A STOCHASTIC SNOW MODEL by Lawrence Ernest Cary A Dissertation Submitted to the Faculty of the DEPARTMENT OF WATERSHED MANAGEMENT In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 1974 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE I hereby recommend that this dissertation prepared under my direction by LAWRENCE ERNEST CARY entitled A STOCHASTIC SNOW MODEL be accepted as fulfilling the dissertation requirement of the degree of DOCTOR OF PHILOSOPHY Date After inspection inspection of the final copy of the dissertation, the following members of the Final Examination Committee concur in its approval and recommend its acceptance: 5 / Atira i? 7V 17-c, ,/y>,- his approval and acceptance is contingent on the candidate's adequate performance and defense of this dissertation at the final oral examination. The inclusion of this sheet bound into the library copy of the dissertation is evidence of satisfactory performance at the final examination. STATEMENT BY AUTHOR This requirements is deposited rowers under dissertation has been submitted in partial fulfillment of for an advanced degree at The University of Arizona and in the University Library to be made available to borrules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: i zzle44„: ,i,e_e 1/./ ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Dr. M. M. Fogel, dissertation director, for furnishing guidance and counsel throughout the program. To Dr, V. K. Gupta for his assistance in the model development, his instruction and availability for consultation, the author is deeply indebted. Dr. Gupta's interest in the project and desire to help were responsible, in great part, for its successful completion. The author is also deeply indebted to Dr. C. C. Kisiel and Dr. L. Duckstein with whom many useful discussions were held and who furnished many valuable suggestions. To Dr. D. B. Thorud, major advisor, the author is indebted for continuous support throughout the graduate program. The author also extends his appreciation to Dr. D. D. Evans and Dr. J. L. Thames for serving on the graduate committee and for reading and critiqueing the manuscript. The National Weather Service is acknowledged for its collection of climatological data, without which, studies such as this could not be carried out. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS LIST OF TABLES viii ABSTRACT ix CHAPTER 1. INTRODUCTION 1 2. A REVIEW OF LITERATURE 5 Probabilistic Precipitation Models in General . • • Some Pertinent Stochastic Precipitation Models • • Probabilistic and Statistical Snow Studies 3. 18 THEORETICAL CONSIDERATIONS Storm Frequency and Magnitude Snowpack Accumulation and Ablation Snowpack Duration and Ablation Frequency 4. 18 22 26 EXPONENTIALLY DISTRIBUTED SNOW STORM MAGNITUDES Snow Storm Precipitation Snowpack Ablation Rate Total Snowfall Time to Snow Disappearance Snowpack Duration The Snow-Free, Snow Cycles The Snow Renewal Process Unconditional Probability Distributions Analysis of Parameters 5. 5 8 10 APPLICATION TO A CLIMATOLOGICAL STATION Station Selection Snow Storms Storm Magnitudes Snowpack Ablation Rate 35 35 36 38 41 42 44 48 49 51 55 55 56 68 73 iv TABLE OF CONTENTS--Continued Page Total Snow Water Equivalent The Snow-Free Periods Snowpack Duration Snow-Free, Snow Cycles The Snow Renewal Process 6. 81 85 88 92 96 106 106 110 SUMMARY AND RECOMMENDATIONS Summary of Results Recommendations for Further Studies APPENDIX A: INVERSION OF THE LAPLACE TRANSFORMS OF Y, T AND Tn . APPENDIX B: PROGRAMMING THE DISTRIBUTION FUNCTIONS OF X(t), Y AND Tn 112 REFERENCES 118 131 LIST OF ILLUSTRATIONS Page Figure 3.1 The snowfall, accumulation and ablation process 19 24 3.2 The snow process generated by a storm of magnitude u . . 3.3 The snow process in general 4.1 The ratio p as a function of y 53 5.1 The observed distribution functions of storm interarrival times for three 40-day periods and the 120-day season . . 60 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 30 The observed and theoretical distribution functions of storm interarrival times 63 The Poisson parameter estimate for six 20-day intervals, three 40-day intervals and the 120-day season 65 Two estimates of the Poisson parameter for the 120-day season using cumulative years of record The observed distribution functions of storm snow water equivalent for three 40-day periods, the 120-day season and the theoretical distribution function 66 69 The parameter of the exponential distribution of storm magnitudes estimated using cumulative years of record . . 72 The empirical distribution of snowpack ablation rates with minimum rate computed using X = .097 The empirical distribution of snowpack ablation rates with minimum rate computed using X = .123 Daily snowpack ablation rates estimated for the eight consecutive 15-day intervals included in the 120-day season The observed and theoretical distribution functions of the 120-day season total snow water equivalent vi 76 77 80 83 vii LIST OF ILLUSTRATIONS--Continued Figure 5.11 5.12 Page The observed and theoretical distribution functions of snow-free periods 86 The observed distribution and computed distributions of snowpack duration (Y) 5.13 The observed and theoretical distribution functions of the snow-free, snow cycles 5.14 The observed and theoretical probability mass functions of the number of renewals in a 20-day interval 5.15 5.16 5.17 The observed and theoretical probability mass functions of the number of renewals in a 40-day interval The observed and theoretical probability mass functions of the number of renewals in a 60-day interval 90 94 98 99 100 The observed and theoretical mean value functions of the snow renewal process 103 5.18 The observed and theoretical variances of the snow renewal process 104 LIST OF TABLES Table 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Page Mean monthly precipitation and estimated water equivalent of mean monthly snowfall at Flagstaff, Arizona 57 Mean and variance of the number of storms and the storm interarrival times at Flagstaff, Arizona 62 A summary of the linear regression analyses of snowpack water equivalent on days after the termination of storms / at Flagstaff, Arizona 74 A summary of the linear regression analysis of snowpack water equivalent on days for eight 15-day periods within the winter snow season at Flagstaff, Arizona 79 The observed and theoretical means and variances of total seasonal snow water equivalent at Flagstaff, Arizona 84 The observed and theoretical means and variances of the snow-free periods at Flagstaff, Arizona 87 The observed and theoretical means and variances of the snowpack duration at Flagstaff, Arizona 91 The observed and theoretical means and variances of the snow-free, snow cycles at Flagstaff, Arizona 95 Chi-squared tests of the probability mass functions of the snow renewal process 101 viii ABSTRACT The purpose of this study was to develop a stochastic model of the snowfall, snow accumulation and ablation process. Snow storms occurring in a fixed interval were assumed to be a homogeneous Poisson process with intensity X. The snow storm magnitudes were assumed to be independent and identically distributed random variables. The magnitudes were independent of the number of storms and concentrated at the storm termination epochs. The snow water equivalent from all storms was a compound Poisson process. In the model, storms then occurred as positive jumps whose magnitudes equaled the storm amounts. Between storms, the snowpack ablated at a constant rate. Random variables characterizing this process were defined. The time to the occurrence of the first snowpack, generated by the first storm, was a random variable, the first snow-free period. The snowpack lasted for a random duration, the first snowpack duration. The alternating sequence of snow-free periods followed by snowpacks of random duration continued throughout the fixed interval. The snow-free periods were independent and identically distributed random variables as were the snowpack durations. The sum of each snow-free period and the immediately following snowpack duration formed another sequence of independent and identically distributed random variables, the snow-free, snow cycles. The snow-free, snow cycles represented the interarrival times between epochs of complete ablation, and thus defined a secondary ix renewal process. This process, called the snow renewal process, gave the number of times the snowpacks ablated in the interval. Distribution functions of the random variables were derived. The snow-free periods were exponentially distributed. The distribution function of the snowpack durations was obtained using some results from queueing theory. The distribution function of the first snow-free, snow cycle was derived by convoluting the density function of the first snowfree period and the first snowpack duration. The distribution of the sum of n snow-free, snow cycles was then the n-fold convolution of the first snow-free, snow cycle with itself. The probability mass function of the snow renewal process was evaluated numerically, from a known relationship with the sum of snow-free, snow cycles. The snowpack ablation rate was considered to be a random variable, constant within a season, but varying between seasons. The snowpack durations and snow-free, snow cycles were conditioned on the ablation rate, then unconditional distributions derived. An application of the model was made in the case where snow storm magnitudes were exponentially distributed. Specific expressions for the distribution functions of the random variables were obtained. These distributions were functions of the Poisson parameter X, the exponential parameter of storm magnitudes, Ne l , and the snowpack ablation rate. The snow model was compared with data from the climatological station at Flagstaff, Arizona. Snow storms were defined as sequences of days receiving 0.01 inch or more of snow water equivalent separated xi from other storms by one or more dry days. Snow storms occurred approximately as a homogeneous Poisson process. Storm magnitudes were exponentially distributed. Empirical distributions of snowpack ablation rates were obtained as the coefficients of a regression analysis of snowpack ablation. Two methods of estimating the Poisson parameter were used. The theoretical distribution functions were compared with the observed. The method of moments estimate generally gave more satisfactory results than the second estimate. CHAPTER 1 INTRODUCTION Often information is sought on future snow occurrences. The planning of new dams on snow fed streams and the optimum operation of present water supply systems depends upon estimates of these future occurrences. Watershed, forest and range management plans for land areas within snow zones can be affected by future snow events. Structural engineers require snow load data for snow zone construction. Snow data has also been increasingly sought by planners and operators of winter sports areas. Many hydrologic phenomena are probabilistic in nature, since, given the present, future events cannot be predicted with certainty. Recently, the theory of stochastic processes has been used to describe hydrologic processes and obtain the distribution functions of the defined random variables. Several investigators have applied stochastic process theory to the modeling of the rainfall process, but few attempts have been made to model the snowfall, snow accumulation and ablation process. Snow, when viewed as a stochastic process, poses some diffi7 culties in addition to those encountered in modeling rainfall. Unlike rain, snow may accumulate on the ground where it remains for a variable length of time. The snowpack may then be comprised of snow from one or 1 2 more storms. Snowpack ablation may occur soon after snow falls or after a lag of weeks or months. The objective of this study is to consider the process of snowfall, snow accumulation and ablation as a stochastic process. The process is then analyzed under a set of assumptions that are largely motivated phenomenologically. The basic process of the occurrences of snow storms and their magnitudes is first considered. Random variables that characterize the snowpack resulting from these storms are defined to be the snowpack durations and the length of the snow-free periods. The distribution functions of these random variables are derived. The resulting stochastic model is then compared with historical data from a climatological station. In a search of the literature, few references could be found that dealt directly with the stochastic modeling of the snow process. However, some of the literature regarding the stochastic modeling of precipitation pertain to the present study. Therefore, a review of literature is included in Chapter 2 in which probabilistic precipitation models in general are discussed. Next, references that consider the precipitation process as a stochastic process and that contain information pertinent to the present study are discussed. In the last section, papers that treat various aspects of the snow process probabilistically or statistically are reviewed. The theoretical considerations are presented in Chapter 3. The basic process of snow storm terminations is considered to be a 3 homogeneous Poisson process. A sequence of random variables representing storm magnitudes are defined. The storm magnitudes are assumed to be concentrated at the storm termination epochs, independent and identically distributed and independent of the number of storms. The total snow water equivalent in an interval is then a compound Poisson process. The snowpack ablation rate is initially assumed to be constant through the winter season. This is later generalized by considering the snowpack ablation rate to be a random variable, varying between seasons. The snow process is then one of positive jumps and negative drifts, with the slope of the drift equal to the snowpack ablation rate. The time from the onset of the winter season to the first jump is defined to be the first snow-free period, a random variable. The snowpack initiated by the first storm remains on the ground for a random duration, called the first snowpack duration. The epoch marking the end of the first snowpack also initiates the second snow-free period of the season, which is terminated by the formation of the next snowpack. The sequence of snow-free periods are independent and identically distributed random variables, as is the sequence of snowpack durations. The sums of each pair, the snow-free period and the subsequent snowpack duration, form another sequence of random variables, termed snow-free, snow cycles. The snow-free, snow cycles are independent, identically distributed, positive-valued, and represent the interarrival times between occurrences of zero snowpack water equivalent. They are therefore the interarrival times of a secondary renewal process, called the snow renewal process. Based upon some results by Prabhu (1965), expressions for the distribution functions of these random variables are derived. 4 In Chapter 4, specific distribution functions, as well as expressions for the means and variances of the random variables, are obtained. The sequence of independent and identically distributed random variables representing snow water equivalent per storm are assumed to be exponentially distributed after the model was developed in general in Chapter 3. The gamma distribution representing the k-fold convolution of an exponential distribution is substituted into the general expressions for the distribution functions of the random variables defined in Chapter 3. In Chapter 5, data from an Arizona climatological station is used to obtain parameter estimates. The assumptions of a homogeneous Poisson process of storm terminations and of exponentially distributed storm magnitudes are investigated. Snowpack ablation rates are estimated from the daily snow records by regression analysis. Empirical distributions of snowpack ablation rates are next obtained. Unconditional distributions of snowpack durations, snow-free periods and snow-free, snow cycles are derived by discretizing the empirical distribution functions of snowpack ablation rates, then numerically integrating. The unconditional theoretical distribution functions are compared with the distribution functions of observed values of the random variables. Probability mass functions of the secondary renewal process are determined from the distribution functions of the sums of snow-free, snow cycles. The theoretical and observed probability mass functions of the secondary renewal process are compared. A summary of the results obtained in Chapter 5 is presented in Chapter 6. This is followed by specific recommendations for further studies. These include possible extensions of the model. CHAPTER 2 A REVIEW OF LITERATURE The literature contains numerous papers devoted to the application of probability theory and statistics to the general field of hydrology, and particularly, to the study of precipitation phenomena. The purpose of the present study is to apply specific stochastic models to the snowfall, accumulation and ablation process. Therefore, a general review of the application of stochastic processes to hydrology has not been undertaken. The review has been organized into three sections. In the first section a brief review of precipitation models in general has been presented. This has been included to indicate the numerous approaches which have been taken in the probabilistic or statistical modeling of precipitation. Certain references, although not directly concerned with snow modeling, contain information which is pertinent to the present study and are discussed in the second section. Those papers which are devoted to the probabilistic or statistical modeling of the snow process are presented in the last section. Probabilistic Precipitation Models in General In discussing stochastic models, Gupta (1973) classified them as being either (1) empirical or purely statistical models, 5 6 (2) phenomenological or process oriented stochastic models, or (3) physically based stochastic models. Empirical or purely statistical models have been used extensively in hydrology. Precipitation frequency analyses are examples of such use. The use of empirically derived distribution functions to describe the probability of precipitation magnitudes and of precipitation magnitudes of a given duration has served as a useful tool in precipitation analysis. Discussions of frequency analysis and empirical model fitting can be found in Chow (1964), Weisner (1970), Viessman, Harbaugh and Knapp (1972), and Yevjevich (1972), as well as in earlier texts. According to Gupta (1973) there have not been many physically based stochastic models to date. It seems that present probabilitistic modeling of hydrologic phenomena such as precipitation is at an intermediate level, having left the state of purely empirical models and not yet at the point where extensive use of chemical, physical and biological laws is made to relate random variables. This intermediate level is represented by the phenomenological, or event based, approach. Phenomenological approaches have been frequently used to model precipitation in the last 15 years. Although any such classification is somewhat arbitrary, there appears to have been two general approaches. In the first, sequences of wet and dry periods are modeled. Commonly such sequences have been modeled as first order Markov processes (Gabriel and Neuman 1962, Weiss 1964, Feyerherm and Bark 1965), although higher order Markov processes have been used when first order processes appeared to be inadequate (Hopkins and Robillard 1964, Wiser 1965, 7 Feyerherm and Bark 1967). Alternatives to the use of Markov processes have been proposed to describe wet-dry sequences (Green 1965, 1970; Chatfield 1966). Other investigators have used a Markov process to describe the wet-dry sequence, and also modeled the amount of precipitation and considered the seasonal variation in model parameters (Ison, Feyerherm and Bark 1971; Crovelli 1972; Woolhiser, Rovey and Todorovic 1972). In the second approach, precipitation events rather than sequences of wet and dry days are considered. How an event is defined is dependent upon the type of precipitation process, the form of the data and the mathematical approach taken to analyze the process (Gupta 1973). The underlying probability law governing arrivals of events is usually determined. Then other random variables such as intensity, duration and magnitude of events, interarrival times and total seasonal precipitation are frequently considered. Gupta (1973) included a review of phenomenological stochastic models used in hydrology in his dissertation. A theoretical study of the precipitation process that belongs in this category was completed by Todorovic and Yevjevich (1969). Applications of event based approaches in modeling convective thunderstorms have been made by Fogel and Duckstein (1969), Fogel, Duckstein and Kisiel (1971), Duckstein, Fogel and Kisiel (1972), Lane and Osborn (1972) and Osborn, Mills and Lane (1972). Kao, Duckstein and Fogel (1971) used an event based approach in modeling winter precipitation in the Southwest. Gupta (1973) developed a space-time phenomenological model of precipitation and demonstrated its use with summer rain storm data. Using what 8 could be considered an event based approach, Epstein (1966) studied point-area precipitation probabilities using a storm cell model. Some Pertinent Stochastic Precipitation Models A common feature of the studies reviewed in this section is that a rainfall event is defined as a sequence of periods (minute, hour, day) during which precipitation occurs, preceded and succeeded by no rainfall. The precipitation events and the event arrivals are independent random variables. Although these criteria are satisfied by several convective rainstorm models, these models are excluded. Only those models which have been developed for winter season precipitation or for an entire year are considered. Thom (1959) described the number of excessive rainfall events during a year by means of a Poisson distribution. His events were magnitudes of hourly rainfall data at three stations in the mid-western United States. The interarrival times of the events he found to be exponentially distributed. His analysis indicated the usefulness of a Poisson distribution in the analysis of precipitation events. In a rainfall simulation and runoff model developed for a river basin in Japan, Ishihara and Ikebuchi (1972) used a Markov chain to describe the probabilities of wet and dry days at other stations given the events at a base station. It was determined that the simulated record did preserve the statistical characteristics of the historical record. For periods during which similar monthly frequencies of wet and of dry days occurred, they found that the lengths of dry periods 9 were approximately exponentially distributed while the number of precipitation days were Poisson distributed. A theoretical study of the intermitten precipitation process was conducted by Todorovic and Yevjevich (1969) in which a storm was defined as continuous rain between two non-rainy intervals. A stochastic process (C t > 0; t > 0) was defined to be the precipitation intensity at a station. Recognizing that precipitation intensity is seldom measured, they then defined six stochastic processes which were characteristic of the precipitation intensity. Two random events, the first being the event that exactly n complete storms occur in an interval and the second, the event that the total precipitation from the n storms in the interval is an amount x, were defined. It was shown that the density and distribution functions of the six families of random variables could be expressed in terms of the probabilities of these two random events. Only two parameters, common to all distributions, were important. These were X i , the average number of storms per unit time interval and X 2 , the inverse of the average amount of precipitation per storm. For sufficiently small time intervals and sufficiently small precipitation amounts, it was then shown that both random events were Poisson distributed. Comparison of the theoretically derived distributions with daily and hourly precipitation records led them to conclude that their results were satisfactory. However, in using each rainy hour and each rainy day as a storm, the true number of storms was apparently overestimated. Conversely, use of uninterrupted sequences of hours or days underestimated the true number of storms in the interval. 10 An event based approach was used in the analysis of winter (October through March) precipitation at Tucson, Arizona, by Kao et al. (1971). A storm group was defined as a sequence of wet days uninterrupted by more than one dry day. A wet day was one receiving an amount of precipitation equal to or greater than a stated threshold value. The storm group interarrival times were exponentially distributed while the numbers of storm groups were Poisson distributed. The storm group duration was found to be described by a negative binomial distribution. When rainfall amounts per storm were expressed as integral one-half inch units, the amounts were geometrically distributed. Under the assumption that the rainfall amounts per storm group were independent and identically distributed, and independent of the number of storm groups, total precipitation was described by a compound Poisson distribution. Kao et al. (1971) applied the model to daily rainfall data from San Francisco. Storm interarrivals, defined as the length of time between the beginnings of successive storms, were no longer exponentially distributed. The density function of the interarrival times was then determined from the sum of the (assumed) independent random variables, storm duration and length of dry spell. Using a derivation by Gupta (as reported in Kao et al. 1971) the density function for the number of occurrences per season was obtained. Probabilistic and Statistical Snow Studies A search of the literature has revealed few attempts to consider the complete snowfall, accumulation and ablation process in an integrated 11 manner. Therefore, in this section, literature pertaining to each aspect of the snow process is discussed by topic. Depending upon the purpose of the investigation, two approaches have been used in the modeling of snow accumulation and ablation. In the first, snow depth and snow water equivalent were of primary interest. Snow depth and water equivalent on a specified date, maximum seasonal values and total seasonal values were random variables frequently of interest. A second empirical approach has been the investigation of the time seasonal snowcover first forms, length of time snow remains on the ground and the time snow melts in the spring. In some studies, probability distributions were fitted to the random variable of interest. In others, relative frequencies and recurrence intervals without regard to a descriptive probability distribution were used. Areal extension of point information has frequently been accomplished by extending the point information as isolines on maps. Frequently, because of a lack of water equivalent data at many climatological stations and a short period of record at other stations, analyses of snowstorms have used snow depth. The average number of days during the winter months receiving small amounts of snow were most common while the average number of days receiving a given amount of snow decreased rapidly with increasing daily snowfall (Lautzenheiser 1968, Miller and Weaver 1971). As part of their study of snow in Ohio, Miller and Weaver used a Fisher-Tippet Type 1 distribution as a model of extreme 24-hour snowfall. 12 In a proposed stochastic snow model, Fogel, Duckstein and Kisiel (1973) suggested a geometric distribution (with integral one-half inch units) of water equivalent per storm event. Their proposal used definitions similar to those used in an earlier paper by Kao et al. (1971). The number of storm groups were distributed as a Poisson random variable and the total seasonal water equivalent was compound Poisson distributed. A simulation model of daily snowfall records was developed by Bolduc (1970) using a Markov chain. In order to utilize the fact that daily precipitation records were more common than daily snowfall records, Bolduc constructed a transition probability matrix of wet days following wet days based upon daily precipitation records. A ratio, r, the average ratio of snowfall to total precipitation, was then used to transform the transition probability matrix into another matrix of snowfall transition probabilities. A Beta distribution was used to model monthly precipitation amounts. The depth of snow on the ground or its water equivalent is of considerable interest. Information commonly required includes maximum seasonal values, depth or water equivalent for a particular time during the winter season and total seasonal values. Since snow accumulates, at least temporarily on the ground, the depth of snow or its water equivalent at any point in time may be the result of one or more storm events. An empirical model of snow on the ground then requires data in addition to precipitation records. Both snow depth and water equivalent have been treated as random variables. Thom (1966) found that the maximum accumulated seasonal water 13 equivalent for 140 National Weather Service stations were log normally distributed. The stations that did not receive snow every season were described by a mixed binomial, log normal distribution to account for the atom at the origin corresponding to the relative frequency of no snow. Areal extension was then accomplished by using contour lines on a map to indicate equal values of the distribution parameter estimates. The log normal distribution of maximum seasonal water equivalent was later used to develop snow load design criteria for the United States (Thom 1970). The water equivalent data were expressed in terms of weight per unit area (load, in pounds per square foot). Quantiles (values associated with a specified probability) were then obtained. Gumbel's extreme value distribution has been used in Canadian studies of snow depth frequencies (McKay and Thompson 1968, Lutes 1970). Since snow depth information was more common in the records, McKay and Thompson first modeled snow depth, then obtained estimates of water equivalent by using an assumed snow density. Lutes used Gumbel's distribution to develop design snow loads. The United States Weather Bureau (1964) selected the log normal distribution to describe the maximum snow water equivalent on the ground for the first and second halves of March in the North Central United States. For the analysis, 61 first order stations and 463 second order stations were used. Water equivalent was estimated from relationships between water equivalent and total precipitation data developed for that region. Maps of the region showing probabilities of maximum water equivalent were then constructed. 14 In a Russian study, Kovzel (1969) found that the areal distribution of snow storage (in terms of water equivalent) was approximately normally distributed. Kovzel expressed the snow storage characteristics of an area in terms of mean areal water equivalent, the skew coefficient and coefficient of variation. Kuz i min (1969), in characterizing snowcover thickness, used the coefficient of variation of thickness and defined a modular coefficient as being the ratio of snow accumulation at a given point to the average accumulation for different values of the coefficient of variation. A major source of data for water equivalent of snow on the ground is the United States Soil Conservation Service. Three recent papers have attempted to obtain additional information from snow course records by fitting empirical probability distributions. For selected snow measurement dates and snow courses with five or more years of record in Oregon and Utah, Vance and Whaley (1971) fitted a log Pearson type three distribution to water equivalent and snow depth. Measurement dates with entries of no snow were not included in the analysis. The relative frequency of snow at each snow course was considered by weighting the distributions of depth and water equivalent values by the relative frequency. In a similar empirical analysis of snow course water equivalent for Arizona, Cary and Beschta (1973) found that a mixed binomial, log normal probability distribution fitted the observed water equivalent at most of 22 snow courses for the six common measurement dates. The binomial distribution adequately described the relative frequency of no snow at the various courses. Regression analysis was used to relate 15 parameter estimates to elevation. Engelen (1972) considered snow depth at snow courses in New Mexico and Colorado to be normally distributed with two sets of parameters, one for the winter accumulation period and one for the spring melt period. Engelen also considered monthly mean values of snow depth and water equivalent as well as the coefficients of variation as affected by time, latitude, elevation, and local influences. Snow continues to be a difficult hydrologic variable to measure as has been indicated by numerous studies (United States Army Corps of Engineers 1956, United States Weather Bureau 1964, Anderson 1972, Peck 1972, Rechard 1972). One exception is the occurrence of snow. Dates of formation and melt and length of time it remains on the ground are well defined and relatively easy to measure (Thom 1966, McKay and Thompson 1968). The time, measured from an arbitrary reference date, to the occurrence of the first snowfall of the season or to the first snowfall to exceed a given threshold value, has been found in several studies to be approximately normally distributed (Thom 1957a, 1957b; Miller and Weaver 1971; Cehak 1972). Cehak also determined that the date of the first snowcover (first time snow remained on the ground) and the date of winter cover (date of snowfall after which snow remained on the ground for the remainder of the winter season) were approximately normally distributed. Similarly, in Alberta, McKay and Thompson (1968) found the date of formation of the winter snowpack to be normally distributed, except at some stations in the southern prairie. 16 In regions where snow cover persists throughout the winter season, duration is relatively easy to characterize. Where snowpacks form, melt and reform several times during the winter and spring, quantification of duration is somewhat more difficult. Clapp (1967) related the relative frequency of days per month with one inch or more of snow to other meteorologic variables. Dickson and Posey (1967), using climatologic data from several countries, developed maps showing the probability of snowcover, one inch or more in depth at the end of each month (September through May) for the northern hemisphere. Using snow depth and duration as parameters, Potter (1965) divided Canada into seven regions based upon the homogeneity of snowcover. For selected stations within each region, he presented relative frequency curves of snowcover versus month. McKay and Thompson (1968) found that the duration of snowcover at four stations in Alberta could be considered normally distributed. For a large number of stations, Cehak (1972) used a normal distribution of snowcover duration as a first approximation in his study of snow conditions in the Austrian Alps. The date by which snow can be expected to be melted has also been treated as a random variable. As with the variables already discussed, it requires a specific definition when used because it can be defined in various ways. McKay and Thompson (1968) considered the date of loss of seasonal snowcover as being the last day of cover, following which snowcover did not recur for a continuous sequence of seven days. For the period of record, they found that the loss date was normally distributed For the last day with snowcover and the last day with uninterrupted 17 snowcover, Cehak (1972) again used a normal distribution as a first approximation. In the studies of rainfall reviewed here, the theory of stochastic processes was used to develop models of storm occurrences, storm amounts and the total precipitation occurring in an interval. However, in describing the stochastic nature of the snow processes, reliance has been placed on the empirical fitting of distributions to data. Random variables such as storm amounts, snowpack depths, snowpack durations, date of snowpack formation, date of last snowmelt and number of days with a given amount of snow were defined and distributions fitted to their observed values. The theory of stochastic processes is used in the present study to model the processes of snowfall, snow accumulation and ablation in a unified manner. A model of the basic process of storm occurrences and their magnitudes is developed. Subsequently, random variables characteristic of the snowpack are defined based upon considerations of the process. CHAPTER 3 THEORETICAL CONSIDERATIONS In this chapter, a generalized model is developed based upon phenomenological considerations of the actual process of snowfall, snow accumulation and ablation. The basic stochastic process of snow storms is described and random variables arising from the basic process are defined. The mathematical formulations for the random variables are then given in general. Storm Frequency and Magnitude A hyetograph of daily snowfall water equivalent and the accompanying snow accumulation and ablation is shown in Figure 3.1(a). It is assumed that the water equivalent of each storm can be concentrated on the last day of a multiday storm. Then the snow process can be represented as a process of the form shown in Figure 3.1(b). Let (0,t] be an arbitrary time interval of interest, such as the winter season. Denote the termination epoch of the i th storm by T i . Then the collection of termination epochs forms a discrete parameter stochasticprocessfT.;i = 1,2,...,4. Another discrete parameter processisthecollectionofindividualstormmagnitudesfX.;i = 1,2,...,4. It is assumed that X.; i > 1 are mutually independent, _— identically distributed random variables. The assumption of independence 18 19 0 20 40 60 80 100 Time, days (a) 0 20 60 40 Time, days (b) 80 900 Figure 3.1. The snowfall, accumulation and ablation process. -- (a) An observed snowfall hyetograph (solid lines) and snowpack water equivalent (dashed lines). (b) A conceptualization of the process. 20 between individual storm magnitudes has been made in the modeling of winter precipitation in the southwest (Kao et al. 1971) and proposed for magnitudes of individual storm events (Fogel et al. 1973). The assumption that they are identically distributed can be tested (Chapter 5) using a k-sample Kolmogorov-Smirnov test derived by Kiefer (1959). Let fN(t); t > 0 1- be a counting process denoting the number of - complete snow storms occurring within the interval (0,t]. The random variable N(t) can be defined in terms of fT il as N(t) = sup{i; It is assumed that Ti < tl. N(t) follows a homogeneous Poisson process with a constant (with respect to time) intensity function X (Parzen 1962). Todorovic and Yevjevich (1969) have described the process N(t) by a nonhomogeneous Poisson process with intensity X(t) a continuous function of time. They found that for sufficiently small intervals, X(t) could be assumed constant. The assumption of X(t) E X within (0,t] is made as a simplifying assumption, although it should be pointed out that it can be relaxed under certain conditions without loss of generality, Parzen (1962, p. 126) presents a transformation that can be used to transform a nonhomogeneous process into a homogeneous process. It is next assumed that {x i ; i = 1,2,...,4 is independent of the stochastic process fN(t); t > 01. That is, the individual storm magnitudes are independent of the number of storms within an interval. This assumption is made as dependence makes analytical approaches almost intractable. It has been shown to be valid in other studies of winter and yearly precipitation (Todorovic and Yevjevich 1969, Kao et al. 1971, Fogel et al,) 1973). The assumption of independence between storm 21 magnitude and numbers of storms may not hold in regions where storm durations are "long" and snowfall is uniform. Under the above assumptions, the total water equivalent within (0,t] resulting from a random number of storms is defined as X(t) = (3.1) Then X(t) is a compound Poisson process with the distribution function of X(t) given as FX(t)(x) = P(X(t) =< x) e -Xt (Xt)n F n(x), n! X (3.2) n=0 where F n(x) = P(Xn < x) X and X n =X +X + 1 2 + X . n Since the distribution function appearing in equation (3.2) is not differentiable over its entire range, the density function of the random variable X(t), denoted by dFX(t)(x) is defined only for x > O. An atom of magnitude e -Xt occurs at the origin (x = 0). The theoretical con- siderations involved here are covered in Feller (1971, Volume 2). In order to maintain generality in this chapter, the notation dF (x) is X used to denote the probability density function of X, which can be of any form. 22 Snowpack Accumulation and Ablation The snow process as defined in this study includes snowfall, snow accumulation and ablation. Since only snowfall is considered in equation (3.2), additional processes must be defined to take this into account. Snow accumulation in the model is assumed to occur as jumps whose magnitudes equal total storm snow water equivalent and which occur at the storm termination epochs, Ti. Snowpack ablation is the decrease in snowpack water equivalent due to melting and evaporation. The ablation of a snowpack is the effect of available energy which, in turn, depends upon many components and meteorologic factors. The ablation rate changes through the season, generally increasing in the early spring. The ablation rate will also exhibit short term variations including diurnal variations depending upon the relative magnitudes of the several energy components (United States Army Corps of Engineers 1956). However, since the purpose of the present study is to develop a stochastic snow model, simplifying assumptions regarding the ablation rate are deemed warranted. It is assumed that the ablation rate, c, is linear. This assumption has frequently been made for daily, or longer, periods of melt in empirical equations with reasonable results. Examples of such equations can be found in Snow Hydrology (United States Army Corps of Engineers 1956). In an initial regression analysis of ablation rates, a constant rate, c, was hypothesized. The results of the regression analysis are discussed further in Chapter 5. A generalization will be to view c as a value of a random variable. At present, c is viewed as a constant and in the 23 next chapter, the randomness in c is incorporated by taking the derived results as conditional probabilities with respect to c. Another stochastic process is now defined as {Z(h); 0 < h < where Z(h) is the point snowpack water equivalent at time 0 < h < t. In particular Z(t) can be expressed as, Z(t ) = z (0) + X(t) - ct + cj L[Z(h)]dh, 0 (3.3) where L[Z(h)] = 1 if x = 0 = 0 otherwise. The index function LEZ(h)1 insures that the water equivalent, Z(t), will not become negative, a physical impossibility. The process as defined by equation (3.3) is illustrated in Figure 3.2. Since c in equation (3.3) has been assumed a constant, it can be rewritten as Z(t) C Z(0) C X(t) t+ j L[Z(h)]dh 0 Or Z (t) = u + x(t) - t + j L[Z (h )]dh (3.4) Note that the process {Z(t)1 given by equation (3.4) has an ablation rate unity. In subsequent development it is assumed that c = 1. Division of depths of snow water equivalent by the ablation rate results in each term in equation (3.4) having units of time. To avoid introducing new notation, Z(t) is written as Z(t) = Z(t)/c and X(t) = X(t)/c. 24 - A - o o 11 ( 54 `(4)Z) lioDdhlOuS Jo luOIDA!nbj aolOM 25 Prabhu (1965) has shown that when u = 0, equation (3.3) can be alternately expressed as Z(t) = supfX(h) - hl. (3.5) 0 < h < t Another random variable, T(u), is defined to be T(u) = inffh > Z(h) = 0, Z(T i) = (3.6) Equation (3.6) is interpreted as follows. If the water equivalent of the snowpack due to a storm terminating at T 1 is u, then T(u) is the random variable which describes when the snowpack will ablate to 0 for the first time. However during this time a random number of storms can occur, adding more snow water equivalent to the pack. The accumulation and ablation processes just described are analogous to a single server queue (M/G/l) with customer arrivals a Poisson process, a general and as yet unspecified service time distribution and waiting time decreasing at a constant rate, c, between customer arrivals. The waiting time is the total water equivalent of the snowpack and service times the water equivalent from each storm. Unlike more conventional applications of queueing theory, the analog of the number of customers waiting service loses its physical significance as it is the number of storms which have contributed water equivalent to the snowpack and awaiting ablation. The waiting time (water equivalent of the snowpack), busy period (time snow remains on the ground) and busy 26 cycle (time between occurrences of no snow) become the random variables of concern. Prabhu (1965, pp, 72-73) has derived the general expression for the distribution function of T(u). This derivation is therefore omitted and only the final expression given. The distribution function of T(u) arises from the fact that X(t) is a compound Poisson process and that the sequence was initiated by the occurrence of the first event, followed by n-1 events in the interval (O, t] and is FT (t) = P[T(u) t1Z(71)= e -XT (XT) = T= n-1 xu n! dF n(T-u). X (3.7) The density function is given as co dF T(u) e = r-i-Xt n-1 Xu (X° n! dF x n(t-u). (3.8) n=0 Given an initial value, u, dF T(u) (t) gives the probability the snowpack will persist for a length of time t. Snowpack Duration and Ablation Frequency A random variable is defined to be Y 1 = inffh > T i ; Z(h) = 01. Y 1 is interpreted as the length of time a point snowpack, generated by the occurrence of one or more snow storms, persists (i.e., snowpack duration). Another random variable, V 1 , is defined as V 1 = Tl. 27 V 1 is the first snow-free period, the time from the onset of the winter season to the formation of the first snowpack. The second snow-free period, V a , is then the snow-free period commencing at the end of Y 1 and lasting until the foLmation of the second snowpack of the season. The second snowpack will endure for a time, Y a , where Y a is defined as Y a = inffh > V 1 + Y 1 + V a ; Z(h) = 01. The random variables V. and Y. i > 1, are illustrated in Figure 3.3. the first storm magnitude; 1 is initiated by this has been denoted by X 1 with density function dF x (u). Hence, the Recall that Y density function of Y 1 is the unconditional density function of T(u), for all values of X 1' and is obtained as (y) = dF Yl 0 Substituting for dF dF T(u) (Y) dF x (u) 1 (3.9) T(u) (y) Y LdF x n (y-u) dF x (u) e-XY 1L dF (y) = j n! Y 1 0 n0 1 = - (y) = dF Y1 - n-1 y NY (NY)j udF n (y-u) dF x (u). X n! 1 0 Applying the identity (Prabhu 1965, p. 72) (x) xm dB j v dBm (v) dB n (x-v) = m+n m+n 0 (3.10) 28 to the integral in equation (3.10), where m = 1 since F (u) is the X1 distribution function of the magnitude of the first event while Fn(x) is the distribution function of the sum of n events following the first n-1 711' 4 dF x n+1(y) dFY (y) =e 1 n=0 co dF e-XY ( )7+1 (n- I)-1! dFX 111- 1 (y). (y) = 1 n=0 Letting k = n + 1, CO dF (Y) = Y 1 e -XY k-1 dF k (y). ( 1Xj) x (3.11) k=1 The distribution function of Y 1 CO F (y) = r Y e Y 1 0 k=1 is then k-1 k! dF k (w), X (3.12) where w is a dummy variable of integration. The total time that snow remains on the ground during the winter season is then the sum of independent, identically distributed Y's K(t) K(t) _ Y Y.. i=1 (3.13) 29 The number of times the snowpack ablates is a random variable, K(t), therefore YK(t) is the sum of a random number of random variables. An expression for the distribution function of Y K(t) is deferred until the next chapter. It is obtained as the K(t)-fold product of the Laplace transform of Y 1 (or Y since the Y 's are as Y), which depends upon the distribution function of water equivalent per storm. The waiting time to the occurrence of a Poisson event is exponentially distributed. Since V 1 is the time from the onset of the winter season to the occurrence of the first snowpack (generated by the first storm), it is exponentially distributed with parameter X. Due to the memoryless property of the exponential distribution of storm interarrival times, the second snow-free period, V 2 , measured from the epoch of complete ablation of the first snowpack to the formation of the second snowpack, is independent of the first snow-free period and the first snowpack duration. Similarly, V 3 ,V 4 ,... are independent of preceding snow-free periods and snowpack durations. Also on account of the Markov- ian property, the snow-free periods, which are analogous to the idle periods in an M/G/1 queue, are distributed identically as V 1 (Prabhu 1965, p. 111). The snow-free periods are illustrated in Figure 3.3 and the distribution function of V 1 given by (v) = 1 - F (3.14) V1 As is indicated in Figure 3.3, let T 1 = V 1 Y . 1 (3.15) 30 (5q t(q)z) 5ioDdmous JO uolomnb2 Joi.om 31 T 1 is termed the first snow-free, snow cycle. The entire sequence of random variables {T 1 } are similarly defined and form a sequence of snowfree, snow cycles. V and Y 1 1 are independent random variables and since their density functions are known, the density function of T I can be obtained by the convolution of the density functions of V 1 and Y 1 t dF T = 1 dF v (t y) dF (Y). - y1 1 '0 Substituting the specific density functions and simplifying gives the density function of T 1 as co (t) = dF Tl S 0 r dF T ' .1 t -X l: et (t) 1 „k-1 - X(t - Y) - XY ( 07) dF k (y) X ee x k! 0 k=1 y k-1 k! The distribution function of T 1 dF xk (Y). (3.16) is obtained by integrating equation (3.16) F T = 1 S tS 0 k e -XT k! Y k-1 dF k(y)• x (3.17) The total number of snow-free, snow cycles in the interval (0,t] is a random variable, K(t). The sum of K(t) snow-free, snow cycles gives the total length of the cycles in the interval as 32 T K(t) K(t) T.. (3.18) i=1 As with Y functions of T K(t) , the expression for the density and distribution t) are obtained from the K(t)-fold convolution of the LaplacetransformofT1(orTsincetheT:s are i.i.d. as T) and are determined in Chapter 4. The last process to be considered is a secondary renewal process generatedbytherandomvariablesv.and Y i , i > 1, and is shown in Figure 3.3. Starting from some initial time, Z(h) is positive for a random duration (Y) and then reaches zero level. It stays at this level . for some random duration (V) until a new storm occurs, then it again becomespositive.sinoefT„i > represent the interarrival time b > 1 } forms a renewal process. A renewal process is one in which the interarrival times between events are positive-valued,'independent and identically distributed random variables (Parzen 1962). If K(t) is the number of complete renewals, that is the number of times the snowpack melts completely, is reformed, then remelts in the interval (0,t], then the probability mass function of the secondary renewal process dF K(t) (n) = P[K(t) = n] (3.19) gives the probability that the snowpack completely ablates n times within (0,t]. The expected number of renewals, the mean value function of the renewal process, is given as (Parzen 1962, p. 171) 33 cx \Ln ndF K(t) (n). m(t) = E[K(01 = (3.20) a=1 The number of renewals in the interval (0,t] is defined as, K(t) = supfn; Y T i < tl. (3.21) i i=1 The probability that there will be n renewals in the interval (0,t] is related to the cumulative probabilities of the random variables T. as n n+1 P[K(t) = n] = P(Y T. < t) - P T < t), i _, 1 1=1 i=1 (3.22) Based upon (3.21) and (3.22) it can be shown that m(t) = P n=1 (ET. i=1 < = FTn(t). (3.23) n=1 If the Laplace transform of both sides of equation (3.23) is taken, the right hand side is the sum of a geometric series (Parzen 1962, p. 178), Or L T (s) Lm(t) (s)1-LT(s) (3.24) The Laplace transform could then be inverted to yield the mean value function. 34 Commonly in some areas (e.g., high elevations, high latitudes, northern exposures and protected sites) a snowpack forms and remains throughout the winter season, resulting in only one renewal. Although the probability that there will be but one renewal at such locations is high, it is not equal to one. Therefore the secondary renewal process is of importance in characterizing the stochastic nature of these snowpacks as well as snowpacks that form and ablate several times during a winter season at lower elevations. To study the stochastic nature of snowpacks, the secondary renewal process is used in conjunction with the random variables defined above. CHAPTER 4 EXPONENTIALLY DISTRIBUTED SNOW STORM MAGNITUDES In this chapter an application of the general expressions obtained in the last chapter is provided. In particular, an exponential distribution function is assumed for the snow storm magnitudes. The exponential distribution of snow storm magnitudes is substituted into the general expressions, yielding specific distribution functions. Expressions for the means and variances are obtained. Snow Storm Precipitation The selection of a probability distribution to describe the amount of snow water equivalent per storm was based upon two considerations. A distribution was required that would adequately fit the data but would not unduly complicate the analysis. Storm precipitation amounts have been found to be approximately exponentially, or geometrically, distributed in some studies (Todorovic and Zelenhasic 1968, Kao et al. 1971). In others, a gamma distribution was found to be an adequate model of precipitation amount per storm (Todorovic and Yevjevich 1967, Ison et al. 1971). Woolhiser et al. (1972) used an exponential distribution to describe daily rainfall amounts. The total rainfall for n days was then described by a gamma distribution. Similarly, Dingens and Steyaert (1971) proposed a modified exponential distribution for 35 36 daily rainfall and a gamma distribution for k-day totals. However, for seven day precipitation amounts, they found that the shape parameter was equal to one which reduces the gamma distribution to an exponential distribution. In view of the fact that an exponential distribution has been shown to fit precipitation per storm in some earlier studies, it is selected as the model for snow water equivalent per storm in the study. Recall from Chapter 3 that the precipitation amounts per storm X., i > 1 are mutually independent and identically distributed random variables. Thus the distribution function of X. is given as 1 F x (x) = 1 - e - Y 1 xx > 0, all i > 1. (4.1) The density function of the total precipitation from k storms is the k-fold convolution of the exponential density function f (x), which X. is a gatialta density function k k-1 y1 x dF k(x) = f k(x) = x x e - y x 1 (k-1)! (4.2) Specific distribution functions of the random variables defined in Chapter 3 can now be obtained by substitution of equation (4.2) for dF k(x). X Snowpack Ablation Rate Snowpack ablation is a stochastic process. However, it is assumed to be deterministic with a rate that is a random variable, C, which assumes one value each winter season. 37 The snowpack ablation rate is incorporated into the model as follows. Recall from equation (3.4) that X(t) — X(t) (4.3) where X(t) was defined to be the sum of a random number of individual storm magnitudes in equation (3.1). N(t) (X./c) X(t) = (4.4) i=1 Substituting Xi /c for X i in equation (4.1) F X. (x) = P(X./c < x) — = P(Xi < cx), i > 1 = 1 - e -yx . > 1, , (4.5) where y = cy l . Thus the random variables X., i —> 1 can now be treated as exponentially i distributed with parameter y, and the snowpack ablation rate is incorporated into the model by use of the parameter y. After the specific distribution functions have been derived, the unconditional distribution functions and moments are obtained by integrating over the density function of C. In all subsequent discussion, equation (4.5) will represent 38 the distribution function of X., i > 1. Note that in this light, — equation (4.2) now represents a gamma density function with parameters k and y, i.e., kxk-1e -yx d Fxk(x) = f k(x) - Y x (k-1)! Total ' Y = cYl. (4.6) Snowfall Recall that the distribution function of total snowfall water equivalent in an interval (0,t] is given by equation (3.2). Substituting the distribution function of Xk into equation (3.2) leads to the distri- bution function of the compound Poisson process of total snowfall in the Interval, or total seasonal snowfall, given by CO F X(t) (x) = k x k k-1 -Yw e-Xt (Xt)r Y w edw k! J (k-1)! 0 k=1 co -Xt k k x ey k-1 -yw w e dw. k! (k-1)! J 0 (4.7) k=1 The above distribution function is absolutely continuous in the interval 0< x < m, with an atom at x = 0 corresponding to the probability of no snow, i.e., CO dF X(t) (x) = f X(t) e-X t (Xt) k! (x) = k k k-1 y x e-Yxdx, x > 0 (k-1)! k=1 (4.8) 39 and dF (0) = e X(t) -Xt (4.9) The integral in equation (4.7) is the incomplete gamma function for which tables are available. However it can also be integrated by substitution of the series form of e -Yw , which allows isolation of the variable of integration. Substituting (-1 e Yw = - ) 3 3 3 Y w (4.10) j! j=0 into equation (4.7) yields F X(t) j j k k -Xt (Xt) y 7 (-1) y j e k ! (k-1)! j! (x) = j=0 k=1 w j+k-1 dw. 0 Integrating, and redefining the inner index of summation to begin at 1 results in CO F X(t) e (x) = -Xt k k (10 y (-1) k! (k-1)! k=1 L, j j y (j-1)! x k+j-1 (k+j-1) • (4.11) j=1 Laplace transforms of random variables are commonly used in probability theory since the transform of a distribution function is unique and often moments can be more easily obtained from the transform than from the density function (Feller 1971). The Laplace transform of a compound Poisson distribution is (Feller 1971) 40 (s) = e Xt(Lx(s) - 1) L (4.12) X(t) where L (s) is the Laplace transform of X, i.e., the precipitation X magnitude per event and s is the Laplace variable. The Laplace transform of an exponential distribution is (4.13) . L (s) - X y + s Substituting equation (4.13) into equation (4.12), L X(t) (s) = e Xt ( y - 1). y + s (4.14) The mean can then be obtained by differentiation of the Laplace transform as can higher moments E [ x( t) ] = ( 1 ) E[X(t) 2 (4.15) Lx(t)(s )I s= 0 = - 2 2 Lx(t) (s)l s. 0 ] = Os 2 2Xt 2 (Xt) 2 + 2 E[X(t) ] - (4.16) The variance of X(t) is then obtained from (4.15) and (4,16) as Var[X(t)] = E[X(0 2 1 - E[X( 01 -9 = 2Xt 2 (4.17) 41 Time to Snow Disappearance If, on a given date, the point snowpack water equivalent was observed to be Z(0), a random variable that may be of interest in certain applications is the length of time from the observation date to the time the snowpack ablates to 0. This random variable, T(u), was defined by equation (3.6) and the general expression for its density function given by equation (3.8). Substitution of equation (4.6) into equation (3.8), gives the density function of T(u), namely CO f k -Xt-yt eyu V (Xy) [t(t-u)] k-1 = ue k! (k-1)! k=1 T(u) (4.18) When t = u, the density function reduces to (Prabhu 1965, p. 95) f T(u) (u) = e -Xu , (4.19) which gives the probability that the snow remains on the ground for u = t days when no more storms occur. In the equations involving u, it is implicit that u = Z(0)/c, and thus the right hand side becomes dimensionally correct. An expression for the distribution function of T(u) is obtained by substituting the infinite series form of e-t(k y) into equation (4.18) and integrating, m F T(u) (0 = ue" 1 (XY) m k (-1)j-1(X+Y)j-1 (j-1)! k! (k-1)! k=1 t . 2 k+J-„k-1, T T=l1 j= kT U) - UT (4.20) 42 The Laplace transform of the distribution of T(u), from which the moments may be obtained is (Prabhu 1965, p. 95) L T(u) (s) s , expi-uf (s+?c-Y) +2 "4s+X-Y)2 +4 1] (4.21) The mean and the second moment of T(u) are obtained as E[T(u)] = (-1)L Os T(u) E[T(u) 2 (s)I s=o 2 ] - L OsT(2 u) (s)1 = (4.22) y X - 2uXy y 22 u s=o (Y - X) 3 (y-X) 2 (4.23) The variance of T(u) using equations (4.22) and (4.23) is given as 2 Var[T(u)] = E[T(u) ] - E[T(u)] 2 - 2uXy 3 (Y-X) (4.24) Snowpack Duration Recall from Chapter 3 that the duration of the first snowpack, Y 1, was defined as the length of time the snowpack initiated by the first snowstorm of the season would persist. At higher elevations, northerly exposures or otherwise protected sites, it would be anticipated that the snowpack would remain for the entire winter season, ablating completely only once, in the spring. At lower elevations, southerly exposures or sites which receive higher net energy, the snowpack may form and ablate one or more times. The density function of Y 1 is given by equation 43 (3.11) for the general case. Substituting equation (4.6) into equation (3.11) yields the density function of Y 1 , e - XY (xy) k-lykyk-le -y y f (Y) = Y 1 k! (k-1)! (4.25) • k=1 The distribution function of Y 1 is next obtained by integrating equation (4.25) F y (Y) = 1 k k-1 f y 2(k Y X w k! (k_.1)1 0 k=1 - 1) ' e _ w ( x+y) d w. (4.26) Substituting the infinite series form of e-w(X+y) and carrying out the indicated integration results in 00 , F (y) = k k-1 Yk!X(k-1)! y1 k=1 00 (-1) j-1 j-1 2k+j-2 (VEY) Y (j-1)! (2k+j-2) . (4.27) j= The Laplace transform of the density function of Y1 is (Prabhu 1965, p. 96) s + X + y -,/(s+X+y) L (s) Y 2X 1 2 - 4Xy (4.28) As before, the mean and variance of Y 1 can be obtained by differentiating the Laplace transform. This gives E[Y1 ] = (y-x) (4.29) 44 E[Y 2 2y ] 3 (y - X) Var[Y 1 ] - (Y 0 (4.30) (4.31) 3 - The mean and variance of Y 1 (which is analogous to the busy period in an M/M/1 queue) as given by equations (4.29) and (4.31) are identical to the mean and variance of the busy period as obtained by Feller (1971, p. 199). The Snow-Free, Snow Cycles The first snow-free, snow cycle of the winter snow season ' T 1 , was given as T 1 = V + Y 1 1 The density function of T I was given in general by equation (3.16). Sub- stituting the exponential distribution of snow water equivalent per storm into equation (3.16) m k -Xt 2k-2 k -yw t 7 dw ce w xe f T (t) = j k! (k-1)! 0 1 k=1 Substituting the infinite series form of e m m f 1 (t) = T k=1 j=1 integrating (-1) j-l kyk+j-l e -Xt t 2k+j -2 7 k! (k-1)! (j-1)! (2k+j-2) . (4.32) 45 The distribution function of T 1 is next obtained by integrating equation (4.32). Using the substitution of the series form of e -Xt to isolate the variable of integration yields CO F T 1 (t) = = CO j-1 k k+j-1 e-xww2k+1-2dw (-1-) L, k! (k-1)! (j-1)! (2k+j-2) f 0 k=1 j=1 LJ CO m co (-1)3+m-2xk+m- 1 yk+j- 1 t2k+j+m-2 k! (k-1)! (j-1)! (m-1)! (2k+j-2)(2k+j+m-2) k=1 j=1 m=1 (4.33) The mean and variance of T 1 are obtained using the fact that T is the sum of two independent random variables ' V 1 and Y 1, therefore the mean and variance are the sums of the means and variances of V E[T i ] = E[Y 1 ] + E[V i ] — x(y...20 + (Y - X) Var[T 1 ] = Var[Y i ] + Var[V i ] — x 2 (x) 2 3 X (y - X) The Laplace transform of T Laplace transforms of Y 1 and V 1 1 Y 1 and 1 (4.34) 3 (4.35) is obtained as the product of the since they are independent 1 46 L (s) = L Tl (s) yl L v (s) 1 (s+X+Y) As+X+Y) 2 4 XYX 2X (s+X-FY) X+s Asd- X+Y) 2 - 4 XY 2(X+s) (4.36) Inversion of equation (4.36) (see Appendix A) gives f T where I 1 -1 = L 1T 1 / a (s) = e -Xtj e -yw n wXY, 0 1- 1' 2w(Xy)']dw (4.37) is the modified first order Bessel function. Equation (4.37) can be shown to be identical to equation (4.32) by substitution of the infinite series forms of e -Yw and of I [2w(Xy) 2 ] 1 and carrying out the indicated integration. The distribution function of T 11 ,can be obtained by inverting its Laplace transform. Since the random variables, T i , i 1 are mutually independent and identically distributed, the Laplace transform of T n is given by the n-fold product of the Laplace transform of T 1 with itself L n (s) = [(s+x y) T +y)2 - 4Xy] (4.38) '/(/ The inverse of the transform (see Appendix A) was found to be f n(t) = L n -1 (s) = ne -Xt r s n/2 j 0 0 I n [2w(xy) e -yw (XY) k ](dw) n (4.39) 47 The indicated integrations are carried out by again using the infinite series forms of e 'Y w and of the nth order modified Bessel function, the substitutions of which permit isolation of the variable of integration 00 fTn(t) = ne -Xt ON) n): 00 (-1)i-1(Xy)k (k-1)! (k+n-1)! (j-1)! k=1 j=1 t t . 0 w2k+j+n-4(dw)n 0 ( _ 1) j-l xkyk+j-l t 2k+j+2n-4 = n(X)nt f T n(t) k=1 j= (k-1)! (k+n-1)! (j-1) ! i n (2k+j+2n-i-3) 1=1 (4.40) The distribution function of T n is co 00 t (-1)j-lxkyk+j-lw2k+j+2n-4 dw n(t) = f nay)ne-xwy F T ' --- (k-1)!(k+n-1)!(j- 1''"° In1 (2k+j+2n-i-3) 0 k=1 j=1 i=1 When the infinite series form of e -2n.w is substituted into the distribu- tion function of T n , the expression integrated and the inner index of summation redefined to begin at one, the expression for the distribution function becomes 48 CO 03 (...1) j+m-2 x k+m-2 k+j-2 t 2k+j+2n+m-4 F n(t) = n(Xy) nY T ( k- 1) ! ( k+n- 1) ! ( j -1) ! (m-1) I ( 2k+j+2n+m- 4) k=1 j=1 m=1 1 (4.41) 1 I (2k+j+2n-i-3) i=1 The mean and variance of T n can be obtained by successive differ- entiation of the Laplace transform, or more simply by noting that since n . T is the sum of n independent, identically distributed random variables, the mean and the variance are the sums of the means and variances of T., E[T n ] = E[T i ] + E[T 2 1, + + E[T n. ] E[T n i — (4.42) Var[T n i = Var[T i l + Var[T 2 ] + „. + Var[T] ( 107 r,N Var[T n ] — n L AAi ' ` Y- " / -1 , for all n > 1. x 2 (x) 3 (4.43) The Snow Renewal Process The snow renewal process and the mean value function of the renewal process were introduced in Chapter 3 and their significance indicated. The mean value function could be obtained by substitution of the Laplace transform of T, given by equation (4.36), into equation (3.24) 49 and then inverting. However a relationship exists between the probability mass function, mean value function and higher moments and the distribution function of T n which allows them to be obtained directly from the distribution function. For n > 1, the probability mass function is given by eauation (3.22). The probability that no renewals occur is given by (Parzen 1962) P[K(t) = 0 1 = 1 - FT (t). / (4.44) The mean value function and second moment are m(t) = E[K(t)] = n[FTn(t) - F T n+1(01, (4.45) n=i and CO E[K(0 2 ] = ): n2 EF n(t) - F n+1(t)]. T T n=1 (4.46) The variance of the snow renewal process is then Var[K(t)] = E[K(t) 2 ] - [m(t)] 2 . (4.47) Unconditional Probability Distributions Although not stated in the eauations because of notational convenience, the random variables T(u), Y and Tn were all conditioned upon the snowpack ablation rate C. 50 The unconditional distribution function of Y is obtained by integrating the product of the conditional distribution function of Y given by equation (4.27) and the probability density of C over all values of C, or co Fyle(y1c) f (c) dc, a I F(y) = (4.48) where a=c . . minimum The lower limit of integration, a, is explained in the next section and is based upon theoretical considerations. An infinite ablation rate would indicate instantaneous ablation. Instantaneous ablation would be characteristic of sleet or snow that melted upon contact with the ground, or rain. The unconditional mean of Y can be easily obtained as E[Y] = f E[Ylc] f c (c) dc. (4.49) a The unconditional variance has been shown to be (Parzen 1962) Var[Y] = E[Var[Ylc]] + Var[E[Ylc]]. (4.50) The unconditional distribution functions, means and variances of T(u) and T n are similarly obtained. 51 Analysis of Parameters The probability distributions of the random variables selected for study are all functions of two parameters, X and y, where y was defined as Y = y i is the parameter in the exponential distribution of precipitation per storm and is eaual to the inverse of the mean precipitation per storm. Since y is the product of the snowpack ablation rate, c (inches day -1 ) and y 1 (1n. -1 ), it is the inverse of the time required for the mean snow water eauivalent per storm to ablate to O. The ratio defined as P ablation time of mean snow per storm mean storm interarrival time X (4.51) is a aualitative measure of snowpack characteristics. If p > 1, there is a positive probability (Prabhu 1965, p. 16) that snow will remain on the ground indefinitely. The storms are occurring at a rate that is greater than the time reauired to ablate the mean snow water equivalent per storm to O. A value of p > 1 (y < X) would then be characteristic of perennial snowpacks. Such a value of p would be expected for arctic and antarctic regions, and in temperate climates, only at a few high elevation sites. For most temperate region snow zones, p < 1 (y > X). The value of p would, in general, increase with latitude and with elevation. On a local basis p would be expected to be a function of local topography, vegetation and climate. Small values of p would be associated with 52 shallow snowpacks, infrequent storms and high ablation rates. For the case where p = 1 (y = X), the snowpack will eventually ablate, however all moments of the distributions of T(u), Y and Tn are infinite. The specific model that has been presented in this chapter is based upon the restriction that p < 1. This is then a snow model for temperate regions where the snowpack eventually ablates entirely. Under this restriction, the probability distribution functions of T(u), Y and n T are proper since the Laplace transforms of each are equal to 1 when evaluated at s = 0, and the moments are finite and positive. Since y > X in this model, a lower bound exists on snowpack ablation rates, cy i > X C > yl , (4,52) The snowpack ablation rate, c, is thus restricted to the interval X < c < m. A probability distribution of c then becomes a shifted yl distribution, the shift being equal to the lower bound on c. y was expressed in terms of multiples of X. Values of p were then computed and plotted in Figure 4.1. For a given value of X, p is very sensitive to increasing values of y up to 4X . The value p = 0.5 is reached when y = 2X. Thereafter increasing y values have relatively little affect on P' For a given storm interarrival time, the mean snowpack duration decreases rapidly with decreasing storm magnitude and increasing snowpack ablation rate. For values of p = 0.5 the snowpack 53 1 - Figure 4.1. The ratio p as a function of y. 54 duration will be the dominant feature of the snow process. When p decreases' below 0.5, the snow-free period of the winter season becomes the major feature. For values of y > 4, the increasing snowpack ablation rate and/or decreasing magnitude per storm exert relatively little influence on the fraction of the winter that snow remains on the ground. CHAPTER 5 APPLICATION TO A CLIMATOLOGICAL STATION In this chapter, data from a climatological station in Arizona are used to obtain parameter estimates. The basic assumptions are investigated and the model is compared with data. Station Selection Daily precipitation measurements as well as other climatologie data are available at first and second order stations, and at cooperator stations. However, only at first order stations are daily measurements of snowpack water equivalent made. Except for the estimation of snowpack ablation rates for which snowpack water equivalent measurements were necessary, all the information required by the model for parameter estimation (daily precipitation and daily snowfall) and for comparison of the derived random variables with observed values, is available at all climatological stations. To obtain the necessary estimates of snowpack ablation rates, a first order climatological station was selected. Although other climatological data are available at many stations for 60 years or more, snowpack water eauivalent measurements were initiated at many first order stations during the period 1952-1953 (Thom 1966), The daily precipitation, snowfall and snow depth records corresponding to the longest usable record of snowpack water equivalent were therefore used. 55 56 A station that exhibited variable snowpack characteristics from season to season was desired so that a range in observed values of the random variables would be obtained. The climatological station at Flagstaff, Airzona, met these criteria since it is located at a reasonably low elevation (6993 feet) within an important snow zone, the Central Arizona Highlands, and is a first order station. Although snowpack water equivalent measurements were initiated in 1953, numerous missing data in the first few years resulted in the selection of the 15-year period 1957 through 1971 water years as the length of record to use in the model comparison. The winter precipitation season in Arizona includes the period November through April and at higher elevations 75 percent or more of the precipitation occurs as snow (Sellers 1964). At Flagstaff snow may occur as early as October and as late as May (excluding occasional summer and early fall snow showers). Based upon a summary of climatologic data for Arizona by Smith (1956), Table 5.1 shows the estimated percent of mean monthly precipitation that occurs as snow at Flagstaff. If the criterion of 50 percent or more of the mean monthly precipitation occurring as snow is used as the basis for defining the winter snow season at Flagstaff, the winter snow season is the period December through March (approximately 120 days). Snow Storms A storm has been defined by Todorovic and Yevjevich (1969) as continuous precipitation between two non-rainy intervals. However, precipitation data are commonly reported as totals for a specified time 57 Table 5,1. Mean monthly precipitation and estimated water equivalent of mean monthly snowfall at Flagstaff, Arizona. Length of Record Nov. Dec. Jan. Feb. Mar. Apr. Mean monthly precipitation (inches) 1898-1953 1.24 1.72 1,92 1.91 1.86 1.28 Mean monthly snow water equivalent** (inches) 1897-1953 .60 1.33 2.00* 1.49 1.41 .56 78 76 Percent of precipitation occurring as snow 48 77 104 43 * Mean snow exceeding mean precipitation is probably due to snow density averaging less than 0.10 during January as well as measurement errors. ** Snow water equivalent estimated using snow density of 0.10. interval such as total hourly or daily precipitation. Therefore a working definition should be one that permits a storm to be identified from such data. Each rainy hour or day has been considered as a storm event as has each uninterrupted seauence of rainy hours or days (Todorovic and Yevjevich 1969). Since freauently only daily data are available and cyclonic precipitation events often last for more than one day, a storm is defined as an uninterrupted seauence of wet days, each day receiving an amount of snow water equivalent equal to or greater than .01 inch, Each storm is separated in time from other storms by at least one dry day. A dry day is one receiving no precipitation or an amount 58 less than .01 inch. This definition is similar to that used by Kao et al. (1971) and that proposed by Fogel et al. (1973). During the winter season occasional rain events do occur. Since only snow is considered in the model, rain events were not included. Rain events were isolated by comparison of the daily precipitation records with new fallen snow records. When a mixed event occurred, a new snow density of 0.10 was assumed and multiplied times the depth of new fallen snow. Although the density of new fallen snow may vary from as low as 0.01 to 0.15, an average density of 0.10 is commonly used in the United States (Garstka 1964, United States Weather Bureau 1969). For precipitation events comprised entirely of snow, daily precipitation was used as an estimate of new snow water equivalent. The counting process fN(t); t > 01 was assumed to be a homogeneous Poisson process with parameter X in the development of the snow model. A Poisson process is characterized by independent, exponentially distributed interarrival times with parameter X representing the mean number of arrivals per unit of time (Parzen 1962). In order to determine the reasonableness of the assumption of a homogeneous process, the winter snow season was divided into three intervals of approximately 40 days. The interarrival times (I) as measured between storm termination epochs, for each interval then formed a sample. The three observed distribution functions were tested against the observed distribution function of interarrival times for the entire season. In plotting observed distribution functions in this study, the Weibull plotting positions were used (Chow 1964). The k-sample analog of the Kolmogorov-Smirnov test 59 derived by Kiefer (1959) was used to test the hypothesis that the interarrival times during each of the three 40-day periods were identically distributed as the interarrival times of the entire 120-day winter snow season. At a confidence level of a = 0.05, the hypothesis was not rejected (fT = 1.112, cp 2 -1 (.95) = 1.584). T was computed by the method given by Kiefer and the critical value of the statistic is from Kiefer's Table 2. Although such a test is not conclusive it does indicate that the assumption of identically distributed interarrival times is acceptable. The observed sample and pooled distribution functions are shown in Figure 5.1. Two parameter estimates of the Poisson process were obtained. The first estimate was made by the method of moments (total storms/total days = 174/1800 = .097 storms-day -1 ). A second estimate was made using an equation for a homogeneous Poisson process derived by Gupta (1974) EEI max ] (Xt)i (-1) i+1e -Xt i X i! (5.1) i=1 where I is the maximum storm interarrival time per time interval, max t is the length of the time interval, and X is the Poisson parameter. The sample mean of the observed maximum interarrival time for each season (26.67 days) was used as an estimate of EEI max I. The parameter estimate -1 obtained from equation (5.1) was X = .123 storms-day . 60 I.0 0 0.9 a° 4 A A 0.8 X ! A 07 A A 0.6 1--n it ).< AA 4 r• In A )1( k• i X I : *• :>< 0 0 0 0 A 0.3 .ç 0 0 0° 0 ! AX0 : A ' ,.; 1 * iI 0 / o • iô i R I 0 0 I A Ile 0 • i o 2 k6 ' 0 • o Legend: - Pooled Sample xxxx- Dec.1 -Jan.9 0000 - Jan.10 - Feb.I8 AALIA.— Feb.I9 - Mar.'30 0 • 0.4 A o A xo o i A Ao 0.5 A x x • A0 Kiefer's K-sample K-S Test: Nri----1.112<T: (.95)=1.584 --- -Maximum Deviation O o 4 o 8 • 02 A X x 0.1 0.0 10 20 30 Storm Interorrival Time; I (days) Figure 5.1. The observed distribution functions of storm interarrival times for three 40-day periods and the 120-day season. 40 61 Two estimates of the Poisson parameter were used because under certain conditions, one or the other may not be a sufficient statistic. Storms as defined in this study represent the lumping of one or more bursts of continuous snowfall into daily snowfall and the snowfall from consecutive days into a single storm. As more bursts of snowfall are included in a given storm, the method of moments estimate of X would further underestimate the true storm intensity. Under such conditions, this estimate would no longer be sufficient since it would no longer contain all possible information regarding the Poisson parameter. That is, the estimate would be conditioned upon the number of bursts included in the storms. In the second estimate of X, if storms were infrequent, the maximum interarrival time would be long and little affected by storm occurrences. If storms increased in frequency, the maximum interarrival time would be shorter and increasingly dependent upon storm frequency. Under these conditions, the second estimate of X would no longer be sufficient. Since the interarrival times of a Poisson process are exponentially distributed, the hypothesis that storm interarrival times were exponentially distributed was tested using a chi-squared goodness of fit test. At the ce = .05 confidence level, the null hypothesis that storm interarrival times were exponentially distributed with parameter X = .097 was not rejected (x 2 2 calculated = 13.241, x . 05, 7 d.f. oN 14.067). Similarly the distribution with X = .123 was not rejected (x 2 2 calculated = 3.399, x .05, 7 d.f. = 14.067). The observed and theoretical distribution functions of storm interarrival times are 62 shown in Figure 5.2. The sample means and variances of the number of storms and of the storm interarrival times were computed. Table 5.2 shows these sample statistics as well as the theoretical means and variances. Since X = .097 was estimated by the method of moments, the mean of the Poisson distribution of the number of storms is equal to the observed mean number of storms. Table 5.2. Mean and variance of the number of storms and the storm interarrival times at Flagstaff, Arizona. Observed Number of Storms (Dec. 1 Mar. 30) Storm Interarrival Time (Days) Mean Variance X = .097X = .123 • Mean Variance Mean Variance 11.64 10.97 11.64 11.64 14.76 14.76 8.54 64.85 10.31 106.28 8.13 66.10 That the storm termination epochs can be considered as Poisson events is evidenced by the exponentially distributed interarrival times and the ratio of the sample mean and variance of the numbers of termination epochs. The ratio of the mean and variance of a Poisson process is one. For Flagstaff the ratio of the sample mean and variance was 1.06. In the application of the model to a specific short time interval, use of the Poisson parameter estimated for the entire season may give 63 ( !51)d 64 less satisfactory results. This is indicated in Figure 5.3 where X was estimated for six 20-day intervals and three 40-day intervals. In this case the larger fluctuations in parameter estimates in the 20-day intervals than in the 40-day intervals is probably the result of smaller sample sizes. However, a greater storm frequency later in the 120-day period is indicated but is not large enough to result in the rejection of the assumption of homogeneity. The length of the interval within which the Poisson parameter can be assumed constant would be a function of the regional climate. In Arizona this may be up to 180 days while in other regions, more or less. When all the random variables defined on the process are considered (see below), the method of moments estimate of X generally gives better results than does the estimate obtained from equation (5.1). Since equation (5.1) was derived from the consideration of a homogeneous Poisson process, this may be the result of a non-homogeneous Poisson process over the 120-day period. However, in Figure 5.4 the two parameter estimates are shown as estimated from cumulative years of record, 1 through 15 years. An apparent downward trend and considerable fluctuation in the value of X as estimated by equation (5.1) is shown. This may indicate that an insufficient number of years of record was used to estimate X using equation (5.1). The lower curve in Figure 5.4 shows the method of moments estimate of X. After the third year's data had been included, the estimated value of x stabilized. Addition of more years had little apparent influence upon the parameter estimate. 65 LO 1" 1 (\I -7 4 .1048LUDiDcj co o UOSSIO cl eq 1 0 0 o a4ow4s3 66 (.0 3 c‘i xr 'Jo laumiod LIOSSIOd 941 co o d 0 o o d 67 Figure 5.4 also indicates that useful estimates of the Poisson parameter may be obtainable from as little as three to five years of data at Flagstaff. Fogel et al. (1971) found that useful estimates of the Poisson parameter of the number of thunderstorm events per season could be obtained with as few as five years of record and a model calibrated with 10 to 15 years. Since thunderstorm events are generally more variable than other storm types, it is reasonable to expect that more stable Poisson parameter estimates could be obtained for winter cyclonic storms with the same period of record. Arizona lies essentially in a single climatic region (Sellers 1964). It is anticipated that similar behavior of the estimates of X would be found for other stations in Arizona. However, some variation in the parameter estimates may arise due to the change in the frequency of snow events with elevation. In other climatic regions the number of years of record required may vary. The over or underestimation of X would influence the model re,. sults. For example, if X > X more than the true mean number of storms would be predicted. For a given probability, more total snow water equivalent would be predicted by the distribution function of X(t) using X than with X. The snow-free periods would be shorter and the snowpack durations would be longer for a given probability level. The number of snow-free, snow cycles and therefore the number of renewals A may then be fewer than would be predicted using X. If X < X the opposite effect would be seen. 68 Storm Magnitudes Snowstorm amounts were assumed to be independent, identically distributed random variables. In order to test the reasonableness of the assumption that storm magnitudes were identically distributed, the observed distribution function of storm magnitudes occurring in the three periods used above for storm interarrival times were compared with the observed distribution function of all storms during the December through March winter period. Using Kiefer's test for the three samples and a confidence level of 0.05, it was concluded the hypothesis that the storm magnitudes for each of the three periods were identically distributed as the observed distribution function of storms for the entire season could not be rejected (/T = 0.566, (43, 2 -1 (.95) = 1.584). The observed distribution functions of the three periods and of the total winter snow season are exhibited in Figure 5.5. The parameter of the exponential distribution of storm magnitudes, y i , was estimated by the method of moments using all storms dur- ing the 15 years of record. The estimated value was found to be y i = 1.80 storms-inch -1 . The hypothesis that snow storm magnitudes were distributed ex. ponentially with parameter y l = 1.80 was not rejected at the .05 level 2 of significance (x 2 calculated = 2.675, x .05, 3 d.f. = 7.815). Figure 5.5 shows the theoretical and observed distribution functions of the snowstorm magnitudes. A model assumption was that storm magnitudes could be concentrated at the storm termination epochs. When applied using daily data, the 69 A 0.9 0.8 117 116 Legend: • •• Pooled Sample • xxx Dec. I - Jon. 9 000 Joni° - Fe0.19 AAA Feb.19- Moc30 04 - Goodness of FIT Tests: (lifer's K-sample Test: IY ..566< 1•;' (.95)•1.594 xa Test: Q3 - 1(lcole. • 2 E 75 c'''Jas,3d.r. ' 18115 Q2- - Q0 00 114 0.6 08 ID 1.2 lA 1.6 1.8 22 2.4 2.6 2.8 3.0 Storm Snow Water Equivalent, X(inches) Figure 5.5. The observed distribution functions of storm snow water equivalent for three 40-day periods, the 120-day season and the theoretical distribution function. 3.4 3.5 7.0 7.2 70 working assumption was that the total storm precipitation could be concentrated on the last day of a storm. The ablation rate in the model does not change during the storm duration. Therefore in regions where storms may last a number of days, during which meteorological conditions do not favor ablation, the model may underestimate the snowpack duration and overestimate the snow-free periods. However, in Arizona, storms seldom remain within the state for very long, seldom resulting in more than two days of continuous, widespread precipitation (Sellers 1964). The mean storm duration at Flagstaff for storms as defined in this study is 2.06 days. In regions where storms persist but a short time such as Arizona, the predicted snowpack behavior may not be significantly affected by this assumption. Although an exponential distribution was not rejected as a model of storm magnitudes, inspection of Figure 5.5 indicated that for small values of X, the computed distribution function gives smaller probabilities than were observed. Since the model as applied in this study was specific for exponentially distributed storm magnitudes, application to regions where magnitudes were found to be described by'some other distribution would require the substitution of this distribution for the exponential distribution in Chapter 4. Conceptually this poses no problem. Depending upon the specific distribution, the derivation of the distributions of the random variables could become mathematically C omplex. To determine the number of years of record required for an approximately stable estimate of the exponential distribution parameter, 71 the parameter was estimated using cumulative years of data and plotted in Figure 5.6. It can be seen that there is considerable variation in the estimated value and a general downward trend. The parameter estimate appeared to stabilize after the 12th year had been included. Unlike the occurrence of storms, the determination of a stable distribution requires considerable data. Up to 50 years of data are recommended for stable frequency distributions in mountainous areas (Weisner 1970). However, unlike traditional precipitation frequency analyses, individual storm amounts are considered in an event based stochastic model. This may result in a larger effective sample size. Although a stable parameter estimate of the distribution of storm magnitudes may not have been obtained with 15 years of record, Figure 5.6 indicated that at least the length of record was adequate to dampen the large fluctuations in the estimate. Inaccuracy in the measurement of snow water equivalent could have a significant affect upon the general goodness of fit of the model. Precipitation, including snow, is usually measured in precipitation gages at National Weather Service stations. Even with Alter shields installed to reduce turbulence at the orifice, a rain gage measurement of snowfall may be considerably less than the true snowfall (United States Army Corps of Engineers 1956, Rechard 1972). The estimate of y i was based upon precipitation gage measurements of snow water equivalent. Since precipitation gage snow catches underestimate the true amount of snowfall with the magnitude of the underestimation increasing with wind speed during snowfall, the parameter y i would be 72 uo!invisla 10Ru3uodx3 Btu o J919WD1Dd ell/ ;0 a ICIMIS3 73 expected to be smaller than its estimate y i . The estimate of y l has a direct influence upon the adequacy of the computed distributions of X, X(t), Y, T and the renewal process. Since y i is the inverse of the mean storm water equivalent, y i < y i would have a similar affect upon the distributions of these random variables as would X > X. Snowpack Ablation Rate In the model the snow ablation rate was assumed to be a random variable, C, varying between seasons. The estimation of snowpack ablation rates was made as follows. For each winter season the snowpack water equivalent was extracted from the records for all days between storms, beginning the first day after a storm and ending the day prior to the conmencement of the next storm, or when the snow water equivalent reached the minimum value recorded. Within each winter season, the water equivalents were adjusted to the maximum water equivalent observed during the Season. To each sequence of water equivalent values for days between storms, the difference between the maximum observed water equivalent and the water equivalent on the first day after a storm was added to all water equivalent values of that sequence. In this way the sequence of observed water equivalent values for the interstorm periods were corrected to a common value. Snowpack water equivalent was regressed on days after the end of storms. An estimate of the seasonal average daily ablation rate was then the negative of the regression coefficient. A summary of the regression analyses is presented in Table 5.3. 74 Table 5.3. A summary of the linear regression analyses of snowpack water equivalent on days after the termination of storms at Flagstaff, Arizona. Yéar Sample Size 1957 21 .082 in.-day 1958 26 1959 Ablation Rate (=-b) Standard Error -1 .011 in. -.86 52.25** .171 .020 -.87 71.93** 24 .082 .034 -.46 5 • 97* 1960 60 .048 .003 -.90 247.30** 1961 20 .053 .016 -.62 11.14** 1962 72 .084 .003 -.97 1038.01** 1963 24 .054 .006 -.88 78.74** 1964 28 .190 .040 -.68 22.53** 1965 46 .134 .015 -.79 75.52** 1966 56 .068 .007 -.80 93.82** 1967 23 .051 .015 -.60 12.04** 1968 50 .041 .011 -.46 13.01** 1969 57 .171 .029 -.62 35.50** 1970 16 .117 .021 -.83 30.44** 1971 36 .053 .007 -.79 57.13** * Significant at a = .05. ** Significant at a = .01. 75 By equation (4.52) the theoretical minimum ablation rate must be greater than Y-- Yi Therefore the distribution function of the random variable C is shifted. The magnitude of the shift, c, minimum as 0.001 inch-day -1 was taken greater than the value of the ratio /— . For Yi Flagstaff, c „ corresponding to X = .097 was .054 and that correminimum sponding to X = .123 was .068. In determining the observed distribution function of ablation rates, the rates from Table 5.3 were plotted. An ablation rate less than the minimum rate was assumed to be equal to the minimum. Thus each observed distribution function possessed an atom at the origin. The distribution function corresponding to X = .097 is shown in Figure 5.7(a) while that corresponding to X = .123 is shown in Figure 5.8(a). In both cases, the observed distribution function was divided into three intervals of width equal to one-third of the range. The interval probability was determined and the distribution function discretized by assuming the interval probability was concentrated at the interval midpoints. In subsequent use conditional and unconditional distribution functions were obtained using the interval midpoints and associated probabilities. Unconditional distribution functions and moments were obtained by weighting the conditional distribution functions and moments by each interval probability of C, then summing over all three values. The intervals, midpoints and probabilities are also shown in Figures 5.7(b) and 5.8(b). 76 0.0 0.00 .02 .04 .06 .08 .10 .12 .14 .16 Snowpack Ablation Rate, C(inches- day 0 .18 .20 .18 .20 - (a) 1.0 0.8 1.7 c.) 0.6 0.4 0.2 0.0 0.00 .02.04 .06 .08 .10 .12 .14 .16 Snowpack Ablation Rate, C(inches-day ') - (b) Figure 5.7. The empirical distribution of snowpack ablation rates with minimum rate computed using X = .097. -- (a) The distribution function. (b) The discretized density function. 77 1 .0 0.8 a. 0.4 0.2 0.0 0.00 .02 .04 06 08 .10 .12 .14 .1 6.18 .20 Snowpack Ablation Rate, c (inches-day) (a) 1.0 0.8 0.2 0.0 0.00 .02 .04 .06 .08 .10 .12 .14 .16 .18 Snowpack Ablation Rate (inches-day) (b) Figure 5.8. The empirical distribution of snowpack ablation rates with minimum rate computed using X = .123. -- (a) The distribution function. (b) The discretized density function. .20 78 Since a constant value of C within seasons was required in the model, the affect of this assumption upon the results is of interest. Using the same regression technique that was applied in estimating seasonal snowpack ablation rates, the ablation rates for the eight 15-day periods included within the 120-day winter snow season were estimated. In the analyses, the snowpack water equivalents within each 15-day period for the 15 years of record were adjusted to the maximum water equivalent observed during the 15 years. The difference between the maximum observed value and the value on the first day after a storm was added to all observed vàlues for that particular intrastorm period. The snowpack water equivalent for each 15-day period was then regressed on days after the ends of storms to yield an estimate of the average ablation rate (the negative of the regression coefficient). The results of the regression are summarized in Table 5.4. The snowpack ablation rates were plotted in Figure 5.9. The average seasonal ablation rates computed from the empirical distribution of seasonal snowpack ablation rates corresponding to X = .097 and X = .123 are shown as are the corresponding minimum values and the largest estimated rate. Except for the ablation rate for the first period, the ablation rates follow the seasonal trend of total net energy, decreasing from December to a low in January, then increasing with a sharp increase in late March. The empirical seasonal distribution of the ablation rate um = .054 to .190. All but corresponding to X = .097 extends from c minim the first and last 15-day period estimates fall within this range. This indicates that the assumption of a single ablation rate for the entire 79 Table 5.4. Period A suuflary of the linear regression analysis of snowpack water equivalent on days for eight 15-day periods within the winter snow season at Flagstaff, Arizona. Sample Size Ablation Rate 19 .028 in.-day 2 73 3 Standard Error -1 .0104 in, -.55 7.28* .186 .0281 -.62 29.55** 100 .067 .0034 -.89 391.93** 4 72 .054 .0061 -.73 80.89** 5 86 .110 .0103 -.76 112.94** 6 64 .078 .0191 -.46 16.82** 7 63 .114 .0141 -.72 65.29** 8 34 .444 .0495 -.85 80.17** * Significant at = .05. ** Significant at a = .01. 80 0.40 o Q; 0.30 6 cc c o :4--0 4 ‹t 0.20 cmax.=.190 0.10 0.00 15-Day Time Intervals Figure 5.9. Daily snowpack ablation rates estimated for the eight consecutive 15-day intervals included in the 120-day season. 81 season could be expected to yield acceptable results even though the ablation rate may be over or underestimated for certain short time intervals. Since the range of the empirical distribution corresponding to X = .123 is more narrow, less satisfactory results might be obtained. The estimation of the minimum ablation rate by equation (4.52) can also influence the model. A larger c , . results in higher minimum ablation rates while a smaller c . . results in lower ablation rates minimum as is indicated in Figure 5.9. Although a random variable, the consistent over or underestimation of C would have a similar affect upon the random variables Y and T as did X and y i . If the values of C were overestimated, the unconditional distribution functions of Y and T would reflect a higher ablation rate. For a given probability the predicted snowpack duration would be less than for lower rates of ablation. Since T = V + Y, the effect would be less direct since the length of a snow-free, snow cycle depends upon the snow-free period also. Errors in the measurement of the daily snowpack water equivalent as well as in the regression technique employed to estimate values of C could influence the results. However, these are probably small when used in conjunction with the assumption of a single seasonal mean daily ablation rate. Total Snow Water Equivalent The sum of the individual magnitudes of storms occurring during each winter snow season was used to obtain the observed distribution 82 function. The sample mean and variance were calculated. The compound Poisson distribution function given by equation (4.11) was evaluated using both Poisson parameter estimates. A Fortran program was written for the evaluation of the distribution function. The programming is explained in Appendix *B and a listing of the program provided. The observed and theoretical distribution functions were plotted in Figure 5.10. Although an atom exists at the origin for each theoretical distribution function, in both cases they were of such small magnitude that they were not shown. P[X(t) = 0] = 9 x 10 -6 and P[X(t) = 0] = 4 x 10 -7 when X = .097 and .123, respectively. The distribution function computed using X = .097 only approximately describes the observed distribution function according to Figure 5.10 while the distribution function computed using X = .123 is unsatisfactory. This observation was substantiated by the chi-squared goodness of fit test. At the a = .05 level, the compound Poisson distribution computed using not rejected (x 2 calculated 7 3.038, x2 . 05, 2 d.f. a = .097 was = 3.841). The compound Poisson distribution evaluated using X = .123 was rejected (x 2 calculated = 14,826, x 2 .05, 2 d.f. = 3.841). The sample mean and variance and the means and variances computed from equations (4.13) and (4.15) have been presented in Table 5.5. The individual storm magnitudes were assumed to be independent of the number of storms. The total storm snow water equivalent was therefore a compound Poisson process. This independence assumption may be reasonable where the storm duration is very short in comparison to c• 83 0 • O 4-1 9-1 0 4J4-) 0 (111 4-1 0 Cr' a.) CY) II ti44-4 4-1 3-1 (1) • CO 3-1 4-1 CO ' (1) '0 0 1-1 cr r-4 c.) cti • 4-) 4-0 W O 0 4)0 U) 4-) -0 co cti 10 '0 C) ' • nJ • r..1 0 84 Table 5.5. The observed and theoretical means and variances of total seasonal snow water equivalent at Flagstaff, Arizona. Observed Mean (inches) 2 Variance (inches ) = .097 X = .123 6.44 6.47 8.20 8.67 7.19 9.11 the time between storms. In some areas, storm durations may be of such length that they give rise to a stochastic dependence between the number of storms and their magnitudes (Gupta 1973). The mean storm duration at Flagstaff was 2.06 days for the 15 years of record used. The mean time between storms (measured as the time between the last day of a storm and the first day of the next storm) was 6.47 days. The mean interstorm period is approximately 3.2 times as long as the mean storm duration. Although probably not large enough to strictly meet the requirement of independence, it is thought to be sufficiently large so that the assumption is reasonable. Without it, the derivations of the distribution functions of the random variables would become mathematically intractable. In spite of the possible inaccuracies in the data and the several assumptions made, the compound Poisson distribution with X = .097 was found to adequately describe the 120-day total snow water equivalent at Flagstaff. There is a close agreement between the observed mean and the mean computed from equation (4.13). The observed variance is greater than the theoretical variance but the magnitude of the difference is quite small (Table 5.5). 85 The Snow-Free Periods The snow-free periods are independent random variables, identically distributed as V with the same (exponential) distribution as the storm interarrival times. An investigation of the snow-free periods was undertaken since the distribution of the snow-free periods is potentially useful information and is another check upon the applicability of the model. Observed values of V were obtained from the daily record of the depth of snow on the ground as the number of days between occurrences of measurable snow depth (one inch or more). To compare the observed snowfree periods with the computed exponential distributions, all snow-free periods for the 15 years of record were counted. If a measurable amount of snow was reported on December 1, V 1 was recorded as 0 days in length. For the 15 years of winter snow seasons, 81 snow-free periods were counted. The observed distribution function and the two computed distribution functions were plotted and are shown in Figure 5.11. The hypothesis that the snow-free periods were exponentially distributed with X = .097 was not rejected at the a goodness of fit test (X 2 = .05 level by a chi-squared calculated = 3.379, X2 ., 05 4 d.f. = 9.488). The exponential distribution function with a = .123 was rejected, however (x 2 calculated = 11.641, x 2 .05, 4 d.f. = 9.488). The sample mean and variance were computed and are compared with the theoretical means and variances in Table 5.6. That the snow-free periods are exponentially distributed is also indicated by the near equality of the observed mean and standard deviation (10.32 days). There is a close correspondence 86 87 Table 5.6. The observed and theoretical means and variances of the snow-free periods at Flagstaff, Arizona. A Observed Mean (days) 2 Variance (days ) X = .097 X = .123 10.65 10.31 8.13 106.45 106.28 66.10 between the sample mean and variance and the theoretical mean and vanA ance computed using X = .097. The snow-free periods (V) and the interarrival times between storm termination epochs (I) in the model are identically, exponentially distributed, with parameter X. The hypothesis that V was distributed exponentially with parameter X = .097 and the hypothesis that I was exponentially distributed with parameter X = .097 were not rejected at = .05. This may indicate that V and I are identically distributed. A A similar conclusion cannot be drawn for the parameter estimate X = .123 since the hypothesis that the snow-free periods were exponentially distributed with parameter X = .123 was rejected. The length of the snow-free periods as observed from data depends upon the definition of snowpack duration. Traces of snow depth were considered as being snow-free when extracted from the climatological records. If a trace of snow on the ground were not so considered, then the lengths of these periods would be somewhat less than observed. The significance of this possible source of observational error is not apparent in Figure 5.11. 88 Snowpack Duration The random variable snowpack duration (Y) was obtained as the unconditional distribution of the random variable T(u), the time to snow disappearance, by integrating the product of the density function of T(u) and the density function of X 1 over all values of X 1 (equation 3.9) Therefore only the random variable Y was selected for further exposition. The distribution function of Y given by equation (4.27) was programmed. The Fortran programming is described in Appendix B and a listing provided. I are identically distributed as Y) it was found that the number of terms required for convergence depended upon the size of the value of the random variable and also upon the snowpack ablation rate, C. Therefore, if, after a predetermined number of terms, the equation failed to converge, calculations were terminated. This did not occur until the unconditional distribution function reached or exceeded a probability of .90. Therefore all features but the upper tail of the distribution were obtained. The observed snowpack durations, Y, were determined from the daily records of snow depth as the number of days snow depth was reported as being one inch or more. Less than one inch of snow is reported as a trace and is considered here to be snow-free. Eighty-one observations of snowpack duration were obtained from the 15 years of record. The total length of time that snow remains on the ground Y fl was defined by equation (3.13). As was indicated in Chapter 4, Y n is of less practical value than Y since the number of times the snowpack ablates to 0 is also a random variable K(t). The distribution function of Y describes the probabilistic characteristics of the individual snowpack 89 durations. When used in conjunction with the random variable T and the secondary renewal process (discussed below) information as to the probabilistic nature of snowpack duration and the number of renewals can be obtained. The theoretical distribution functions of Y for both parameter estimates X = .097 and X = .123 and the observed distribution function of Y have been plotted in Figure 5.12. Unlike earlier cases, changing X parameter estimates has only a small apparent affect upon the theoretical distribution functions. The two theoretical distribution functions are quite similar. Both yielded smaller cumulative probabilities than is indicated by the observed distribution function. Neither theoretical distribution function was rejected as a model for snowpack duration at the cy = .05 level. For X = .097, x2 calculated = 4.275 while for X = 2 .123, x calculated = 5.005 and x2 . 05, 2 d.f. = 5.991. The observed mean and variance and the theoretical means and variances computed using equations (4.29) and (4.31) have been compared in Table 5.7. The observed mean was considerably smaller than the theoretical means while the observed variance was much smaller than the theoretical variances. Both theoretical distribution functions gave somewhat lower probabilities than were observed (Figure 5.12). If this is a result of parameter estimation in the model, it would indicate that X > X or y i < y l . Either the mean number of storms was overestimated or the mean snow water equivalent per storm was overestimated. In view of the be. havior of X and y discussed above and the other results, it is felt 1 90 (A 5A) d 91 Table 5.7. The observed and theoretical means and variances of the snowpack duration at Flagstaff, Arizona. Observed Mean (days) 2 Variance (days ) X = .097 X = .123 9,90 17.89 21.17 254.39 1263.64 4698.72 that the parameter estimation does not explain the fit of the theoretical distribution functions to the observed. The fit is probably the result of the assumptions regarding the ablation rate or to errors in the observed values of Y. Since the theoretical distribution functions were not rejected, the differences between the theoretical and observed distribution functions are not statistically significant (at cy = .05). The observed mean snowpack duration was considerably smaller than either theoretical mean (Table 5.7). The large number of short duration snowpacks shown in Figure 5.12 explains some of this difference. The very large observed and theoretical variances overshadow the difference in means and show that large fluctuations in the snowpack duration must be expected. In view of the large variances, the means are of relatively little importance in the characterization of the snowpack duration. The difference between observed and theoretical variances can be attributed, at least in part, to the calculations of variances conditioned on small values of C. Recall from Chapter 4 that when c = the moments of Y are infinite. When values of C near the theoretical 92 minimum value are used, the resultant conditional variances are very large (Figures 5.7 and 5.8). Since low ablation rates occurred more frequently, the variance conditioned on c 1 received greater weight in the calculation of the unconditional variance. When the observed values of V and Y were determined, a snow-free day was one with either no snow on the ground or a trace. In the monthly climatological data, the depth of snow was reported as a trace when it decreased below a depth of one inch. Snow density increases rapidly soon after snowfall and then continues at a slower rate. During melt the snowpack density maY be as high as .45 to .50 (Garstka 1964). The amount of snow water equivalent remaining on the ground when the snow depth is reported as a trace may be significant. The actual snowpack duration may then be underestimated from records when days with only a trace of snow are considered to be snow-free. If the days when a trace of snow contained appreciable snow water equivalent could be determined and included in the observations of Y, then the observed distribution of Y may be closer to the computed distribution than was indicated in Figure 5.12. Snow-Free, Snow Cycles n Unlike the random variable Y ,the random variable giving the total length of snow-free, snow cycles (i.e., the sum of n cycles) dur- ing a season, T 'a , assumes additional importance since the probability mass function, mean value function and variance of the secondary renewal process can be directly obtained from the distribution function of T. n The distribution function of T, given by equation (4.41) was programmed 93 (see Appendix B). As in the evaluation of the distribution function of Y, a convergence problem existed for large values of T and C. If the distribution function did not converge after a given number of terms, calculations were terminated. However, the probability of 0,90 was equaled or exceeded in the unconditional distribution before the equation no longer converged, so all but the upper tail was obtained. The observed values of T were determined as the sums of the paired observations of V and Y. The resultant 81 values of T for the 15 years of record were then plotted to form the observed distribution function. The theoretical distribution functions for each estimate of X and the observed distribution function have been presented in Figure 5.13. Up to approximately t = 35 days, the theoretical distribution functions bracket the observed distribution function. Thereafter both theoretical distribution functions gave lower cumulative probabilities than estimated by the cumulative relative frequencies. The chi-squared goodness of fit test was used to test the hypotheses that each of the theoretical distribution functions fit the observed values of the random variable. At the oe = .05 level, neither theoretical distribution func- tion was rejected. For the distribution function computed using X = .097, 2 2 = 12.592. For the distribution = 7.133 while X .05 , calculated X 6 d.f. ^ 2 calculated = 10.247 and the critical function computed using X = .123, value was the same as in the first test. As before the sample mean and variance of T and the theoretical means and variances computed using equations (4.34) and (4.35) were placed in Table 5.8. The sample mean was somewhat less than the 94 a) a) 0 • 95 Table 5.8. The observed and theoretical means and variances of the snow-free, snow cycles at Flagstaff, Arizona. Observed Mean (days) 2 Variance (days ) X = .097 X = .123 20.56 28.20 29.30 331.10 2370.06 4466.22 theoretical means. The variances computed from equation (4.35) were an order of magnitude greater than the sample variance, indicating that, at least for the 15 years of record at Flagstaff, the length of the snow-free, snow cycles was much less variable than would be predicted by the model of the snow process. As with the snowpack duration, the distribution of the snow-free, snow cycles depends upon the two parameters X and y as well as the distribution of snowpack ablation rates. Since T = V + Y, and V depends only upon x, the affects of the estimates of x and y i on T and the distribution of C would be generally similar to the affects the estimates have upon Y. The observed values of T were determined as sums of pairs of V and Y values. The possible underestimation of Y and overestimation of V values from the daily snow depth records would result in changing observed values of T. The observed mean of the snow-free, snow cycles was somewhat less than either theoretical mean but due to the inclusion of the 96 snow-free periods, this difference was relatively less than that noted for Y. The lack of agreement between observed and theoretical variances of T can be attributed, at least in part, to the calculations of conditional variances, conditioned on small values of C as in the calculations of the variances of Y. The large variances also indicate that at Flagstaff, large fluctuations in the length of the snow-free, snow cycles must be anticipated. However the model shows much greater fluctuations than is indicated in the historical record. In view of the large variances, the mean is of relatively little value in describing the behavior of the snow-free, snow cycles. The Snow Renewal Process The relationships between the probability mass function of the snow renewal process, {K(t); t > 0} , and the distribution function of n T , equations (3.22) and (4.44), were used to evaluate the probability mass function. Then equations (4.45) and (4.47) were used to compute the mean value and variance. The distribution functions of T n for n = 1,2,...,10 were computed for each estimate of x. For n = 10, the probabilities were of such small magnitude (P[T i° < 60] = .0008 when X = .123 and .0002 when X = .097) that calculations were terminated. The largest value of T for which con. vergence was obtained was t = 60 days when X = .123 and 70 days when X = .097. The 120-day period was divided into twelve 10-day intervals and the number of times the snowpack depth decreased to less than one inch in each interval was recorded. The probability mass functions for 20, 97 40 and 60 days were determined using each 20-day period, each 40-day period and each 60-day period beginning December 1. The 120-day winter snow season then consisted of six 20-day, three 40-day or two 60-day periods. These were pooled to yield a single sample for the 20, 40 and 60-day periods. The probability mass functions for t = 20, 40 and 60 days and for each estimate of X were then evaluated. The distribution function of the sum of ten snow-free, snow cycles was the largest calculated, therefore the probability mass function of a maximum of nine renewals could be determined. This was adequate, however, since the maximum number of renewals observed during any 60-day period at Flagstaff was seven. The observed and theoretical probability mass functions for the 20, 40 and 60-day periods are shown in Figures 5.14, 5.15, and 5.16, respectively. The results are variable. One theoretical probability mass function does not seem to give a consistently closer fit to the observed relative frequencies than the other for a given time period, nor is there a consistent pattern between the three time intervals. Both theoretical probability mass functions generally follow the observed mass function. A chi-squared test was used to compare the goodness of fit of the observed relative frequencies with the theoretical probability mass functions. A summary of these tests has been presented in Table 5.9. The probability mass function computed using the method of moments estimate of X was rejected (at 0 = .05 level) for the 20-day , period, but not for the 40 or 60-day periods. The probability mass function computed using x estimated by equation (5.1) was rejected for the 20 and 60-day periods. 98 q-1 o X 0 0 H v - 6 ro 6 cv 6 (u= (o z)>1)d _ 6 0 6 0 O 99 CD acr rn ch > N n n .0 0 4:e.C4s*C I 4:51 X X 0 X 0 x o O r°. (u= ot7)>Od ( 100 X 0 10 0 Cf) (0 o X 0 I .r-i 0 0 ;-1 a , IX re) 0 • y•-1 Ct1 C.) 5 -4 9-1 0 4-1 0 14 0 0 w .0 (0 '0 0 0 c0 0 0 4•4 $-4 CI) 01 ctl OX 0 3 0 0 H )0- 006. ttl CNJ O (u=(09))Od a) 101 Table 5.9. Chi-squared tests of the probability mass functions of the snow renewal process. Time Interval Critical Values Calculated X = .097 2 20 days x2 Values X = .123 = 3841 6.46 4.72 40 days 2 = 3 841 ' x .05, 1 d.f. 3.40 3.12 60 days 2 = 5991 X .05, 2 d.f. 2.41 6.05 X .05, 1 d.f. The mean value function and the variance of the snow renewal process were computed using equations (4.50) and (4.52). The observed mean value function and variance were computed for 10, 20, 30, 40, 50 and 60-day periods. All of the consecutive 10 through 60-day periods within the 120-day winter snow season were pooled to yield a single estimate of the mean value function and variance for each period. One exception to the above procedure was the 50-day period. Since the 120day season could not be evenly divided into 50-day perios, the two periods December 1 through January 19 and January 30 through March 20 were used. Over all time periods, the observed mean value function and the mean value function corresponding to X = .097 are in closer agreement than are the observed and that corresponding to X = .123. However, for A the first two time periods, 10 and 20 days, X = .123 gives a better fit 102 (Figure 5.17). The observed and theoretical variances have been presented in Figure 5.18. The variance of the observed snow renewal process is consistently larger than the variances computed from equation - (4.52). The difference increases with the length of the time period. That the computed probability mass function (X = .097) was re- jected (at ce = .05) for 20-day intervals but not for longer intervals may indicate that in short time intervals, the snow model may give results which inadequately describe the probabilities of the number of times the snowpack ablates. Since the snow renewal process was the last step in the model development, it is dependent upon all of the assumptions made throughout the development. The over or underestimation of parameters and the assumption of a single seasonal daily snowpack ablation rate are reflected in the probability mass functions of the renewal process and in the mean value function and variance. The manner in which the number of renewals within an interval were observed may have influenced the relative frequencies. As in the observation of Y, when the snowpack depth decreased to a trace, complete ablation was assumed. If the traces contained significant snow water equivalent and were considered, the resultant relative frequencies of the number of renewals may have been altered. The renewal process is asymptotically normally distributed (Parzen 1962). As the time interval becomes large, the probability mass function of the renewal process approaches the normal density function. Although a 60-day interval is probably not of adequate length for the number of renewals to be approximately normally distributed, comparison 103 3.0 2.0 1.0 0.0 10 20 30 40 50 60 Time Intervals (Days) Figure 5.17. The observed and theoretical mean value functions of the snow renewal process. 104 10 40 20 30 Time Intervals (Days) 50 60 Figure 5.18. The observed and theoretical variances of the snow' renewal process. 105 of Figures 5.14, 5.15 and 5.16 reveals that as the interval length increases, the observed and computed probability mass functions become less skewed. In Chapter 4, the ratio p was defined by equation (4.51). The ratio then represented the expected fraction of time that the ground was snow covered. When p was computed using X = .097, a value of 0.59 was obtained. For X = .123, p = 0.67. The ratio of the sample means of Y and T yielded a sample estimate of = 0.48. For the 15 years of record, the ground was snow covered an average of 48 percent of the 120-day season. The average amount of time the ground was snow covered as estimated from the ratio was somewhat greater than observed. In view of the large variances of Y and T observed in the Flagstaff data and computed, the ratio p would be expected to vary over a considerable range. Therefore the ratio p could only be used in a very qualitative description of the characteristics of the Flagstaff snowpack. CHAPTER 6 SUMMARY AND RECOMMENDATIONS In Chapter 6 a summary of the results of the snow model application to an Arizona climatological station is presented. Recommendations for possible future studies are made. Summary of Results A stochastic model of the snowfall, accumulation and ablation process was developed. Random variables characteristic of the snow process of snow storm occurrences and storm magnitudes that were independent and identically distributed. Distribution functions of the random variables were derived in general, and then in the case where storm amounts were exponentially distributed. An Arizona climatological station was selected. Parameter estimates were made, basic assumptions investigated and the model results compared with data. The climatological station at Flagstaff, Arizona, was used since it is located within an important snow zone and the snowpack observations necessary for model application were available. During the 120day long period December 1 through March 30, snow storms, uninterrupted sequences of days receiving .01 inch or more of snow water equivalent preceeded and followed by dry days, occurred approximately as a homoge- nous Poisson process. The snow storm magnitudes were exponentially 106 107 distributed with parameter y l . Storms were usually of short duration, therefore the assumption of independence between storm occurrences and magnitudes was thought to be reasonable. Two estimates of the Poisson parameter X were used. The first was determined by the method of moments while the second was developed from the consideration of the maximum interarrival times of a homogeneous Poisson process. The method of moments estimate generally gave better results than did the second estimate of X. The difference between the two parameter estimates could be due to non-sufficiency of the estimates, a non-homogeneous Poisson process or small sample size. It was concluded that the sample was of insufficient size for a stable estimate of x using the second method. The compound Poisson distribution with Poisson parameter estimated by the method of moments was found to adequately describe total snow water equivalent at Flagstaff. The compound Poisson distribution with the Poisson parameter estimated by the second method did not. The observed snow-free periods were exponentially distributed with parameter x estimated by the method of moments. In theory, the snow-free periods and the storm interarrival times are identically, exponentially distributed. This was also indicated in the observed distributions, where the snow-free periods and storm interarrival times were both described by an exponential distribution with parameter X estimated by the method of moments. The exponential distribution of snow-free periods with the parameter estimated by the second method was rejected, therefore a similar conclusion could not be drawn. 108 Thetheoret icaldistributionsofthesnowpackciuratio ns (Y ) and . ofthesnow-free,snowcycles (T. )fit the observed distributions at the chosen level of significance. These distributions were functions of the Poisson parameter x, the parameter y i from the exponential distribution of storm magnitudes and the snowpack ablation rate, C. Both Poisson parameter estimates were used in each of the theoretical distributions. Both estimates yielded similar results for the snowpack durations and for the snow-free, snow cycles. For large values of the random variables and high ablation rates, the infinite series in the expressions for the distribution functions did not converge when an acceptable number of terms were used. Cumulative probabilities of at least .90 were obtained before this occurred, so all but the upper tails were obtained. The distributions of Y, T and T n were conditioned upon the snow- pack ablation rate which appeared in the distribution functions as y = cy l . Estimates of the mean daily ablation rate for each season were obtained from a linear regression analysis of snowpack water equivalent data. The empirical distributions of ablation rates were formed from the regression coefficients. The unconditional distributions were subsequently found by discretizing the empirical density functions of the snowpack ablation rates and numerically integrating. Even though a single ablation rate was assumed to apply through each winter season, the model results were generally satisfactory. One exception was that theoretical variances much larger than the observed variances were obtained when ablation rates near the theoretical minimum were used. 109 The probability mass functions of the snow renewal process, giving the probabilities of the number of times the snowpack ablated completely in an interval, were determined from the distribution func- tion0fT.'s for 20-day, 40-day and 60-day intervals. When the distributionfunction o fT . I s with X estimated by the method of moments was used, the computed probability mass functions for the 40 and 60-day intervals adequately fit the observed relative frequencies. The computed probability mass function of the 20-day interval was rejected. The theoretical probability mass functions with X estimated by the second method did not fit the relative frequencies of the observed number of renewals in either the 20 or the 60-day periods. The theoretical mean value functions and variances of the snow renewal process were computed from the distribution function of T.'s. 1 The mean value function computed using the method of moments estimate of X showed close agreement with the observed mean value function. The theoretical mean value function computed using X estimated by the second method generally yielded larger values than the observed. The theoretical variances of the snow renewal process were somewhat lower than the observed variances and the magnitude of the difference increased with the length of the time interval. It was concluded that the snow model developed in this study could give useful results when the underlying assumptions were met, at least approximately. Applications to other regions would require verification of the adequacy of the assumptions. 1 10 Recommendations for Further Studies The model developed in this study possesses several shortcomings that warrant further research. In the derivation of the distribution functions of Y, T and T n , the infinite series forms of e the modi- fied Bessel function were used. For large values of these random variables, at high ablation rates, the infinite series in the distribution functions failed to converge with an acceptable number of terms. Further investigation is needed into alternate methods of determining the distribution functions. Two methods of estimation of the Poisson parameter were used. Under certain conditions, the estimates may not be sufficient. The conditions under which they are not could be the subject of further study. Alternate methods and alternate storm definitions could be included. The model is sensitive to the snowpack ablation rates. The empirical distributions of snowpack ablation rates were determined by regression analysis. The regression coefficients were then considered to be realizations of the albation rate. The use of regression coefficients as values of a random variable may be misleading since different results may be obtained when other methods of estimating the ablation rate are used. In future applications, the assumption of a single ablation rate might yield acceptable results. This single ablation rate could be estimated by regression analysis. Other methods of determining ablation rates and ways of including variable (within season) ablation rates need to be explored. 111 In the study, the model was compared with data from only one station. The applicability of the model to other stations in Arizona should be checked since the frequency of snow storms, storm magnitudes and ablation rates may vary from station to station. In addition, the space extension of the model would require parameter estimates and estimates of the ablation rates. When applying the model to stations outside the climatic region in which it was developed, consideration should be given to the several assumptions made in its development. A potential use of the model is in the determination of the stochastic properties of streamflow. A possible approach to obtaining the distribution function of discharge, Q(t), would be the application of unit hydrograph theory. The snowpack water equivalent, Z(t), is analogous to a rainfall hyetograph. Convoluting Z(t) with the kernal function representing ordinates of the unit hydrograph would define a new random variable, Q(t), the discharge or ordinate of the streamflow hydrograph. A significant difference between this approach and classical unit hydrograph theory is the snowmelt lag time, which would require investigation. APPENDIX A INVERSION OF THE LAPLACE TRANSFORMS OF Y, T AND T n The Laplace transforms of the density functions of Y, T and Tt are inverted. The inversions were carried out using Table of Laplace Transforms (Roberts and Kaufman 1966). All references with respect to Laplace transform pairs and operations with Laplace transforms are from this table. Inversion of the Laplace Transforms of Y The Laplace transform of Y was given by Prabhu (1965) as 2 _ LY(s) - (s+X+y) - As+X-Py) - 4XY 2X (A.1) From the table of operations, Part II of Roberts and Kaufman, inverse transform pair (1) ag -1 (s) = af(t) (A.2) where a= I. X The numerator of equation (A.1) is nearly of the standard form (inverse transform pair 84, p. 215) /22 s - is - a (A.3) 112 L 113 Applying the shifting theorem given by transform pair 3, p. 169 g -1 (cs - b) = c -1 b/c t e f(t/c) (A.4) Let b = + y) c= 1 t = y Then equation (A.4) becomes g-1(cs - b) = ef(y) (A.5) Rewriting equation (A.1) in the form given by equation (A.2) 1 LY (s) =-5-[(s+x+y) -/(s+y+x) 2 ,. - 4] (A.6) Applying the results of the shifting theorem given by equation (A.5) to the portion of the right hand side of equation (A.6) in brackets - (s) Y = Y(X+y) e 2X [s -/s 2 - (A.7) where a = 2(Xy) 2 . Using transform pair 84, p. 215, the inverse of equation (A.7) e -Y(x+Y)2(NY, 1/2 k 1 [2(XY) Y] Y , L'(s) - 2X Y where 1 1 (u)is the modified Bessel function of order 1. (A.8) 114 Inversion of the Laplace Transform of T and T n The transform inversion of L (s) differs somewhat from the inverT sion of the transform of Y in that the Laplace variable appears in the denominator, The Laplace transform of T was given as L (s) _ (s+x+y) - 1(s+x+y) 2 2(X+s) - 4 XY (A.9) Applying transform pair (3), p. 169 from the table of operations g -1 (as-b)= a -1eb/a t f(t/a) (A.10) Letting a= 1 b = Then g -1 e;.xt (A+s) — (A.11) f(s) 2 L -1eXt (s) — L - [ s+N - - 4 NY] (A.12) From transform pair (13) in the table of operations L -1 1 f(u) (du) 1' g(s) = (A.13) 0 When n = 1 L - 1 _g_(2) . f(u) du (A.14) But f(u) = L -1 g( s ) (A.15) 115 and g(s) = (s+y) - As+y) 2 - 4xy (A.16) Again using transform pair (3), p. 169 h(y) e -yt (s v/S2 4xy) (A.17) where a= 1 b=-y Then L e - Y t L -1 (s -/s 2 - 4xy) (A.18) By transform pair (84), p. 215 L -1-(s 470 2( ) - - r )2 I 1 [2t(xy)] (A.19) Substituting the right hand side of (A.19) into (A.18) i _ -1 . . ... L g(s) - e -yt 2(Xy) I [2t(Xy) 2 ] t 1 (A.20) Substituting equation (A.20) into equation (A.14) t g(s) 0 e-Yu 2(xy)15E2u(Xy)1/21du 1 (A.21) and subsequently, substituting equation (A.21) into equation (A.12) L -1 T (s) = e -Xt f 0 e (xy) I [2u(Xy) 1/2 ]du 1 (A.22) which is the inverse of the transform of T as given by equation (A.9). 116 The Laplace transform of T n was given as A 2 s+X+y) - 4 Xy L n (s) — [(s+X+Y) -2(x+s) T (A.23) As in the inversion of the transform of T, two shifts are necessary to put the transform into a standard form. The first shift is carried out as it was in equations (A.10) through (A.12) resulting in L n -1 e -Xt -1[(s+y) - j(s+y) (s) — 2 - 4Xy] n (A.24) 2n By equation (A.16), L -1 1 g(s) n s j f(u) (du) n — (A.25) 0 But f(u) = L and -1 g(s) • g(s) = Us+Y) - As+0 2 4 XY1 n . (A.26) Using transform pair (3), p. 169 h(s+y) = e - Y t (s -/s 2 - 4Xy) n where a= 1 b=-y. Then C ig(s) = e - Y t L -1 (s - i/s 2 - 4Xy) n(A.27) 117 By transform pair (36), P. 226 L-1 (s - - 420;) n n2(Xy)k In[2(Xy)kt]. (A.28) Substituting equation (A.28) into equation (A.25) 2n(W)2 [2(Xy)'ul(du)n. un L -1 h g(s) = j sn 0 (A.29) Substituting equation (A.29) into equation (A.24) L n -1 (s) = n e -Xt T e -yu ( xy , ) n /2 in [2u(xy) 1/2 ](du)n. 0 (A.30) APPENDIX B PROGRAMMING THE DISTRIBUTION FUNCTIONS OF X(t), Y AND T n Due to the complex nature of the expressions derived for the distribution functions of X(t), Y and T n , the University of Arizona CDC 6400 computer was used to obtain numerical results. Computer programs for each distribution function were written in FORTRAN. The programming techniques are briefly explained below, followed by a listing of each of the three programs. The Distribution Function of X(t) The distribution function of X(t) was written in the following form (yitx) F X(t) (x) = k 00 v - '1 — k! (k-1)! L (j - l)! (k+j-1) k=1 i=1 (B.1) In order to minimize roundoff errors due to the potentially large magnitudes of the numerators and denominators, the following approach was used. Beginning with the inner summation, the first term was stored. Each succeeding term was computed as the product of the immediately preceding term and a multiplier. For example, the first term of the inner summation is - y i (B.2) 118 119 When j = 2, the term is -Y 1 • (j-1) 2 Yl x (B.3) When j = 3, the term is (B.3) times • Similarly, the terms for j = 4,5, ... were computed as the preceding terms times this multiplier. After all terms of the inner summation were computed and stored (up to the predetermined maximum number of terms) the outer summation was computed. Again, the terms were computed by determining the multiplier, then computing the terms for k = 1,2,... in the same way they were computed for the inner summation. The terms of the outer summation were also stored in a temporary storage location. The distribution function was then evaluated by multiplying the outer temporary storage variable corresponding to k = 1 times every inner storage variable corresponding to j = 1,2,3, ... . The results were summed and the procedure repeated for k = 2. A cumulative sum was used to sum over all k. The product (k+j-1) in the denominator of the inner summation was included in the last step since it depended upon k. Upon termination of the summation at the maximum k, the exponential e -xt was included. The number of terms required for convergence (to 4 decimal places) depended upon the value of the random variable, X(t). The approximate number of terms required for convergence was determined experimentally for selected values of the random variable. These results formed an algorithm which was used to compute the number of terms in the inner and outer summations for a given value of X(t). 120 Double precision variables were used to further minimize errors due to roundoff. In the last step the mean and variance were computed. The Distribution Function of Y In programming the distribution function of Y, a similar approach was used. The distribution function of Y was written as CO F (y) = Y 1 k k-1 2k-1 Y X Y k! (k-1)! (-1)j-1(X+Y)j-iYi-1 (j-1)! (2k+j-2) ( B. 4) k=1 The first term of the inner summation was stored. The second term was computed as the product of the first and the multiplier. Each succeeding term was similarly calculated. The outer summation was computed in the same way. The remainder of the evaluation was carried out in the same manner as was the evaluation of X(t). Double precision variables were used to further minimize roundoff errors. The unconditional distribution function was computed by weighting the conditional probabilities by the probability of the snowpack ablation rate, C, and summed over all values of the ablation rate. The number of terms in the outer and inner summations required for convergence depended upon the value of Y. In addition, the number of terms also depended upon the snowpack ablation rate, C. The number of terms required for convergence were computed experimentally for selected values of Y and each value of the snowpack ablation rate. The results formed an algorithm for the calculation of the required number of terms. 121 In the last steps of the program, the conditional and unconditional means and variances were computed. The Distribution Function of T n The evaluation of the distribution function of T n was accomplished using the same prograuning techniques employed for X(t) and Y. The distribution function was written as k-1 2k+2n-2 F n(t) = n(Xy)n 7 ON) t T L, (k-1)!(k+n-1)! k=1 j= (-1)1-1(yt)j-1 I (2k+j+2n-i-3) 1=1 (...1)m-1(x)m-1 • (m-1)! (2k+j+2n+m-4) (13.5) m=1 The presence of a third summation required the employment of additional temporary storage. Otherwise, each summation was evaluated by computing each term as the product of the immediately preceding term and the multiplier. The product in the denominator of the middle summation was determined by computing the product for a given value of n over all values of j and k. The results were stored in a temporary two-dimensional array. The array was then called inside the do loops used to evaluate the distribution function. Double precision variables were again used to further minimize roundoff errors. The conditional and unconditional distribution functions were evaluated as they were in the program for Y. The conditional and unconditional means and variances were calculated. 122 Discussion The time required for compilation and execution of the programs increased with the increasing complexity of the program. The program for the evaluation of X(t) (TPRECIP) required approximately 6.4 seconds and 31060B core storage. Program SEVREV, which evaluated the distribution function of Y required approximately 19 seconds and 31603B core storage. Program SIX (the distribution function of Tn) required 1636 seconds and 64747B core storage. The time required increased tremendously in program SIX since the distribution functions of Tn (n = 1,2,...,10) were computed for 19 values of T and 3 values of C, and required the evaluation of a triple summation for each value of T and C. The number of terms required in the summations in Y and Tn increased nonlinearly as functions of the value of Y (or T) and of the snowpack ablation rate. For large values of Y (approximately 40 days) and T (approximately 60 days) and high ablation rates (c = .17 inchday -1 ) the probabilities no longer converged within acceptable computing time limits. Calculations were terminated when this occurred. In both programs, for each estimate of X, cumulative probabilities equal to or greater than .90 were obtained. In spite of the failure of convergence, all of the distributions except the upper tails were evaluated. 123 PROGRAM TPRECIP (INPUT,OUTPUTITAPE1=INPUT,TAPE2=OUTPUT) C PROGRAM TO EVALUATE THE DISTRIBUTION FUNCTION OF THE COMPCUND C POISSON PRCCESS OF TOTAL ACCUMULATED WATER EQUIVALENT TO TIME T C C C C C C C C C C C C C C C C C C LIST OF PROGRAM VARIABLES, PARAMETERS AND INDICES GAM - PARAMETER FPCM THE EXPONENTIAL DISTRIBUTION OF WATER EQUIVPER STORM - INNER SUMMATION INDEX - MAXIMUM NUMBER OF TERMS IN INNER SUMMATION - OUTER SUMMATICN INDEX - MAXIMUM NUMBER OF TERNS IN OUTPR SUMMATION - CCMPUTED NUMBER OF TERMS IN OUTER SUMMATION - COmPUTED NUMBER OF Tr_RMS IN INNER SUMNATICN - NUMBER OF VALUES CF THE RANDOM VARIABLE, X - LENGTH OF THE TIME INTERVAL, DAYS - TOTAL WATER EQUIVALENT TN THE INTERVAL OF LENGTH T - PARAMETER FROM THE POISSON DISTRIBUTION OF THE NUMBER OF STORMS IN THE INTERVAL T THE COMPUTED MEAN OF TOTAL SNOW WATER EQUIVALENT IN THE X MEAN INTERVAL T XVAR - THE COMPUTED VARIANCE OF THE TOTAL SNOW WATER EQUIVALENT IN THE INTERVAL T ALENT J JJ K KK LIOI) LJ(I) N T X(I) XLAM - DOUBLE PRECISION DFX(20),TEMPB,E,GX,TLOTEMP(150),ATEMP(150),XG,TL 1XG,Z DIMENSICh LK(20), LJ(20),X(20) C SET JJ AND KK EQUAL TO DESIRED UPPER LIMITS JJ = 150 KK = 50 C READ IN PARAMETERS, LENGTH OF INTERVAL,T, AND NUMBER OF VALUES OF X READ (1,100) XLAM,CAM,T,N 100 FORMAT(3F10.0,I3) C INITIALIZE ARRAYS 00 77 I=1.N X(I)=0. OFX(I)=0. 77 CONTINUE READ(1,101) (X(I),I=1,N) 101 FORMAT(8F10.0) TL = XLAM*T C = OEXP(- TL) C 00 LOOP TO ITERATE X VALUES 00 10 I=10 IF(X(I) .GT. O.) GO TO 999 OFX(I) = E GO TO 10 ARE CALCULATED C THE NUMBER OF TERMS OF THE OUTER AND INNER SUMMATIONS * 6 999 LJ(I) = 20 + LK(I) = 15 + X(I) . LK(I) ARE LESS THAN OR C IF THE COMPUTED NUMBER OF TERMS, LJ(I) AND TERMS, THE TOTAL NUMBER OF C EQUAL TO THE MAXIMUM ALLOWABLE NUMBER OF C TERMS ARE SET EQUAL. TO THE COMPUTED NUMBER KKKK = KK 124 IF( LK(I) .LE. KK) KKKK=LK(I) JJJJ = JJ IF(LJ(I) .LE. JJ) JJJJ = LJ(I) GX = GAM*X(I) XG = -GX TLXG=TL*GX C 00 LOOP TO COMPUTE ELEMENTS OF INNER SUMMATION OF DISTRIBUTION FUNCTION BTEMP(1) = 1 00 43 J=2 1 JJJJ BTEMP(J) = BTEMP(J-1 ) * (XG/(J-1)) 43 CONTINUE C DO LOOP TO COMPUTE ELEMENTS OF OUTER SUMMATION OF DISTRIBUTION FUNCTION ATEMP(1) = TLXG DO 53 K=2,KKKK KTK = K * (K-1) ATEMP(K) = ATEMP(K-1) * (TLXG / KTK) 53 CONTINUE C 00 LOOP TO ITERATE K DO 12 K=1,KKKK KKK = K-1 TEMPS= DO 13 J=1,JJJJ b. TEMPB = TEMPB + BTEMP(J) / (KKK+J) 13 CONTINUE OFX(I) = OFX(I) + ATEMP(K)*TEMPB 12 CONTINUE DFX(I) = OFX(I) * E • 10 CONTINUE C C COMPUTING MEAN AND VARIANCE OF TOTAL SEASONAL WATER EQUIVALENT XMEAN = (XLAM*T)/GAM XVAR = (2.*XLAM*T)/(GAM*GAM) PSTM= XLAM*T C PRINTOUT OF RESULTS WRITE(2,400) (LK(I),LJ(I),I=1,N1 400 FORMAT(1H ,I3,5X,13) WRITE(2,200) XLAM,GAM,PSTM,T,XMEAN,XVAR 200 FORMAT(1N1,15X,*CUMULATIVE DISTRIBUTION FUNCTION OF TOTAL SEASONAL ,20X,*GAMMA 1 I WATER EOUIVALENT4/1H0,20X,*LAMB0A*,14X,*=*,F11.4/1H 1*,13X,*=*,F11.4/1H ,20X,*MEAN STORMS/SEASON =*,01.1.4/1H ,20X,*SEA 4 ,011.4/1H t2 3 30 N LENGTH*,7X,*=*,F10.3 1 1H ,20X,*MEAN SNOW WE*,8X,*= ,011.41 4 SNOW WE = 40X,*VARIANCE OF WRITE (2,201) 201 FORMAT(1H0,22X1*WE,INCHES * ,4Xt * CUMULATIVE * / 36X,*P R OBABILITY*) WRITE(2,202) (X(I),OFX(I),I=1,N) 202 FORMAT(1H0,20X,F5.215X, 013 . 6) STOP END 125 PROGRAM SEVREV(INPUT,OUTPUT,TARE1=INPUT,TAPE2=OUTPUT) C PROGRAM TO EVALUATE THE DISTRIBUTION FUNCTION OF THE SUN OF N C PACK DURATIONS, Y C LIST OF VARIABLES, PARAMETERS, AND INDICES C NUNN = NUMBER OF RENEWALS CF THE SNOW PROCESS C NUMC = NUMBER OF INTERVALS SELECTED FROM THE DISTRIBUTION C NUMY = NUMBER OF VALUES OF THE RANDOM VARIABLE Y = MAXIMUM NUMEER OF TERMS IN THE OUTER SUMMATION OF KK C JJ = MAXIMUM NUMBER OF TEPMS IN THE INNER SUMMATION OF COMPUTED NUMBER OF TERMS IN THE INNER SUMMATION C JUJJ COMPUTED NUMBER OF TERMS IN THE OUTER SUMMATION C KKKK C XLAM = PARAMETER FROM THE POISSON PROCESS OF STORMS = PARAMETER FROM EXPONENTIAL DISTRIBUTION OF STORM C GAM C Y(I) = VALUES OF THE RANDOM VARIABLE OF C THE OF OF Y THE OF OF Y - MAGNITUDE DOUBLE PRECISION OFYN(24,3),UNYN(24),TEMPK(60),TEMPJ(250),NFAC,SUM 1K,SUMJ,SLOPE,AVAL DIMENSION Y(25),C(5),PC(5),YMEAN(5),YVAR(5),GL(5),GPL(5),CG( 5 ) C SET MAXIMUM NUMBERS OF TERMS IN SUMMATIONS, NUMBER OF SNOWPACK C DURATIONS, NUMBER OF VALUES OF SNCWPACK ABLATION RATES, AND NUMBER C OF VALUES OF THE RANDOM VARIABLE Y NUMN = 1 NUMY = 24 NUMC = 3 KK = 60 JJ = 250 C C C C C C INPUT PARAMETERS, VALUES OF Y, SNOWPACK ABLATION RATES, AND ABLATION RATE PROBABILITIES REA0(1,100) XLAM I GAM 100 FORMAT(2F10.0) READ(1,101) (Y(I),I=1,NUMY) READ(1,111) (C(I),I=1,NUMC) READ(1,101) (PC(I),I=1,NUMC) 101 FORMAT(8F10.0) COMPUTE VALUES WHICH ARE USED IN THE PROGRAM DO 55 L=1,NUMC CG(L) = GAM 4 C(L) GL(L) = XLAM*CG(L) CG(L) GPL(L) = XLAM 55 CONTINUE DO LOOP TO ITERATE THE NUMBER OF RENEWALS, N DO 999 N=1,NUMN INITIALIZE ARRAYS DO 11 I=1,NUMY UNYN(I) = 0.0000 DO il j=1,NUMC OFYN(I,J) = 0.0000 11 CONTINUE CALCULATING N FACTORIAL NFAC = 1 IF( N .LE. 1) GO TO 87 DO 9 J=2,N 9 NFAC = NFAC*J 87 CONTINLE 126 C DO LOOP TO ITERATE Y VALUES DO 13 I=100 U9Y YSO = Y(I) *Y(I) DO LOOP TO ITERATE C VALUES DO 19 L=1,NUmC C COMPUTE Nut-19ER OF TERMS IN SUMMATIONS C 777 778 22 27 C KKKK = 10 + 40*C(L) + Y(I)*.2 IF(Y(I) .GT. 40.) GO To 777 ZJ = ALOG(Y(I))+ C(L)*ALOG(Y(I)) GO TO 778 ZJ = ALoG(Y(I))*1.04 + C(L)*ALOG(Y(I))*1.7 JJJJ = (EXP(ZJ)/2) * 2 IF(KKKK .GT. KK) KKKK = KK IF(JJJJ .GT. JJ) JJJJ = JJ CGY = Y(I) *CG(L) YSQLG = YSO*GL(L) SUMK = 0. 00 21 K=1,KKKK IF(K .GT. 1) GO TO 22 TEMPK(1) = 1. / NFAC GO TO 27 TEMPK(K) = TEmPK(K-1)*(YSQLG/((K-1)*(K+N - 1))) KW = 2*K+N-3 YLG = Y(I) * GPL(L) SUmJ = 1. / (KN+1) TEMPJ(1) = 1.0000 DO 23 J=2,JJJJ TEMPJ(J) = TEMPJ(J-1)*((-YLG)/(J - 1)) SUMJ = SUMJ + (TEMPJ(J) / (KN +J)) 23 CONTINUE SUNK = SUNK + TEMPK(K) * SUMJ 21 CONTINUE OFYN(I,L) = N*(CGY**N) *SUNK UNYN(I) = UNYN(I) + OFYN(I,L)*PC(L) 19 CONTINUE 13 CONTINUE COMPUTING mEANS AND VARIANCES XN = N UNPIN = 0. DO 33 I=1,NUMC DIF = CG(I) - XLAN YMEAN(I) = XN / DIE YVAR(I) = XN*((CG(I)+XLAM)/(CIF * DIF * DIF)) UNMN = UNMN + ymEAN(I)*PC(I) 33 CONTINUE C COMPUTE UNCONDITIONAL VARIANCE UNVAR = 0.0 DO 99 I=1,NUMC UNVAR = UNVAR+YvAR(I)*PC(I) +(YMEAN(I)-UNMN)*PC(I) 99 CONTINUE C PRINTOUT RESULTS WRITE(2,211) N 211 FORMAT(1H1,20X,*NUMBER OF RENEWALS =*,I3) wRITE(2,200) 200 FORHAT(1H0,7X,*CON0ITICNAL AND UNCONDITIONAL PROBABILITIES OF SNOW AN I REMAINING ON THE GRCUND*//15x,*SEAS0NAL SNCwPACK ABLATION RATE 20 PROBABILITY.) WRITE(2,202) (C(J),J=1,NUMC) 202 FORMAT(1H 1 5X 1 *RATE*,7X, 4 (F6.4, 5 X) 0 WRITE(2,203) (PC(J),J=1,NUMC) 127 203 FORMAT(tH ,5)(,*PRO3*;7X,4(F6.4,5X)) WRITE (2,209) 205 FORMAT(1H ,25X,*CONDITIONAL*,14X,*UNCONOITIONAL * /5X, * Y,DAYS * ,1 2 X 1PROFIABILITY OF Y*,10X,*PRO3ADILITY OF Y*) WRITE(2,210) (Y(I),(OFYN(I,J),J=1,NUMC),UNYN(I);I=1,NUMY) 210 FORMAT(1H ,5X,F4.0,6X,4011.4) WRITE(2,220) (YMEAN(I),I=1,NUMC),UNMN 220 FORMAT(1111,5X1*MEAN*,4X14F11.4) WRITE(2,230) (YVAR(I),I=1,NUMC),UNVAR 230 FORMAT(1110,5X1*VARIANCE*15F11.4) 999 CONTINUE STOP END ,* 128 PROGRAM SIX(INPUT,OUTPUT,TAPEi=INPUT,TAPE2=OUTPUT) C PROGRAM TO EVALUATE THE CONDITIONAL AND UNCONDITIONAL DISTRIBUTION C FUNCTION OF THE SUM OF N SNOw-FREE, SNOW CYCLES, T C LIST OF VARIABLES, PARAMETERS AND INDICES C M C J - INDEX OF INNER SUMMATION - INDEX OF MIDDLE SUMMATION C K - INDEX OF OUTER SUMMATION - MAXIMUM NUMBER CF TERMS IN INNER SUMMATION C MM C JJ - MAXIMUM NUMBER OF TERMS IN MIDDLE SUMMATION C KK C HUNT C NUMC C N C XLAM - MAXIMUM NUMBER OF TERMS IN OUTER SUMMATION NUMBER OF VALUES OF THE RANDOM VARIABLE, T NUMBER OF VALUES OF SNOWPACK ABLATION RATE NUMBER OF SNOW-FREE, SNOW CYCLES PARAMETER FROM THE POISSON DISTRIBUTION OF THE NUMBER OF C STORMS - PARAMETER FROM THE EXPONENTIAL DISTRIBUTION OF WATER EQUIVC GAM C ALENT PER STORM C C(I) - SNOWPACK ABLATION RATE ARRAY C PC(I) - ARRAY FOR SNOWPACK ABLATION RATE PROBABILITIES C T(I) - ARRAY FOR VALUES OF THE RANDOM VARIABLE T COMPUTED NUMBER OF TERMS IN INNER SUMMATION C MMMM C JJJJ COMPUTED NUMBER OF TERMS IN MIDDLE SUMMATION C KKKK - COMPUTED NUMBER OF TERMS IN OUTER SUMMATION DIMENSION C(5),PC(5),T(25),CONMN(5),CONVAR(5) DOUBLE PRECISION OFTN(20,5),UNDF(20),PROD(200,30,NFAC,TEMPA,SUMA, ITEMPB,SUMB,TEMPC,SUMC C SET MAXIMUM NUMBER OF TERMS AND NUMBER OF VALUES OF C, Tf AND N MM = 150 JJ = 200 KK = 36 NUMC = 3 NUMT = 19 NN = 10 C INPUT LAMBOA,GAMMA,C1PROBABILITY OF C READ(11100) XLAM,GAM 100 FORMAT42F10.0) READ(1,101) (C(J),J=1,NUMC) READ(1,101) (PC(J),J=1,NUMC). READ(1,101) (T(J),J=1,NUMT) 101 FORMAT(8F10.0) C CO LOOP TO ITERATE N VALUES 00 999 N=1,NN C PRINT HEADINGS WRITE(2,209) N 209 FORMAT(1H1,20X,*NUMBER OF RENEWALS = 41 ,13) WRITE (2,200) 200 FORMAT(1H0,7X,*CONDITIONAL AND UNCONDITIONAL PROBABILITIES OF THE 1SNOW RENEWAL PROCESS*//15X,*SE4SONAL SNOWPACK ABLATION RATE ANO FR 10BABILITY*) WRITE(2,2021 (C(J),J=1,NUMC) 202 FORMAT (1H ,5X,*RATE*,7X,4(F5.3,6X)) WRITE(2,203) (PC(J),J=1,NUMC) 203 FORMAT(1H ,5X,*PRO3*,7X,4(F5.316X)) WRITE(2,210) 210 FORMAT(1H ,25X,*CONDITIONAL*,14X,*UNCONDITIONAL*/6X,47,0AYS*,12X1* 1PRO8ABILITY OF T*,10X.*PROBABILITY OF T*) 129 00 121 I=1,NWIT UNDF(I) = 0 DO 121 J=1,NUMC OFTN(I1J) = 0 121 CONTINUE C COMPUTE AND STORE N FACTORIAL NFAC = IF(N.E0. 1) GO TO 91 DO 88 II=21N 88 NEAC = NFAC II 81 CONTINUE C COMPUTE AND STORE N- FOLD PRODUCT (2K+2I+J-4) 00 77 J=1,jj DO 77 K=11KK KLN = 2 4 K+J+2*N PROD(J,K) = KLN -4 IF(N.EQ. 1) GO TO 77 DO 87 II =20 PROD(J 1 K) = PROD(J,K)*(KLN-II-3) 87 CONTINU P 77 CONTINUE C DO LOOP TO ITERATE T VALUES CO il I=1 1 NUNT TL = XLAM * T(I) TIN = TL**N C DO LOOP TO ITERATE C VALUES DO 22 L=1,NUMC CG = GAM*C(L) C C C COMPUTE NUM2ER OF TERMS IN THE SUMMATIONS IF COMPUTED NUMDERS OF TERMS ARE LESS THAN THE SPECIFIED MAXIMUM THE TOTAL NUMBERS OF TERMS ARE SET EQUAL To THE COMPUTED NUMBER KKKK = 8+L+T(I)*.25 MMMM = 10 + T(I)*.4 IF(T(I) .GT. 40.) GO TO 777 ZJ =ALOG(T(I))+(C(L)/3.1)*ALOG(T(I)) GO TO 779 777 IF(T(I).GT.60. .ANO.C(L).GT. .16) GO 10 789 ZJ=ALOG(T(I))+(C(L)/1.6)*ALOG(T(I)) GO TO 778 789 ZJ = ALCG(T(I)) + C(L) * ALOG(T(I)) 778 JJJJ = (EXP(ZJ)/2)*2 IF(KKKK .GT. KK) KKKK = KK IF(JJJJ .GT. JJ) JJJJ = JJ IF(MMMM .GT. MM) MMMM = MM TGAM = T(I)*CG TGN = TGAM**N TGLN = N*TGN*TLN C DO LOOP TO ITERATE OUTER SUMMATION SUmA = 0.0 00 33 K=1,KKKK IF(K .GT. 1) GO TO 37 TEMPA = 1. / NEAC GO TO 38 37 TEMPA = TEMPA*I(TGAM*TL)/((K - 1) * (K 4 N - 1))) 38 CONTINUE - C DO LOOP TO ITERATE MIDDLE SUMMATION SUMB = 0.0 130- DO 44,J=1,JJJJ IF(J .GT. q) GO TO 60 TEMPB = 1.0000 GO TO 61 60 TEMPO = TEMP8*((-TGAM)/(J-11) 61 CONTINUE C DO LOOP TO ITERATE INNER SUMMATION KJN = 2*K +J+2*N-4 TEMPC = i.00ao SUNG = TEMPC/(KJ)441) DO 55 M=2,MMM4 TEMPC = TEMPC*((-TL)/(M-1)) SUNG = SUMC +(TEMPC/(KJN+m)) 55 CONTINUE SUMB = SUMO + SUMC*(TEMPB/PROD(J,K)) 44 CONTINUE SUMA = SUMA + TEMPA*SUMB 33 CONTINUE OFTN(I l L) = SUMA*TGLN UNDF(I) = UNOF(I) + OFTN(I,L)*PC(L) 22 CONTINUE C PRINTOUT FINAL RESULTS WRITE(2,205)(T(I), (OFTN(I,L), L=1,NUMC),UNDF(I)) 205 FORMAT(1H0,5X 1 F4.0,6X,5011.4) 11 CONTINUE C COMPUTE UNCONDITIONAL AND CONDITIONAL MEANS AND VARIANCES UNMEAN = 0 XN = N DO 66 J=1,NUMC CG = C(J) * GAM OIE = CG - XLAM ODIF = DIF*DIF*DIF SUM = CG + XLAM CONMN(J) = XN* (CG/(XLAM*DIF)) CONVAR(J) = XN* t(XLAM*XLAM*SUM+ODIF)/(XLAM * XLAM*Q0IF)) (JNMEAN = UNMEAN + CONMN(J) * P0(J)• 66 CONTINUE C COMPUTE UNCONDITIONAL VARIANCE UNVAR = 0 Do 67 J=1,NUMC TOIF = CONMN(J)-UNMEAN SQ = TOIF*TDIF UNVAR = UNVAR+CONVAR(J)*PC(J) + SQ*PC(J) 67 CONTINUE WRITE(21206)((CONMN(J),J1,NUMC) , UNMEAN ) 206 FORMAT(1H0,5X.*MFAN*,6X,5F11.4) WIITE(2,207)((CONVAR(J),J=1,NUMC),UNVAR) 207 FORMAT(1H0,5X.*VARIAhCE*.2Xl5F11. 4 ) 999 CONTINUE STOP ENO REFERENCES Anderson, H. 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