Processing digital terrain models by regularized spline

Proceedings of the Open source GIS - GRASS users conference 2002 - Trento, Italy, 11-13 September 2002

Processing digital terrain models by regularized splinewith tension: tuning interpolation parameters for

different input datasets

Tomas Cebecauer*,**, Jaroslav Hofierka*,***, Marcel �úri*,**

* GeoModel s.r.o., Hlaváčiková 26, 841 05 Bratislava, Slovak Republic, tel. +421 905 441 409, fax +421 26531 5915, e-mail [email protected], [email protected], [email protected]

** Institute of Geography, Slovak Academy of Sciences, �tefánikova 49, 841 73 Bratislava, Slovak Republic,tel. +421 2 5249 2751, fax +421 2 5249 1340

*** Department of Geography and Geoecology, Faculty of Humanities and Natural Science,University of Pre�ov, 17. novembra 1, 081 16 Pre�ov, Slovak Republic

1 IntroductionDigital terrain models (DTM) are widely used in geographical information systems (GIS)for different purposes, ranging from simple data analysis and visualization tosophisticated and complex modeling. The measured terrain data are often processed toDTM using spatial interpolation. Different methods of interpolation are currentlyavailable in GIS, however, most commonly used are: inverse distance weighted averageinterpolation, interpolation based on a triangulated irregular network (TIN), variationalmethods and geostatistical methods (kriging). Details about these methods can be foundin literature, e.g. [1].

The quality of DTMs depends mainly on the quality of input data and interpolationmethod. The selection of appropriate interpolation method is important because for aparticular type of data one method performs better than the other [2] (e.g. contours vs.GPS data). Hence, the quality of interpolation method is also determined by itscapabilities to process different types of data.

The regularized spline with tension (RST) interpolation method is one of the bestinterpolation methods available in current GIS [3, 4, 5]. RST is a variational methodbased on the assumption that the interpolation function should pass through (or close to)the input data points and should be at the same time as smooth as possible [6]. Thespecific feature of RST is the existence of regular derivatives of all orders needed forsurface geometry analysis [6]. Tension and smoothing parameters are useful to change theshape and smoothness of surface according to the modelled phenomenon. The accuracyof interpolation by RST can be assessed by various statistical methods evaluatinginterpolation errors (residuals) in input points or predictive error of interpolation in areasoutside input points using cross-validation technique [5, 6, 7]. Unfortunately, thesequantitative measures usually describe only a statistical aspect of interpolation quality,but applicability of the interpolated DTM for a specific purpose can be still limited (e.g.hydrologically correct DTM, correct representation of undersampled features) [c.f. 7 page317, 8].

The paper identifies common problems in application of s.surf.rst on different types ofterrain data. A careful selection of RST parameters and, if necessary, pre-processing ofinput data may substantially improve interpolation results. Application examples includeelevation data taken from contours and elevation points, photogrammetric measurementsat various scales and volume of datasets. The results of interpolation using different

mailto:[email protected]

mailto:[email protected]

Processing digital terrain models...2

parameters are presented incorporating both conventional accuracy assessment methodsas well as expert evaluation of results. The solution of interpolation problems associatedwith insufficient data or specific terrain features [9, 10] is presented.

2 MethodsThe RST interpolation method is implemented in GRASS GIS as s.surf.rst and s.vol.rstcommands (2- and 3-dimensional versions, respectively). In older GRASS releases the2D RST was called s.surf.tps. In this paper we deal specifically with s.surf.rst. A specificfeature of this method and its implementation in GRASS is a set of interpolationparameters providing flexibility in data processing. These parameters give the user acapability to process �difficult� datasets (e.g. photogrammetric, GPS and LIDAR data),that can be hardly effectively processed by other conventional methods. As we show inthe paper, the RST is capable of processing such datasets, but it usually requires goodknowledge of the method and its parameters and pre-processing of input data is alsoneeded.

The physical interpretation of RST (thin steel plate minimizing its bending energy)explains the behavior of interpolating surface in areas with insufficient data density andsteep changes in gradient. For example, RST can create overshoots (local depressions)which reflect improper selection of parameters and insufficient data. Therefore it isimportant to know the behavior of the function in these areas (stiffness or elasticity) andhow to minimize interpolation artefacts by optimization of interpolation parameters ordata pre-processing.

The most common problems using RST in GRASS can be summarized as follows:• existence of segments,• overshoots,• presence of non-realistic artificial surface features,• incorrect representation of artificial features.

2.1 Tension and smoothing

The s.surf.rst command is controlled by a set of parameters described in detail by itsGRASS manual page. The behavior of interpolation function is primarily controlled bytension parameter (tension) and smoothing (smooth, smatt). Spatially variable smoothing(defined by smatt) is a new feature. It allows to define �freedom� or precision ofinterpolation in different areas within given dataset according to the character andreliability of the input point data and/or user needs. At the same time it defines asmoothness of resulting surface and influences level of detail or scale to some limitedextent.

The RST method is scale dependent and the tension works as a rescaling parameter. Thismeans that high tension "increases the distances between the points" and reduces therange of impact of each point, low tension "decreases the distance" and the pointsinfluence each other over longer range. Surfaces with tension set too low behave like astiff steel plate and overshoots can appear in areas with rapid change of gradient. Surfaceswith tension set too high behave like a membrane (rubber sheet stretched over the datapoints) with peak or pit ("crater") in each given point and everywhere else the surfacegoes rapidly to trend. If digitized contours are used as input data, high tension can causeartificial waves along contours. Lower tension and higher smoothing is suggested forsuch case.

Strong gradients in data occur, for example, where two or more neighboring points havevery different elevation values. This situation is quite common in areas where steepslopes are rapidly changing to flat areas or when interpolating artificial features in terrain.

http://grass.itc.it/gdp/html_grass5/html/s.surf.rst.html

Tomas Cebecauer, Jaroslav Hofierka, Marcel Suri 3

This often leads to sharp overshoots. The algorithm identifies extreme overshoots andinforms the user about the magnitude and position of the overshoot by warning message.However, small overshoots may still remain uncovered during computation. Manyapplications need a DTM with minimized overshoots and local depressions. This problemcan be in most cases resolved by increase of booth smoothing and tension parameters.

It is worth to note that unless the tension parameter is used without –t option (dnormindependent tension) the tension is influenced also by npmin and dmin parameters viachange in distances among points. Anisotropic features can be better modelled usingtheta and scalex parameters that change the tension in a specified direction and scale,respectively.

2.2 Segmentation parameters

If the number of input points in the dataset is greater than segmax (default value is 40)segmented processing in s.surf.rst is used. The segmentation is based on a local behaviorof RST method. The region is split into rectangular segments, each having less thansegmax points and interpolation is performed on each segment of the region. Surface ineach segment is computed using points in the segment as well as nearby points outsidethe segment that are in the rectangular window surrounding the given segment. Thenumber of all points (in the segment, as well as those nearby) taken for surfaceinterpolation within a particular segment is controlled by npmin, the value of which mustbe larger than segmax. However, the actual number of points is based on internalsegmentation analysis. The user can save output vector files treefile and overfile whichrepresent the quad tree used for segmentation and overlapping neighborhoods from whichadditional points for interpolation on each segment were taken. The size of segments iscontrolled by segmax. The smooth interconnection of segments is controlled by npminparameter. In spite of this fact, sometimes the input data distribution is so heterogeneousthat output surface contains visible segments (steps). This artefact is visible especially inflat areas with low density of input points that are close to areas with high data densities(figure 3a).

The segmentation procedure is also indirectly influenced by dmin parameter. The dminparameter defines a minimum distance between points, extra points are excluded fromcomputation. In case of extremely unevenly sampled points (contour lines in lowlandswith sparse interval, areas of rocks without elevation points) some areas can beinsufficiently covered by input data, while others are oversampled. The result is a creationof steps between adjacent segments, instead of their smooth interconnection. The densityof input points in oversampled areas can be decreased by dmin parameter, and the numberof points used in segments can be increased by higher npmin. The drawback of increasingnpmin is that the speed of computation slows down dramatically. Therefore, theincreasing of the npmin parameter has some limitations. For huge datasets comprising ofmillions of input points the time of processing may be the crucial restriction. Moreover,in some areas even considerable increase in npmin parameter may not overcome unevenspatial distribution of the input points and �steps� between segments still remain visible.

2.3 Data pre-processing

Usually, the input data sources do not contain sufficient information on a modelledphenomenon. For example, contours on topographic maps usually do not coversatisfactorily mountainous rocky areas, alluvial planes, in some situations contours aretoo much generalized to preserve enough information for creation of hydrologicallycorrect DTM. The input data important for interpolation, such as breaklines, are often notavailable in the form needed for interpolation. In such cases additional pre-processing ofinput data is needed to get the highest-possible quality of interpolation.


In mountainous rocky areas with sparse information about terrain elevation a user mustoften add supplemental data representing ridges, passing through peaks and saddlepoints. This preserves shapes and creates realistic picture of the extreme type of terrain,which is not possible to obtain solely by using elevation points and contours fromtopographic maps. Unfortunately, precise elevation of rocky terrain is rarely availableeven in detailed maps and a user is constrained to use data from GPS or less reliablesources such as schematic sketches or expert evaluation. Moreover, different parts ofterrain require different smoothing and interpolation precision. Peaks need low smoothingto keep elevation as precise as possible, slopes represented by contour lines needintermediate smoothing, while inaccurate data points from less reliable sources needhigher smoothing. Therefore, pre-processing comprises of analysis and preparation ofsupplemental elevation points and/or selecting subsets of data with variable smoothing.Appropriate smoothing parameters can be identified by exploring preliminary results ofinterpolation or by statistical evaluation of residuals.

Contour lines provide only a generalised representation of terrain. It is a commonproblem that interpolation of contour data leads to DTM with local depressions in valleys.Problems originate predominately in narrow valleys where undersampling of input databetween contour lines may cause the emergence of artificial step-like features andmicrodepressions dramatically influencing the overland flow routing. Themicrodepressions can be minimised by addition of supplemental points placed inundersampled valley areas. A terrain “skeleton” comprising important features such asvalley bottoms, ridges and lines of discontinuities can supplement contour lines. Theterrain skeleton preserves the shape of terrain in areas important for many applications.The construction of the terrain skeleton is based on expert evaluation of contour data andanalysis of the preliminary interpolation results of original data.

Overriding default parameters controlling the behaviour of interpolation function and/orsegmentation overcomes problems with overshoots and segmentation only partially. Inthese cases a user has to add supplemental points to the dataset in the undersampled areasor areas with overshoots based on assumption that terrain surface is smooth. Also, it isappropriate to add supplemental points with higher smoothing to give more freedom tointerpolation function in these areas.

Existence of oversampled input data along linear features (e.g. contour lines) iscommonly perceived as unsuitable [11]. However in specific cases addition ofsupplemental points along linear features may help to better represent real shape ofterrain surface. This is especially useful in cases of modelling of artificial (human-made)features in urban areas, where faults, embankments and other features with breaklinesdominate. The artificial features are usually sampled densely and are in the form of lines,but s.surf.rst does not support explicit definition of such linear features. Usually, onlyvertexes of linear features are treated as input points for interpolation, which is rarelysufficient. To preserve the shape of such features, a higher density of points is necessary.Fortunately, it is not difficult to derive vector data representing these features in order toadd supplemental points along linear features. Such additional points help theinterpolation function to hold the shape of the surface as close to the reality as possible.

3 Examples

3.1 DTM of urban area from photogrammetric data

Photogrammetric elevation data from urban areas are usually processed using TINinterpolation methods. The main reason is an existence of many artificial terrain featureslike roads, excavations, breaklines etc., where TIN methods proved to be quite effective.However, raster-based methods can be successfully applied too, especially where area ofinterest covers both natural and urban landscapes. Photogrammetric data are usually


spatially heterogenuous. Smooth areas (e.g. rural, forests) are poorly covered, whereasartificial features (roads, breaklines) have high data density. Spatial interpolation of suchareas requires a high resolution DTM and data pre-processing to preserve all technicalfeatures in DTM. The following example shows processing of photogrammetric datacovering urban area of Bratislava (Slovakia).

To create a DTM for technical applications using RST, the user must set approppriateinterpolation parameters and pre-process input data according to different type offeatures. In our cas the photogrammetric data were collected in separate layers (terrainpoints, breaklines and streets) where linear features were 3D entities having elevation as athird dimension for each vertex. The interpolation parameters were chosen with thefollowing values:

! to highlight artifical features the tension was set to 55;

! every layer of terrain feature can hold different smoothing according to required levelof interpolation accuracy. For example, rural areas represented predominantly byterrain points usually can be smoothed while roads and breaklines need a preciseinterpolation. At the same time it must be kept in mind that too low smoothing maycause overshoots. Therefore the smoothing of 0.5 was given to terrain points whilevalue of 0.25 was used for streets and breaklines.

Moreover, the density of data representing linear features was increased by pre-processing.

Table 1 shows results of interpolation where only points and vertexes of lines fromoriginal photogrammetric data were used. It is obvious that artificial features are notcorrectly represented (figure 1a). By adding points for linear features (increasingsampling density of linear features using step 2.5 m; dmin=2 m), a substantialimprovement in the linear features representation was achieved (figure 1b). However,several issues still remain unsolved in the neighborhood of artificial features (overshoots,smooth transition between feature and surrounding terrain). This was improved by addingsupplemental points in problematic parts identified by visual inspection (figure 1c).

Table 1 Comparison of datasets with different preprocessing for technical DTM

data pre-processing # of points RMS errorterrain points and only vertexesof the linear features (original data)

9 209 0.481

terrain points and densified linearfeatures (step 2.5 m)

83 328 0.241

terrain points, densified linear features(step 2.5 m) and supplemental data points

107 857 0.189


Figure 1 Comparison of interpolation results with different pre-processing a) use ofterrain points and line vertexes, b) use of terrain points and densified linear features, c)use of terrain points, densified linear features and supplemental points (visualized in nvizusing a vertical exaggeration factor = 2)

c)

b)

a)


3.2 Undersampled mountainous area

Contour lines do not represent rocky terrain features in mountains and instead specificcartographic symbols are used to depict irregular, rugged terrain on topographic maps.However, for whatever reason, it is intended to produce DTM with elevation coveringthese areas as well.

In the second example we have used contour lines and elevation points from topographicmaps for High Tatras region (Slovakia) with large part of rocky terrain. Interpolatingthese areas without supplemental data leads to �peaky� surface that is far from reality(figure 2a). The peaks emerge in undersampled areas where isolated input elevationpoints are present. Addition of supplemental points representing specific terrain linearfeatures (ridges - shown in blue) can substantially improve quality of DTM (figure 2b).The supplemental linear features connecting isolated elevation points were prepared byexperts using additional supporting materials (different maps and schemes). Because ofdifferent reliability and importance of the input data, lines and points have variablesmoothing (peaks 0.05, ridges 5.0, contours 1.0).

Figure 2: Influence of supplemental data points in mountainous rocky terrain oninterpolation results: a) data from topographic maps, b) data from topographic mapssupplemented by ridges (High Tatras, Slovakia).

a)

b)


Table 2 Comparison of datasets with different preprocessing for DTM in mountains

data pre-processing # of points RMS errorcontours and elevation points 134 297 2.945contours, elevation pointsand supplemental data points

137 616 4.547

It is apparent from the table 2 that the subjective increase of interpolation qualityperceived by a user may not be in correspondence with magnitude of residualsrepresented by RMS error. In this case the increase in RMS error is a consequence ofusing supplemental input points with high smoothing (high freedom of interpolationfunction).

3.3 Visible segments in undersampled flat areas

In this example we focus on visible segments in DTM due to unevenly distributed inputpoints. The problem was solved by different combination of segmentation parameters(npmin, dmin) and by adding supplemental contours to the original dataset.

The selected area represents terrain with uneven distribution of input data in western partof Ko�ice basin (Slovakia) and surrounding mountains. Input data were derived fromcontours from topographic maps at scale 1:50 000. The interpolation with defaultsegmentation parameters led to DTM with visible segments (figure 3a). Therefore theincrease of dmin parameter to 20 m (causing reduction of oversampled data alongcontours) and increase of npmin to 300, 350 and 400 was analyzed from the point of viewof segment elimination and reasonable computing time.

Figure 3: Elimination of visual segments. Results of interpolation using: a) defaultparameters (npmin= 200), b) npmin=300, c) npmin=400, d) supplemental contours andnpmin=300. Western part of Ko�ice basin in Slovakia (visualized in nviz using verticalexaggeration factor = 4)

a) b)

d)c)


As the results show, increasing dmin parameter had only subtle effect on removingsegments and change of computing time. The effect of increasing the number of usedpoints for each individual segment by altering npmin parameter was evident (figure 3b,3c), but using even high npmin values (400) did not eliminate all visible segments. Thedisadvantage of this approach is the dramatic increase of the computing time (table 3).The change of npmin from 200 to 300 increased computing time 4.2 times and in the caseof npmin=400 the computing time was 11.5 times longer.

Table 3 Comparison of interpolation results with changing npmin and pre-processing

used data npmin dmin # of points comp. timehh:mm:ss

% of ref.time*

RMSerror

contours 200 12.5 38 730 0:09:08 100 1.268contours 200 20 37 216 0:09:12 101 1.291contours 300 20 37 216 0:38:45 424 1.527contours 350 20 37 216 1:04:42 708 1.639contours 400 20 37 216 1:45:32 1 155 1.747contours andsupplemental contours

300 20 37 955 0:39:58 438 1.499

*as reference time interpolation with default parameters (npmin=200, dmin=12.5) was used

For interpolation of small datasets comprising tens of thousands input points the increaseof computing time is usually not limiting condition for processing, but in the case of hugedatasets (millions of points) the compromises are necessary. Therefore in some cases itmay be unfeasible to use too high values of npmin parameter and pre-processing of inputdata have to be used to overcome problems of visible segments. Combination of thesetwo approaches may be also used � intermediate values of npmin (e.g. 300) mayovercome majority of segment issues with acceptable increase of computing time andadding of supplemental points may be used only in areas with extreme segmentationproblems.

In case of Kosice basin the use of supplemental contours with combination ofintermediate npmin has proved as good approach for overcoming visible segmentsproblem (figure 3d). The supplemental contours were added by expert taking into accountassumption of smooth surface. Because of the lower reliability of such contours, the highvalue of smoothing was used (5.0 for supplemental contours in comparison with 1.0 fororiginal contours).

3.4 Hydrologically correct DTM

Models of environmental processes associated with overland flow (e.g. erosion, pollutiondispersion) require DTM allowing correct rooting of the mass movement. Importantissues of development of hydrologically correct DTM from contour lines in agriculturallandscape are described on the example from Mochovce basin (Slovakia).

The input data for interpolation were derived from contour lines at scale 1:10 000 withvertical step of 5 m. Using parameters, tuned for eliminating problems of �waves� nearcontour lines [c.f. 17 page 317], the original DTM was prepared with cell resolution of5 m. For checking of continuity of surface flow the network of flowlines was generatedusing the r.flow command that is based on the D-infinity algorithm [12].

As expected the visual inspection of flowlines has shown that the generated DTM usingonly original contour lines does not fulfill the condition of flow continuity (figure 4a)because of the presence of artificial depressions. Such data are not suitable for correctoverland flow modeling.


Figure 4: Influence of data preprocessing on flow routing: a) flowlines generated usingcontour based DTM, b) terrain skeleton as a source of supplemental points, c) flowlinesgenerated using enhanced DTM by use of supplemental points

The post-processing approaches for filling these artificial depressions are widely used,and are based on the D-8 algorithm [e.g. r.fill.dir command]. Their application is limitedand results can not resolve all problems arising when more precise D-infinity algorithm isrequired. Therefore we have adopted an approach of depression elimination that is basedon use of supplemental points derived from terrain skeleton (figure 4b). As results on afigure 4c demonstrate, this approach can overcome these problems.

Figure 5: Influence of data preprocessing on slope gradients in valleys: a) slopegenerated using contour based DTM, b) terrain skeleton as a source of supplementalpoints, c) slope generated using enhanced DTM by use of supplemental points

Another issue, influencing mainly the intensity of overland flow, is associated withemergence of artificial changes in slope gradient in bottoms of narrow valleys. They aregenerated during interpolation in areas near contour lines (figure 5a) and are result ofoversampling the input points along lines and undersampling in between them. The

a) b) c)

a) b) c)


interpolation enforcement by the points of terrain skeleton (figure 5b) resolved thisproblem resulting in gradient in valley bottoms having smooth character (figure 5c).

3.5 Merging DTM regions: data post-processing

Currently, huge datasets containing tens of millions of elevation data points are becomingquite common (e.g. from LIDAR measurements or when computing national high-resolution DTMs). In these cases the internal segmentation procedure used in the methodis not sufficient because of computer memory limits (RAM and swap space). Thes.surf.rst uses memory to read all input data and to compute output grid files. The onlysolution is to divide original dataset to smaller, overlapping regions interpolatedseparately and then merge them to the region covering the whole study area. However,merging separate regions of interpolated grids may produce steps and discontinuities inelevations and other parameters (e.g. slope gradients, aspects) on edges of these regions.The differences in the DTM�s interpolated in the common overlapping area of twoadjacent regions originate in the slight distinctions of internal segmentation results andinfluence of dmin parameter in separately processed regions. These problems arisepredominately in areas with extremely unevenly distributed input points such as alluvialplains and rocky areas.

The problem can be resolved by two approaches:

! merging of adjacent grids in a transitional belt, where influence of values from oneregion are continuously decreasing and on the contrary influence of the values ofsecond region are increasing;

! finding the areas with local minimal differences between the adjacent regions andmerging the grids at the line passing these areas.

Whereas the first approach is easy to implement using several equations in r.mapcalccommand, second one allows better to control differences in values for adjacent regionsand therefore problems as overshoots in one region may be eliminated.

The second approach was successfully applied during the creation of DTM of SlovakRepublic with resolution of 25 m (16 000 x 8 000 cells) computed from 13 million inputpoints. Prior the interpolation, the area was divided into 14 overlapping regions and eachregion was subsequently interpolated separately. Local minima of differences between thepairs of adjacent regions were identified by visual inspection of the results and �seam�lines were drawn and consequently used for merging. The magnitude of differences in theclose surroundings of seam lines did not exceeded 0.5 m for all segments covering wholeSlovakia.

4 ConclusionsThe regularized spline with tension implemented in GRASS GIS as s.surf.rst commandhas proven its robustness, flexibility and interpolation accuracy in many tasks and fordifferent datasets. We have described most common problems occuring duringinterpolation as well as their possible solution. A number of parameters give a userflexibility to manage and eliminate dificult interpolation situations in which othermethods often fail. To get best possible outputs, it is important to know the behaviour ofinterpolation method, parameters and possible solutions. Some problems can be easilysolved by change of parameters, other require data pre-processing (variable smoothing,supplemental data points). Future development should focus on automatization of tuningprocedure of parameter selection using optimization techniques (e.g. crossvalidation) thatcould make the use of s.surf.rst command more convenient even for less experencedusers.


5 AcknowledgementsThis work was carried out in the frame of activities of the GeoModel s.r.o. company andproject No. 2/7050/20 "Hydrogeographical regional types of Slovakia: a problem ofextrapolation of hydrological response and rational use of their water recources"supported by the Scientific Grant Agency of the Ministry of Education and SlovakAcademy of Sciences.

References[1] Mitas, L., Mitasova, H., 1999, Spatial interpolation. In: Longley, P., Goodchild,

M.F., Maguire, D.J., Rhind, D.W., eds., Geographical Information Systems:Principles, Techniques, Management and Applications, Wiley: 481 � 492.

[2] Kemp, K.K., 1996, Managing spatial continuity for integrating environmentalmodels with GIS. In: GIS and Environmental Modeling: Progress and ResearchIssues. M. F. Goodchild, L.T. Stayaert, B.O. Parks, C. Johnston, D. Maidment, M.Crane, S. Glend inning, eds. Fort Collins, CO, GIS World Books: 339-343.

[3] Mitasova, H., Mitas, L. 1993. Interpolation by regularized spline with tension: I.Theory and implementation. Mathematical Geology, 25, 641 � 655.

[4] McCauley, J.D., Engel, B.A., 1997, Approximation of noisy bivariate traverse datafor precision mapping. Transactions of the American Society of AgriculturalEngineers, 40: 237-245.

[5] Hofierka J., Parajka J., Mitasova H., Mitas L., 2002, Multivariate Interpolation ofPrecipitation Using Regularized Spline with Tension. Transactions in GIS, 6: 135-150.

[6] Mitasova, H., Mitas, L., Brown, W.M., Gerdes, D.P., Kosinovsky, I., Baker, T.,1995, Modelling spatially and temporally distributed phenomena: new methods andtools for GRASS GIS. International Journal of Geographical Information Systems,9: 433 � 446.

[7] Neteler, M., Mitasova, H. 2002. Open Source GIS: A GRASS GIS Approach, KluwerAcademic Publishers, p. 434.

[8] Cebecauer, T., Hofierka, J., �úri, M., 2000, Vplyv kvality údajov na modelovaniepovrchového toku vody. Zborník referátov z 1. konferencie ASG pri SAV: 21 � 28.

[9] Hofierka, J., �úri, M., Cebecauer, T., 1998, Rastrové digitálne modely reliéfu a ichaplikačné mo�nosti. Acta Facultatis Studiorum Humanitatis et Naturae UniversitatisPrešoviensis, Prírodné vedy, 30, Folia Geographica 2: 208 � 216.

[10] �úri, M., Hofierka, J., Cebecauer, T., 1997, Tvorba digitálneho modelu reliéfuSlovenskej republiky. Geodetický a kartografický obzor, 43: 257 � 262.

[11] Weibel, R., Heller, M., 1991, Digital terrain modelling. In: Maguire, D.J.,Goodchild, M.F., Rhind, W.D., eds., Geographical information systems: principlesand applications. Longman, London, vol. 1: 269 � 297.

[12] Mitasova, H., Hofierka, J., 1993, Interpolation by regularized spline with tension: II.Application to terrain modeling and surface geometry analysis. MathematicalGeology, 25: 657 � 671.

Documents

Processing digital terrain models by regularized spline