Hysteretic dynamics of seasonal snow depth distribution in the Swiss Alps
 For many snow-hydrological studies, it is essential to estimate the spatio-temporal distribution of snow within a given catchment. One approach is to consider all the different processes that entail a heterogeneous spatial distribution of snow, such as differential accumulation, redistribution and ablation. These processes and their complex interactions with, e.g., topography, vegetation and meteorology have been intensively studied and described in a large number of publications [e.g., Elder et al., 1991; Blöschl, 1999; Liston, 1999; Grayson and Blöschl, 2000; Deems et al., 2006]. Snow hydrological models attempt to reproduce the snow cover development by accounting for these processes, using either deterministic or conceptual techniques. A main objective of snow-hydrological modeling is to estimate the total amount of snow water equivalent (SWE) and to predict the snow melt water run-off originating from the snow. There exists a large range of snow model types, from physical deterministic spatial distributed models [e.g., Lehning et al., 2006; Liston and Elder, 2006] and lumped snowmelt models including the concept of the areal depletion curve [Luce et al., 1999] to pure statistical models using sums of correlated gamma distributed variables [Skaugen, 2007].
 In this study, we attempt to understand the spatio-temporal distribution of snow depths as one stochastic process that unites all different physical processes that drive the observed snow pattern. In this context, we consider the formation/depletion of the snow cover over the Swiss Alps as a growing/diminishing surface. This approach allows transferring analytical methods to snow distribution modeling that have actually evolved in statistical physics to characterize the variability of surfaces or interfaces and their development [see Barabási and Stanley, 1995]. In this paper, focussing on the spatial variability of snow depth during the course of snow accumulation and depletion, we derive a statistical description that should ideally a) allow forecasting characteristics of the snow depth distribution that are relevant to catchment-scale models and b) identify similarities in the seasonal progression of such characteristics between years. In contrast to previous snow studies, we describe the spatial variability of the snow depth distribution in terms of the mean snow depth instead of date, which enables us to achieve the above-mentioned premises a) and b) at the same time. This approach has already been applied to soil moisture distribution studies [e.g., Crow and Wood, 1999; Famiglietti et al., 2008]. So far our analysis is restricted to snow depth data from a set of flat alpine field sites. However, we argue that the conceptual framework outlined here can be applied to a broader range of length scales and to different types of terrain.
2. Data and Methods
 This study is based on snow data from the Swiss network of alpine automatic weather stations IMIS [Rhyner et al., 2002]. Snow depth was measured using the ultrasonic SR50 sensor (Campbell Scientific Inc.), which features an absolute accuracy of ±0.03 m under real field conditions. All data has been aggregated to daily measurements by selecting the snow depth values at midnight, checking for plausibility, and then correcting single missing/faulty values by interpolation where necessary. We only used data from those 77 IMIS stations that provided complete data sets on daily snow depth readings for six consecutive winter seasons between 2001 and 2007. The stations are spread over the entire range of the Swiss Alps, an area of approximately 100 km by 300 km (http://www.slf.ch/alpen-info/alpeninfo.html). The selected sites cover an elevation range from 1610 m to 2990 m a.s.l., where altitudes are roughly normally distributed with a median of 2310 m a.s.l. and where about 75% of the stations are located above the regional treeline. IMIS station sites were carefully selected in flat, open terrain with as little wind influence as possible. A thorough evaluation procedure ensured that sites were as representative as possible with respect to regional snow conditions in flat terrain. This selection process implies that our analysis does not cover snow depth variation with respect to terrain effects depending on aspect and slope.
 Let xi(t) be a snow depth reading at station i for day t. We then characterized the spatial variability of snow depth as the standard deviation:
where denotes the mean of xi at a day t and i indexes the number of measurements at the 77 IMIS stations. The methodological framework attempted to describe the evolution of the snow cover as a growing surface. Our approach was inspired by concepts presented by Barabási and Stanley  and by a preceding study of the snow surface roughness on a micro scale (L = ∼10−1 m)) [Löwe et al., 2007]. Although the processes forming the spatial distribution of snow depths at micro and macro scale are completely different, we tried to adopt the descriptive framework of growing surfaces that was successfully applied to the micro scale. As we only had data from 77 stations, a reliable scale analysis was infeasible. However, given daily data over many years, we could investigate β, the exponent of growth, which determines the dynamic evolution of the growing surface at the given scale of the entire Swiss Alps (L = ∼105 m). For Ballistic Deposition at micro scale, β is generally related to the time of growth t as:
for times where the variability of the surface is increasing before saturating at a constant value [Barabási and Stanley, 1995]. Assuming that the deposition or melting rate of snow is constant, increases or decreases linearly with time, i.e.
This assumption is partly applicable to snow surface growth at the micro scale [Löwe et al., 2007], but it is certainly not true for the development of the snow cover at the scale of the Swiss Alps, where assumedly precipitation/melting rates are not constant with time. However, for the accumulation phase we hypothesize that the form of equation (2) can be conserved if we replace t by t′ = , i.e. σ(xi, ) ∼ β. This led us, in contrast to many previous studies, to analyze the evolution of snow cover variability σ(xi, t) in terms of instead of t:
 Figure 1 displays σ() over from first of September 2006 to end of August 2007. Each point of the trajectory represents one day of the year. Corresponding plots for the other 5 winter seasons show a similar behavior although the maximum mean snow depths ranged from 1.53 m to 2.42 m. The characteristics of the trajectories for each season are summarized in Table 1 and are discussed in section 3.1 and 3.2.
Table 1. Parameterizations of the σ(i) Trajectories During Accumulation and of the Uniform Melting Curve Periods for Six Winter Seasonsa
|2006/07||0.839||0.995||0.63 · (1 − e)||0.988||1.04||0.99|
|2005/06||0.819||0.988||0.73 · (1 − e)||0.983||1.04||0.99|
|2004/05||0.844||0.987||0.64 · (1 − e)||0.972||1.08||1.03|
|2003/04||0.837||0.992||0.86 · (1 − e)||0.966||1.07||1.01|
|2002/03||0.838||0.971||0.67 · (1 − e)||0.975||1.19||1.08|
|2001/02||0.869||0.987||0.81 · (1 − e)||0.976||1.04||1.0|
 The trajectory of σ() shows a quasi-linear course during the period of accumulation (Figure 1, circles) until maximum is reached (peak of winter). After the peak of winter, the trajectory turns by 180° and retreats with decreasing along the same path until a distinct turning point (σ(xi)M) is reached (Figure 1, vertical gray bar). From σ(xi)M on, the trajectory describes a clearly different path. During this time of the winter, snow depletion causes a decrease in while σ() remains approximately constant, resulting in an almost horizontal path of the trajectory (black dots). However, later in season the trajectory bends downward to ultimately reach the origin (complete melt-out). Note, that this general behavior was also found with data subsets, e.g restricted to a small number of stations, to meteorological subregions or to particular elevation bands. However our data set limited to 77 stations did not allow for any further respective analysis. In the following discussion, we will refer to the phenomenon of two separate paths of the σ() trajectory during accumulation and depletion period as the hysteresis effect of Alpine seasonal snow depth distribution.
 Corresponding trajectories for soil moisture distribution did not show a distinct hysteresis effect, but instead feature a convex upward form [cf. Famiglietti et al., 2008; Pan and Peters-Lidard, 2008].
3.1. Accumulation Period
 Snow depths during the accumulation period are dominated by both snow precipitation and settling. Following the concept of growing surfaces [Barabási and Stanley, 1995], the time series of σ(i) during accumulation were fitted to the equation σ(i)acc. = (Figure 1, solid line). The exponent βacc. of the best-fit power law ranges from 0.869 to 0.819 (Table 1) and is thus surprisingly constant between years. Correlation coefficients close to 1 (Table 1, Racc.2) indicate how well the data follow a power law. During the accumulation phase, βacc. is practically identical for periods with/without snowfall. During snowfall, σ(i) moves along the trajectory from the origin outwards (Figure 1, circles with vertical bar). In periods without precipitation, however, σ(i) moves in the opposite direction due to snow settling (Figure 1, circles with horizontal bar).
3.2. Ablation Period
 We separated ablation and accumulation periods by means of the turning point σ(xi)M. σ(xi)M can be easily determined from the data by seeking the first point of the trajectory with decreasing and blocal < 0.02, where blocal is the local slope of (t), i.e. the slope of a linear fit through 7 consecutive points (t) to (t + 6). The path of the trajectory during the ablation period resembles the respective course for uniform melting, which corresponds to the simplistic assumption of a constant melting rate for all sites. For comparison, we simulated the hypothetical trajectory for ideal uniform melting conditions (σunif.m). Starting from the initial distribution of xi at σ(xi)M, we removed 0.01 m of snow per time step at every snow station until all sites were snow-free (Figure 1, gray stars, labeled as A. At the beginning of the ablation period, the observed trajectory (σmeas.m) follows σunif.m quite well. However, for all 6 years σmeas.m diverges to a path above σunif.m at some stage during the melting process. We quantified this offset by calculating the ratio:
where the integral is calculated from 0 to σ(xi)M. In 5 out of 6 years, 1.04 < rtotal < 1.08, indicating that the melting behavior at our set of stations is not far from the uniform melting assumption (Table 1). However, in 2002/03, the offset is significantly larger (rtotal = 1.19).
 Applying the concept of growing/diminishing surfaces revealed a simple method to clearly distinguish between different phases during the formation and depletion of a seasonal alpine snow cover that are similar between years. While precipitation and settling are obvious features during the accumulation periods, a distinct and abrupt transition between snow settling after the peak of winter and the onset of melting may not be obvious from snow depth data alone. Moreover, analyzing the σ(xi)M trajectories reveals incidences during which the distribution of snow depth clearly differs from the idealistic behavior of uniform melting. In the following sections we discuss the different features of the trajectories in greater detail.
4.2. Ablation Period
 We have compared the σ(i) trajectories during the ablation periods to a hypothetical trajectory for uniform melting conditions. Values for rtotal > 1 indicate that the assumption of one common melt rate for all sites is too simple to characterize the snow depth distribution during melt-out (Table 1). Our data reveal that at the beginning of the ablation period σmeas.m follow σunif.m quite well. However, at some point during the melting process σmeas.m escalates to values distinctly above σunif.m. Such incidents can be attributed to substantial precipitation events. Compared to the local slope for snowfall during the accumulation period, precipitation events during the melting period may feature a local trajectory with a much steeper, almost vertical path. A possible explanation is mixed precipitation in spring with rain at lower sites and snow at higher sites. We tested whether after such events the trajectories again follow a path corresponding to the assumption of uniform melting. By recalculating the hypothetical trajectory for uniform melting conditions starting from after the last significant precipitation event before melt-out (Figure 1, gray stars labeled as B). In analogy to rtotal (equation (5)), we determined the ratio rlast between the recalculated hypothetical trajectory and corresponding observed data. For 4 out of 6 years, this ratio was within 0.99 and 1.01 (Table 1). It thus appears as if the uniform melting assumption is at least temporarily appropriate, i.e. in precipitation-free sub-periods. While for 2004/05 rlast = 1.03 is yet reasonable close to 1, it is again winter season 2002/03 that features the highest deviation (rlast = 1.08). However, even in this particular winter we obtain a rlast of 1.00 if it is calculated from the final 30 days before melt-out only.
5. Extending the Hysteresis to a Broader Range of Terrain Conditions
 It is important to note that these above-discussed results were deduced from snow depth data at flat sites over a limited range of elevations (1610 m to 2990 m a.s.l.). It may be astonishing that the snow depth distribution during ablation period at these sites partly follows the characteristics of uniform melting. However, the observed characteristics will certainly become more complex, if additionally including data from sloped terrain and/or from a larger range of elevations. The data from melting period 2002/03 to a certain degree sticks out of the rest of the data. In particular rtotal = 1.19 is significantly higher compared to other years because the path of the trajectory right after σ(xi)M is slightly bent upwards instead of being horizontal as is the case in the other years (specific trajectory not shown). The spring season in 2003 featured extraordinarily warm temperatures [Schär et al., 2004]. We speculate that these conditions may have contributed to significant differences in melt rates between sites, which would cause an increase in σ(i) after the onset of melting (i.e. after σ(xi)M) rather than σ(i) remaining at a constant level. This specific behavior may help predict what trajectories to expect if extending the analysis to snow depth data from a larger range of terrain conditions, especially with respect to elevation, slope, and other length scales (typical distance between sites). Such an extension would allow characterization of the snow distribution within an entire mountainous catchment.
 If snow depth data from sloped terrain were included, we would certainly face a larger range of melt rates between sites. The consequences have been outlined above where we discussed the trajectory of the spring season 2003. Under such conditions, we expect a typical trajectory to look like that displayed in Figure 1 (gray triangles). It seems possible to find a parameterization or a numerical procedure of melting that describes such a trajectory even if the assumption of uniform melting no longer holds. Data presented by Pomeroy et al.  and the theoretical considerations of Essery and Pomeroy  support this view. In the same context but contrary to corresponding soil moisture distribution studies that indicate significant scale effects [Crow and Wood, 1999], we do not expect a change of the general hysteretic form of trajectories for snow distribution on catchment scale.
 It may be more challenging to include data from lower elevations (<1500 m a.s.l.). At such elevations, a clear separation into one accumulation phase and a subsequent melting phase per season is no longer typical [Rohrer, 1992]. Instead snow at such elevations can experience temporary melting periods any time during the winter, such that accumulation and ablation periods may alternate several times during the snow season.
 In summary, the conceptual framework outlined in this paper allowed characterizing the seasonal progression of the snow depth distribution in the Swiss Alps in a statistical way that showed a clear link to the underlying physical phenomena. We are thus confident, that the method can be extended to data sets covering a broader range of length scales and topography that better represent the characteristics of single mountainous catchments, at least of catchments that do not extend to below subalpine elevations. Future studies could verify this hypothesis and apply the approach used here to incorporating stochastic means in modeling snow distribution in Alpine terrain.
 We thank H. Löwe, C. Manes and M. Guala for helpful discussions and the reviewers for their constructive comments.