The spatial variability of the snow cover is driven by wind-induced snow transport processes. The main aim of this study is to investigate the link between the scaling behavior of snow depths and the wind-induced processes driving the spatial structure of snow depths. We used very high-resolution atmospheric and snow transport simulations to compare their spatial characteristics with snow depths obtained from airborne and terrestrial laser scans. The directions of the strongest autocorrelations of snow depths and wind velocities were predominantly perpendicular to the modeled local flow direction, independent of whether more homogeneous lee slope loading in one area was considered or the formation of drifts in another area. In the case of homogeneous lee slope loading, the same direction of anisotropy was only found in the modeled preferential deposition of precipitation as for both wind and snow distribution. This deposition process appears to dominate homogeneous lee slope loading. In contrast, saltation-driven processes seem to dominate the formation of drifts and snow dunes. Furthermore, the fractal analysis suggests similar scale breaks for the modeled wind velocity field and measured snow depths defining the upper scale of landscape smoothing through snow.
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 The spatial heterogeneity of winter snow cover is governed by wind-induced snow transport processes, consisting of saltation, suspension, and the preferential deposition of precipitation [Lehning et al., 2008]. The driving force for these processes is the air flow within the atmospheric boundary layer, which is shaped by the local terrain. The topography is continuously modified by snow deposition during the winter. While saltation acts on a small scale of meters, suspension and the preferential deposition of precipitation act on larger scales of tens to hundreds of meters [Lehning et al., 2008; Dadic et al., 2010a; Mott and Lehning, 2010]. Thus the different processes shape the spatial distribution of the snow depths and modify the terrain on various scales. Information on the spatial variability of snow cover is crucial for estimating and assessing the magnitude and timing of snowmelt [Pomeroy et al., 1997; Luce et al., 1998; Grünewald et al., 2010], as well as for forecasting avalanche danger after severe storm events [Birkeland et al., 1995; Schweizer et al., 2003]. However, the spatial structure caused by process interactions on a variety of scales is difficult to analyze.
 One approach to describe the spatial structure of snow depths is fractal analysis, which is based on the self-similarity of properties over multiple scales. This approach was introduced into snow science after investigations of fractal characteristics of the spatial distribution of snow depths from manual probings of snow in low-relief terrain of Russia lowlands, Canadian prairies, and arctic environments [Shook and Gray, 1996; Kuchment and Gelfan, 2001]. Other studies used a more detailed data set of snow depths gained from airborne laser scans [Deems et al., 2006; Trujillo et al., 2007] to analyze the fractal characteristics of snow depths on Colorado mountain sites. These studies observed a scale-break for snow depth in the order of tens of meters and a stronger autocorrelation before the scale break than after it. They found an agreement between the scaling characteristics of the snow cover and vegetation due to a break in scaling behavior at a similar distance. Additionally, Trujillo et al.  found a link between the scale break distance and the degree to which areas are wind-dominated. Deems et al.  and Trujillo et al.  also noticed that the anisotropy obtained for snow depth structures are mostly perpendicular to the prevailing wind direction. A recent study (M. Schirmer and M. Lehning, Persistence in intra-annual snow depth distribution: 2. Fractal analysis of snow depth development, submitted to Water Resources Research, 2011) used a set of high-resolution snow depth measurements obtained from terrestrial laser scans (TLS) in an Alpine catchment (Wannengrat area, Switzerland) to describe snow depth structures caused by individual storms throughout one winter season. They showed that fractal parameters are able to distinguish snow depth structures in wind-protected and wind-exposed areas and to describe the structure of snow depth change during more and less wind-influenced snowfall periods. They found that the snow depth structure at the time of peak accumulation inherited the spatial characteristics of the snow depth distribution caused by one type of storm. Furthermore, they interpreted the scale break distances as the roughness scale of summer terrain, which is modified by the overlying snow cover but still remains dominant for driving snow-deposition processes. In the same study (M. Schirmer et al., Persistence in intra-annual snow depth distribution: 1. Measurements and topographic control, submitted to Water Resources Research, 2011), a very high interannual consistency of the snow depths was found at the time of peak accumulation for 2007/2008 and 2008/2009. Such a strong interannual consistency for two winter seasons was also found by Deems et al. .
 However, all these earlier studies on fractal properties based their analysis of the strong relationship between the spatial structures of snow depths and the air flow on data obtained from single weather stations. None of them used atmospheric and snow drift modeling for a more precise scaling analysis of the flow features and wind-induced snow transport processes. In this study, in addition to examining the relationship between anisotropic scaling in snow depth fields, we examine the spatial pattern of wind speed modeled at very high resolution. While working on the same field site as Schirmer and Lehning (submitted manuscript, 2011), additional measurements from airborne laser scans are presented in this study.
Mott et al.  studied the individual flow features in the Wannengrat area acting as the driving forces for snow transport processes and therefore the formation of diverse snow-deposition patterns, but no scaling analysis was attempted. The drift simulations indicated that the smaller-scale deposition patterns are mainly governed by saltation/suspension processes, while the more homogeneous lee slope loading is mainly caused by the preferential deposition of precipitation. In the current study we aim to confirm the findings of Mott et al.  and complete the results of Schirmer and Lehning (submitted manuscript, 2011) by providing a process-oriented scaling analysis of the seasonal snow cover. We therefore examine modeled flow fields and snow transport processes [Mott et al., 2010] for fractal scaling characteristics as major driving factors for snow accumulation and redistribution and relate them to snow depths at time of peak accumulation and to snow depth changes after a specific storm event, using Airborne and Terrestrial Laser-Scanning data. Since the local flow field is strongly modified by the terrain and influences the snow depth structure, the flow field can be expected to show similar scaling characteristics as snow depth. Our investigation area is above the tree line, so we did not need to consider vegetation effects [Deems et al., 2006], and could focus on analyzing the wind-induced processes as the main driving processes.
2. Methods and Data
 Measured and modeled data sets used for the scaling analysis are summarized in Table 1. In the following we give a brief discussion about measurements, models, and scaling analysis methods used in the current study.
Table 1. Data Input for the 2-D Autocorrelation and Variogram Analysis
 The study site is in the Wannengrat area near Davos, Switzerland (Figure 1). The site is located above the local tree line at altitudes ranging from 2000 to 2658 m a.s.l. and is equipped with a high-density network of meteorological stations consisting of seven permanent weather stations and additional 17 mobile stations [Mott et al., 2010] during winter 2009/2010. Two Airborne Laser-Scanning (ALS) campaigns were conducted in the area at the time of peak accumulation in 2007/2008 (HSpeak 0708) (Figure 1 and Figure 2e) and 2008/2009 (HSpeak 0809) (Figure 2a) using a helicopter-based technology [Skaloud et al., 2006]. Furthermore a series of 12 Terrestrial Laser-Scanning (TLS) measurements were performed to obtain snow depth changes (dHS) after snowstorm events during winter 2008/2009 and 2009/2010 [Mott et al., 2010]. The accuracy of the ALS data set was checked by comparing it with tachymeter survey and TLS measurements [Grünewald et al., 2010]. A detailed discussion of the setup and accuracy of the TLS in the Wannengrat area can be found in the work of Grünewald et al. . Compared to the tachymeter survey, they found a mean deviation in z-direction of less than 5 cm for the ALS and less than 4 cm for the TLS. A comparison between TLS and ALS gave a mean deviation in z-direction of 10 cm. Although Grünewald et al.  showed that differences between the ALS and TLS measurement techniques are locally high (tens of centimeters), results from a variogram analysis are nearly identical if ALS and TLS data obtained at the same date (time of peak accumulation 2008/2009) are analyzed (not shown). Therefore the spatial variation in accuracy of ALS and TLS data do not have a significant influence on the scaling analysis.
 A general discussion of how to measure snow depth with TLS is given by Prokop , Prokop et al. , and Schaffhauser et al. . The two ALS data sets (HSpeak) were used for a spatial autocorrelation analysis (Table 1), which was performed for two subareas, the Bowl (0.18 km2) and the NE slope (0.11 km2) (Figure 1). One data set obtained from TLS after the first storm event of the accumulation season 2008/2009 (P10809) (Figure 3a) and one ALS data set (HSpeak 0809) were subjected to a variogram analysis, which was done only for the NE slope (Table 1). Owing to the limited field of view caused by topographic shadowing, the area measured by TLS is restricted to 76% of the NE slope (Figure 3a). Therefore only the area covered by TLS measurement points was used for the variogram analysis.
 A recent study (M. Schirmer et al., submitted manuscript, 2011) analyzed the interannual consistency of the HSpeak data sets obtained from TLS for 2007/2008 and 2008/2009. For the NE slope and the Bowl correlations of r = 0.97 and r = 0.93 between the two HSpeak situations were found. Therefore the snow depth distributions of these 2 years feature very similar spatial structures at the time of peak accumulation.
2.2. Model Setup of the Flow Field and Drift Simulations
 As input for the current scaling analysis we used results from atmospheric and snow transport modeling presented by Mott et al. . In the following we will give a summary of the model setup. A more detailed discussion of setup and validation of the models Advanced Regional Prediction System (ARPS) and Alpine3D is given by Mott et al. , where the snow transport processes during the individual storm periods of the accumulation seasons 2008/2009 and 2009/2010 are also discussed.
 Mean flow fields were calculated with the nonhydrostatic and atmospheric prediction model ARPS, which is based on compressible Navier-Stokes equations describing the atmospheric flow [Xue et al., 2001]. The ARPS model was developed at the Center for Analysis and Prediction of Storms (CAPS), University of Oklahoma, and configured according to Raderschall et al. . Since snow deposition patterns are mainly shaped by the NW storms in the Wannengrat region and permanent weather stations clearly show a strong time consistency of prevailing wind directions during those storms [Mott et al., 2010], mean flow fields were only computed for synoptically induced NW situations. In total five wind fields were modeled, which were shown to be representative for NW storms [Mott et al., 2010]. The model validation showed that the spatial distributions of wind directions and wind velocities measured at 24 meteorological stations were well captured by the atmospheric model [Mott et al., 2010]. The three-dimensional and terrain-following ARPS grid has a horizontal grid resolution of 5 m and a vertical grid resolution ranging between 0.5 m at ridges and 1.1 at flatter terrain for the first level above ground. The domain size was set to 2.5 × 2.8 km.
 The grid of wind velocities was used as an hourly input for the Alpine3D model [Lehning et al., 2008; Mott and Lehning, 2010] to drive the snow transport module. Within the modular frame of Alpine3D, the snow transport module is coupled to the energy balance module [Helbig et al., 2009, 2010] and the SNOWPACK module [Lehning and Fierz, 2008]. The snow transport module captures three wind-induced snow transport processes, i.e., saltation, suspension [Clifton and Lehning, 2008], and preferential deposition of precipitation [Lehning et al., 2008]. While saltation and suspension processes describe the redistribution of already deposited snow, preferential deposition of precipitation is defined as the spatially variable deposition of precipitation due to topography-induced near-surface modification of flow field [Lehning et al., 2008; Dadic et al., 2010a, 2010b; Mott and Lehning, 2010]. The transport module further allows preferential deposition of precipitation to be computed separately from saltation and suspension. Snow depth changes calculated with all wind-induced transport processes are labeled dHSall in this paper, and those caused only by preferential deposition of precipitation dHSprec. All snow transport simulation runs were performed on a horizontal grid resolution of 5 m using the ARPS grid. The work of Mott et al.  could demonstrate that most of the deposition patterns found at the time of peak accumulation and after NW storms can be explained and modeled with the snow transport model using a few mean wind fields. Nevertheless, the snow transport model did not capture the locally reduced snow deposition which was measured after all NW storm events on the top of the NE slope (Figure 3). This reduced snow deposition area might be explained by very low deposition rates or enhanced erosion caused by a strong local flow, which is not captured by the ARPS model. In addition, deposition immediately after the Chüpfenflue ridge was overestimated because of overestimated speed-up effects in the ARPS simulations [Mott et al., 2010].
 Since one typical storm event (here, northwest (NW) storms) was found to consistently shape the local snow distribution toward the typical peak of winter distribution, we only analyze simulated mean flow fields for a synoptically induced NW situation and snow depth changes caused by a typical NW storm. In the current study, we only refer to dHS calculated for the first storm period in the accumulation season 2008/2009 (P10809) (Table 1). Since the scaling analysis showed similar results for all simulated wind fields, we will discuss the scaling properties for one wind field, initiated with a moderate wind velocity of 4 m/s at 2400 m a.s.l. (vwini=4) (Table 1).
2.3. Autocorrelation and Fractal Analysis
 The measured snow depths (HSpeak 0708, HSpeak 0809) (Figures 2a and 2e), modeled wind velocities (vwini=4) of the first level above ground (Figures 2b and 2f), and modeled snow depth changes (dHSall and dHSprec) (Figures 2c, 2d, 2g, and 2h) were subjected to a spatial autocorrelation analysis (Table 1). Furthermore, based on a variogram analysis, the scale break and fractal dimensions were determined for the measured snow depths (HSpeak 0809), measured and modeled snow depth changes after the first storm period in winter 2008/2009 (dHS and dHSall) (Figures 3a and 3b) and the modeled wind velocity (vwini=4) (Table 1). The spatial autocorrelation analysis was performed for the NE slope and the Bowl, whereas the variogram analysis was done only for the area covered by the TLS measurement points on the NE slope (Figure 3a).
 A detailed description of the variogram analysis can be found in the work of Schirmer and Lehning (submitted manuscript, 2011). How to estimate fractal parameters is discussed by Xu et al.  and Sun et al. .
 Variograms were estimated following equation (1) using 50 log width bins,
where N(h) is the set of point pairs (i,j) in each distance class h [Webster and Oliver, 2007]. Subsequently, a piecewise log-log linear model was fitted to the estimated variogram with model parameters scale break distance L, the slope α before and after L and ordinal intercept β,
where the continuity constraint is
 In the discussion part we only address the scale break distance L and the short-range and long-range fractal dimension Ds and Dl, respectively. The fractal dimensions are calculated following [Sun et al., 2006]
3. Results and Discussion
3.1. Two-Dimensional Autocorrelation Functions
 In Figures 4a–4d, the 2-D autocorrelation functions for snow depths measured at time of peak accumulation 2008/2009 (HSpeak 0809), the modeled change in snow depth (dHSall and dHSprec) and the wind velocity field (vwini=4) are illustrated for the wind-exposed NE slope. The autocorrelation functions for HSpeak 0809 indicate a high anisotropy of snow depth patterns (Figure 4a). The anisotropy exhibit the strongest autocorrelation structure in the northeast-southwest direction with a autocorrelation of r = 0.8 obtained at distances of 14 m. A very similar directional dependency was found for the simulated field of 3-D wind velocity (Figure 4b) and for dHSall (Figure 4c). However, the structure of vwini=4 showed autocorrelations over a longer distance (Figure 4b) than for HSpeak (Figure 4a). For the simulated dHSall (Figure 4c), the pronounced anisotropy coincides with the modeled and observed cornice-like drifts (Figures 2a and 2c). These dominant drifts and the associated anisotropy in the snow depth structure developed perpendicular to the northwesterly wind direction in the NE slope, as modeled by Mott et al. . As discussed above, the model strongly overestimates the snow deposition on the upper parts of the NE slope. These snow deposition patterns do not significantly influence the anisotropy but provoke a strong autocorrelation over a longer distance than shown for HSpeak.
 In contrast, the autocorrelation function of dHSprec (Figure 4d) demonstrated an indistinct anisotropy (Figure 4d). Thus the autocorrelation functions suggest that it is not the preferential deposition of precipitation but rather the saltation and suspension processes that drive the spatial structure of snow deposition on the NE slope.
 The autocorrelation functions for the wind-protected Bowl are illustrated in Figures 4e–4h. We used HSpeak 0708 instead of HSpeak 0809 since a huge avalanche occurred in winter 2008/2009, which substantially altered the spatial structure of snow depths within the Bowl. Owing to the strong interannual persistence of the snow depth data over these 2 years (see section 2.1.) the mixed use of both data sets was thought to be adequate.
 For the wind-protected Bowl the autocorrelation function for HSpeak 0708 (Figure 4e) exhibited the strongest autocorrelation structure in the northwest-southeast direction, but it was still considerably lower than on the wind-exposed NE slope: at a relatively short distance of 3 m, the autocorrelation had already decreased to 0.8 and at distances of 14 m the autocorrelation is only 0.4. Similar anisotropies are found for vwini=4 (Figure 4f) and for dHSprec (Figure 4h). These anisotropies once more developed approximately perpendicular to the modeled prevailing wind direction, which is southwesterly in the western part of the Bowl during NW storms [Mott et al., 2010]. The stronger autocorrelation found in the model can be explained by the horizontal model resolution of 5 m. The resolution of 5 m is still insufficient to resolve the large number of very small-scale narrow channels which control the near-ground flow field and therefore the snow deposition processes which appear to be active on a lower scale than 5 m. The agreement in anisotropies of HSpeak 0708 and modeled dHSprec (Figures 4e and 4h) is probably the result of homogeneous preferential deposition of precipitation in the wind-sheltered Bowl, with snow filling the narrow channels of the summer terrain (Figure 2e), which are mostly aligned normal to the local southwesterly wind direction. Results from Mott et al.  show that zones of decreased wind velocities and downdrafts (negative surface-normal wind velocities) additionally promote deposition of precipitation within the channels due to higher deposition velocities of snow.
 In contrast, for dHSall concentric lines of autocorrelation values indicate no anisotropy (Figure 4g) as no clear drifts in any particular direction were formed because the wind speeds in this area were too low except near the ridge. These results support the findings of Mott et al.  that preferential deposition of precipitation is the dominant process driving the spatial structure of the snow deposition in the Bowl.
 In contrast to Trujillo et al.  who analyzed snow depth structures in the presence of vegetation, our results suggest that there is an anisotropy in the autocorrelation function of snow deposition patterns even for areas with low wind speeds and no snow redistribution but with a locally prevailing wind direction perpendicular to significant topographical features (e.g., narrow channels and ridges) of the summer terrain.
3.2. Omnidirectional Variograms
 In Figure 5, four variograms for measured HSpeak 0809, measured dHS, modeled dHSall and wind velocity are shown for the NE slope. One of the main findings is that the distances of the scale breaks are of the same order of magnitude for all data sets, with values ranging from 16 m to 20 m. The scale breaks of HSpeak 0809 can be interpreted as the upper scale at which the roughness of the summer terrain is persistently influencing the snow depth structure (Schirmer and Lehning, submitted manuscript, 2011). In other words, the smoothing of the snow cover stops at this scale and significant local flow features continue to be formed by the terrain at this scale. Therefore the wind velocity field must show a similar scaling effect as snow depth. Our model results are able to reproduce this important spatial property.
 The variance of modeled dHSall and measured dHS are similar for all scales, especially at the scale break distance. The similar fractal dimensions as seen from the slopes of the variogram curves (Table 2) also indicate that the scaling characteristics of the modeled and measured data are strongly related. The fractal dimensions before the scale break are small for all data sets (Table 2), with Ds values ranging from 2.2 to 2.3. This indicates a low-frequency variation and a dominance of long-range processes below the scale break. After the scale break the fractal dimension range from 2.7 to 2.9, indicating a high-frequency variation of wind velocity and snow depths at larger scales. Thus the flow features are spatially persistent at scales below the scale break and provoke the formation of spatially persistent snow deposition structures at these scales, i.e., the formation of the cornice-like drifts (e.g., Figure 2a). The agreement of the fractal parameters indicate that the spatial fluctuations found in the modeled flow field and the measured snow depth are dominantly influenced by the same terrain roughness scale, which can be identified with the scale break.
Table 2. Break Distances, L, and Fractal Dimensionsa
Short-range fractal dimensions are Ds and long-range fractal dimensions are Dl.
 The scaling characteristics found in this study have an important ramification for snow drift modeling. To capture the whole range of wind-induced snow transport processes, the model resolution has to be chosen according to the scale break of these processes. Modeled and measured data sets suggest that for a wind-blown landscape above tree line the driving processes are active on scales below the scale break. Therefore an adequate model resolution for snow drift modeling would be considerably below the scale break.
 We have explored the links between the scaling behavior of snow depths and the wind-induced processes driving the spatial structure of snow depths. Using results from atmospheric and snow transport modeling, we examined the scaling characteristics of the flow field and the wind-induced snow transport processes in more detail. The results from 2-D autocorrelation functions suggest that, for both subareas, the wind velocity distributions have similar anisotropies to those of the measured snow depth. Mott et al.  investigated the processes driving the spatial patterns of snow deposition found after NW storms by modeling different wind-induced snow transport processes. What could be shown by their model results and is now supported by the 2-D autocorrelation functions is that, for the NE slope, saltation and suspension processes dominate the formation of the two main drifts, i.e., the cornice-like drifts. In contrast, in the Bowl, it was the preferential deposition of precipitation that appeared to drive the spatial structure of the snow depths but again influenced by the local flow field.
 The direction of the strongest autocorrelation of both snow depth and wind velocity was perpendicular to the local prevailing wind direction, as Trujillo et al.  and Deems et al.  also found. Unlike Trujillo et al. , however, we found a directional dependency of snow depth structures even in areas with very low velocities and no formation of drifts, so long as the prevailing wind direction was perpendicular to the narrow channels on the summer terrain. Results from modeling suggest that in the presence of local downdraft zones, the preferential deposition of precipitation led to these surface depressions becoming filled and then dominating the spatial structure of the snow deposition.
 Since topographically modified flow fields strongly influence snow depth structure, the flow field was expected to show a similar scaling pattern to snow depth. This was supported by the model results. Thus wind fields, calculated snow depths, and measured snow depths seem to have a similar fractal behavior, as expressed by the fractal dimensions and scale break distances of the respective variograms. Our snow transport model, implemented in Alpine3D and driven by several atmospheric flow fields, performed well in reproducing the scale break and the variances of the snow depths measured at all scales.
 The scaling analysis showed that a model resolution of 5 m is still insufficient to capture the whole range of scales where driving processes are active on. A higher spatial resolution of atmospheric and snow transport modeling would certainly improve the capability of a scaling analysis.
 The wind simulations were made using the Advanced Regional Prediction System (ARPS) developed by the Center for Analysis and Prediction of Storms (CAPS), University of Oklahoma. Part of the work was funded by the Swiss National Science Foundation and the European Community (FP7 project HYDROSYS). We thank our colleagues who helped during the field work and model development, especially Thomas Grünewald, Nick Dawes, Luca Egli, Mathias Bavay, and Vanessa Wirz. We are grateful to Sylvia Dingwall for language editing and the three anonymous reviewers who helped to improve the paper.