1. Introduction

Evaluation of the maximum runout distance of snow avalanches is one of the most important problems in avalanche zoning. It is also one of the most difficult and controversial subjects in avalanche research. This is especially true in avalanche-prone districts where there may be a need for housing, so that calculation of maximum runout distance has many implications concerning possible future accidents and damage.

Models that will predict avalanche runout distance are therefore much in need. Such models should be based on a small number of parameters. These parameters should be as objective as possible, and not be based on subjective judgment from the various avalanche experts handling the problem.

The purpose of this paper is to study extreme avalanche runout distance based on (1) topographic parameters from about 200 avalanches, (2) calculation of runout distance from the same collection of avalanches based on a numerical /dynamical model, and (3) combination of the two models.

2. Runout Distance based on Topographical Parameters

Evaluation of maximum runout distance of avalanches based on topographical parameters is described by Lied and Bakkehøi (1980). These parameters could be easily identified on a topographic map, so that the subjective judgment of the numerical values belonging to the different parameters could be neglected. In the present work, the number of avalanches studied is increased from 111 to 206; the maximum extent of all these is known and listed in Table I, The avalanches chosen for this study have quite different paths, ranging from steep slopes, with an abrupt transition to the valley floor, to long, gently inclined paths with a gradual transition from track to runout zone. The extent of rupture area and degree of confinement in the path varies greatly.

Table 1. List Of Avalanches (For Explanation Of Column Headings, See Text)

The parameters which may be used to evaluate maximum avalanche runout are the average gradient of the avalanche path a, the average inclination of the avalanche track B, the inclination of starting zone Q (measured on map), the second derivative y" of the slope function y = ax² + bx + c, the vertical drop Y, and the height H from the starting point to the vertex of the parabola. These parameters are illustrated in Figure 1, and are derived as follows. The angle a, which describes the total reach of the avalanche, is determined on the path profile by the line connecting the end points for the largest known event. Tne angle ß is determined by the line connecting the top point, and the point on the profile where the slope angle is 10°. The reason for using this line, or angle, is to generate a simple description of the main inclination of the track. Slope gradient values around 10° are thought to represent the transition zone between the track and the runout zone for big, dry avalanches (de Ouervain 1972, Buser and Frutiger 1980).

Fig. 1. Topographical variables for the calculation of α.

The inclination of the starting zone 9 is measured on a map as the average gradient of the uppermost 100 m of the path, vertically measured. The avalanche profiles are expressed by the equation y = ax² + bx + c (see Fig.l). H is defined as the height difference from the starting point to the vertex of the calculated parabola. The second derivative y" = 2a describes the curvature, and by multiplying with H the theoretical profile is made dimensionless. H is usually different from the vertical drop Y of the avalanche.

The second derivative y" of the slope function is a shape factor. The shape of the path is essential to α, and our experience is that avalanche paths that are just steep enough to trigger the avalanche, and keep it moving, must have longer runout distances (lower values of a) than steeper paths. This is the reason for introducing y" which describes the whole profile in more detail than β. If the slope really had been a parabola, we can derive the connection but the 10° point found on the map differs fron this theoretical number.

In the present paper we discuss a classification of the avalanche paths in an attempt to obtain better accuracy in the runout calculation. As indicated by Lied and Bakkehøi (1980), a tendency was found for confined avalanches to obtain lower values of a than unconfined avalanches, but it was not possible to confirm this tendency in the statistical analysis.

The avalanche paths are classified as obviously confined (47 avalanches) and obviously unconfined (77 avalanches). Classification is also performed on the basis of values of ß, y", H, and θ. On steep slopes with high values of ß, y", and θ, avalanches tend to start with less snowfall than on gently inclined slopes. If avalanche velocity and runout is dependent on avalanche mass, a difference between steep and gently inclined paths concerning relative runout distance would be expected. As Indicated by both Körner (1980) and Laatsch and others (1981), the model of Lied and Bakkehøi (1980) does not explicitly take mass dependency into consideration. A classification based on different values of ß will implicitly consider a mass dependency.

The classification based on the above mentioned parameters is obtained in the following way:

2.1. Results

The regression analysis based on 206 avalanches gives the equation:

with R = 0.92 and SD = 2.28°. In this equation, [H] represents the numerical value of H. If, as an example, we choose β = 30", θ = 35°, H = 1 000 m and y" = 3x10⁻⁴ this will illustrate the importance of each term in the equation:

Introduction of more variables in the regression equation will not give a significant increase in the accuracy, β has a t-value (Daniel and Wood 1971: 353) of 19.0. The variable Hy" θ has a t-value of 1.3. which is significant to the 0.10 level, while H is significant to the 0.12 level. Using β as the only free variable, this equation is obtained: α = 0.96β 1.4°, with SD = 2.3° and R = 0.92. This clearly demonstrates the dominating effect of β. The result indicates that, even if the total number of avalanches is increased, a higher accuracy in the prediction of avalanche runout by this method cannot be obtained. In the equation presented by Lied and Bakkehøi (1980) R = 0.95 and SD = 2.3°.

One regression equation for both confined (47 avalanches), and unconfined paths (77 avalanches) is computed. If avalanches in the confined paths (wide rupture zone and narrow track) obtain longer runout distances, this tendency should be found statistically by the regression analysis. No such tendency was found. By running the equation for the confined paths on unconfined avalanches and vice versa, it was not possible to identify any tendency for confined avalanches to obtain lower a values than unconfined avalanches. The result may be partly explained by the inaccuracy in the model, but mainly by the fact that when an avalanche is forced into a confined path from a wider rupture zone, the snow masses must be compressed so that sintering and den-si fication takes place. This process must lead to higher friction and consequently reduce the velocity increase.

Regression analysis is performed, based on a classification of β values. For tracks with β <30° we found this equation:

with R = 0.84 and SD = 1.49°. Computed with the equation for all avalanches, SD = 1.51°.

In our opinion, this is a high accuracy, and especially valuable on these gently inclined paths where differences in degrees amount to the biggest difference in metres concerning runout distance.

For 30°<β <35° the equation

with R = 0.53 and SD = 2.50°, was found. The same sample computed with the equation found for all avalanches gave SD = 2.56°.

Correspondingly, the equation for β > 35° is

with R = 0.62 and SD = 2.67. This sample computed with the equation found for all avalanches gave SD = 2.69.

This last example of classification does not appear to increase the accuracy of the model.

A classification based on H, y", ô and s is presented in Table II.

Table 2. "Classification Based On H, Y", 9, And 6

These results indicate that B is the best scaling factor of the examined parameters. A classification based on 9 or y" does not increase the accuracy of the model.

In combination with β, H seems to be an important variable. For H > 900 m and β <30°, SD = 1.02° and R = 0.90. The equation for all the avalanches used on this sample gives SD = 1.07°. This last result is of great value, because it improves the predictive accuracy for a group of avalanches where erroneous runout calculations have the greatest consequences. For an avalanche with H = 1 000 m, β < 30° and α = 25°, one standard deviation gives a possible longer horizontal reach Δl1 = 91 m and a possible shorter reach Δl2 = 85 m. The regression equation for this group is

3. Runout Calculation based on a Dynamic Model

Perla and others (1980) presented a two-parameter model of snow avalanche motion expressed by the equation

where M is the avalanche mass, θ the slope angle, y the coefficient of μ the coefficient of friction, and D a coefficient of dynamic drag. The equation is solved numerically by dividing the entire avalanche path into small segments where θ are considered to be constant in each segment. Each segment is assigned an angle θ_i, a length L_i , a friction value μ_i, and a mass/drag value (M/D)_imentum loss is corrected at the segment transitions. Avalanche velocity is computed for each segment, progressively along the path, until the stopping position. The model was first applied on 25 avalanche paths in north-west USA. The usefulness of this model is dependent on a knowledge of the values of μ and M/D. These values can vary within wide limits, with endless possibilities of combination, and still satisfy a given runout distance. The model presented by Reference VoellmyVoellmy (1955) is encumbered with the equivalent problem.

In a later work the same model was tested statistically on 136 extreme avalanche paths from the northwest USA and Norway (Bakkehøi and others 1981). The range of M/D values was here related to the scale factor Y, i.e. the total vertical drop of the avalanche. M/D values Y/100 > M/D > 0.01Y were considered. Ry computing the corresponding values of p and velocity v, it was found that, for M/D values between 1 000 Y and 10 Y, both y and v are near constant.

Based on the general knowledge and measurements of avalanche speed (see section 6) and of assumed values of p in dry avalanches, it seems realistic to choose M/D values in the range 10Y > M/D > Y/10 (Bakkehøi and others 1981). This is still a wide range, and applied to runout calculations it means a fairly low accuracy in predicting runout distance.

4. Statistical analysis of the Dynamic Model

The dynamic model introduced by Perla and others (1980) is applied on the sample of avalanches already described in this paper. The runout distance in terms of a is computed for all the avalanche paths with different sets of values of p and M/D, and the avalanches are classified for different values of β, β < 30°, 30°< β < 35°, β > 35°. The values of p and M/D are computed for the most probable pairs, and the best fits are presented in Table III.

Table 3. Values Of μ And (M/D)Y Related To ß (α: Mean Of Observed Values, α _C: Mean Of Calculated Values)

One general conclusion is that steep tracks need higher friction values than gentle tracks to obtain the best correlation. For β > 35°, μ = 0,30 and for β < 30°, μ = 0.20, for identical values of M/DY. The standard deviation of residuals is between 3.2° and 3.8°, which is a wider deviation than found by the topographic model where the smallest values of SD with the same classification varied between 1.5° and 2.6°, The accuracy of the model seems to increase slightly, although not much, due to the classification, as SD = 3.2° for 30° < β < 35°, and SD = 3.3° for β > 35°, compared to SD = 3.5° for all paths. The best fitted pairs for all paths are μ = 0.25, M/DY = 0.5, with R = 0,83 and SD = 3.5°.

The reason why higher friction values must be applied on steeper paths in the model may be as indicated in section 2. Avalanches in such slopes contain less mass of snow, and should thereby obtain a shorter runout distance.

5. Combination of the Topographic and Dynamic Models

In our consulting work, we are frequently faced with the question of potential avalanche runout distance, and we therefore need to improve the accuracy of the existing models. The model presented by Reference VoellmyVoellmy (1955), and since then widely used in alpine countries is difficult to handle because the hydraulic radius R, flow height h, coefficient of friction p, and drag-coefficient, vary within wide limits. Years of experience are necessary to sort out empirically the right values of these constants which in turn can give realistic runout distances. This is clearly demonstrated by Föhn and Meister (1982), who documented that a 10% error in rupture width and slab height may cause ±100 a difference in runout distance.

We have therefore developed the statistical method described earlier in this paper. On the basis of this, it is possible to predict the most probable extreme runout distance. This probablistic way of handling the problem excludes most of the subjective judgement in fixing the values of the constants. By combining the topographic and dynamic models we see a possibility of increasing the accuracy of runout. prediction and velocity estimates. The procedure is as follows. A given position in a potential runout zone is analyzed with regard to possible reach of avalanche debris. The profile of the avalanche path is drawn from a map with a scale 1:50 000 and with contour lines 20 m apart. The lowest part of the path is usually supplied with data from a detailed map of scale 1:5 000. A set of 10 to 20 x- and y-coordinates are sufficient to describe the profile. The B point is identified and ß found. The inclination of the rupture zone is measured on the map, and the most probable vertical drop Y is evaluated from the map also. The real profile is transformed to a parabolic function of the type y = ax2 + bx + c, with the best fit found by the method of least squares, o is then computed by the best fitted empirical equation (see section 2.1). The avalanche speed and runout distance are calculated by using the dynamic method described in section 4. Different pairs of values of u and M/D are preloaded in the computer program, and avalanche runout distance is calculated for the best fits depending on $ (see Table III).

With the aid of a map and a computer, the whole procedure is easily done within half-an-hour. Central processing unit time for the calculations is less than 1 s.

6. Velocity Measurements

In the work of Bakkehøi and others (1981) it is emphasized that values of the avalanche speed are needed to evaluate what regimes of μ and M/D are applicable. Velocity measurements have been performed at the Ryggfonn avalanche at the avalanche research station of the Norges Geotekniske Institutt (NGI). The avalanche is triggered by explosives. Total vertical drop is ~1 000 m, and path length about 2 000 m. The avalanche volume ranges from the order of 104 to 105 m3 snow.

We have two reports of velocity measurements from this avalanche. In the first, the avalanche volume was ~105 m3. The air temperature was below 0°C in the entire path. The velocity profile is presented in Figure 2. In the second (Fig. 3), the avalanche consisted of ~5xlO4* m3 snow. The 0°C isotherm was at 1 100 m a.s.l. Therefore, in the lowest part of the path, from ~1 000 to -600 m a.s.l., the snow was wet. This again resulted in a marked reduction of avalanche speed in this part because of hiqher friction in the wet snow (velocity profile, see Figure 3). The avalanches were photographed by cine photography at three different points and the frontal speed was calculated by comparing the pictures with points in the path where × and y coordinates are known.

Fig. 3. Velocity, measured and calculated in the Ryggfonn avalanche, February 1975.

Fig. 3. Velocity, measured and calculated in the Ryggfonn avalanche, April 1982.

The interesting facts arising from these two field measurements are that maximum speed is as high as -60 m s⁻¹ (frontal speed), and that both the avalanches obtained almost the same maximum speed despite the difference in mass. This observation gives support to the view that avalanche speed is more a function of snow quality in the path than of avalanche mass. The snow cover in the main part of the track consisted of light, dry snow in both cases. Moreover, these measurements illustrate that the computer program is applicable and makes it possible to choose a realistic regime of y and M/D when both velocity and stopping position of the avalanche are known. The best fitted pair of values of μ and M/D from these measurements corresponds with the values found by the statistical analysis of the numerical model discussed in section 3.

Conclusion

In this work we have presented a topographical and statistical model which predicts avalanche runout distance based on four parameters. The parameters are all easily measured or calculated from topographic maps. By classification of the avalanche paths we have increased the accuracy of the model. The classification of the avalanches seems to confirm that confinement of the path has no significant influence on the runout distance. The main reason for this is thought to be that compression, sintering, and density increase takes place during the confinement. These processes increase the friction and reduce the velocity increase created by the confinement. Avalanches on steep paths seem to obtain a relatively short runout distance related to α. This may be caused by a relatively greater loss of energy due to the dependent friction term v² when avalanches are running on steep slopes with high speed. In addition, steep slopes tend to contain less snow than gentle slopes when avalanches are triggered, and these avalanches therefore have less mass and volume. This again may lead to relatively lower speeds and shorter runout distance.

The results obtained by the numerical/dynamical model which is applied to avalanche population, seems to lead to the same conclusion, i.e. that avalanches on steep slopes need greater friction values to fit the observed runout distances.

By the topographical method we have obtained a prediction accuracy of SD = 1 to 2° of the runout distance α,, depending on the path topography. This is in the order of ± 100 to ± 200 m for an avalanche with a vertical drop of 1 000 m and α =25°. In combination with the numerical/dynamical model presented by Perla and others (1980), it is possible to predict speed and runout distance with an accuracy which is applicable for practical consulting work. To increase the accuracy of the model, measurements of speed and runout distance from full-scale experiments will be particularly important.