Introduction
Avalanches provide a qualitative clue to snowpack stability. Fresh avalanches strongly affect snow safety decisions and stability evaluations, lend credibility to avalanche warnings, and serve as test criteria for the performance of predictive models. However, no quantitative guidelines for measuring regional snowpack stability exist at present, and estimation of the mechanical stability of a regional snowpack with shear frames is impractical. Since very little is known about the complex relationship between avalanches seen in a limited area and overall snowpack stability, researchers have not found any generally acceptable means of using avalanches to measure model performance. Indices for such purposes have been restricted to the “yesno” of Obled’s “avalanche day” (Bois and Obled, 1973), to the summed mass of debris in the runout zone (Salway, 1979), or to simple combinations of number and size Perla, (1970; Judson and Erickson, 1973; Judson and others, 1980). A better means of indexing regional snowpack stability with avalanches will produce more accurate stability evaluations and avalanche prediction techniques. This paper presents a semiquantitative method of linking snowpack stability with natural slab avalanches. The approach uses probability theory and a sequential probabilityratio testing procedure to evaluate existing daily snowpack stability. Results are referenced to daily avalanche evaluations made by forecasters from the U.S. Forest Service Avalanche Warning Center in Colorado.
Observations and Data
Avalanches were observed on 185 paths in Colorado’s Front Range during the nine winters 1974–75 through 1982–83. These selfconsistent, highquality records were collected daily by experienced Forest Service avalanche personnel with help from the Colorado Department of Highways and the UradHenderson Mines. Good access was available by highway, oversnow vehicle, and skis. When poor visibility prevented an adequate view of avalanches in the 200 km^{2} sample area, the record was reconstructed on the next goodviewing day. Dates for prior events were estimated by comparing their snow surface features with those from avalanches whose occurrence time was known. The overall record had comparatively few errors, and those that did occur did so mainly during rare cases of extended poor visibility. Only natural slab avalanches that traveled more than 50 m slope distance were retained for analysis.
Most of the starting zones lie at or above timberline between 3500 and 3800 m above sealevel. Catchments are located on all aspects in multiform terrain. Winter climate is continental with mostly small but frequent snowfalls, frequent wind transport of snow, and cold temperatures (Judson, 1977). Resulting avalanches occur mainly in cold dry snow from November through March, but may involve dry, damp, or wet snow in April. Both wet and dry slab avalanches are included in this research.
Theory
Previous studies on the complex relationship between avalanches and snowpack stability are limited. Recent studies by Judson (1983) and Judson and King (1983) indicate that the majority of of starting zones produce avalanches independently of one another due to spatial variations in stability. The degree of independence changes from one winter to the next and is related to terrain features unique to each starting zone, avalanche frequency, snow conditions, prior avalanche patterns, and other factors. Over a period of several winters, it appears that the probability of an avalanche on a given day, is more or less unique for every path, but similar mean stability levels may be common to groups of “similar” release zones when the general snow properties are about the same on individual paths of the group. An indication of a possible structure is seen in Judson’s (1983) study, where he found higher joint probabilities of avalanches between avalanche paths in separate groups than among individual paths. Groups in that study were based on proximity and terrain similarity. The authors suspected that a group structure based on avalanche frequency and snow conditions would give better results. For example, the probability of joint avalanche release between groups with equivalent frequency will be higher than among groups with different frequencies (Judson, 1983). Probabilities of joint avalanche release should further improve if one compares frequency by snow conditions. Therefore, it seems reasonable to assume that the probability of an avalanche on paths with similar snowpack properties and like frequencies is approximately equal, but varies with time. These two simplifying assumptions form the basis of the present study.
To avoid depleting the nineyear data base, days with like snowpack properties were approximated in a general manner by dividing the NovemberApril avalanche season into bimonthly segments. This allowed for the shallow and weak snow conditions of early winter, the midwinter transition period, and the deeper snowpacks of spring. Within each bimonthly period, the 185 paths of the sample were ranked by number of avalanches and assigned to classes of high, moderate, and low frequency.
Class distinctions were made arbitrarily because there were no obvious demarcȧtion points in the frequency distribution. Paths in the first decile were assigned to the highfrequency class. The top 25% of the remaining paths were classed as moderatefrequency, and the rest of the paths became the lowfrequency class. The choice of three frequency classes was subjective, based on our perception of the frequency structure. More or fewer classes could be used if dictated by patterns in the data.
The development of a regional snowpack stability estimator is as follows: let X, Y, and Z denote the number of avalanches in a sample of n_{x}, n_{y}, and n_{z} paths, where X, Y, and Z refer to the high, moderate, and lowfrequency classes for each day in the bimonthly periods. X can take on values between 0 and n_{x}, Y between 0 and n_{y}, and Z from 0 to n_{z}. If one assumes statistical independence among individual paths, the number of paths that fail in any frequency class should be a binomially distributed random variable. The joint probability of a realization of X, Y, and Z is:
where p_{x}, p_{y}, and p_{z} are the probabilities of an avalanche occurring on individual paths in each group on a given day. A relationship between p_{x}, p_{y}, and p_{z} is then hypothesized such that for each day within a bimonthly Period:
Since the majority of paths lie in the lowfrequency class, further simplification by setting a_{z} ≡ 1 produces
a_{x} and a_{y} are estimated from the data. Significance of an avalanche is then an inverse function of frequency, which follows the reasoning of field personnel who assign less weight to failures on highfrequency paths and more weight to events on lowerfrequency paths. This is done because the highfrequency starting zones often respond in small volume when the snowpack on most other areas is generally stable. As will be discussed later, the data support setting a_{z} to one. Parameter p is the regional snowpack stability index for the population of individual paths within the Colorado Front Range. The value of p, which varies between 0 and 1, reflects the degree of snowpack stability in the region. Letting b_{x} = l/a_{x} and b_{y} = l/a_{y},
For a realization of X, Y, and Z, the maximum likelihood estimator of p for the region is
Hence, when avalanches are visible, p can be estimated regardless of whether failures occur on high, moderate, or lowfrequency paths. A more complete picture of snowpack stability in the population could be given after p is estimated by obtaining
_{x} and
_{y} from Equation (4).
Parameter Estimates and Goodness of Fit
Numerical maximum likelihood estimation (ṀLE) techniques were used to estimate b_{x}, b_{y}, and b_{z} for each snowpack category. The daily log_{e} (In) likelihood,
was summed within each snowpack category and across years:
where L_{ij}, x_{ij}, y_{ij} and p_{ij} represent the likelihod function, the number of observed avalanches by frequency classes, and regional probability of avalanches for the j^{th} day in the i^{th} year for a particular season; N_{i} is the number of days with avalanches in the i^{th} year. In L represents a composite likelihood for a given snowpack category for the period of record. A quasiNewton minimization method using subroutine ZXMIN (IMSL, [^{c}1982]) estimated b_{x} and b_{y} given b_{z} = l, such that In L was maximized, considering p as estimated by conditioned on and . Allowing b_{z} to vary produced no improvement in the fit, thus supporting the decision to hold b_{z} constant at one.
A simplified procedure for obtaining b_{x} and b_{v} given b_{z} = 1 was tried for comparative purposes. The parameter b_{x} was obtained by normalizing the mean number of avalanche days for the highfrequency paths to the mean frequency (avalanche days) of the lowfrequency class in the respective snowpack categories. b_{y} was similarly obtained. This set of parameters is referred to as mean estimated parameters in later discussion. Estimated parameters for all snowpack categories are listed in Table I.
Table 1. Parameter Estimates and Goodness of Fit for the 1974–82 Front Range Aalanche Data
Goodness of fit of the model to the data was assessed by forming daily ×^{2} statistics for goodness of fit and accumulating across days within a snowpack category and years. In notation similar toEquation (6), if x_{ij}, y_{ij}, and z_{ij} are estimated as = , and , respectively, where is the estimated stability parameter p for the i^{th}–j^{th} day, then
Goodness of fit with MLE and mean estimated parameters was evaluated both on a daily and on a pooled basis . Individual were evaluated as having two degrees of freedom and the pooled was given degrees of freedom Significance was assessed at α = 0.05
Considering all days in each bimonthly period, there is significant disagreement between model and data (Table I.) However, by partitioning the pooled x^{2} into two components, one for days exhibiting significant lack of fit on an individual basis and another for the remaining days, it appears that the serious disagreement is concentrated in only 10% of the days in each snowpack catgegory. For the remaining 90% of the days on which avalanches occurred, the model is an adequate representation.
For the days exhibiting significant lack of fit on an individual basis, two patterns were apparent: one was characterized by three or four avalanches from highfrequency paths with no avalanches in the other classes; the other involved days with many avalanches on lowfrequency paths with successively fewer than expected avalanches on paths with higher frequency. The former pattern occurred on 40% of the poorfit days (4% of the total avalanche days), contained small softslab surface avalanches, and was strongly concentrated in the March and April data. Threeday precipitation totals preceding such avalanches varied from 7 to 40 mm, and all but one of the 24 h amounts were less than 25 mm. The occurrence of small to moderatesized storms on a generally stable spring snowpack often produces scattered surface instability on the more active paths. Such episodes are not viewed with as much concern because they involve comparatively few avalanches on any particular day, and the avalanches are of small magnitude. The second pattern appeared on 30% of the poorfit days (3% of the total avalanche days). Accompanying avalanches were generally bigger than with the first pattern, and these avalanches were concentrated in January and Febuary. Threeday precipitation totals prior to these avalanches varied from 20 to 63 mm and daily totals often exceeded 25 mm. The deep newsnow layers accompanying significant storms may violate the structure of the model, but there were other large storms that produced avalanches from all frequency classes where the goodness of fit was acceptable. Missed observations associated with poor visibility produced by certain synoptic weather patterns and the effects of prior avalanching on highfrequency paths both decrease the number of observed avalanches on active paths, but have little effect on the number of avalanches from the more numerous paths (Table I) in the moderate and lowfrequency classes. A better explanation and refinement of the model to account for such days will require further study. No clear pattern appeared on the remaining 3% of the avalanche days.
Goodness of fit associated with the mean estimated parameters was similar to that observed for the MLE parameters. This indicates that the mean estimated parameters, which are more readily obtainable, could be employed without appreciable loss of sensitivity. They could also be used at locations with a less extensive avalanche record than was used in this study.
Sequential Probability Ratio Testing
Statistical sequential analysis is any statistical procedure in which the number of observations needed to satisfy the sampling objective depends, as the sampling proceeds, on the results of prior observations. In particular, sequential probability ratio testing is a methodology useful for distinguishing between two or more statistical hypotheses. In brief, observations are taken in sequence until all but one hypothesis are rejected by the data. Wald (1947) summarized the methodology and a more recent treatment was given by Wetherill (1975).
Our use of the methodology is to provide a consistent mechanism for attaching physical meaning to the regional snowpack stability parameter p. Using avalanche hazard categories similar to those employed by operational avalanche warning centers (Williams, 1980), we wish to distinguish among the following hypothesized conditions:

1) high stability H_{1}: p = p_{1},

2) moderate stability H_{2}: p = p_{2},

3) low stability H_{3}: p = p_{3},
where p_{1} < p_{2} < p_{3}. Competing sequential methods to evaluate this set of hypotheses are given by Sobel and Wald (1949) and Armitage (1950). Although Wetherill (1975) prefers the approach of Sobel and Wald, in this study Armitage’s technique was used because it offers greater flexibility in specifying error probabilities of making the wrong decision. Evaluation of our data by both methods produced only minor differences in results. Rarely were different decisions indicated.
Details of the methodology can be found in Armitage (1950), but a brief description of the technique is given here. Let L_{1}, L_{2}, and L_{3} (here the log_{e} transformation is not used) denote the value ofEquation (6) evaluated for the data in hand on any given day, letting p = p_{1}, p_{2}, and p_{3}, respectively. The hypotheses H_{1}, H_{2}, and H_{3} are evaluated based on the ratios L_{1}/L_{2}, L_{1}/L_{3,} and L_{2}/L_{3}. If L_{1}/L_{2} is greater than one, there is reason to think H_{1} is more likely than H_{2} to be true. Conversely, if L_{1}/L_{2} is less than one, H_{2} is more likely. To reach a decision requires consistency among all three ratio values. For instance, to accept H_{1} requires both L_{1}/L_{2} and L_{1}/L_{3} to be sufficiently larger than one to satisfy test criteria determined by preestablished error probabilities. These error probabilities, denoted by π_{ij} are the probability of accepting H_{i} when in fact H_{j} is true. They partially protect the user from making a wrong decision. The degree of protection is varied according to the consequences of making a particular error. If the data in hand produce ratio values that are inconsistent with each of the three hypotheses (or are insufficiently different from one), then the testing procedure cannot reach a decision, and more observations must be obtained. Sampling continues, with intermediate pauses to carry out a hypothesis testing exercise, until all hypotheses but one are rejected.
Probabilities of a natural slab avalanche in each of the three hazard categories had to be specified in terms of what might be reasonably expected from a large sample of uncontrolled paths at undeveloped sites in the mountains. After some reflection on this problem, it was decided that the chances of release in the low through high avalanche hazard categories could be approximated by setting p_{1}, p_{2}, and p_{3} equal to one in a thousand, one in one hundred, and one in twenty, respectively. The choice of these values was mainly experiential and was made without direct reference to warningcenter estimates. π_{ij} error probabilities were similarly selected, with the rationale that, where public risk is involved, slight overwarning is better than underwarning. To declare the snowpack stable when it is unstable (π_{13}) is a serious error that involves greater risk than the reverse situation of π_{31}. It was also felt that to err between low and moderate hazard (π_{12} or π_{21}) involved less risk than an error between moderate and high hazard (π_{23} or π_{32}) Guided by these general concepts, we chose the following π_{ij} error probabilities:

π_{12} = 0.10, π_{21} = 0.20,

π_{13} = 0.0001, π_{31} = 0.01,

π_{23} = 0.001, π_{32} = 0.10.
Since optimum values for these probabilities are unknown, further refinement or adjustments are not planned until more is known about the relationship between avalanches and regional snowpack stability.
Based on applying this methodology to the 1974–82 data and a limited simulation of sequential sampling from a hypothetical population containing 1000 avalanche paths with the proportion of paths in each frequency category similar to the observed data, some insight can be obtained into how the methodology functions in practice. First, it must be understood that the 1974–82 data were not recorded with the objective of applying a sequential testing procedure. Therefore, estimation of snowpack stability and evaluation of the hazard hypotheses was based on the whole sample. It was not possible to add more samples when test results were inconclusive; conversely, if a decision was reached, it was not possible to estimate whether the decision could have been based on fewer observations, although entry of data into the sample could have been simulated. The testing procedure is more sensitive to observations in the high and, then, moderatefrequency categories. If failures are observed in the higher frequency categories, smaller samples are required to reach a decision because proportionally larger increments are added to the likelihood functions and a higher hazard estimate occurs. Intuitively, we would rather assign more weight to observations in the lowfrequency class because this group represents the bulk of the population and avalanches here may represent greater instability than those from more frequent paths. However, likelihood ratio testing of the frequency distribution given by Equation (1) dictates otherwise, and the result is a testing procedure which, if it errs, will generally do so by declaring a hazard higher than true, not lower.
As noted by Wetherill (1975) and Corneliussen and Ladd (1970), the sample size needed to reach a decision using a sequential testing procedure is highly variable and poorly estimated by Wald’s (1947) theory. However, sequential sample sizes will generally be smaller than those needed for testing procedures with fixed sample size because early detection of avalanches in a sequential sampling exercise can produce a quick decision. Hypothesis testing of our sample of 185 paths resulted in a number of inconclusive decisions. This is consistent with our simulation results which indicated average sample numbers (ASN) of about 150 to make a decision when actual p = 0. As p was increased toward p = 0.01, ASN increased to about 225 with subsequent gradual decrease back toward 150 and below as p was further increased toward p = 0.05. Reaching a decision in the neighborhood of p = 0.01 requires more extensive sampling because all three hypotheses are realistically possible, whereas for p near 0 or 0.05, it is really only necessary to distinguish between two hypotheses. Conceptually, ASN would continue to decrease with increasing p, but it is difficult to provide specific estimates because the possible mix of failures among the three frequency categories becomes increasingly variable for larger p. The above figures are only averages, and the variability was large, ranging in our limited simulation from 5 to about 600.
If the sample sizes needed for this testing procedure seem unreasonably large, it must be realized that even when snowpack stability is low, avalanches are comparatively rare. In a population of 1000 paths in which the majority of paths fall in the lowfrequency class, the difference between the hypothesized high and moderate snowpack stability conditions is the appearance of avalanches on only an additional nine paths (1 versus 10 in a 1000), two of which would be expected to appear in a random sample of 200. The difference between moderate and low stability is more distinguishable (10 versus 50 avalanches in 1000 paths), and generally results in smaller expected sample sizes as p increases beyond 0.03. Mathematically, a decision could be made with only two paths if both were in the highfrequency group and both had failed. Realistic application of the method to a large area should require a minimum sample of about 100 paths, distributed among the frequency classes, in order to represent the population properly.
Comparison with Operational Forecasts
Likelihood ratio test results for the avalanche seasons of 1979–82 are plotted with daily estimates of avalanche hazard from the Colorado Avalanche Warning Center in Figure 1. For clarity, the likelihood ratio test results are referred to as “the stability index” in the remainder of this paper. The two systems are not directly comparable because the Warning Center estimates potential hazard from natural and triggered avalanches, and the plotted Warning Center estimates apply to some areas in the northern Colorado mountains that are not represented by our sample. Nonetheless, the daily Warning Center snowpack stability evaluations provide a valuable independent reference for the model. Center forecasters appraise the potential for release on a fourpoint scale that corresponds to low, moderate, high, and extreme hazard, but the extreme category occurs so infrequently that only the first three hazard levels are displayed. The 1.50 and 2.50 scale values are used when the stability index was inconclusive, and for Warning Center hazard estimates between low and moderate or moderate and high. A hazard rating of three is the threshold value for issuance of public warnings in the U.S.
Fig. 1. Comparison of sequential test results with Warning Center hazard estimates and avalanche control information for winters 1979–82. A indicates the daily sequential test results, which are connected by a line. W refers to daily Warning Center stability estimates. T indicates a control attempt (try) without an avalanche, and H (hit) refers to a control attempt that caused an avalanche.
Natural avalanches fell on 30% of the winter days during the four test winters. Following the first avalanches of early winter, the stabilityindex produced hazard estimates somewhat lower than those from the Warning Center. A reversal in this trend appeared in spring when instability was confined to the surface layers of the snowpack, a case in which avalanche activity from highfrequency paths dominated the procedure. Snowpack stability indicated by natural avalanches is a rapidly fluctuating parameter with a marked lack of persistence. The stability index indicates that rapid stabilization from high to low hazard often occurs in one or two days, whereas slower stabilization rates and strong persistence are estimated by conventional methods. There were only three cases in the four winters when the stability index indicated high hazard persisting for two or more consecutive days compared to 21 such cases for Warning Center estimates. Similarly, the maximum number of consecutive days with a stability of 1.5 or more was 10, as indicated by the stability index compared to 85 by Warning Center personnel. These differences in trend, response, and persistence in snowpack stability result from a combination of real differences and a necessary safety margin used by Warning Center personnel.
SnowPack Stability and Artificial Release
The combined results of daily avalanche control by the highway department, a mining operation, and a ski area are plotted in the lower part of the seasonal displays in Figure 1. Each T indicates a “try” without release, and an H represents a hit, which is a “try” with release. The control attempts were limited to cold, dry snow conditions, which are common to the winter months, and are also prevalent on many days in spring. There is good overall agreement between snowpack stability estimates and control results, but poor agreement obtained on several individual cases such as the one on 5 April, 1983, nullifies the relationship on a quantitative basis. The correlation coefficient between the stability index and shooting success, defined as a ratio of the number of hits to the number of “tries,” was r = 0.58. Average shooting success, based on days with 10 or more attempts, was 35% on days with no natural avalanches versus 51% on days with snowpack stability 1.5 or greater. This increased to 76% when the stability index indicated high hazard. There were fewer “tries” on stable days, which reflects the lower hazard perceived by control personnel. Some problems associated with using control results as an index of regional snowpack stability include: the very small sample size of controlled avalanches, the tendency for control efforts to be concentrated in a few isolated pockets within the sample area, the fact that information from heavily controlled sites is mostly limited to the stability of surface layers, and the tendency for control teams to vary the number of shots in accordance with the reaction of the snow to the first few control attempts. Moreover, control efforts are not equally distributed across all frequency classes, but are concentrated in the highfrequency group. And since snow removal and traffic considerations generally take precedence over avalanche control along most highways, snowpack stability information based on such control results is frequently unavailable when most needed. Further research is needed to determine the degree of residual instability in the snowpack after the last natural avalanche occurs.
Summary and Discussion
A framework has been proposed for summarizing and interpreting the information available from avalanche occurrence on individual avalanche paths within generalized snowcondition categories contained in the three bimonthly periods of early, middle, and late winter. Avalanches on paths having heterogeneous probabilities of release were pooled into a composite regional stability index denoted as parameter p of the statistical distribution, chosen to approximate the probability of failure in the population. The stability index offers more resolution than the avalanche day, and estimation of p is possible even when poor visibility or other factors preclude observation of an entire sample. An analysis of goodness of fit suggests that there is adequate agreement between observations and fitted distribution about 90% of the time across the three snowcondition categories. The disagreement is somewhat patterned, suggesting that improvement might result if a more sophisticated algorithm incorporating weather, snowpack stratigraphy, antecedent failure history, and other factors was developed to partition the population into the X, Y, and Z categories on a continuing basis rather than assuming constancy within twomonth periods. We suspect that the distinctions among X, Y, and Z groups become quite muddled under certain weather patterns and hope to explore this avenue of investigation. A testing procedure incorporating more than three frequency classes might offer further improvement.
A stability index (sequential testing procedure) was also proposed as a mechanism to interpret consistently the magnitude of p as it applies to avalanche hazard estimation. Application of the testing methodology to our data produced reasonable results in general agreement with independently generated hazard estimates from an operational avalanche warning program and information from control efforts. Successful release of snow during control attempts appeared to be strongly related to the degree of hazard indicated by the stability index. Average sample sizes needed to reach a sequential testing decision appeared large, ranging from 150 to 225 paths, although a decision could be quickly reached after observation of only a few avalanches with low snowpack stability (high avalanche hazard). These tests would provide realtime evaluation potential if a series of dispersed observers were available to convey the avalanche information rapidly to a central location.
We feel our initial efforts suggest a promising new approach to systematically relating snowpack stability with avalanche occurrence, although further refinement of the present methodology is indicated. As already noted, a more sophisticated approach for assigning a particular path to the high, medium, or lowfrequency classes based on snow texture and stress would improve the reliability of the methodology. Presently, there is some difficulty in specifying the potential population, especially in the lowfrequency class. Previously unrecorded events which occur on new paths every year increase the size of the Z group and complicate the definition of sample size. An underestimate of n_{z}, would cause overestimation of p and reduced sensitivity of the sequential testing procedure to information from the z group. Although the proposed methodology requires only a simple “yesno” avalanche observation for each path, without any detailed attempt at classifying each avalanche event, a subjective valuejudgement is needed for the size of avalanche that constitutes a hazard; the values of p_{1}, p_{2}, and p_{3}; and the π_{ij} error probabilities; these would depend on the purpose of the investigation. After these judgments have been made for a particular application, consistent nonsubjective evaluation of observed failure frequency is possible.
Acknowledgements
The authors extend sincere appreciation to Dr Charles Obled of the Institute de Mécanique, Grenoble, France, whose stimulating discussion and useful ideas helped guide our initial attack on this intricate subject. The authors also thank the referee whose helpful suggestions improved the clarity of the presentation.
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