Analysis of Mapping Techniques for Mountain Precipitation : A Case Study of Alpine Region , Austria

Truly representative precipitation map generation of mountain regions is a difficult task. Due to poor gauge representativity, complex topography and uneven density factors make the generation of representative precipitation maps a very difficult task. To generate representative precipitation maps, this study focused on analyzing four different mapping techniques: ordinary kriging, spline technique (SP), inverse distance weighting (IDW) and regression kriging (RK). The generated maps are assessed through cross-validation statistics, spatial cross-consistency test and by water balance approach. The largest prediction error is produced by techniques missing information on co-variables. The ME and RMSE values show that IDW and SP are the most biased techniques. The RK technique produced the best model results with 1.38mm and 72.36mm ME and RMSE values respectively. The comparative analysis proves that RK model can produce reasonably accurate values at poorly gauged areas, where geographical information compensated the poor availability of local data. Keywords-mountain regions; poor gauge representativity; spatial interpolation techniques


INTRODUCTION
Scientists generally agree that the earth is undergoing critical climate changes [1][2][3][4][5].Climatic changes based on the hydrology, development and management of water resources have been under major attention over the years.The distributed hydrological models are gaining huge standing in analyzing and investigating the overall impacts on mountain regions and their environment [6][7][8].Distributed hydrological models require input variables like estimates of climatic variables at regular and continuous intervals as pre-requisites for their proper functioning [5,6,9,10].The amount of rainfall is the most vital parameter for any distributed hydrological model.Nevertheless, the amount of rainfall is a matter of various uncertainties like measurement errors, systematic errors during applying interpolation and stochastic errors resulting from the random nature of rainfall.The performance of the models depends heavily upon the accurate estimation of precipitation over specific area and time.The results can be highly compromised [11].This challenge of reliable and accurate rainfall estimates increases in mountain regions where the geography is complex and measuring stations are scattered over vast areas and concentrated in the valleys [12][13][14].The measurement of accurate data for the mountainous range is a very difficult task, resulting in poor representation in the model that analyzes the various rainfall patterns.Resultantly, in these types of situations, when no single method is optimal nor the accuracy of a specific interpolation technique is proven, the performance relies on the variable under study, spatial configuration, and the assumptions used in the estimation [15][16][17].The accuracy of measured data under a certain technique can be verified by comparing and analyzing by applying different techniques to the same data.In order to achieve this objective, the current study analyzes the range of stochastic and deterministic mapping techniques to estimate the values at ungauged locations.

A. Study Area
The study was carried out at the Alpine area of Kitzbühl Ache region situated in the Austrian Eastern Province of Tyrol with an area of about 2000km 2 .Complete details about the study region are given in [18].The available 30 year time series  of mean annual precipitation of 14 gauge stations are taken into consideration for spatial interpolation.The catchment topography is highly rugged with elevation that ranges from 400m to 2400m above m.s.l as shown in Figure 1.The catchment shows strong seasonal precipitation behaviors.to deterministi que includes in for interpolati e includes ord out on the ba [22][23][24][25].The m intrinsic hypo or climatic ma fly presented o [20,29,30].
n gauge stations ue that interp in the neighbo ed that the var ight with incre a is given as: (1) (2) n x, λ i = the wei bserved value point to interp oice of powe esults.High p and results in more weighta hich represen surfaces [34].ns to certain nu the whole su while passing a ber piece acro g the entire su are regularize m changing su = in known smooth ution of a syst own point to nction, C= the efficients found

OK Techniqu
This techniqu d works the s ghboring mea ed on distance known points.kriging, after D.L. Krige: = th served value a ues at the i th l number of ation and spat und the pre imator varianc own as "best ights sum to u 1 ( , (  where matrix A contains the semivariances of all data point pairs, b is a vector containing semivariances between the location of interest and observed point.λ i is the weight to be calculated [20,[37][38][39][40][41].

F. RK Technique
Authors in [17] conclude that ordinary kriging does not produce representative precipitation values in mountain regions all the time.Alternative techniques might fully utilize the relationship between predicted variable and co-variables for variability analysis.Two of these techniques are cokriging and regression kriging.Due to poor cross-covariance between precipitation and any topographical variable, the former application did not produce good results.However, in such situations regression kriging seemed the natural choice, which is commonly used in hydro-sciences [42,43].The technique defines the relationship between target and co-variables in order to predict the values at grid nodes through linear regression.The auxiliary variables are easy to measure, they provide an alternative to target variable at the under sampled locations to model and quantify the existing patterns.To quantify the existing trend of the variable and its variability in a regression model, we preferred to use multiple linear regressions -the further extension of straightforward linear regression with the variety of descriptive variables.As altitude alone flopped to signify the variation of precipitation sufficiently, we tested other variables like slope, topographic index, aspect, hill shade, curvature etc.These extra explanatory variables were obtained from the elevation model, in order to make better predictions of the target variables at grid nodes of the DEM.We adopted stepwise procedures to select the most crucial variables and the subsequent regression equation to predict the target variable at un-sampled locations.The regression led to a three parameter equation, significant at 5% level, explaining 62% of variability of precipitation.
The resulting regression residuals (ε) are further kriged at grid nodes by fitting variogram models, and then finally both values are summed up to predict the target variable values.

III. VALIDATION METHODS
The mapping techniques performance was assessed by following three steps:

A. Cross-Validation Statistics
The cross validation method depends on eliminating one sample location (measurement station) from the data set at a time and calculating the value of the removed sample with the remaining data points.This routine was followed for each measurement station.The comparative indices were then used as a measure of prediction quality by the ME and the RMSE, which are defined as follows: where n is the number of validation points, and ˆ& T T i i are the predicted and observed values at location i.The ME criterion is used to check the conditional bias property, while the RMSE criterion assesses the precision quality.A smaller value of RMSE indicates higher accuracy and vice versa.Cross validation statistics can be used to find the optimal mapping technique, however, the presence of short range correlations in data may raise questions regarding the reliability of its statistical results [35].

B. Cross-Consistency
A second step to analyze reliability and consistency of predictions, spatial cross-consistency approach was adopted [44].All statistical parameters of different calculated precipitation mapping estimates were compared with a referenced precipitation map (RPM).This RPM was carefully produced during a 4 year project (www.waterpool.org) in which different experts from different institutes were involved, and results were consistent with water balance estimates.

C. Water Balance Approach
Finally all calculated precipitation maps including the RPM were evaluated by means of a general water balance approach.
Gridded discharges were calculated for each mapping technique as a result of subtraction of actual evapotranspiration grid estimates from interpolated mapping precipitation grid estimates.The actual evapotranspiration values are obtained from the hydrological Atlas of Austria.Storage changes can be ignored, as for long-range mean annual water balances; it was assumed that there is no sensible change in the water contents of different reservoirs, e.g.groundwater, snow cover [44,45].The difference between the calculated discharges with observed discharges gives a measure for the reliability and consistency of the precipitation.

IV. RESULTS AND DISCUSSION
The ME values clearly indicate the superiority of RK technique over the other techniques, showing almost 42% less bias than OK.Whereas IDW and SP techniques produce higher bias, almost 2 to 5 times higher than OK technique.The RMSE results also reveal the primacy of RK technique over the other ones.However, all techniques yield high uncertainty in calculated values.Thus, considering co-variables into account certainly improves the performance by decreasing RMSE values from 111 to 72.The spatial cross-consistency tests are also conducted by computing precipitation maps with ref ado value at lo i = the coeffic equations, r i = point, K 0 = the qual to 0.577 n of a system o ivariogram for IDW works, However, we pend on total s 35] developed the theory in t d value at loc and λ i = the w λ i depends on ints, distance hips among th tions.Krigin es the ocation x, f(x i , cients found = the distance e modified B 7215, and α i = of linear equat r spatial predi which weight eights are not spatial arrange d a general for the mining ind (6) cation x, T(X i ) weight of obse n: Fitting mod to the predi he measured Fig stud

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