Causes and consequences of error in digital elevation models*

Progress in Physical Geography 30, 4 (2006) pp. 467–489

© 2006 SAGE Publications 10.1191/0309133306pp492ra

I IntroductionThe digital record of land surface elevationswas one of the first widely available forms ofgeographical information. Such digital recordsare often distributed in the form of a digitalelevation model or DEM, and their deriva-tives are frequently employed throughoutphysical geography for applications rangingfrom geomorphometry (Pike, 2000) to hydro-logical modelling (Kenward et al., 2000) and

the physiographic correction of digital satelliteimagery (Goyal et al., 1998).

DEMs come in a number of forms, but allusually consist of files containing a large numberof records (often more than 105 records) whereeach record represents a statement or estimateof the elevation at a point in space. From theoutset, it is important to be aware that theDEM is usually the end result of a number ofmodelling and processing steps as typified by the

Causes and consequences of error in digital elevation models*

Peter F. Fisher1** and Nicholas J. Tate2

1Department of Information Science, City University, Northampton Square,London EC1V 0HB, UK2Department of Geography, University of Leicester, University Road, Leicester LE1 7RH, UK

Abstract: All digital data contain error and many are uncertain. Digital models of elevation surfacesconsist of files containing large numbers of measurements representing the height of the surface of theearth, and therefore a proportion of those measurements are very likely to be subject to some level oferror and uncertainty. The collection and handling of such data and their associated uncertainties hasbeen a subject of considerable research, which has focused largely upon the description of the effectsof interpolation and resolution uncertainties, as well as modelling the occurrence of errors. However,digital models of elevation derived from new technologies employing active methods of laser and radarranging are becoming more widespread, and past research will need to be re-evaluated in the nearfuture to accommodate such new data products. In this paper we review the source and nature oferrors in digital models of elevation, and in the derivatives of such models. We examine the correctionof errors and assessment of fitness for use, and finally we identify some priorities for future research.

Key words: digital elevation model, digital surface model, error modelling, fitness for use,uncertainty, visualization.

*Part of this paper has been previously published in French as Tate and Fisher (2005) and is included herewith agreement of Hermès Publishers.**Author for correspondence.

at PENNSYLVANIA STATE UNIV on March 4, 2016ppg.sagepub.comDownloaded from

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468 Causes and consequences of error in digital elevation models

flow chart in Figure 1. The progression fromconceptual model through to digital modelrequires not only the selection of suitable proce-dures, but also the application of suitable meas-urement and statistical processes. Ourunderstanding of these processes and theirassociated errors tell us that there is a very smallprobability (effectively no chance) that all digitalrecords in any DEM are correct. DEMs there-fore contain endemic error, and this error canpropagate through to products derived from theDEM, for example to hydrogeomorphic param-eters such as catchment size and stream net-work characteristics (Walker and Willgoose,1999) and surface flow dispersal areas (Endrenyand Wood, 2001). In this respect, DEMs are nodifferent from any other type of digital geo-graphical data, all of which contain some errorthat can propagate to dependent operations/products. However, models of elevation are dis-tinct for four reasons: 1) they were one of thefirst forms of digital geographical informationwhich became available; 2) they are now widelyused; 3) they are closely associated with themathematical concepts of surface modelling;and 4) they represent a tangible, directlyobservable phenomenon of which all peoplehave direct experience: the surface of the earth.

This paper reviews research on the under-standing, modelling and propagation of error

in DEMs, with the aim of presenting a comprehensive statement of the issues sur-rounding these topics. We therefore start bydefining key terms and methods used todescribe error. Next, we examine varioussources of error that can accumulate during theprocess of DEM construction. Statistical mod-els of error, and other approaches that havebeen suggested for examining error, are thefocus of the next section, followed by a consid-eration of the propagation of the error intoderived information. We review some of themethods suggested for error visualization andcorrection and, finally, we examine research todetermine fitness for use of DEM products. Inconclusion, we identify a number of areas whichrequire further research in the near future.

II Definitions

1 Digital elevation model (DEM)The term digital elevation model has two gen-eral meanings. First, it is any set of measure-ments that record the elevation of the surfaceof the earth, such that the spatial proximity of,and spatial relationships between, those meas-urements can be determined either implicitlyor explicitly; a simple list of elevations is not aDEM. On the other hand, a list of triplet meas-urements of elevation together with Eastingand Northing to give location does constitute aDEM, because the spatial relationships of theelevations can be recreated from the locationinformation. Second, and more specifically, aDEM is a set of elevation values which arerecorded on a regular grid – most commonly ina square form, less frequently in a triangular orrectangular form. Since the dimensions of thegrid are known and the number of observa-tions in each row is known, the implicit spatialrelationships between elevation values can bedetermined. The DEM in grid form is the mostwidely used data model for the distribution ofdigital elevation data by data providers (forexample, successive USGS DEMs), and hasbeen the format on which the vast majority ofresearch on error and uncertainty has beenbased. In discussion below we therefore use

Figure 1 A flow diagram showing theprocess of construction of a DEM throughthe intermediary of a contour map



Peter F. Fisher and Nicholas J. Tate 469

‘DEM’ synonymously with ‘gridded DEM’although much of what we state is equallyapplicable to other formats. Digital terrainmodel (DTM) is sometimes used instead ofDEM, but is more correctly used to describe aset of digital records related to terrain not justelevation (Burrough, 1986).

2 Error and uncertaintyIn this paper, we adopt the convention thatconsiders error to be the objective or formalproblems with measurement/estimation andother, less tangible issues to be uncertainty(Hunter and Goodchild, 1993; Gahegan andEhlers, 2000).

People can conceive of the surface of theearth relatively clearly, and we can thereforesay that it is possible to measure the height ofthis surface above a defined datum at a set ofpoints in space. If we believe firmly in this con-ceptualization, and we are able to revisit eachpoint and repeat the measurement, we canassert that any difference encounteredbetween the two measurements is due toerror in one or, more likely, both measure-ments. If the two data sets are obtained by dif-ferent methods, for example one by opticaltheodolite and the second by laser levelling,then there may be a justified belief in one set ofmeasurements being more accurate (the latter)and so one (the first) being in relative error.

If, in addition to a set of point measure-ments (however derived), we have a model ofthe form of the surface of the earth, whichcan be expressed mathematically, then it ispossible to estimate the elevation at unknownlocations from those at measured locations bya mathematical process of interpolation (as isoften the case in DEMs and is discussedbelow). Differences between the estimatedelevation and the measured elevation are amatter of the fidelity of the mathematicalmodel. These are still regarded as errors in thedigital elevation model, however.

Generalizing these two instances, there-fore we can say that error of a given set ofpoint measurements of a surface can onlyproperly be determined by comparison with

another set of known, usually more accuratemeasurements. Such data are often termed‘reference’ data and assumed to be error free(Gens, 1999; Kyriakidis et al., 1999).

Other forms of doubt about the quality ofthe digital elevation model constitute aspectsof uncertainty, and are largely related torepresentation. For example, if we do notvary the quality of measurement/estimation,differences due to data gathered at contrast-ing resolutions should more correctly beregarded as an aspect of the uncertainty of themodel representation.

3 A typology of errorErrors in DEMs can clearly occur in both theelevation or vertical (Z) and planimetric orhorizontal (XY) coordinates, but the focus isusually on the former because planimetric errorwill produce elevation error. Many commercialdata suppliers only report elevation error. Errorsin DEMs are usually (Cooper, 1998; Wise,2000) categorized into three groups: gross errorsor blunders, systematic errors due to determinis-tic bias in the data collection or processing, andrandom errors.

Gross errors or blunders can be the result ofuser error or equipment failure: such errors areinfrequent in commercial DEMs but they dooccur. They are evident with higher frequencyin non-commercial products. Systematicerrors can be defined as the result of a ‘deter-ministic system which if known may be repre-sented by some functional relationship’(Thapa and Bossler, 1992: 836). Conceptualexamples of simple systematic errors are por-trayed in Figure 2, A and B. Real examples ofsystematic errors include the contour line‘ghosts’ identified in many DEMs derived fromcontour data (Guth, 1999) as terracing, andthe distinctive parallel striping artifacts found insome USGS 7.5-minute DEMs (Brown andBara, 1994; Garbrecht and Starks, 1995) and the Canadian Terrain Resource InformationManagement (TRIM) DEM data product(Albani and Klinkenberg, 2003).

In contrast, random errors in a DEMaccrue from a great variety of measurement/




operational tasks in producing the DEM.These may be represented conceptually asrandom variations around the true referencevalue, but a number of models of the spatialdistribution are possible, and two alternativesare shown in Figure 2, C and D.

4 Describing errorAs with the definition of error, the measure-ment of error in DEMs is somewhat confused.

The most common descriptor is the root meansquare error (RMSE; Li, 1988; Shearer, 1990;Desmet, 1997; Hunter and Goodchild, 1997):

where zDEM � the measurement of elevationfrom the DEM, and zRef � higher accuracymeasurement of elevation for a sample of n

RMSEz z

nDEM=

−∑( )Ref2

Figure 2 Comparisons of a profile through a DEM and the occurrence of error. (A)The occurrence of error with bias; (B) the occurrence of systematic error; (C) theoccurrence of spatially autocorrelated error (the normal situation); (D) the occurrenceof random error (no spatial autocorrelation). In each instance the upper diagram showsthe ground surface as a thick line and the ground surface with the error as a thin line,and in the lower diagram the error aloneSource: Modified from Shearer (1990); also published in Tate and Fisher (2005).




points. This measure has the property that it isalways positive. Some authors use n – 1 as thedenominator acknowledging the similaritybetween this equation and a standard deviation(Shearer, 1990; Desmet, 1997). Indeed, theRMSE is equal to the standard deviation of theerror if the mean error is (or is assumed to be)zero. It should be noted that RMSE is a widelyused measure of conformity between a set ofestimates and the actual values, and hasbecome a standard measure of map accuracy.

RMSE is usually reported as a single aspatialglobal statistic per DEM based on comparisonwith a limited sample of points. For example,for the USGS 7.5-minute DEM product theRMSE calculation requires a minimum of only28 points (USGS, 1990). On the other hand,the Ordnance Survey of Great Britain does noteven claim to determine the error for each tileof the DEM, but asserts a single statement ofthe RMSE in the product specification, imply-ing that it is the same for the whole nationaldata set, although the report does give a rangeof values for different land surface slopes.

In a number of studies, however, the meanerror has not been found to be equal to zero(Li, 1988; Monckton, 1994; Fisher, 1998), andso the RMSE is not necessarily a gooddescription of the statistical distribution ofthe error. Therefore other researchers havesuggested the use of a more complete statis-tical description of errors by reporting themean error (ME) and error standard deviation(S) (Li, 1988; Fisher, 1998):

ME can be either negative or positive, andrecords systematic under- or overestimation ofthe elevations in the DEM, otherwise knownas bias. Figure 2C shows a conceptual modelof error in a DEM without bias and Figure 2Ashows the same pattern of error with positive

Sz z ME

nDEM=

− −−

∑[( ) ]Ref2

1

MEz z

nDEM=

−∑( )Ref

bias. Shearer (1990) and Desmet (1997) advo-cate use of the mean absolute error by replacingzDEM �zRef with the modulus, |zDEM�zRef|.This is similar to RMSE. Both ME and S arepreferred, however, as these will allow the esti-mation of bias. S records the dispersion, asdoes the RMSE, but if ME is relatively largethen S and RMSE may be very different.

None of these descriptive statistics(RMSE, ME, S) reports more than a globalsummary statistic for a data set. Crucially, allfail to describe the spatial pattern of error, andin a DEM the error is likely to vary spatially(Figure 2C, as compared with Figure 2D).However, in spite of improved understandingabout the types of error within DEMs, thereis still relatively little known about the spatialstructure of that error (Monckton, 1994;Hunter and Goodchild, 1997; Liu and Jezek,1999). As a response, there have been a vari-ety of studies that have attempted to describethe pattern of DEM errors spatially by meansof both geostatistical variograms (Fisher,1998; Kyriakidis et al., 1999; Liu and Jezek,1999; Holmes et al., 2000; Weng, 2002;Zhang and Goodchild, 2002; see section V)and Fourier-based analysis (Liu and Jezek,1999). In a study of the errors produced froma comparison of low accuracy elevation datawith higher accuracy reference data, Liu andJezek (1999) employed both methods todescribe the spatial pattern of error for51.2 km by 51.2 km DEM of the McMurdoDry Valley region in the TransarcticMountains of Antarctica. Their analysisrevealed a highly anisotropic and scale-dependent pattern of error, closely correlatedto the characteristics of the terrain surface. Incontrast, Holmes et al. (2000) observed thatfor a low accuracy–high accuracy comparisonfor the Los Olivos quadrangle, Santa BarbaraCounty, California, the correlation betweenerror and various terrain indices was poor.

A further important weakness of RMSE hasbeen noted by Hunter and Goodchild (1996:15) who observed that ‘while containing usefulinformation about the final [DEM] product, [it]says nothing about the numerous contributing




factors that may have played a role in the over-all process’ giving rise rise to the error.

III Sources of error in DEMsThree main sources of error in DEMs are usu-ally identified (Shearer, 1990; Li, 1992; Li andChen, 1999; Gong et al., 2000):

1. those derived from variations in the ‘accu-racy, density and distribution’ (Li andChen, 1999: 203) of the measured sourcedata as determined by the specific methodof data generation;

2. those derived from the processing andinterpolation used to derive the DEM from the source data;

3. those resulting from the characteristics of the terrain surface being modelled inrelation to the representation of the DEM.

The first two are quite clearly errors. Thethird, however, should be considered a matterof uncertainty. Alternatively, we can think of1 as data-based, being strictly concerned withthe source data, while 2 and 3 are model-based being concerned with how well theresulting DEM approximates the real physio-graphy (Theobald, 1989; Shortridge, 2001).

1 Method of source data generationHistorically, DEMs encountered by the scien-tist/academic have most frequently beensourced by digitizing contour lines and spotheights from paper maps. Other sources mayhave included imagery such as stereo aerialphotographs using various types of pho-togrammetry, or less frequently point meas-urements derived direct from land survey.

The first step in the construction of such aDEM from contours is therefore the creation ofthe source map. Source map error is generallydefined as arising from the processes of collec-tion, recording, generalization, symbolizationand production inherent in the ‘cartographicprocess’ (MacEachren, 1985; Muller, 1987).Even though the spatial elevation data used toproduce a contour map may have been pro-duced photogrammetrically, the transformationto a contour map will introduce inaccuracy,

from both cartographic generalization and mapproduction (Fryer et al., 1994). It is difficult togeneralize about the errors introduced into con-tours/spot heights from these sources, butFryer et al. (1994) suggest that the photogram-metric errors alone might reasonably beexpected to be about 2 or 3 per 10,000 units offlying height. To the error in the contour lines isadded error from the digitizing process.Digitized contours can be stored in their ownright, but more usually they are interpolated toproduce the gridded DEM (see section III.2).

DEM construction directly from manual andsemi-automated photogrammetry introducesboth random and systematic errors (Petrie,1990; Shearer, 1990; Fryer et al., 1994).Random errors may accrue through the lack ofprecision in the identification of target points onthe photograph as part of the process of aerialtriangulation, and systematic errors may accruefrom changes in the film media, instrumenterrors and from operator fatigue (Fryer et al.,1994). Measurement methods include themanual collection of elevation points along pro-files using a stereoplotter, or the more auto-mated collection of elevation points from adigital stereomodel. In the former, a well-known example is the ‘Firth Effect’ (Hunterand Goodchild, 1995: 534) which occurs when,collecting data in profiles, elevations are under-estimated when moving in an upslope directionand overestimated in a downslope directionproducing a distinct ‘herringbone’pattern in theelevations. Hunter and Goodchild (1995) alsonoticed edge discontinuities that they attrib-uted to interpolation errors from the automatedphotogrammetric system used by the USGS.

The recent availability of computer-baseddigital photogrammetric systems (DPS), some-times as a component of image processing soft-ware, means that bespoke DEM constructionhas become more widespread. Such systemsare usually based on some form of hierarchicalstereo image correlation, and are able to pro-duce gridded DEMs in a fully automatic mannerwith optional manual editing. It is difficult togeneralize about the characteristics of such sys-tems, but Gong et al. (2000) have suggested




that, in the absence of any editing, the fullyautomated mode may produce data of lowaccuracy relative to traditional photogrammet-ric methods. Interestingly, Davis et al. (2001)have modelled the relationship between theestimated stereo-correlation quality within theDPS and the RMSE obtained of the resultingDEM when compared with a higher accuracykinematic GPS survey. For 1:40,000 scaleimages of an urban area, they were able to pre-dict DEM RMSE to within 8%.

Recently, there has been an increase in theuse of active airborne sensors for DEM cre-ation. Such active systems include LiDAR(light detection and ranging, also known aslaser altimetry; Dubayah et al., 2000) andInSAR (interferometric synthetic apertureradar; Goyal et al., 1998). LiDAR uses theemission and reflection of light pulses usuallyfrom an airborne scanner. The quality of ele-vation information obtained is a function ofthe sensor and scanning system, the natureand quality of the positioning/orientation sys-tem in the aircraft, aircraft speed/flyingheight, and the characteristics of the terrainsurface (Huising and Gomes-Pereira, 1998;Wehr and Lohr, 1999). For example, whenworking over land cover types such as forest,it can be difficult to determine whether thelight pulse has penetrated to the ground(Dubayah et al., 2000). The systematic errorfor laser altimetry has been estimated torange from a minimum of 5 cm in flatpaved/barren areas to a maximum of 200 cmin grass and scrubland, and random errorsfrom 10 cm in flat areas to 200 cm in hillyareas have been noted (Huising and Gomes-Pereira, 1998). Similarly to Laser-LiDAR, thequality of the elevation information obtainedfrom InSAR is related to sensor and terrainsurface characteristics (Goyal et al., 1998).

DEMs produced by these active systemshave the potential to be of much higher accu-racy than some of the traditional methods ofDEM construction discussed above, althoughBaltsavias (1999: 90) has noted that there aremany more sources of error in active systemsthan in photogrammetry, which make both

the assessment and propagation of error morecomplicated.

The construction of DEMs using fullyautomated photogrammetry and active sys-tems of data capture constitute whatLemmens (1999) has termed a process of‘blind sampling’ of terrain. He identifies fourprimary terrain surface characteristics whichwill influence the quality of DEM data: theexistence of micro-relief which make elevationmeasurement points spatially unrepresenta-tive; non-selective spatial coverage of the sen-sor; the presence of sloping ground alteringsignal reflection; and signal attenuation/falloutdue to the varying reflectivity of different landcover types. In fact, for all these methods theelevation ‘surface’ identified by the sensor maynot be the surface of the ground but a com-posite surface of other features includingbuildings and vegetation. Indeed, measure-ment accuracies as high as 5 cm, togetherwith problems of the penetration of vegeta-tion, mean that the certainty that the surfaceof the earth is the height being measured hasbeen lost for these systems.

Irrespective of the method of DEM con-struction, the error in a DEM can also beinfluenced by the density and distribution ofthe measured point source data. For example,for each of a selection of DEMs constructedby manual photogrammetry, Östman (1987)observed that an increased point densityreduced the RMSE.

2 Processing and interpolationThe degree of processing and interpolationrequired to derive a regular gridded DEMfrom a set of measurements will depend onthe method of measurement and the natureof the data model. Thus if resources allowed,data collection by field survey, for example,could be tailored to the specifications of agridded DEM by recording height measure-ments at all grid intersections. Data collectionin photogrammetry and active systems canalso generate a gridded model by direct meas-urement. If the distribution of the source datais irregular or not at the desired spacing, then




some degree of processing/interpolation ofvalues at grid intersections is required and thiscan itself be a source of error.

There are a great variety of interpolationmethods available for terrain surface interpo-lation. Watson (1992) has identified twoclasses of interpolation methods for terrainsurfaces: fitted functions and weighted averag-ing. In contrast, Hutchinson and Gallant(1999) have identified three classes based on:triangulation, local surface patches and locallyadaptive gridding. For further details aboutthese methods of interpolation, and alterna-tive more generic schemes of classification,the reader is referred to Lam (1983),Burrough (1986), Watson (1992), Hutchinsonand Gallant (1999) and Mitas and Mitasova(1999). Less generic interpolation methodscan be identified for specific types of sourcedata. For example, when interpolating con-tour data to a DEM, Robinson (1994) has dis-tinguished between those generic methods ofinterpolation mentioned above, and morepurpose-designed methods which make someuse of the extra information supplied by thecontours; for example, the construction oflines of steepest descent between contours(Leberl and Olson, 1982). The crucial point isthat since different methods of interpolationproduce different estimates for height valuesat the same point, these methods will alsoproduce different quantities of error in theDEM.

A variety of empirical work has looked atthe effects of different methods of interpola-tion on DEM error (MacEachren andDavidson, 1987; Desmet, 1997; Yang andHodler, 2000; Rees, 2000; Wise, 2000)usually by means of observing the results ofdifferent interpolators on sample DEMs.Estimates of error can be obtained by com-paring interpolated values with a higher accu-racy reference surface (section II.2; Fisher,1998; Holmes et al., 2000; Davis et al., 2001),or with a subset of original points withheldfrom the interpolation process (MacEachrenand Davidson, 1987; Desmet, 1997; Yang andHodler, 2000). Accuracy description can be

summarized statistically, or more qualitativelyusing visual descriptions (Declercq, 1996;Carrara et al., 1997; Desmet, 1997; Yang andHodler, 2000). We consider the role of visual-ization in error identification more explicitly insection VI below.

In general, there seems to be no singleinterpolation method that is the most accuratefor the interpolation of terrain data.Geostatistical kriging is attractive from a sta-tistical standpoint, since it is the best linearunbiased estimator and the error introducedby the processes of estimation can be directlydetermined (Oliver and Webster, 1990).However, kriging variance is directly propor-tional to the distance of an interpolated valuefrom an input observation. Furthermore, thesuccess of a given interpolation method appar-ently depends on the nature of the terrainsurface (smooth or rough) and the distributionof the measured source data (irregular or reg-ular). This may result in no clear interpolationmethod being preferred (Wise, 2000). Wheninterpolating an existing DEM to a higher res-olution DEM, Rees (2000: 17) suggests thatthe RMS accuracy (RMSE) of the interpolatedDEM, r, is directly proportional to the stan-dard deviation of the height differencebetween adjacent points in the DEM, �. Theconstant of proportionality is dimensionlessand in the cases studied varied between 0.21and 0.6 depending on the interpolator used,the factor by which resolution is increased,and the fractal dimension. For a variety ofDEMs Rees observed that ‘the RMS accuracyof the interpolated DEM has very little sensi-tivity to the choice of interpolation methodbetween the bilinear, bicubic and krigingapproaches’ (Rees, 2000: 18). For a DEM rep-resenting smooth undulating agricultural ter-rain Desmet (1997) found that although splineinterpolation appeared preferable ‘the extrap-olation of this conclusion must be done withcare as the study area was extremely smooth’(Desmet, 1997: 579). It is clearly difficult tomake any general conclusions.

The data collected by active systems maynot only require interpolation, but also




considerable processing to obtain the result-ant DEM. For example, Gens (1999) observesthat the error introduced into InSAR-derivedDEMs depends very much on the details ofthe interferometric processing applied. Thisincludes the processes of registration of theradar image, formation of the interferogram,‘phase unwrapping’ and reconstruction of theDEM (de Fazio and Vinelli, 1993, as cited inGens, 1999). Processing may also influencethe final form of the data and hence thedegree of error in the DEM. For example, areduction in precision by rounding off eleva-tions to the nearest metre can introduce

sufficient error into the DEM to generate sig-nificant error in terrain derivatives (Figure 3;Carter, 1992; Nelson and Jones, 1995).

3 Terrain representationRecall from section III.1 that terrain surfacecharacteristics can directly influence the qual-ity of elevation measurements, particularlywith the more active systems of data capture.However, terrain surface characteristics alsointeract with the resolution of the model indi-rectly reflecting the fact that DEMs consti-tute a discrete sample of a continuousvariable. In section II.2 we noted that the

Figure 3 The SK40 20�20 km tile of the Ordnance Survey 50 m resolution DEM ofBritain. (A) Standard grey scale view; (B) histogram with the diagnostic cyclic peaksindicating over-representation of the contour linesand; (C) slope map showing steepslopes at both the contour positions (very steep slopes – white lines) and integer rounding in low relief areas (moderate slopes – grey lines) Source: © Crown Copyright/Database right 2005. An Ordnance Survey/EDINA supplied service.




issues of resolution do not map comfortablyinto error and error alone, but are more anissue of the wider concept of uncertainty.Many authors have viewed resolution as aquestion of error, however, and for complete-ness issues relating to resolution are consid-ered here.

The resolution (or sampling interval) ofthe DEM is a function of both the source dataand any interpolation process carried out onthat source data to derive the gridded DEM.In conceptual terms, we might think of thechoice of resolution of the DEM to be akin tothe discrete sampling of a continuous func-tion: and information will clearly be lost forthose distances that are smaller than the sam-pling interval, but information will also bealtered at distances up to twice the samplinginterval, known as the Nyquist CriticalFrequency (Press et al., 1989: 386). Both ofthese can cause error in the DEM. On thisbasis, it might be expected that the larger theresolution of the DEM, the less well themodel will approximate the underlyingcontinuous real terrain and the greater thepotential for error (Gao, 1998). However, res-olution is intimately related to the character-istics of the terrain surface, since at a givenresolution error can also be increased in theDEM by increasing the complexity of the ter-rain surface. Clearly, the accuracy to whichany given terrain is approximated by the DEMwill depend on the match (or mismatch)between the resolution and the spatial char-acteristics of the terrain: some landforms willbe approximated well, others less so(Theobald, 1989).

A variety of empirical work has confirmedthe relationship between resolution and errorin real DEMs. For example, in an analysis ofphotogrammetrically derived DEMs createdfrom the ISPRS DEM evaluation exercise(Torlegård et al., 1986), Li (1994) observed apositive relationship between resolution anderror standard deviation (S). In an analysis ofhigh-resolution DEMs derived from digitizedcontours, Gao (1987) observed that errors interms of RMSE increased linearly with spatial

resolution, and that the accuracy of represen-tation was inversely correlated with terraincomplexity. Similar trends have been observedby Fisher (1998) and Gong et al. (2000).

An attractive strategy for assessing qualityof a DEM is to compare the data with a for-mal model of that data, which, followingFrench use, is termed a terrain nominal(Duckham and Drummond, 2000). For exam-ple, Duckham and Drummond (2000)employed a fractal model of physiography as aformal model for the analysis of river networkcharacteristics. A similar fractal model wasalso used by Polidori et al. (1991) to detectsmoothing due to interpolation in DEMs. Inboth cases statements about quality are con-ditional on the selection of an appropriate for-mal model, but the appropriateness of such amodel may itself be uncertain (Goodchild andTate, 1992).

IV Error modelsModelling the error in continuous variablescan take two possible routes: derivation ofthe error analytically, and simulation of theerror stochastically (Shortridge, 2001; Zhangand Goodchild, 2002). Stochastic simulationis further subdivided into: unconditioned andconditioned.

1 Analytical error modelsHunter and Goodchild (1995) utilized a simplemodel of error based on the RMSE of theDEM. For any given pixel, error was assumedto follow a Gaussian distribution around themeasured elevation value, and the globalRMSE for the DEM was taken to be an esti-mate of the local error variance around thisvalue. In this manner, it is relatively simple tomap per-pixel probabilities across a DEM inrelation to a particular given elevation value, acontour line. This model is embedded in thePclass operation of the Idrisi GIS and has beenused by Eastman et al. (1993) in assessing therisk of sea-level rise leading to flooding. Thisapproach is also exploited in the work of Huss and Pumar (1997) to determine theprobability of the visible area.




2 Unconditioned error simulation modelsUnconditioned error simulation models relyon stochastic simulation of realizations of ran-dom functions (RF). They are informed byproperties of the error distribution but theydo not honour any actual estimates of error.At their most basic, they comprise an algo-rithm to select independent and uncorrelatedvalues drawn from a normal distributionwhich can be added to the original DEMoften using a Monte Carlo method of simulat-ing a number of independent realizations (eg,Fisher, 1991; 1992). Such models may beoptionally constrained to a description of thepattern of spatial dependence in the error.This can be achieved in a variety of ways, thatinclude using a random cell-swapping algo-rithm which iterates towards a desired valueof Moran’s I (Fisher, 1991), Geary’s C(Veregin, 1997) or � (rho) (Hunter andGoodchild, 1997); a spatially autoregressiveprocess with a target correlation (Zhang andGoodchild, 2002: 107) or by using a variogramto characterize estimates of error (Englund,1993; Fisher, 1998).

The problem with unconditioned simula-tion is that it still makes the assumption thatthe pattern of error is uniform over the studyarea or a wider region. This is not necessarilythe case, as is demonstrable from studies ofthe distribution of actual errors (Monckton,1994; Fisher, 1998). If error is spatially auto-correlated, then it is generally larger in onearea and smaller in another, it is not the sameeverywhere (compare Figure 2, C and D). Toaddress this regionalization of the error, themodel needs to be conditioned.

3 Conditioned error modelsConditioned error models directly honourobservations of error at the sample locations.Such observations, for example, might havebeen obtained by comparison between theDEM and a higher accuracy reference dataset collected from within the area of theDEM. Geostatistical methods of conditionalsimulation are popular (Fisher, 1998;Kyriakidis et al., 1999; Holmes et al., 2000).

A practical approach to geostatistical condi-tional simulation (Delhomme, 1979; Zhangand Goodchild, 2002) is to create an uncondi-tional realization of a RF with a covariancewhich matches the observed sample data.Then, the values of the unconditional realiza-tion for the sample locations are kriged. Sinceboth unconditionally simulated and krigedestimates are available for the RF realization,this allows direct estimation of the krigingerror. Since kriging is an exact interpolator,this error will be zero for locations correspon-ding to the sample locations, but non-zeroelsewhere. This estimate of the kriging errorcan then be added to the kriged surface of theobserved sample data to produce a simpleconditional simulation (Delhomme, 1979:272).

Individual unconditional realizations can beaccumulated in a Monte Carlo methodologyto estimate error statistics as considered insection V below.

4 Fuzzy elevation modelsRecently, a number of researchers have chal-lenged the assumption that the concept of the‘land surface’ is well defined. They argue thatthe definition of what is measured in a digitalelevation model is vague, and so it is suitableto a treatment by vague or fuzzy mathemat-ics (Santos et al., 2002; Lodwick and Santos,2003). Fuzzy set theory and fuzzy logic havebeen widely researched in GeographicalInformation Science (Fisher, 2000), but fuzzymathematics has rarely been employed.Instead of the values stored in a DEM beingregarded as an estimate of the actual eleva-tion of the land surface at a point, a fuzzyDEM assumes that any elevation stored isone of a number of possible elevations.Furthermore, the possibility distribution ofthe elevations reflects, not the error in theDEM, but the uncertainty in the conceptual-ization of the surface given that the valueintended to be measured is vague. With thevertical precision of LiDAR remote sensingfor DEM creation, this is a very real problem:is the sensor measuring the top of the crop in




the field or the top of the ground and, if it isbare ground, is it the crest of the plough fur-row, or the trough? Indeed, comparable ques-tions, at a different precision, can be asked ofearlier DEMs and even of contour maps: is itthe top of the vegetation (trees) that is meas-ured or the ground surface? The intention isclear, but the actuality is in doubt.

V Error propagationFor the study of error distributions in data tohave any meaning, it is important to studytheir propagation into subsequent products orpredictions. Digital elevation models are verywidely used in making planning decisions andin environmental models, and it is the propa-gation of errors to these types of models thatare of greatest interest. Unfortunately, how-ever, such propagation is also complicated,and studies are rare.

Heuvelink et al. (1989) and Heuvelink(1998) have shown that the Taylor series ofequations can be used to evaluate the propa-gation of errors into the derivatives whenmeasurements at a location are compared.This method can, for example, be employed iftwo DEMs of an area are being used to mon-itor the progressive accumulation of materialin landfill sites where the volume of the fill isan important derivative and margins of errorin estimation are an important potential cost.Even simpler propagation formulae areembedded in the Idrisi GIS for simple overlayactions (Eastman et al., 1993), based on stan-dard error propagation formulae (Taylor,1982). As soon as information that is local tothe area of concern is used, including slope oraspect calculation, the local dependence oferror must become part of the equation andthe formulae become complex. Therefore,Heuvelink (1998), and many others, recom-mend working with Monte Carlo simulation.

1 Slope and aspect derivativesSlope and aspect are important componentsfor the determination of hydrological flowpaths (Veregin, 1997) and indices employed byrainfall-runoff models such as TOPMODEL

(Wolock and Price, 1994; Brasington andRichards, 1998).

The evaluation of the accuracy of eleva-tion derivatives has usually been obtained bydirect comparison with measurements fromhigher accuracy reference surfaces (Changand Tsai, 1991) or the real land surface(Bolstad and Stowe, 1994). Florinsky (1998)has argued, however, that such comparisonsare invalid since they imply the existence ofhigher accuracy reference surfaces that arelocally smooth and differentiable. Real terrainpossesses the fractal characteristic of non-differentiability, and therefore zooming in tolarger scales will reveal different surfaces withdifferent gradients/aspects and different morphometry (Skidmore, 1989; 1990).

We can identify three components to theerror budget of the derivatives of elevation.First, error, which occurs in measured orinterpolated elevation values, results in errorin the derivatives. Second, uncertainty can beintroduced owing to the resolution of theDEM. Various empirical studies have beenundertaken to examine the effects of DEMresolution on the accuracy of slope, gradientand aspect, with some variability of observa-tions. For example, Chang and Tsai (1991)observed that error in all three is positivelyrelated to DEM resolution, although Gao(1998) found that gradient is the most sensi-tive to resolution change. These first twocomponents collectively constitute whatFlorinsky (1998: 49) has termed ‘the accuracyof the initial data, that is the DEM’. The thirdcomponent concerns the precision of the cal-culation method (Florinsky, 1998) where erroris introduced by the specific method of deriv-ative calculation. Making use of Evans’s(1980) polynomial representation, Florinsky(1998) derived RMSE error expressions forgradient, aspect, horizontal and vertical pro-file curvatures obtained from a DEM. Heshowed that error in the derivatives is directlyproportional to elevation RMSE and inverselyproportional to DEM resolution. That finerresolutions introduce larger errors is a reversalof the more intuitive positive relationship




between DEM resolution and error discussedabove, whereby coarser resolution DEMsintroduce greater error as a result of poorerapproximation of the real terrain surface.

The consequences of DEM error on slopeand aspect have also been examined byHunter and Goodchild (1997). They simulatederror in a variety of simulations containing dif-ferent degrees of positive autocorrelation andadded them to the DEM. They explored therelation to slope and aspect derived from theDEM. This result was previously suggested byGoodchild (1996) on the basis that, if the land-scape is smooth (showing positive spatialautocorrelation) and the DEM is smooth, thenthe error must also vary smoothly from placeto place. If the error has high positive spatialautocorrelation, then slopes over short distances (normally cell-to-cell in a GIS calcu-lation) will have less error than if the positiveautocorrelation over the same distance is closeto zero (ie, the error is white noise). ManyDEMs are not smooth, however, and showvarious systematic errors of their creation as spatial discontinuities, as discussed in section III.1.

2 Visibility and other productsIn a series of papers, Fisher (1991; 1992; 1993;1996a; 1996b; 1998) has explored the propa-gation of DEM error into visibility determina-tion from a DEM (the viewshed) using MonteCarlo simulation. He has shown that as spa-tial autocorrelation in the error is increased,the area determined as being visible increases(on the same basis as Goodchild’s argumentfor slope determination). Furthermore, thedistribution of the probability of being visibleis more polarized, with more locations havinghigher probabilities. Fisher (1998) alsoreported that if the error model is conditionedon the distribution of empirically determinederrors in the DEM, then the probabilities maybe lower than in an unconditioned errormodel with the same degree of cell-to-cellautocorrelation. Fisher (1998) argued that theanalyst’s confidence in propagating the condi-tioned error into the visibility problem should

be greater than using unconditioned errorbecause the model is using as much informa-tion as is available on the distribution of error.On the other hand, Ehlschlaeger et al. (1997)explored the sensitivity of predicted paths toDEM uncertainty related to the change inresolution between 0.5 minute arc and 30 m.They used the latter scale data as high-qualityinformation to parameterize and model theuncertainty in the former. The same scaletransformation was studied by Kyriakidis et al.(1999) and Holmes et al. (2000), usingsequential simulation. Holmes et al. (2000)showed the sensitivity of various simple andcomplex derivatives from the DEM, includinghillslope failure. They suggested that workingwith the original DEM might seriously under-estimate the area at risk of such failure.

Several studies have examined the sensi-tivity of landslide risk estimation to DEMerror. Murillo and Hunter (1997) used MonteCarlo simulation of DEM error in the USPacific Northwest to propagate the error intoa simple model of landslide susceptibilityinvolving only the DEM and a geological dataset which was treated as correct. Davis andKeller (1997a; 1997b) went one step furtherwhen they used sequential simulation basedon the variogram for Monte Carlo simulationof DEM error. The realizations of the errormodel were combined in a model with fuzzymemberships of soil types to model theboundary uncertainty in the soil database.

A further development is that of sensitivityand uncertainty analysis advocated byCrosetto and Tarantola (2001). They pre-sented a framework where the uncertainty ininputs and sensitivity in modelling can both beexamined and the influence of each assessed.Unfortunately, although their example appli-cation includes a DEM as input, they do notparameterize the errors in the DEM andassess its contribution or importance in theoverall uncertainty of the model. The GLUEmodel (generalized likelihood uncertaintyestimation), proposed and investigated exten-sively by Beven (2002), is primarily intendedfor analysis of parameter sensitivity, and has




been used in a number of applications wheredigital records of elevation are important,such as hydrological modelling. It has notbeen used to examine the model sensitivity toparameters of DEM error models.

Anile et al. (2003) have considered theconsequences for visibility calculation whenthe DEM is treated as fuzzy. They basicallyuse the fuzziness to predict whether a loca-tion is in view, could be in view or is not, giv-ing a three-valued outcome instead of theusual binary solution (in-view or out-of-view).

The propagation of DEM error into a num-ber of environmental and planning models hasbeen explored. Future research must focus onjudging when the DEM error is critical to anapplication and how much uncertainty (fromwhatever source) is possible in the other databefore the DEM error is relatively unimpor-tant in determining the possible variety ofoutcomes.

VI Visualization of errorOne of the most diagnostic methods forinvestigating errors is visualization. The mostcommon view of a DEM as a contour map oras a colour (or grey scale) image (Figure 3A),is only good for detecting the most extremeerrors, however; values that differ dramati-cally from the elevations in the vicinity (seesection II.3 regarding gross errors/blunders)can sometimes be detected by this method,but even then, it is not the most diagnosticapproach.

Hunter and Goodchild (1995) have sug-gested visualizing uncertainty around a par-ticular contour line (see also Kraus, 1994).They take the RMSE to be as a standard devi-ation from a normal distribution, and calcu-late the probabilities of any elevation beingless than or greater than a threshold elevation(the contour value) that can be estimated andvisualized (Hunter and Goodchild, 1995).

The most diagnostic visualization methodsrely on either summary graphs or mappingDEM derivatives like slope and shaded relief.These are effective for recognizing system-

atic errors, as opposed to identifying randomerror. For example, ghost contours are a sys-tematic error in many DEMs interpolatedfrom contour data. They are caused by over-representation of elevations equal to the digi-tized contours (Wood, 1994; Guth, 1999),and can be detected very simply by the cyclicarrangement of peaks in a histogram of theDEM (Figure 3B). They are also detectable asalignments of steep slopes in the DEM due tothe relatively sudden changes from one con-tour value to another, and are detectable inslope maps derived from the DEM (Figure3C). A similar terracing (alignments ofsteeper slopes) can be discernable in areas ofvery gentle relief, due to storage of the DEMas integers. This terracing can also be visiblein slope and shaded relief maps derived fromthe DEM (Wood and Fisher, 1993). Othersystematic errors that can be detected inshaded relief maps are the triangular facetsthat result from TIN-based interpolation, andpiecewise reformatting in georeferencing(Hunter and Goodchild, 1995).

Fisher (1997) and Ehlschlaeger et al. (1997)used animation to visualize uncertainty. Inboth studies, time for the viewer is used as ametaphor for uncertainty, so the longer anitem is unchanged the more certain it is.Ehlschlaeger et al. (1997) used serial animationto show the uncertainty in land inundatedwhen sea levels rise. The land areas exposedfor the longest period are most likely to be dryland for a particular amount of sea-level rise inBoston Harbour (see also Eastman et al.,1993). Fisher (1997), on the other hand, useda continuously varying random selection ofgrid cells in the elevation model, and changedthe elevation in them according to a stochasticmodel of the occurrence of error. The displaywas therefore continuously changing in aprocess he calls random animation. These ani-mations have advantages and disadvantages,but to experienced users both methods arequite intelligible.

Hunter and Goodchild (1996) have dis-cussed the need for a general model for visu-alization and management of uncertainty.




They have advocated the generation of multi-ple stochastic realizations of equally probablemappings of all data (including any DEMs)and, when asked to display a particular themefor an area, made random selection of one todisplay. Crosetto and Tarantola (2001)employed exactly this approach to uncer-tainty and sensitivity modelling (especially ofland cover). However, although they used aDEM in their model, they do not consider theuncertainty in it.

VII Error correction and fitness for useThe correction of the errors in a DEM is a logi-cal progression from their identification, detec-tion, measurement and propagation (Veregin,1989; Li and Chen, 1999; Lòpez, 2002) and ispart of the process of error reduction.

1 Error reductionCertain types of DEM error can be detectedand corrected relatively easily. Systematicerrors, as depicted in Figure 2D, can often beidentified visually and corrected using somevariant of appropriate low-pass filtering andvalue adjustment (Albani and Klinkenberg,2003). Remarkably few methods exist foreither the detection of other errors from can-didate elevation values or their correction.Lòpez (1997; 2000) and Felicìsimo (1994)have presented methodologies for the identi-fication of blunders using a variety of statisti-cal criteria to distinguish locally extremevalues. Felicìsimo (1994) suggested that localdeviations from multiples of standard devia-tions could be used, while Lòpez (1997) pro-posed a method based on PCA transformedsubsets of DEM values. Lòpez (2002) subse-quently employed both methods in an analy-sis of the DEMs generated as part of theISPRS DEM evaluation exercise (Torlegårdet al., 1986). Although Lòpez (2002) sug-gested that an expert should assess and cor-rect each blunder identified, he also proposedthat corrected values could simply beobtained by linear interpolation from the localvalues in the neighbourhood of extreme

points. Using this procedure, he noted thatreductions in RMSE of up to 8% were possiblealthough the outliers identified comprised lessthan 1% of values in each DEM tested.

Other methods for error identification andcorrection rely on the introduction of contex-tual information to determine whether or notan elevation point contains error. For exam-ple, the presence of isolated depressions/sinksin a DEM which make little sense hydrologi-cally (Jenson and Domingue, 1988) and flatregions characteristic of rounding errors(Nelson and Jones, 1995) have respectivelyled to the development of methods for theremoval of spurious pits and the smoothing ofDEMs. These are relatively simple and com-mon error correction procedures, and theiruse is often motivated by specific hydrologicaluses of the DEM, such as drainage networkderivation (Wise, 2000). Wood (2002), how-ever, has argued that pits can occur as a logi-cal consequence of the process of DEMcreation, and are common at the confluenceof two channels. However, if the model isrequired for hydrological modelling, the flowthrough the confluence must be preserved,and so pit removal is essential.

2 Fitness for useThe determination of whether or not a DEMis of sufficient quality for a certain applicationis a more difficult question. While consider-able progress has been made in describing andmodelling error, comparatively little progresshas been made in determining the minimumdata quality requirements for specific applica-tions, or the development of methods toassess what Chrisman (1983) termed ‘fitnessfor use’ (Frank, 1998; Veregin, 1999; de Bruinet al., 2001). This may be partly explained by alack of choice in data supply. Historically, few,if any, alternative sources of DEM data wereavailable for specific applications, and whereno alternative data exist the process of assess-ing fitness for use can be considered unneces-sary (Agumya and Hunter, 2002). However,with increasing choice in data supply, this situ-ation is becoming the exception rather than




the rule, at least in developed nations. Indeed,Veregin (1999) has identified increased datasupply by the private sector as one of the mainreasons for an increased interest in data qual-ity issues. In order to discriminate betweenthe increasing number and variety of DEMdata products of often contrasting quality thatare now available for individual locations,appropriate strategies must be developed.Since error will influence the quality of infor-mation obtained from any DEM product, theuser ideally needs to be in a position to answerthe questions identified by Agumya andHunter (1999b: 42): ‘Of the available informa-tion [ie, several candidate DEMs] that can beafforded, which is the most suitable?’ and ‘Isthis information [a specific candidate DEM] fitfor my application?’. Data quality is funda-mental to both these questions along withother issues of suitability and fitness such asperhaps resolution and cost. The second ques-tion requires an assessment of the quality ofthe DEM, along with propagation of the qual-ity to derivatives required by an application.Data can effectively be considered to be fit foruse when the quality of a data set is betterthan the ‘worst acceptable quality’ required bythe application (Frank, 1998: 7). While thisnotion is conceptually simple, in practice, the‘worst acceptable quality’ for a specific DEMapplication is at best difficult to determine in arobust and verifiable manner and at worstunknown or unknowable. In such a situation, asimple, although cost-ineffective and far fromoptimal solution, would be to avoid any assess-ment of fitness for use at all, and obtain thehighest-quality data available. Such a situationis clearly untenable, because such data maynot be fit for use for all conceivable applica-tions, for reasons such as resolution and cost.

As noted by Agumya and Hunter (1999a),the usual approach to assess fitness for use isstandards-based; for example, the userasserts a threshold of acceptable RMSE for aDEM. In addition to the shortcomings ofRMSE noted in section IV above, and moregeneric problems of standards, such as theirstatic nature and implementation difficulties

(Veregin, 1999), the crucial observation,made by Devillers et al. (2002) among others,is that fitness for use can only be assessed rel-ative to an intended use. Therefore, absolutestandards-based statements such as theRMSE of a DEM are on their own of littlepractical use to the data user, who will oftennot know a limiting value of RMSE for theirintended application. Agumya and Hunter(1999b: 35) have observed that the use ofsuch standards-based statements for theassessment of fitness for use is severely lim-ited by the lack of any ‘quantitative estimationof the consequences’ of error on the decisionsmade using the data. For example, given thepropagation of a specific RMSE of a DEM,what are the consequences of the resultingerror in the derivatives on decisions madeusing the data as part of an application? In anattempt to develop methods to help answerthis question, and to provide an alternative toa standards-based statement, Agumya andHunter have developed a risk-based strategyfor determining the fitness for use of digitalgeographic data including DEMs (Agumyaand Hunter, 1997; 1999a; 1999b; 2002). Thekey component of this strategy is an appraisalof the consequences of being exposed to risksof error in the data (by using the data to makedecisions), set against the degree of risk that isconsidered to be tolerable. The overall riskstrategy encompasses risk identification, risk analysis, risk exposure, risk appraisal, riskassessment and risk response (Agumya andHunter, 1999b: 40), who present an exampleconcerning the selection of a DEM for floodextent estimation. This strategy offers anobjective procedure to address the conse-quences of error in decision making, as well asa framework which requires the assessmentof error as a matter of course. However, the potential user of a DEM is faced withadditional problems, including making thecomparison between an overall estimate of potential risk exposure and the acceptable/tolerable risk which is com-pounded by the variety of units (eg, lives,injuries, money) in which risk exposure may




be expressed (Agumya and Hunter, 1999b),and determining just what degree of risk istolerable in any situation.

De Bruin et al. (2001) have approached thefitness for use problem by the estimation ofwhat they term the ‘expected value of con-trol’ (EVC) within an explicit framework ofprobabilistic decision analysis. This enablesthe choice of one DEM for a given locationfrom a selection of candidate DEMs. Thiswas achieved by the estimation and compari-son of the expected ‘loss’ incurred, if eachDEM is used for a specific decision-makingtask where loss can be expressed in a numberof ways including economic loss. The esti-mate of the error in each DEM forms a keyelement of the loss. In practice, this error isestimated, and then propagated probabilisti-cally into a loss function from which losses areobtained. In this manner, de Bruin et al. (2001)obtained and compared the expected lossesfor two candidate DEMs of differing originand resolution that were available for a spe-cific construction task in The Netherlands,where loss could be expressed directly as themonetary costs incurred to correct any errorin the volume estimated from each DEM.The expected value of control translates asthe ability of the user to minimize these

losses/costs by the selection of one DEMrather than another. In the construction DEMexample, there was little difference betweenthe final estimates of loss/cost, and bothDEMs were deemed equally suitable for thetask at hand. As noted by de Bruin et al.(2001: 459), the outcomes of the decisionframework need to be assigned values, andalthough losses/costs in monetary termswere calculated for the example used, it maybe impossible to calculate such objectivequantities in other contexts.

VIII ConclusionFrom this paper, it can be seen that the focusof the vast majority of the research on error(and uncertainty) in DEMs is concerned withits identification, description, visualizationand modelling. Such work is often only con-cerned with subsets of steps in the overallprogression from conceptual model to digitalrecord, as compared with the full process(Figure 1). Most prevalent are comparisonsbetween the conceptual model of the landsurface and the contour model, and the con-ceptual model and the final DEM (Figure 4).Complete inventories of errors accumulatingfrom the beginning to the end of this processare missing from the research reviewed.

Figure 4 Typical studies of DEM error and uncertainty relate to only part of theconceptualization of the process of production of DEMs; no research seems to havestudied the whole process




A large number of studies have looked atthe process of interpolating a DEM from asparse scattering of points or from contourlines (with intense sampling of points alongthe line but none elsewhere). A number ofingenious approaches to this problem havebeen advanced, and knowledge of the sys-tematic errors introduced by different inter-polation methods is available. This knowledgerelates to a very specific step in the creationof a DEM, however, ignoring the errors thatmay occur in previous steps in the process(Figure 1). Furthermore, such studies fre-quently consider interpolation methods avail-able to academics, and actually tell usrelatively little about the processes of com-mercial DEM production even if the methodsare clearly stated by the producers. It mightbe argued that this whole corpus of researchis of increasingly little relevance, however,because the method of DEM creation isincreasingly by active methods described insection III.1, and measurements may even bemade at a greater density than the grid of thederived DEM.

Error description, as expressed in stan-dards of spatial data quality, has always beenbased on the RMSE, in spite of the increasingresearch literature on error modelling whichhas highlighted the statistical shortcomings ofthis measure, and demonstrated that muchmore can be achieved with a description ofthe spatial distribution of error, either as a var-iogram or by actually including more accuratevalues of elevation along with the DEM.Researchers have used these measures oferror as a basis of modelling and propagatingthe error into a number of standard deriva-tives of DEMs.

An interesting and novel approach to sur-face creation has been the treatment of theelevation values as fuzzy numbers. Thisapproach, however, has been introduced toorecently to be evaluated in any detail in thisreview.

The assessment of DEM fitness for use isof increasing importance given the widerchoice in DEM data supply that now exists for

many geographical locations. However, rela-tively little work has been done in this area.This must change, for it is arguably only whenthe link is made between DEM quality andapplication-quality requirements that the realrelevance of DEM error is apparent.

In general, the studies reviewed abovehave only considered elevation in isolationfrom other types of data. However, the errorin the DEM is just one type of uncertaintythat may enter a particular model. Clearly,uncertainties can accrue from any error anduncertainty in other data in the model, specif-ically, uncertainty in the conceptualization ofthe model itself, and uncertainty in the algo-rithm used to implement the model. Amongthose authors who have examined uncer-tainty in spatial data other than in DEM appli-cations, very few have examined theconsequences of DEM uncertainty in com-parison with the uncertainty of the other datalayers in the analysis.

Whilst there is an increasing tendency tocollect larger volumes of elevation data withseemingly ever-improved precision and accu-racy, we have no evidence that this improve-ment and the associated costs areworthwhile. Very little work has been done todetermine the minimum data requirementsfor specific applications of DEMs. The centralquestion in a modelling process suffused withuncertainty is: are the errors which may bepresent in one type of data input to the modelsignificant in terms of the sensitivity of themodel? In certain situations they may be crit-ical, but in others they may not. If the DEM iscombined with other data in contexts likehydrological and diffuse pollution modelling,for example, the effect of the error may bediluted, and be unimportant compared toerrors in other data and uncertainty in themodel itself. So far, not only has this questionbeen unanswered, it has been unaddressed.

AcknowledgementsThis paper is the outcome of many years workand therefore owes much to many people, toonumerous to name individually. We would like




to thank all those we have discussed DEMerror with over the years. Nicholas Tate wouldspecifically like to thank the University ofLeicester for financial support during a periodof study leave/sabbatical while working onthis paper, as well as logistical support fromMichael Goodchild /NCGIA(UCSB) andKaren Kemp (University of Redlands) duringthis period. The text also benefited from someuseful comments by Claire Jarvis.

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Causes and consequences of error in digital elevation models*

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