Juraj Parajka, Experimental Hydrological Station, Institute of Hydrology, Slovak Academy of Sciences, Ondrašovecká 16, 031 05 Liptovský Mikuláš, Slovakia.

Jaroslav Hofierka, GeoModel, s.r.o., J. Grešáka 22, 085 01 Bardejov, Slovakia

Helena Mitášová, Department of Geography, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.

Keywords: precipitation, digital terrain model, cross-validation, spatial interpolation

Spatially distributed measurements of precipitation have gained renewed interest in connection with climate-change impact studies, determination of water budgets at different temporal and spatial scales, the validation of atmospheric simulation models and hydrological models. Frequently, precipitation values are available at a number of meteorological stations and spatial estimates of precipitation fields are obtained by interpolation techniques. Such estimates are especially difficult to obtain in mountainous environments where can be large changes in these fields. In these areas measured data are sparse and are usually restricted to lower elevations.

Doplnit o dalsie moznosti hodnotenia interpolacie (nestandardne metody): porovnanie s rucnou mapou a mapu vodneho odtoku

This study focuses on the spatial interpolation of the long-term mean annual precipitation (MAP) station (point) data on the territory of Slovakia using Regularized Spline with Tension (RST). Its main goals are:

1. to find factors (parameters) affecting quality and accuracy of precipitation interpolation in mountainous areas
2. to find out a set of optimal RST parameters minimizing interpolation errors (such as CV RMSE),
3. to test the quality and accuracy of RST in interpolation of MAP data for Slovakia using selected evaluation procedures,
4. to propose an optimized approach for precipitation modeling with respect to time and spatial variability and dependancies in precipitation data.

The RST method has been described by Mitas and Mitasova 1993 (an earlier version called Completely Regularized Spline) and Mitasova et al. 1995 (the general d-dimensional formulation with applications for 2D,3D,and 4D data). Therefore, here we briefly recall only the basic principles and focus on new developments and issues specific to interpolation of precipitation data.

(the rest will be added later by Helena and Lubos)..........

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One of the advantages of the RST method is its flexibility (adjustability to various types of phenomena) within the single radial basis function (as opposed to the rather complex and often subjective task of selecting the proper semivariogram in kriging). While most applications require only slight adjustment of tension parameters there are applications (such as precipitation in mountanious regions) when a wider set of tunable parameters can improve the predictive accuracy. We investigated the impact of the following parameters on the predictive accuracy of RST:

• horizontal uniform tension ,
• uniform smoothing w,
• anisotropy: rotation and scale (theta,s),
• vertical tension c (vertical scaling)
• level of detail (smoothing) of 3-rd variable

The parameters can be selected empirically, based on the knowledge of the modelled phenomenon, or automatically by minimization of the predictive error estimated by a cross-validation procedure (Mitasova et al 1995).

Napiste ako ste to robili - toto je text z nasho IJGIS clanku 1995

We have used an ordinary cross-validation to find both optimal tension and smoothing. If we assume that $\{w_j\}$ are the same for all data points then the smoothing parameter is $w=w_j/w_0$.

Let $S_{N,\varphi,w}^{[k]}$ be the smoothing spline with tension using all the data points except the k-$th$. We can take the ability of $S_{N,\varphi,w}^{[k]}$ to predict the missing data point $z_k$ as a measure of predictive capability of the spline with the given tension $\varphi$ and smoothing $w$, which can be represented by a mean error $V_0$

........

Then the optimal values of tension $\varphi_O$ and smoothing $w_O$ are found by minimizing $V_0(\varphi,w)$. The cross-validation error computed for each point can also be used for the analysis of spatial distribution of predictive error and location of areas where this error is high and denser sampling is needed.

Maybe include some comments on: many of the geostatistical concepts can be exploited within the spline framework.

Try to fit the RST RBF to variogram? -see whether Roling has done that.

There several methods optimizing a spatial interpolation. Among them the cross-validation is the most widely used. Alternatively, comparison with other maps such as expert hand-drawn precipitation maps and maps of derived hydrological processes such as runoff can be used.

Duro

Duro

Zhodnotenie evaluacnych metod

- it would be good to have both annual and daily for both sets ????????

(Jaro)

Long-term mean annual precipitation values (MAP) were obtained from 423 meteorological stations and from 12 storage gauges of the Slovak Hydrometeorological Institute network. Length of records was 30 years (1951-80) for all meteorological stations and for 4 storage gauges and 20 years (1961-80) for 8 storage gauges. The elevation data set was represented by digital elevation model of Slovakia with resolution 1x1 km (Fig. XX). For the cross-validation purposes we used digital interpretations of manually produced (expert) isoline maps of the MAP and evapotranspiration, as well as long-term mean annual runoff data from 57 gaged sites. The sites represent basins (Fig. XX) ranging in size from 40 to 3800 km².

(Duro, Jaro, Hela)

nie je dobre mat daily pre jedno uzemie a annual pre druhe - aspon pre jedno by sme mali mat oboje
do kazdeho by bolo dobre dat graf zavislosti CV od tenzie, zmult, vyhladenia terenu a ukazat ktore parametere su nakolko
dolezite (a ktore sa oplati optimalizovat a ktore staci len odhadnut)
Sem dat aj tabulku porovnania s inymi metodami s odcitovanim povodnych publikacii (svajc a Durov paper/PhD)

Resulted MAP precipitation grid map for Slovakia is presented in Figure XY. The precipitation estimates by TPS3D method show patterns fairly related to topography with minimum precipitation at lowlands and increasing MAP values in the mountains.

To test the accuracy of mapping methods used for spatial interpretation of MAP over the territory of Slovakia, so called ”jack-knife” analysis was conducted. The analysis consists of setting aside one precipitation sample, using remaining precipitation observations to compute an interpolated surface, and then analyzing the difference between values interpolated from the surface and actual precipitation values at the withheld sample sites. Results of this quantitative uncertainty analysis are presented in Table XW.

The most noticeable differences among the mapping methods are statistical parameters of range and sample variance. Although TPS3D estimates produced the highest range of differences, the lowest sample variance statistics indicate that this method gives most satisfactory results.

The spatial cross-consistency of resulted precipitation map has been investigated with respect to the digital interpretation of manual contoured precipitation (PMAN) map. Similar approach was chosen in Parajka (1999), where the cross-consistency between kriging, cokriging and PMAN maps has been evaluated. The difference between computed TPS3D map and the PMAN were expressed as grid map of absolute percentual differences (Fig. XY2) and frequencies of each category of difference were tabulated (Table XY).

Table XY shows that TPS3D precipitation map is more similar to PMAP then kriging or cokriging estimates. The differences between TPS3D grid map and PMAP are at 87% of the territory of Slovakia within the range of 10%. The differences are observed in some mountainous regions with sparse observations, but there is questionable not only the accuracy of TPS3D estimates, but the accuracy of hand-drawn isoline map as well.

The cross-consistency have also been tested within the framework of hydrological balance. For each grid precipitation map – PMAP, TPS2D, TPS3D, kriging, cokriging - we constructed long-term mean annual runoff map using water balance equation (R=P-ET), where spatial representation of evapotranspiration was represented by digital interpretation of expert-drawn evapotranspiration map. The differences between computed basin averages of mean annual runoff and gaged runoff values for 57 basins were calculated and statistical comparison of these differences is listed in Table XY2.

The smallest range, as well as the lowest standard error and sample variance values of the average basin runoff differences presented in Table XY2 indicate, that average basin runoff values computed from TPS3D estimates come nearer to the measured values. In comparison with the other precipitation models (kriging, cokriging, TPS2D, PMAP), TPS3D model fulfils the best the balancing constraints of the hydrological balance equation.

Detailed water balance check in regions with the highest differences between PMAP and TPS3D (presented in Table XXX) has shown that estimates provided by TPS3D method are more accurate with respect to measured runoff than PMAP.

The acceptable results of the cross-validation procedures demonstrate that TPS3D method offers repeatable and sufficiently accurate estimates of the MAP and could be used to describe the spatial variability of MAP over the territory of Slovakia.

(Jaro, Hela, Duro)

Tu je najdolezitejsie zhrnut co sa urobilo a co je z toho nove
  (zavery o moznostiach, pristupoch a limitoch pri modelovani zrazk. dat, moznosti zlepsenia)
 
 

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