1
STATISTICAL MODELLING OF SNOW DISTRIBUTION ON A
CENTRAL SPITSBERGEN GLACIER
combining terrain based parameters and wind field modelling
BSc Thesis by : Chris van Diemen1 31 July 2015
Coordinators: John van Boxel1, Emiel van Loon1, Andy Hodson2
1
University of Amsterdam, The Netherlands
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Abstract
Glaciers are widely known to be indicators of climate change, but factors such as snow redistribution by wind complicate the anthropogenic signal. The central Spitsbergen area is an area that is especially vulnerable to redistribution by wind. The area is subject to high wind speeds and low precipitation throughout the winter. In this study a statistical model will be presented that makes use of modelled wind fields and topographic parameters to predict snow depth. The use of binary regression trees is found to be superior to multivariate linear regression models. Modelled wind fields do not increase predictive power as expected, mainly due to error in sampling plan. The use of topographic parameters such as a parameter indicating sheltered areas produce increased model fit. Models are presented that reach a coefficient of determination that is similar to other studies and among the best snow depth models available today.
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Content
List of figures ... 4 List of tables ... 4 1 Introduction ... 5 2 Study Area ... 7 3 Methods ... 8 3.1 Data collection ... 9 3.1.1 Metrological data ... 9 3.1.2 Elevation data ... 103.1.3 Snow depth data ... 10
3.2 Modelling ... 11
3.2.1 Wind Field modelling ... 12
3.2.2 Statistical modelling ... 12
4 Results ... 15
4.1 Wind Model and Sx calculation ... 15
4.2 statistical modelling ... 16
5 Discussion ... 17
6 Conclusions ... 18
7 Acknowledgements ... 18
8 References ... 19
Appendix A: Distribution test description ... 22
Appendix B: Metrological data ... 23
Appendix C: Complete table of results ... 24
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List of figures
Figure 1: Overview map op Advent valley location of weather stations. ... 5
Figure 2: Overview of Foxfonna with Lower and Upper Fonna indicated . ... 7
Figure 3: Impressions of the study area. ... 8
Figure 4: Wind roses visualizing wind speed and wind direction. ... 9
Figure 5: Distribution of snow depth sampling locations for the years 2012 and 2014. ... 11
Figure 6: Sx calculation schematic (Adapted from Winstral and Marks (2002)) ... 12
Figure 7: Visualization of the results from the three types of wind modelling ... 14
List of tables
Table 1: Variable test design ... 12Table 2: Summary statistics for modelling results of different Sx parameters. ... 15
Table 3: Summary statistics for snow depth of different data sets. n = number of observations. ... 15
Table 4: Summary results form BRT and MLR models. ... 16
5 Figure 1: Overview map op Advent valley location of weather stations with data available for public use. A) indicates location of Foxfonna and map section shown in figure 2.
1 Introduction
Glaciers are widely known to be indicators for global climate change. An often heard statement is that almost all glaciers are shrinking due to anthropogenic causes. In 1994 86 glaciers were studied and found to have a mean 1.3 m/year terminus retreat indicating global glacier retreat (Oerlemans, 1994). This loss of length corresponds to a loss of 30 % of area and 38 % of volume during the last century (Mernild et al., 2013). More recent research has shown that glacier retreat during 1991 - 2010 is largely caused by anthropogenic causes. As much as 69 ± 29 % has been attributed to human activities (Marzeion et al., 2014). Future scenarios paint a similar picture for the coming years where an additional 29 - 41 % volume loss is expected at 2100 (Radić et al., 2014).
Even though glaciers represent less than 1% of the global ice mass (Marzeion et al., 2014) they have an influence on our planet that cannot be neglected. Glaciers have contributed to a sea level rise (Marzeion et al., 2014) and continuous sea level rise will contribute to an increase in flood risk in the future (Nicholls et al., 1999). In parts of the world that are densely populated, glaciers provide water and food security. Water supply is expected to change in this decade due to change in glacier extend, consequently threatening the food security of 60 million by 2065 (Immerzeel et al., 2010).
Estimates about the future change in glacier volume and size are important when considering the changes stated above. Possible mitigations could benefit from accurate predictions especially in areas that rely on glaciers and snow for their water supply (Barnett et al., 2005) Therefore it is paramount to understand glacier dynamics.
Glaciers represent an equilibrium between accumulated snow at high elevations and ablation of snow or ice at lower elevations(Benn & Evans, 2014). As mentioned above most glaciers are currently not in equilibrium and anthropogenic influences cause destabilization. However, the growth and shrinking of glaciers is also dependent on many other factors that are less well understood and create noise in the signal of climate change in glaciers and render predictions about the future less accurate. One example of these less understood factors is the dependence of glaciers on snow accumulation due to wind. Machguth et al. (2006)pointed out that snow distribution can vary over the surface of a glacier, emphasising the need for a comprehensible modelling technique. Glaciers in areas with relatively low amount of winter precipitation and high wind speeds such as central Spitsbergen are especially reliant on windblown snow for a positive mass balance (Jaedicke & Gauer, 2005).
6 With the emerging possibilities of Geographic Information Systems and increasing computer power modelling techniques using spatial statistics and differential equations are widely used tools for understanding and predicting surface processes. The modelling of wind fields has developed and results are approached with optimism (Finardi, Tinarelli, Faggian, and Brusasca, 1998; Forthofer, 2014). The use of binary regression trees is a tried and tested technique when predicting snow accumulation in alpine areas (Elder, Rosenthal, and Davis, 1998; Winstral and Marks 2002; Molotch, Colee, Bales, and Dozier, 2005; Prokop et al., 2013). The technique has also been tested on large Antarctic ice caps, (Frezzotti et al., 2004;Frezzotti et al., 2005) in forested areas (Gelfan Pomeroy, and Kuchment, 2004), and even on the larger Spitsbergen glaciers (van Pelt et al., 2014). The work done by Jaedicke and Gauer (2005) tried to explain snow distribution in the central Spitsbergen area on mesoscale. However, a detailed investigation has not yet been conducted on a small glacier, let alone the unique Arctic location like that of the Foxfonna. The Foxfonna glacier poses an unique opportunity of investigation. Spitsbergen is relatively easy to access since regular flights are available and infrastructure is present. Many data resources are freely available online and weather stations are relatively well distributed over the area.
This report aims at presenting a statistical model that predicts local snow distributions due to modelled wind direction in order to better understand the impact and quantity of these phenomena. It also aims at testing the presented technique in an environment different than the ones studied by Elder, Rosenthal, and Davis, (1998) and Molotch et al. (2005) and enhance the predictive power by incorporating promising wind field modelling techniques in the process. A binary regression tree (BRT) is used as a statistical model next to a multivariate linear regression model (MLR). The model is built with local snow distribution data on the Foxfonna glacier and local meteorological and elevation data. The focus area of the research is central Spitsbergen and in particular the Foxfonna glacier.
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Figure 2: Overview of Foxfonna with Lower and Upper Fonna indicated (50 m contour lines, blue line is glacier outline).
2 Study Area
The study area is the small glacier called Foxfonna located in the central part of Svalbard at 78°7’41"N, 16°11’01"E (maps use UTM zone 33X projection and WGS84 datum). The glacier consists of two main parts. The first part is an ice cap that spans approx. 2 km2 (Upper Fonna on figure 2) and a part that is more characteristic of a small valley glacier with an area of around 1.6 km2 (Lower Fonna on figure 2). The Upper Fonna reaches from an elevation of 700 to around 818 m a.s.l. and has slopes that do not exceed angles of 12°. The Lower Fonna consists of two tongues that reach from 330 m to 660 m a.s.l., the first with a dip direction of 350° at max 15° slope (figure 3c) and the second with a dip direction of 10° at around 20° slope in the steepest areas. The two parts of the glacier are connected by an ice flow over a 25°-45° slope that is crossed by several crevasses indicating movement of ice over topography (see cover page and figure 2). Boreholes have indicated that the ice is at maximum 100 meters thick and entirely cold based and no signs of surging have been found (Christiansen et al., 2005). The glacier equilibrium line is observed at around 600 m a.s.l. (Ole Humlum, Personal communication, May 2015). The Upper Fonna has been observed since 2007 and has lost an accumulated 2.05 cm water equivalent (w.e.) between 2007 and 2014 at a rate that fluctuates between +0.3 cm w.e./year and −0.9 cm w.e./year (Andy J. Hodson, Personal communication, 3 May 2015). The surrounding area can be characterized as a high relief periglacial landscape subject to continuous permafrost and a lack of any high vegetation (Humlum et al., 2003).
Lower Fonna
8 Approximately 15% of the surrounding area is covered with glaciers (Jaedicke and Gauer, 2005) and completely covered by snow during winter. Mean annual temperature around the start of the 21st century is measured at -5° Celsius (Humlum et al., 2003). The archipelago is influenced by cold and dry arctic air masses from the north and moist, relatively warm air masses that are transported from the south by the North Atlantic cyclone track. The cyclicity of these air masses is typical for the high Arctic location of Svalbard (Serreze et al., 1995) and most of these air masses pass by Svalbard in the south causing the general wind direction over the island to be from the east and south-east (Humlum et al., 2003). The contrast between these two systems is strongest in winter and causes snow storms that can precipitate and displace large amounts of snow (Eckerstorfer and Christiansen, 2011). In favourable conditions large warm air masses can be transported up north and cause temperatures above zero and snow melting in the middle of winter. Warm spells with above zero temperatures cause peaks in precipitation and seem to increase in frequency (Hansen et al., 2014). Topography can cause the wind direction to vary on a local scale and deviate from the general east south-east direction (Humlum, 2002). The study area is located in the driest part of Svalbard with an annual mean precipitation of 200 mm during the last century (Førland et al., 1997). These meteorological factors are important when modelling wind speeds and snow distribution and will be examined more closely in the chapters below.
x
a) Breinosa weather station b) Snow depth probing c) Lower west Fonna
Figure 3: Impressions of the study area.
3 Methods
The methods can be divided into two main parts. The data collections and the data processing, where processing is called modelling in this case. Data is collected from a number of sources. Meteorological stations in the area provide information about multiple environmental factors and are needed to determine the period for which wind fields are modelled and boundary conditions for wind field models. Snow depth and density data is collected during multiple field work campaigns at the end of period in which snow accumulates. The data from the field and the wind modelling will be combined with elevation data in order to make statistical models. The separate steps in the process stipulated here will be described in detail below.
9 3.1 Data collection
The data collection consists of three main parts. The snow depth and density data collected in the field, the meteorological data and the elevation data gathered from freely available online resources .
3.1.1 Metrological data
Meteorological data is gathered from two sources. The first is the Norwegian Meteorological Institute (DNMI), eKlima service (DNMI, 2015). The second source are the weather stations that are runned by UNIS personnel (UNIS, 2015). From these sources information can be gathered on several environmental factors over the course of many years. The DNMI provides information that is important when determining the time window for which wind fields will be modelled. Snow will only fall when temperatures are below or close to zero. Data provided on snow depth makes it possible to approximate the moment at which snow starts accumulating (see Appendix B for explanation). These dates are determined to be the beginning of October 2011 for 2012 measurements and the beginning of September 2013 for the 2014 measurements. The end date for the time window is determined on the date that the depth measurements are taken. This is the 28th of April for the 2012 data and the 18th of May for the 2014 data. When the time frame is established the UNIS data can be used to determine the wind speed and direction during this period. These data are used as initial conditions for the wind speed models (figure 4). Only relatively high wind speeds can transport significant amounts of snow. Fresh snow is most susceptible to being transported by wind. Svalbard’s dry snow at around −10°C is transported at a wind speed of around 6 m/s or higher (Li and Pomeroy, 1997). Wind speeds below the 6 m/s threshold are unlikely to transport any meaningful amount of snow are therefore omitted from the record. The remaining data is used to determine general wind speed and direction for the four weather stations that are in proximity of the Foxfonna glacier (see figure 1 and 4). One should keep in mind that data from DNMI are regularly checked and a comprehensive scheme to check quality is in place where data gathered at UNIS are run by a minimal staff and are only occasionally calibrated. Data is considered to be accurate enough to be used in wind modelling.
Figure 4: Wind roses visualizing wind speed and wind direction over 6 m/s for the four meteorological stations (see figure 1 for locations of weather stations, for Breinosa weather station see figure 3a).
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3.1.2 Elevation data
Elevation data is provided by the map service of the Norwegian Polar Institute (Npolar, 2015). Elevation is generated by stereo-photogrammetric models on aerial photos. A model with a 20 meter resolution is used that has a standard deviation of approx. 2 to 5 meters. The standard deviation might be slightly higher on glaciated areas (Npolar, 2015).
3.1.3 Snow depth data
Over the years 2012-2015 UNIS students have been working together with Andy Hodson to measure the snow depth on the upper and lower parts of the Foxfonna glacier. Data is collected as a part of a research project that has been running since 2006 (RIS, 2015).
The method that is used for snow depth measurement is probing. A 2 meter stick with centimetre markings is used to measure snow depth at multiple sites on the glacier. By pushing the probe through the snow until it reaches the glacier ice (figure 3b). Three readings are taken in a square meter at each measurement point. The GPS locations are taken using a Garmin gpsmap c62. In some instances a GPS snow depth measure device (i.e. MagnaProbe) was used and resulted in a higher depth observation count (e.g. Lower Fonna in 2012 and complete glacier in 2014 see Figure 5a and b).
The snow depth data from the year 2012 is collected by Tørstad et al. (2012), 2013 data by Farsund (2013), 2014 data by Braasch et al. (2014); Bergset and Allaart (2014). The current project has also included the fieldwork that produced the 2015 data.
Two data sets are used for the current project. Because different students have been working on the glacier the datasets differ in the number of observations and also in the spatial distribution of these observations (Figure 5). The usability of the data largely depends on how representative it is for the situation on the Foxfonna glacier. It is therefore important that the observations are uniformly distributed over the area of the glacier. A χ2
– test and clustering test are used. Data that is randomly distributed or has a uniform distribution is preferred over data that is clustered. Both tests point to data from 2012 and 2014 as the most uniformly distributed and least cluster forming of the 4 data sets (see Appendix A).
11 a) 2012 sampling points b) 2014 sampling points
Figure 5: Distribution of snow depth sampling locations for the years 2012 and 2014.
3.2 Modelling
Multiple studies have confirmed that wind is an important factor in the distribution and redistribution of snow (Winstral and Marks, 2002; Feldman and O’Brien, 2002; Frezzotti et al., 2004, 2005; Molotch et al., 2005; Jaedicke and Gauer, 2005; Machguth et al., 2006). There have been several attempts at capturing this phenomena in a numerical model (Winstral and Marks, 2002; Molotch et al., 2005; Jaedicke and Gauer, 2005; Liston et al., 2007). Snow depth is considered to be the most important factor when snow distribution is concerned. The use of a terrain based parameter called maximum upwind slope, or Sx, developed by Winstral and Marks (2002) seems to be the most accurate way of predicting snow depth available today.
This parameter assigns a degree of shelter to terrain as a function of the wind direction and the upwind terrain (Winstral and Marks, 2002). The Sx factor is calculated for every cell in a spatial grid by measuring the highest angle (see angle α in figure 6) between the cell of interest and the upwind terrain along multiple vectors (see figure 6 on the following page R1 to Rn+i). The range of angles (R1
to Rn+i) is determined by the mean wind direction and the distribution of wind. Data on mean wind
direction and spread of wind directions is gathered from meteorological stations as mentioned above. Winstral and Marks (2002) and Molotch et al. (2005) use a mean wind direction over the entire area to determine the prevailing wind direction for each grid cell. Jaedicke and Gauer (2005) has shown that it is likely that the local topography has influence on the wind direction and on snow deposition. Replacing the mean wind direction over the entire area with modelled local wind direction could benefit the results from the parameterized model. For the development of the enhanced parameters three wind models are used that are described in more detail in the section below. The enhanced parameters and original parameters are used together with other terrain based parameters to build a statistical model.
12 Figure 6: Sx calculation schematic (Adapted from Winstral and Marks (2002))
3.2.1 Wind Field modelling
Three wind models are used to calculate the wind direction over the study area. The calculated wind direction will be used in the Sx calculation as described above. The first and simplest technique is assuming a uniform wind direction over the complete area. A uniform wind field is also is applied by Winstral and Marks (2002); Molotch et al. (2005). The other two techniques are represented by readymade wind models. The second of these models is the windninja model that was developed for the prediction of forest fire spread. This model uses a mass consistent wind model (Forthofer, 2007) and will be referred to as ’mass’ in the sections below. The last model that is used is the windstation model. This model makes use of differential equations and is called computational fluid dynamics (CDF) in the sections below. The CFD model is specially developed for the simulation of wind fields over complex terrains (Lopes, 2003).
For both wind models first a large grid is calculated that incorporates all meteorological stations as mentioned above. Using the large grid as boundary conditions a smaller grid with higher resolution that only incorporates the glacier is calculated. The small grid is used as input of the Sx calculation.
Table 1: Variable test design. Different combinations of variables are represented by a model number. Model numbers with asterisk include variable combinations with all three Sx variables (small crosses).
Variable \ Model number 1 2 3 4 5 6 7* 8* 9* 10* 11* 12* 13* Sxmean (°) x x x x x x x x x Sxmass (°) x x x x x x x x x SxCFD (°) x x x x x x x x x Precipitation (mm) x x Elevation (m) x x x x x x x x x x Slope (°) x x x x x x Northness (-) x x x x x x x Eastness (-) x x x x x x
3.2.2 Statistical modelling
Two statistical techniques are used to test a variation of terrain based variables and Sx on its predictive value for snow depth. A binary regression tree (BRT) and a regular multivariate linear regression (MLR) model is fitted. The same evaluation parameters are used for the BRT and MLR in order to compare the two methods.
A multitude of terrain based parameters are calculated for sampling locations and are used as independent variables to build the BRT and the MLR. The snow depth is used as the independent variable and is considered to be the most representative parameter for snow distribution.
13 It is useful to divide the independent variables that are used into two categories in the light of the current investigation. The variable Sx could differ a lot when considering the two different years and has to be calculated separately for each dataset. The other factors, elevation, slope, aspect, and northness are stable over long periods of time and only need to be calculated once for both years. Molotch et al. (2005) indicates that northness can be a substitute for solar radiation indicating that radiation does not vary so much over the years. Therefore northness is used as a substitute of solar radiation in this study and is defined as the product of the cosine of the aspect and the sine of the slope (Molotch et al., 2005). Northness is also considered to be a stable variable and is calculated once for both years. Aspect is a factor that is unusable in MLR as well as in BRT models because it is given in degrees where 0 is the same as 360 and the methods cannot recognize the proximity of the angles on a circle. Therefore the aspect is translated to its orthogonal components and a parameter called eastness is used. Eastness is the cosine of the aspect and can therefore indicate if a sample is taken on a slope facing east between -1 to 1 where -1 is a slope facing west 0 is a slope that faces either north or south and 1 is a slope facing east. Both northness and eastness parameters are without unit. The parameters elevation and slope are expected to be self explanatory.
Three different Sx parameters are calculated from the three wind models mentioned above and are called Sxmean, Sxmass, and SxCFD. The three Sx parameters are used separately in different models
(Table 1).
In this case the software MATLAB (2014) is used to build regression tree. A regression tree is build up of nodes and leafs. A node is defined as the a point where a split is indicated by the corresponding values of a independent variable. A leaf is the end point that assigns a value to the combination of values in the independent variables after it has passed through the corresponding nodes.
To build a BRT an algorithm bins independent variables in subsets that are approximate homogeneity ever closer. Several steps are taken by the software to minimize the mean square error MSE in each split of the data. First all the data is examined on all possible binary splits for every predictor. The splits that reach lowest MSE are selected. Splitting of data stops when either the minimum observations per leaf (MOPL) indicated by the user has been reached or when the MSE drops below the product of MSE in the dependent variable in entire data and the tolerance on the quadratic error per node (MATLAB, 2014). These steps are taken recursively for each leaf until any of the stopping criteria is reached. Ten-fold cross validation is applied using 10 different bins versus the minimum number of observations per end leaf between 1 and 100. The general version of the coefficient of determination (R2 = Explained variation / Total variation), the root mean square error (RMSE) and the mean absolute error (MAE) are averaged over all iterations as a means of evaluating the fit of the model. The model predicts depth in cm and error measures RMSE and MAE are given in cm as well. R2 is a dimensionless parameter and is given in a fraction between 0 and 1 and can also be translated to percent. The optimal number of minimum observations per end leaf was determined by selecting the minimum RMSE and minimum MAE. The MAE was only used if no conclusion could be drawn from the other two parameters. If one model was found to have a higher MAEbut a lower RMSE than the model with the lowest RMSE is chosen as the best model. The model with the optimal number of observations per end leaf was selected as the best model for the corresponding set of independent variables. The technique described above is an adaptation of the technique described by Molotch et al. (2005) which is based on similar studies by Elder et al. (1998), Balk and Elder (2000), and Winstral (2002).
By indicating the MOPL the user can control the number of leafs, or depth, of the BRT. In the procedure mentioned above the optimal number of leafs is determined by the number that produces the model with the best fit. However, a model with a higher or lower amount of leafs can be produced by fixing MOPL on a set number. Say 700 observations are available and a tree with 10 leafs is desired than MOPL will be set at 700/10 = 70 observations per leaf. This procedure will be used to compare the models between studies and is elaborated upon in the discussion.
14 .
a) Mean wind field
b) Sx
meanc) Mass wind field
d) Sx
mean- Sx
masse) CFD wind field
f) Sx
mean- Sx
CFDFigure 7: Visualization of the results from the three types of wind modelling. a) The mean wind direction for the whole area. Arrows indicate mean wind direction for corresponding location. c) the result of the mass-consistent model. e) the result of the CFD model. b) Sx value for the mean wind direction over the whole field. d) Difference between the mean wind modelled Sx and the mass consistent model Sx, green areas indicate a negative change. f) Difference between the mean wind model Sx and the CFD Sx, green areas indicate negative change
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4 Results
4.1 Wind Model and Sx calculation
The simplest wind model that only uses the mean wind direction is unable to take account of any topography (figure 7a). The mass-consistent model takes some account of topography as it can be recognized that air flow is somewhat changed in a more south south-east direction when the air flow moves from the valley on the east side of the glacier up the plateau (figure 7c). Lee side turbulence is badly accounted for in the mass-consistent model as no lee side wind direction change is recognizable. The CFD model seems to react to both, topographic streamlining and lee side turbulence (figure 7e). Streamlining is most recognizable over the long axis of the lower west tongue of the glacier where wind direction is almost parallel to the long axis and the strike of the steep slope to the south west of this tongue. Lee side turbulence is most striking at the east side of the Lower Fonna. Here the air masses are pushed up from the 170 meter valley floor to an altitude of 780 meter (see figure 2 for topographic reference). At the west side of this topographical high the glacier surface is at about 650 meter a significant drop. The wind direction at this point is chaotic.
When Sx is calculated for the mean wind direction grid one can distinguish quite clearly the areas with shallow slopes and the steep areas (figure 7b). As expected the steep slopes on the lee side result in the highest Sx values and are most prone to snow deposition. The difference in Sx between the mean and mass wind model are most pronounced on the Upper Fonna. The slight direction change causes a higher Sx on the east and lower on the west side of the Upper Fonna (figure 7d). The difference in Sx between the mean and CFD models is more prominent on the Lower Fonna. Here Sx is lower on areas of the lower west tongue and much higher on the lee side of the topographic high on the east side (figure 7f). The extend of the differences are generally small when summary statistics are calculated for sampling locations (table 2). Spread increases slightly for 2012 from Sxmean to Sxmass and
Sxmean to SxCFD at 0.5 and 2 % respectively. Differences are bigger for 2014 where spread changes
from Sxmean to Sxmass and Sxmean to SxCFD at 0.3 and -36%. Subsequently the three different Sx values
are tested on correlation in statistical modelling in the next section.
Table 2: Summary statistics for modelling results of different Sx parameters.
2012 2014
Parameter mean min max St. dev. mean min max St. dev.
Sx_mean 15.23 -2.0 37.57 8.25 14.76 -1.8 38.09 8.18
Sx_mass 15.28 -0.7 37.57 8.29 14.73 -3.4 38.09 8.20
Sx_CFD 15.22 -3.1 39.00 8.48 9.32 -7.2 28.48 5.25
Table 3: Summary statistics for snow depth of different data sets. n = number of observations. 2012 Snow depth (cm) 2014 Snow depth (cm) Both years Snow depth (cm) Minimum 95 45 45 Maximum 287 220 287 Mean 155 114 122 Std. dev. 34 17 27 n 147 581 725
16 Table 4: Summary results form BRT and MLR models. Only the best fit models for the years 2012, 2014 and data from both years toghether is visualized for all Sx parameters used. The best model of three is selected as the model with the lowest RMSE and the hightes R2. * Best model ** Overall best model.
Year(s)
2012
2014
BOTH years
variables Sx_mass + height + Eastness Sx + slope + height + Eastness Sx+ northness + precip
R2 RMSE leafs R2 RMSE Leafs R2 RMSE Leafs
BRT
Sxmean 0.480 21.34 6 0.208* 15.21 20 0.503** 18.43 27
Sxmass 0.482 20.83 10 0.188 15.29 21 0.496 18.54 26
SxCFD 0.488* 21.73 11 0.186 15.44 23 0.502 18.57 27
variables 1 + Sx + height + Eastness 1 + Sx + northness 1 + Sx + northness + precipitation
MLR
Sxmean 0.349 24.89 N/A 0.031 16.93 N/A 0.375 21.03 N/A
Sxmass 0.351 24.92 N/A 0.034* 16.89 N/A 0.379 20.97 N/A
SxCFD 0.356* 24.85 N/A 0.019 17.04 N/A 0.411* 20.41 N/A
4.2 statistical modelling
The BRT performed higher than MLR as expected. BRT models gave a mean R2 of 0.48 and RMSE of 21.7 cm over all models where MLR reached R2 of 0.34 and RMSE of 25.2 cm (table 4) of data with a mean std. dev. of 27 cm. RMSE can be compared to the standard deviation of the data that is used as presented in table 3. For both years the BRT technique leads to models with a better fit that MLR. The difference between BRT and MLR ranges from an order of magnitude no difference at all and is greatest for 2014 data..
Models that used the Sx parameter added between 26 and 9% of explanatory power. Where 9% gain was achieved in case of the best fit model (table 4 **).
The BRT model using the variables Sxmean, northness, and precipitation had the best fit although
differences are neglectable (table 4). Difference is highest with the data from the year 2014 where Sxmean has a 2.2% advantage over the model that uses SxCFD. In case of the overall best fit model where
50.3% of variation was explained the model using Sxmean had a 0.1% gain over SxCFD. The difference
in RMSE was 0.14 cm between SxCFD and Sxmean and 0.03 between SxCDF and Sxmass. The model that
best fitted the data was model 13 (table 1). This model uses Sx, northness, and precipitation. The model performed better than all the other variable combinations. Ten-fold cross validation procedures determined tree sizes for the best model around 27 leafs.
The MLR models show a different tendency and point to the model using SxCFD as the best model
for 2012 data and models build on data from both years. Differences are again neglectable with only fractions of centimetres difference between the different Sx parameters used. Models build on the year 2014 data show a fit that is particularly bad for all variable combinations.
Explanatory power suffers when the number of leafs is lowered. When BRT model 13 is lowered to 7 leafs R2 drops to 0.33. At 10 leafs R2 = 0.45 and at 18 R2 = 0.47 (table 5).
When data from only 2014 is considered the fit is considerably worse than the best fit for both years together. This fit is achieved with the variables Sx, slope, height, and eastness. The best fit for 2012 is higher using variables Sx, slope, height, northness, and eastness.
Models for the years 2012 and 2014 using eastness have better fit than models using northness as a variable representing slope aspect. When both years are taken together the variable northness produces a better fit.
17 Table 5: Comparison of model results between similar studies and the results in this study at 3.6 km2. Best model in this study (see text) is modified to match number of leafs in other studies in order to make a more insightful comparison possible. Limitations of this comparison are emphasized in text.
Other studies: Area (km2) Leafs R2 This study*
R2
Elder (1995) and Balk and Elder (2000) 6.9 18 0.59 0.47
Erxleben et al. (2002) 1 12 0.30 0.45
Elder (1995) - 10 0.40 0.45
Molotch et al. (2005) 19 7 - 8 0.37 0.33
5 Discussion
Efforts using differential equations to model snow redistribution rendered large errors when point wise comparison was applied (Prasad et al., 2001). Sx is found to be a significant predictor of snow distribution in multiple studies (Molotch et al., 2005; Prokop et al., 2013) as well as in this study.
BRT models are capable of dealing with independent variables that correlate to the dependent variable in a nonlinear manner. Regular regression models such as MLR were expected to perform badly on variables that correlate in a nonlinear way. This study confirms that BRT models produce better fit than MLR models. Sometimes the difference is small (see table 4 and appendix C) but generally larger than a 10% gain in model fit in the best models based on the R2 parameter.
In former investigations a uniform wind field was used to calculate for Sx. Forthofer et al. (2014) has shown that these models lack accuracy in the leeward side of slopes. Over the last decades fluid dynamics has been used for a large range of applications. Subsequently possibilities in the field of nonlinear flow modelling have improved. This development can be contributed to computer power and theoretical understanding (Wood, 2000). Mass consistent models are simpler and have been around for longer (Sherman, 1978) but are still used in everyday applications such as the prediction of forest fire spread (Forthofer et al., 2014). In this study the modelling software windstation and CFD is used. The results of windstation are not very impressive when boundary conditions are unknown (Lopes, 2003). The accuracy of these models is regarded with mixed enthusiasm. Finardi et al. (1998) is quite optimistic about the capabilities of both techniques. A comparison between both aforementioned techniques by Forthofer et al. (2014) revealed that both techniques work relatively well on the windward slopes of a mountain or hill and less well on the leeward slopes.
Incorporating mass-consistent and CFD wind field models as a factor was expected to increase the predictive power of the models. The small differences in the models reflect the great similarity in wind field model outcome over the sampling area as can be appreciated when sample locations (figure 5 a and b) are compared to wind field outcome (figure 7a, c, and d). There are some differences but these mainly occur outside of the measured area. The surface of the glacier is generally smooth, this causes wind flow over the glacier to be very similar to the mean wind direction over the entire area. The size of the study area is therefore an influence on the similarity between Sx parameters and general model outcome (Molotch et al., 2005). The glacier has a relatively even aspect all over its surface, especially the Lower Fonna. which leads to the fact that the study area does not incorporate many slopes with different aspects. Choosing a larger study site could overcome these problems and lead to more representative models.
Comparison between studies is hard because locations are often hard to compare climatically as well as geographically and the parameter used for model fit may vary between studies. The use of BRT further complicates the comparison for an increased number of terminal nodes will increase R2 and lower RMSE which are generally used as measures of fit. Still it can be insightful to look at other studies when limitations of comparison are understood and conclusions are read with coution. Research by (Elder, 1995; Balk and Elder, 2000) in the Tokopah Basin rendered considerable successes. They studied an area of 6.9 km2 that was characterized by high elevations between 3091 and 4003 meters. The site consisted of glacially scoured bedrock. Using the Sx factor they modelled snow distribution with an R2 of 0.59 using a regression tree with 18 leafs. A study conducted by (Molotch et al., 2005) in the same but a larger area (19 km2 ) using the same techniques averaged at R2
18 = 0.37 but used 8 or 7 terminal nodes. In an earlier study by (Elder, 1995) in 1993 and 1994 in the same area a R2 of 0.40 was found using a regression tree with 10 leafs. Erxleben et al. (2002) used the same technique in a forested part of the Colorado Rocky Mountains and reached an R2 of 0.25. The model fit in this study can be regarded as generally similarly accurate to those in the studies mentioned above, with two studies reaching higher and two reaching lower explanatory values (table 5).
As mentioned above, there are many reasons for these discrepancies, the most important ones will be mentioned here. First of all there is the difference in precipitation. It is difficult to compare the 1500 mm rainfall that falls on the areas in rocky mountains (ESRL, 2015) with the 182 and 103 mm that fell during the same period in Spitsbergen during the years 2012 and 2014 respectively (see weather data). The snow depths in the studies mentioned above reach up to 7 to 8 meters of snow while 2 – 3 in this study. Slopes in the current study are far less complex as the glacier surface is generally smooth with only a small area that harbours slopes over 15° where slopes in the studies mentioned above are more irregular and slopes often exceed 30°. Also the area that was modelled in this study was completely free of vegetation as opposed to the area in the Erxleben et al. (2002) study. The Spitsbergen area is subject to instable winter temperatures with regular warm spells. Such events disrupt a regular snow depositional environment by covering existing layers of snow with ice layers that potentially reach thicknesses of up to 20 cm (Hansen et al., 2014). All these factors could potentially either improve or decrease the added explanatory value of the Sx parameter and explain the difference between values in other studies.
6 Conclusions
It can be concluded that a model with higher predictive accuracy and higher resolution is produced than was available in literature for the Spitsbergen area. The use of the topographic parameters has produced a model that is comparable in explanatory power to the models that are created in other environments and are among the best models available at the moment. The use of wind modelling seems to increase explanatory power only for MLR models but this increase can be neglected in this study. The lack of increased model fit is probably thanks to the location of sampling points and the small area that was studied. The variables for height and the octagonal component of slope aspect produce the best models. The tested technique using BRT modelling did produce models with more predictive power than the MLR techniques on all possible variable combinations and points to the superiority of the BRT technique in snow depth modelling.
7 Acknowledgements
The author wants to thank Andy Hodson who has provided the data that made it possible to write this thesis and John van Boxel and Emiel van Loon who gave valuable feedback during the process. Thanks goes out to Gunnar Malom, Hanna Starf Breivik, Markus Richter, Mabel Gray and all the people at UNIS logistics for helping out with the field work.
19
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Appendix A: Distribution test description
Two tests are used to test the spread of snow depth sampling data:
The χ 2 tests first bin the data. The observations are binned in a spatial grid with a number of bins that approximates the square root of the number of observations. The number of observations per bin is compared to the number of observations that would be in one bin was the distribution uniform using a χ 2 distribution.
A technique described by Swan and Sandilands (1995) and Trauth et al. (2007) subjects a spatial distribution to a nearest neighbour criterion testing it for clustering. This technique tests data to the null hypothesis that the data is random using the Z statistic. The random distribution calculates the Z statistic. If the value of the test statistic is lower than -1.96 the null hypothesis is rejected, this indicates that the data is clustered. If the Z statistic is higher than 1.96 the hypothesis is also rejected, the data can be considered to be uniformly distributed. If Z is between -1.96 and 1.96 than the hypothesis is accepted and the data can be regarded as randomly distributed (Trauth et al., 2007). The hypothesis that the data has the same distribution wi a 95% significance can only be accepted for the 2014 data where the χ 2 data value exceeds the χ 2 test value. This test alone only indicates one data set that is useful for analysis. The results from this test render the expected values for the 2014 data that is uniformly distributed. It also shows us that the data from the year 2012 is least clustered with a Z value of -4.7814 (table 1) . Using these two tests the data sets from 2014 and 2012 are selected for the respectively building and cross verification of the model.
Table 1: Test results from χ2 test and nearest neighbour creterion test for data from 2012 to 2015
2012 2013 2014 2015 Z statistic -4.78 -6.16 21.00 -5.10 χ 2 data value 122.46 12.14 695.07 37.03 χ 2 test value 174.10 74.47 633.99 107.52
23
24
Appendix C: Complete table of results
2012 mean = , std = , n = 147
R2 RMSE MAE Leafs Function / included variables Note
1 BRT 0.46913 21.2024 17.1049 9 Sx_mean + slope + dem + northness 2 BRT 0.46455 21.2524 17.2133 9 Sx_mass + slope + dem + northness
3 BRT 0.47855 21.2251 16.8365 8 Sx_CFD + slope + dem + northness CDF is best
4 BRT 0.48001 21.3353 16.8342 6 Sx_mean + dem + Eastness
5 BRT 0.48156 20.8326 16.4029 10 Sx_mass + dem + Eastness
6 BRT 0.48826 21.7294 17.0112 11 Sx_CFD + dem + Eastness CDF is best
7 BRT 0.47806 22.432 16.7552 19 Sx_mass + slope + dem + Eastness
8 BRT 0.51264 21.7326 16.1327 19 Sx_mass + dem 9 BRT 0.44093 23.9806 17.9727 32 Sx_mass + Eastness 10 BRT 0.51142 22.4132 16.6151 25 Sx_mass + northness 11 BRT 0.55564 21.0196 15.4548 15
Sx_mass + slope + dem + northness + Eastness
12
BRT Same as 1
13
BRT Same as 10
1 MLR 0 1103.3529 1085.0759 Depth ~ 1 + Sx_mean + slope + dem + northness Overfit
4 MLR 0.35136 24.9626 18.1301 Depth ~ 1 + Sx_mean + dem + Eastness
5 MLR 0.35749 24.9434 18.1835 Depth ~ 1 + Sx_mass + dem + Eastness 6 MLR 0.35906 24.9039 18.1796 Depth ~ 1 + Sx_CFD + dem + Eastness
7 MLR 0 604.6015 596.9727 Depth ~ 1 + Sx_mass + slope + dem + Eastness Overfit
8 MLR 0.3313 25.388 18.5602 Depth ~ 1 + Sx_mass + dem
8 MLR 0.95753 32.3314 23.0669 Depth ~ Sx_mean + dem
No intercept / constant
9 MLR 0.33781 25.3589 18.4682 Depth ~ 1 + Sx_mass + Eastness
10
MLR 0.32707 25.4752 18.3583 Depth ~ 1 + Sx_mass + northness
11
MLR 0 4662.1641 4590.6913
Depth ~ 1 + Sx_mass + slope + dem + northness +
Eastness Overfit
12
MLR Same as 1
13
25
2014 n = 581
R2 RMSE MAE Leafs Function / included variables Note
1 BRT 0.063337 16.9634 12.2215 29 Sx_mean + slope + dem + northness 2 BRT 0.070839 16.7493 12.0643 29 Sx_mass + slope + dem + northness
3 BRT 0.10072 16.2386 11.9972 24 Sx_CFD + slope + dem + northness CFD is best
4 BRT 0.19148 15.2066 10.9774 21 Sx_mean + dem + Eastness
5 BRT 0.1964 15.2613 10.8248 25 Sx_mass + dem + Eastness mass is best
6 BRT 0.18491 15.4215 11.0004 23 Sx_CFD + dem + Eastness
7 BRT 0.20798 15.2098 10.9561 20 Sx_mean + slope + dem + Eastness mean is best
8 BRT 0.1394 15.8558 11.3512 21 Sx_mean + dem mean is best
9 BRT 0.11676 15.9542 11.9176 28 Sx_mean + Eastness mean is best
10 BRT 0.10072 16.2386 11.9972 24 Sx_CFD + northness CFD is best 11 BRT 0.18968 15.2533 11.0966 21
Sx_mean + slope + dem + northness + Eastness
mean is best 12 BRT Same as 1 13 BRT Same as 10 1 MLR 0 422.8657 411.5805
Depth ~ 1 + Sx_mean + slope + dem +
northness Overfit
4 MLR 0.020929 16.9528 12.4051 Depth ~ 1 + Sx_mean + dem + Eastness
5 MLR 0.023795 16.9222 12.3535 Depth ~ 1 + Sx_mass + dem + Eastness
6 MLR 0.02935 16.9061 12.4907 Depth ~ 1 + Sx_CFD + dem + Eastness CFD is best
7 MLR 0 647.2405 630.1542 Depth ~ 1 + Sx_mean + slope + dem + Eastness Overfit
8 MLR 0.030668 16.9267 12.5082 Depth ~ 1 + Sx_CFD + dem CFD is best
8 MLR 0.96655 20.9605 15.0298 Depth ~ Sx_mass + dem
No intercept / constant, mass is best
9 MLR 0.015465 17.0127 12.7018 Depth ~ 1 + Sx_CFD + Eastness CFD is best
10
MLR 0.03418 16.8909 12.3485 Depth ~ 1 + Sx_mass + northness mass is best
11
MLR 0 331.9183 296.4848
Depth ~ 1 + Sx_mass + slope + dem + northness
+ Eastness Overfit
12
MLR Same as 1
13
26
both years mean = , std = , n = 725
R2 RMSE MAE Leafs Function
1 BRT 0.069967 26.0846 18.1297 108
Sx_mean + slope + dem + northness
2 BRT 0.073407 26.1361 18.3662 88 Sx_mass + slope + dem + northness
3 BRT 0.27783 22.5732 15.9784 31 Sx_CFD + slope + dem + northness CFD is best
4 BRT 0.11113 25.1523 17.5136 19 Sx_mean + dem + Eastness 5 BRT 0.085139 25.6997 17.9636 41 Sx_mass + dem + Eastness
6 BRT 0.32258 21.8728 15.6325 34 Sx_CFD + dem + Eastness CFD is best
7 BRT 0.33787 21.5571 15.2816 33 Sx_CFD + slope + dem + Eastness CFD is best
8 BRT 0.29992 22.219 15.8789 33 Sx_CFD + dem CFD is best 9 BRT 0.29515 22.328 16.223 33 Sx_CFD + Eastness CFD is best 10 BRT 0.25694 22.9797 16.4367 33 Sx_CFD + northness CFD is best 11 BRT 0.34474 21.4235 15.2914 33
Sx_CFD + slope + dem + northness
+ Eastness CFD is best
12
BRT 0.5141 18.3873 12.9117 28
Sx_mean + slope + dem +
northness + precip Mean is best
13 BRT 0.51512 18.3388 13.1496 25 Sx_CFD + northness + precip CFD is best 1 MLR 0 550.2661 536.3668
Depth ~ 1 + Sx_mean + slope +
dem + northness Overfit
4
MLR 0.04136 26.4186 18.1315
Depth ~ 1 + Sx_mean + dem + Eastness
5
MLR 0.046287 26.3431 18.0707
Depth ~ 1 + Sx_mass + dem + Eastness 6 MLR 0.26005 23.0046 16.4897 Depth ~ 1 + Sx_CFD + dem + Eastness CFD is best 7 MLR 0 271.0859 262.6816
Depth ~ 1 + Sx_mass + slope + dem
+ Eastness Overfit
8
MLR 0.22596 23.545 16.8342 Depth ~ 1 + Sx_CFD + dem CFD is best
8
MLR 0.067613 27.149 19.5336 Depth ~ Sx_mean + dem
No intercept / constant, CFD is best, others order of magnitude worse R2
9
MLR 0.23331 23.3941 17.007 Depth ~ 1 + Sx_CFD + Eastness CFD is best
10
MLR 0.22268 23.5811 17.0023 Depth ~ 1 + Sx_CFD + northness CFD is best
11
MLR 0 282.1304 230.6968
Depth ~ 1 + Sx_mass + slope + dem + northness + Eastness Overfit
12
MLR 0 369.0246 311.4961
Depth ~ 1 + Sx_mean + slope +
dem + northness + precip Overfit
13
MLR 0.37557 20.926 14.7379
Depth ~ 1 + Sx_mean + northness + precip
