Brief communication: Weak control of snow avalanche deposit volumes by paths morphological characteristics

Snow avalanches are a major component of the mountain cryosphere, but little is known about the factors 10 controlling the variability of their deposit volumes. This study investigates the influence of morphological characteristics on c. 1500 deposit volumes recorded between 2003 and 2018 in 77 snow avalanche paths of the French Alps. Different statistical techniques show a slight but significant link between deposit volumes and path mean elevation and orientation, with contrasted patterns between winter and spring seasons. The weakness, partially non-linear nature of this control may result from the combined influence on the genesis of deposit volumes of mean path activity, climate conditions and 15 mechanical thresholds determining avalanche release.

to advanced (deep learning) statistical techniques, we present a first detailed quantification of how paths morphological characteristics impact snow avalanche deposit volumes. Specifically, we show that the control of deposit volumes by path morphology is weak but significant.

Avalanche deposit volumes 35
This research was conducted on the two hillsides of the upper part of the Maurienne valley in the Northern French Alps between the municipalities of Lanslevillard and Bonneval-sur-Arc (Fig. 1). Because of its important number of active avalanche paths, this study area is particularly relevant for our analysis (e.g., Eckert et al., 2009;Favier et al., 2014;Zgheib et al., 2020). The dataset of snow avalanche deposit volumes used in this study is primarily based on the Permanent Avalanche Survey (EPA) which was created at the beginning of the 20 th century to document avalanche events as 40 exhaustively as possible on more than 3,000 avalanche paths in the French Alps. For each single avalanche event, the size of the deposit is documented, based on a visual estimate carried out by devoted survey operators from the EPA network. For each deposit, the length, the width and the mean depth is estimated, which eventually provides a volume estimate. For each path, EPA operators systematically use the same predefined observation point, so as to maximize the accuracy of the estimation. However, a further correction and completion work was carried out to develop a more comprehensive and error-45 free snow avalanche deposit database (Kern et al., 2020). Input errors or outliers introduced within the EPA when old records registered on paper archives were converted to numerical data were spotted and corrected. In addition, few other snow avalanche events (less than 1% of the total number of events) that were not included in the EPA were added from other sources: operational services in charge of hazard management and a citizen science dataset: data-avalanche.org.
From the entire deposit volume data set available since the beginning of the 20 th century (Kern et al., 2020), our study only 50 uses data covering the period from November 2003 to June 2018 (15 "full" avalanche seasons). This limits the biases and inaccuracies induced by the estimation method which were much higher earlier due to less sharp topographical references available to the EPA operators and a less standardized observation protocol until the 2000s (Kern et al., 2020). Thus, our study includes 1491 single avalanche events and associated deposit volumes registered in 77 distinct paths (Fig. 1).
To analyse the possible links between path morphology and deposit volumes, we first computed the interannual mean 55 deposit volume in each of these paths. Then, the same operation was done for both the winter (avalanches that occurred between 1 st of November and 29 th of February) and spring (avalanches that occurred from1 st of March to 31 th of May) subseasons. Eventually, in order to investigate the potential influence of avalanche activity on deposit volumes, we evaluated the interannual mean number of avalanches per year and path, including within the computation snow avalanche records for which deposit volumes could not be estimated. Seasonal (Nov-Feb and Mar-Jun) occurrence rates were also evaluated. 60 The data from two weather stations handled by Météo-France and located at elevations of 1715 m a.s.l. and 2740 m a.s.l. in Bessans, respectively ( Fig. 1), was analyzed in order to determine climate conditions having locally prevailed over the study period. This showed that the depth of the local snowpack regularly exceeds 50 cm at 1715 m a.s.l. and 200 cm at 2740 m a.s.l.. The winter (Nov-Feb.) season is characterized by a cold mean air temperature (-4°C at 1715 m a.s.l., -5.5°C at 2740 m a.s.l.), with heavy precipitation that nearly only fall in the form of snow but the depth of the snowpack remains relatively 65 thin (90 cm at 2740 m a.s.l.). By contrast, the spring season is characterized by higher mean air temperatures (3.5°C at 1715 m a.s.l., -2°C at 2740 m a.s.l.) and the occurrence of significant daily warm spells (daily mean air temperature up to 25°C at 1715 m a.s.l), which favors the occurrence of rain on snow events and wet snow avalanches. The mean daily fresh snowfall is twice lower than during the winter season, but, on average, the snowpack remains thick (170 cm).

Avalanche paths morphology and related volume samples
For each studied EPA path (Fig. 1), a large set of morphological variables was calculated from a metric accuracy DEM. We first defined the presumed preferential avalanche flow path (PPFP) within the path. The PPFP is the simplified thalweg of 80 each path. For each PPFP, the length, the min, max and average slope were calculated as well as the min, max and average elevation and the vertical drop. From the whole extent of each EPA path, surface area, min, max and average slope were calculated (Supplementary Table 1). In addition, the general orientation of each path was evaluated as a categorical variable

Statistical analyses
One-way analyses of variance (ANOVA) were first conducted to evaluate the significance of the partition into two subsamples (winter & spring deposits, and "high frequency paths" with more than 2 events per year versus "low frequency 95 paths" with less than 2 events per year). In other words, we investigated whether or not the variability of deposit volumes by path morphology varies i.) according to the season, and hence, prevailing snow and weather conditions and related types of avalanche activity, ii.) according to path's mean activity.
To shed light on the control of deposit volumes by paths morphological characteristics, Spearman correlation coefficients (r) were first calculated between each descriptive variable of path morphology and the annual (Nov-Jun.), winter (Nov-Feb.) 100 and spring (Mar-Jun.) deposit volume data series. This coefficient was chosen rather than the more classical Pearson one because the statistical distributions of deposit volumes are asymmetric, with extreme values strongly departing from the mean (Fig. 2). With a dataset of 77 individuals (one mean deposit volume per avalanche path), the relationship is significant at the 0.05 level if the Spearman coefficient is greater than 0.25 in absolute value.
Stepwise linear regressions were undertaken in order to determine the combination of morphological variables that best 105 explain the variability of mean deposit volumes. This was done first using the complete database of 77 paths and 1491 https://doi.org/10.5194/tc-2021-103 Preprint. Discussion started: 15 April 2021 c Author(s) 2021. CC BY 4.0 License. deposit volumes. Afterwards, distinct linear models were fitted i) on the 649 snow avalanche deposits recorded in 68 of these paths during the Nov-Feb. winter season, and ii) on the 842 snow avalanche deposits recorded in 73 of these paths during the Mar-Jun. spring season. With a stepwise procedure, the set of predictive variables retained is selected by an automatic sequence of Fisher F tests. Starting from an initial null model with no covariates and then comparing the explanatory power 110 of incrementally larger and smaller models, it combines forward selection and backward elimination. Forward selection tests the variables one by one and includes them if they are statistically significant based on the p-value of the F statistics, while backward elimination starts with all candidate variables and tests them one by one for statistical significance, deleting any of them that are not significant on the basis of the p-value of the F statistics. We used the classical 0.05 and 0.01 probability thresholds for forward selection and backward elimination, respectively. However, before running the stepwise regression, a 115 variable preselection was carryout out. This was made to avoid too much redundancy within potential predictors, which can lead to masks and numerical instabilities during the stepwise selection. To this aim, Pearson's correlation ρ was calculated between all pairs of potential morphological variables (Supplementary Table 2). Among the strongly correlated variables (ρ > 0.8 and p < 0.001), we kept as potential predictor only the one with the highest marginal correlation with deposit volumes.
Eventually, in order to account for potential nonlinear relationships between morphologic variables and snow avalanche 120 deposits, more flexible neural networks models were constructed, again both for the full data set and the winter/spring subseasons. For the three data sets, the full set of 16 morphological variables previously presented was used as potential covariates (8 quantitative variables and the 8 binary variables corresponding to orientations). Both standard 3-layers and advanced 8-layers (deep learning) neural networks were developed. Models were trained using 70% of the data randomly selected from the analyzed sample of paths/mean deposit volumes with the Levenberg-Marquard algorithm (Moré, 1978). 125 carried out with 15% of the data and model testing was carried out with the remaining 15% of the data. This Validation was typical machine learning approach allows both progressive improvement of the model with cross validations and limitation of overfitting. To account for the variability of obtained relations, a 100 bootstrap iterations was conducted, varying data partition into calibration/validation/test subsamples and initial conditions for the Levenberg-Marquard algorithm.

High spatiotemporal variability of deposit volumes and avalanche activity
An important spatial variability in deposit volumes is observed, with the path mean deposit volume over the study period varying from 1400 to 49,800 m 3 , the "mean of the mean" path deposit volumes being 15,100 m 3 (Fig. 2). If one looks further 135 in the distribution of mean deposit volumes, the sample mean and dispersion is significantly higher (one-way ANOVA p = 0.010) for winter season (path mean deposit volume = 18100m 3 ) than for spring season (path mean deposit volume = Bonneval-sur-Arc and Bessans show more avalanches than those at Lanslevillard, in the lowest elevation part of the study area. Avalanche activity is stronger in spring season (860 avalanches with documented deposits) than in winter season (631 avalanches with documented deposits). Considering the frequency indicates that the high frequency paths show larger mean deposit volumes (16800m 3 ) than low frequency paths (12900m 3 ). This observation is validated by a one-way ANOVA (p = 0.029). A significant relationship exists between winter deposit volumes and the mean annual frequency of each path (r = 150 0.35; p < 0.001).

Relationships between path morphology and deposit volumes
Avalanche deposit volumes are significantly correlated with several morphological variables and with a South-East orientation. For the full (annual) data set, the best pairwise correlations are with path mean elevation (r = 0.51), surface area (r = 0.48) and max elevation (r = 0.46). However, a clear distinction between the two seasons is observed (Table 1). For the 155 winter season, the correlations are significant (r > 0.25) between deposit volumes and seven morphological variables among which mean elevation and surface area are the best predictors. The winter deposits reveal a significant correlation with an East orientation. In addition, deposit volumes are also influenced by frequency, through the positive correlation of frequency with min slope (r = -0.24 p < 0.05) and max slope variables (r = 0.27; p < 0.05). By contrast, correlations between path morphological variables and deposit volumes are significant in spring season for 5 variables only, and correlations are lower. 160 Slopes variables are not significantly correlated with deposit volumes for the spring season.
Stepwise linear regressions (Fig. 2)  non-significant correlation values. By contrast, all selected variables are relative to the path surfaces: min elevation, mean elevation, min slope, max slope, mean slope, surface area and orientation. For the annual data set, the retained model 165 includes three positive significant morphological variables increasing the deposit volumes: mean elevation and North and South-East orientation. However, R 2 remains low with only 30% of the deposit volume variability explained by these variables. The seasonal stepwise linear regression shows interesting differences between the two seasons. The retained models include four significant morphological variables increasing the deposit volumes for winter season: mean elevation and east, south-east and west orientation. Only one positive variable is retained in the model for spring season: mean 170 elevation. Resulting R 2 is higher for deposits in winter (R 2 =0.41) than for spring deposits (R 2 = 0.15), which remains particularly low (Fig. 2).
Neural networks significantly enhance the predictive power with higher R 2 values between the full set of morphological variables and deposit volumes, for both annual and seasonal data sets. With the 3-layer models, depending on the bootstrap iteration, best 2.5% of the models reach R² of 0.46 (Supplementary Table 3), and, again, best fit is obtained for winter season 175 (R²=0.57 for the 2.5% best models, versus 0.37 for the spring season). Switching to the even more flexible deep-learning based 8-layer models even enhances these values to R²=0.76 and 0.54 for the 2.5% best models in the winter and spring season, respectively. However, on average on the 100 bootstrap iterations, retained neural network models do not reach high predictive power. For instance, the median R² value among the 8-layer models fitted on the annual sample is 0.28 only. By contrast, as soon as a reasonably good agreement between observations and prediction is obtained (Figure 2 on which 180 models providing R²=0.61, R²=0.58 and R²=0.34 are showed), discrepancies are low all over the calibration, validation and test samples, with a nearly unbiased, Gaussian-like, distribution of residuals (Supplementary figure 2). This all confirms the weak but significant control of deposit volumes by morphological variables. The increment in predictive power with regards to linear models also suggests that this control grounds at least partially on non-linear relationships. Linear regression did not improve the relationships much, with no more than 30% of the annual deposit volumes variability explained by a combination of morphological variables, increasing to 41% for winter deposit volumes variability but 215 decreasing to 15% of the spring deposit volumes variability. In the three cases mean elevation is retained as a relevant predictor. Orientation variables are only retained by the annual and winter deposits model. Winter deposits show a strong positive relationship with East, Southeast and West path orientations. This indicates how important the solar radiation and/or the path positioning in respect to the prevailing wind direction may be for the properties of the snowpack, later influencing the volume deposits. However, there is no reliable data on wind direction or speed at the scale of a massif, so it is not 220 possible to precisely characterize the wind contribution to our study. The use of a more flexible neural network approach leads to significant improvements, notably with some deep learning-based models, but, overall, the power of morphological variables to predict snow avalanche deposit volumes remains somewhat limited. Grounding on these results, we suggest that path's morphology controls deposit volumes significantly but weakly, and at least partially on the basis of non-linear relationships. This could be confirmed (or not) with further studies in different mountain areas where topography and/or 225 avalanche activity regime is different.
Mean avalanche frequency appears as an important factor to explain these results. Indeed, slope variables partly influence the annual frequency and indirectly the deposit volumes. High frequency paths (> 2 events per year) present a steeper slope than low frequency paths (< 2 events per year) paths: 40° and 37°, respectively. Also, high frequency paths show larger path's mean deposit volumes (16,800m 3 ) than low frequency paths (12,900m 3 ). These somewhat counterintuitive results are in line 230 with those of Sovilla et al. (2010) that highlighted complex relationships between slope, avalanche activity and deposit volumes.
We interpret the weak relationship between mean path deposit volumes and morphological variables to be partly due the predominant control of avalanche activity by snow mechanical behavior. This especially occurs trough the mechanical thresholds involved in avalanche triggering processes (Gaume et al., 2012, Li et al., 2020, which are primarily related to 235 snow depth and stratigraphy in the release area as well as to the slope and ground roughness in the release area. This may explain why snow avalanche deposit volumes do not seem that much affected by avalanche path size, for example. Also, mechanical release thresholds may explain the significant variations we observed in the control of winter or spring deposits by path morphology, since, from one season to another, different snow depths and stratigraphy may lead to release for different slopes / elevations as soon as the critical stress value is exceeded. Note that we did not take into account in our 240 study the roughness of the ground, which was not possible to accurately document over the full sample of paths, but this could be an insightful perspective for further work. More widely, we speculate that the weak relationship between volume and morphological variables may be due to an important control by climate conditions since variations in snowpack characteristics determine avalanche triggering and flow properties (Steinkogler et al., 2014, Kölher et al., 2018, and notably snow entertainment and deposition during the flow, which ultimately determines deposit volumes. Recent climate change thus impacts snow avalanches frequency, magnitude, seasonality and localization, leading, e.g., to an increasing proportion of wet snow avalanches documented in the French Alps between 1958 and 2009 (Naaim et al., 2016). Our approach should therefore now be extended to simultaneously take into account the control of deposit volumes by morphological and meteorological variables, and how these controls evolve as climate change goes on. Such an approach combining morphological and climatic variables has, for example, already been 250 applied in Svalbard (Eckerstorfer and Christiansen., 2011) or for debris-flows in the French Alps (Jomelli et al., 2019).

Data Availability
The whole EPA avalanche data is freely availed at https://www.avalanches.fr/. The dataset of mean deposit volumes and morphological variables analyzed in this study can be requested to HK.