11
88 Volume 44, Number 1, February 1992 port of the Committee on Transpiration and Evaporation, 1943-44. Transactions, American Geo- physical Union 2;:638-93. tionships between land-surface processes and clio mate, and the statistical analysis of large-scale clio mate fields. JOHANNES FEDDE1\IA is AssistantProfessor of Geography at the University of California, Los An- geles. He was formerly a Ph.D. student at the Uni- versity of Dela\vare. His research interests include \vater balance climatology and climate/teleconnec- cion processes. CORT WILLMOTT is Chair of the Department of Geography and Professorof Geography and Ma- rine Studies at the University of Dela\vare. He also is a member of the University:s Center for Climatic Research.His research interests include the rela- On the Issues of Scale, Resolution, and Fractal Analysis in the M,apping Sciences* Nina Siu-Ngan Lam Dale A. Quattrochi Lo/lisiana State University NASA Science and Technology Laboratory, Stennis Space Center Scale and resolution have long been key issues in geography. The rapid development of anal:-.ticalcartography, GIS; and remote sensing {the mapping sciences} in the last decade has forced the issues of scale and resolution to be treated formally and better defined. This paper addressesthe problem of scale and resolution in geographical studies, \vith special reference to the mapping sciences. The fractal concept is introduced, and its use in identifying the scale and resolution problem is discussed. The implications of the scale and resolution problem on studies of global change and modeling are also explored. Key words: scale, resolution, fractals, mapping sciences. Scale, Resolution, and Geography areas. Some geographers rely on data obtained from satellites, yet others depend on data re- T he concept of scale is central to geography garding pollen counts and soil particles ob- (Harvey 1969; Meentemeyer 1989; Wat- tained through electronic microscopes. Diver- son 1978; Woodcock and Strahler 1987). It is sity within the discipline results in the need one of the main characteristics that portrays to address spatial problems from multiple geographic data and provides a unique percep- scales and resolutions. tion of spatial attributes as they relate to form, This variation in scales can be regarded both process, and dimension. Geographers often as a strength and weakness of the discipline. deal with spatial phenomena of various scales. Analyzing geographical phenomena using a For example, geomorphology encompasses range of scales offers a special view and meth- stUdies ranging from patterns of river net- ' odology that other disciplines seldom employ, \vorks, river basins, and coastline changes to enhancing geography's strength. To the con- potholes, cave, and gully formation based on trary, the massive amount of data needed for international, national, regional, or local analysis of spatial phenomena at various scales, scales. Climatologists study upper air circula- coupled with the possibility of applying an tion around the globe as \vell as effects of local inappropriate methodology, often leads to a climate on agricultural production and health. meaningless study. This invites criticism and Urban geography includes studies ranging confusion from within the discipline and from from analyzing the urban systems in an in~er- other related disciplines. national, national, or regional context to as- This paper does not attempt to solve issues sessing the impact of facility location on local of scale and resolution, but rather brings to- .\\.e thankGrego~. Carter, Lee De Cola, the anonymousrevie,vers, and the editor for improving this paper; Clifford Duplechin a~d Mary Lee Eggart, and the LSl: graduate students for preparing the graphics; and the NASA, john C. Stennis SpaceCenter, Director's Discre- tiona~. Fund for supporting in part the development of this paper. Profession.! Geogr.pher.-1-1(1) 1992, pages 88-98 tC Cop.vright 1992 b~. Association of American Geographers Initial submission, April 1991; final acceptance, August1991

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Page 1: On the Issues of Scale, Resolution, and Fractal Analysis ... · overview of the use of fractals in geography has been presented by Goodchild and Mark (1987) and recently by Lam and

88 Volume 44, Number 1, February 1992

port of the Committee on Transpiration andEvaporation, 1943-44. Transactions, American Geo-physical Union 2;:638-93.

tionships between land-surface processes and cliomate, and the statistical analysis of large-scale cliomate fields.

JOHANNES FEDDE1\IA is Assistant Professor ofGeography at the University of California, Los An-geles. He was formerly a Ph.D. student at the Uni-versity of Dela\vare. His research interests include\vater balance climatology and climate/teleconnec-cion processes.

CORT WILLMOTT is Chair of the Departmentof Geography and Professor of Geography and Ma-rine Studies at the University of Dela\vare. He alsois a member of the University:s Center for ClimaticResearch. His research interests include the rela-

On the Issues of Scale, Resolution, and Fractal Analysis in theM,apping Sciences*

Nina Siu-Ngan Lam Dale A. QuattrochiLo/lisiana State University NASA Science and Technology Laboratory,

Stennis Space Center

Scale and resolution have long been key issues in geography. The rapid development of anal:-.tical cartography,GIS; and remote sensing {the mapping sciences} in the last decade has forced the issues of scale and resolution tobe treated formally and better defined. This paper addresses the problem of scale and resolution in geographicalstudies, \vith special reference to the mapping sciences. The fractal concept is introduced, and its use in identifyingthe scale and resolution problem is discussed. The implications of the scale and resolution problem on studies ofglobal change and modeling are also explored. Key words: scale, resolution, fractals, mapping sciences.

Scale, Resolution, and Geography areas. Some geographers rely on data obtained

from satellites, yet others depend on data re-

T he concept of scale is central to geography garding pollen counts and soil particles ob-

(Harvey 1969; Meentemeyer 1989; Wat- tained through electronic microscopes. Diver-son 1978; Woodcock and Strahler 1987). It is sity within the discipline results in the need

one of the main characteristics that portrays to address spatial problems from multiple

geographic data and provides a unique percep- scales and resolutions.

tion of spatial attributes as they relate to form, This variation in scales can be regarded both

process, and dimension. Geographers often as a strength and weakness of the discipline.

deal with spatial phenomena of various scales. Analyzing geographical phenomena using a

For example, geomorphology encompasses range of scales offers a special view and meth-stUdies ranging from patterns of river net- ' odology that other disciplines seldom employ,

\vorks, river basins, and coastline changes to enhancing geography's strength. To the con-

potholes, cave, and gully formation based on trary, the massive amount of data needed for

international, national, regional, or local analysis of spatial phenomena at various scales,

scales. Climatologists study upper air circula- coupled with the possibility of applying an

tion around the globe as \vell as effects of local inappropriate methodology, often leads to a

climate on agricultural production and health. meaningless study. This invites criticism and

Urban geography includes studies ranging confusion from within the discipline and from

from analyzing the urban systems in an in~er- other related disciplines.national, national, or regional context to as- This paper does not attempt to solve issuessessing the impact of facility location on local of scale and resolution, but rather brings to-

.\\.e thank Grego~. Carter, Lee De Cola, the anonymous revie,vers, and the editor for improving this paper; Clifford Duplechin a~d MaryLee Eggart, and the LSl: graduate students for preparing the graphics; and the NASA, john C. Stennis Space Center, Director's Discre-tiona~. Fund for supporting in part the development of this paper.

Profession.! Geogr.pher. -1-1(1) 1992, pages 88-98 tC Cop.vright 1992 b~. Association of American GeographersInitial submission, April 1991; final acceptance, August 1991

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Scale, Resolution, and Fractal Analysis 89

SCALE Closely related to scale is the concept ofresolution. Resolution refers to the smallestdistinguishaoleparts in an object or a sequence(Tobler 1988), and is often determined by thecapability of the instrument or the samplinginterval used in a stUdy. A map containingdata by county is considered of finer spatialresolution than a map by state. In reference tothe three meanings of scale, small-scale stUdiesusually employ data of finer spatial resolutionthan large-scale studies. Small-scale maps, onthe contrary, often contain lower or coarserresolution data than large-scale maps. Finally,phenomena operating at a smaller scale, suchas potholes and caves, require data of finerresolution. Because large-scale studies (geo-graphic or observation scale) involving fine res-olution are rather uncommon, scale and reso-lution are often blended together and looselyreferred to as "scale," although it is recognizedthat a large-scale stUdy does not necessarilymean that it has a coarser resolution, and viceversa. In this paper the notion of the resolutionis generally implied whenever the term scaleis used.

-

~ ~ ~"" SPATIAL -SPATIo- TEMPORAL -TEMPORAL

~ ~,

~~

GEOGRAPHIC OPERATIONAL(or Observation)

Figure 1: Meanings of scale.

CARTOGRAPHIC(or Map)

gether important points regarding scale andresolution in light of recent developments, andproposes the use of fractals for analyzing spa-tial phenomena encompassing various scales.With appropriate methodology and sufficientattention to the problem of scale, we believethat the geographical perspective derived fromanalysis using differing scales can contributein many \vays to an understanding of variousspatial phenomena. This is particularly im-portant in relating earth processes or otherspatial and environmental phenomena withinthe context of global dynamics (e.g., the In-ternational Geosphere/Biosphere Program).

Controversies over the exact definition andmeasurement of scale and resolution exist(Woodcock and Strahler 1987). Some clarifi-cation of terminology, therefore, is necessary.The term scale may include all aspects-spa-tial, temporal, or spatio-temporal (Fig. 1). Inthis paper \ve focus on spatial scale. There areat least three meanings of scale. First, the termscale may be used to denote the spatial extentof a study (i.e., geographic scale or scale ofobservation). For example, a spatial analysisof land use across the entire United States isconsidered a large-scale study, as compared toa land use plan for a city. This definition ofscale is quite different from the second usecalled cartographic scale, where a large-scalemap covers smaller area but generally withmore detail, and a small-scale map coverslarger area \\1th less detail. The third usage ofscale refers to the spatial extent at which a

particular phenomenon operates (i.e., opera-tional scale). For example, mountain-buildingprocesses operate at a much larger scale thanthat of river pothole formation. These threemeanings of scale, however, are often mixedand used vaguely in the literature.

Methodological Issues Related toScale and Resolution

Several methodological dilemmas surroundingthe scale and resolution issue can be summa-rized here. First, different spatial processesoperate at different scales, and thus interpre-tations based on data of one scale may notapply to another scale (Harvey 1968, 1969;Stone 1972). The noted "ecological fallacy" inspatial analysis, making inferences about phe-nomena observed at differing scales or imply-ing finer resolution from coarser resolutiondata, attests to this problem. A simple exampleby Robinson (1950) sho\vs that correlationsbetween t\vo variables, measured IQ and race,decreased from 0.94 to 0.73, and finally to0.20, as resolution increased from census re-gion, to state, and to individual study scales(Openshaw 1984).

Because spatial patterns are usually scalespecific, inferring spatial process from spatialpattern is perplexing; this is illustrated by thewell-known dilemma of different processesleading to the same spatial pattern (Harvey1969; Turner et al. 1989b). Moreover, a spatial

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90 Volume 44, Number 1, February 1992

and landscape patterns, and to utilize fullyremote sensing and GIS.

In an integrated GIS and remote sensingsystem, data of various types, scales, and res-olutions are used. How will the analyticalmethods and results be affected by differentscales and resolutions? What is the relationshipbetween scale and accuracy? The selection ofan appropriate scale is also influenced by thetechniques used to extract information fromremote sensing data. The factors of scale andresolution, therefore, have become a major re-search direction in GIS and remote sensing.

pattern may look clustered at one scale butrandom at another. For example, the mortalitypattern of leukemia cancer in China appearsrandomly distributed if based on county-leveldata; if the data are aggregated and reportedby province, a clustered geographical patternresults (Lam et al. 1989). Depending upon thescale of observation, therefore, processes thatappear homogeneous at a small scale may be-come heterogeneous at a larger scale. This maybe exemplified by patterns of coniferous for-ests infested \vith pine bark beetle blight. Ata small spatial scale, the patterns of infectedindividual trees or groups of trees \\.ithin theforest are not evident, because the pattern ofinsect damage becomes integrated as part ofthe spatially homogeneous coniferous forest.At a large geographic scale, ho\vever, groupsof trees infected \\.ith pine bark beetle blightappear as patches of dead trees and can beeasily distinguished from other trees. Thus,the pattern of insect infestation becomes het-erogeneous at larger scales. How do we kno\\.\vhat scale and resolution we should use? Andhow do wekno\v \vhen the results are mean-ingful and valid given the scale and resolutionof the data?

Rapid development in the mapping sci-ences, particularly GIS and remote sensing,in the last decade has to a great extent "for-malized" the scale and resolution problem. Animmediate question in designing a GIS andremote sensing system is: what scale and res-olution should \ve use for a specific applica-tion? For example, shall we use a 1:24,000 or1:250,000 scale map? Will Landsat TM im-agery (with a pixel size of 30m) be more ap-propriate than Landsat MSS imagery (80mpixel)? Shall \ve sample or update the dataevery five or 10 years? The concern here isboth theoretical and practical. Increased res-olution will increase data storage and process-ing time, \vhereas decreased resolution leadsto inaccuracy that compounds quickly whenseveral types of data are overlaid and analyzed.What is the optimum resolution or does anoptimum really exist? Ultimately, the "best"resolution depends upon the study objectives,the type of environment, and the kind of in-formation desired. Hence, much more \vorkis required to understand the effects of scalein interpreting variability in earth processes

A Brief Description of Fractals

Fractals are now widely used for measuring,as well as simulating, forms and processes andare attractive as a spatial analytical tool in themapping sciences. Since Mandelbrot coinedthe term in 1975 (Mandelbrot 1977), the con-cept has been further developed and expandedacross virtually every major discipline. Worksby Mandelbrot (1977, 1983), Peitgen andSaupe (1988), Feder (1988), and more recentlyFalconer (1990) articulate some of the thought-provoking issues related to fractal analysis. Anoverview of the use of fractals in geographyhas been presented by Goodchild and Mark(1987) and recently by Lam and De Cola

(forthcoming).Fractals were derived mainly because of the

difficulty in analyzing spatial forms and pro-cesses by classical geometry. In classical ge-ometry (i.e., Euclidean geometry), the dimen-sion of a curve is defined as I, a plane as 2,and a cube as 3. This is called topologicaldimension (Dr). In fractal geometry, the fractaldimension D of a curve can be any value be-t\veen 1 and 2, and a surface between 2 and3, according to the complexity of the curveand surface. Coastlines have fractal dimensionvalues typically around 1.2, and relief dimen-sions around 2.2. Fractal dimensions of 1.5and 2.5 for lines and surfaces, respectively, aretoo large (i.e., irregular) for modeling earthfeatures (Mandelbrot 1983, 260 and 264-65).

The key concept underlying fractals is self-similarity. Many curves and surfaces are self-similar either strictly or statistically, meaningthat the curve or surface is made up of copies

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91Scale, Resolution, and Fractal Analysis

(a)m=4.r=1/4

D = -log (4)/log (1/4) = 1

(b)

EBm=4.r= 1/2

D = -log (4)/log (1/2) =:l

(c)~_../\- m=4.r=1/3~"LA- D = -log (4)/log (1/3) = 1.2618

Figure 2: Relationships between fractaldimension (0), number of copies (m), and scalefactor (r).

of itself in a reduced scale. The number ofcopies (m) and the scale reduction factor (r)can be used to determine the dimensionalityof the curve or surface, where D = -log(m)/log(r) (Falconer 1990). For. example, in Figure2a, a line segment is made up of four copiesof itself, scaled by a factor V4 (i.e., one-fourthof the line length), and the dimension D =-log4/log(V4) = 1. A square (Fig. 2b) is madeup of four copies of itself and scaled by a factorV2 (i.e., half of the side length), thus yieldingD = -log4/log(V2) = 2. Similarly, a von Kochcurve is made up of four copies of itself \vitha scaled factor V3 and having dimension D =-log4/log(V3) = 1.262 (Fig. 2c).

Practically, the D value of a curve (e.g.,coastline) is estimated by measuring the lengthof the curve using various step sizes. The moreirregular the curve, the greater increase inlength as step size decreases; D can also becalculated for a curve by the regression equa-tion logL = C + BlogG and D = 1 -B,where L is the length of the curve, G is thestep size, B is the slope of the regression, andC is a constant. The 1;) value of a surface canbe estimated in a similar fashion and severalalgorithms for measuring surface dimensionhave already been developed (e.g., Shelberg etal. 1983). A scatterplot illustrating the rela-tionship between step size and line length inlogarithmic form (i.e., fractal plot) becomesthe source of the derivation of fractal dimen-sion. An example of a fractal plot using theLouisiana coastline and its corresponding map

Figure 3: (a) Louisiana coastline and its

subdivisions; coasts A to F are: Chenier plain,

Sale-Cypremort subdelta, Teche and Lafourche

subdeltas, Plaquemines and Balize subdeltas,St. Bernard subdelta, and Lake Ponchartrain and

Lake Borgne; (b) corresponding fractal plot; the

bars on each curve show the range of points

(i.e., steps) used in the regression and thusdefine the self-similarity range; the number of

steps used for coasts A to F and the overall

coastline (LA)-are 9,7,5,6,5,6, and 11,

respectively. (Adapted from Diu 1988 and Lam

and Diu, forthcoming).

is shown in Figure 3 (Lam and Qiu, forthcom-ing; Qiu 1988). Previous stUdies have dem-onstrated that fractal plots of empirical curvesand surfaces are seldom linear, with many ofthem demonstrating an obvious break (Butten-field 1989; Goodchild 1980; Mark and Aron-son 1984). This indicates that true fractalswith self-similarity at all scales are uncommon.

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92 Volume 44, Number 1, February 1992

in the fractal literature, such as the conceptsof self-affinity, random fractals, and multi-fractals, have expanded fractal applications tomany phenomena \vhere true fractals withstrict self-similarity do not exist. At the sametime, fractals have generated criticisms fromresearchers in various disciplines, and indeedthere are limitations in effectively applyingfractals.

Fractals in the Mapping Sciences

There is a relative paucity of research thatemploys fractals in the mapping sciences.Goodchild (1980) in a pioneer article demon-strated that the fractal dimension can be usedto predict the effects of cartographic general-ization and spatial sampling, a result that mayhelp in determining the appropriate resolutionof pixels and polygons used in studies relatedto GIS and remote sensing. Muller (1986) pro-posed the use of fractal dimension as a guidingprinciple for future implementation of gener-alization algorithms in automated cartography.Fractal surfaces have been used as test datasets to examine the performance of variousspatial interpolation methods (Lam 1982,

The fact that self-similarity exists only over alimited range of scales could be utilized posi-tively, ho\vever, to summarize scale changes.Thus, fractals are potentially useful tools forinvestigating the issues of scale and resolution.

Based on the self-similarity concept, curvesand surfaces of various diinensionalities can begenerated, and it is the simulation capabilityof fractals that makes this technique a favora-ble tool for spatial analysis. Moreover, variousobjects and structures, such as planets, clouds,ocean floors, trees, particle growth, and urbanmorphology can be simulated using fractals(Peitgen and Richter 1986; Peitgen and Saupe1988). Several methods have been developedto simulate fractal curves and surfaces, includ-ing the shear displacement method, the mod-ified Markov method, the inverse Fouriertransform method, and the recursive subdivi-sion method (e.g., Carpenter 1981; Dutton1981; Fournier et al. 1982). Figure 4 sho\vssome sample surfaces with different D valuesgenerated using a shear displacement algo-rithm (Goodchild 1980; Lam 1990).

The foregoing description of fractals israther elementary and intentionally made sim-ple. There are many other ways of definingfractal dimensions and hence, the applicationsof fractals vary extensively. Recent additions

2.2

:1i..

c/

"'-"'. ...

.-0 .Z.O0 .Z.O

Figure 4: Examples of fractal surfaces generated using a shear displacement algorithm.

Source: Lam 1990

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Scale, ResolutiOIl, alld Fractal Analysis 93

tal conditions over extensive areas of the earth.O'Neill et al. (1988) sho\\ed that the fractaldimension has a highly significant correlation\vith the degree of human manipulation of thelandscape. Landscapes dominated by agricul-ture tend to have simple polygons and lowfractal dimensions (negative correlation), andlandscapes dominated by forest tend to havecomplex shapes and high fractal dimensions(positive correlation). Thus, the application offractals allo\\.s not only a different and con-venient \vay of describing spatial patterns, butalso the generation of hypotheses about thecauses of the patterns. The above suggestionsopen new directions for the potential applica-tion of fractals.

Fractals, Scale, and Resolution

It is apparent that the fractal model is a usefultool for simulation and modeling. Fractal anal-ysis of spatial forms and processes, however,can be limited by problems at both the theo-retical and technical levels (Lam 1990). Thefirst and foremost problem relates directly toscale and resolution. The self-similarity prop-erty underlying the original fractal model as-sumes that the form or pattern of the ~patialphenomenon remains unchanged throughoutall scales, implying that one cannot, throughfractal analysis, determine the scale of the spa-tial phenomenon from its form or pattern.This kind of strict self-similarit}' is consideredunacceptable in principle, and hence, has gen-erated criticism about the spatial applicationof fractals- Empirical studies have shown thatmost real-world cur\-es and surfaces are notpure fractals \\"ith a constant D at all scales.Instead, D varies across a range of scales(Goodchild 1980; Mark and Aronson 1984).These findings, however, can be interpretedpositively. Rather than using D in the strictsense as defined by Mandelbrot (1983), it ispossible to use the D parameter to summarizethe scale changes of the spatial phenomenon.This latter use of fractals is supported by theresults from Burrough's (1981) work, \vhere itis suggested that ~Iandelbrot's D value can beused as an indicator over many scales for nat-ural phenomena. Through interpretation of Dvalues, therefore, it may be possible to relateor separate scales of variation that might bethe result of particular natural processes.

1983) and to assess the efficiency of a quad treedata structure (Mark and Lauzon 1985). Frac-tal curve generation, on the other hand, hasbeen used as an interpolation method and asthe inverse of curve generalization by addingmore details to the generalized curve (Carpen-ter 1981; Dutton 1981; Jiang 1984).

The use of fractals in remote sensing is rel-atively ne\v and seems to have great potential.De Cola (1989) applies fractal analysis to landcover patterns derived from Landsat ThematicMapper (TM) data and concludes that self-similarity, fractal dimension, and Pareto sizeparameter are useful measures for analyzingdigital images. His results also show that ur-ban land cover gives rise to more complicatedspatial patterns than does intensive agricul-ture. In a study comparing urban, rural, andcoastal areas using Landsat TM imagery, Lam(1990) has demonstrated that different landcover types have different fractal dimensionsin different bands. Urban land cover \vasfound to have the highest dimension, followedclosely by coastal and rural land cover types.

In addition to the two types of applicationsmentioned above, fractal surfaces have beensuggested as a null-hypothesis terrain or normwhereby further simulation of various geo-morphic processes can be made (Goodchildand Mark 1987). Similarly, Loehle (1983), ina paper examining the applications of fractalconcepts in ecology, suggested using the frac-tal dimension as a parameter summarizing theeffect of a certain process up to a particularscale. He also proposed the use of the self-similarity property as a null hypothesis. Inother work by Burrough (1981), it was sho\vnthat many environmental variables are fractalswith varying fractal dimensions, and that theexamination of D values would be useful forseparating scales of variation that might be theresult of natural processes.

Furthermore, fractals can help formulatehypotheses concerning the spatial scale of pro-cess-pattern interactions (Krummel et al.1987). It was also concluded that fractal tech-niques should be particularly applicable toanalysis of remotely sensed data, since it pro-vides D as a simple measure that indicates thescale at which processes are occurring (i.e.,operational scale). Changes in D computedfrom remote sensing data, therefore, have pri-mary implications for changes in environmen-~

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94 Volume 44, Nttmber 1, Febrttary 1992

The second problem of applying fractals re-lates to the technical aspect of fractal mea-surement, The fact that self-similarity existsonly within certain ranges of scales ~akes itdifficult to determine a breaking point for self-similarity; i.e., a point at \\'hich self-similarityends for a curve or surface (Fig. 3b). Thisaffects the final D value, \vhich is then usedto characterize the curve or surface (Butten-field 1989). Furthermore, there is some ques-tion as to whether maps or map products pro-vide sufficient detail to accurately measure thefractal dimension (Carstensen 1989). The cal-culation of the fractal dimension for surfacespresents an added technical problem. Roy etal. (1987) have shown that for the same sur-face, different D values could result from us-ing different algorithms, with a range as lowas 2.01 to a high of 2.33. Finally, all the ex-isting methods have so far been applied onlyto regular grid data, such as digital terrainmodel data or Landsat TM data. Fractal mea-surement of many other socio-economic phe-nomena, such as population and disease dis-tributions, or environmental process-responsephenomena, presents another challenge. Forexample, identification of the patterns of pinebark beetle infestation\vithin a forest may re-veal insight into possible causes-responses offorest landscape dynamics as affected by insectinfestation. These data are typically reportedin an aggregate polygonal form, with irregularboundaries and possibly holes (in the forms oflakes or islands) or missing data. The existingalgorithms used to delineate such features willhave to be modified and extra steps will haveto be taken before actual measurement takesplace (Krummel et al. 1987).

variables influencing the process at differentscales must be understood. Third, the appro-priate methods for translating the results fromone scale to another must be de\"eloped. Fi-nally, the methods and results must be testedacross scales.

Several methods have been employed to ex-amine issues related to the scale and resolutionproblem, including for example, spectral anal-ysis (Moellering and Tobler 1972), movingstandard deviation measures (Woodcock andStrahler 1987), and variance and spatial cor-relation measures (Carlile et al. 1989). Withconsideration of the conceptual and technicaluncertainties associated \\.ith their use, fractalscould contribute to all four steps noted above.Fractals could be used to identify the spatialand temporal scale of the process. The fractalplot of step size against line length or areameasurement can be used to determine thedimension values and their corresponding self-similarity ranges. These values can be utilizedto help identify the underlying processes cre-ating the patterns. Since most natural curvesare not strictly self-similar, the derivations ofthe fractal dimension and associated self-sim-ilarity range are often based only on the rangeof points sho\ving linearity (Fig. 3b).

For example, a study of the Louisiana coast-line by Qiu (1988) has demonstrated thatcoasts of different origins and developmentstages are distinguishable by their fractal di-mension values and associated self-similarityranges (Lam and Qiu, forthcoming). Coastsdominated by deltaic processes generally havehigher fractal dimensions (about 1.3) with anarro\ver scale range of 0.~12.8 km, \vhilecoasts dominated by marine processes havelo\ver dimensions (about 1.1) and a wider self-similarity range of 0.2-51.2 km (Fig. 3a andb).

In analyzing China's cancer mortality pat-terns using data by commune for the TaihuRegion, Lam and Qiu (1990) have demon-strated that distinct self-similarity ranges existin the three leading cancer mortality patternsin China (stomach, esophagus, and liver).Fractal dimensions and self-similar ranges forthese cancer surfaces were computed by thevariogram method (Mark and Aronson 1984).Figure 5 is the variogram plot in logarithmicform illustrating the relationship be~veen dis-

A Research Agenda

As mentioned previously, the basic questionsurrounding the issues of scale and resolutionis: can a study predicated on or founded atone scale be used to make inferences to thesame phenomena under observation at differ-ent scales? To ans\ver this question, severalinterrelated steps are involved (Turner et al.1989a). First, the spatial and temporal scale ofthe process must be identified. Second, theimportance or changes in importance of the

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Scale, Resoltttion, and Fractal Analysis 95

scales and so does the importance of control-ling factors. Cancer mortality stUdies can againprovide an example in this context (Lam et al.1989). It is possible that, as a first step, thebroad environmental context fo,certain cancertypes can be determined by the typical GIS-overlaying methodology using large scale,coarse resolution data (e.g., climate, relief).This is then followed by a small scale, finerresolution study investigating local factors(e.g., industrial locations, \vater supply). An-other example that requires data of variousscales is in veterinary medicine and parasitol-ogy. Coarse resolution satellite data, such asthose obtained from the A VHRR (AdvancedVery High Resolution Radiometer) with anaverage pixel resolution of 1 km, could be usedto monitor and identify broad areas of largeanimal parasite habitats (e.g., snails, mosqui-tos). The broad areas of habitat could then berefined using finer resolution remote sensingdata, such as airborne scanner data with pixelresolution of five meters or less. Fractals, alongwith other statistical measures, can be used todetermine the complexity and spatial cluster-ing of these patterns, thus providing clues onthe existence of certain factors related to thedistribution of parasite habitatS.

The capability of fractal models in simula-tion, or rather, interpolation or extrapolation,provides another method for translating resultsacross scales. If the phenomenon of interestdisplays a self-similarity property with a rel-atively stable dimensionality, one can invertthe process and re-generate the pattern basedon the fractal dimension value. The issue ofline generalization and line generation, whichhas been a subject of intense research in ana-lytical cartography, is basically an issue ofttanslating results across scales. Fractals, al-though with relatively less success, haveplayed an important role in this realm of anal-ysis (e.g., Dutton 1981; Jiang 1984; Muller1986). More stUdies in this area, however, areneeded.

Finally, fractal analysis can be used as ameans to test the results across multiple scales.In addition to the cancer stUdies mentionedearlier, research is now underway in whichdigital remote sensing images of different res-olutions are being tested and examined bymeans of fractals (Quattrochi and Lam 1991).

tance and variance (or squared difference) forall three cancer patterns. Fractal dimensionsof these patterns can be determined by theequation D = 3 -(B/2), where B is the slopeof the individual regression line through therespective points in the variogram. In thisstudy the mortality pattern of liver cancer wasfound to have the highest dimension (D =2.86), followed by stomach (D = 2.76) andesophageal (D = 2.71). The linearity exhibitedon the variogram plot for all three patternsindicates that self-similarity exists within cer-tain scale ranges. The patterns behave differ-ently, however, when reaching a certain scaleor in this case, a distance range (i.e., the breakpoint in the variogram plot). Esophageal andliver cancers have similar distance limits ofabout 5-150 km, while the limit is approxi-mately 5-90 km for stomach cancer. This sug-gests that if the communes are aggregated toa distance range as large as the limit, the pat-terns would look very different and wouldresult in a different interpretation of the pro-cesses underlying the patterns as defined bythat level of scale and resolution. Furthermore,it is suspected that whatever the underlyingcontrolling factors (e.g., climate, topography,pollution), they are likely to operate in scaleranges similar to those of the cancer mortali-ties.

Different processes operate at different

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Literature CitedThe results will provide insights \vith regardto scale, resolution, accuracy, and landscaperepresentation. Fractals may offer a useful per-spective on predicting spatial, environmental,and ecological system dynamics by revealingthe self-similarity of phenomena at differentscales, in rum, providing better models in ahierarchical frame\vork for the analysis of scale(O'Neill et al. 1989).

The issues of scale and resolution are no\\"emerging as important aspects of the mappingsciences that will have a significant place inglobal change and global modeling studies(NASA 1988; Wickland 1989). It is recognizedthat more than a single scale of observationmay exist, thereby necessitating measurementat several levels of resolution. One of the majoranalytical challenges in studying global pro-cesses is to develop measurement samplingtechniques that \vill elucidate how these pro-cesses range over broad spatial and temporalscales. Fractal analysis may become a majortool for measuring, and ultimately predicting,ho\v landscape patterns and distributions, asimportant contributors to land-atmosphereglobal processes, change through a hierarchyof scales. Fractals may also be useful in iden-tifying the break points in a pattern or distri-bution at a particular scale where the processescontributing to these patterns become unstable(i.e., a homogeneous pattern becomes hetero-geneous and vice versa). This is important tothe isolation of selected interactive variables,such as the relationship of precipitation distri-bution with other meteorological and hydrol-ogical variables at a local or regional scale fordevelopment of more reliable hydrometeor-logical models at the global scale. Moreover,it may not be economically or logistically fea-sible to measure the phenomenon at the mostappropriate scale, especially in the case of so-cio-economic data that are already collectedby pre-defined areal units (e.g., census tracts).Knowledge of the appropriate scale for a par-ticular phenomenon or a process will, there-fore, lead to efficient sampling that yields themost information about the phenomena. With-out a clear understanding of the effects of scaleand resolution on various models, and how theresults are translated to other scales, interpre-tations based on the results and predictionscould be invalid. .

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Scale, Resoltttion, and Fractal Analysis 97

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NINA SIU-;..;GAN LA.\I is Associate Professor ofGeography at Louisiana State University, BatonRouge, LA 70803. Her current research interests

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are fractals, scale, and error issues in the mappingsciences and spatial modeling in medical geography.

DALE A. QUATfROCHI is a Geographer withthe National Aeronautics and Space Administra-tion, Science and Technology Laboratory. iocated

at the John C. Stennis Space Center, MS 39529-6000. His research interests focus on the integrationof multiple scaled remote sensing data with GIS,spatial analysis in landscape ecolog}., and thermalremote sensing.