3.1. Feature extraction from DEM
It is essential to characterize the topographic factors properly in order to study the related orographic precipitation patterns. The Swiss DEM with a resolution of 250 × 250 m2 used for feature extraction is illustrated in Figure 2. A variety of topographic indices can be computed from it using convolution filters (Wilson and Gallant, 2000). The set of features is computed on the 1 × 1 km2 grid of the radar.
The extraction of a set of baseline topographic descriptors at different characteristic length scales (degrees of smoothness) was selected. A resulting set of correlated variables characterizes the geometric properties of the surface in the vicinity of every particular spatial location. Two sets of features have been considered (Foresti et al., 2011b):
the first set of features is computed by evaluating terrain gradient at a number of different spatial scales. The set of terrain gradients is computed from the DEM by gradually increasing the bandwidths of the smoothing Gaussian kernels. Gradient vectors will be combined with flow direction to compute flow directional derivatives (FD, see Section '3.2. Motion vector field and flow derivatives'). They are expected to explain stable upslope ascent and convection due to mechanical lifting;
the second set of topographic features was created using a combination of Gaussian smoothing filters. By subtracting two smoothed DEM surfaces obtained with different smoothing bandwidths, the ridges and valleys of different characteristic length scales are highlighted. These features are referred below to as Differences of Gaussians, DoG. DoGs are finite difference approximations of the Laplacian operator (Marr and Hildreth, 1980) which enables the computation of terrain convexity at different spatial scales. These features are expected to explain orographically triggered thunderstorms at the top of mountains.
Two sample features computed at different scales from each group are shown in Figure 3 along with the bandwidths σ of the convolution filter. Table 1 depicts all spatial scales at which features are extracted. Dubbed 2σ values are listed since they better represent the geographical area over which the DEM values are averaged.
Figure 3. Norm of terrain gradient computed at (a) small (2σ = 1 km) and (b) medium scales (2σ = 10.2 km). Differences of Gaussians computed at (c) small (2σ1 = 1.3 km/2σ2 = 1.6 km) and (d) medium scales (2σ1 = 5.4 km/2σ2 = 16.3 km). Feature values are standardized to zero mean and unit variance
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Table 1. Length scales at which DoGs and FD features are derived
|Feature type||Bandwidth of Gaussian filter (2σ, (km))|
3.2. Motion vector field and flow derivatives
Orographic precipitation is a dynamic process mostly driven by the flow field. It cannot be completely described by the static features listed in Section '3.1. Feature extraction from DEM' The motion vector field (flow direction and velocity) derived from subsequent radar images is a way of considering the spatial dynamics of precipitation which is commonly used in Lagrangian extrapolation-based nowcasting schemes (Germann and Zawadzki, 2002; Bowler et al., 2004).
Most of the motion vector field estimation techniques, also referred to as optical flow algorithms, are limited by two constraints (Sun et al., 2008): brightness constancy and spatial smoothness. Brightness constancy constraint assumes that optical features (for example a precipitation cell) persist over time and that their intensity is invariant by translation. Spatial smoothness constraint forces neighbouring pixels to have similar motion characteristics since they are likely to be governed by similar physical forces. The first constraint is violated if precipitation cells are dissipated or created, for example, in convective situations. The second one is violated if two close precipitation cells are moving in completely different directions, which is rare. Despite these constraints, optical flow algorithms provide a basis for modelling the displacement of precipitation on radar images.
The model used in this study was recently developed and explained in Sun et al. (2008). It defines a probabilistic model of optical flow (motion vector field) as:
where R1 and R2 are two consecutive radar images, U is the motion vector field composed of the two components of the displacement vector (u,v), p(R2|U, R1;ΩD) is the data term responsible for brightness constancy, p(U|R1;ΩS) is the spatial term describing spatial smoothness and Ω acts as a regularization parameter that controls the smoothness trade-off. Several values of Ω were tested and the one which provided a good compromise between smoothness and precision was chosen. To increase the robustness of the approach, optical flow was computed on radar images which were thresholded at 10 mm h−1 and smoothed with a Gaussian filter (2σ = 2 km).
Optical flow was then combined with terrain gradient to compute directional derivatives with respect to flow direction (referred to as FD, flow derivatives). Such flow varying derivative is an extremely important attribute for highlighting the upwind flank of mountains (see Figure 4) since orographic precipitation has a strong directional dependence (Weston and Roy, 1994; Panziera and Germann, 2010). Flow derivative is computed as follows:
where ∇z(xgeo) is the gradient vector of the altitude field z computed at location xgeo = (X, Y) and u(xgeo, t) is the vector with cosine (west to east, u) and sine (south to north, v) components of the motion vector field computed at xgeo = (X, Y) at time t. Flow derivatives are derived at a number of different spatial scales. Samples are shown in Figure 5(a) and (b) (the corresponding flow field is depicted in Figure 5(c) and (d): vectors are only displayed at the location of precipitation cells).
Figure 4. Flow derivative is positive if the precipitation cell is located on the upwind side, around 0 if moving parallel to the mountain ridge or over flatlands and negative if descending on the downwind side
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Figure 5. (a) Flow derivative computed at a medium scale (2σ = 2 km) and the largest scale (2σ = 102.2 km) for prevailing southwesterly flows. The large scale derivative is displayed in a panel in the upper left corner but has the same spatial extension of the small scale derivative. Values are standardized to zero mean and unit variance. (b) Same as (a) but with northeasterly flows. (c) Example of a dynamic precipitation pattern: thunderstorms are moving over Swiss plateau with southwesterly winds; in the south of Grisons (see Figure 2 for details) there are typical isolated showers triggered by thermal ascent over mountains. (d) Example of a blocking precipitation pattern from northeast: precipitation cells move towards the Prealps but the southern leeward side of Alps is sheltered and no precipitation is observed. (e) and (f) Corresponding precipitation anomalies computed with Differences of Gaussians from which cells are detected. This figure is available in colour online at wileyonlinelibrary.com/journal/met
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In summary, the space of features is composed of 10 fixed input features (X, Y, Z and 7 DoG) and 10 varying input features (7 FD, u, v and the rainfall intensity as estimated by radar). These features will be used for solving different tasks. Clustering will be performed in the 3D space composed of (u,v) and the largest scale flow derivative which basically describes the cells' position relative to the main Alpine divide (windward and leeward). All features excepting (u, v) and X, Y coordinates will be used in a future study to characterize orographic enhancement using non-parametric classification models. X, Y coordinates are registered for visualization purposes but are not directly used as explanatory variables.
3.3. Why different spatial scales?
The reasons why topographic and flow related features are extracted at different scales are of different origins. The first is due to the dynamics of airflow which is delayed in space and in time in relation to topographic forcing. Topography is not directly affecting precipitation but provokes saturation by forcing the uplift of an air mass. Due to cloud micro-physics (see a thorough description in Houze (1993)) there is a spatial and temporal delay between air saturation and precipitation. Also, numerical studies have demonstrated that the trajectory of hydrometeors is a function of their size and composition (snow flakes, rain drops) which in turn determines the fall speed (Hobbs et al., 1973). Hydrometeor fall speed and wind intensity are just two among many controlling factors characterizing the spatial disagreement between the place of air uplift, condensation and the location of precipitation fallout at the ground plane.
Heavy precipitation is more likely to occur on the upwind flank at a certain spatial scale, and is the subject of linear models of orographic precipitation (Smith and Barstad, 2004). However, a too small-scale flank of a ridge perpendicular to flow direction is often not enough to trigger the mechanism of condensation and precipitation and hence is a poor explanatory variable. Larger scale features are expected to be more informative since the uplift of air needs a slope of a certain size and length to take place significantly.
Airflow speed and air instability are also controlling factors for the spatial distribution of precipitation. Airflow speed is proportional to the activity of a cold front approaching a topographic barrier which in turn increases precipitation quantities on upwind slopes (Johansson and Chen, 2003). Unstable air associated with strong turbulent flows increases the efficiency of particle growth and fallout causing higher precipitation rates on upwind slopes (Houze et al., 2001; Panziera and Germann, 2010).
The relation between wind speed and the spatial distribution of precipitation is even more complex. High precipitation rates can be found at the top of hills and ridges (described by DoG) as modelled by Smith and Barstad (2004) and not on upwind slopes depending on the scale of analysis and the wind speed. There are also cases when precipitation is enhanced on the leeside of a mountain due to lee-side convergence behind small topographic obstacles (Cosma et al., 2002), to spillover behind narrow mountains (Smith and Barstad, 2004), to gravity waves and lee-side cold air pools (Zängl, 2005), to the time elapsed between rain drop nucleation and falling to the ground (Roe, 2005; Zängl, 2007) which can be important when associated with seeder-feeder processes (Zängl, 2007). These effects are less frequent and need small scale explanatory features to be described. Extensive numerical simulations analysing the relative positioning of maximum rainfall rates with respect to a mesoscale ridge are illustrated in Miglietta and Rotunno (2009, 2010). The wind speed, air stability, height and width of the ridge determine the spatial distribution of high rainfall rates. Two real case studies, including the event of August 2005, are studied in Zängl (2007) using numerical simulations and observed rain gauge measurements. These show that windward and leeward accumulations of precipitation vary from region to region and depend on wind speeds, freezing levels and the presence of seeder-feeder mechanisms.
In the present paper, topographic characteristics describing the neighbourhood of a precipitation cell are considered by computing delocalized features which describe them at different spatial scales (Section '3.1. Feature extraction from DEM'). An alternative approach would be to integrate topographic data surrounding the location of interest for different distances and directions, but the number of input dimensions of features would increase dramatically.
Finally, Roe (2005) concludes that ‘precipitation maximizes over the windward slopes, whereas for smaller hills the maximum tends to occur nearer the crest’ and that there is ‘evidence of a close association between orography and precipitation patterns at spatial scales of a few kilometres’. The problem of analysing at which spatial scale these statements become valid is crucial for a better understanding of orographically induced precipitation.
The range of scales considered in this paper (see Table 1) is aimed to capture most of precipitation variability in the Alps due to topographic forcing. The minimum and maximum ranges of smoothing bandwidths are chosen to be wide enough to have an extensive feature set from which the statistically relevant ones can be selected. Thus, small scale features are expected to be explanatory variables for enhancement effects due to triggered convection. On the other hand, large scale features can explain widespread precipitation patterns due to blocking by the Alps.
3.4. Precipitation cells detection
Radar images illustrating the proposed methodology of orographic precipitation pattern analysis concern the Swiss Alps region in the period of 18–23 August 2005. The northern flank of the Alpine chain was affected by severe flooding due to persistent thunderstorms and blocking situations (Rotach et al., 2006).
Meteoswiss operates a network of three C-band Doppler weather radars located at the top of Monte Lema, La Dôle and the Albis (see Figure 2). Radar-derived composite instantaneous rainfall rates at a 1 × 1 km2 grid resolution and with 5 min of temporal resolution are available. They are derived with the pre-processing steps described in Joss et al. (1998) and Germann et al. (2006), including the corrections for vertical profiles in sheltered regions, application of ground clutter elimination algorithms, bias correction with respect to rain gauges and corrections for the bright band effect. It is currently the most reliable radar product available from Meteoswiss. However, profile corrections (Germann and Joss, 2002) cannot solve all visibility problems of Swiss radars in regions which are shielded by mountainous relief or by obstacles close to the antenna. In such places the lowest elevation scans are too high in the atmosphere for a reliable estimate of rainfall rates at the ground. Only convective precipitation presenting a sufficient vertical extension can be detected, while stratiform precipitation is often unseen. The visibility of Swiss radars can be evaluated by geometric approaches to correct radar-rain gauge biases using the distance from the radar, the height of the lowest visible scan and the height of the ground in linear or non-linear regression models (Gabella and Perona, 1998; Gabella et al., 2001, 2005; Golz et al., 2005). The effective radar visibility can then be observed by evaluating the relative rainfall detection rates of radar and rain gauges in the spatial domain (Wüest et al., 2010). In the Valais and Grisons regions (see Figure 1) the height of the lowest visible scans varies between 4000 and 6000 m above sea level which does not allow detecting and extracting precipitation patterns (see Golz et al., 2005, a detailed map of the height of the lowest scans of the Monte Lema radar). The analysis is restricted to the places monitored by low level scans such as the Prealps which were also the most touched by orographic precipitation during August 2005, but low level scans are more prone to ground clutter mainly due to the backscattering of beam by orographic features which eventually overestimates radar reflectivity. This has to be considered when extracting and analysing precipitation fields along with the interpretation of results.
The first step in the analysis of orographic enhancements is the extraction of precipitation cells from radar images. There are well-established algorithms for convective cell detection and tracking (Dixon and Wiener, 1993; Lakshmanan et al., 2009). The operational precipitation cell tracking approach adopted in Switzerland (Thunderstorms Radar Tracking—TRT) is described in Hering et al. (2004). It is specifically targeted to extract and track convective cells characterized by significant vertical and spatial extensions.
The approach proposed in the present paper considers the extraction of precipitation cells by finding points of extrema, above a certain threshold, of filtered radar images. Compared to TRT, it also allows extraction of non-convective cells, including the ones due to orographic enhancement effects. Rain rates of radar images were first filtered using DoGs with small and large scales (2σ = 2 km and 2σ = 20 km). The resulting image describes precipitation anomalies in the spatial domain such as isolated cells or precipitation enhancements inside spread precipitation areas. Points of extrema exceeding 5 mm h−1 of the field of precipitation anomalies have been extracted (Figure 5(e) and (f)). Filtering of the radar image is also performed to shrink the effects of clutter and ground echoes. While Germann et al. (2009) report that 98% of all cluttered pixels are eliminated, one should still be aware of the presence of noise and outliers provoked by beam and shadow effects of the radar. A rough check for ground clutter for the locations touched by many precipitation cells was carried out with respect to the accumulated precipitation field constructed by integrating all radar images. Only a few of the detected cells were located in places showing anomalously high values (due to the summation of clutter effects) compared to the climatology of the event.
The effective spatial resolution of the measured precipitation field is decreasing as a distance to the radar. The scanned volume is re-sampled to provide precipitation data on a regular grid of 1 × 1 km2. However, the real resolution of the precipitation field, and in particular for places far from the radar, is better captured by working with filtered radar images. Notice that under the considered setting we only aim at identifying the location and topographic shape below precipitation cells without requiring the reliable quantitative precipitation estimates.
The full set consists of 28 758 cells detected in 1728 radar images captured over 6 days. The instances in this dataset, along with their topographic and optical features (dimensions) result both from orographic and atmospheric effects. Some outliers are present in the dataset due to instrumental errors.