Introduction
This paper examines the use of a conceptual approach to predetermining the run-out distance of rare avalanches (i.e. those with a large period of return, basically 100 years or more). Conceptual approaches are common in hydrology, notably in the problems related to the predetermination of discharge in a given watershed from rainfall data. To our knowledge, approaches of this kind have never been fully attempted in the study of avalanche extension, although they can provide a more robust alternative to the statistical and deterministic (physical) approaches so far used for determining the run-out distance (point of furthest reach) of rare avalanches.
In the statistical approach, the basic ideas were expressed in the pioneering work of Reference Lied and BakkehoiLied and Bakkehoi (1980). These authors assumed a regional homogeneity in avalanche behaviour for a given mountain range. This allowed them to pool the data from various paths in a common database. Using regression techniques, they thus obtained a relationship between the run-out distances and various key variables of the path profile. This methodology has been applied to different mountain ranges throughout the world (see BakkehøReference Bakkehøi, Domaas and Liedi and others, 1983; Reference McClung and LiedMcClung and Lied, 1987; Reference Fujisawa, Tsunaki and KamiishiFujisawa and others, 1993; Adjel, 1995) and extended to introduce the period of return as a new parameter (Reference McClungMcClung, 2000, 2001). In Alpine countries, where the avalanche paths of the same mountain range most often exhibit no similarity in their shape, the fundamental assumption of avalanche homogeneity is questionable (Adjel, 1995).
In the deterministic approach (for a review, see Reference HarbitzHarbitz, 1999; Reference Ancey, Balmforth and ProvenzalleAncey, 2001), avalanche features are deduced by solving the equations of motion (mass and momentum equations). Deterministic models (sliding-block and depth-averaged models) introduce a friction law, reflecting the interaction between the avalanche and the path. In most models, the friction law includes two empirical frictional parameters, which have been fitted from field observations (Reference SchaererSchaerer, 1975; Reference Buser and FrutigerBuser and Frutiger, 1980; Reference Dent and LangDent and Lang, 1980). The resulting values were proposed as default values in engineering guidelines (Swiss guidelines on the so-called Voellmy–Salm–Gubler method (Reference SalmSalm and others, 1990) or the U. S. Geological Survey handbook (Reference MearsMears, 1992)). Though it aims primarily at providing a physical picture of avalanche motion, the deterministic approach involves too many ad hoc assumptions to be considered a true physical approach. Indeed, a number of basic physical processes (snow entrainment or release, turbulent suspension and transformation into an airborne avalanche, etc.) occurring in the avalanche course are either unknown or neglected in the avalanche-dynamics models. Furthermore, very little is known on the bulk rheological behaviour of snow, and therefore, despite a number of attempts to find physical justifications for their expressions (see, e.g., Reference SalmSalm, 1993), the friction laws used so far remain speculative and empirical. In this respect, it is not surprising that, in a recent benchmark of avalanche-dynamics models (see Reference Barbolini and SaviBarbolini and others, 2000), a significant mismatch was found in the frictional parameter values fitted from field data: this merely means that these parameters do not represent the physical properties of snow (such as snow viscosity), but, on the contrary, they reflect the interrelated influences of snow, path profile and model assumptions on the computations. Such a result is sufficiently disturbing to cast doubt on the use of avalanche-dynamics models to paths where no field data are available.
This diversity in the approaches is also encountered in hydrological sciences. As a result of more extensive practice and intense debates within a large and growing community, the ideas in hydrology on modelling are probably more precise: a clear distinction is made between statistical, physical and conceptual models (see, e.g., Reference Betson, Ardis and KirkbyBetson and others, 1989; Reference BevenBeven, 1989; O’Reference O’Connell and TodiniConnel and Todini, 1996; Reference Bates and CampbellBates and Campbell, 2001; Reference Vanderkwaak and LoagueVan der Kwaak and Loague, 2001). Most of the time, selecting one of these approaches depends on the level of knowledge of basic physical processes involved in the problem together with the number, quality and type of available data. Both conceptual and physical approaches represent the catchment response to a rainfall event as the result of basic processes (infiltration, storage, runoff, etc.) but in very different ways. In the physical models, the elementary processes are assumed to be known from scale-down experiments in the laboratory; the parameters introduced in the models represent physical properties that can be measured accurately and independently. On the other hand, in the conceptual models, the processes believed to be dominant in the hydrological response of a basin are idealized in the form of mathematical operators; the objective is to mimic the natural processes and not to explain them. In this case, the model parameters are purely empirical functions or values, which must be calibrated using the observable physical variables that are the object of the prediction.
It is helpful to draw a parallel between the avalanche run-out problem and the rainfall-runoff transformation. In both cases, one has input and output variables, between which one tries to find a causal/functional relationship using a physical/conceptual model. A seemingly slight difference, at first glance, does exist, however: for avalanches, the only available output information is the run-out distance, which has the drawback of being path-dependent, while in flood hydrology the selected output variable is the flow discharge, which is a truly physical variable, that is, independent of the river section at which it is measured. This difference in the type of output variable has substantial implications in developing the model since, in the latter case, it is mathematically permissible to extrapolate the probability distribution to estimate flood discharges with a long period of return, whereas, in the former case, such an extrapolation is not licit without further information. This also leads to the model parameters remaining constant for all flood events, whereas for avalanches they should vary from one event to another.
The objective pursued in this paper is to provide a proper method of extrapolating the probability distribution of run-out distances observed in a given avalanche path. Since the physical approach is questionable for the reasons given above, we have developed a conceptual model. The basic idea is to determine the dependency between the probability distributions of input and output variables for a given path. Here, this is done numerically using Monte Carlo simulations. As the input variable, we will use the starting altitude and the snow volume involved in the avalanche. An avalanche will be idealized as a sliding block and we will use a Coulomb-like model (a one-parameter model) and a Voellmy-like model (a two-parameter model) as the mathematical operators for mimicking the main features of avalanche motion. Unlike other works relying on Monte Carlo simulations (e.g. Reference Keylock, McClung and MagnUssonKeylock and others, 1999; Reference Barbolini and SaviBarbolini and Savi, 2001), the present method need not use questionable assumptions (such as regionalization of data) to specify the probability distributions of variables and parameters but, on the contrary, uses only the observed field data to deduce consistent probability distributions. It then introduces an important constraint: extreme avalanches are expected to have the same behaviour as observed events. If this does not seem acceptable, the method should not be used.
Methodology
If we have a time series of field data including the run-out distance x stop, it is quite easy to deduce its empirical probability distribution. The point is that usually the time series cover a narrow period, typically a few decades. Therefore the largest period of return that can be evaluated in this way cannot exceed a few decades, while avalanche zoning requires determining the run-out distance of long-return-period avalanches, typically avalanches whose return period equals or exceeds 100 years. Because the empirical probability distribution P(x stop) is not smooth and x stop depends on the path profile, it is not permissible to directly extrapolate P(x stop) to determine the quantiles associated with low probabilities. Here we develop a model that uses available field data at best in order to properly extrapolate P(x stop)
Figure 1 depicts the general framework used to reach our objective. The basic idea is to assume that there is a single functional relationship G between the run-out distance and other field data. This means that we need to have distinctive types of field data at our disposal to apply our method. These other field data include the snowfalls preceding the avalanche, the starting-point elevation, the released volume of snow, and so on. For the moment, we do not specify the type and number of these data but merely refer to them generically as the input variables Θ. The functional relationship G relates the run-out distance x stop of a given (fictitious or real) event to the input variables Θ. Obviously there is not a one-to-one universal function linking x stop to O: indeed, it is expected that G also depends on the topographical features of the path and on a set n of internal or structural parameters, reflecting the diversity and variability of snow consistency and avalanche motion. We express this complex relationship to take the following form: x stop = G(Θ |Π; path). Here, in order to take the path influence into account, we assume that the functional G is a mathematical operator resulting from the integration of a momentum equation along the path profile y = f (s) (see below); in the following, we will use the short-cut notation: x stop = G(Θ|Π). If the input variables Θ and/or the internal parameters Π are random, then the run-out distance is also a random variable. If we are able to adjust the internal parameters Π for the computed run-out distances to match the observed run-out distance, then it is possible, using Monte Carlo simulations, to create a large number of fictitious events coherent with the observations. If the run-out distance sample is large enough, we can fit an empirical probability distribution and then accurately determine the quantile related to a low occurrence probability (e.g. as low as 0.01 or 0.005).
Fig. 1. Conceptual diagram of the approach.
Contrary to the deterministic models, the conceptual model gives results that, by definition, can be different from the measured data. To this structural error it is necessary to add the uncertainties of the data as well as of the internal parameters. Taking this into account leads to the following and more complete formulation of the model: x stop = G(Θ|Π,E), where E is the uncertainty vector. The variables are taken into account through their probability distributions, within a more complete but large model. In hydrology, a more simple theoretical model is generally used, considering only an additive error ε, leading to x stop = G(Θ|Π) + ε. However, in hydrology, internal parameters of conceptual models are constant for each event, which is not the case for avalanche models. Further research is needed to determine whether the use of a simply additive error is consistent in our field. Yet we already know that the indirect linkage between the probability distributions of the internal parameters and the distribution of x stop gives results similar to the direct inversion method used in the first step of the methodology below (Reference Ancey, Balmforth and ProvenzalleAncey and others, 2003). So, we will develop the entire methodology using this direct method and will not use the complicated error formulation in this first attempt at the conceptual approach.
After outlining the general principles, we will explain how the method can be applied in practice. The method can be broken down into four steps:
The first step selects the input variables among all the available field data and fits Π. Usually the avalanche database has various types of information available (snowfall, volume, etc.), but not all the information can be used. For instance, redundant data must be set aside. Similarly, data whose time series is not complete or not consistent with other time series cannot be retained as input variables of the model. Next, the different values of Π are adjusted for each documented event. Different strategies can be used to solve this inverse problem. In hydrology, a current practice is to use Bayesian inference to deduce p(Π) from p(Θ) and p(x stop), where p(X) is the probability density function (PDF) of the random variable X. Deterministic methods can be used equally for this purpose (see, e.g., Reference Ancey, Balmforth and ProvenzalleAncey and others, 2003). Here we directly compute Π for each event and then we adjust a probability distribution to the resulting sample of values Π. All these methods have a common impediment in that we have a single type of output variable, whereas the dimension of Π may be larger than unity. In the case where Π = {π 1···πn} with n ≥ 2, only one parameter πi can be determined, provided that other parameters πj with j ≠ i are known; therefore, we have: πi = G −1(x stop, Θ|πj). In practice, using propagation operators involving a large number of internal parameters leads to substantial complications.
In the second step, attention is paid to obtaining the PDFs of the input variables p(Θ) and of the internal parameters p(Π) (if the latter has not been determined in the first step). Since the data are available, this merely means that we try to adjust usual probability distributions (Gumbel, Pearson, normal, etc.) from the selected field data and the internal parameters.
In the third step, Monte Carlo simulations are performed. For each fictitious event, random realizations of Π and Θ are generated from their respective probability density function.
In the fourth step, the run-out distance is computed by applying the propagation operator to these random vectors and then it is stored. Avery large sample of x stop can be obtained in this way. We then deduce the empirical PDF of x stop. This function must approximately match the empirical distribution adjusted from the recorded distances but, since it results from a much larger sample, it extends over a wider range of probabilities. This allows us to accurately compute the quantile associated with a low non-exceedance probability. For instance, the run out distance whose period of return is 500 years (corresponding to a non-exceedance probability of 0.002) can be determined by generating 1000 years or more.
Propagation operator
In this paper, we examine a propagation operator involving one or two internal parameters. Here an avalanche is idealized as a solid mass sliding along a curvilinear path and experiencing a frictional force F, possibly depending on θ and/ or u (θ is the local slope, u is the local velocity). In a first approximation, we assume that the structure of this frictional force is identical whatever the path and the avalanche; only its parameters can vary from one event to another, but they remain constant during the entire course of an avalanche. The general expression of the momentum equation is written (m is the avalanche mass, t is time):
As initial conditions, we use u(x
start) = 0, where x
start is the starting-point abscissa. The momentum equation is integrated along the path profile y = f(s), where y denotes the elevation and s the abscissa along a horizontal axis; x is a curvilinear abscissa taken from an arbitrary origin on the path profile: 
After integrating the equation numerically, we look for the position of the stopping point at which the avalanche velocity vanishes. We refer to this point as the run-out point or distance x
stop. In Equation (1), two expressions of the frictional force have been tested:
One-parameter expression (Coulomb-like model): the force F assumed to be slope-dependent F = μmg|cos θ, where Π = {μ} is the internal parameter.
Two-parameter expression (Voellmy-like model): the force F is split into a slope-dependent term and a velocity-dependent term: F = μmg cosθ + ku
2, where Π = {μ, k} are the two internal parameters. The former contribution makes it possible to control the avalanche extent while the latter mainly influences the maximum velocity that the avalanche can reach. Moreover, it has often been recognized that the avalanche mass or volume often influences the force: the larger volume V is, the lower its bulk friction is. Thus, parameter k must be a function of the avalanche volume. For convenience, here we assume that this dependency can be written in the following form: k = g/(ξH), where ξ is a friction coefficient and 
is a typical length assumed to give an estimate of the mean flow depth of the avalanche. Using heuristic arguments presented in the Appendix, we will use the following ad hoc relationship between H and V: H = 2.5 + 5 × 10-3V1/2.
Application
Selected avalanche path and data
For the case study, we have selected the Entreme’ ne avalanche path, which is situated on the left side of the Arve river in the Chamonix valley, France. This choice was motivated by two different considerations: first, avalanche activity is substantial and regular; thus we have a long time series of avalanche data on this path (approximately 100 years); second, the upper part of the profile is sufficiently smooth and open for the avalanche dynamics to be simple and similar for each event. In the lower part of the profile, there is a sharp transition in the path slope since, at an elevation of approximately 1000 m, the path is very close to horizontal (Fig. 2). This transition significantly affects avalanche motion since several avalanches stopped in the transition zone. In order to test the influence of the profile on the distribution of extreme run-out distances, we will also consider a modified path profile, for which the slope discontinuity was smoothed. Both profiles are depicted in Figure 2.
Fig. 2. Slope profile of the Entreme’ne avalanche path.
With the conceptual approach, the quality of the results is largely dependent on the quality of the data; it is necessary to have the best and the longest dataset possible, containing exactly the variables that are needed by the model: in our case, because we want to use the Voellmy model, it is neces sary to include the heights of the avalanche events, deduced from the volume deposits, in order to restrict the sample size to the period where the deposit volumes have been estimated.
The avalanche database includes 59 events since 1905. Not all of these events have been recorded; notably, the avalanches stopping in the upper part of the site were not taken into account. For the period 1905-70, the observed data were the starting and stopping elevations and the deposit volume; in Table 1, 30 events are presented but, for three of them, no information on the deposit volume is available. In Figure 2, we have reported the different starting and stopping points of the 27 avalanche events for the period 1905— 70. Since 1971, the deposit volume has no longer been estimated. For the recent period (1970—2002), 29 events have been recorded and will not be used here. This is a pity since we will use only half of the available events from the entire recorded period, and it is obvious that large samples give more accurate results. However, it does not seem possible to easily generate the missing volume data of the recent period in order to increase the size of samples. The Monte Carlo simulation may provide a solution. Furthermore, the number of events in the period used is simlar to that in the shorter recent period. There is seemingly a problem of stationarity, which should be studied for an engineering application, but this will not be dealt with in this methodological paper. Likewise, the influence of the sample size on the result accuracy will not be studied.
Table 1. Avalanche data on the Entremitnepath and μ values for the Coulomb-like model
Uncertainty on the run-out elevation varies with time.
At the beginning of the 20th century, it probably exceeded ±100 m, while nowadays it is expected to be much lower (±25 m). If we use distance rather than elevation to describe avalanche run-out, then uncertainty on the stopping-point position is amplified for the path parts where the slope is low, typically here in the nearly horizontal part of the profile. Uncertainty on the volumes is very large: on this path, it has been estimated as high as 50% according to the deposit dimension measurements. We must also add the uncertainty due to the shape of the deposit. This large uncertainty may have consequences for the methodology results. A specific study could improve this point, but this present study will investigate only the confidence limits which are generated by the method itself.
In conclusion, the analysis of the available dataset is fundamental in the conceptual approach; consequently, important statistical questions arise, replacing the deterministic questions occurring in previous studies.
Procedure with the Coulomb-like model and results
For the first step, we chose the input and output variables: the starting and run-out distances x start and x stop. By fitting the model, we obtained the sample of friction parameters (seeTable 1).
For the second step, we first fitted the sample of x start values. Since x start is bounded, it is very convenient to use the beta distribution as the probability distribution. Figure 3 shows that the frequencies of the middle classes in the experimental values are higher than the fitted distribution. The experimental histogram is very irregular (e.g. for some classes, empirical frequency is zero). We fitted a Gumbel distribution to the a sample, as shown in Figure 4. The Gumbel distribution was used in a first approximation; other possibilities will be explored below, such as the normal distribution, which is included in Figure 4 and whose adequacy seems better than the Gumbel distribution.
Fig. 3. Statistical distribution of the avalanche starting distances.
Fig. 4. Probability distribution of the values (Coulomb-like model).
In the third step, we created large samples of x start and μ values using a random-number generation routine. As the experimental sample contains 27 events for 60 years, the simulated sample for 1000 years should contain 450 events. Figures 3 and 4 show the simulated values for x start and μ, respectively; these samples are very close to the theoretical distributions.
In the last step, we used these two samples to generate 450 fictitious avalanches. The run-out distance of each event was stored. As shown in Figure 5, there is little difference between the empirical probability distribution of the simulated run-out distances computed using the Coulomblike model and the Voellmy-like model respectively (see below). The simulated run-out distances match the field data well, except for the last four identical points (run-out distance at 2100 m). At this critical point, the path profile is nearly horizontal (river bed) and, as stated previously, uncertainty on the stopping-point position is high. For an operational study, it would be necessary to evaluate the genuine run-out distances with an historical analysis. Fitting the distributions will be more convenient for these four experimental points. However, in the conceptual approach all the experimental points are considered in the evaluation of the fitness quality and not only the extreme points. Thus, we conclude that on the whole, the model correctly describes the past avalanche activity in the Entremène path.
Fig. 5. Run-out distance statistical distributions obtained with the Coulomb-like model and the Entreme'ne real profile.
The 500 year run-out distance obtained is 2269 m. This result is unique since the model has only one parameter and we used a single distribution for x start and μ We will see below what happens with different distributions.
Procedure with the Voellmy-like model and results
In the first step, we computed the input and output variables together with the friction coefficients. The Voellmy-like model involves two input variables: the initial condition x start value and the avalanche depth H. We verified that they are not correlated. The model has two internal parameters (μ, ξ). Since we have only one output variable, the value of one of them must be kept constant for the inverse problem to be solved. In the following, we will use the notation (ξ|μ) when it is considered that the parameter • is free while the other parameter μ is held constant for all the events occurring in the path. Figure 6 provides the values of the friction parameter ξ that were deduced from the field data when the value of the other friction parameter μ is held constant for all events. A very similar figure (not reported here) was obtained when computing the μ values for a varying ξ parameter. In either case, the figure exhibits outliers from the same events, which we will ignore later on. It is also worth noting that:
the curve μ(ξ) is horizontal for low and large values of ξ.
This gives a first argument for reducing the range of ξ values: μ being fairly constant for the lowest and highest ξ values, we can consider that ξ ranges from 500 to 10 000 ms2 without loss of generality. Other arguments below can be used to narrow this range further.
the data scattering defines two event families (denoted as the first and second group in Figure 6): for a given ξ, there is no continuum in the μ values, but, on the other hand, two narrow ranges of possible values can be observed. This clear separation in the fitted μvalues has also been observed on other paths when the Voellmy model is used (Reference Ancey, Balmforth and ProvenzalleAncey and others, 2003), and no convincing explanation has been found. On the Entremène path, the separation of the μ values corresponds to a separation of the stopping altitudes, below and above 10701100 m a.s.l.
Fig. 6. Computed ξ values with fixed μ.
In the second step, we first verified that the input variables were independent and necessary. Most authors who used a sliding-block model (or, in fact, any avalanche-dynamics model) did not consider the starting-point position as a reliable dynamics parameter since the initial conditions would affect neither the steady-state solution nor the run-out distance. We examined this point more closely: for all the experimental events, we computed the runout distances with several fixed values of ξ, the value of μ (fitted to the observed events) and a fixed x start value. The mean difference between the recorded and simulated runout distances was found to vary as a function of the ξ value. For ξ values as high as 1000 m s-2, the mean difference is close to zero and, in this case, the starting-point position need not be considered as an input parameter. However, for ξ> 1000 m s-2, this no longer holds. The mean difference increases substantially with • and can exceed 40 m for ξ > 5000 m s-2. In short, surprisingly enough and contrary to common belief, it is necessary to include the input variable x start in the computations since the run-out distance may depend on x start for sufficiently large values of ξ. A beta distribution was used to fit the x start values. For the avalanche depth H, a Gumbel distribution fits the data well (to the naked eye), as shown in Figure 7.
Fig. 7. Statistical distribution of the avalanche heights.
Determining the statistical distributions on the (ξ|μ) and (μ|ξ) samples turns out to be intricate since it is not permissible to use a one-peak distribution fitted to the data. The simplest approximation involves taking the sum of the two probability distributions fitted on each group. In this way, we obtain the one-variable conditional distribution of each parameter. In a previous investigation, we found that a probability distribution made up of two beta distributions fitted the (μ|ξ) values well (Reference Meunier, Ancey and NaaimMeunier and others, 2001). However, since in the following we are also interested in determining the (ξ|μ) distribution, we find it simpler to fit a statistical model resulting from superposing two normal distributions on the (μ|ξ) and (lnξ|μ) values; here we use (ln ξ|μ) rather than (ξ|μ) because the range of possible • values covers several orders of magnitude. As shown in Figure 6 for ξ, we have explored the entire range of possible values for the two parameters, even though they may have no physical meaning for the practitioner. Figure 8 shows the cumulative distribution functions fitted to the (ln ξ|μ) values. These figures call for several remarks:
The existence of two event groups is fully conditioned by the μ value: for μ < 0.4, the second group does not exist because, in this case, the computed run-out distances x stop (ξ|μ) never match the field data.
For extreme μ values (i.e. when μ ≤ 0.225 or μ ≥ 0.5), agreement between empirical and computed frequencies is poor: the empirical distribution is step-shaped while the fitted distribution tends to smooth abrupt variations in the ξvalue.
For these extreme μ values, the fitted sum of two normal distributions is no longer parallel to the others, which implies that the curves intersect. This point is disturbing: if we keep them, inconsistent values will be generated when we apply the Monte Carlo method. Obviously this undesirable behaviour originates from the fact that the curve μ(ξ) flattens out when ξ « 10ms-2: in this case, low variations in the a value induce large variations in the ξ value.
Fig. 8. One variable conditional probability distribution of (ξ|μ) withfixed a (Voellmy-like model).
This motivated us to reduce the μ range by removing values that provide the non-parallel curves in Figure 8. Since we use a sum of two normal distributions, this also means that we discard the a values as soon as both the mean ξ and the standard deviation oy of the (ln ξ|μ) sample differ significantly from a set of selected values. Translated into a mathematical expression, this condition can be expressed as follows. If the condition:
is not fulfilled, in which k = 20 for the first avalanche group and k = 55 for the second avalanche group, then the μ value must be put aside. Using this criterion, we found that a must range from 0.23 to 0.46 for the first group while μ must fall within the range 0.43-0.55 for avalanches belonging to the second group. The same exercise was done by inverting the role of μ and ξ. The same analysis as previously gives the same conclusions and leads to a similar criterion (see Fig. 9):
where k′= 0.04 for the first group and k′ = 0.025 for the second. Note that here, because the derivatives of the mean and the standard deviation do not have the same order of magnitude, we multiply the derivative of the standard deviation by 6 in order to give the same weight to the two derivatives. In the end, we find that the • range is 900— 5000 m s-2 for the first group of avalanches and 165— 2800 m s-2 for the second group. The resulting ranges of the friction parameters are shown inTable 2.
Fig. 9. Determination of the reduced range of μ for the Voellmy-like model (the arrows indicate the limit of validity of the criterion).
Table 2. Range of practical interest for the friction parameters (Voellmy-like model)
The third step is much easier: for each input variable, a sample of 450 values was randomly generated from its adjusted empirical probability distribution (see Fig. 2 for x start and see Fig. 7 for H). Similarly, 450 values of the internal parameters were created from the different conditional distribution of (ln ξ|μ) or (μ \ ξ.
In the fourth step, the run-out distances were computed and stored. Figure 10 reports the statistical distributions of the run-out distances obtained when using the (μ | ξ.) samples. Let us examine what happens when we consider extreme events that could occur in the future or that could have occurred in the distant past. A key point is that the bundle of curves diverges for run-out distances in excess of the critical point (2100 m). Data scattering is pronounced (200 m for the 500 year return-period run-out distance) when the entire range of ξ is considered. In contrast, when we focused our attention on the limited range of ξ determined previously, we found that the range of 500 year run out distance is narrow: approximately 50 m.
Fig. 10. Run-out distance statistical distributions with the μsamples, different fixed values of ξ (in m s-2), and the real profile (Voellmy-like model
We proceeded similarly with the (ξ.| μ) sample. The results are reported in Figure 11. The range for 500 year run-out distance was similar for the entire or limited ranges of μ (120 m vs 90 m). However, the mean values for the practical range were somewhat different: larger for the (μ \ ξ.) simulations (2228 m) than for the (ξ.| μ) simulations (2206 m).Yet the difference was only 22 m (seeTable 3).
Fig. 11. Run-out distance statistical distributions with the ξ samples (in m s2), different fixed values of μ, and the real profile (Voellmy-like model).
Table 3. Comparison of the 500year run-out distances obtained with the two models andfor the two profiles
To summarize, we conclude that:
-
(i) either the (μ| ξ) or the (ξ|μ) sample can be used;
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(ii) the range of values for the fixed parameter is not completely free, and the choice of the variation range is important: it leads to an uncertainty of 50 m for the (μ| ξ) sample vs 90 m for the (ξ|μ) sample.
If we now compare these results with the 500 year runout distance obtained with the Coulomb-like model (2269 m), we see that the difference is approximately 50 m, the Coulomb-like model providing a larger distance.
Complementary Analysis
Study of the modified profile with the Voellmy-like model
The previous developments used the real profile whose final part was horizontal. In the Chamonix valley path, profiles of this kind are frequent, while in many other valleys in the French Alps, the slope variation is gentler. In order to test the conceptual approach in this case, we defined a modified profile (see Fig. 2) and made the same calculations as with the previous profile.The three first steps were the same, and the fourth step differed only in the path profile used in the propagation operator.
Only the results for the (μ| ξ) possibility are presented (see Fig. 12). The 500 year run-out distances obtained using the alternative possibility are reported inTable 3. Comparing Figures 10 and 12 shows that the bundles of curves are very similar below the key point of the horizontal terminal part of the profile and clearly differ above this point (2100 m). The range of results for the 500 year return-period run-out distances is very wide for the entire range of ξ values (310 m), but very narrow for the limited range (40 m). This means that it is very important to use the ranges of practical values for the internal parameters rather than the entire ranges.
Fig. 12. Run-out distance statistical distributions with the μ samples, different fixed values of ξ. (inm s2), and the modified profile (Voellmy-like model).
Table 3 shows that, once again, the run-out distances obtained using the (μ| ξ) values are larger than those obtained when the (ξ|μ) values were used. The difference (73 m) is greater than for the flat profile (22 m). The Coulomb-like model gives a 500 year return-period runout distance of 2382 m situated between the two evaluations with theVoellmy-like model (2347 and 2420 m). We can consider that in the conceptual approach, the use of each model is nearly equivalent when the Coulomb-like model is used with a Gumbel distribution for the v values.
Study of the friction parameter distribution with the Coulomb-like model
In the conceptual approach, the variability of all the input variables and the friction parameter are reflected in the Monte Carlo simulations by their statistical distributions. Therefore, the choice of these distributions is of great importance, especially for the variables or the parameters that can reach large values (e.g. H or ξ in the Voellmy-like model) or come close to zero (e.g., μ in both models). This problem will not be studied completely, but we will provide a few indications in comparing the 500 year return periods obtained with the Coulomb-like model when the Gumbel and the normal distributions for the friction parameter v are used. Figure 4 provides the non-exceeding probability of μ obtained using a normal distribution for μ instead of a Gumbel distribution. Although the two distributions provide similar results for the centre of the sample, they differ significantly for the extreme values, especially for the lower values of μ. These differences entail substantial modifications (see Table 3): the results are much larger with the normal distribution than with the Gumbel distribution (115 m for the real profile and 211m for the modified profile). These differences are greater than those observed when comparing the Coulomb-like and Voellmy-like models.
The role of the friction parameter distribution turns out to be a strategic one in the conceptual approach. This problem needs further investigation both on this path profile and on other profiles for which a large number of documented data are available.
How large should the simulated samples be?
In the preceding calculations we always retained the same samples of 450 random numbers, which gave either the v sample for the Coulomb-like model or the (μ|ξ) (or the (ξ|μ)) samples for the Voellmy-like model. The results obtained so far depend only on the model and the profile. Figures 10-12 show that the curves vary quite irregularly, implying that the confidence limits are large. In order to estimate the computation accuracy (i.e. taking the random uncertainty into account), the size of the simulation samples required to yield a confident estimation of the 500 year runout distance is sought. To that end, we will exemplify the method proposed for assessing the confidence interval by considering a particular case: in the computation, we will use theVoellmy-like model, the modified path profile (since it was shown that it amplifies data scattering) and the conditional probability of (μ|ξ = 1024ms-2). As stated earlier, when • = 1024 m s-2, the influence of the starting position is sufficiently weak to be ignored in the simulations.
We therefore performed ten Monte Carlo simulations. For each of them, 450 events were simulated (as previously but each time with different random-number samples). Taken as a whole, these computations can be seen as either the simulation of the avalanche activity over a 10 000 year interval or the reproducibility test of the avalanche activity over a 1000 year interval. As an example, the 500 year return period of the height H computed with the 1000 year simulation is 7.5 ± 1 m. The run-out distance distributions are reported in Figure 13. Data scattering is substantial for individual samples and long periods of return. Similarly, the distribution related to the 10 000 year interval is much smoother than individual distributions and provides an average trend around which the individual distributions vary. In this respect, the 10 000 year distribution can be used to determine a quantile accurately while the individual distributions provide an idea of the possible variations around this value, that is, an estimate of the confidence limits. For instance, if we consider the 500 year quantile of the run-out distance, we infer from Figure 13 that the mean value is 2406 m and the confidence interval is approximately 22802650 m. The outcome is not very different if we take a longer interval of time (20 000 years): we obtain 2394 m instead of 2406 m; the absolute difference between the two predictions is only 12 m.
Fig. 13. Determination of the confidence limits of the run-out distances (Voellmy-like model) for the modified profile.
A practical rule emerges from this example: it is highly recommended to compute the desired quantile from a very large sample of simulations. Typically, if one is interested in determining the 500 year return-period avalanche, one should generate a sample corresponding to a continuous period of 10 000 years (or more).
Summary
Table 3 sums up the results for the 500 year return-period run-out distances. The following conclusions can be drawn:
The predicted run-out distances obtained with the Coulomb-like model are larger with the normal distribution
than with the Gumbel distribution. The difference ranges from 100 m with the real profile to 200 m with the modified profile.
The results given by theVoellmy-like model are different depending on the (μ\ξ) or the (ξ\μ) sample. The difference ranges from 20 to 70 m, less than the difference introduced by the choice of the friction parameter distribution.
The results obtained with the two models are very similar except when using the normal distribution for the μ values with the Coulomb-like model.
Conclusion
In this paper we have developed a method to compute the run-out distances of long-return-period avalanches. Since this variable depends on both the path profile and avalanche dynamics, it is not licit to compute high-return-period run-out distances merely by extrapolating an empirical distribution fitted on the observed values.
Here we have suggested using a conceptual avalanche-dynamics model and a four-step methodology. The first step involves choosing the input variables according to the available data and calculating the parameters of the avalanche-dynamics model from the recorded historical events. The second step involves adjusting the statistical distributions of the input variables and of the model parameters. In the third step, Monte Carlo simulations are performed by making use of the previously determined statistical distributions. In this way, in the fourth step, we can generate a series of fictitious avalanches over a period of any duration. By taking a very long interval of time (typically 10 000 years), it is possible to obtain an accurate estimate of the run-out distance of a very rare avalanche (500 year return period). This four-step methodology has been exemplified here using an avalanche path in the French Alps and two sliding-block models: the Coulomb-like and theVoellmy-like model.
We have shown that this methodology can be easily used with the Coulomb-like model, which needs only one frictional parameter. It is more complicated with the Voellmy-like model based on two frictional parameters. We have also demonstrated that the conceptual approach can be used with benefit in the avalanche field to compute a high-return-period event. An important result is that the statistical distribution of the friction parameter plays a central role in the final results. Its choice is crucial, and further investigations are needed to clarify this point. Furthermore, no significant difference was found in the extrapolated quantile x stop when comparing the Coulomb-like and Voellmy-like models, when using the Gumbel law for the Coulomb-like model. Since the normal distribution seems to better fit the data for this model, it is necessary to study this point more thoroughly.
Finally, it seems that this methodology can be very useful for practical studies, but further research is required: applying it to many other paths and comparing the results with similar studies using the deterministic approach and studying the problem of the determination of the statistical distribution of the friction parameters using many different datasets. However, practitioners must be aware of just how important the field dataset is: in the French context, a large preliminary study would be required to assess the validity of the data. Extrapolating the statistical distributions of the internal parameters consistently with the data, the path profile and the model results must also be investigated. These two requirements may be difficult to meet, but this work is well worth the effort, since this is how long-return-period run-out distances can be determined scientifically and consistently. At the very least, the comparison with the results given by empirical methods will always be useful.
Acknowledgements
This study was supported by Cemagref, and funding was provided by the European programme CADZIE (EVG1-CT-1999-0009). The authors are grateful to M. Naaim for providing the numerical code of theVoellmy-like model.
Appendix
The mean flow depth H can be estimated from the recorded avalanche volumes using assumptions similar to the empirical arguments used for debris flows, which were roughly validated by Russian experiments (Reference MeunierMeunier, 1991, p. 208):
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(i) For natural flows, a relationship exists between height and width, for which a power-law function can be used. For water discharge, the exponent is zero for the rectangular section, equal to 1 for the triangular section, and less than 1 for convex sections. Here we assume that the width of the avalanche flow W is a power function of the avalanche height: W / HQ:7. This argument is more doubtful for an open slope avalanche, depending on the rheological behaviour of avalanche snow.
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(ii) Similarly, the length L of the avalanche flow is a power function of the avalanche height, but with a lower exponent: L / H0:3. Thus, we obtain a crude relationship between the avalanche volume and its height: V / H2. This expression holds on average but not necessarily for every event.
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(iii) We assume that the avalanche flow depth ranges from 2.5 to 8 m.
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(iv) The avalanche volume is assumed to be constant during the avalanche run and deposition (snow entrainment and compaction are ignored).
Combining all these relationships and assumptions, we finally obtain the following result linking H and V: H = 2.5 + (5 × 1Q-3)V 1/2.
