Ranking Spatial Interpolation Techniques Using a Gis Thiessen Method

INTRODUCTIONThe analysis and interpretation of spatial datasets forms an important part of geostatistics andis, unfortunately, highly human dependent(Genton and Furrer, 1998). For instance, it is wellknown that different individuals will take differ-ent approaches, yielding a large assortment of dis-tinct solutions. It is often the case where judge-ment and experience play a key role in selectingthe proper spatial interpolation technique for

each individual case (Englund, 1990). This is part-ly due to the variety of available spatial interpola-tion methods, which range from simple intuitivepredictions to more sophisticated and complexprocedures (Cressie, 1991). Estimating both rain-fall at ungaged locations and mean areal rainfallover an area (e.g. a catchment) based on theresults of meteorological observations, motivatedthe development of gridded estimates of precipi-tation to provide inputs to spatially distributed

Global Nest: the Int. J. Vol 6, No 1, pp 1-20, 2004

Copyright 2004 GLOBAL NEST

Printed in Greece. All rights reserved

RANKING SPATIAL INTERPOLATION TECHNIQUES USING A GIS-BASED DSS

1 Department of Civil Engineering,

McMaster University,

1280 Main Street West,

Hamilton, Ontario, Canada, L8S 4L72 Laboratory of Water Resources

Management & Coastal Engineering

Technical University of Crete

Polytechnioupolis, Chania,

73100, Crete, Greece

*to whom all correspondence should be addressed:

Tel:+(30) 2821037799

Fax: +(30) 2821037849

e-mail: [email protected]

ABSTRACTA GIS-based Decision Support System (DSS) was developed to select the appropriate interpolationtechnique used in studying rainfall spatial variability. The DSS used the ArcView GIS platform byincorporating its spatial analysis capabilities, the programming language "AVENUE", and simple sta-tistical methods. The system consists of a series of modules and can be applied in spatial studies ofother hydrological parameters. A test case from the country of Switzerland is used to demonstrate theapplicability of the system. This should aid in better input to hydrological models.

KEYWORDS: ArcView GIS, AVENUE, Mean, Raingages, Spatial Interpolation Techniques, StandardDeviation.

S. NAOUM1

I.K. TSANIS2, *

Received: 15/05/02Accepted: 17/12/02

1naoum.qxd 7/7/2004 12:57 Page 1

hydrologic and management models. Although there are numerous articles have beenwritten that are concerned with spatial interpola-tion, there is little or no agreement among theauthors on the superiority of some techniquesover others. Additionally, the increasing interestin Geographic Information Systems (GIS) withtheir broad usage and popularity, made it crucialto simply investigate the credibility and applica-bility of the different ready-to-use spatial interpo-lation techniques that are embedded in those sys-tems. Generated with that in mind, this work hasalso been inspired by the Journal of GeographicInformation and Decision Analysis initiative'sspecial edition on spatial interpolation (SpatialInterpolation Comparison SIC97).

APPROACH AND PROCEDUREVariability is often a result of changes in conditionsunder which observations are made, differences inthe way people do the work, difference in processvariables, difference in environmental factors, themeasurement system, or sampling. Statistical tech-niques are used to describe and understand vari-ability. To provide a basis of comparison betweenthe different techniques/models in this work, sim-ple statistical methods are adopted. Since themethod is data-driven and fully automated, it doesnot require preprocessing. This could be of value inan emergency situation where rapid, yet justifiable,results are required. At first, the concept of anobjective function has to be established. This isnormally followed by defining the constrained opti-mization of that function. The process initiates byrandomly eliminating some of the available gages.The different interpolation techniques are thenapplied to estimate the "unobserved/missing" val-ues on the basis of the "observed/remaining" ones.The purpose of the random selection of gages,which in this case takes the form of twenty tries, isintended to overcome the problem of outliers, ifthey exist. Experience shows that measured datacontains between 10 to 15 per cent of outlying val-ues due to gross errors, measurement mistakes,and faulty recording. Identifying and rejecting, orremoving, outliers is highly opinion dependent andis not normally recommended since they, beingextremes, represent critical cases or worst case sce-narios. The errors/residuals at theunobserved/ungaged locations are then calculatedas the difference between observed and estimated

values and categorized as positive and negativeresiduals. If the absolute value of the sum of thepositive residuals is greater than the absolute valueof the sum of the negative residuals, it implies thatthe observed values are greater than estimatedones. The model is then said to be underestimating.If the absolute value of the sum of the positiveresiduals is less than the absolute value of the sumof the negative residuals, it implies that the esti-mated values are greater than observed ones. Themodel is then said to be overestimating. At the sec-ond phase, three values are calculated for eachtechnique. The errors at the ungaged (unobserved)locations are grouped in one column as absolutevalues, where the mean and the standard deviationare calculated as follows:

(1)

(2)

where: : the mean of absolute residuals, xobs: theobserved value of rainfall, xest: the estimated valueof rainfall, n: the number of observed/estimatedvalue (sample size), S: the standard deviation, and

: the absolute value of the individual residual.Generally, for a model to be considered satisfac-tory, the mean ( : MeanAbsErr) and standarddeviation (S: StDevAbsErr) of the absolute valuesof residuals are expected to be as low as possibleamong the other techniques. From the rain sur-face (grid), which is generated using the observedvalues, the average value is calculated as the sumof the rain values at each grid cell divided by thetotal number of cells in the grid. This representsthe mean areal precipitation (MeanEst). Theaverage of the observed values at all locations(observed and unobserved) is then calculated(MeanObs). A good model should generate avalue (MeanEst) that matches (or be as close aspossible to) the (MeanObs) and the differencebetween these two values is minimal. The differ-ence is then calculated (Diff). The main criterionfor judging the best model for each run is based onthe minimum value obtained by averaging (Avg)the values (Diff, MeanAbsErr, StDevAbsErr),

2 NAOUM and TSANIS

1naoum.qxd 7/7/2004 12:57 Page 2

assuming equal weights for all. The twelve tech-niques are ranked accordingly from best(MinAvg) to the worst (MaxAvg) in addition tomany other statistics in a report format.

SPATIAL INTERPOLATION METHODS IN A GISThe following is a description of the interpolationtechniques available in ArcView GIS 3.2.

Spline (Regularized & Tension):Spline interpolation consists of the approximationof a function by means of series of polynomialsover adjacent intervals with continuous derivativesat the end-point of the intervals. Smoothing splineinterpolation enables to control the variance ofthe residuals over the data set. The solution is esti-mated by an iterative process. It is also referred toas the basic minimum curvature technique or thinplate interpolation as it possesses two main fea-tures: (a) the surface must pass exactly through thedata points, and (b) the surface must have mini-mum curvature. The reader is referred to Franke(1982) and Mitas and Mitasova (1988) for furtherreading about the technique.

Inverse Distance Weighting (IDW)Inverse Distance Weighting (IDW) is an interpo-lation technique in which interpolated estimatesare made based on values at nearby locationsweighted only by distance from the interpolationlocation. IDW does not make assumptions aboutspatial relationships except the basic assumptionthat nearby points ought to be more closely relat-ed than distant points to the value at the interpo-late location. This technique determines cell val-ues using a linearly weighted combination of a setof sample points. The weight is a function ofinverse distance. IDW allows the user to controlthe significance of known points upon the inter-polated values, based upon their distance fromthe output point. The reader is referred to Tung(1983) and Watson and Philip (1985) for furtherreading about the technique.

KrigingKriging provides a means of interpolating valuesfor points not physically sampled using knowledgeabout the underlying spatial relationships in a dataset to do so. Variograms provide this knowledge.Kriging is based on regionalized variable theorywhich provides an optimal interpolation estimate

for a given coordinate location, as well as a vari-ance estimate for the interpolation value. Itinvolves an interactive investigation of the spatialbehavior of the phenomenon before generating theoutput surface. It is based on the regionalized vari-able theory, which assumes that the spatial varia-tion in the phenomenon is statistically homoge-neous throughout the surface; that is, the same pat-tern of variation can be observed at all locations onthe surface. This hypothesis of spatial homogeneityis fundamental to the regionalized variable theory.Data sets known to have spikes or abrupt changesare not appropriate for the Kriging technique. Insome cases, the data can be pre-stratified intoregions of uniform surface behavior for separateanalysis. The reader is referred to Burrough(1986); Heine (1986); McBratney and Webster(1986); Oliver (1990); Press (1988); and Royle et al.(1981) for further reading about the technique.

Trend SurfaceThe linear trend surface interpolator creates afloating-point grid. It uses a polynomial regres-sion to fit a least-squares surface to the inputpoints. It allows the user to control the order ofthe polynomial used to fit the surface. Trendinterpolation is easy to understand by consideringa first-order polynomial. A first-order linear trendsurface interpolation simply performs a least-squares fit of a plane to the set of input points.Trend surface interpolation creates smooth sur-faces. The surface generated will seldom passthrough the original data points since it performsa best fit for the entire surface. When an orderhigher than 1 is used, the interpolator may gener-ate a grid whose minimum and maximum mightexceed the minimum and maximum of the inputpoints. The most common order of polynomials is1 through 3. The reader is referred to Chidley andKeys (1970); Shaw and Lynn (1972); Lee et al.(1974); and Kruizinga and Yperlaan (1978) forfurther reading about the technique.

Theissen PolygonsAnother choice given in the software for codingbased on the value of a chosen attribute of theseed feature. This is appropriate if we wish todefine the "region of influence" of a point or line.The region of influence is based on "nearestneighbours" to the point or line. The region ofinfluence for a series of points is represented by a

3RANKING SPATIAL INTERPOLATION TECHNIQUES USING A GIS-BASED DSS

1naoum.qxd 7/7/2004 12:57 Page 3

set of polygons encoded with the nominal valuefor each point. These polygons are referred to asThiessen Polygons or collectively as a proximalmap. Theissen (1911) came up with the first tech-nique to estimate areal average precipitation.Theissen polygons are probably the most com-mon approach for modeling the spatial distribu-tion of rainfall. The approach is based on definingthe area closer to a gage then any alternate gageand the assumption that the best estimate of rain-fall on that area is represented by the point mea-surement at the gage. Because the basis of themodel is geometry and gage location, implemen-tation of Theissen polygons in a GIS environmentis not difficult. However, one impact of the use ofTheissen polygons is the development of discon-tinuous surfaces defining the rainfall depth overthe area under study. This effect arises at theboundaries of the polygons where a discretechange in rainfall depth occurs (Ball and Luk,1998). The reader is referred to Whitemore et al.(1961); Rainbird (1967); Hutchinson (1969);Diskin (1969); and Diskin (1970) for further read-ing about the technique.

As a result, and to summarize, the twelve spatialinterpolation techniques employed in this studyare listed in Figure 1.

TEST CASEThe module is applied to a group of raingages inSwitzerland to illustrate its applicability.Switzerland lies at the heart of Western Europeand covers an area of 41,284 km2. A DigitalElevation Model and some interpolated rain sur-faces (interpolated from observed values usingdifferent techniques) are shown in Figure 1. Thedata set being used is related to the period ofChernobyl Nuclear Power Plant accident (April,26th 1986) (Dubois, 1998). During the days fol-lowing the accident, a radioactive plume, led bythe action of atmospheric flow, was crossing manyEuropean countries. Radioactive deposition onthe ground was mainly a function of the rainfall.The primary set of data includes 467 daily rainfallrecords made in Switzerland on May 8th 1986.The collection of the data was carried out by theAir Pollution Group at Imperial College inLondon under financial support of JRC-Ispra.

4 NAOUM and TSANIS

Figure 1. Digital Elevation Model and Interpolated Rain Surfaces in Switzerland

1naoum.qxd 7/7/2004 12:57 Page 4

The location of the data points was also provided.Both Digital Elevation Model (DEM), with a res-olution of around 1 km * 1 km, and the countryborder, used to define the area under study, wereprovided as secondary information. The rainfallmeasurements were in the form of text files, theDEM in the form of a normal ASCII ARC/INFOformat, and country borders were available as anAutoCad Interchange Drawing file.

STRUCTURE AND DESCRIPTION OF THE PROJECT A Geographic Information System can be morepowerful when some added features, representedin this case by statistical methods, are combinedwith its many capabilities, as described above,resulting in the generation of a good decision sup-port system. This GIS module, as shown in Figure2, was developed in the ArcView GIS environ-ment using AVENUE (the ArcView program-ming language). The programming languageAVENUE provides a well-defined mechanism forallowing user-written routines to be called fromwithin the normal user interface of the GIS pack-

age. In addition, this language also provides amenu-driven graphic interface, which makes itpossible to guide a user with prompts and expla-nations throughout the application. The DialogDesigner and Spatial Analyst extensions areloaded to the ArcView project.The project is composed of AVENUE scripts,dynamic link libraries (dlls), and designed dialogs.The scripts were mainly created by the authors. Insome cases, however, they were modified versionsof scripts that existed in the ArcView on-line help.As the name implies, the "makedir.dll" and"deldir.dll" dynamic link libraries were built tocreate and delete directories without the use ofbatch files. A dialog, created for the convenienceof the user, included all spatial interpolation tech-niques available in ArcView but not available inthe normal interface of the program. The devel-oped module allows for one-/multi-process inter-polation. The main module uses four main piecesof information to perform its task and generatethe new network: the location of gages, rainfalldata, region boundary, and a Digital ElevationModel and it consists of five sub-modules:


Figure 2. The Main Module of the Interpolation Engine Project

1naoum.qxd 7/7/2004 12:57 Page 5

Module 1: ReadMeThis module documents the project by informingthe user of the different parts of the main moduleusing many graphic illustrations. It can be viewedas either a Word Perfect Document (help.wpd) oran HTML (help.htm). The advantage of theHTML is that it provides links to all AVENUEscripts and examples of output tables and text files.

Module 2: Data PreparationThis is a five-step module that prepares the pro-ject for modules to follow. The data inputrequirements for this module include informationon the boundary of the region, the DigitalElevation Model, and the data and location ofraingages.

DXF to Shapefile ConverterIn many cases, data is provided as AutoCad draw-ing files (for example: dxf files) which must thenbe converted to ArcView Shapefiles.

Polyline to Polygon ConverterAfter converting the AutoCad drawing file to anArcView shapefile, the user should convert theresultant "polyline" shapefile into a "polygon"shapefile, which will be used in a later step.

Import Grid from ASCII Format A grid, DEM in many cases, will be saved in theASCII ARC/INFO format. In this case, it has to beimported into an ArcView grid format.

Grid ClippingAn area has to be extracted (clipped) from theoriginal imported grid from the previous step inorder to calculate mean areal rainfall or meanelevation over a specific region. The clipping process (as shown in Figure 3) isaccomplished by using the "polygon" shapefilegenerated in step 2 and the imported grid fromstep 3.

6 NAOUM and TSANIS

Figure 3. Clipped DEM Using the Menu Item "Grid Clipping"

1naoum.qxd 7/7/2004 12:58 Page 6

Generate Shapefile from a Database FileA database file that contains information aboutlocation of raingages and amounts of rainfall canbe converted to a "point" shapefile and added tothe project. It should be noted that this module was createdspecifically for the purpose of this study. If allinput requirements are satisfied and the data is inthe proper format, there is no need to go throughthe five steps.

Module 3: InterpolatorsThis is a key four-step module that is responsiblefor executing many tasks as shown in Figure 4,where a number of gages are selected randomlyand a rain surface (grid) is generated. The remain-ing number of gages (the unselected ones) is thenprojected on the generated grid, and estimated val-ues for rainfall at those locations are then extract-ed. A comparison between the observed and esti-mated precipitation values is held and the residu-


Figure 4. Module 3 (Interpolators)

1naoum.qxd 7/7/2004 12:58 Page 7

als/errors are calculated. The results of the com-parison are summarized in a text file and a data-base file from which illustrative charts can be gen-erated if desired. The last item of the module gen-erates Theissen polygons for the selected gages.

Random SelectionIn this step, the user is prompted to enter therequired number of gages that should be random-ly selected. The result is two shapefiles. One rep-resents the selected gages and the other repre-sents the unselected ones.

Spatial Interpolation (ONE)The user is presented with the interpolation dia-log (Figure 5) described earlier so that they mayselect the type of interpolation technique theywill be using to generate the rain surface from theselected gages (as shown in Figure 6).

StatisticsThis task is executed on the shapefile of the uns-elected gages. There is some interaction between

the user and the program. The user is promptedby some messages and a text file and a databasefile are then generated, as shown in Figure 7, inthe working directory.

Theissen PolygonsThis task is executed on the shapefile of theselected gages. The mean areal rainfall will be cal-culated and presented to the user.

Module 4: Spatial Interpolation (MANY)This module is an extended (advanced) version ofmodule (3), where the user interference is mini-mized. It simply generates grids using all the spa-tial interpolation techniques available (includingTheissen polygons) by using the shapefile of theselected gages. Twelve grids are then generatedusing the interpolation techniques and the esti-mated values are compared to the observed val-ues of the unselected gages. The result is a textfile and twelve database tables. Each table repre-sents the results of each interpolation techniquein the same sequence as in the text file. At the end

8 NAOUM and TSANIS

Figure 5. The Developed Interpolation Dialog as Part of Module (3)

1naoum.qxd 7/7/2004 12:58 Page 8

of the text file, the different interpolation tech-niques are ranked from the best to the worstaccording to their performance.

Module 5: Automated Interpolation OperationsThis module is an advanced version of module (4)with the least interference from the user. Threeinputs are required from the user at the start ofthe operation: the number of iterations, the cellsize, and the number of gages to be randomlyselected, and the files DEM, Rain.shp, andCountry.shp as shown in Figure 8. The mainscript, when executed, opens new views that areequal to the number of iterations specified by theuser. It also generates new working directories foreach one of the views inside the main workingdirectory "C:\Inter", sets the properties of theviews (map units and distance units), sets theanalysis properties (extent, cell size, and mask),and copies the necessary themes (Rain.shp,Country.shp, and DEM) from the main view tothe other views. The main script, then, triggersanother script which, in turn, opens each of the

views and performs the random selection of thegages according to the number that was initiallyspecified by the user. A third script is run fromwithin the second, which is responsible for open-ing each of the previously generated views andperforming the spatial interpolation task usingthe different methods and generate the text file,as shown in Figure 9, and the 12 database files.All output files are located in the respective work-ing directory of each view, each of which is namedafter the view (i.e. all work done in view1 is storedin the sub-directory "C:\Inter\View1"). In manycases, the user will choose to terminate the run.For example, if the technique is not suitable for acertain data set or the user would like to run thesame case with different cell size, the run may beterminated. The user will then choose to resumeexecution without generating new views or newshapefiles. At this point, the menu item"Continue Operations ..." may be used to workwith the existing files. This menu item is attachedto the script "Inter_Continue" which will run thescript "Inter_Continue1" or it will run the script


Figure 6. Module 3 - Menu Item: Spatial Interpolation [ONE] (Output)

1naoum.qxd 7/7/2004 12:58 Page 9

"Inter_DifferentCellSize" if the user would like torun the "Automated Operations" using differentcell size on multiple views that had been previ-ously generated.

Project AccessoriesThe project is equipped with more scripts that areautomatically executed upon opening and closingthe project. They perform additional functionswhich are intended to facilitate the project/userinteraction and results presentation.

EXECUTIONSpeculating that the cell size and the number ofavailable gages may influence the ranking of theinterpolation techniques, a total of 12 runs were

performed. The first 4 runs were done using only40% of the available gages (187 out of 467 gages)as observed records with a cell size of 500m,1000m, 5000m, and 10000m. The second set of 4runs used 60% of the available gages (280 out of467 gages) as observed records with a cell size of500m, 1000m, 5000m, and 10000m. The third andfinal set of 4 runs used 80% of the available gages(374 out of 467 gages) as observed records with acell size of 500m, 1000m, 5000m, and 10000m.Each of the 12 techniques was then evaluatedbased on the average value of the 20 tries withineach of the 12 runs. The evaluation was done ona scale of "0" to "10" with "10" being a perfect tech-nique which means the smallest average value forthe 20 (Avg) values; while a "0" is the worst tech-

10 NAOUM and TSANIS

Figure 7a. Module 3 - Menu Item: Statistics (Output)

1naoum.qxd 7/7/2004 12:58 Page 10

nique which means the largest average value forthe 20 (Avg) values.

RESULTSRelying on the multiple random selection ofgages to eliminate the effect of any errors or out-liers, no statistical data preparation or prelimi-nary analysis was done. Descriptive statistics wereemployed to provide inferences for the differentmodels. The module output takes various forms: 1. Visual Grids (Surfaces): as shown in Figure

10, where the user can see the distribution ofselected (solid dots) and unselected gages (x-marked locations). In addition to this, the dif-ferent techniques can be visually compared toeach other.

2. Text Files (Report Format): as shown inFigures 7 and 9, the user is able to obtain per-manent records of the various runs for com-parison purposes with helpful statistics listedfor each technique. In addition, the techniquesare listed in performance sequence.

3. Database Files: as shown in Figure 7, databasefiles are permanently stored and from whichplots can be generated (as shown in Figure 11).This confirms the results previously obtainedin the text (report) format in Figure 9.

Generally, the results are shown in Figure 12,where the horizontal axis represents the interpo-lation techniques in the same order as in Figure 1and the vertical axis represents the 0-to-10 scale.


Figure 7b. Module 3 - Menu Item: Statistics (Output)

1naoum.qxd 7/7/2004 12:58 Page 11

12 NAOUM and TSANIS

Figure 8. Module 5 - Automated Interpolation Operations

1naoum.qxd 7/7/2004 12:58 Page 12


Figure 9. Module 5 - Automated Interpolation Operations (Output: Interp1.txt)

1naoum.qxd 7/7/2004 12:58 Page 13

The following can be concluded:a. the Spline_Regularized and the 2nd Order

Polynomial techniques showed poor perfor-mance in almost all cases.

b. Theissen Polygons and Kriging (Linear;Gaussian; Circular; Universal_2) techniquesfluctuated from one case to the other.

c. the Spline_Tension, IDW, and Kriging(Spherical; Exponential; Universal_1) tech-niques were able to provide reliable estimates.The Kriging_Exponential and Kriging_Universal_1 models are recommended.

Results show that changing the cell size of theinterpolated grid did not significantly affect theclassification/rank of the interpolation techniqueswhen using small number of gages (187 gages), asshown in Figure 13a, except for the Theissen poly-gons method which dropped on the scale signifi-cantly when a cell size of 10000m was used.However, by increasing the number of gages, thecell size started to show a more noticeable influ-ence as some techniques show higher perfor-mance while the others show lower performance.

It should be noted that the 2nd order polynomial(technique 11) did not respond to any changesthroughout the analysis. It was always ranked 12.Figure 13b shows that changing the number ofgages used in the interpolation when using a cellsize of 10000m did not have any effect on the per-formance of the different techniques. It is clearthat increasing the number of gages available forinterpolation enhanced the performance of thetechniques except for three techniques: Spline_Regularized, Spline_Tension, and TheissenPolygons. Because Spline tries to fit a smooth sur-face that passes through the points, increasing thenumber of gages does not help the techniqueespecially if there is abrupt changes in rainrecords, resulting in erratic estimated values. ForTheissen polygons, as the number of gagesincreases, the size of the polygons decreases andtheir count increases resulting in erratic estimat-ed rain values.

DISCUSSIONThis network is a relatively dense network with adensity of one gage/88.4 km2 and it records daily

14 NAOUM and TSANIS

Figure 10. A Visual Comparison between Four Interpolation Techniques

1naoum.qxd 7/7/2004 12:58 Page 14


Figure 11. Residual Plots as well as Observed vs Estimated Rainfall Values for Four Interpolation Techniques

1naoum.qxd 7/7/2004 12:58 Page 15

precipitation. Due to the high variability normallyassociated with daily precipitation records and thehigh density of the network, it is likely that tech-niques such as Trend and Spline_Regularizedwould not provide nice estimates. Trend surfacesare always smooth surfaces which do not normallypass through the original data points but performsa best fit for the entire surface. In other words itprovides an approximate direction of the intensityof rain rather than an accurate description of thespatial variability of rain. On the other hand, sur-faces generated using Spline_ Regularized try topass through the points which, in this case, is notsuitable because of the rapid changes in gradi-ent/slope in the vicinity of the data points.However, Spline_Tension is a more relaxed ver-sion of Spline which could fit a less smooth curves.Kriging is generally a good interpolator. TheOrdinary Kriging is represented in this case by theSpherical, Circular, Exponential, Gaussian, and

Linear methods. With these options, Kriging usesthe mathematical function specified by themethod to fit a line or curve to the semi-variancedate in the semi-variogram. These five models areprovided to ensure that the necessary conditionsof the variogram model are satisfied. TheExponential and Spherical methods seem to bet-ter fit the spatial variation of this data set. TheUniversal Kriging, represented by the Universal1and Universal2 methods, assumes that the spatialvariation across the surface has a structural com-ponent (drift). Drift is a systematic change in thecell values at a particular scale. This scale is relat-ed to the radius of the search area. The goal is tochange the search radius to find the scale at whichthe drift can be detected and the variance is low-est. Universal1 uses a first order polynomial toapproximate the drift and Universal2 uses a sec-ond order. The first derivative (Universal1) wasappropriate for this specific application.

16 NAOUM and TSANIS

Figure 12. Performance of All Interpolation Techniques.

1naoum.qxd 7/7/2004 12:58 Page 16


Figure 13a.The Effect of Changing the Cell Size on the Performance of the Interpolation Techniques.

1naoum.qxd 7/7/2004 12:58 Page 17

18 NAOUM and TSANIS

Figure 13b.The Effect of Changing the Number of Gages Used for Interpolation on the Performance

of the Interpolation Techniques

1naoum.qxd 7/7/2004 12:58 Page 18

Theissen polygons and IDW techniques areknown to provide good results when used for rel-atively dense networks as in this case. However,increasing the number of gages can be problemat-ic for the Theissen polygons technique.It should be noted that repeated runs for differentdata sets is required to verify the results obtained.For example, wet, moderate, and dry conditions;hourly, daily, monthly, and yearly data; short andlong term average; ...etc. The one available dataset used as a test case in this study does not pro-vide enough evidence that certain techniques arebetter than others.

CONCLUSIONNo interpolation technique, no matter howsophisticated, can accurately predict rainfallamounts at ungaged locations and, subsequently,estimate mean areal rainfall. This work establish-es an approach by using GIS and historical datato locate the best technique. It should be notedthat the selection of a given method to be used in

cases of emergency should be based on compar-isons dealing with more than one data set.Rainfall data from different days, months, oryears are likely to behave differently under vari-ous meteorological conditions. Performances ofa method applied to a partial data set are likelyto be different from performances of the samemethod with a larger data set. By taking a sampleof the data set, the problem scale is changed, andthe boundary between large and small scale vari-ation is modified. For that reason, data sets withdifferent sizes are recommended. Finally, it ishighly advisable for any country or region toobtain data from the closest parts of neighboringcountries or at least to have as many data as pos-sible from locations within the country/region.Based on the one data set available for this study,it was clear that the Kriging_Exponential andKriging_Universal_1 models showed consistentperformance and provided reliable estimatesregardless of the number of gages or the cell sizeused in the interpolation.


REFERENCESBall J.E. and Luk K.C. (1998), Modeling Spatial Variability of Rainfall Over A Catchment, Journal of Hydrologic

Engineering, 3, 122-130.

Burrough P.A. (1986), Principles of Geographic Information Systems for Land Assessment, Oxford University

Press, New York.

Chidley T.R.E. and Keys, K.M. (1970), A Rapid Method of Computing Areal Rainfall, Journal of Hydrology, 12,

15-24.

Cressie N. (1991), Statistics for Spatial Data, New York, Wiley.

Diskin M.H. (1969), Thiessen Coefficients by Monte Carlo Procedures, Journal of Hydrology, 8, 323-335.

Diskin M.H. (1970), On the Computer Evaluation of Thiessen Weights, Journal of Hydrology, 11, 69-78.

Dubois G. (1998), Spatial Interpolation Comparison 97: Forward and Introduction, Journal of Geographic

Information and Decision Analysis, 2, 1-10.

Englund E. J. (1990), A Variance of Geostatisticiansm, Journal of Mathematical Geology, 22, 417-455.

Franke R. (1982), Smooth Interpolation of Scattered Data by Local Thin Plate Splines, Journal of Computation

and Mathematics with Applications, 8, 273-281.

Genton M.G. and Furrer R. (1998), Analysis of Rainfall Data by Simple Good Sense: Is Spatial Statistics Worth

The Trouble?, Journal of Geographic Information and Decision Analysis, 2, 11-17.

Heine G.W.(1986), A Controlled Study of Some Two-Dimensional Interpolation Method, COGS Computer

Contributions, 3, 60-67.

Hutchinson P. (1969), Estimation of Rainfall in Sparsely Gaged Areas, International Association of Hydrologic

Sciences Bulletin, 14, 101-120.

Kruizinga S. and Yperlaan G. J. (1978), Spatial Interpolation of Daily Total of Rainfall, Journal of Hydrology, 36,

65-73.

Lee P.S., Lynn P.S. and Shaw E.M. (1974), Comparison of Multiquadratic Surfaces for the Estimation of Areal

Rainfall, International Association of Scientific Hydrology Bulletin, 19, 303-317.

McBratney A.B. and Webster R. (1986), Choosing Functions For Semi-Variograms of Soil Properties and Fitting

Them to Sampling Estimates, Journal of Soil Science, 37, 617-639.

Mitas L. and Mitasova H. (1988), General Variational Approach to the Interpolation Problem, Journal of

Computation and Mathematics with Applications, 16, 983-992.

1naoum.qxd 7/7/2004 12:58 Page 19

20 NAOUM and TSANIS

Oliver M.A. (1990), Kriging: A Method of Interpolation for Geographical Information Systems, International

Journal of Geographic Information Systems, 4, 313-332.

Press W.H. (1988). Numerical Recipes in C, The Art of Scientific Computing, New York, Cambridge University

Press.

Rainbird A.F. (1967), Methods of Estimating Areal Average Precipitation, Reports on WMO/IHD Projects,

Report No. 3, pp. 45.

Royle A.G., Clausen F.L. and Frederiksen P. (1981), Practical Universal Kriging and Automatic Contouring,

Geoprocessing, 1, 377-394.

Shaw E.M. and Lynn P.O. (1972), Areal Rainfall Evaluation Using Two Surface Fitting Techniques, International

Association of Scientific Hydrology Bulletin, 17, 419-433.

Thiessen A.H. (1911), Precipitation for Large Areas, Monthly Weather Review, 39, 1082-1084.

Tung Y.K. (1983), Point Rainfall Estimation for a Mountainous Region, Journal of Hydraulic Engineering, 109,

1386-1393.

Watson D.F. and Philip G.M. (1985), A Refinement of Inverse Distance Weighted Interpolation, Geo-Processing,

2, 315-327.

Whitemore J. S., Van Efden F. J. and Harvey K. J. (1961), Assessment of Average Annual Rainfall Over Large

Catchments, In: Inter-African Conference on Hydrology, C.C.T.A. Publication No. 66, 100-107.

1naoum.qxd 14/7/2004 12:38 Page 20

Ranking Spatial Interpolation Techniques Using a Gis Thiessen Method

Documents

Transcript of Ranking Spatial Interpolation Techniques Using a Gis Thiessen Method