goodchild _1989.pdf - University of Colorado Boulder

The Accuracy of Spatial Databases
Edited by
Michael Goodchild and Sucharlta Gopal
Narional Center for Geographic Infonnation & Analysis
University of California
Sanra Barbara, CA 93 106
t4qq
Taylor & Francis
London . New York . Philadelphia

Chapter 10
Modeling error in objects and fields
Michael F. Goodchild
Introduction
Thc currcnt inr€xEsr in spalial darabases strms ldsely frorn rheiI rolc in suDDonins
geographic informadon syste-ms. and rhe rapidly growing GIS indusny. diss a;
powerful systems for handling sparial data, and in recent ycars they have found
application in fields as different as Eansponadon, foresrry and archaeoiofy. Yet the
power of a CIS to input, srore, alalyze and output geogaphic information oi all kinds is
at rhe same time a maior liabiliry. To rhe databasc, lhc sEucture used to storc a Dolvson
or pixel has almost no connection !o lhc rcal meaning of $e polygon, as a parcel of iind,
object on a topographic mapor.stard of dmbcr.. The analysl making usc of a polygon
oveflay operaEon nas srmlafly utue Fessurc !o De sensrlvc to the tnlerprctation ofthe
data.layers.being overlaid. In realry a_CIS may€ncourage poor analysii by s€paradng
me oara couecEonr comp atron and analysls Nnclons, aod taurng to m8*e $e user aware
of rhe.possible dangers of indiscriminate use of such funcions as scale change,
lecnssll lcanon ano ovcfl av.
Onc of thc more bbvious issucs from this persD€ctive is thc existence of two
Eaditions ofCIS analysis. The disrincrion berwcen rasier and veclor is ofren seen as a
problem of system design. but actually presenls a major issue ofdala inrc4'reladon. To
emphasize this difference, and to sEess the context of data inrerDreErion mther than
system design, we will use the rerms field and object in rhis paper;alrhough rhey are ro
some extenl synonymous with raster and vector respectively. Sotde sDadal databases
represent the world as if it were populaFd by obje.is - poiirs, lines ind areas - with
associated arEibuks. continuing a Eadition developed in cnnography. Orhers reDrcsenr rhe
world as fields. or arrays ofpiiels, again with associated artri--buies'- fie choice bcrween
the two represenhtions has varioDsly b€en seen as depending on lhe method of data
collecdon (salellites_generale fields.canogmplers ge-nemre objecrs),_ fie degree of sparial
r€solunon requreo (oDjecrs appear ro tmply nlgher tevetS ol spattal rcsolutlon, whereas
pixels imply a lev€l which is fixed by fie pixel size). and rhe efficiency of atgorilhrns (for
example, rhe widely held perceprjon thar overlay is faster in nsrer). However we will
argue in !his_paper rhar frelds.and objects represenr fundamenrally differenr tmms of
aDsracEon or geogmpnrcar reallty.
This paper examines the relarionship between fields and objects ftom the
perspective of database error. It is clearly possible o represenr a given ser of data in
either [orm. and lo derive one from rhe olh€r by a simp]e CIS operarion such as
rasLer/vector conversion. But conversion must be sensitive lo the nalrie oI Lhe dara and
its uocertainty if subsequent analysis is to be successful.
107

Errors in objects
Goodchild
Consider the comfilon process of creating a spatial darabase of obiecrs by
digitizing a topographic map. Amibules will tikely be fnrered ftom a keyboird, ani
provided they are keyed correcrly. we can reasonably exped rhem ro be perfeclly
accumle. The localions ofobjects will b€ oblained by digitizing or scanning, and will be
subject to assoned errors. A numb€r of factors may connibule lodistonionbfrhe source
docurnenl includirg folding and stretching, changes in humidity, copying processes etc.,
and the process of map registradon will inEoduce addirional error Ifdigirizing is used,
posidonal accuracy will be affecred by lhe op€rator's precision in posidoning the cursor,
and by the rules used to select poinrs to be djgirized ftom line or polygon objecrs, whether
in poinr or sEeam mode. Finally rhe positional accuacy ofa scanncd line will be
affected by the resoludon of the scanner.
Of all of $ese errors, only cursor positioning has been subject to succsssful
analysis. Keefer, Smith and Gregoire (19E8) have described a model in which each
digitized point on a line or area objecr is disroned ftom irs rrue posidon by a bivadate
disEibudon. Ifeach poinr werc disroned independendy we would reach de rlnreasonable
conclusion rhat the expecled error ar digirized points is greater fian berween dighized
poinrs. bur it is likely thar errors arc posilively correlated bclween adiacenr poinls alone
the digitjzed line. Chrisman and Yandell (1988) a-nd Griffith (rhis volirme) hive obuined
useful esdmates of the expected error in polygon area measures based on this type of
model of digitizing enor. However ir is much more diff:cuh !o devise a reasonable model
of the process of point.selecrion.wbich varies_subslanrially berween digirizer operators
ano rypes or Dnes. and lu(ely conhDutes at leasl as much to error in area esrimates,
Unfonunately any such model woutd have to be sensitive ro the rype of line beins
digilized, as meandering rivers clearly presenr very different problems from ropographii
contours or hlghways! whcreas this seems less imponant for modeling cursor positioning
In the cose of a counry boundary or the oudine of a building, rle darabase obiecr
corresponds directly to a clearly defined objeq in rhe real, geographical world. Buj in
many cases rhe object in the darabase is an absEacr model of reat, conrinuous and
compl€x geographic variarion. Althollgh counries and buildines are frequenrly found in
spatial databases, many CIS appljcadons have bcen developed for absrraated obiecrs. In
lhe forest industry, an invenrory of foresr resources is co;monly mainrained i; diqital
torm by dividjng the foresr into area objecrs or 'stands" wirh descriprions whictiare
a(ributed homogeneously ro lhe entire stand, In reality the boundaries of stands are
tnnsition zones, and the attributgs are heterogeneous, Despite the popularity of vecior
dahbases, th€ object model is often a poor represen ation of geognphic vafiad6n.
Crearer difficulty may arise ifrhe artribures assigned ro-an object are rhemselves
abstractions. For exanple the telm ""old growth"" may be attached to a forest stand, but
ambiguities in the definirion of old growrh may make it impossible ro rcsolve wherher a
panicula.r point wirhin the srand is or is nol covered by old gowlh. Similarlv ir mav be
necessary to observe a significanr area in order to assign a ilassification suci as a;Den
parkland". l*hich is nor sFictly an arniburc of a poinl.bu of an exknded area.
Various lerms have been used to disringuish between fie two t)@es of error
implied by rhis argumenr. ln lhis paper we use lhe term processing error 16 describe rhe
unce(ajnry inEoduced by digirizing and any subsequenl form ofdigiral processing. such
as veclor/raster converslon. I he terrn source error is used to describe lhe dilferenc€s
whict may exist belween rhe obje(L model and $e geogaphical nuth which it represents.
While source error\ may be absent in represenrarioni of counries or buildings,we
conjecture ihal rhey will e\ceed processing errors jn representadons ofsuch seocraihical
variares as vegetarion, soil or land use. Since lhe iesutls ot a CtS anaivsi; w t be
interpreted by
cleate the datal
for the field.
MacDougall, 1
The di!
accumcy of ml
stand can be ol
is independcnt
estimated by s
will be subject
the arEa within
about hete$ge
Maps (
combines two
point samples
rcmotely senst
field, augmcnl
cleady a derivi
canographer ir
area objects wj
likely vary dei
would-expect i
is difficult to a
acls as a low
convening iror
filter, and sugt
oojects.
Wlilc !
and areas, it I
processing ani
Blakemore(19
a band of widd
and source erl
Moreover whil
point in polyl
€sumanng eI'
benchmark GIi
modeling error
Conlou
derived from I
displace it usin
topological in(
derived from z
deriv€d from I
proposidon - t
modeling errot
we explorc this

Eftor in ob|.crs andfeus 109
interFeied by reference to geo$aphic truth rather than to the souce documents used to
deate the dalabase, the existence of large source erro$ is a problem of major significance
for fie field. Several early papers (see for example McAlpine and Cook, 19?1;
MacDougall, 1975) drcw attendon to the impc'rtance of source erors.
The distinction beN€en processing and soulce eftor is critical in cstimating the
accumcy of measdrcs derived ftom a database. The eror in the estimated arca of a forest
stand c6n be obrained by analyzing the enors introduced in processing its ar€a object, and
is independent of source enors implicit in the object. The area of a given species can be
esdmated by summing the areas ofthe objects classified as containing rhe species, bu!
will be subject to source errors if the species 4rEibute do€s not apply homogeneously to
the area wirhin each stand. Since an object database comnronly contains no information
about hetoogeneity, it may be difficult or impossible to estimate sou€e enors.
Maps of forcst stands or soil rypes are compiled by a complex prccess which
combines two forms of dala. Information is first obtaioed on the gound Aom a series of
point samples or Eansects, and then extended spatially using an aerial phologaph or
remorely sensed scenc. or similar image. Thc area objects are compiled by interpreting a
field, augmenred by point attriburcs, so in rhese eramples $e objecr represenration is
clearly a derivative of the field reprcsentation. In compiling the map ftom the image, $e
cartogapher imposcs his or her own oxp€ctations on the data. The boundaries between
area objecrs will be srnoolh genemlizations of complex Eansidon zones whose widft will
likely vary depcnding on rhe classes on eirher side. The holes and islands which one
would expect in and around the transition zone will b€ commonly deleted. Finally, in the
case of forcst stands thc canographer rnay impose sorne concept of minimum size, since it
is difficult to administer stands ofless than, say, l0 hectarcs. In essence the cartogmpher
acts as a low-pass filter, removing much of the detailed g€ographical variation in
convening from field to objects. Unfortunately this removes much of the informalion on
which an enor model might b€ based, since enor is selectively deleted by a low-pass
filter, and sugges$ tbat we might focus on mod€ling enoN in fields rather than derivative
oDJCCIS,
Wlile some progress has been made in developing error models for digitizad lines
and areas. it has Droven much morc difficult to construct comDrehensive models of
processing and s6urce emors for complex geographical objec6. Chrisman (1982),
Blakemore(!984) and others have described the uncenainty in de position of a line using
a band of $ jd rh e r Perkal, 1956), which is a useful modeiof cenain kinds of processi n !
and source errors but does not deal wilh the problem of hetercgeneity of arEibutes,
Moreover while the epsilon band can be used to give a probabilistic interprera(ion ro rhe
poinr in polygon operation (Blakemore, 1984) it is not adequate as the basis for
estimating error in area measures, or tor simulating elfor in object databases to
benchmark GIS error propagation. At this point we have no fully satisfactory means of
modeling enor in complex spatial objects.
Conrours provide another example of the difficulty of modeling e.ror in objects
derived from fields. A model of object distonion mighi lake each contour line and
displace it using a random, aulocorelat€d error process, but it would be easy to prduce
topological inconsistenci€s such as crossinS contours or loops. However contours are
derived fmm an elevatlon field: no matter how elcvations are distoned, the contours
derived from lhem must always be topoloSically consistent. Tlis suggesK a general
proposidon - $ar the solulion lo modeling error in complex spatial objecrs may lie in
modeling elror in the fields ftom which many of th€m are obtained. In the next section
we explorc this possibility in more detail.

IIO
Field models of enor tn area objects
The previous section concluded with the p.oposition that satisfactory models of
enor in complex spatial objects could be formulated as models of error in fields. In this
section we consider two such models,
Goodchild and Dubuc (1987) proposed a model based on an analogy to the effect
ofmean annual temperature aid precipitation on life zones. W€ first g€nerate two nndom
fields, using one of a number of available methods, such as the fractional Brcwnian
prccess, tuming bands or Fourier transforrns (Mandelbmt, 1982). The autocovariance
structur€ of th€ fields can be varied to create a mnge of surfaces from locally smooth to
localy rugged. Imagine that one fi€ld represents mean annual temperature and the other,
annual DreciDitation.
-Holdridge
etdL (1971) have proposed a simple twodrmensional classifierwhich,
given temperaturc and precipitation, yields the co.responding €cological zone. By
applying such a classifier to the two fields, we obtain a map in which each pixel has been
classified into one of the available zon€s. If the pixels are now vectorized into
homogeneous areas, the map salisfies lhe requiremenrs ofmany rypes of area class maps:
the space is exhausted by non-overlapping, imegula y shaped area objects, and edges
meet in predominantly three'valent venices. On the other hand if a single field had been
used with a one-dimensional classifier, fte rcsult would have the unmistakable featu&s of
a contour or isopleth map. To simulate the influence of the cartographer, *hich we have
previously compared to the action of a low-pass filter, Coodchild and Dubuc (1987)
applied a sptne function ro the vectorized edges, ffd selectively removed small islands.
The model has intcresting properties which it shares with many rcal datasets of
this class. Because the unde ying fields arc smooth, classes can be adjacent in the
simulated map only ifthey are adjacent in the classifier, so certain adjacencies are much
more common than otheis. The response of an edge to a change in one of the underlying
fields depends on the geomery of the classifiel it is maximum if the corresponding edge
in rhe classifier space is perpendicula.r ro fie appropriare axis, and minimum (zero) ifrhe
edge is parallel to the axis. Sodislonion or error can be simulared by adding distonion ro
the underlying fields.
The model successfully simulates the appearance of ajea class maps, and is usgful
in crcating datasets under controlled conditions for use in benchmarking spatial databases
and CIS processes, and for Eacking the propagation of enor. On the oder hand tle large
number of Darameters in the model make it difficult or imDossible to calibrate asainst real
datasers. P'arameter values would have to be esrablishea for rhe underlying Felds. the
distoning field(s), the classifier, and !h9 spline funcdon used to smooth vectorized edges.
Goodchild and Wang (1988) describe an altemative model based on an analogy lo
remote sensint. Suppose that the process of image classification has produced an army
ofpixels. and that associated with each pixel is a ve.ctor of probabilitjes ofmembership in
each of the known classes. For example the vector{0.3,0.3.0.4} would indicate
probabilities of 0.3,0.3 and 0.4 of membership in classes A, B and C respectively. ln
practice we would expect the Foponion of non-zero probabilities to be small. allowing
the vectors b be stored efficiently,
We now requir€ 6 process of realizing pixel classes such that (l)the probabilities
of each class across realizations are as specfied, and (2) spatial d€pendence between
pixels in any one rcalization. The degree of spatial depenalence will aletennine the size of
homog€neous patches which develop. With high spatial d€pend€nce, a giv€n pixel will
belong to large patches in each Ealization: in 30% of realizations th€ €xample pixd will
be part of a patch of class A, 3O% B and 404o C.
Goodchild and Wang (1988) described a simple Focess which satisfies only one
of lhe requirements. Initial classes were assigned to each pixel by independent trials, and

y models of
ields. In this
to the €ffect
aI Brownian
ly smooth to
nd the oth€r,
Eifier which,
tl zone. By
ixel hrs been
torized into
r class rhaps:
s, and edges
eld had been
le fea$rcs of
ubuc (1987)
all islands.
Ll datasets of
acent in |he
ies are much
e underlying
|olr(Ung edge
(zero) if the
distortion to
?rnd is useful
ial databas€s
rnd the large
, against rcal
g fields, the
'rized edges.
Ln analogy to
ced an an?y
tmbership in
uld indicate
ectively. In
all, allowing
probabilities
rc the size of
en pixel wilt
rle pixel will
les only one
nt Eials, and
E.rcr in objecls and fields 1
a 3x3 modal fdler was then passed over Lhe army, reolacins lhe vajue ofrhe cenE-dl Dixel
by the modai value of rhe 3ri window. While tliis ens-ures sDdlial dependence. Lhe
poslenorprobabiliries are nor equal to rhe priors excepr in special iases.
More rec€nrly, we have experiniented witil two merhods *hich saaisfv borh
requirements. In rhe firsr. w€ firsr generarc a lafge number of realizarions'usins
lndepenclent. trl al s. On each simulared map. w€ counr lhe numbfr of 4-adjacenciei
oerween u

112 GoodcNld
Discussion
. The_process by wbich a spatia.l dahbas€ is crealed ftom a sourcc map is comDlex.
ano error-ol_vanous types is introduced at each slep. Some of$e ermr corioonenti can
De mooeted,. and progress has been made in analyzing the errors dui to cursor
pos ronlng.. Dur others such as point selecrion are more difficuh, and rhe eoal of a
comprehrnslve rD@et ot spabal dau processing errors is still elusive. yet deapirc fiis.
ror most t)?€s ot spaual daE $e enors inhercnt in $e sourcc document are cleailv more
srgnrrrcant than those inEoduced by processing. This is pa$icularly Eue when the-source
oocumevnr conurns oDJects wtuch are approxjmate absFactions of complex and continuous
. We have argued in rhis paper thar many of rhe more absract ryDes of sDatial
oDJecb nal/c Deen obl8jned or compiled ftom raw dara in rhe foIm of fieFa,
pnocess
and $at rhe
Ol comprlahon olten removCs much Of the information On which a useful mod€l
or uncenarnty or error mighr be based. The role of the caflopraDher in comDilin!
vegeunon or sou maps was compartd to the acdon of a low_paas filter in seleciiveli
rcmovrng tne nrgh spattal tiequencies wbich conhin diagnosric informarion oir
uncenarnry! such as wrggly lines and small islands.
From a-spatial shristical poinr ofview, lhe cenEal poposition of this DaDcr is ftal
uncena$ty Infh€_obJectson a map rcsults ftom different outcomes of a stochistic orocess
oetned lor a held. I lris is subsuntially different from rhe apploach which has underlain
mucn work^on stochas[c pftresses for images. The problem of image rcstoration has the
oDJectrve ot llndrng tie"Eue value for each pixel given some distoned valDe. However
rn rne.gcograpnrcal case $ere js usually no comparable noiion of Eue value. because
sparlat vaflatron_€xtends to all scales, and because class definitions arc fieouendv
amDlguous. srmrtarly. atthough spatial statislics contains extensive litcrature on i-aei
scgmenBuon, tnrs problem is seen as one of a range of possible rhodels for fie
canographic process of foming objecG Fom fields.
trom the perspecrive of sparial darabascs
"
and GIS, rhere are several Dossible roles
lorl modet oJ.enotin spatial data. On the one hand it would be useiul to have a
ca orared model wi)ch coutd be used to describe ellor in a panicular daraset, to track fte
error through cls.p_rocesses, and to repon uncenainry in ttre resutts oi piociiilnpi
nowever such modils may be unobtainable in many cases, due to the lack df adeluaie
rnrornauon torcattbrahon, and to the complexiry of the error process itself. There s;ems
ro De no equlvalent wlth the generality of the Gaussian distribution for comDlex sDalial
ooJecls. Un Ine olher h,Md modets of error are useful for generadng simDlaied daiases '-*_"
under known conditions. for benchmarking GIS pro".r... ind ,ror"q-. .irh;;
Abstritct pornl, tine and area models developed in canographtb€cause ofrhe need
ro represent conplex sparial^variarion on paper using images w:triil6outa Ui;"rcd ;i;
a simple pen (Coodchild, t988). DEMS:ftNs andquad-nees are a few ofrht;;;;;;
mooers whlcn.have been developed for spadal databases to lake advanuqe of the removal
ol caflographlc consra'nts, The conlour was devised as an efficieni wav to disDtav
sparial valiarion ofelevarion on a bpographic map *;tt p.".up"ui. oiarii,in! iir?i.1
fixed width, bur DEMs and TtNs haie ijisrincr advantales " over dieidzed coirours in
rermsofsamplingefficiencyandtheexecurionofvariouianatvr;.op"e.aiGi.i!-etaui
seen ln uts -paper that Lhe selecrion of lhe canographic model alao has lhe effecr ol
removing informarion on uncenainty. since il is e'a.i.. ,o -oo.t .rroii" riiai ir,in i,
oenvarrve oblects. A darabase populaled by 6elds is more usetul for modeline
IracMng.
and
uncenalnly tnan one populated by absnacted objects. From an
perspecnve,
acciracv
lnen. rt woutd be prettrable iI darabases representing spalial varialion
paramerers
o,f
sucn as sort type, vegeration or land use conrained !h€ raw point samDlfs and
lma8es rrom whrch suci maps are usually compiled. lf rhe canograDhjc vie; of ihe
cBraDa\e were requr€d, ir could be generared by inlerpreling areai objects.
inreracrively
eirher
or auromaticalty. But lhii snalegy would avord rlie commori pracrice of
imposing the cr
rcpresents.
References
Blalemorc, M.
131-9.
Chrisman, N. tr
Unpublished
Chdsman, N. I
and MoppinS
Geman, S,, an
Bayesian resl
Intelllgence .
Goodchild, M.
cartography. .
Goodchild, M.
aDDlications
.dSPRs/Ac
Goodchild, M.
PrcceedinSs
Commission
Haining, R. P,
autoconelato
Holdridge, L.
Foresl Enrir(
Keefer, B. J.,
vA, 475-83.
MacDougall, E
McAlpine, J.
Proceedines,
43d Congrcl
Mandelbrot, B.
Perkal, J., 195,
399-403.

Enor in objects andfrelds II3
inposing the carrogaphic view as a filtel betwe€n ihe alatabfi€ atd the teality whicb it
rcprcseints.
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