## 1. Introduction

Recent changes in snow precipitation patterns and snow cover regimes have been investigated in depth (e.g., Beniston 1997; Keller et al. 2005; Fyfe and Flato 1999; Huntington et al. 2004; Dahe et al. 2006). These changes challenge the assumption of the stationarity of snow avalanche activity (Paul 2002; Seiler 2006), which is defined by McClung (2003b) as the stationarity of the weather conditions that generate snow stability. Indeed, climate warming and increased warm winter spells (Beniston 2005) could mean less snow and therefore fewer avalanches as well as a reduction in the mean avalanche size. On the other hand, possibly greater climate variability (Fuhrer et al. 2006; ProClim 1999) could mean more avalanches and/or more intense avalanche cycles. The consequences in terms of avalanche hazards are important, since the exposure level of mountain communities may be modified. This also suggests that snow avalanches could be used as proxy indicators that point out signals of climate change such as, for example, tropical storms (Michener et al. 1997; Webster et al. 2005).

Many authors have investigated the relationship between snow avalanches and climate. Past (Martin et al. 2001) and future (Qu and Hall 2006; Lazar and Williams 2008) snow cover conditions under a given climatic scenario have been modeled, showing that an increase in the proportion of wet snow avalanches compared to dry snow events seems realistic. Smith and McClung (1997a,b) and McClung (2003a) used statistical analyses and classification trees to better understand the physical processes that induce avalanche release and constrain the avalanche mass. Detailed studies of major avalanche cycles have been carried out (Birkeland and Mock 2001; Höller 2009; Birkeland et al. 2001), showing strong correlations with negative pressure anomalies. Keylock (2003) pointed out the influence of the North Atlantic Oscillation on avalanche activity in Iceland. These approaches have permitted the development of advanced climate-based avalanche forecasting models (e.g., Gassner and Brabec 2002; McCollister et al. 2003; Jaedicke and Bakkehoi 2007). They also allowed the concept of “snow and avalanche climatology” to be proposed, which refers to the division of the mountain space into homogenous zones that share common features in terms of avalanche activity. For instance, Mock (1996) and Mock and Birkeland (2000) have shown how in different regions of North America the mean interannual avalanche climate can be characterized, and how possible exceptional winters related to large-scale climatic patterns can be detected. Hägeli and McClung (2003, 2007) have pointed out within the same framework the importance of taking into account large-scale weak layers.

In Alpine areas, several studies have investigated trends in the avalanche data. In the Vallée de la Maurienne, France, Jomelli et al. (2007) found no correlation between the fluctuations of avalanche activity and large-scale atmospheric patterns between 1978 and 2003. Laternser and Schneebeli (2002) highlighted no changes in avalanche activity over the 1950–2000 period in Switzerland. Eckert et al. (2010) studied the variations of avalanche occurrences in the northern French Alps, showing no monotonic trend, but a complex combination of pseudo-periodic fluctuations and abrupt changes. Finally, Schneebeli et al. (1997) investigated possible changes in the number of catastrophic avalanches around Davos, Switzerland, showing no modifications during the twentieth century. This latter approach unfortunately biases the evaluation of the intrinsic variations of the physical processes by the changes in land use and human activities.

Relations between avalanching and climate over a longer period than the longest available avalanche chronicle have been investigated using indirect data from lichenometry (McCarroll 1993) and dendrochronology (Dubé et al. 2004; Stoffel et al. 2006). This is facilitated by the presence of scars in trunks hit by avalanches. Major avalanche winters such as 1950–51 could be detected (Casteller et al. 2007) and correlated with atmospheric and oceanic circulation patterns (Hebertson and Jenkins 2003; Reardon et al. 2008). Germain et al. (2009) have thus shown that the frequency of high-magnitude avalanche winters seems to have increased after 1980 in eastern Canada and have correlated this trend with an increase in climatic variability. Furthermore, Jomelli and Pech (2004) have demonstrated that, at low altitudes, avalanche magnitude has been declining since 1650 in the Massif des Ecrins, French Alps; McCarroll et al. (1995) have obtained similar results in Norway.

Thus, numerous papers provide insight into the climatic control of snow avalanches but provide little evidence of a recent change in avalanche activity. On the other hand, indirect proofs indicate that intense avalanching that occurred during the Little Ice Age no longer exists today. Bridging these two types of approaches would allow us to assess more precisely the evolution of the avalanche magnitude–frequency relationship over the past few decades. The main difficulty is that, as stated by Schaerer (1977), avalanche data are generally sparse and subject to many sources of uncertainty: observation errors, changes in observation protocols, etc. Control measures, when they exist, also cause bias while investigating climate patterns. This lack of homogeneity has major implications in terms of statistical modeling. First, standardization is necessary for comparing data from several paths. Second, the different factors that explain the observed data have to be separated to extract robust indicators of the regional fluctuations of avalanche activity and relate them to climate changes.

In this paper, we therefore reconsider the problem of extracting systematic fluctuations that may be linked to a climatic signal from an avalanche data series. We focus on one of the most critical variables for avalanche hazard mapping, the runout altitude, and take advantage of the existence of a well-documented French avalanche database. In section 2, we introduce a scaled measure of avalanche runout and implement filtering procedures to retain subsamples of relatively reliable data. In section 3, a change-point model derived from previous work in hydrology is adapted to the data structure. Section 4 applies this new framework to France over the last 60 yr, highlighting significant changes in both mean- and high-magnitude avalanche runout altitudes. Section 5 shows that these changes correlate well with the changes in a set of climate variables. Section 6 discusses the advantages and limits of the proposed runout indexes as climate change indicators, the consequences of the results obtained for land-use planning, and the relevance of the chosen probabilistic framework.

## 2. Data studied

### a. French avalanche database

The *Enquete Permanente sur les Avalanches* (EPA) is a chronicle describing the avalanche events on approximately 3800 determined paths in the French Alps and Pyrenees. Avalanche counts have been registered since the beginning of the twentieth century (Mougin 1922), along with different quantitative (runout altitudes, deposit volumes, etc.) and qualitative (flow regime, snow quality, etc.) data (Jamard et al. 2002). The field observations, which were originally devoted to the evaluation of damage to forests, are collected by rangers and stored by the Cemagref. Since the accuracy of the records strongly depends on rangers’ careful data recording, like all avalanche databases, the EPA suffers from serious quality problems. Some local series are poor, at least during the years corresponding to one ranger’s career, with missing events or even occasionally events that were never occurred. During intense avalanche cycles, some events are also missing at a larger scale because the rangers are too busy to record all events or unable to reach certain locations for safety reasons. Finally, confusion among the path names also arises, especially for remote or very low-frequency paths.

However, this database is highly valuable for two reasons. First, the data series are unusually long, so that they are now routinely used for local predetermination using physical modeling (e.g., Ancey et al. 2003), combined statistical–dynamical approaches (Meunier and Ancey 2004; Eckert et al. 2007b), and risk analyses (Eckert et al. 2009). Second, the EPA database’s objective is to be as exhaustive as possible on a sample of paths rather than recording only certain avalanches on all paths. Even if the path selection was originally not based on scientific arguments, it gives a relatively accurate view of the spatiotemporal fluctuations of avalanche activity in France over the last century. Eckert et al. (2007a) have thus highlighted coherent spatial patterns in the northern French Alps. It can therefore be assumed that EPA also offers a good opportunity to investigate large-scale temporal fluctuations related to climate change.

### b. Runout altitude data

For safety reasons, rangers do not really measure runout altitudes, but estimate them, generally using binoculars, from a distant observation point, and then plot these estimations on a map. For extreme events involving very dry noncohesive snow and/or a powder cloud, the point of the farthest reach is sometimes very difficult to locate because of the absence of marked deposits. Moreover, whereas GPS today helps to convert an observation into a coordinate, only raw topographical maps were available at the beginning of the twentieth century, and they were replaced by better ones only progressively. This all makes observation errors a critical problem. To keep them within reason, we focus only on the period of *T*_{obs} = 61 years between 1946 and 2006. This includes 57 473 avalanches from 3840 paths. The uncertainty level remains, however, a minimum of 50 m immediately after World War II and at least 10 m today. Note also that here a year is a winter, with the year 2000 corresponding, for example, to winter 2000/01, which may be confusing with regard to the usual definition of the hydrological year. The full notation (e.g., 2000/01), is therefore also used in the text.

These data were filtered using several deterministic and statistical procedures. For instance, for each path, the runout altitude series was compared to the path’s topography. As soon as an aberrant value such as a runout altitude lower than the path’s lowest altitude was found, the path and its full data series were discarded. Many records strongly impacted by the major human artifacts listed in section 2a such as confounding two paths were thus eliminated. A total of 19 660 avalanches (i.e., 34% of the original sample) and *M* = 2725 avalanche paths were kept in the analysis, with 38–1027 avalanches yr^{−1}, a mean annual avalanche number of 322.3 and 1–170 avalanches path^{−1}. This filtering step may have smoothed certain low-marked temporal patterns that were present in the original dataset. Nevertheless, it is coherent with our objective of investigating robust common signals that can be confidently linked with climate patterns. This would not have been possible with the original series, which included too many anthropogenic perturbations.

The natural runout process can be biased by avalanches that were artificially released for safety reasons. In the present study, considering only the events clearly recorded as natural avalanches would have considerably reduced the sample size, because the release cause was recorded only for 13 441 events out of 19 660 (Table 1). However, 96.5% of the filtered events for which the release cause is known are naturally released ones, so that the effects of artificial releases were considered to be insignificant for the full filtered sample. The presence of avalanche defense structures can also be considered a major source of uncertainty for studying the links between runout altitude variations and climate. However, defense structures are not that numerous in France, mainly because of their high cost. Moreover, most current structures are already very old and therefore should not be responsible for recent changes in runout altitude chronicles. As a consequence, the 2725 paths corresponding to the filtered series were initially kept in the analysis, but a sensitivity analysis has been performed; see section 4e.

### c. Runout altitude indexes

The runout regime on each path depends on the topography, exposure (which constrains snowdrift and snow metamorphism in the release zone), and on the regional climatic forcing, which is related to north–south and east–west gradients weighted by more local orographic effects. This makes the direct comparison of runouts from different paths nearly impossible. Figure 1 shows that more than half of the avalanche sample studied was recorded in the northern French Alps. In part, this stems from the fact that the northern French Alps is by far the most active region in France, but also arises because data collection started later in the other regions. Hence, during the years directly following World War II, data collection was already intensive in the northern Alps, where avalanches often reach low altitudes such as 1200 m, but was just beginning in the rest of the country, where the typical runout altitude is much higher. To limit these strong inhomogeneity problems, we propose using an easy-to-interpret runout altitude index (RAI). Its intrinsic limits are discussed in section 6a.

*z*

_{stopijt}as the runout altitude corresponding to the avalanches

*i*

*N*(

*j*)] recorded in the avalanche path

*j*

*M*] during the winter

*t*

*t*,

_{o}*t*+

_{o}*T*

_{obs}− 1], where

*t*is the first winter considered (here, 1946/47). RAI

_{o}*is obtained by scaling the observation by the valley floor*

_{ijt}*z*

_{minj}, that is, the minimal possible runout altitude [see Eq. (1)]. By definition, our RAI equals 1 if the minimal runout altitude of the path is reached. Otherwise, it is a continuous and strictly decreasing function of the runout altitude belonging to the ]0, 1[ interval:

*z*

_{min}ranges from 500 to 2500 m MSL, with a mean value of 1223.3 m MSL and with most of the values between 1000 and 2000 m MSL (Fig. 2, top left). The global distribution of the runout altitudes analyzed is not very different (Fig. 2, top right), with a mean value of 1373.7 m MSL. This indicates that most of the avalanches analyzed have reached low altitudes in their path. For instance, a total of 3398 avalanches out of 19 660 reached their valley floor, with an overall probability of 0.17. The global distribution of the RAI is therefore strongly skewed to the right (Fig. 2, bottom left).

The RAI’s distribution varies greatly from one year to another. Figure 2 (bottom right) shows three cumulative distributions. For winter 1998/99, high values correspond to a given probability, and the annual percentage of avalanches having reached the valley floor is higher than for the two other winters. This clearly shows that, during this catastrophic winter (SLF 2000; Ancey et al. 2000), many avalanches reached long runouts. On the contrary, winter 2001/02 was characterized by runout altitudes that were higher than usual and by a low percentage of avalanches having reached the valley floor. The interannual fluctuations can be summarized by the mean annual runout index, *m _{t}* = E[RAI

*]; the annual variance*

_{ijt}*υ*= Var[RAI

_{t}*], which quantifies the annual variability; and*

_{ijt}*p*=

_{t}*P*(RAI

*= 1), the annual probability of high-magnitude avalanches reaching the valley floor. The time series (*

_{ijt}*m*,

_{t}*υ*,

_{t}*p*) are thus annual proxy indicators of the runout altitude regime on a mean path at the scale of the area studied. Since they depend a great deal on the annual climatic conditions, it can be assumed that they accurately capture the large-scale mountain winter climate changes investigated.

_{t}## 3. Unsteady modeling of the RAI

### a. Statistical modeling versus empirical estimates

_{empt},

_{empt}, and

_{empt}for

*m*[Eq. (2)],

_{t}*υ*[Eq. (3)], and

_{t}*p*[Eq. (4)] are very easy to obtain. In Eq. (4), the indicator function

_{t}*I*

_{{RAIijt=1}}= 1 if the minimal runout altitude is reached and 0 if it is not:

_{meanempt}, the mean runout altitude associated with

_{empt}, implies inverting Eq. (1). In contrast to

_{empt}, this depends on the path considered. If

*z*

_{minmean}, the mean altitude of the valley floor in the sample studied, is chosen, the altitude representing the mean regional behavior is obtained [Eq. (5)]. If another local value

*z*

_{minj}is preferred, the mean value is adapted to the local topography:

Nonparametrical confidence intervals can be computed for the different estimates using a bootstrap procedure. However, we prefer to employ a parametrical modeling approach to distinguish a possible systematic evolution that may be related to climate change from the interannual fluctuations.

### b. A mixture model adapted to the data structure

_{ijt1}is a discrete random number taking the value 1 if the avalanche

*i*occurring in the year

*t*on the path

*j*reaches its possible minimal altitude, and 0 if not. It is modeled by a Bernoulli distribution whose annual parameter

*p*is the annual probability of reaching the valley floor [Eq. (7)]. In addition, RAI

_{t}_{ijt2}is a continuous and skewed random number modeling all the other avalanches belonging to ]0, 1[. It can be modeled by a beta distribution [Eq. (8)], with an annual parameter pair (

*α*,

_{t}*β*):

_{t}_{t1}and RAI

_{t2}are assumed to be independent, which seems tenable for the dataset studied; see section 4c. Due to the linearity of mathematical expectancy, the annual mean

*m*[Eq. (9)] and the annual variance

_{t}*υ*[Eq. (10)] are then easily obtained from the model’s parameters (

_{t}*p*,

_{t}*α*,

_{t}*β*):

_{t}*p*,

_{t}*α*,

_{t}*β*) fully characterizes the RAI annual distribution. This is also true for the triplet (

_{t}*p*,

_{t}*m*,

_{t}*β*) because

_{t}*α*can be computed from the other parameters [Eq. (11)] under the constraints

_{t}*p*

_{t}*m*[ and

_{t}*m*

_{t}*p*,

_{t}*m*,

_{t}*υ*) cannot be used to parameterize the model, since inverting Eq. (10) has no single solution.

_{t}Finally, we note

### c. Capturing the change point and the monotonic trend

To capture the systematic variations of the mean and high-magnitude avalanche runouts, we model the *m _{t}*’s and the

*p*’s as latent variables, that is, as model unknowns that behave as parameters with regard to the data but whose distributions are indexed by (hyper)parameters. This multilayer construction characterizes a hierarchical framework (Clark and Gelfand 2006; Banerjee et al. 2003). Moreover, we decompose both latent variables into interannual fluctuations and structured signals corresponding to the large-scale changes in the runout altitudes that are investigated. To do so, various time series models already in use in climatology (von Storch and Zwiers 2002) or developed in other fields, mainly econometrics (Booth and Smith 1982), hydrology (Rao and Tirtotjondro 1996), and meteorology (Bloomfield 1992), can be put to use. The time series model has to be chosen with regard to the investigated patterns (Carlin et al. 1992), for instance changes in means (Mearns et al. 1997), variances (Diaz 1982; Menzefricke 1981), or extreme values. Here, we limit our purpose to a model originally developed in the hydrological community (Perreault et al. 2000a; Perreault et al. 2000b). It is relatively simple but flexible enough to capture a monotonic trend as well as various types of changes in the mean and variance.

_{t}*τ*as the year of a possible change point separating two periods of different runout regimes. Before and after the change point, both the

*m*’s and the

_{t}*p*’s are broken down into random noises and linear trends. The latter ones, noted trendm

_{t}*=*

_{t}*a*

_{m.}+

*b*

_{m.}

*t*and trendp

*=*

_{t}*a*

_{p.}+

*b*

_{p.}

*t*, are designed to capture the possible progressive evolution patterns of the mean- and/or high-magnitude events. Depending on the continuity of trendm

*and trendp*

_{t}*around*

_{t}*τ*, the change point can be abrupt, with a clear separation into two runout regimes, or simply a distinction between two different trends. The random noises, with respective variances

*τ*to also be a change point in variance [Eqs. (12) and (13)]. Note that the fraction of the signal captured by the random noises is not truly random, but corresponds to phenomena that are not reducible to a simple linear evolution such as long-memory climatic effects:

_{m.}[Eq. (14)] and frac.struc

_{p.}[Eq. (15)]. They can take any value between 0 and 1 and are high only if the hypothesized linear time trends explain a large part of the interannual fluctuations. Both ratios can take two different values, before and after the change point:

*m*’s and

_{t}*p*’s are nonexchangeable as soon as the time trends

_{t}*b*

_{m.}and

*b*

_{p.}are significantly nonzero, which is not the case in a standard ANOVA scheme. Moreover, the variance decomposition is, with our model, performed on a non-Gaussian and even noncontinuous variable using hierarchy. This is significantly different from Perrault’s original model, which was directly applied to the observed discharges.

### d. Bayesian inference of the proposed model

*p*,

_{t}*m*,

_{t}*β*), a probability distribution function (pdf) p(

_{t}*β*,

_{t}*a*,

_{p}*b*,

_{p}*s*,

_{p}*a*,

_{m}*b*,

_{m}*s*,

_{m}*τ*) that expresses expertise prior to data acquisition and the distribution p(

*p*,

_{t}*m*|data,

_{t}*a*,

_{p}*b*,

_{p}*σ*,

_{p}*a*,

_{m}*b*,

_{m}*σ*,

_{m}*τ*,

*β*) of the latent variables given all of the parameters and observations. The result of the computation is p(

_{t}*p*,

_{t}*m*,

_{t}*β*,

_{t}*a*,

_{p}*b*,

_{p}*σ*,

_{p}*a*,

_{m}*b*,

_{m}*σ*,

_{m}*τ*|data), the joint posterior pdf of all parameters and latent variables [Eq. (16)]. Note that all observations are considered mutually independent [Eq. (17)]. The main advantage of this framework for our problem is that, thanks to its flexibility, the posterior pdf of the latent time series is likely to fit relatively complex temporal patterns even with a relatively simple model [see Fortin et al. (2004) for an example in hydrology]:

*β*’s as well as uniform or poorly informative normal distributions for the trend parameters. The marginal posterior distributions obtained for a few model parameters are shown in Fig. 4. All of them have a much lower variance than do the prior distributions, which indicates that information has been captured from the data. Moreover, all distributions are nicely shaped and unimodal, making the use of point estimates possible for all of the model’s parameters. The posterior means have been considered, and the bounds of the 95% credible intervals have been chosen to quantify the related uncertainty.

_{t}### e. Quantifying runout altitude variations

*, for the estimated probability of reaching the valley floor in the year*

_{t}*t*, and tren̂dp

_{t}, the associated trend, are readily available as soon as the model’s inference has been completed. As for empirical estimates,

_{meanjt}, the estimated mean runout altitude on the path

*j*in the year

*t*, depends on the minimal altitude

*z*

_{minj}of the path considered and must be derived from

*[Eq. (18)]. The associated trend is obtained by considering tren̂dm*

_{t}_{t}instead of

*:*

_{t}*t*+

*δt*knowing only the values

*and*

_{t}_{meanjt}. This is once again easy for the high-magnitude events [Eq. (19)] and slightly more cumbersome for the mean runout altitudes because of the nonlinearity of the RAI [Eq. (20)]:

*τ*

*t*,

*t*+

*δt*]. If not, the change point has to be considered. Moreover, it must be stressed that the estimators

_{p.}and

_{m.}correspond only to a mean regional pattern of behavior at the scale of the area studied; consequently, their local application to paths with various characteristics remains questionable.

Possible negative probabilities can be generated using Eq. (19) if a strong negative value for _{p.} over a long period is considered, especially if * _{t}* is already low at the beginning of the period considered. This highlights an intrinsic limitation in our model. It could be overcome in a future study by testing a more flexible nonlinear trend possibly tending to zero but remaining positive, for example, using a logittransformation of the

*p*’s.

_{t}## 4. Application

### a. Posterior distributions of m_{t}, p_{t}, and υ_{t}

For the sample studied, the mean RAI increases slowly during nearly the first half of the studied period and then decreases slowly to reach its initial state again (Fig. 5, top). The interannual fluctuations are strong enough for the different *m _{t}*’s to be significantly different from each other at the 95% credibility level. The

*p*’s are nearly constant and close to the overall mean

_{t}*p*= 0.175 during the 1946–76 period, increase to around 0.25 over a few years, and then decrease continuously and relatively sharply until 2006 (Fig. 5, middle). Their estimated values are not different from each other during the first part of the study period. On the other hand, the interannual variability of the

_{o}*p*’s increases sharply during the second part of the study period, differentiating the values from each other. Finally, no strong trend in annual variance is visible, with complex fluctuations and values generally not different from each other (Fig. 5, bottom).

_{t}Even if the patterns of behavior of the mean and high-magnitude events are therefore not identical, there is evidence in both time series of *m _{t}* and

*p*that a change point seems to have occurred around 1977. Moreover, the existence of different trends before and after the change point seems likely for both time series. On the contrary, no strong trend in annual variance is visible, which suggests that the annual variability range of runout altitudes has not changed very much in France over the last 60 yr. It can only be noted that the lowest values of

_{t}*υ*are observed for the years around the change point, which corresponds to the highest values of

_{t}*m*and

_{t}*p*. This can be explained by the fact that when most of the events reach the valley floor or stop close to it, the annual variance of the RAI is necessarily small.

_{t}### b. Estimated temporal trend and change point

The preferred year of change is *τ* = 1976 if the posterior mean is considered. More precisely, the posterior pdf of *τ* shows 2 yr of high and close posterior probabilities, 1975 and 1976, and very small posterior probabilities for all other years between 1969 and 1993 (Fig. 4). This nice shape indicates that the data clearly support the existence of a significant change point. Table 2 shows that three linear trends out of four are significantly nonzero at the 95% credibility level (i.e., all of them except *b*_{p1}). This confirms that the mean RAI increased before the level shift and then decreased to its initial state. This also confirms that the probability of reaching the valley floor has remained nearly constant before the level shift and has substantially decreased since then. Finally, *σ*_{m1} is not different from *σ*_{m2} at the 95% credibility level, but *σ*_{p2} is nearly twice as high as *σ*_{p1}. This confirms that *τ* is also a change point in variance for the probability of high-magnitude events, but not for mean events.

Both mean trends, trendm* _{t}* and trendp

*, are nearly continuous around*

_{t}*τ*, indicating that

*τ*is much more of a separation between two different trend pairs (

*b*

_{m1},

*b*

_{p1}) and (

*b*

_{m2},

*b*

_{p2}) than an abrupt change between two different runout regimes (Fig. 6). The variance ratios indicate that the two trends explain 37%–66% of the variability of the mean- and high-magnitude events, depending on the period considered (Table 2). These values show that the underlying linear trends are relatively well supported and significantly quantify the variations of the mean- and high-magnitude avalanche runout altitudes in France during the period studied. Moreover, the higher values after the change point give us confidence in the existence of a decreasing trend for mean- and high-magnitude events between 1977 and 2006.

### c. Quantification of mean runout altitude variations

Figure 7 compares the annual empirical estimates _{meanempt} computed using Eq. (5) with the annual estimates and the mean trend provided by the model using Eq. (18). The 95% credible intervals for the two model components are also shown. The minimal altitude considered is *z*_{min} = *z*_{minmean} = 1223.3 m, the mean altitude of the valley floor in the sample studied, so as to obtain the pattern of behavior of a mean EPA path. The interannual variability of the empirical estimates is clearly captured by the annual estimates provided by the model, but with a smoothing effect due to the capacity of the hierarchical structure to transfer information from one year to another (shrinkage effect). The extraction of the mean trend can be considered to be a second smoothing that distinguishes the predominant pattern from the interannual variability. The width of the credible intervals is logically smaller for the mean trend than for the annual estimates. These considerations suitably illustrate the value of our hierarchical model for deconstructing the data in two steps and extracting the large-scale mean behavior. They also indicate that the independence of RAI_{t1} and RAI_{t2} is not to be rejected for this dataset. Indeed, since no assumption is necessary for computing the empirical estimates, the model would have provided biased estimates if RAI_{t1} and RAI_{t2} had been strongly linked.

More precisely, the mean trend decreased up to the change point *τ* from a little more than 1400 m MSL in 1946 to nearly 1350 m MSL in 1977. Since the change point, the mean trend increased again, up to nearly 1400 m MSL in 2006. The difference between the mean runout altitudes in 1946 and 1977 (i.e., 55 m) is twice as large as the spread of the 95% credible interval around the mean trend, which is about 25 m wide throughout the study period. This indicates that the estimated trend is statistically significant.

### d. Quantification of high-magnitude avalanche variations

*T*was computed [Eq. (21)]. A constant mean avalanche rate,

_{t}*f*= 0.293 avalanches per year, and path were considered. These results were obtained by studying avalanche activity over the same time period on the EPA paths (Cemagref 2008). The 95% credible interval for

*T*was obtained by considering the bounds of the 95% credible interval for tren̂dp

_{t}_{t}, which does not consider the uncertainty related to

*f*:

### e. Influence of countermeasures

A rough evaluation of the number of perturbed EPA paths has been made by cross-checking the EPA database with the French avalanche map. Only 92 avalanche paths were detected. This low number justifies the statement made in section 2 that there is a low proportion of avalanche paths equipped with defense structures in France. Most of those paths are situated in a few townships such as Chamonix and Val d’Isère, where significant destructive avalanches occurred during the twentieth century. Figure 9 shows that the mean trends obtained for the subsample of 2633 paths and 18 941 avalanches classified as unperturbed is not different with the mean trends extracted from the whole dataset (Fig. 5). This demonstrates that enhanced defense structures are insufficient for explaining the recent systematic fluctuations of runout altitudes in France, for example the clear increase engaged for nearly 30 yr.

### f. Model choice: Are the trends and change point necessary?

Three simplified models were compared with the proposed model, noted M_{2}, to check whether a simpler model could be as efficient for our case study. Model M_{0} is exchangeable and nonhierarchical, which means that the *p _{t}*’s and the

*m*’s are modeled as exchangeable parameters with no possible systematic evolution. Model M

_{t}_{1}assumes that a change point

*τ*exists, but that there is no structured linear trend before and/or after the breakpoint. The

*p*’s and the

_{t}*m*’s are therefore sampled from two Gaussian distributions whose constant means characterize the two runout regimes investigated. On the contrary, model M

_{t}_{1′}assumes linear trends for the

*p*’s and the

_{t}*m*’s over the whole period.

_{t}Rather than the Bayes factor (Carlin and Chib 1995; Kass and Raftery 1995), the deviance information criterion (DIC) was used. For a given dataset, the model with the lowest DIC is the most efficient at capturing the data structure (Spiegelhalter et al. 2002). In this study, the simplest model with no time structure is the worst of the four competing models (DIC = −16 931). On the contrary, model M_{2} is the best (DIC = −16 956). Considering a breakpoint and a linear trend for both *m _{t}* and

*p*is therefore well adapted to the data structure. Model M

_{t}_{1′}is slightly better (DIC = −16 950) than model M

_{1}(DIC = −16 947), indicating that the breakpoint is a bit more useful than the linear trends to fit the data.

## 5. Link with climate fluctuations

### a. Considered climatic covariates

To investigate the climatic relevance of our annual runout altitude indicators, we compared them with a few climatic series. We chose to use real data rather than outputs of climatic simulations because of the difficulty of taking into account in climate models the strong spatial and altitudinal gradients that exist in mountain spaces. Furthermore, we focused on datasets already considered to be reliable indicators of climatic fluctuations by the French National Observatory for Climate Change (ONERC). They include winter climatic series at Col de Porte and two glacier mass balance series, Saint Sorlin and Sarennes. These sites are located approximately at the center of the French Alps (Fig. 1). Observation periods are not fully coherent (since 1960 at Col de Porte, 1957 at Saint Sorlin, and 1948 at Sarennes), but cover most of the period of interest on each site.

The data series at Col de Porte offers one of the most reliable climate chronicles in France at midaltitudes (1326 m MSL), since the measurements are directly taken by the Snow Study Center of Météo-France. It is consistent with climate reanalyses recently performed in the French Alps (Durand et al. 2009), and with a broad range of data/results from different countries of the Alpine space which have been synthesized within the framework of the Climate Change, Impacts and Adaptation Strategies in the Alpine Space (CLIMCHALP) project (ONERC 2008). We chose three variables: the winter temperature *T _{t}* and the mean winter snow depth WSD

*, which are mean values between 1 December and 30 April, and the number of days where the snow cover is deeper than 1 m, DWS > 1*

_{t}*, which quantifies the duration of harsh winter conditions. The Saint Sorlin and Sarennes mass balance series, bStS*

_{t}*and bSar*

_{t}*, are measured by the Laboratoire de Glaciologie et Géophysique de l’Environnement (LGGE) and Cemagref, respectively. A preliminary separation of the spatial effects has been performed (see Thibert et al. 2008). Mass balances result from winter and spring accumulation, and summer ablation. Even if they are indirect climate series, they are known to be excellent integrated indicators of climate variability at high altitudes (Vincent 2002), where avalanche starting zones are most commonly located but where direct measurements of meteorological variables are more infrequent and highly uncertain due to orographic and wind effects.*

_{t}We processed the different climatic series using Perrault et al.’s (2000a,b) nonhierarchical change-point model to extract the underlying trends in a fashion similar to how they were extracted from the mean- and high-magnitude RAIs in this paper. This means *y _{t}* ~ N(

*a*

_{1}+

*b*

_{1}

*t*,

*σ*

_{1}

^{2}) for

*t*

*t*,

_{o}*τ*] and

*y*~ N(

_{t}*a*

_{2}+

*b*

_{2}

*t*,

*σ*

_{2}

^{2}) for

*t*

*τ*+ 1,

*t*+

_{o}*T*

_{obs}− 1], where

*y*is either

_{t}*T*, WSD

_{t}*, DWS > 1*

_{t}*, bStS*

_{t}*, or bSar*

_{t}*. For each variable, (*

_{t}*a*

_{1},

*b*

_{1},

*a*

_{2},

*b*

_{2},

*σ*

_{1},

*σ*

_{2}) are the parameters characterizing the underlying trends and interannual fluctuations before and after a possible change point,

*τ*. Figures 10 and 11 show the five data series; the estimated trends tren̂dT

_{t}, trendŴSD

_{t}, trendDŴS > 1

_{t}, trendb̂StS

_{t}, and trendb̂Sar

_{t}; the corresponding credible intervals; and scatterplots with RAI annual estimates. Note the complex shapes of the mean trends, which fit the data well even if only one change point separating two linear trends is postulated. This is caused by the ability of the Bayesian paradigm to cope with the uncertainty related to the year of change. It was less clear in Fig. 6 because the uncertainty around

*τ*was smaller in the avalanche data. Finally, empirical correlations between the different runout altitude indexes and the covariates are provided in Table 3.

### b. Comparison with Col de Porte climate data

Although some interannual fluctuations obviously exist around the estimated trends, the winter snow depths and the number of days with more than 1 m of snow have clearly been decreasing at Col de Porte since around 1970. Over the same 1960–2006 period, the mean winter temperature at Col de Porte has increased (Fig. 10). These results are not surprising, since it is well established that mean temperatures have been rising since the end of the Little Ice Age in the Alps, decreasing the mean snow supply at low and middle altitudes. It is also increasingly accepted that these changes have been accelerated during the last few decades, presumably because of the human impacts on climate warming. This acceleration is well corroborated by the three Col de Porte data series, with a substantial temperature increase around 1986–87 and a significantly sharp decrease in the two snow data series at the beginning of the 1980s. The fact that the estimated preferred year of change for the two snow data series precedes the preferred year of change in temperatures suggests that a change in snow precipitation also occurred at Col de Porte in the early 1980s.

These patterns extracted from local midaltitude data series are coherent with the fluctuations of our large-scale runout altitude indexes. For instance, the recent evolution through less snowy and cold winters logically creates less favorable conditions for low runout altitudes. Empirical correlations are therefore high, negative between temperatures and the runout altitude indexes, and positive between the two snow data series and the runout altitude indexes (Table 3). Both in terms of absolute values and levels of significance, correlations are better with the two snow data series than with the temperatures. Annual snow data series are therefore better predictors of annual runout altitude indexes than annual winter temperatures, which is not surprising since temperature has only an indirect influence on runouts (see section 5d). The correlation with the probability of high-magnitude runouts is slightly better for the annual number of days with more than 1 m of snow than for the total winter snow depth, which also makes sense, since high snow depths are generally necessary to reach long runouts. Finally, it must be noted that correlations are generally much better with and between the extracted trends than with and between the data series themselves (Table 3), thus confirming the existence of similar coherent patterns of behaviors, and also the pertinence of the change-point models chosen for analyzing the different series.

### c. Comparison with Saint Sorlin and Sarennes mass balance series

For the two glacier mass balances, the trend is slightly positive from 1950 to 1975–80, with a few years of high values clustered between 1975 and 1980, especially at Sarennes (Fig. 11). This corresponds to a short period of positive or nearly positive balances, which has been observed on several French glaciers. It differs from the general context of glacial retreat that takes place since the end of the Little Ice Age (roughly 1850–70) because of low ablation values. After 1980, the two annual mass balance series sharply decrease because ablation has significantly risen (Vincent 2002).

These patterns (i.e., the slightly positive trend, the concentration of high values around a change point, and the strong decreasing trend from the change point until now) also exist in our runout altitude indexes, which induces excellent correlations between the two glacier mass balance series and the annual runout indexes (Table 3). As for the Col de Porte data, the correlations are even better between the trends than between the data series, with correlation coefficients reaching around 0.9 between the trends in mass balances and the trend in the annual probability of high-magnitude avalanches. This is surprisingly high, especially when considering that mass balances also depend on ablation that occurs far later in the season than the latest avalanches. One possible explanation is the relevance of the mass balance for capturing climate variability at altitudes corresponding to avalanche starting zones. Another explanation is that avalanche runout altitudes and glacier mass balances both depend in a complex manner on similar climate parameters, for instance in spring (late accumulation and early snowmelt), so that they give a similar integrated vision of the mountain climate at high altitudes.

### d. Response of runout altitudes to climate fluctuations

The fluctuations of our large-scale annual runout altitude indexes seem to follow the variations of different local climatic variables quite well, both at mid- and high altitudes. A significant link between our results and recent climatic fluctuations can therefore be reasonably postulated. Discussing in details the impacts of climate change on the physical processes that control avalanche release and flow (snow accumulation, snowpack transformation, snow transformation during the flow, etc.) is outside of the scope of this paper. However, simple physical explanations of our results can be postulated.

Even if it remains debated in the avalanche community, it is generally admitted that large volumes generate long runouts because of the reduction of friction during the flow (Dade and Huppert 1998). The most obvious physical explanation of the observed runout altitude fluctuations is therefore changes in the amount of snow available, for instance high accumulations of cold snow around 1980 (Fig. 11), and a decreasing amount of snow cover since 1980, at least at midaltitude (Fig. 10, top and bottom). In the latter case, several physical processes may be involved: lower precipitation totals, a higher snowmelt because of higher temperatures, and even a higher proportion of rain with respect to snow because of the rise of the mean freezing level.

A complementary physical explanation of our results is a change in the snow quality (density, humidity, grain size, etc.) in the release zone. Indeed, snow quality controls friction during the flow, with higher friction associated with higher temperatures (Casassa et al. 1989), for instance when wet snow is involved. Since Martin et al. (2001) have already stated that the number of wet snow releases is going to increase with regard to dry snow avalanches, this could partially explain how the trend in winter temperatures (Fig. 10, middle) impacts avalanche runouts.

Finally, changes in the soil cover could also be suggested to explain our results, with possible origins in the modification of agricultural and forestry practices (McClung 2001) or as an indirect effects of climate change through the rise of the tree line resulting from climate warming (Theurillat and Guisan 2001; Dullinger et al. 2004). This point was not considered in our study since wooded paths are not very numerous in our database, but it could be investigated in further work focused on specific paths.

### e. Influence of avalanche flow regime

Since 1973, the EPA database distinguishes avalanches with a powder part (i.e., purely powder snow avalanches and mixed avalanches) from purely dense snow avalanches. The annual proportions of the avalanche flow regimes have been processed using the same nonhierarchical model as the climatic covariates to extract the underlying trend. Only trend̂PSA_{t}, the trend for the proportion of avalanches with a powder part, PSA* _{t}*, is provided, since purely dense snow avalanches evolve in a symmetrical manner. Note also that the reliability of the flow regime information is lower than the reliability of the corresponding runouts, so that the results must be interpreted with care.

Globally, there is a significant decreasing trend in the proportion of avalanches with a powder part throughout the period studied (Fig. 12). There is also a relatively strong positive correlation between the runout altitude indexes and the proportion of avalanches including a powder part (Table 3). For instance, the correlations between trend̂PSA_{t} and the runout altitude indexes are all significantly high, thus highlighting that the evolution of the flow regime’s annual proportions and annual runout indexes are globally coherent over the last 30 yr. This joint evolution of avalanche runout altitudes and preferred flow regime is consistent with the previous statement of a recent change in the quality of the flowing material under climate warming. Indeed, even if *dry snow avalanche* and *powder snow avalanche* are not synonyms (dry or wet snow refers to the humidity of the snow, mainly in the release zone, whereas dense or powder snow avalanche refers to the flow dynamics), these two qualitative descriptions are strongly linked because powder snow avalanches nearly always involve dry snow. More wet snow avalanche releases as postulated by Martin et al. (2001) could therefore imply less powder snow avalanches with regard to purely dense snow avalanches and, therefore, shorter runouts. Since long runouts often correspond to powder snow or mixed avalanches, this is presumably especially true for extreme events. A good argument for this is that PSA* _{t}* is significantly correlated only with

*(Table 3).*

_{t}It is noteworthy that the preferred discontinuity in the underlying trend illustrated in Fig. 12 corresponds to the higher proportion of powder snow avalanches recorded during the 1997/98 and 1998/99 winters rather than to any of the years of the 1975–85 period that appeared critical for all the other data series. This presumably comes from the shortness of the flow regime series, which makes a possible change around 1980 undetectable. As stated in section 2c, 1998/99 was a very harsh winter both in terms of the number and magnitude of the recorded avalanches. On the contrary, 1997/98 saw a low avalanche number because of late snowfalls and a warm spring, but the recorded events showed relatively long runouts. This highlights that our approach is focused on avalanche magnitude, but says nearly nothing about the number of avalanche occurrences.

## 6. Discussion

### a. Limits of the defined runout altitude indexes

This study has introduced the RAI as a scaled measure of avalanche magnitude. In some ways, it is similar to the Norwegian runout ratio, which is nothing more than the runout abscissa normalized by the beginning of the runout zone and which is often used for computing high-return-period avalanches (McClung and Lied 1987). Like the runout ratio, our RAI can compare runouts from different paths. Both measaures also remain a simplified vision of the avalanche magnitude, which cannot be reduced to the maximum distance traveled. This is especially true for unchanneled paths, where lateral spreading and/or different trajectories are possible depending on the properties and volumes of the flowing material.

However, by employing altitudes instead of abscissas, our approach dictated by the available data introduces additional limitations. First, for V-shaped valleys, high-magnitude avalanches are sometimes able to climb opposite slopes. When detailed in the database, an RAI of 1 was attributed to the avalanche, saying that the valley floor was reached, but this remained somewhat an underestimation of the true magnitude of the event, and this level of detail was often not available. Second, distances are measures of the true stopping positions, which are much more precise than altitudes, since runout zones are generally relatively flat. This makes the fluctuations of the RAI small, even between events that significantly differ from each other, making only well-marked fluctuations detectable.

Another limitation of our RAI is that the mean annual runout altitude depends on the chosen minimal altitude *z*_{min}. To this point, the mean behavior was analyzed. Equation (18) makes it possible to investigate the sensitivity to other choices. Figure 13 shows the results obtained for minimal altitudes of 500 (top) and 2000 m MSL (bottom), which corresponds roughly to the variation range of the paths from the EPA database. The variation of the mean runout altitude and the associated uncertainty is proportional to *z*_{min}. Moreover, the global shape always remains similar, since it corresponds to the underlying trend *â*_{m.} + _{m.}*t*. For a minimal altitude of 500 m MSL, the altitude difference equals 23 m ± 6 m between 1977 and 2006, whereas it equals 90 m ± 24 m for a minimal altitude of 2000 m MSL. In other words, the runout altitude increase for the 1977–2006 period is greater for paths where avalanche activity is located at a high altitude than for avalanche paths for which avalanche activity is located at a low altitude. One part of this high sensitivity to *z*_{min} may be related to a physical reality. Indeed, paths situated at high altitudes are more exposed to climate variability (heavy snowfalls, etc.) than paths situated at lower altitudes. It is therefore plausible that they react more radically to climate trends. However, a part of this high sensitivity is also undoubtedly a direct consequence of the definition of the RAI.

Finally, there is also sensitivity to the difference between the mean runout altitude and *z*_{min}. As an illustration, different mean runout altitudes *τ* + 1) were considered for *τ* = 1976, and the mean runout altitude for the year *τ* + *δ _{t}* = 2006 was computed using Eq. (20) with a constant

*z*

_{min}= 1223.3 m MSL (Table 4). Due to the estimated trend

_{m2},

_{mean}(

*τ*+

*δ*) is much higher than

_{t}_{mean}(

*τ*+ 1) for any value of

_{mean}(

*τ*+ 1). However, the difference,

*, increases slightly with*

_{z}_{mean}(

*τ*+ 1) −

*z*

_{min}, ranging from 47.5 m for

_{mean}(

*τ*+ 1) = 1250 m MSL to 75.4 m for

_{mean}(

*τ*+ 1) = 1800 m MSL. This is also true for the related 95% credible interval. This is a model artifact that has no physical reality since it is related to the choice of a minimal runout altitude,

*z*

_{min}. This analysis and the previous one strongly challenge the information provided by a local version of the regional trend. It must therefore be kept in mind that the advantage of our approach lies in its ability to extract a mean pattern of behavior on a mean path rather than in a local quantification of temporal fluctuations.

### b. Runout altitude indexes as climate indicators

In sections 2 and 4e, we have shown that artificial releases and defense structures have no significant effects on the results obtained. Changes in the EPA protocol are other human sources of data inhomogeneities. A good summary can be found in Burnet (2006). For instance, in 1976, the maps available to the rangers were on a different scale, and notebooks were replaced by forms directly sent to the Cemagref. It is impossible to fully exclude that these changes partially explain the presence of a change point in the data in 1976, but we believe this is insufficient to explain all the temporal patterns highlighted in section 4, especially given that more substantial changes also affected the chronicle in the past, such as modifications of the list of paths being surveyed.

The systematic observation errors discussed in section 2 are another major source of concern. The filtering procedures implemented obviously kept in the analysis certain aberrant values in addition to the lack of precision inherent to avalanche runout data collection. This presumably affects the annual variance series, so that it should be considered with caution. On the other hand, there is no real reason for assuming that this could be responsible for a change point in the middle of the period studied for the mean runout altitude accompanied with nearly linear modifications before and after the change point. In our opinion, the trends in *m _{t}* and trendm

_{t}should therefore be considered physically significant in addition to their statistical significance. A good argument is that the 55-m difference between the mean runout altitude in 1976 and 2006 is higher than the systematic observation error in the recent years. For high-magnitude events, one part of the large decrease since the change point may be related to the increase in the precision of the runout altitude records over the same period. Indeed, before the availability of precise maps, the minimal altitude was sometimes considered to be the runout altitude even if the avalanche truly stopped a few meters higher, which is no longer the case. However, the decreases in

*p*and trendp

_{t}*are so great and continuous that they necessarily also reflect a physical reality worth considering, especially given their high importance for land-use planning.*

_{t}Beyond these considerations, the best argument in favor of a climatic explanation for our results is the strong correlations between our annual runout altitude indexes and the climatic chronicles at different altitudes presented in section 5. Although correlation is not causality in general, it is clear, as discussed in section 5d, that the avalanche runout process is climatically controlled through the amount and quality of the available flowing material. It is therefore not surprising that an effect of the already well-documented systematic changes of the winter climate in the French mountains can be detected in an avalanche data series with intrinsic deficiencies, but also strong points (length of the series, exhaustiveness on certain paths, etc.). Particularly convincing are the consistency of the decreasing trend in runout indexes over the last 30 yr within the general context of an accelerated climate warming, the presence of a change point in all climate and avalanche series between 1975 and 1985, and the correspondence between high runout indexes and positive mass balances around 1980 because of harsh winters at high altitudes. All of this is, for us, sufficient to consider our indexes reasonable and robust new indicators of mountain climate fluctuations.

Various points though remain unclear. For instance, the increasing trend in runout altitudes during the 1946–76 period is well correlated with the high-altitude mass balance data, but less well with the midaltitude climatic data from Col de Porte. Moreover, even if there seems to be evidence in all the series studied (except the flow regimes) of a change around 1980, the level of significance of the change point found in each data series is different, and its date varies from one series to another: the beginning of the 1980s at Col de Porte, earlier in the mass balance series, and for the runout indexes. There is therefore much space for further developments at different scales. A first promising perspective would be to compare our indexes with the results of climate models and with synoptic climate indicators, such as in Keylock (2003), so as to link the observed trends in avalanche activity with large-scale patterns. A complementary approach would be to perform a more detailed joint analysis and modeling of our well-documented avalanche chronicles and systematically recorded climatic data such as snow depths, wind speeds, and temperature series at different altitudes and locations, so as to look for local and/or specific differences in the trends and change points. This would help define more homogenous spatial scales in terms of snow and avalanche climate, thus bridging our data-oriented work and the work of Mock and Birkeland (2000). Finally, a specific study could be undertaken of the most well-documented paths of the database, so as to investigate in detail the correlations and physical relations between changes in climate forcing, avalanche runout altitudes, and other quantitative and qualitative data characterizing avalanche activity (volume, snow humidity, etc.). This would allow us to evaluate the respective contributions of the different explanatory factors listed in section 5d.

### c. Consequences for hazard assessment

The decreasing trends in runout altitude indexes over the last 30 yr highlighted in this paper and their relations to climate variability and changes are crucial for avalanche hazard mitigation. Indeed, our results strongly challenge the assumption of an underlying stationary process generating high-magnitude events that is generally made for avalanche mitigation. Moreover, if there is a real link between the changing climatic variables and runout indexes, as our results tend to show, these changes may continue and even be amplified in the upcoming years due to the acceleration of climate change in the French mountains.

However, the belief in decreased exposure of French mountain communities to avalanche hazard must be tempered for several reasons. First, obviously, a high uncertainty level remains because of the inherent limits of the indexes discussed above in terms of artifacts and data quality. Second, even though likely because of the general context of climate warming, the persistence of the highlighted trends beyond 2006 remains hypothetical, and further investigation linking future climate modeling and simulation of avalanche activity should be undertaken to be more conclusive. Third, section 4 has established that there has been an increase in the interannual variability of the probability of high-magnitude events since 1976. This indicates that even if, on the average, high-magnitude avalanches are less frequent, some exceptional winters such as the winter of 1998/99 with many more high-magnitude avalanches than predicted by the global decreasing trend are still to be expected. Finally, the return periods that could be computed using our indexes range from 20 to 40 yr only. High-magnitude avalanches as defined in this paper are thus truly rare events, but not the extreme ones that are generally taken as reference scenarios for hazard mapping, and for which the typical return period ranges from 100 to 300 yr. Truly quantifying the recent changes affecting such extreme events and their relation to climate change would therefore require further investigations (i.e., transforming the actually observed runout altitudes into distances), so as to be able to assign different return periods for all the avalanches that attained the same valley floor but reached different runout positions. This is very difficult, however, because of the above-mentioned indetermination of the true stopping position for flat or nearly flat runout zones.

### d. Relevance of the chosen hierarchical Bayesian framework

Inspired by previous hydrological studies, a hierarchical change-point modeling framework has been adapted in this study to the available data. Reliable mean patterns of behavior for both mean- and high-magnitude events were thus extracted from the residual interannual variability described as a random signal. Applying similar models to different climatic covariates also showed the relevance of this approach for investigating climate trends by allowing most of the correlations to be emphasized, thus showing apparently strong and coherent links between the evolution of mountain winter climate and the defined runout indexes.

Given the limited quality of the available data, the aim of the study was to extract rough but robust indicators of the response of snow avalanches to changes in climatic constraining factors. This guided the choice of a relatively simple model, which, as is shown in the detailed analysis of section 4, is well supported by the data. Section 4f also shows that simpler models would have been less well adapted. Therefore, our model is arguably a reasonable compromise between model complexity and robustness. Indeed, there is no real evidence in the point estimates of Fig. 5, suggesting that several change points or strong periodicities could exist in the data studied. Moreover, our annual series only include 61 values, so that robust estimates for more highly parameterized models cannot really be expected. Multiple change-point models allowing several breakpoints and/or autoregressive integrated moving average (ARIMA) models taking into account more properly long-memory climatic effects using overlapped cycles would therefore be highly advantageous options only for studying longer data series, such as the runout data available since 1898. This interesting perspective for further work devoted to the search for climate proxies over a longer time period was found to be too ambitious for a first attempt because of greater data homogeneity problems.

Hierarchical modeling should not necessarily be associated with Bayesian inference. However, as shown in several of our figures, Bayesian inference of the latent time trend has the advantage of capturing relatively complex underlying trends with a relatively simple model because the posterior distribution can take into account model uncertainty to a certain degree. Moreover, the inference of a hierarchical model under the classical paradigm is often tricky. On the contrary, Bayesian simulation tools are very convenient for complex hierarchical models (Wikle 2003) because of the strong duality existing between Markov chain Monte Carlo (MCMC) simulation methods and hierarchy. Indeed, both approaches are based on conditional probabilities. MCMC schemes were therefore used to sample the joint posterior pdf’s of all the model’s unknowns. Convergence was checked by comparing different chains starting at different points in the parameter space (Brooks and Gelman 1998). The posterior pdf’s of the other quantities of interest, for instance, the *υ _{t}*’s and the ratios frac.struc

_{m.}and frac.struc

_{p.}, were obtained by computing their values at each iteration. A general explanation of the principle of MCMC methods can be found in various sources (Brooks 1998; Gilks et al. 2001), whereas application to avalanche modeling has been described in Ancey (2005) and Eckert et al. (2008), and is therefore not detailed here.

Hence, the hierarchical Bayesian choice was made in this study for practical reasons rather than for its ability to include expert *prior* information in the analysis. In fact, the old controversy against Bayesian theory has even been fully avoided by choosing poorly informative priors, which give asymptotically the same inferential results as a frequentist approach (Berger 1985). This was done because the sample size was assumed to be largely sufficient to let the data speak for themselves (Bernardo and Smith 1994) and thus obtain unbiased indicators of the response of snow avalanches to changes in winter climate.

## 7. Conclusions

In this paper, we used an advanced statistical framework to define robust annual avalanche runout indexes and correlate them to climate variability. We showed that the mean avalanche runout altitude is not different now than it was 60 yr ago in France. We also demonstrated that a change point exists in the winter of 1976/77 that separates two periods with significantly different trends and that an accelerated retreat of avalanche runouts has been occurring for nearly 30 yr, especially for high-magnitude events. Strong similarities with the behavior of several climatic datasets and consistency with the evolution of the preferred flow regime suggest that our indexes may be usable as indicators of climatic fluctuation at high altitudes, where direct climatic series remain uncommon and subject to many uncertainty sources. All these results are crucial in terms of hazard assessment. As a consequence, the priority now is to confirm the temporal patterns obtained using a more detailed joint analysis of the avalanche and climatic data at different spatial and temporal scales. This will for instance make it possible to assess the chances in the persistence of the highlighted trends, so as to extrapolate the changes in the mean and extreme runout altitudes beyond 2006 within the current context of an accelerated pattern of climate change.

## Acknowledgments

We are grateful to Météo-France (Snow Study Centre) and Y. Durand for providing the Col de Porte climatologic data, to LGGE and C. Vincent for providing the Saint Sorlin mass balance data, and to E. Thibert for providing the Sarennes mass balance data. We also thank C. J. Keylock for his feedback during data analysis, and C. Ancey and two other anonymous referees for their useful comments.

## REFERENCES

Ancey, C., 2005: Monte Carlo calibration of avalanches described as Coulomb fluid flows.

,*Philos. Trans. Roy. Soc. London***363A****,**1529–1550.Ancey, C., F. Rapin, E. Martin, C. Coleou, M. Naaim, and G. Brugnot, 2000: L’avalanche de Péclerey du 9 février 1999.

,*Houille Blanche***5****,**45–53.Ancey, C., M. Meunier, and D. Richard, 2003: Inverse problem in avalanche dynamics models.

,*Water Resour. Res.***39****,**1099. doi:10.1029/2002WR001749.Banerjee, S., B. Carlin, and A. E. Gelfand, 2003:

*Hierarchical Modeling and Analysis for Spatial Data*. Chapman and Hall, 472 pp.Beniston, M., 1997: Variations of snow depth and duration in the Swiss Alps over the last 50 years: Links to changes in large-scale climatic forcings.

,*Climatic Change***36****,**281–300.Beniston, M., 2005: Warm winter spells in the Swiss Alps: Strong heat waves in a cold season? A study focusing on climate observations at the Saentis high mountain site.

,*Geophys. Res. Lett.***32****,**L01812. doi:10.1029/2004GL021478.Berger, J. O., 1985:

*Statistical Decision Theory and Bayesian Analysis*. 2nd ed. Springer-Verlag, 617 pp.Bernardo, J. M., and A. F. M. Smith, 1994:

*Bayesian Theory*. Wiley, 586 pp.Birkeland, K. W., and C. J. Mock, 2001: The major snow avalanche cycle of February 1986 in the western United States.

,*Nat. Hazards***24****,**75–95.Birkeland, K. W., C. J. Mock, and J. J. Shinker, 2001: Avalanche extremes and atmospheric circulation patterns.

,*Ann. Glaciol.***32****,**135–140.Bloomfield, P., 1992: Trends in global temperature.

,*Climatic Change***21****,**1–16.Booth, N. B., and A. F. M. Smith, 1982: A Bayesian approach to retrospective identification of change-points.

,*J. Econometrics***19****,**7–22.Brooks, S. P., 1998: Markov chain Monte Carlo method and its application.

,*Statistician***47****,**69–100.Brooks, S. P., and A. Gelman, 1998: General methods for monitoring convergence of iterative simulations.

,*J. Comput. Graph. Stat.***7****,**434–455.Burnet, R., 2006: Cartes et bases de données d’avalanche.

*Dynamique des Avalanches,*C. Ancey, Ed., Presses Polytechniques et Universitaires Romandes, 167–169.Carlin, B. P., A. E. Gelfand, and A. F. M. Smith, 1992: Hierarchical Bayesian analysis of changepoint problems.

,*Appl. Stat.***41****,**389–405.Carlin, J. B., and S. Chib, 1995: Bayesian model choice via Markov chain Monte Carlo methods.

,*J. Roy. Stat. Soc.***57B****,**473–484.Casassa, G., H. Narita, and N. Maeno, 1989: Measurements of friction coefficients of snow blocks.

,*Ann. Glaciol.***13****,**40–44.Casteller, A., V. Stöckli, R. Villalba, and A. C. Mayer, 2007: An evaluation of dendroecological indicators of snow avalanches in the Swiss Alps.

,*Arct. Antarct. Alp. Res.***39****,**218–228.Cemagref, 2008: Calcul des fréquences avalancheuses communales moyennes. Rapport Technique à la DPPR, 13 pp. [Available online at http://www.avalanches.fr/].

Clark, J. S., and A. Gelfand, 2006:

*Computational Statistics: Hierarchical Bayes and MCMC Methods in the Environmental*. Oxford University Press, 205 pp.Dade, W., and H. Huppert, 1998: Long-runout rockfalls.

,*Geology***26****,**803–806.Dahe, Q., L. Shiyin, and L. Peiji, 2006: Snow cover distribution, variability, and response to climate change in western China.

,*J. Climate***19****,**1820–1833.Diaz, J., 1982: Bayesian detection of a change of scale parameter in sequences of independent gamma random variables.

,*J. Econometrics***19****,**23–29.Dubé, S., L. Filion, and B. Hétu, 2004: Tree-ring reconstruction of high-magnitude snow avalanches in the northern Gaspé Peninsula, Québec, Canada.

,*Arct. Antarct. Alp. Res.***36****,**555–564.Dullinger, S., T. Dirnböck, and G. Grabherr, 2004: Modelling climate change-driven treeline shifts: Relative effects of temperature increase, dispersal and invisibility.

,*J. Ecol.***92****,**241–252.Durand, Y., M. Laternser, G. Giraud, P. Etchevers, L. Lesaffre, and L. Mérindol, 2009: Reanalysis of 44 years of climate in the French Alps (1958–2002): Methodology, model validation, climatology, and trends for air temperature and precipitation.

,*J. Appl. Meteor. Climatol.***48****,**429–449.Eckert, N., E. Parent, L. Belanger, and S. Garcia, 2007a: Hierarchical modelling for spatial analysis of the number of avalanche occurrences at the scale of the township.

,*Cold Reg. Sci. Technol.***50****,**97–112.Eckert, N., E. Parent, and D. Richard, 2007b: Revisiting statistical–topographical methods for avalanche predetermination: Bayesian modelling for runout distance predictive distribution.

,*Cold Reg. Sci. Technol.***49****,**88–107.Eckert, N., E. Parent, M. Naaim, and D. Richard, 2008: Bayesian stochastic modelling for avalanche predetermination: From a general system framework to return period computations.

,*Stoch. Environ. Res. Risk***22A****,**185–206.Eckert, N., E. Parent, T. Faug, and M. Naaim, 2009: Bayesian optimal design of an avalanche dam using a multivariate numerical avalanche model.

,*Stoch. Environ. Res. Risk***23A****,**1123–1141.Eckert, N., E. Parent, R. Kies, and H. Baya, 2010: A spatio-temporal modelling framework for assessing the fluctuations of avalanche occurrence resulting from climate change: Application to 60 years of data in the northern French Alps.

, doi:10.1007/s10584-009-9718-8, in press.*Climatic Change*Fortin, V., L. Perreault, and J. D. Salas, 2004: Retrospective analysis and forecasting of streamflows using a shifting level model.

,*J. Hydrol.***296****,**135–163.Fuhrer, J., M. Beniston, A. Fischlin, C. Frei, S. Goyette, K. Jasper, and C. Pfister, 2006: Climate risks and their impact on agriculture and forests in Switzerland.

,*Climatic Change***79****,**79–106.Fyfe, J. C., and G. M. Flato, 1999: Enhanced climate change and its detection over the Rocky Mountains.

,*J. Climate***12****,**230–243.Gassner, M., and B. Brabec, 2002: Nearest neighbour models for local and regional avalanche forecasting.

,*Nat. Hazards Earth Syst. Sci.***2****,**247–253.Germain, D., L. Filion, and B. Hétu, 2009: Snow avalanche regime and climatic conditions in the Chic-Choc Range, eastern Canada.

,*Climatic Change***92****,**141–167.Gilks, W. R., S. Richardson, and D. J. Spiegelhalter, 2001:

*Markov Chain Monte Carlo in Practice*. Chapman and Hall, 486 pp.Hägeli, P., and D. McClung, 2003: Avalanche characteristics of a transitional snow climate—Columbia Mountains, British Columbia, Canada.

,*Cold Reg. Sci. Technol.***37****,**255–276.Hägeli, P., and D. McClung, 2007: Expanding the snow-climate classification with avalanche-relevant information: Initial description of avalanche winter regimes for southwestern Canada.

,*J. Glaciol.***53****,**266–276.Hebertson, E. G., and M. J. Jenkins, 2003: Historic climate factors associated with major avalanche years on the Wasatch Plateau, Utah.

,*Cold Reg. Sci. Technol.***37****,**315–332.Höller, P., 2009: Avalanche cycles in Austria: An analysis of the major events in the last 50 years.

,*Nat. Hazards***48****,**399–424.Huntington, T. G., G. A. Hodgkins, B. D. Keim, and R. W. Dudley, 2004: Changes in the Proportion of Precipitation Occurring as Snow in New England (1949–2000).

,*J. Climate***17****,**2626–2636.Jaedicke, C., and S. Bakkehoi, 2007: Climate database for avalanche consulting and warning in Norway.

,*Cold Reg. Sci. Technol.***47****,**171–179.Jamard, A. L., S. Garcia, and L. Bélanger, 2002: L’enquête permanente sur les Avalanches (EPA): Statistique descriptive générale des événements et des sites. DESS Ingéniérie Mathématique Option Statistique, Université Joseph Fourrier, Grenoble, France, 101 pp. [Available online at http://www.avalanches.fr/].

Jomelli, V., and P. Pech, 2004: Effects of the Little Ice Age on avalanche boulder tongues in the French Alps (Massif des Ecrins).

,*Earth Surf. Processes Landforms***29****,**553–564.Jomelli, V., C. Delval, D. Grancher, S. Escande, D. Brunstein, B. Hetu, L. Filion, and P. Pech, 2007: Probabilistic analysis of recent snow avalanche activity and climate in the French Alps.

,*Cold Reg. Sci. Technol.***47****,**180–192.Kass, R. E., and A. E. Raftery, 1995: Bayes factors.

,*J. Amer. Stat. Assoc.***90****,**773–795.Keller, F., S. Goyette, and M. Beniston, 2005: Sensitivity analysis of snow cover to climate change scenarios and their impact on plant habitats in alpine terrain.

,*Climatic Change***72****,**299–319.Keylock, C. J., 2003: The North Atlantic Oscillation and snow avalanching in Iceland.

,*Geophys. Res. Lett.***30****,**1254. doi:10.1029/2002GL016272.Laternser, M., and M. Schneebeli, 2002: Temporal trend and spatial distribution of avalanche activity during the last 50 years in Switzerland.

,*Nat. Hazards***27****,**201–230.Lauritzen, S., 1996:

*Graphical Models*. Oxford University Press, 312 pp.Lazar, B., and M. Williams, 2008: Climate change in western ski areas: Potential changes in the timing of wet avalanches and snow quality for the Aspen ski area in the years 2030 and 2100.

,*Cold Reg. Sci. Technol.***51****,**219–228.Martin, E., G. Giraud, Y. Lejeune, and G. Boudart, 2001: Impact of climate change on avalanche hazard.

,*Ann. Glaciol.***32****,**163–167.McCarroll, D., 1993: Modelling late-Holocene snow-avalanche activity: Incorporating a new approach to lichenometry.

,*Earth Surf. Processes Landforms***18****,**527–539.McCarroll, D., J. A. Matthews, and R. A. Shakesby, 1995: Late-Holocene snow-avalanche activity in southern Norway: Interpreting lichen size-frequency distributions using an alternative to simulation modelling.

,*Earth Surf. Processes Landforms***20****,**465–471.McClung, D., 2001: Characteristics of terrain, snow supply and forest cover for avalanche initiation caused by logging.

,*Ann. Glaciol.***32****,**223–239.McClung, D., 2003a: Magnitude and frequency of avalanches in relation to terrain and forest cover.

,*Arct. Antarct. Alp. Res.***35****,**82–90.McClung, D., 2003b: Time arrival of slab avalanche masses.

,*J. Geophys. Res.***108****,**2466. doi:10.1029/2002JB002299.McClung, D., and K. Lied, 1987: Statistical and geometrical definition of snow-avalanche runout.

,*Cold Reg. Sci. Technol.***13****,**107–119.McCollister, C., K. Birkeland, K. Hansen, K. Aspinall, and R. Comey, 2003: Exploring multi-scale spatial patterns in historical avalanche data, Jackson Hole Mountain Resort, Wyoming.

,*Cold Reg. Sci. Technol.***37****,**299–313.Mearns, L., C. Rosenzweig, and R. Goldberg, 1997: Mean and variance change in climate scenarios: Methods, agricultural applications, and measures of uncertainty.

,*Climatic Change***35****,**367–396.Menzefricke, U., 1981: A Bayesian analysis of a change in the precision of a sequence of independent normal random variables at an unknown time point.

,*Appl. Stat.***30****,**141–146.Meunier, M., and C. Ancey, 2004: Towards a conceptual approach to predetermining high-return-period avalanche run-out distances.

,*J. Glaciol.***50****,**268–278.Michener, W. K., E. R. Blood, K. L. Bildstein, M. M. Brinson, and L. R. Gardner, 1997: Climate change, hurricanes and tropical storms, and rising sea level in coastal wetlands.

,*Ecol. Appl.***7****,**770–801.Mock, C. J., 1996: Avalanche climatology of Alyeska, Alaska, U.S.A.

,*Arct. Alp. Res.***28****,**502–508.Mock, C. J., and K. W. Birkeland, 2000: Snow avalanche climatology of the western United States mountain ranges.

,*Bull. Amer. Meteor. Soc.***81****,**2367–2392.Mougin, P., 1922:

*Les avalanches en Savoie*. Ministère de l’Agriculture, Direction Générale des Eaux et Forêts, Service des Grandes Forces Hydrauliques, Paris, Tech. Rep., 175–317.ONERC, 2008: Changements climatiques dans les Alpes: Impacts et risques naturels. ONERC Rapport Technique 1, 86 pp. [Available online at http://www.risknat.org/docs/Technical%20Report%20N%B01.pdf].

Parent, E., and J. Bernier, 2007:

*Le Raisonnement Bayésien: Modélisation et Inférence*. Springer, 380 pp.Paul, P., 2002: Reconstitution d’anomalies de paramètres climatiques et de fréquences de catastrophes naturelles (crues, sécheresses, tempêtes) au cours des 500 dernières années en Europe Centrale.

,*Houille Blanche***6/7****,**111–114.Perreault, L., J. Bernier, B. Bobée, and E. Parent, 2000a: Bayesian change-point analysis in hydrometeorological time series. Part 1. The normal model revisited.

,*J. Hydrol.***235****,**221–241.Perreault, L., J. Bernier, B. Bobée, and E. Parent, 2000b: Bayesian change-point analysis in hydrometeorological time series. Part 2. Comparison of change-point models and forecasting.

,*J. Hydrol.***235****,**242–263.ProClim, 1999: De pareils hivers à avalanches sont-ils encore normaux?

*Climate-Press*, No. 5, ProClim—Forum for Climate and Global Change, Bern, Switzerland. [Available online http://www.proclim.ch/Products/ClimatePress/ClimatePress05F.pdf].Qu, X., and A. Hall, 2006: Assessing snow albedo feedback in simulated climate change.

,*J. Climate***19****,**2617–2630.Rao, A., and W. Tirtotjondro, 1996: Investigation of changes in characteristics of hydrological time series by Bayesian methods.

,*Stoch. Env. Res. Risk***10A****,**295–317.Reardon, B. A., G. T. Pederson, C. J. Caruso, and D. B. Fagre, 2008: Spatial reconstructions and comparisons of historic snow avalanche frequency and extent using tree rings in Glacier National Park, Montana, U.S.A.

,*Arct. Antarct. Alp. Res.***40****,**148–160.Schaerer, P. A., 1977: Analysis of snow avalanche terrain.

,*Can. Geotech. J.***14****,**281–287.Schneebeli, M., M. Laternser, and W. Ammann, 1997: Destructive snow avalanches and climate change in the Swiss Alps.

,*Eclogae Geol. Helv.***90****,**457–461.Seiler, W., 2006: Klimawandel im der Alpenraum Auswirkungen und Herausforderung: Les changements climatiques dans l’espace alpin: tendances, retombées et défis. Deuxième Manifestation Thématique: Changement du Climat dans L’Espace Alpin—Effets et défis. 31st Réunion du Comité Permanent, Galtür, Austria, 7–19. [Available online at http://www.cenat.ch/ressources/planat_product_fr_786.pdf].

SLF, 2000:

*Der Lawinenwinter 1999: Ereignisanlyse*. Institute for Snow and Avalanche Research, Davos Dorf, Switzerland, 588 pp.Smith, A. F. M., 1975: A Bayesian approach to inference about a change-point in a sequence of random variables.

,*Biometrika***62****,**407–416.Smith, M. J., and D. M. McClung, 1997a: Avalanche frequency and terrain characteristics at Rogers’ Pass, British Columbia, Canada.

,*J. Glaciol.***43****,**165–171.Smith, M. J., and D. M. McClung, 1997b: Characteristics and prediction of high-frequency avalanche runout.

,*Arct. Alp. Res.***29****,**352–357.Spiegelhalter, D. J., N. Best, B. Carlin, and A. Van der Linde, 2002: Bayesian measures of model complexity and fit (with discussion).

,*J. Roy. Stat. Soc.***64B****,**583–640.Stoffel, M., M. Bollschweiler, and G. R. Hassler, 2006: Differentiating past events on a cone influenced by debris-flow and snow avalanche activity—A dendrogeomorphological approach.

,*Earth Surf. Processes Landforms***31****,**1424–1437.Theurillat, J. P., and A. Guisan, 2001: Potential impact of climate change on vegetation in the European Alps: A review.

,*Climatic Change***50****,**77–109.Thibert, E., R. Blanc, C. Vincent, and N. Eckert, 2008: Glaciological and volumetric mass balance measurements: An error analysis over 51 years, Glacier de Sarennes, French Alps.

,*J. Glaciol.***54****,**522–532.Vincent, C., 2002: Influence of climate change over the 20th century on four French glacier mass balances.

,*J. Geophys. Res.***109****,**4375. doi:10.1029/2001JD000832.von Storch, H., and F. W. Zwiers, 2002:

*Statistical Analysis in Climate Research*. Cambridge University Press, 494 pp.Webster, P. J., G. J. Holland, J. A. Curry, and H. R. Chang, 2005: Changes in tropical cyclone number, duration, and intensity in a warming environment.

,*Science***309****,**1844–1846.Wikle, C., 2003: Hierarchical Bayesian models for predicting the spread of ecological processes.

,*Ecology***84****,**1382–1394.

Empirical distributions of the data. (bottom left) Corresponds to the interannual distribution. (bottom right) Corresponds to three specific winters. The winter 1998/99 is statistically different from the two others at the 95% significance level (Kolmogorv-Smirnov test). The winters 2001/02 and 2002/03 are not statistically different from each other.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Empirical distributions of the data. (bottom left) Corresponds to the interannual distribution. (bottom right) Corresponds to three specific winters. The winter 1998/99 is statistically different from the two others at the 95% significance level (Kolmogorv-Smirnov test). The winters 2001/02 and 2002/03 are not statistically different from each other.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Empirical distributions of the data. (bottom left) Corresponds to the interannual distribution. (bottom right) Corresponds to three specific winters. The winter 1998/99 is statistically different from the two others at the 95% significance level (Kolmogorv-Smirnov test). The winters 2001/02 and 2002/03 are not statistically different from each other.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Direct acyclic graph of the proposed model. Arrows express conditional dependence, circled nodes represent stochastic variables, and rectangles indicate observed values.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Direct acyclic graph of the proposed model. Arrows express conditional dependence, circled nodes represent stochastic variables, and rectangles indicate observed values.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Direct acyclic graph of the proposed model. Arrows express conditional dependence, circled nodes represent stochastic variables, and rectangles indicate observed values.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Marginal posterior distributions of model unknowns.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Marginal posterior distributions of model unknowns.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Marginal posterior distributions of model unknowns.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Box plots of the mean RAI and of the probability of reaching the valley floor. For each year, the box plot provides the posterior mean, the interquartile range, and the 95% credible interval.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Box plots of the mean RAI and of the probability of reaching the valley floor. For each year, the box plot provides the posterior mean, the interquartile range, and the 95% credible interval.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Box plots of the mean RAI and of the probability of reaching the valley floor. For each year, the box plot provides the posterior mean, the interquartile range, and the 95% credible interval.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Annual fluctuations of the mean runout altitude. The mean altitude *z*_{minmean} of the valley floor is used.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Annual fluctuations of the mean runout altitude. The mean altitude *z*_{minmean} of the valley floor is used.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Annual fluctuations of the mean runout altitude. The mean altitude *z*_{minmean} of the valley floor is used.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Annual probability of reaching the valley floor and the associated return period. The return period is computed by considering a mean avalanche rate of 0.293 avalanches per winter and path.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Annual probability of reaching the valley floor and the associated return period. The return period is computed by considering a mean avalanche rate of 0.293 avalanches per winter and path.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Annual probability of reaching the valley floor and the associated return period. The return period is computed by considering a mean avalanche rate of 0.293 avalanches per winter and path.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Temporal trends for the paths with no defense structures.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Temporal trends for the paths with no defense structures.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Temporal trends for the paths with no defense structures.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Covariation with climatic variables at Col de Porte.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Covariation with climatic variables at Col de Porte.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Covariation with climatic variables at Col de Porte.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1

Covariation with two glacier mass balances.

Citation: Journal of Climate 23, 12; 10.1175/2010JCLI3312.1