Erdogan ESPL 2009

EARTH SURFACE PROCESSES AND LANDFORMSEarth Surf. Process. Landforms 34, 366–376 (2009)Copyright © 2009 John Wiley & Sons, Ltd.Published online 16 January 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/esp.1731

John Wiley & Sons, Ltd.Chichester, UKESPEarth Surface Processes and LandformsEARTH SURFACE PROCESSES AND LANDFORMSEarth Surface Processes and LandformsThe Journal of the British Geomorphological Research GroupEarth Surf. Process. Landforms0197-93371096-9837Copyright © 2006 John Wiley & Sons, Ltd.John Wiley & Sons, Ltd.2006Earth ScienceEarth Science99999999ESP1731Research ArticleResearch ArticlesCopyright © 2006 John Wiley & Sons, Ltd.John Wiley & Sons, Ltd.2006

A comparision of interpolation methods for producing digital elevation models at the field scaleInterpolation methods for producing digital elevation models

Saffet ErdoganAfyon Kocatepe University, Faculty of Engineering, Department of Geodesy and Photogrammetry, Ahmet Necdet Sezer Campus, Gazligol Road, 03200 Afyonkarahisar, Turkey

Received 21 August 2007; Revised 26 May 2008; Accepted 14 June 2008

* Correspondence to: Saffet Erdogan, Afyon Kocatepe University, Faculty of Engineering, Department of Geodesy and Photogrammetry, Ahmet Necdet Sezer Campus,Gazligol Road, 03200 Afyonkarahisar, Turkey. E-mail: [email protected]

ABSTRACT: Digital elevation models have been used in many applications since they came into use in the late 1950s. It isan essential tool for applications that are concerned with the Earth’s surface such as hydrology, geology, cartography,geomorphology, engineering applications, landscape architecture and so on. However, there are some differences in assessing theaccuracy of digital elevation models for specific applications. Different applications require different levels of accuracy fromdigital elevation models. In this study, the magnitudes and spatial patterning of elevation errors were therefore examined, usingdifferent interpolation methods. Measurements were performed with theodolite and levelling. Previous research has demonstratedthe effects of interpolation methods and the nature of errors in digital elevation models obtained with indirect survey methods forsmall-scale areas. The purpose of this study was therefore to investigate the size and spatial patterning of errors in digital elevationmodels obtained with direct survey methods for large-scale areas, comparing Inverse Distance Weighting, Radial Basis Functionsand Kriging interpolation methods to generate digital elevation models. The study is important because it shows how the accuracyof the digital elevation model is related to data density and the interpolation algorithm used. Cross validation, split-sample andjack-knifing validation methods were used to evaluate the errors. Global and local spatial auto-correlation indices were then usedto examine the error clustering. Finally, slope and curvature parameters of the area were modelled depending on the errorresiduals using ordinary least regression analyses. In this case, the best results were obtained using the thin plate spline algorithm.Copyright © 2009 John Wiley & Sons, Ltd.

KEYWORDS: digital elevation models; interpolation; geostatistic; GIS

Introduction

The shape of the Earth’s surface, with valleys, plains, hills andso on, is usually referred to as topography. The representationof topography has always been an important and complextopic for surveyors and cartographers because topographyvaries continuously over space and has to be flattened to atwo-dimensional map (Hu, 1995). Digital elevation models(DEMs) have been used as a tool to represent the Earth’ssurface in many applications, such as hydrological modelling,precision agriculture, civil engineering, large-scale mappingand telecommunications. A DEM is a numerical data file thatcontains the elevation of the topography over a specifiedarea, usually at a fixed grid interval over the surface of theEarth. DEMs can be generated using different methods thatdepend on collection procedures and techniques. Photogram-metric methods, satellite-based techniques and field surveyingare direct methods of DEM production. Meanwhile, DEMsare sometimes generated by digitizing existing topographic

maps. However, any DEM derived from digitized topographicmaps is an approximation of an approximated real world(Carter, 1988). Since many applications rely on DEMs, thequality of DEMs and information about the spatial structure oferrors within DEMs are particularly important.

The quality of a DEM is a result of individual factors. Thesegenerally can be grouped into three classes: (i) accuracy,density and distribution of the source data; (ii) the inter-polation process; and (iii) characteristics of the surface (Gonget al., 2000). The first two factors are clearly errors whereasthe third should be considered a matter of uncertainty (Fisherand Tate, 2006). The accuracy of source data varies withtechniques such as: map digitization; active airborne sensorsincluding interferometric synthetic aperture radar and airbornelaser scanning techniques; photogrammetric methods; andfield surveying. The density of data changes with differentsampling intervals and is also one of the factors affectingaccuracy (Chaplot et al., 2006; Weng, 2006). Another factorrelated to the source data is distribution, which may be

Copyright © 2009 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms, Vol. 34, 366–376 (2009)DOI: 10.1002/esp

INTERPOLATION METHODS FOR PRODUCING DIGITAL ELEVATION MODELS 367

regular, random, progressive, selective, or composite, and so on.Characteristics of surfaces such as flat, hilly or mountainousare the second main factor affecting the accuracy of DEMs(Chaplot et al., 2006). The last factor is the interpolationalgorithms that are used to generate DEMs. Many interpola-tion algorithms exist but there is no definite rule to indicatewhich algorithm is most suitable for a particular surface.

A number of previous studies have examined the effects ofinterpolation methods based on applications in a range ofdisciplines (Desmet, 1997; Zimmerman et al., 1999; Priyakantet al., 2003; Hofierka et al., 2005; Robinson and Metternicht,2005; Fencik and Vajsablova, 2006; Yilmaz, 2007). However,quite a few studies, examining the accuracy of interpolationtechniques in comparison of sample density and landformtypes, have been made.

Nonetheless, many users need high-resolution DEMs ofsmall areas for specific applications such as landslide proneand hydrological risk areas. Therefore, field surveying with atheodolite and levelling was used to generate high qualityDEMs of the whole topographic surface in this study. Fourcommonly used spatial interpolation algorithms (inversedistance weighting (IDW), ordinary Kriging (OK), multi-quadratic radial basis function (MQ), and thin plate spline(TPS)) were examined and compared for the level of modelerrors.

The main objectives of this study were: (1) to evaluate theeffects of (a) the density of raw data and (b) interpolationtechniques on the accuracy of DEM generation for a rocky hill;and (2) to examine methods for quantifying the uncertainty ofDEMs in this region using spatial measures. By quantifyingthe amount and distribution of the error introduced by sampleinterval and interpolation, some statistical expressions ofaccuracy such as mean error and root mean square errorwere calculated using different validation techniques. Globaland local spatial autocorrelation indices were also employedto quantify the spatial pattern of the uncertainty. Meanwhilethe relationship between DEM error and morphometric charac-teristics of the hill, such as slope and curvature, were examinedusing the correlation coefficients of linear ordinary least square(OLS) regression analyses.

Interpolation

Interpolation is a topic of interest to many disciplines includ-ing mathematics, earth science, geography and engineeringbecause measurements can be time-consuming, expensiveand laborious in many environmental applications. Interpola-tion is a procedure used to predict values at a location forwhich there is no recorded observation. It can also be definedas the procedure of estimating the values of properties atunsampled sites within the area covered using existing pointobservations (Algarni and Hassan, 2001). Interpolation methodscan be classified in many ways including local/global, exact/approximate and deterministic/geostatistical methods. Globalinterpolators determine a single function that is mappedacross the whole region, whereas local interpolators apply analgorithm repeatedly to a small portion of the total set ofpoints. Exact interpolators honour the data points on whichthe interpolation is based, whereas approximate interpolatorsare used when there is some uncertainty about the given surfacevalues. Nevertheless, deterministic/geostatistical methods arethe most widely used. Deterministic interpolation methodsare used to create surfaces from measured points based oneither the degree of similarity (e.g. IDW) or the degree ofsmoothing (e.g. radial basis functions (RBF)). Geostatisticalinterpolation methods are based on statistics and are used for

more advanced prediction surface modelling, which includeserror or uncertainty of prediction (Gong et al., 2000).

There are many routines available for interpolation and thesehave been widely tested and documented over the years (Isaaksand Srivastava, 1989; Desmet, 1997; Smith et al., 2005).Although there are many variants, four interpolation methodsare widely used and popular in GIS software, and these fourwere examined. Each of the methods produces different heightvalues across characteristics of the surface. An IDW algorithmdetermines cell values using a linearly weighted combinationof a set of sample points. All points or a specified number ofpoints within a specified radius are used to predict the outputvalue for each unsampled location (Burrough and McDonell,1988).

The general formula of IDW is

(1)

where Z(s0) is the value predicted for location s0, N is thenumber of measured sample points surrounding the predictionlocation, λi are the weights assigned to each measured point,and Z(si) is the observed value at the location si. The weightsare a function of inverse distance. The formula determiningthe weights is

(2)

The power parameter p in the IDW is the significance of thesurrounding points upon the interpolated value (Priyakant et al.,2003). When the distance (d ) between the measured locationand the prediction location increases, the weight that themeasured point has on the prediction decreases (Burrough andMcDonell, 1988). So a higher power results in less influencefrom distant points.

The RBF methods are a series of exact interpolation algorithmsthat a surface must go through in each measured sample loca-tion. There are several types of RBF including thin plate spline,spline with tension, completely regularized spline, multiqua-dratic function and inverse multiquadratic spline. Multiqua-dratic function is considered by many to be the best (Yang etal., 2004). The RBFs are conceptually similar to fitting a rubbermembrane through the measured sample values while mini-mizing the total curvature of the surface. The selected RBFalgorithm determines how the rubber membrane will fit betweenthe values (Burrough and McDonell, 1988). Because an RBFmethod is always an exact interpolator, it can be introducedas a smoothing factor to all the methods in an attempt toproduce a smoother surface (Yang et al., 2004). This methodis a linear combination of the different basis functions

(3)

where φ (r) is a radial basis function, r = |si – s0| is euclideandistance between the prediction location s0, and si wi: i = 1,2, . . . , n are weights to be estimated. In MQ function

φ (r) = (r 2 + σ 2)1/2 (4)

and in TPS

φ (r) = (σ.r)2 ln (σ.r) (5)

Here, σ is the optimal smoothing parameter, which iscalculated by minimizing the root mean square errors usingcross validation (Bishop, 1995).

Z( ) ( )s Z si ii

N

01

==∑λ

λii

p

ip

i

Nd

d

=−

−

=∑

0

01

z( ) ( )s w s s wi ii

n

n0 01

1= − +=

+∑ φ


368 EARTH SURFACE PROCESSES AND LANDFORMS

The other interpolation method, Kriging, is a powerful geo-statistical method which depends on mathematical and stati-stical models for optimal spatial prediction. In classic statisticsit is assumed that observations are independent and that thereis no correlation between observations. However, in geostatisticalmethods, information on spatial locations allows distancesbetween observations to be computed and autocorrelation tobe modelled as a function of distance (Burrough and McDonell,1988). Kriging uses the semivariogram, which measures theaverage degree of dissimilarity between unsampled values andnearby values, to define the weights that determine the contri-bution of each data point to the prediction of new values atunsampled locations (Krivoruchko and Gotway, 2004). Theimportant step in Kriging is adjustment of the experimentalmodel to the appropriate type of variogram model. There aremany models and each has its own basic structure, which isthe function of the distance among data (Fencik and Vajsablova,2006). There are several types of Kriging, such as simple,ordinary, universal, indicator, disjunctive and probability Kriging,of which simple, ordinary and universal are linear predictors.The difference between these methods is in the assumptionsabout the mean value of the variable under study (Krivoruchkoand Gotway, 2004). The general Kriging model is based on aconstant mean μ for the data and random errors ε (S) withspatial dependence.

Z(s) = μ(s) + ε(S) (6)

where Z(s) is the variable of interest, μ(s) is the deterministictrend and ε(S) is the random, autocorrelated errors. Variationson this formula form the basis of all the different types ofKriging. Ordinar Kriging assumes a constant unknownmean and estimates mean in the searching neighbourhood,whereas simple Kriging assumes a constant known mean.Thus these two methods model a spatial surface as deviationsfrom a constant mean where the deviations are spatiallycorrelated.

Study Area and Topographic Characterization

To compare the effects of interpolation process and data densityon DEM accuracy, a rocky hill near the campus of AfyonKocatepe University, Turkey, was chosen as a study area(Figure 1). The study area is approximately 120 000 m2 withan average elevation of 1005 m above sea level. In this area

approximately 24 000 locations were sampled at regular 2, 4,10 and 20 m intervals, giving a mean density of 2500, 625,100 and 25 points ha–1. These points were determined withthe coordinates of X, Y, Z using a level (Pentax AL-320 with aprecision of ±0·8 cm) and electronic distance meter (ZeissElta RS 45 with a precision of ±1 cm). The coordinates werecalculated using the National Horizontal and Vertical ControlNetwork and input to the database using ArcGIS 9.2 software.

Methodology

Many software packages require that original data (survey data)are interpolated onto a regular grid for visualization and ana-lysis of the elevation model (Robinson and Metternicht, 2005).ArcGIS, which is an integrated collection of GIS softwareproducts for building complete organizations, was used inthis study. The Geostatistical Analyst module of ArcGIS 9.2 isa set of models and tools developed for geostatistical analyses.This study was evaluated using Geostatistical Analyst in twosteps. The first step was examining geostatistical data withexploratory data analysis tools for dependency, stationarityand distribution of input data. If data are independent, itmakes little sense to analyse them; if data are not stationary,they need to be made so, usually by data detrending andtransformation. Geostatistical analyses work best when dataare Gaussian; if not they need to be made close to theGaussian distribution (Krivoruchko, 2005). In the secondstep, after receiving information on dependency, stationarityand distribution of the data, four interpolation methods wereperformed on this regularly sampled data.

All the interpolation methods’ parameters were optimizedusing cross validation because cross validation is the techniquemost commonly used as an exploratory procedure to find themost suitable model among a number of models (Davis, 1987;Smith et al., 2005). The parameters used in the evaluationsare shown in Table I.

Evaluation of DEM Accuracy

Error predictions may provide important information about thedeficiencies of a method, so may be an important input whenusing and comparing methods for particular applications (Smithet al., 2003). The accuracy of interpolation methods can beevaluated from different aspects. The most straightforward is

Figure 1. Study area. This figure is available in colour online at www.interscience.wiley.com/journal/espl



to predict some error indices such as mean error, meanabsolute error, root mean square error, and so on, thatcharacterize the interpolation accuracy via different validationtechniques. There are a number of validation methods. Severalauthors recommend cross validation for the evaluation of theaccuracy of interpolation methods (Kravchenko and Bullock,1999; Webster and Oliver, 2001; Smith et al., 2003). Thecross-validation method involves using all the raw data forcomparison. The main advantage of this method is a clearlydefined and user independent formulation that can be imple-mented on the surface. The method is more reliable for surfaceswith a sufficient number of representative input points (Hu,1995). The most common form of cross validation is the‘leave one technique’. This technique involves omitting onepoint before the interpolation process; performing the inter-polation then predicts the value of the omitted point and thedifference between the predicted and actual values of theomitted point is then calculated. This process is repeated forall samples. Another validation method used in this study wasthe split-sample method. This method can be used to assessthe stability of the interpolation algorithm (Declercq, 1996;Smith et al., 2005). In this method some raw data are omitted,interpolation is performed, and the difference between thepredicted and measured values of the omitted values is calcu-lated. This difference is used as a measure of the stability ofthe interpolation algorithm (Declerq, 1996; Smith et al., 2005).Another method is the use of an independent set of sampledata that is never used in the interpolation process (Desmet,1997; Robinson and Metternicht, 2005). For each point thedeviation between the actual and predicted values is calcu-lated, and accuracy is tested according to these values.

Mean, minimum, maximum, mean absolute, root meansquare errors, and so on, are the statistical means that areusually employed to evaluate the overall performance ofinterpolation methods. The measure most widely used as asingle aspatial global statistic is the root mean square error(RMSE), which measures the dispersion of the frequencydistribution of deviations between the original points andinterpolated points. The main attraction of RMSE lies in itsstraightforward concept and easy computation (Weng, 2006),mathematically expressed as:

(7)

where; (xi) is the predicted value, z(xi) is the observed value,and N is the number of values. The RMSE expresses the degreeto which the interpolated values differ from the measuredvalues, and is based on the assumption that errors arerandom with a mean of zero and normally distributed aroundthe true value (Desmet, 1997). In a number of studies, themean error has not been found to equal zero and thereforesome researchers have recommended the use of mean absoluteerror and standard deviation indices (Desmet, 1997; Fisherand Tate, 2006). Meanwhile it must be noted that thesedescriptive statistics are single summary indices and assumeuniform values for entire DEM surfaces. This assumption isnot always true, and many authors have suggested that thedistribution of errors will show some form of spatial pattern(Fisher and Tate, 2006; Weng, 2006). An important way toexamine the distribution of error is to create prediction errormaps. These maps have the advantage of clearly indicatingwhere serious errors occur. Another method is the usage ofglobal (Morans I, Getis-Ord General G) and local spatialautocorrelation (LISA, Getis Ord Gi*) measures to examinethe extent of error clustering.

Moran’s I is a global measure of spatial autocorrelation whichis produced by standardizing the spatial autocovariance bythe variance of the errors. The range of possible values ofMoran’s I is taken as –1 to 1, since positive values indicatespatial clustering of similar error values and negative valuesindicate clustering of dissimilar error values. Moran’s Iindicates clustering of high or low error values, but it cannotdistinguish between these situations. The General G statisticis usually used to understand clustering of high or low errorvalues. A large or larger than expected value for the G statisticmeans that high error values are found together, while converselya low value for the G statistic means low error values arefound together (Ord and Getis, 1995).

These global spatial data analyses show error clustering butthey do not show where the clusters are. To investigate thespatial variation as well as the spatial associations it is possibleto calculate local versions of Moran’s I and the General Gstatistic. The LISA (local indicators of spatial association:Anselin’s formed I value, 1995) were used to detect localpockets of dependence that may not show up when usingglobal spatial autocorrelation methods. Lastly, the relationshipbetween DEM error and morphometric characteristics (slopeand curvature) of the hill was examined via linear ordinaryleast square regression analysis. For each DEM, correlation

Table I. Method parameters

Methods Intervals Power Model Major range Partial sill Lag sizeNumberof lags

Number ofneighbours

Number of divisions

OK 2 – Spherical 187·6 62·873 31·9 12 7 8OK 4 – Spherical 193·8 61·474 33·1 12 7 1OK 10 – Spherical 198·4 62·333 33·9 12 10 4OK 20 – Spherical 195·42 61·721 31·9 12 25 8TPS 2 0 Circular 128·64 17 1TPS 4 0 Circular 128·07 7 4TPS 10 0 Circular 127·73 10 1TPS 20 0 Circular 121·81 10 1IDW 2 3·3943 Circular 128·64 8 1IDW 4 4·1166 Circular 128·07 8 1IDW 10 6·0657 Circular 127·73 8 1IDW 20 8·934 Circular 121·81 4 1MQ 2 0 Circular 128·64 44 1MQ 4 0 Circular 128·07 21 1MQ 10 5·6403 Circular 127·73 22 1MQ 20 23·919 Circular 121·81 25 1

RMSE = −{ }=∑1 2

1Nz x xi i

i

N

( ) ( )z



coefficients were calculated for slope and curvature parametersof the terrain. Then a multivariate ordinary least square linearregression model was used to identify relationships betweenerror and slope-curvature characteristics.

Results

It has been demonstrated that DEM error can vary to a certaindegree with different interpolation algorithms and data density(Desmet, 1997; Weng, 2006). The level of this error is import-ant for many specific applications. Therefore, four of the mostwidely used interpolation algorithms were compared withdifferent data densities, which were determined regularly.DEMs generated from surveyed data with the 2 × 2 sampleinterval are shown in Figure 2.

The main purpose of this study was to quantify the amountand distribution of error introduced by interpolation methodsand data density for this rocky hill. As a first step, interpolationmethods and data density were validated by cross validation,split-sample and jack-knifing methods as mentioned above.To compare the accuracy of interpolation methods and datadensity statistically, mean error, maximum error, minimumerror and root mean square error indices were calculated viavalidation methods. Cross-validation values are shown in

Table II. Cross validation is a useful indicator of the generalcharacteristics of interpolation methods, but it cannot be usedas a measure for the robustness of algorithms (Smith et al.,2003). Therefore the split-sample method and jack-knifingwere used to assess the model’s overall accuracy and stability,as measures of the robustness of the methods. The split-samplemethod was performed using 95, 75 and 50 per cent of theraw data for investigation. The results of the split-samplevalidation investigation are shown in Table III. Jack-knifing byusing an independent set of the sample was performed with173 irregularly surveyed test points. These test points werecompared with the estimates obtained by interpolation algo-rithms. The results of this validation are shown in Table IV. Tomeasure the precision of methods, RMS is usually used. SmallRMS values are required, because it is desirable that thepredicted data should be close to the raw data.

When we compared the results of the validation methods, theIDW algorithm provided the worst interpolation and producedthe greatest overall error of all the methods. The MQ and OKmethods had nearly similar results, but TPS was the best inter-polator according to accuracy indices produced by validationmethods. To compare the differences between methods’interpolation predictions by considering the data density, theprediction values were plotted against measured values. Thescatter points for the TPS at 2, 4, 10 and 20 m intervals are

Figure 2. Digital elevation models generated from surveyed data with 2 × 2 sample interval with contours. This figure is available in colouronline at www.interscience.wiley.com/journal/espl



Table II. Results of cross-validation with different sample intervals

MethodsSample

interval (m)Mean

error (m)Maximum

(+) error (m)Minimum

(–) error (m)Root mean

square error (m)

OK 2 –0·000110 2·24 –1·81 0·1239OK 4 0·000468 2·50 –2·25 0·2111OK 10 0·001126 3·46 –2·09 0·4141OK 20 0·027790 4·56 –4·19 0·8098TPS 2 –0·000160 1·55 –1·77 0·1216TPS 4 0·000525 2·06 –2·66 0·1957TPS 10 –0·006440 3·15 –2·56 0·3932TPS 20 –0·014540 4·32 –3·55 0·7340IDW 2 –0·000120 3·02 –2·38 0·1461IDW 4 0·003684 3·02 –2·78 0·2686IDW 10 0·017150 3·31 –2·61 0·5078IDW 20 0·075340 4·80 –4·21 0·9823MQ 2 –0·000140 2·24 –1·81 0·1238MQ 4 0·000316 2·48 –2·24 0·2108MQ 10 0·000971 3·15 –4·23 0·4293MQ 20 –0·004190 4·19 –2·91 0·7096

Table III. Results of split-sample validation with different sample intervals

Table IV. Results of independent data set validation with different sample intervals

ModelPercentageraw data

Depth(m)

Maximumerror (m)

Maximum(–) error (m)

Mean(m)

RMSE(m)

Depth(m)

Maximumerror (m)

Maximum (–) error (m)

Mean(m)

RMSE(m)

OK 0·95 20 –1·00 0·53 0·05 0·36 10 0·82 –1·07 –0·04 0·29TPS 0·55 –0·73 0·06 0·33 1·01 –1·07 –0·01 0·28IDW 0·75 –2·19 0·04 0·67 0·86 –1·39 –0·04 0·40MQ 0·99 –1·52 0·03 0·57 0·90 –1·11 –0·02 0·27OK 0·75 4·54 –4·50 0·03 1·02 2·22 –2·24 –0·02 0·40TPS 4·30 –4·03 0·02 0·96 2·32 –2·20 –0·02 0·39IDW 4·24 –6·46 0·01 1·38 2·38 –2·75 0·00 0·66MQ 4·25 –3·57 0·07 0·87 2·35 –2·21 –0·02 0·39OK 0·50 3·20 –5·47 0·20 1·19 2·20 –2·34 –0·02 0·50TPS 1·69 –1·95 0·00 1·15 2·31 –2·63 –0·03 0·50IDW 6·72 –7·27 –0·07 1·68 2·95 –3·74 0·00 0·82MQ 4 3·37 –5·89 0·09 1·23 2 2·34 –2·16 –0·02 0·48OK 0·95 1·86 –1·29 0·01 0·25 1·03 –0·75 0·00 0·11TPS 1·72 –1·05 0·00 0·22 1·20 –0·81 0·00 0·11IDW 1·93 –1·46 0·20 0·28 1·14 –0·95 0·01 0·13MQ 1·90 –1·19 0·01 0·24 1·04 –0·75 0·00 0·11OK 0·75 1·92 –2·43 –0·01 0·26 1·59 –1·19 0·00 0·14TPS 1·33 –2·30 –0·01 0·23 1·90 –1·33 0·00 0·13IDW 1·94 –2·76 0·00 0·33 2·01 –1·99 0·00 0·18MQ 1·63 –2·16 –0·01 0·24 1·59 –1·19 0·00 0·14OK 0·50 2·33 –2·85 –0·01 0·29 2·28 –1·95 0·00 0·15TPS 2·20 –2·50 –0·01 0·24 1·69 –1·95 0·00 0·15IDW 2·72 –4·04 0·00 0·41 2·90 –2·37 0·00 0·22MQ 2·34 –2·58 –0·01 0·28 1·99 –1·60 0·00 0·15

MethodsSample

interval (m)Mean

error (m)Maximum

(+) error (m)Maximum

(–) error (m)Root mean

square error (m)

OK 4 –0·01 0·77 –1·46 0·18OK 10 0·01 1·64 –2·55 0·42OK 20 0·05 3·15 –4·17 0·69TPS 4 –0·01 0·95 –1·38 0·18TPS 10 0·01 2·31 –2·33 0·41TPS 20 0·01 2·5 –4·14 0·66IDW 4 –0·01 0·97 –1·44 0·22IDW 10 0·01 2·61 –3·28 0·65IDW 20 0·09 3·54 –4·03 1·12MQ 4 –0·01 0·77 –1·46 0·18MQ 10 0·00 2·18 –2·15 0·39MQ 20 0·03 2·91 –4·32 0·72



shown in Figure 3. The spread of the plots increased generallyas sample density decreased. For all densities and interpolationmethods, linear correlation coefficients were over 0·9. TheTPS method had the best correlation coefficients for alldensities compared with the other methods (0·994 – 0·999).

Accuracy indices produced by validation methods assumesingle uniform error values for the entire region. However,several researchers have identified this assumption as beinginvalid (Fisher, 1998; Carlisle, 2005). Therefore, it is importantto investigate the spatial variation and distribution of errorsconsidering the stationarity. A way to do this is to show theerrors using choropleth symbol maps. The spatial distributionof errors was examined by plotting the location and magnitudeof errors. The resultant plots of errors introduced by the inter-polation methods using the data sampled with 20-m intervalsare shown in Figure 4.

The size of the points represents the magnitude of errors.Red points represent errors larger than ±1 m. It can be shownthat errors larger than ±1 m occurred more often in the IDWmethod. Locations of errors larger than ±1 m were concentratedin similar places in TPS and OK. Similarly when we examinedthe distribution of errors, it was observed that the errors werelargely coincident with the rocky parts of the surface andthese errors were larger in the low-density samples. The errorsin TPS with 4-m sample intervals are shown in Figure 5.

The size of the points shows the magnitude of errors. Theblack points represent errors larger than ±0·3 m. However,interpretation of the choropleth error maps can be difficultwhen there is too much data. Another way to observe andanalyse the distribution of errors is to create error maps.These maps show where serious and anomalous errors areoccurring and clustering. Therefore, prediction error surfaceswere created to show the spatial pattern of errors resultingfrom interpolation algorithms. Comparison of such surfacescan be extremely informative with respect to the occurrenceand magnitude of errors in relation to terrain slope-curvatureand distribution of input data (Shearer, 1990; Weng, 2006).

An error surface created to show the spatial pattern of absoluteerrors resulting from TPS with a 4-m interval is shown inFigure 6. It is clear from the maps that errors tend to cluster inrocky regions of the study area where slope and curvaturechange rapidly, especially around the crest of the hill.

Another comparison was the test of spatial autocorrelation.All methods with different surfaces had significant values ofMoran’s I, indicating a considerably high degree of clustering(Table V). Similarly, significant General G statistic values ofsurfaces with 2, 4 and 10-m intervals were larger than expected,indicating that high error values were found together (hotspots). However, a detailed look at the autocorrelation valuesrevealed that some interpolators did a better job than others.The TPS method produced the least amount of clustering interms of revealing the systematic errors that resulted from under-representation of rocky areas. Although Kriging is the bestlinear unbiased estimator, the OK method produced reasonablylow values of clustering similar to the MQ method. On theother hand, IDW produced the highest values of clustering.

Global spatial autocorrelation indices show clustering butthey do not show where the clusters are. Therefore, to investigatethe spatial variation as well as the spatial associations, it ispossible to calculate local versions of Moran’s I, for eachlocation (Anselin, 1995). The LISA investigates those clustersof error values with similar values and those clusters of errorvalues with different values. A high value of I means that thepoint is surrounded by features with similarly high or lowerror values, whereas a low value of I means that the point issurrounded by features with dissimilar error values. The zscore shows the statistical significance of the I value for thedistance specified.

All interpolation methods generated clusters around thepeak of the hill. The TPS method produced the least amountof clustering whereas IDW produced the most. Error surfaceand LISA results of the TPS algorithm with 4-m intervals areshown in Figure 6. Red points show the clusters of high-highvalues.

Figure 3. Measured values versus predicted values and correlation coefficients using the thin plate spline method.



Figure 4. Spatial distribution of errors with 20-m sample intervals. This figure is available in colour online at www.interscience.wiley.com/journal/espl

Table V. Global spatial autocorrelation indices of Moran’s I and Getis-Ord General G with mean absolute errors of DEMs

Interpolationmethods

Moran’sindex

Expectedindex z score

ObservedGeneral G

ExpectedGeneral G z score

IDW2 0·046 –0·00004 335·1 0·0112 0·00944 62·4IDW4 0·068 –0·00016 158·3 0·0113 0·00940 39·7IDW10 0·095 –0·00099 33·7 0·0107 0·00936 11·2IDW20 0·138 –0·00369 8·2 0·0103 0·00984 0·9MQ2 0·042 –0·00004 302·8 0·0114 0·00944 61·0MQ4 0·061 –0·00016 153·5 0·0114 0·00940 37·2MQ10 0·079 –0·00099 28·3 0·0112 0·00936 12·3MQ20 0·085 –0·00369 5·3 0·0096 0·00979 –0·4TPS2 0·040 –0·00004 292·9 0·0114 0·00944 59·5TPS4 0·061 –0·00016 151·7 0·0115 0·00940 37·0TPS10 0·069 –0·00099 25·7 0·0119 0·00936 14·1TPS20 0·090 –0·00369 5·7 0·0096 0·00979 –0·4OK2 0·042 –0·00004 302·8 0·0114 0·00944 61·2OK4 0·062 –0·00016 153·9 0·0114 0·00940 37·4OK10 0·079 –0·00099 28·3 0·0107 0·00936 10·1OK20 0·105 –0·00369 6·6 0·0103 0·00979 1·0



Finally, relationships between DEM error and the slope-curvature morphometric parameters of the hill were examinedfor each DEM using correlation coefficients that were calcu-lated for each terrain parameter and elevation error. Never-theless, it was expected that both terrain parameters acting incombination would influence the spatial variation in DEMerror (Carlisle, 2005). Therefore, multivariate ordinary leastsquares linear regression analysis was used to model anysuch relationship. Table VI summarizes the values of multivariateordinary least squares linear regression model correlationswith slope and curvature parameters.

According to the results, the curvature parameter had agreater effect than slope parameter on residuals. Residual errorsfrom the MQ algorithm showed the strongest correlationswith terrain parameters. Correlation values of OK were foundto be close to values of MQ. Residual errors from the IDWalgorithm had weaker correlations with terrain parametersthan the other three methods. Significant correlations indicatethat there is a relationship between DEM errors and terraincharacteristics. The curvature parameter gives a good indicationof the amount of error while sample density increases. Theslope parameter was found to have the greatest values whenthe data interval was 10 m. Meanwhile a combination ofparameters could give a better signal about the amount oferror than a single parameter could.

Conclusion

The magnitude of uncertainty from the interpolation is subjectto many factors. The research revealed that the magnitudeand distribution of errors in a DEM of the hill were strongly

related to the varying characteristics of the terrain, samplingdensity and interpolation algorithm. This study demonstratedthat the IDW algorithm produced the greatest overall error.This probably arose because of its inability to model the steepsurfaces that are common in hill areas. In the IDW surfaceswith 10- and 20-m intervals the bull’s eye effect was toostrong. As well as introducing error, the stepped appearanceof IDW means that it did not produce the most realisticlooking representation of the hill areas. The method intro-duced a little more error than TPS and MQ. Although the

Figure 5. Spatial distribution of errors with 4-m sample intervals and location of these errors over the hill. This figure is available in colouronline at www.interscience.wiley.com/journal/espl

Table VI. Correlation coefficients of regression model results

Methods Slope CurvatureCombination of slope

and Curvature

IDW2 0·460 0·816 0·829IDW4 0·515 0·643 0·700IDW10 0·538 0·414 0·586IDW20 0·397 0·287 0·420OK2 0·393 0·898 0·899OK4 0·458 0·724 0·746OK10 0·526 0·497 0·619OK20 0·487 0·268 0·512TPS2 0·375 0·851 0·852TPS4 0·430 0·680 0·701TPS10 0·522 0·529 0·636TPS20 0·425 0·398 0·492MQ2 0·394 0·898 0·899MQ4 0·457 0·724 0·746MQ10 0·486 0·476 0·582MQ20 0·444 0·434 0·524



model parameter choices, such as search radius, sill, range,nugget and minimum/maximum number of data for OK wereoptimized using cross validation, it should be noted that theresults were related to the parameters rather than to OK itself.The disadvantage of OK was the length of the process time forhigh volume data.

However, it is interesting to note that, where the density ofdata is low, OK seemed to exhibit a larger and more general-ized approximation than the DEMs produced by IDW. TheTPS method was the most appropriate and effective with MQ.Normally, as shown in Table II, as distances between thepoints in the raw (elevation) data increased, the RMS valuesalso increased (the relationship was close to linear). With theexception of the IDW method, the other methods showedalmost the same RMS values. In OK, while distances betweenthe points increased in the raw data, the smoothness of thesurface increased more according to the TPS and MQ methods.

Split-sample validation was performed using sample sizesof 95, 75 and 50 per cent of the raw data for all the methodsto assess the stability of the methods given a smaller input ofraw data. It was found that the OK, TPS and MQ methodswere relatively stable. Similar values were obtained for thesethree methods while the quantity of raw data decreased.Meanwhile, IDW was also found to be stable, but the valuesobtained for this method were rather larger than for the otherthree methods as the quantity of raw data decreased.

The difference in errors created at varying sample densitieswas assessed using the split-sample, jack-knifing and cross-validation methods. It was found that OK, TPS and MQproduced relatively parallel increasing RMS error values.When the spatial distribution of errors was investigated, itwas found that all the large errors (±) were clustered around

the steep surface of the hill area as shown in Figure 7. Errorindices such as RMS, mean absolute error, and so on, alonewere insufficient to indicate DEM uncertainty. To understandthe DEM uncertainty, spatial distributions of the model residualsfrom the four modelling algorithms were investigated usingerror surfaces and the global and local Moran coefficients.Global Moran’s I indices indicated that errors were concen-trated with all methods and all sample densities. The Getisand Ord General G method showed that high error valueswere concentrated with DEMs obtained from input data with2-, 4- and 10-m intervals. Then LISA was utilized to investigatespatial distribution and heterogeneity in model residuals usingfour interpolation algorithms with ordinary least squares (OLS)as the benchmark. According to OLS regression results, residualerrors with interpolation algorithms showed strong correlationsboth alone and in combination with terrain parameters ofslope and curvature. For the study area, highly adjusted regres-sion coefficients changing between 0·4 and 0·9 indicated thatthe spatial distribution of DEM error could be modelled withOLS regression modelling of terrain parameters with a highdegree of success.

Meanwhile, OLS regression is a global technique in that asingle regression model is created that best fits the whole residualdata set over the entire study area (Carlisle, 2006). This studyarea had a highly variable terrain character and DEM errors havea high spatial autocorrelation that show spatial non-stationarity.Therefore, relationships between residual errors and terrainparameters would not show spatial stationarity. The limitationsof OLS regression as a consequence of its assumption of spatialstationary were discussed in another study (Fotheringham et al.,2002). This limitation will be examined in a further studyusing a geographically weighted regression model.

Figure 6. Error surface and local indicators of spatial association (LISA) results of error values. This figure is available in colour online atwww.interscience.wiley.com/journal/espl



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