Accuracy Assessments of Geographical Line Data Sets, the Case of the Digital Chart of the World *

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1 Accuracy Assessments of Geographcal Lne Data Sets, the Case of the Dgtal Chart of the World * Håvard Tvete Department of Surveyng Agrcultural Unversty of Norway, P.O.Box 5034, 1432 Ås, Norway fax: phone: emal: lanht@nlh.no Sndre Langaas UNEP/GRID-Arendal c/o Dept. of Systems Ecology, Stockholm Unversty, S Stockholm, Sweden fax: phone: emal: langaas@grda.no Abstract To be able to utlse geographcal data for analyss, one should know somethng about the qualty of the data. In present geographcal data standardsaton proposals (SDTS, CEN TC287), several aspects of geographcal data qualty have been descrbed, such as lneage (data collecton and processng hstory), spatal accuracy, attrbute accuracy, completeness, logcal consstency and currency. Methods for quanttatve assessments of dfferent aspects of spatal accuracy for data sets of lnear geographcal features, such as shape fdelty and postonal accuracy are descrbed. For these assessments, ndependent data sets of better (and preferrably known) accuracy wll have to be used. In order to be able to do automatc assessments, data set completeness must be taken nto consderaton. The method has been appled for assessng the spatal accuracy for some themes of the Dgtal Chart of the World (DCW) (scale of orgnal maps (ONCs): 1: ), usng the Norwegan mappng authorty s natonal N250 map seres (scale 1:250000) ** as a reference data set. Key words: Accuracy, geographcal, dgtal, data, lne, buffer, overlay, DCW 1 Introducton The avalablty of qualty nformaton s a prerequste for the utlsaton of geographcal data sets. * Ths work has been partally funded by the Norwegan Research Councl under the Geographcal Informaton Technology programme. ** Many thanks to the Norwegan mappng authorty for gvng us access to excerpts of the dgtal N250-data set for these purposes.

2 Tradtonal geographcal maps have conveyed qualty nformaton ndrectly through the qualty constrants and mappng rules that apples to the relevant map seres and mplctly through the (presentaton) scale of the maps. The professonal map users have hopefully been aware of the many aspects of tradtonal map qualty, whle most casual map users probably have used the scale of the map as the only qualty ndcator. Wth the advent of dgtal geographcal nformaton, presentaton scale as such s no longer a useful measure of geographcal data qualty snce dgtal geographcal nformaton n theory can be presented at any scale. The avalablty of dgtal geographcal data and geographcal nformaton systems (GIS) also gves new opportuntes for easy combnaton of geographcal data sets of any scale. The results of analyss on combnatons of data sets depend on the qualty of all the partcpatng data sets. In order to be able to determne the qualty of the results of geographcal data analyss, t s mperatve that qualty measures are avalable for all the nvolved data sets. The ncluson of qualty measures for dgtal geographcal data sets has been mpeded by the lack of standards. There has been some research actvty on spatal data qualty, and some sgnfcant contrbutons nclude: Chrsman 1984, Goodchld and Gopal 1991 (book of artcles), SDTS 1990 (US spatal data transfer standard). The research presented n ths artcle s a part of the ongong project * «Issues of Error, Qualty, and Integrty of Dgtal Geographcal Data: The Case of the Dgtal Chart of the World (DCW)» (Langaas and Tvete 1994). Untl now, we have been nvestgatng methods for qualty assessments, and are now startng to apply the methods on our data sets (DCW and N250). The rest of the paper s structured as follows. In chapter 2, lnear geographcal phenomena are dscussed. In chapter 3, dfferent ways of measurng geographcal lne qualty are presented, and our method for quanttatve assessment of geographcal lne qualty on the bass of data of hgher geometrc accuracy s ntroduced. Chapter 4 rounds t all up wth conclusons and an outlne of future work. 2 Lnear geographcal phenomena The geometrc lne abstracton can be used to represent many geographcal phenomena. Some examples: Roads and ralways Admnstratve (state, muncpalty) and economcal (property) borders Utlty lnes (powerlnes, telephone lnes, water and sewage tubes) Rvers and streams Natural boundares (e.g. vegetaton, sol) Shorelnes * The project presently has a WWW page: URL:

3 Some of these phenomena are nature gven and some are human «constructons» (constraned by nature). There are many ways of provdng qualty measures for lnear features. The choce of a qualty measure depends to some extent on the type of lnear feature we are consderng. 2.1 «Scale» and fractal behavour The «scale» of a lne data set can to a certan extent be determned on the bass of the geometry of the lne alone. Geometrc accuracy s n many cases closely related to «scale». Good ndcatons on scale are: The number of sgnfcant dgts n the representaton of ponts n the data set s the crudest measure of «scale» / spatal accuracy of a data set. Ths s not a useful measure when the orgnal data have been manpulated (e.g. transformed to a new projecton), as most software do not consder accuracy n ther calculatons. Dstance between neghbourng ponts. The ntended scale of the data set can normally be derved from the lowest dstance between neghbourng ponts. Ths s not true f the data set has been manpulated, for nstance by nsertng new ponts on the lnes usng some sort of nterpolaton method. Frequency of curvature change. For curvng phenomena whch change curvature at a hgher frequency than can be captured usng the assumed geometrc accuracy n the data set of nterest, the maxmum rate of curvature change s a good ndcaton of the «scale» of the data set. Such phenomena are phenomena that show fractal behavour (Barnsley 1988) up to larger scales than what can be expected by the data set under consderaton. Most features n nature seem to exhbt fractal behavour over a large spectrum of scales. Examples of such phenomena are: rvers/streams, roads, shorelnes and other natural boundares. The fractal behavour of natural phenomena, and to a certan extent also human-made lnear objects, s often nfluenced by the sol/geology/geomorphology of the area Fractal behavour of nfrastructure When one gets to a large enough scale, nfrastructure wll cease to exhbt fractal behavour. A road wll normally not change curvature more frequently than each 100 meter (1000 meters for a modern motorway, whle perhaps meters for a small older road). The same apples to ralways, powerlnes, telephone lnes and other utltes. When you come to a certan pont, they wll cease to exhbt fractal behavour. The fractal behavour of nfrastructure s, n addton to cultural/hstorcal ssues, also nfluenced by the geomorphology of the area. 3 Methods for assessng the qualty of lnes In the followng sectons, we wll be presentng and dscussng methods for calculatng and quantfyng the geometrc accuracy of lnes.

4 For our assessments, we assume that we have two ndependent data sets, X and Q, coverng the same lne theme and the same area (and collected at about the same pont n tme). One of the data sets, Q, should have a known geometrc accuracy. The geometrc accuracy of Q should be better (preferrably at least an order of magntude) than the expected geometrc accuracy of the data set X. It s also expected that the completeness and consstency of data set Q s sgnfcantly better than that of data set X. Lnes The geometrc accuracy of a lne can be decomposed nto two components: Postonal pont accuracy: Postonal accuracy can easly be gven for well defned ponts on the lne (e.g. the end-ponts). For the rest of the lne, t s dffcult to say anythng about postonal accuracy and to quantfy t. Shape fdelty: To be able to say somethng about the accuracy of a lne, t s useful to talk about ts shape fdelty as compared to another lne. The shape fdelty should ndcate to what extent the curvature of two lnes are smlar. The type of spatal «errors» that can occur for lnear data sets could also be classfed nto categores. E.g.: Scale-dependent errors (generalsaton). These are errors that result from reducng the samplng frequency when collectng data on the lnear phenomena of nterest. Generalsaton/samplng: A lne-representaton that has been generated by samplng a lne of hgh geometrc accuracy represents a specal case. Each pont of the lne s very accurately specfed, but between the represented ponts, there can be large devatons between the nterpolated lne and the orgnal poston of the lnear feature. Ths s closely related to scaledependent errors. Achevable accuracy of fuzzy lnes. The poston of most lnear phenomena get fuzzy as the scale gets larger, and t s generally mpossble to gve them an exact locaton. Rver centrelnes and sol and vegetaton boundares are good examples of fuzzy natural phenomena, but also human constructons can be dffcult to measure wth extremely hgh accuracy (t s dffcult to determne the centrelne of a road wth mllmetre accuracy). «Random» errors. Errors that result from erroneous samplng and data processng. It would be desrable to be able to separate these when descrbng the spatal accuracy of the geometrc representatons of lnear geographcal features. 3.1 Pont measures It s straghtforward to calculate the geometrc accuracy of ponts. For sngle ponts one can measure the devaton vector (e) of the pont representaton (P) as compared to another representaton of the same pont wth better (and known) geometrc accuracy (Q).

5 e = P - Q = (P x -Q x, P y -Q y, P z -Q z ) for 3D space The absolute value of ths devaton vector ( e = ex + ey + ez for 3D space) s a useful measure for further (standard) statstcal calculatons. For multple ponts one has to resort to statstcal measures to determne qualty parameters. Standard devaton or varance can be used whenever the pont-errors of the data sets have no bas and can be consdered normally dstrbuted. The mean error vector (spatal bas) s (P and Q are correspondng ponts n the two data sets): mean(e) = mean(p-q) = 1 ( P -Q) N = 1 N In the case of no pont error bas ( mean(e) = 0), the varance and standard devaton of the pont errors ( e ) are: var( e ) = 1 N e 2 ( 1.. N) SD( e ) = + 1 N e 2 ( 1.. N) (= E RMS ) Both of these measures are acceptable quantfcatons of the spatal accuracy of ponts End-ponts Lne end-ponts can be used to provde a smplfed measure of the geometrc accuracy of the lnes. End-ponts could be cross-roads and dead ends n a road network, rver meets and lakes n a rver/watercourse system or jonts and end-ponts n a tube network. If one s able to dentfy correspondng end-ponts n the reference data set and the data set of unknown spatal accuracy, t wll be straghtforward to compute a statstcal measure of the geometrc accuracy of the end-ponts usng the formulas presented above. Prevous work on quanttatve qualty assessment on the DCW was performed usng 40 evenly dstrbuted cross-roads n the road and ralroad network n the area covered by ONC G18 (the south-west coast of USA.), and usng 1: scale topographcal data (US DLG) as reference data sets (1:24000 data were used for testng vertcal accuracy). Ths work s descrbed n a DMA report (DMA 1990) Intermedate ponts As long as ntermedate ponts are not well-defned features, the only way of fndng correspondng ntermedate ponts s to search for the closest pont on the other lne. A method for determnng spatal accuracy of a lne as compared to a lne of better accuracy could then be to traverse the lne, and at regular ntervals (spacng ε) along the lne take out sample ponts, and on the bass of each of these ponts do a search for the

6 closest pont on the reference lne. At each sample pont, the dstance vector, e, to the closest pont on the reference lne s an ndcaton of the spatal accuracy of the lne at that pont, and an overall measure of lne accuracy can be calculated statstcally usng e as n the formulas presented above. Ths method has to be appled for all lnes that have correspondng lnes n the reference data set, arrvng at an overall measure of the postonal accuracy of the lnes n the data set. The choce of spacng ε could be based on the spatal accuracy of the reference data set. Snce the lnes we are nterested n do not exhbt completely random behavour, ths mples that the smaller ε that s chosen, the more strongly wll the e s of neghbourng pont samples be correlated. To get an overall statstcal measure for the data set, ε should therefore be chosen so large that the e s of neghbourng ponts can be consdered not correlated (Cov(e, e +1 ) 0). ε could be chosen to be of a hgher order of magntude than the accuracy of the reference data set. It could also be nterestng to do several calculatons based on dfferent ε s to gve an assessment of the stablty of the calculated spatal accuracy. To determne separate measures for the lne end-ponts and the nteror of the lnes, a transformaton wll have to be performed on each ndvdual lne pror to the traversal of the lne, n such a way that the end-ponts of the correspondng lnes match exactly. 3.2 Calculatng the geometrc accuracy of a lne usng bufferng The method proposed below uses bufferng of lnes and subsequent overlay analyss to gve a quanttatve assessment of the geometrc accuracy of a lne relatve to another lne (of hgher accuracy). The method should be teratve, because t wll not be possble to determne an optmal buffersze n advance (we do not yet know the spatal accuracy of the lne data set under consderaton). The sze of the frst buffer can be determned on the bass of the known spatal accuracy of the reference data (e.g. the standard devaton, SD, f that s avalable). For each teraton, the sze of the buffer could then be ncreased. 4-5 teraton wll probably be suffcent, and the process should be termnated when the results seem to stablse. Before startng the teratve process t s useful to do some statstcal calculatons on the lnes. The nterestng measure at ths pont n the process s the total length of the lnes The teratve process: For each buffersze bs : bs, { 1,2, 3,..., n} (bs s the wdth of the buffer) perform the followng 3 steps: Frst step - lne bufferng Perform a buffer operaton on each of the two lnes, X and Q, usng the buffer sze bs (resultng n a buffer 2 x bs wde). Call the resultng polygons for Xbs and Qbs.

7 Second step - overlay Perform an overlay of the two polygons Xbs and Qbs, the result beng a new polygon data set: XQbs. Thrd step - statstcs Calculate statstcs (total area, number of polygons, total permeter, permeter/area for each polygon) on XQbs for the followng stuatons: areas outsde Xbs and outsde Q bs (A( Xbs Qbs )) areas outsde Xbs and nsde Qbs (A( Xbs areas nsde Xbs but outsde Qbs (A( Xbs Qbs )) Qbs )) areas nsde Xbs and nsde Qbs (A( Xbs Qbs )) Arrvng at a measure for the geometrc accuracy of lnes The statstcs calculated n the above steps can be used to gve measures of devaton of the lne X from the lne Q. Average dsplacement A Xbs DE = bs ( Qbs) AXbs ( ) DE s the lower bound of the average dsplacement of a lne relatve to another lne (of greater accuracy n our case). Oscllaton O = ( Qbs ) # A Xbs ( ) Length X Where #A(...) s the count of areas. O s an ndcaton of the oscllaton of the lnes X and Q relatve to one another. Ths measure s most useful for «randomly» oscllatng phenomena, where t could be used as an ndcaton of bas (there would probably be a bas f the oscllaton, O, s low for randomly oscllatng lnes of dfferent accuracy). Oscllaton could also be found drectly usng X and Q, by countng the number of nodes ntroduced when overlayng the two lne data sets. O s also a measure of relatve scale for «randomly» (that s random appearance at the relevant scales) oscllatng lnear phenomena.

8 3.3 Calculatng the geometrc accuracy of lne data sets The bufferng method for calculatng the geometrc accuracy of lnes can also be appled to lne data sets. To apply the method on the data set level, all lnes must exst n both data sets (the completeness crteron). If there are lnes that only are present n one of the data sets, these wll ntroduce errors n the calculatons. In conjuncton wth spatal accuracy assessments on real lnear data sets, t s therefore mportant that an assessment of the relatve completeness of the data sets s made and used as a correcton n the method Calculatng completeness for lne data sets usng bufferng Usng an approxmate measure of geometrc accuracy of a data set (X), t s possble to make an assessment of the completeness / number of mscodngs of the X data set, as compared to the Q data set. An approxmate measure of the geometrc accuracy can be obtaned by applyng the method presented above on the complete data sets (gnorng the lack of completeness measures). The method outlned below use a combnaton of bufferng, overlay and selecton (and thnnng). Frst step - buffer Perform bufferng on both lne data sets, X and Q, usng a buffer dstance, BD, whch could be about twce as large as the geometrc accuracy measure found for data set X (for the lne-polygon alternatve presented below, a buffersze that s four tmes as large as the geometrc accuracy measure found for data set X should be used to obtan the same statstcal effect). It s necessary to choose the buffer dstance larger than the statstcal measure of the spatal accuracy (could be SD), snce SD s a sort of weghted mean. When choosng a buffer dstance twce as large as the SD for both lne data sets, we capture all errors wthn 4SD s of the reference lne. The result of ths bufferng s the data sets XB and QB. Second step - overlay Do two lne-polygon overlays: Overlay X wth QB and XB wth Q, resultng n the new mxed data sets XQB and XBQ. Thrd step - statstcs Usng XBQ, calculate the sum of the length of the lnes outsde XB and compare t to the total length of lnes n Q: length( XBQ) Completeness(X) = % length( Q) A more «exact» measure can be obtaned by usng the dentty of the lnes that are not n X, and calculate the length of the complete lnes, as opposed to the part of the lnes that do not fall wthn the buffer.

9 Usng XQB, calculate the sum of the length of the lnes outsde QB and compare t to the total length of lnes n X. Ths s a measure of the amount of mscodngs n X as compared to Q. Ths can also be done n a more «exact» way usng n the same method as descrbed above Ensurng completeness To prepare for the spatal accuracy assessment to come, all mscoded lnes n X and all lnes n Q that are not n X should be removed from the lne data sets. The lnes to be removed can be found n XBQ and XQB, descrbed above. The resultng data sets should be used n the rest of the process Assessment of the spatal accuracy of lne data sets The process for calculatng geometrc accuracy of lne data sets s exactly the same as for ndvdual lnes. It s useful to start out wth calculatng the total length of the lnes n both coverages. The (teratve) process s exactly as descrbed for sngle lnes above: 1. Lne bufferng 2. Overlay 3. Statstcs Arrvng at a measure for the geometrc accuracy of lne data sets The statstcs calculated n the above steps can be used to gve measures of the devaton between the lnes of the X and the Q data set. A lower bound on average dsplacement for complete lne data sets A Xbs DE = bs ( Qbs) AXbs ( ) DE s a lower bound on the average dsplacement of a qualty lne data set relatve to a lne data set of less accuracy. The choce of reference data set wll nfluence DE. We have chosen to use the data set wth the smallest expected total lne length as reference. If the data sets operated on s the orgnal data sets, as opposed to the completeness adjusted data sets, the results must be corrected usng the completeness measures determned above, gvng an approxmate lower bound on average dsplacement for ncomplete lne data sets. ( ) A( Xbs ) A Xbs Qbs ( 1 Completeness( X)) Qbs DE = bs ( 1 Mscondng( X))

10 3.3.5 Oscllaton O = ( Qbs ) # A Xbs ( ) Length X Where #A(...) s the count of areas. Ths s an ndcaton of the oscllaton of the lnes X and Q relatve to one another. O s most useful for «randomly» oscllatng phenomena, where t could be used as an ndcaton of bas (there would probably be a bas f the oscllaton, O, s low for randomly oscllatng lnes of dfferent accuracy). Oscllaton could also be found drectly usng X and Q, by countng the number of nodes ntroduced when overlayng the two lne data sets. 4 What s next? In ths paper we have outlned a method for quanttatvely assessng the spatal accuracy of the representaton of geographcal lnear features. The method utlses the standard GIS operatons buffer and overlay to arrve at a polygon data set that can be analysed usng smple statstcal measures (e.g. sum and count). At the tme of ths wrtng, we are about to start our accuracy analyss of the DCW data set usng these methods. The results of these practcal exercses wll become avalable to the publc n the project report. References Barnsley, M., 1988, Fractals everywhere (Academc Press). Chrsman, N., 1984, The Role of Qualty Informaton n the Long-Term Functonng of a Geographc Informaton System. Cartographca, vol. 21, no. 2/3, pp DMA, 1990, Dgtal Chart of the World - DCW Error Analyss. Prepared by Envronmental Systems Research Insttute, Inc, USA for Defense Mappng Agency, USA. Goodchld, M., and Gopal, S., 1991, Accuracy of Spatal Databases (Taylor &Francs). Langaas, S. and Tvete, H., 1994, Project Proposal: Issues of Error, Qualty, and Integrty of Dgtal Geographcal Data: The Case of the Dgtal Chart of the World. URL: «fle://lm425.nlh.no/pub/gs/dcw/qualty.ps». SDTS, 1990, Spatal Data Transfer Standard, verson 12/90. USGS.

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