 7 3 1 1 . 8 9 . 6 1 1 114.3 7 1 . 2 ( 1 2 , 3 , 1 ) SML 1 9 6 5 - 7 3 18.33 2 . 0 5 9 . 2 0 0 6 . 0 9 3 6 5 . 5 1 1 205.7 144 .1 I 9 6 5 - 6 6 1 2 . 6 1 . 3 2 8 8 .541 1 5 4 . 1 105.0 1966-67 2 1 . 7 1 . 5 2 6 250.2 1 7 2 . 3 1 9 6 7 - 6 8 1 5 . 8 9 6 .401 . 3 0 8 9 . 2 7 2 9 . 4 7 9 2 6 3 . 3 192.0 89 Table XII continued Transfer Function Stochastic Noise Overall SD Year W.E W.E1 W.E2 W.E3 NSE1 NSE2 NSE3 R2 SE 1968-69 18.07 .488 165.5 118.4 1969-70 17.23 .2477 .408 146.5 U3 .3 1970-71 18.05 .2582 .477 224.3 163.0 1971-72 19.81 .574 240.1 156.7 1972-73 19.37 .580 191.0 123.8 (1,1,1) ML 1965-73 .7428 .06691 .06649 .2120 .479 8.60 6.21 1965-66 .4910 .2880 .475 6.34 4.62 1966-67 .9493 .538 10.80 7.36 1967-68 .6350 .2505 .2589 .2478 .428 10.60 8.07 . I968-69 .7146 .442 6.88 5.14 1969-70 .7518 .2411 .343 6.84 5.57 1970-71 .7495 .3497 .486 9.72 '7.01 1971-72 .8007 .529 10.07 6.91 1972-73 .6013 .489 6.61 4.73 90 Table XII continued Transfer Function Stochastic Noise Overall SD Year SWH :.SWH1 SWH2 SWH3 NSE1 NSE2 NSE3 R 2 SE 3. SWH Models (SWH/IO^) (12,1,1) SML 1965-73 8.307 .1136 .0850 .624 103.3 63.5 1965-66 6.122 . .675 79.3 45.2 1966-67 9.770 -1.250 •. 2092 .729 126.1 65.9 1967-68 13.02 3.683 .756 120.0 59.2 1968-69 9.418 .619 78.4 48.4 1969-70 13.99 .2166 .722 80.0 42.4 1970-71 7.445 .596 114.2 72.7 1971-72 6.181 -1.440 .658 117.5 68.7 1972-73 11.77 .666 114.3 66.0 (12,3,1) SML 1965-73 16.53 .1202 .09559 .633 205.7 124.9 1965-66 12.20 .683 154.1 86.7 1966-67 19.51 -2.702 .722 250.2 131.9 1967-68 26.42 8.359 .691 263.3 146.3 91 Table XII continued Year Transfer Function Stochastic Noise SWH SWH1 SWH2 SWH3 NSE1 . NSE2 NSE3 Overall R 2 SD SE 1968-69 20.17 .651 165.5 97.8 1969-70 24.78 .2765 .671 146.5 84.5 1970-71 14.63 .629 224.3 136.7 1971-72 12.73 -2.485 .690 240.1 133.6 1972-73 19.56 .650 191.0 112.9 (1,1,1) ML 1965-73 .6777 .1227 .606 8.60 5.40 1965-66 .4747 .633 6.34 3.84 1966-67 .8493 -.1220 .2023 .741 10.80 5.54 1967-68 I.069 . 3000 .669 10.60 6.09 1968-69 .8184 .633 6.88 4.17 1969-70 1.104 .2981 .596 6.84 4.37 1970-71 .6274 .637 9.72 5.86 1971-72 .5170 -.1248 .643 10.07 6.01 1972-73 .5920 .530 6.61 4.53 92 Table XII continued Transfer Function Stochastic Noise Overall SD Year SWHT SWHT1 SWHT2 SWHT3 NSE1 NSE2 NSE3 ' R SE 4. SWHT Models (SWHT/105) (12,1,1) SML 1965-73 4.088 .1207 .0764 .0881 .580 103.3 67.0 1965-66 3.136 .4013 .470 79.3 58.1 1966-67 4.216 -.7841 .664 126.1 73.1 1967-68 5.727 .724 120.0 63.I 1968-69 4.479 .507 78.4 55.1 1969-70 6.322 .2656 .605 80.0 50.6 1970-71 3.286 .620 114.2 70.4 1971-72 3.204 -.7935 .1799 .479 117.5 85.2 1972-73 4.777 .621 114.3 70.3 (12,3,1) SML 1965-73 8.011 .1314 .09506 .09460 .575 205.7 134.4 1965-66 5.844 .3752 .425 154.1 117.6 1966-67 8.352 -1.647 .659 250.2 146.1 1967-68 12.07 2.402 .732 263.3 136.3 93 Table XII continued Transfer Function Stochastic Noise Overall SD Year SWHT SWHT1 SWHT2 SWHT3 NSE1 NSE2 NSE3 R 2 SE 1968-69 9.554 .526 165.5 114.0 1969-70 10.98 . 3035 .537 146.5 100.4 1970-71 6.395 .670 224.3 128.9 1971-72 6.179 .1983 .441 240.1 180.3 1972-73 8.022 .651 191.0 112.8 (1,1,1) ML 1965-73 .3323 .1283 .0746 .559 8.60 5.72 1965-66 .2107 .3237 .342 6.34 5.17 1966-67 .3678 -.0760 .679 10.80 6.14 1967-68 .4928 .691 10.60 5.87 1968-69 .3855 .503 6.88 4.85 1969-70 .4797 .3083 .448 6.84 5.11 1970-71 .2700 .678 9.72 5.52 1971-72 .2668 -.0758 .1978 .455 10.07 7.46 1972-73 .2555 .624 6.61 4.05 95 The following outlines the steps involved i n obtaining t h i s model. 1) Correlation functions decrease to in s i g n i f i c a n c e at or before the s i x t h l a g . Therefore, s t a t i o n a r i t y i s assumed and no di f f e r e n c i n g i s applied. Since auto, p a r t i a l and cross c o r r e l a t i o n c o e f f i c i e n t s were calculated up to nine lags, data f o r i n d i v i d u a l years were separated by nine sets of zero values. This prevents any overlap of data between years from a r t i f i c i a l l y i n f l u encing the values of these c o e f f i c i e n t s . Hence, the number of observations, N = 824. For the AVAL s e r i e s , the corrected sum of squares, SS T Q T(AVAL) - 8782950, and hence, the standard deviation, SD(AVAL) = 103.3. For the SWH s e r i e s , the corrected sum of squares, SS T Q T(SWH) = 81708.7, and SD(SWH) = 9 . 9 6 4 . 2) Inspection of autocorrelation and p a r t i a l autocorrelation functions suggests a ( 2 , 0 , 0 ) model f o r the SWH ser i e s . Hence, CP1 = 0 . 4 0 9 6 , <PZ = 0.1047, that i s , SWH = 0.4096 SWH1 + 0.1047 SWH2 + rearranging, (1 - 0.4096 B - 0.1047 B 2) SWH = a.-3) Thus, the prewhitening operator i s (1 - 0.4096 B - 0.1047 B 2) As before, the ' t ' subscript notation has been dropped f o r sim-p l i c i t y . 96 and, a - (1 - 0.4096 B - 0.1047 B 2) SWH, B - (1 - 0.4096 B - 0.1047 B 2) AVAL. 4) Transfer Function obtained by least squares i s , g - 8.307 a. 5) Hence, NSE *= AVAL - 8.307 SWH, i s the noise series, For the noise series, the corrected sura of squares, SS T 0 T(NSE) = 3382620. 6) At this point, transfer function estimation can be terminated and the stochastic noise series identified and estimated, as i s indicated i n Table XII, p. 90. Inspection of autocorrelation and par t i a l autocorrelation functions suggests a (2,0,0) model for the NSE series. Hence, <t>± = 0.1136, 0 2 = 0.0850, that i s , NSE = 0.1136 NSE1 + O.O85O NSE2 + a, which can be rewritten, (1 - O.II36 B - 0.0850 B 2) NSE = a, and the sum of squares residual, s sR E S ( a ) - 3306470. 7) Hence, the complete model can be written, AVAL = 8.307 SWH + a , (1 - 0.1136 B - 0.0850 B ) as indicated in Table XII, or, by multiplying throughout by 97 (1 - 0.1136 B - 0.O850 B 2), AVAL = 0.1136 AVAL1 + O.O85O AVAL2 + 8.307 SWH - 0.9^37 SWH1 -0.7061 SWH2 + a, which can be re-estimated more e f f i c i e n t l y using multiple regression techniques, 8) Partial correlation coefficients, calculated after the effect of SWH was subtracted out, suggest that W.E*TMI might be a good secondary variable. 9) For the W.E*TMI series, the corrected sum of squares, SSTQT(W.E*TMI) = 23408700, and, SD(W.E*TMI) = 168.7. 10) Inspection of autocorrelation and pa r t i a l autocorrelation functions suggests a (1,0,0) model for the W.E*TMI series, Hence, <P± = 0.3336, that i s , W.E*TMI = 0.3336 W.E*TMI1 + a. Rearranging, (1 - 0.3336 B) W.E*TMI = a. 11) Thus, the prewhitening operator i s ( l - O.3336 B). a = (1 - 0.3336 B) W.E*TMI, and, B = (1 - 0.3336 B)(AVAL - 8.307 SWH), 12) The transfer function obtained by least squares i s , B = 0.07211 a. 13) Hence, NSE « AVAL - 8.307 SWH - 0.07211 W.E*TMI, i s the noise series. For the NSE series, the corrected sum of squares, SS T Q T(NSE) = 3235040. 14) Inspection of autocorrelation and partial autocorrelation functions suggests a (2,0,0) model for the NSE series. Hence, <PX = 0.0941, 0 2 = 0.0954, that i s , NSE = 0.0941 NSE1 + 0.0954 NSE2 4 a , which can be rewritten, (1 - 0.0941 B - O.O954 B 2) NSE = a, and the sum of squares residual, S S R E S ( a ) = 3171022. 15) Hence, the complete model can be written, AVAL = 8.307SWH + 0.07211W.E*TMI * * , (1 - 0.0941B - 0.0954B ) or, multiplying throughout by (1 - 0.0941 B - 0.0954 B 2 ) , AVAL = 0.0941 AVAL1 + 0.0954 AVAL2 + 8.307 SWH - 0.782 SWHl -0.792 SWH2 + 0.07211 W,E*TMI - 0.00679 W.E*TMI1 -0.00688 W.E*TMI2 + a. 16) This equation was re-estimated more e f f i c i e n t l y using multiple regression techniques, resulting in, AVAL = 0.0824 AVALl + 0.0819 AVAL2 + 7.027 SWH - 1.020 SWHl -0.313 SWH2 + 0.1220 W.E*TMI + p.0198 W.E*TMI1 -O.O365 W.E*TMI2. 99 17) Since the SVTH2 and W.E*TMI1 terms are Insignificant, the model i s re-estimated, giving, AVAL - 0.0947 AVAL1 + O.O589 AVAL2 + 6.85O SWH - 0.937 SWH1 + 0.1310 W.E*TMI - 0.0350 W,E*TMI2, 2 which has an R - value of O.65I and a standard error of estimate, SE of 61.3. 18) Tests were applied to the residual errors to confirm model adequacy. The residuals were found to be uncorrelated and unbiased and the model therefore adequate. Figure 3 shows predicted values obtained with this model, plotted along with actual values of the avalanche a c t i v i t y index, for the entire period, 1965-73, and an excellent agreement i s demonstrated. This model can be used to predict avalanche a c t i v i t y for any winter.which resembles the class of eight, defined by the period, 1965-73. Confidence Limits The variance of the predicted values of avalanche a c t i v i t y can be estimated for any set of values of the independent variables, using the following expression, V(Y) = s2U'Q C X Q). (51) where V(Y) i s the variance of the predicted value Y, s i s the standard error of estimate for the regression, X^ i s a set of X (in matrix notation) and C = (X'X) - 1 (Draper and Smith, 1966:121). Thus 1 - a confidence limits on the true mean value of Y at X Q are given by, Y ± t ( v , l ' - k ) • s V X • 0 X 0 (52) where v i s the number of degrees of freedom upon which s i s based. HGURE 3 : m : 4-4 -U-U_LJ_U_|_UJ i Q589AVAL2 -f 6-85SWH J J ..L4. _,_r T EDICTI.ON i i i 4 - ; - i 3 t w . i r M _ ! _ l J_LL -1100 f{-{-H-H44 T"> I " ' ; r i " ; 1 T, , . T M ~ T " r r t — n ~ ~ r 1 PROFILES FOR 19651-73 -r-r}-r-rrr\'\ -} ! I I ! i i -937SWH-1 u j ! | i: E^Tfyi i H-Q35 w - i E -*T-ry| i -2 -33 TT T T f T T T T i r r j i i i i AvAlr rt T T J J _u UX i i ;-4-0e-: - 2 0 0 n _i_L i I fT i i -!-L LT i i i i ! I L ! I i 4 U_L i I ! J-_U_!_J_L J _ U _ M M I I i ! i I I c o n t i n u e d , • j ' jTPT iT ' " 1 ' rft-_LJ_ . i i J _ L I i i I ! i T T J _ L L i i I _L -;ro2 I T T J L I L i r I ! T T I I J_LI_I i i i ! F>X'I _L±. i i NOV -DE-ei IdA-N-i i i LI_L_L I ! I i I I I I I i I j T T I I M I I i I T T T T r f i I 1 I I i M M I I T T T I [ M M T T T -AVA4r I i M I M M ^l:970-^H-h M M ! i I I. I I T T ±J ! i J_L ! i I 1 4001 200 M i ! i i I i I T T I I TIT" _L_ : i i M _ i i r T T T i i i ! I i i i i HTTTJi • M i , ; i 1 M l M M t ; i ! 1 j ! ; 1 M • M M ' i i 1 . i 1 • M : - M . M M M 1 | : ' i i' i i I > M . M i M i 1 i i 1 i M M iL±_!_ _! 1 i 1 i 1 i M 1 i M i 1 I M 1 1 1 I • i M i l 1 I.I. J I-l i i I i u ! i | 1 II i ! ! I ! 1 I-l II 1 • i i i | D E C j I I I I i ! I 'i-i I I i i M I I I i T T T i i i I i I i -U-i-T I T T~i i i T T T T T T ! I T i " T " T T | I T T I i i L t fr— - H - - 2 0 0 -I 1 ! i . 1 1 _ U _ ' _ ! _ f— » 1 ! 1 1 ! ! I i ! 1 l • ! i i h • 1 : 1 i t i ! i i 1 1 — i — r -' f I ; : i i . i n i 1 1 i f i. i - f ! 1 I P 1 i ' i i • 1 I 1 1 1 1 1 1'— ii i 1 J—i r i i i 1 f l I . irhb 104 For g future observations, confidence limits on the predicted values can be obtained from, Y + t ( v, 1 - k a) . s Vl/g + X 0 G X Q (53) (Draper and Smith, 1966:122). Hence, estimates can be obtained, of the r e l i a b i l i t y of the pre-dictive model to forecast future values of the dependent variable. Assuming that the sample of observations from which the model was develo-ped was large and truly representative of the population of past and future observations of avalanche a c t i v i t y and weather factors, the 95% confidence limits for a single future forecast are, Y + (1.96) . s VI + X ' C X Q , (54) which has minimum lim i t s of Y ± (1.96) . s (55) at the mean values of a l l X Q . Thus, assuming complete knowledge of a set of future X Q values, that i s a perfect weather forecast, a future forecast of avalanche a c t i v i t y w i l l have a 95% probability of lying between the limits, Y + (1.96) . s, i f the set of X ^ are mean values. These limits w i l l of course become wider as the set of X Q departs from i t s mean values, as described by expression (54). The standard error of estimate after f i t t i n g the SWH,W.E*TMI model for the period, 1965-73, was 61.3 ( see page 99). Hence, 95% probability limits for future forecasts of avalanche a c t i v i t y for weather conditions which are not extreme, w i l l be of the order of + 120. Simulated Forecast The computation of confidence limits certainly provides a measure 105 from which the forecasting capabilities of a model can be assessed. How-ever, a more direct assessment may be obtained by dividing the data into two sets, one of which can be used for model development and the other for model testing in a simulated forecasting situation. This procedure i s only satisfactory i f the data can, i n fact, be divided into two samples, which are each representative of the same population. . Using avalanche a c t i v i t y data based on (12,1,1) SML weights, a l l sites and a r t i f i c i a l and natural avalanche events, together with Rogers Pass meteorological data, the following SWHtW.E*TMI model was developed for the four year period, 1965-69, AVAL = 0.1341 AVAL1 + 0.05^3 AVAL2 + 8.585 SWH - 1.2?3 SWHl + 0.0700 W.E*TMI, 2 which has an R - value of O.683 and a standard error of estimate of 57.8. This model i s quite similar to that developed for the entire period, 1965-2 73, described on pages 85 to 99, except that i t has an even higher R -value. Thus, predictions based on the 1965-69 model for the period, 1965-69 would be even better than those depicted in Figure 3. If the model developed for the period, I965-69 i s now applied to the four year period, 1969-73, in a simulated forecasting situation, some 2 interesting results are obtained. As expected, the R - value for the forecasts i s decreased somewhat from O.683 to 0.579, but the forecasts are, nevertheless, s t i l l quite accurate. The standard error of estimate for these simulated forecasts i s 68.8, indicating that predictions based, on the I965-69 model for the period, 1969-73, would not be much less accurate than those depicted in Figure 3, for the period 1969-73, using the I965-73 model, which has a standard error of estimate of 61.3. 106 Hence, the two four-year samples for the periods, 1965^69 and 1969-73, are indeed quite similar. In fact, standard deviations for avalanche a c t i v i t y are 102 and 106 respectively, (SD for the period, I965-73 i s 103), and values range up to 672 and 637 i n each case. Thus, in a real forecasting situation, the model developed for the period, I965-69, could have been applied during the period, 1969-73, -with considerable success, assuming that weather forecasts were s u f f i c i e n t l y accurate. Decomposition of Avalanche Activity Since these SWH, W.E*TMI models are capable of producing reliable forecasts of avalanche a c t i v i t y i n terms of the (12,1,1) SML ac t i v i t y index, i t would be highly advantageous i f such forecasts could be decom-posed into forecasts of individual site a c t i v i t i e s . In order to f a c i l i t a t e this decomposition, a technique has been devised based on probability considerations. After dividing the range of avalanche a c t i v i t y into forty levels, the number of occurrences for each particular site at each level was computed, for the eight year period. These figures were then divided by the t o t a l number of times that level was realized, to obtain probabili-ties of occurrence for each site at each l e v e l . Hence, sites can be arranged in order of probability of occurrence for each level and tabu-lated. Such a table can be updated as more data becomes available. In an actual forecasting situation, the model provides a forecast of the a c t i -vity level, after which forecasts of individual s i t e a c t i v i t y can be obtained from this table. Of course, some very active sites w i l l have high probabilities at almost every level, hence, this approach should be supplemented with a certain element of interpretative experience. For example, i f a s i t e has avalanched recently, then i t s probability may be somewhat diminished. Avalanche a c t i v i t y decomposition tables should be computed for each type of avalanche a c t i v i t y weighting scheme used. Domain Analysis Finally, a new concept i s under investigation, whereby the time series techniques, which have been discussed, can be employed in the development of more accurate models based on observations which are un-equally spaced i n time. During periods of high precipitation and conse-quently high avalanche act i v i t y , observations should be, and often are more frequent. On the other hand, periods of low ac t i v i t y may result in more widely spaced observations. Therefore, i f the data i s transformed from the time domain into the snow domain, for example, in which obser-vations are separated by equal snowfall increments, the theory can s t i l l be applied and the observations exploited to the greatest advantage. A domain transformation routine has been incorporated into the main analytical computer program and i n i t i a l results seem promising. However, in order to obtain a significant improvement over the normal time series models, i t w i l l be necessary to make more frequent meteorological measure-ments and avalanche observations during storm periods, than have been made i n the past. 108 Chapter VIII CONCLUSIONS It has been shown that avalanche a c t i v i t y , for the Rogers Pass area, expressed in terms of the avalanche a c t i v i t y index, can be accu-rately described in terms of certain composite meteorological variables, in the form of linear transfer function and stochastic time series processes. These composite meteorological variables are SWH (the product of snow accumulation, water equivalent and humidity) and SWHT (the product of snow accumulation, water equivalent, humidity and minimum a i r tempera-ture) . SWH and SWHT can be regarded as the most significant meteoro-logi c a l terms to evolve from this study in their relationship with avalanche ac t i v i t y for the Rogers Pass area of Br i t i s h Columbia. The possible physical reasons for the presence of water equivalent of new snow, snow accumulation, humidity and minimum a i r temperature in these composite terms was discussed in some d e t a i l . Water equivalent of new snow, the best of the simple meteoro-log i c a l variables, was f e l t to be more important than depth of new snow, according to the following reasoning. Water equivalent, the product of new snow depth and the density of the new snow, i s a more direct measure of slope loading or 'shear weight' application than new snow depth alone. Furthermore, density contains a temperature effect, a high density often being related to high temperatures and the presence of free water. Finally, high densities may be an indication of developing 'slab' con-ditions in the upper zones. The importance of the snow accumulation term, as a major factor 109 modifying water equivalent, was thought to be the result of the greater participation, in the avalanching, of the deepening snowpack, presumably as a consequence of an increase in the available amount of avalanchable snow. There may also be a delayed action effect, due to snow accumulation, associated with the formation of 'soft slab' conditions. A further possible effect may be related to the influence of settlement rates on snow accumu-lation values, a high value indicating less settlement and hence, greater i n s t a b i l i t y . Relative humidity i s most probably dire c t l y associated with 'soft slab* formation, which i s thought to be the result of the condensation of atmospheric water vapour onto snow crystals or crystal fragments, brought together by moderate winds, and their subsequent cementing together. The appearance of minimum a i r temperature as a minor factor modifying the S'rfH term i s probably also related, to 'soft slab' conditions. Such conditions are a frequent cause of major avalanching at the Rogers Pass. The three best avalanche a c t i v i t y weighting schemes were found to be the (12,1,1) SML, (12,3,1) SML and (1,1,1) ML in that order, indicating that terminus i s a better measure than size and, that small avalanches are not s t a t i s t i c a l l y important. Models were developed for individual years and for the entire period, 1965-73, of the study. Although the models obtained for individual years 2 have somewhat higher R - values than those developed for the total period, in an actual forecasting situation, i t would be d i f f i c u l t to know which one to apply. Variations i n precipitation, temperature, and wind patterns lead to a different model for each year. Similarities do exist between some years, but i t w i l l be necessary to examine data for further years, before 110 such differences and simil a r i t i e s can be thoroughly evaluated. Data for 1973-74 and 1974-75 w i l l be analysed as soon as i t becomes available. A simulated forecast for the period, 1969-73, using a model deve-loped from 1965-69 data produced accurate results. The models obtained for the tot a l period, 1965-73, indicate a high degree of forecasting precision. These models can be directly applied to future winters, since they represent a type of averaging over eight seasons. If a future winter f i t s into this class of eight, accurate forecasts whould be possible. It should be noted however, that, in spite of the lagged terms in the models, the weather forecast i s s t i l l and always w i l l be an essential feature of the avalanche forecasting process. It i s hoped that, ultimately, more reli a b l e mountain weather forecasts w i l l become available for the Rogers Pass area. Perhaps i t w i l l be possible to establish a more local weather forecasting system. As LaChapelle (1970:108) states, " the mountain weather forecast problem i s an important one to solve, for many adminis-trative decisions are based on the short-term hazard forecast." These models can be improved, not only as further data becomes available, but also i f more frequent measurements are made, particularly during storm periods. Avalanche events should be recorded as precisely as possible, for such records are undoubtedly the most important limiting factor in the development of accurate models. Wind measurements may be improved, perhaps by the establishment of another remote wind station, possibly on the north side of the Pass. Since humidity appears to be an important meteorological parameter in the formation of slab conditions, such measurements should be emphasized and more carefully monitored. I l l BIBLIOGRAPHY Armstrong, R.L., 1974: Avalanche hazard evaluation and prediction i n the San Juan mountains of Southwestern Colorado. Symposium of Advanced Concepts and Techniques in the Study of Snow and Ice Resources, December 1973» Monterey, California. National Academy of Sciences. Armstrong, R.L., LaChapelle, E.R., Bovis, M.J., Ives, J.D., 1974: Develop-ment of methodology for evaluation and prediction of avalanche hazard in the San Juan mountain area of Southwestern Colorado. INSTAAR 14-06-7155-3. Atwater, M.M, 1952: The relationship of precipitation intensity to avalanche accurrence. Western Snow Conference, April 1952, Sacramento, California. Bois, P., Obled, C., Good, W., 1974: Multivariate data analysis as a tool for day by day avalanche forecast. International Symposium on Snow Mechanics, Ap r i l 1974, Grindelwald, Switzerland (IUGG - LASH). Bovis, M.J., 1974: S t a t i s t i c a l analysis. Ch, 6 of Development of metho-dology for evaluation and prediction of avalanche hazard i n the San Juan mountains of Southwestern Colorado. INSTAAR 14-06-7155-3. Box, E.P., and Jenkins, G.M., 1970: Time series analysis, forecasting and control. Wiley. Draper, N.R., and Smith, H., I966: Applied regression analysis. Wiley. Freese, F., 1964: Linear regression methods for forest research. USFS research paper FPL, Frutiger, H,, 1964: Snow avalanches along Colorado mountain highways. USFS research paper RM-7. Gardner,.; N.C., and Judson, A., 1970: A r t i l l e r y control of avalanches along mountain highways, USFS research paper RM-61. Judson, A., 1965: The weather and climate of a high mountain pass i n the Colorado Rockies. USFS research paper RM-16.' Judson, A., 1967: Snow cover and avalanches in the high alpine zone of Western United States. Physics of Snow and Ice, Proceedings of International Conference on Low Temperature Science, 1966, Sapporo, Japan, Vol I, part 2. Judson, A., and Erickson, B.J., 1973: Predicting avalanche intensity from weather data: a s t a t i s t i c a l analysis. USFS research paper RM-112. 112 LaChapelle, E.R., 1966: Avalanche forecasting—a modern synthesis. IUGG-IASH International Symposium on S c i e n t i f i c Aspects of Snow and Ice Avalanches, April I965, Davos, Switzerland. LaChapelle, E.R., 1969: Field guide to snow crystals. University of Washington Press, Seattle. LaChapelle, E.R., 1970:. Principles of avalanche forecasting. Conference on Ice Engineering and Avalanche Forecasting and Control, Oct. 1969, Calgary, Alberta, NRC Subcommittee on Snow and Ice, Technical Memo-randum No. 98. LaChapelle, E.R., 1970b: The ABC of avalanche safety. Colorado Outdoor Sports Co. Denver, Colorado. LaChapelle, E.R., 1970c: Instrumentation for snow, weather and avalanche observations. Snow safety guide number 2, Alta Avalanche Study Centre. LaChapelle, E.R., and Fox, T., 197^: A real-time data network for avalanche forecasting i n the Cascade Mountains of Washington State. Symposium of Advanced Concepts and Techniques i n the Study of Snow and Ice Resources, December 1973, Monterey, C a l i f . National Academy of Sciences. Lokhin, V.K. and Matviyenko, V.S., 1969: Experimental wind tunnel study of a i r flow past a model of mountain terrain. National Technical Information Service, Springfield, Va. AD720076. Losev, K.S., 1966: Avalanches i n the USSR. CRREL. AD 715 090. Magono, C., and Lee, C.W., I966: Meteorological c l a s s i f i c a t i o n of natural snow crystals. Journal of the Faculty of Science, Hokkaido University, Series VII (Geophysics), Vol II, No. 4. Marin, Y.A., I968: Role of certain natural factors i n the formation of snow avalanches. National Technical Information Service, Springfield, Va. AD 720059. Martinelli, M., 1971: Physical properties of alpine snow as related to weather and avalanche conditions. USFS research paper, RM-64. Martinelli, M., 1974: Snow avalanche sites, their identification and evaluation. USFS Bulletin 360. Mellor, M., 1964: Properties of snow. CRREL, 111-Al. Mellor, M., I968: Avalanches. CRREL, l l l - A 3 d . Mellor, M., 1974: A review of basic snow mechanics. IUGG - IASH, Inter-national Symposium on Snow Mechanics, Ap r i l 1974, Grindelwald, Switzerland. 113 Miller, L., and Mil l e r , D., 1974: The computer as an aid i n avalanche hazard forecasting. Symposium of Advanced Concepts and Techniques i n the Study of Snow and Ice Resources, December 1973i Monterey, C a l i f . National Academy of Sciences. Moskalev, Y.D., 1966: Avalanche mechanics. National Technical Information Service, Springfield, Va. AD 715 049. Perla, R.I., 1970: On contributory factors in avalanche hazard evaluation. NRC Canadian Geotechnical Journal, Vol VII, No. 4 . Perla, R.I., 1971: The slab avalanche. USFS Alta Avalanche Study Center Report Number 100. Perla, R.I. ed., 1973: Advances i n North American avalanche technology. 1972 Symposium. USFS General Technical Report RM-3. Perla, R.I., and Judson, A., 1973: Study plan fading memory analysis of avalanche contributory factors. USFS Line Project No. F3-RM 1601, unpub. Perla, R.I., 1974: Stress and progressive fracture of snow slabs. IUGG-IASH International Symposium on Snow Mechanics, A p r i l 1974, Grindel-wald, Switzerland. Perla, R.I., and Martin e l l l , M., 1976 Avalanche handbook. Agriculture handbook No. 489, U.S. Government printing of f i c e , Washington, D.C. Rao, C.R., 1952: Advanced s t a t i s t i c a l methods i n biometric research. Wiley. Roch, A., 1956: Mechanics of avalanche release. Armed Services Technical Information Agency, Dayton, Ohio. AD 102 284 Roch, A., I966: Les variations de l a resistance de l a neige. IUGG-IASH International Symposium on the S c i e n t i f i c Aspects of Snow and Ice Avalanches, A p r i l 1965» Davos Switzerland. Schaerer, P.A., 1962: The avalanche hazard evaluation and prediction at Rogers Pass. NRC Division of Building Research technical paper no. 142. Schaerer, P.A., 1971: Relation between the mass of avalanches and charac-t e r i s t i c s of terrain at Rogers Pass, B.C., Canada, IUGG Symposium on Interdisciplinary Studies of Snow and Ice i n Mountain Regions, August 1971, Moscow. Schaerer, P.A., 1973: Terrain and vegetation of snow avalanche sites at Rogers Pass, B r i t i s h Columbia. NRC Division of Building Research research paper no. 550. 114 Schaerer, P.A., 1974: Friction coefficients and speed of flowing avalanches. IUGG - IASH International Symposium on Snow Mechanics, April 1974, Grindelwald, Switzerland. Schleiss, V.G., and Schleiss, W.E. , 1970: Avalanche hazard evaluation and forecast, Rogers Pass, Glacier National Park. Conference on Ice Engineering and Avalanche Forecasting and Control, October 1969, Calgary Alberta, NRC Subcommittee on Snow and Ice, Technical Memo-randum No. 98. Seligman, G,, 1936: Snow structure and ski f i e l d s . Macmillan, Shimizu, H., 1967: Magnitude of avalanche. The Physics of Snow and Ice, International Conference on Low Temperature Science, I966, Sapporo, Japan, Vol I, Part 2. Schmidt, R.A., 1972: Sublimation of wind transported snow—a model. USFS research paper RM-90. Sommerfeld, R.A., 1969: The role of stress concentration in slab avalanche release. Journal of Glaciology, Vol VIII No. 54. Sommerfeld, R.A., and LaChapelle, E.R., 1970: The c l a s s i f i c a t i o n of snow metamorphism. Journal of Glaciology, Vol IX, No. 55. Sommerfeld, R.A., 1971: The relationship between density and tensile strength i n snow. Journal of Glaciology, Vol. X, No. 60. USFS, I96I: Snow avalanches, a hand book of forecasting and control measures. USDA Forest Service Handbook no. 194. Van De Geer, J.P., 1971: Introduction to multivariate analysis for the social sciences. Freeman, Yule, G.U., 1927: On a method of investigating periodicities in disturbed series with special reference to Wolfer's sunspot numbers, P h i l . Trans. Roy. Soc. 226, 267-298. 116 Appendix B THE AVALANCHE SITES Name o f S l i d e (I968) Code Mileage (from east) S i t e Weight Heather H i l l 1 1.13 2' 'Water Tank Heather H i l l l a 1 ,45 2 Beaver E a s t 2 4 . 4 8 2 Beaver West 3 4.68 2 D i v e r t i n g Dam 4 4.90 2 * Connaught 5 9.00 8 *• Unnamed 6 9.58 5 * Stone Arch 7 9,88 4 P o r t a l South 7a 10.07 2 MacDonald G u l l y No. 1 8 10.18 2 MacDonald G u l l y No. 2 9 10.23 2 •* P o r t a l - * 10 10.30 3 *-* MacDonald G u l l y No. 3 11 10.38 7 *-Tupper Timber 12 ' 10.50 2 MacDonald G u l l y No. 4 13 10.68 6 MacDonald G u l l y No. 5 14 10.78 4 Tupper No. 3 15 10.80 3 MacDonald G u l l y No. 6 16 1 0 . 8 8 6 A t l a s 17 10.90 3 MacDonald G u l l y No. ? 18 10.98 2 Tupper No. 2 19 11,05 10 MacDonald G u l l y No. 8 20 .11.18 3 Tupper No. 1 21 11.30 5 P i o n e e r 22 11.40 1 MacDonald G u l l y No. 9 23 11.43 3 Tupper C l i f f s 24 11.50 1 Tupper Minor 25 11.60 1 MacDonald G u l l y No. 10 26 11.63 3 Lens 27 11.70 17 MacDonald G u l l y No. 11 ** 28 11.75 MacDonald G u l l y No. 12 ** 29 11.83 Benches Unconfined •X-30 12.00 Double Bench * 31 12.10 S i n g l e Bench 32 - 12.30 Crossover •** 33 12.35 Mounds 34 12.40 T r a c t o r Shed E a s t 35 12.60 Lone Pine 36* 12.70 T r a c t o r Shed West 37 12.80 T r a c t o r Shed No. 3 38 • 13.30 G r i z z l y 39 13.40. G r i z z l y West 40 13.80 MacDonald West Shoulder No. 1 41 14.38 MacDonald West S h o u l d e r No". 2 42 14.48 Cheops No.. 2 43 14.50 MacDonald West Shoulder No. 3 44 14.53 MacDonald West S h o u l d e r No. 4 45 14.68 Cheops No. 1 46 15.20 Avalanche C r e s t No. 1 4? 15.97 Avalanche C r e s t No. 2 48 16.08 Avalanche C r e s t No. 3 49 16.38 Avalanche C r e s t No. 4 50 16.53 Abbott Observatory 51 17.30 Abbott No. 1 52 17.55 Abbott No.2 53 17.63 Abbott No. 3 54 17.68 Abbott No. 4 55 17.75 J u n c t i o n East 56 18.70 J u n c t i o n West 57 19.00 Cougar Creek E a s t 58 19.33 Cougar Creek West 59 . 19.73 Cougar Corner No. 4 60 19.78 Cougar Corner No. 3 61 . 19.85 Cougar Corner No. 2 62 19.93 Unnamed Cougar Corner No. 3 63 20.02 Unnamed Cougar Corner No. 2 64 20.08 6 Cougar Corner K i t t e n 65 20.23 5 Ross Peak 66 20.28 14 Cougar Corner No; 1 67 20.47 3 Unnamed Cougar Corner No, 1 68 20.68 3 Gunners No. 3 68a 20.90 6 R.R. Gunners 69 21.05 11 Gunners E a s t 70 21.15 3 Gunners West 71 21.35 3 Unnamed Gunners 72 21.?0 3 Mannix 73 21.90 6 Mannix 'West 74 22.20 3 Moccasin F l a t s 75 22.50 8 Generals 76 22.80 2 Smart S l i d e 77 23.20 4 Camp West 78 23.20 8 F i d e l i t y 79 26.00 6 Park One 80 26.90 6 F o r t i t u d e 81 26.90 8 Boundary 82 ' 27.40 1 L a u r i e 83 27.56 11 Lanark 84 • 27.63 11 Twins 85 27.88 12 N e l l i e ' s Jack MacDonald 87 28.75 3 B a i r d 87a 29.21 6 Downie No. 3 90 32.95 2 * Designates the Tupper G u l l i e s as used i n Chapter IV #* Designates the MacDonald G u l l i e s as used i n Chapter IV. Appendix,C STORM PERIODS Start Finich Start Finish 27/H/65 08/12/65 19/01/70 04/02/70 18/12/65 14/01/66 14/02/70 19/02/70 18/01/66 19/02/66 04/03/70 09/03/70 06/03/66 20/03/66 21/03/70 24/03/70 16/10/66 25/10/66 04/04/70 10/04/70 12/11/66 21/12/66 30/11/70 08/12/70 29/12/66 21/02/67 23/12/70 01/01/71 06/03/67 27/03/67 06/01/71 01/02/71 20/10/67 31/10/67 09/02/71 16/02/71 01/12/67 11/12/67 22/02/71 • 27/02/71 21/12/67 20/01/68 : 07/03/7I 12/03/71 29/01/68 07/02/68 23/03/71 03/04/71 18/02/68 24/02/68 13/12/71 25/12/71 12/03/68 19/03/68 30/12/71 26/01/72 26/03/68 30/03/68 •"• 06/02/72 10/03/72 18/11/68 ' 23/11/68 04/04/72 09/04/72 26/11/68 18/12/68 23/11/72 02/12/72 21/12/68 17/01/69 13/12/72 05/01/73 31/01/69 .I3/O2/69 11/01/73 25/01/73 15/03/69 23/03/69 30/01/73 05/02/73 O8/I2/69 15/12/69 10/02/73 22/0 2/73 19/12/69 24/12/69. 09/03/73 23/03/73 09/01/70 15/01/70 

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