U. S. DEPARTMENT OF AGRICULTURE
FOREST SERVICE
WASATCH NATIONAL FOREST
ALTA AVALANCHE STUDY CENTER
Miscellaneous Report No. 10
ENCOUNTER PROBABILITIES FOR AVALANCHE DAMAGE
Edward R. LaChapelle
Avalanche Hazard Forecaster
Wasatch National Forest
March 1966
A common problem in selecting sites for mountain construction or development is determining the probability of avalanche damage. Prudent planning dictates that sites should be completely free of avalanche danger if at all possible. This should be the inflexible standard for buildings and lodges in recreation developments, but a slight exposure to avalanche hazard is sometimes acceptable for ski lifts or parking areas. In the case of mining construction or other industrial enterprises where the character of hazard exposure can be more strictly controlled, a more substantial risk of damage may sometimes be acceptable. This risk is known as encounter probability.
The problem normally is not posed by large avalanches which run frequently (annually or oftener). These present such obvious prospects of repeated damage and destruction that they must either be avoided entirely or else defended or eliminated by what may beprohibitively expensive construction. More often a proposed site lies within or adjacent to a slide path where normal avalanche activity is limited and nondestructive, but which shows evidence of infrequent avalanches of potentially, destructive proportions.
The situation is similar to that posed by other geophysical hazardsearthquakes, floods, tidal waves, hurricanes which recur in destructive size at long and irregular intervals. The concept of a "20year flood" or "100 year flood" is familiar to the hydrologic engineer. The avalanche specialist is faced with the similar problem of evaluating prospective damage from a "20year avalanche" or a "100year avalanche". It often is undesirable or too costly to avoid completely the prospects of damage from an avalanche which may fail only once a century. In some circumstances, such as the exposure of a large number of people, the only acceptable hazard may be zero and there is no choice but to seek another and safer location. There sometimes is economic justification for accepting a limited amount of risk for buildings or other installation, especially mining or other enterprises where a definite and limited building life can be projected. Such risks can logically be taken only if their size can be reasonably estimated. This report presents methods for making such estimates, or encounter probability.
Large, infrequent avalanches, like large, infrequent river floods, are the product of unforeseeable weather and climate. For purposes of statistical analysis they are considered to occur at random even though there may be some evidence for their association with shortterm (in the geologic sense) climate cycles. The average time between a number of such random events is called the return interval. For the kind of avalanches under discussion, typical return intervals might be 25, 50 or 100 years. Most installations are designed for a useful estimated life which depends on such factors as economics, construction materials and rate of obsolescence. When such an installation is exposed for its estimated life to the threat of damage from an infrequent avalanche of a given return interval, there is a definite encounter probability which describes the chance that the avalanche will damage the installation during its estimated life.
Table 1 and 2 enumerate these encounter probabilities for the given values of return interval and estimated life. They are derived from a paper by Borgman (1) which treats the subject of geophysical risks inconsiderable depth. The design engineer seeking a more sophisticated treatment is referred to this paper.
In using these tables, it is important to consider the restrictions imposed on their construction. First, and in general, statistics treat the relations between numbers or groups of numbers. These relations may of may not describe physical reality. They predict probable consequences but do not assign causes. Second, both tables are based on the assumption that occurrences of the event in question (in this case major avalanches) are random and independent. This means that the encounter probability does not change because the event occurs. This is another way of stating the gambler's maxim: "The laws of chance have no memory."
Table 1 is calculated on the assumption that the events occur only at integers on the time scale. This may seem an arbitrary and impractical restriction, but in the case of avalanche hazard it has some useful applications. if most avalanches of a damaging size are known to occur at a given site in, say, January. then such events will fall close to the time scale integers if the convenient time unit of a year is chosen.
Table 2 removes this restriction, allowing the events to occur at any point on the time scale. The following mathematical restrictions, however, have been observed in calculating Table 2: The process is stationary, possesses independent increments, and has a timeindependent average. Two or more events cannot occur simultaneously.
Allowing the events to occur at any point on the time scale gives a more realistic flexibility to the calculations, but does raise another problem when dealing with avalanches in time units of years. Avalanches do not occur at any point on such a time scale; they occur only in the winter. This difficulty may be circumvented if the chosen time unit is winter months for both the return interval and the estimated life. The estimate of encounter probability is then based on a continuous time scale made up of years consisting of those four or live winter months when avalanche damage is possible. The balance of each year when avalanche occurrence is zero is ignored along with that same portion of the estimated life.
Note that for long return intervals and low encounter probabilities the two Tables give very similar or identical figures.
The biggest difficulty in applying these figures to evaluation of avalanche hazard is determining average values of return intervals. The historical record in North America simply is not long enough to provide data from which averages can be calculated for avalanche return intervals greater than two or three decades. The data that are available for a given avalanche path, whether from historical records or inferred from indirect evidence, usually give at best the length of a single return interval. In many cases there is evidence of a major avalanche at some time in the past which has not recurred since. Evidence from tree growth can usually establish when the avalanche last fell, so all that is then known is a minimum possible length of a single return interval. If this is all there is to go on, it will have to serve. But encounter probabilities based on uncertain return intervals obviously will themselves be uncertain. Experienced judgment on the part of the avalanche specialist can take into account terrain, climate and character of the avalanche path, thus placing the estimate of return intervals on a better basis than guesswork. Such improvement in accuracy is real and often considerable, but it is essentially subjective and thus difficult to account for in statistical analysis.
This approach to estimating risks from avalanche hazard does not offer advice an whether risks should be taken. This question is properly one to be answered by administrative or operational decision. What is available is a rational method of determining just what the risk might be. The decision of whether to assume it can then proceed on a sounder basis.
Example No. 1.
An avalanche of unusual size destroyed five houses at Twin Lakes, Colorado, in 1962. Historical records showed that a previous avalanche of this size had fallen at this site in 1882. The houses had been constructed in the avalanche
path during the interim, three of them quite recently. The average age of the other two at the time of destruction was about 40 years. To what risk had they been exposed?
:Here the return interval and estimated life are sufficiently long that there is very little difference between the figures given by Table I and 2. Lacking an average, we must use instead the single return interval of 80 years. Judging from climate and terrain, this is a reasonable approximation. The time unit is years and the estimated (in this case actual) life is 40 years. From Table I the encounter probability is 0.395. The two older houses experienced approximately two chances out of five of destruction. If their normal life anyway is 40 years, they were exposed to this same encounter probability from the day they were built, although the figures would not have been available to calculate it at that time. These figures tell, in other words, that they experienced quite a good chance of eventual destruction at this site.
Example No. 2
Construction of a major ski development has been proposed in an area of known avalanche activity For the purpose of utilizing available private real estate, the developers wish to locate the main center of lodges, ski shops and other buildings adjacent to two major avalanche paths. Historical records show that avalanches large enough to cause damage to the proposed development fell at this site in 1878, 1880, 1906 and 1955. Others may have fallen between 1906 and 1955, but there are no records.
Return intervals from the available data are 2, 26 and 49 years. The average is 252/3 years. Estimated life of the development is 25 years. In this case it would be more realistic to use Table 2. with winter avalanche months as the time unit. Snow data from this area suggest that the four months of January through April are the time of principal avalanche danger. For the purpose of analysis, each year consists of these four months. To the nearest whole months, the return interval is 103 months and the estimated life 100 months. Entering Table 2 at the nearest round numbers for these values, the encounter probability is found to be 0.632. Chances are thus better than six out of ten that the ski area center will experience at least some avalanchedamage during its life.
Such chances are quite unacceptable for facilities intended to house and entertain large numbers of visitors and the use of such a site must be rejected.
The real chance of damage is actually greater than that indicated by Table 2. Because part of the record is missing, the return interval may actually be shorter. Because two separate avalanche paths are involved, there is a chance that the same unusual snow and weather conditions which could create a damaging avalanche on one could also create a damaging avalanche on the other. Thus there is a possibility of two nonindependent events, which violates one of the restrictions on Table 2. But the Table still gives at least the general magnitude of the risk.
References Cited
( 1 ) Borgman, Leon E., The title of the paper is Risk Criteria, Source  journal of the Waterways and Harbors, Division Proceedings of the American Society of Civil Engineers, August 1963, Pages 1  35.
