On the Issues of Scale, Resolution, and Fractal Analysis in the M,Apping Sciences*

88 Volume44, Number1, February1992

port of the Committee on Transpiration and tionships between land-surface processes and clio Evaporation, 1943-44. Transactions,American Geo- mate, and the statistical analysis of large-scale clio physicalUnion 2;:638-93. mate fields.

JOHANNES FEDDE1\IA is Assistant Professorof CORT WILLMOTT is Chair of the Department Geography at the University of California, Los An- of Geography and Professorof Geography and Ma- geles. He was formerly a Ph.D. student at the Uni- rine Studies at the University of Dela\vare. He also versity of Dela\vare. His researchinterests include is a member of the University:s Center for Climatic \vater balance climatology and climate/teleconnec- Research. His research interests include the rela- cion processes.

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

Nina Siu-Ngan Lam Dale A. Quattrochi Lo/lisianaState University NASA Scienceand TechnologyLaboratory, StennisSpace 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 to be treated formally and better defined. This paper addresses the 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 thank Grego~. Carter, Lee De Cola, the anonymous revie,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 tCCop.vright 1992 b~. Associationof AmericanGeographers Initial submission,April 1991;final acceptance,August 1991 Scale,Resolution, and Fractal Analysis 89

SCALE Closely related to scale is the concept of SPATIAL~ -SPATIo-~ TEMPORAL~"" -TEMPORAL resolution. Resolution refers to the smallest distinguishaoleparts in an object or a sequence (Tobler 1988),and is often determined by the ~ ~, capability of the instrument or the sampling interval used in a stUdy. A map containing data by county is considered of finer spatial resolution than a map by state. In referenceto GEOGRAPHIC OPERATIONAL CARTOGRAPHIC the three meaningsof scale,small-scale stUdies (or Observation) (or Map) usually employ data of finer spatial resolution Figure 1: Meanings of scale. than large-scalestudies. Small-scale maps, on the contrary, often contain lower or coarser resolution data than large-scalemaps. Finally, gether important points regarding scale and phenomenaoperating at a smaller scale, such resolution in light of recent developments, and as potholes and caves, require data of finer proposes the use of fractals for analyzing spa- resolution. Because large-scale studies (geo- tial phenomena encompassing various scales. graphic or observation scale)involving fine res- With appropriate methodology and sufficient olution are rather uncommon, scale and reso- attention to the problem of scale, we believe lution are often blended together and loosely that the geographical perspective derived from referred to as "scale," although it is recognized analysis using differing scales can contribute that a large-scale stUdy does not necessarily in many \vays to an understanding of various mean that it has a coarser resolution, and vice spatial phenomena. This is particularly im- versa. In this paperthe notion of the resolution portant in relating earth processes or other is generally implied whenever the term scale spatial and environmental phenomena within is used. the context of global dynamics (e.g., the In- ternational Geosphere/Biosphere Program). Controversies over the exact definition and Methodological Issues Related to measurement of scale and resolution exist Scale and Resolution (Woodcock and Strahler 1987). Some clarifi- cation of terminology, therefore, is necessary. Severalmethodological dilemmas surrounding The term scale may include all aspects-spa- the scale and resolution issue can be summa- tial, temporal, or spatio-temporal (Fig. 1). In rized here. First, different spatial processes this paper \ve focus on spatial scale. There are operate at different scales,and thus interpre- at least three meanings of scale. First, the term tations based on data of one scale may not scale may be used to denote the spatial extent apply to another scale (Harvey 1968, 1969; of a study (i.e., geographic scale or scale of Stone 1972).The noted "ecological fallacy" in observation). For example, a spatial analysis spatial analysis,making inferences about phe- of land use across the entire United States is nomenaobserved at differing scalesor imply- considered a large-scale study, as compared to ing finer resolution from coarser resolution a land use plan for a city. This definition of data, atteststo this problem. A simple example scale is quite different from the second use by Robinson (1950) sho\vs that correlations called cartographic scale, where a large-scale between t\vo variables,measured IQ and race, map covers smaller area but generally with decreased from 0.94 to 0.73, and finally to more detail, and a small-scale map covers 0.20, as resolution increased from census re- larger area \\1th less detail. The third usage of gion, to state, and to individual study scales scale refers to the spatial extent at which a (Openshaw 1984). particular phenomenon operates (i.e., opera- Because spatial patterns are usually scale tional scale). For example, mountain-building specific, inferring spatial process from spatial processes operate at a much larger scale than pattern is perplexing; this is illustrated by the that of river pothole formation. These three well-known dilemma of different processes meanings of scale, however, are often mixed leading to the same spatial pattern (Harvey and used vaguely in the literature. 1969;Turner et al. 1989b). Moreover, a spatial

-~ 90 Volume44, Number1, February1992

pattern may look clustered at one scale but and landscape patterns, and to utilize fully random at another. For example,the mortality remote sensingand GIS. pattern of leukemia cancer in China appears In an integrated GIS and remote sensing randomly distributed if based on county-level system, data of various types, scales,and res- data; if the data are aggregatedand reported olutions are used. How will the analytical by province, a clustered geographicalpattern methods and results be affected by different results (Lam et al. 1989).Depending upon the scalesand resolutions?What is the relationship scale of observation, therefore, processesthat betweenscale and accuracy?The selection of appearhomogeneous at a small scale may be- an appropriate scale is also influenced by the comeheterogeneous at a largerscale. This may techniques used to extract information from be exemplified by patterns of coniferous for- remote sensingdata. The factors of scale and ests infested \vith pine bark beetle blight. At resolution, therefore, have become a major re- a small spatial scale, the patterns of infected searchdirection in GIS and remote sensing. individual trees or groups of trees \\.ithin the forest are not evident, becausethe pattern of insect damage becomesintegrated as part of A Brief Description of Fractals the spatially homogeneousconiferous forest. At a large geographic scale, ho\vever,groups Fractals are now widely used for measuring, of trees infected \\.ith pine bark beetle blight as well as simulating, forms and processesand appear as patches of dead trees and can be are attractive as a spatial analytical tool in the easily distinguished from other trees. Thus, mapping sciences. Since Mandelbrot coined the pattern of insect infestation becomeshet- the term in 1975(Mandelbrot 1977),the con- erogeneousat larger scales.How do we kno\\. cept has beenfurther developedand expanded \vhat scaleand resolution we should use?And acrossvirtually every major discipline. Works how do wekno\v \vhen the results are mean- by Mandelbrot (1977, 1983), Peitgen and ingful and valid given the scaleand resolution Saupe(1988), Feder (1988),and more recently of the data? Falconer(1990) articulate some of the thought- Rapid development in the mapping sci- provoking issuesrelated to fractal analysis. An ences, particularly GIS and remote sensing, overview of the use of fractals in geography in the last decade has to a great extent "for- has been presented by Goodchild and Mark malized" the scaleand resolutionproblem. An (1987) and recently by Lam and De Cola immediate question in designing a GIS and (forthcoming). remote sensing systemis: what scaleand res- Fractalswere derived mainly becauseof the olution should \ve use for a specific applica- difficulty in analyzing spatial forms and pro- tion? For example, shall we use a 1:24,000or cessesby classicalgeometry. In classical ge- 1:250,000scale map? Will Landsat TM im- ometry (i.e., Euclidean geometry), the dimen- agery (with a pixel size of 30m)be more ap- sion of a curve is defined as I, a plane as 2, propriate than Landsat MSS imagery (80m and a cube as 3. This is called topological pixel)? Shall \ve sample or update the data dimension(Dr). In fractal geometry,the fractal every five or 10 years?The concern here is dimension D of a curve can be any value be- both theoretical and practical. Increasedres- t\veen 1 and 2, and a surface between 2 and olution will increasedata storageand process- 3, according to the complexity of the curve ing time, \vhereas decreasedresolution leads and surface.Coastlines have fractal dimension to inaccuracy that compounds quickly when values typically around 1.2, and relief dimen- severaltypes of data areoverlaid and analyzed. sions around 2.2. Fractal dimensions of 1.5 What is the optimum resolution or does an and 2.5 for lines and surfaces,respectively, are optimum really exist? Ultimately, the "best" too large (i.e., irregular) for modeling earth resolution depends upon the study objectives, features (Mandelbrot 1983,260 and 264-65). the type of environment, and the kind of in- The key conceptunderlying fractals is self- formation desired. Hence, much more \vork similarity. Many curves and surfacesare self- is required to understand the effects of scale similar either strictly or statistically, meaning in interpreting variability in earth processes that the curve or surfaceis made up of copies Scale, Resolution,and Fractal Analysis 91

(a) m=4.r=1/4 D = -log (4)/log (1/4) = 1

(b)

m=4.r= 1/2 EB D = -log (4)/log (1/2) =:l

Figure 2: Relationships between fractal dimension (0), number of copies (m), and scale factor (r). of itself in a reduced scale. The number of copies (m) and the scale reduction factor (r) can be used to determine the dimensionality of the curve or surface, where D = -log(m)/ log(r) (Falconer 1990).For. example, in Figure 2a, a line segment is made up of four copies of itself, scaled by a factor V4(i.e., one-fourth of the line length), and the dimension D = -log4/log(V4) = 1. A square (Fig. 2b)is made up of four copies of itself and scaledby a factor V2(i.e., half of the side length), thus yielding D = -log4/log(V2)= 2. Similarly, a von Koch Figure 3: (a) Louisiana coastline and its curve is made up of four copies of itself \vith subdivisions; coasts A to F are: Chenier plain, a scaled factor V3and having dimension D = Sale-Cypremort subdelta, Teche and Lafourche -log4/log(V3)= 1.262(Fig. 2c). subdeltas, Plaquemines and Balize subdeltas, Practically, the D value of a curve (e.g., St. Bernard subdelta, and Lake Ponchartrain and coastline)is estimated by measuringthe length Lake Borgne; (b) corresponding fractal plot; the of the curve using various step sizes.The more bars on each curve show the range of points irregular the curve, the greater increase in (i.e., steps) used in the regression and thus length as step size decreases;D can also be define the self-similarity range; the number of calculated for a curve by the regressionequa- steps used for coasts A to F and the overall tion logL = C + BlogG and D = 1 -B, coastline (LA)-are 9,7,5,6,5,6, and 11, where L is the length of the curve, G is the respectively. (Adapted from Diu 1988 and Lam step size, B is the slope of the regression,and and Diu, forthcoming). C is a constant. The 1;)value of a surface can be estimated in a similar fashion and several algorithms for measuring surface dimension is shown in Figure 3 (Lam and Qiu, forthcom- have already beendeveloped (e.g., Shelberget ing; Qiu 1988). Previous stUdies have dem- al. 1983). A scatterplot illustrating the rela- onstrated that fractal plots of empirical curves tionship between step size and line length in and surfacesare seldom linear, with many of logarithmic form (i.e., fractal plot) becomes them demonstrating an obvious break (Butten- the source of the derivation of fractal dimen- field 1989;Goodchild 1980; Mark and Aron- sion. An example of a fractal plot using the son 1984). This indicates that true fractals Louisianacoastline and its correspondingmap with self-similarity at all scalesare uncommon. 92 Volume44, Number1, February1992

The fact that self-similarity exists only over a in the fractal literature, such as the concepts limited range of scalescould be utilized posi- of self-affinity, random fractals, and multi- tively, ho\vever, to summarize scale changes. fractals,have expanded fractal applications to Thus, fractals are potentially useful tools for many phenomena \vhere true fractals with investigating the issuesof scaleand resolution. strict self-similarity do not exist. At the same Based on the self-similarity concept,curves time, fractals have generated criticisms from and surfacesof various diinensionalitiescan be researchersin various disciplines, and indeed generated,and it is the simulation capability there are limitations in effectively applying of fractals that makes this technique a favora- fractals. ble tool for spatial analysis. Moreover,various objects and structures, suchas planets,clouds, oceanfloors, trees, particle growth, and urban morphology can be simulated using fractals (Peitgenand Richter 1986; Peitgenand Saupe Fractals in the Mapping Sciences 1988). Several methods have been developed There is a relative paucity of research that to simulate fractal curves and surfaces,includ- employs fractals in the mapping sciences. ing the shear displacementmethod, the mod- Goodchild (1980)in a pioneer article demon- ified Markov method, the inverse Fourier strated that the fractal dimension can be used transform method, and the recursive subdivi- to predict the effects of cartographic general- sion method (e.g., Carpenter 1981; Dutton izationand spatial sampling, a result that may 1981; Fournier et al. 1982). Figure 4 sho\vs help in determining the appropriate resolution some sample surfaces with different D values of pixels and polygons used in studies related generated using a shear displacement algo- to GIS and remote sensing. Muller (1986)pro- rithm (Goodchild 1980; Lam 1990). posedthe useof fractal dimension as a guiding The foregoing description of fractals is principle for future implementation of gener- rather elementaryand intentionally made sim- alizationalgorithms in automatedcartography. ple. There are many other ways of defining Fractal surfaceshave been used as test data fractal dimensions and hence,the applications sets to examine the performance of various of fractals vary extensively. Recent additions spatial interpolation methods (Lam 1982,

2.2

:1 i ..

"'. ... "'-

0.-0 .Z.O .Z.O

Figure 4: Examples of fractal surfaces generated using a shear displacement algorithm. Source: Lam 1990

c/ Scale,ResolutiOIl, alld Fractal Analysis 93

1983)and to assessthe efficiency of a quadtree tal conditions overextensive areas of the earth. data structure (Mark and Lauzon 1985).Frac- O'Neill et al. (1988)sho\\ed that the fractal tal curve generation, on the other hand, has dimension has a highly significant correlation been used as an interpolation method and as \vith the degreeof human manipulation of the the inverse of curve generalization by adding landscape.Landscapes dominated by agricul- more details to the generalizedcurve (Carpen- ture tend to have simple polygons and low ter 1981; Dutton 1981;Jiang 1984). fractal dimensions (negative correlation), and The use of fractals in remote sensingis rel- landscapesdominated by forest tend to have atively ne\v and seemsto have great potential. complex shapesand high fractal dimensions De Cola (1989)applies fractal analysis to land (positive correlation). Thus, the applicationof coverpatterns derived from Landsat Thematic fractals allo\\.s not only a different and con- Mapper (TM) data and concludes that self- venient \vay of describing spatialpatterns, but similarity, fractal dimension, and Pareto size also the generation of hypotheses about the parameter are useful measures for analyzing causesof the patterns. The abovesuggestions digital images. His results also show that ur- open new directions for the potential applica- ban land cover gives rise to more complicated tion of fractals. spatial patterns than does intensive agricul- ture. In a study comparing urban, rural, and Fractals, Scale, and Resolution coastalareas using Landsat TM imagery, Lam (1990) has demonstrated that different land It is apparent that the fractal modelis a useful cover types have different fractal dimensions tool for simulationand modeling. Fractal anal- in different bands. Urban land cover \vas ysis of spatial forms and processes,however, found to have the highest dimension, followed can be limited by problems at both the theo- closely by coastaland rural land cover types. retical and technical levels (Lam 1990). The In addition to the two types of applications first and foremost problem relates directly to mentioned above, fractal surfaces have been scaleand resolution. The self-similarity prop- suggestedas a null-hypothesis terrain or norm erty underlying the original fractal model as- whereby further simulation of various geo- sumes that the form or pattern of the ~patial morphic processes can be made (Goodchild phenomenonremains unchanged throughout and Mark 1987). Similarly, Loehle (1983), in all scales,implying that one cannot, through a paper examining the applications of fractal fractal analysis,determine the scaleof the spa- concepts in ecology, suggestedusing the frac- tial phenomenon from its form or pattern. tal dimension as a parametersummarizing the This kind of strict self-similarit}' is considered effect of a certain process up to a particular unacceptablein principle, and hence,has gen- scale. He also proposed the use of the self- erated criticism about the spatial application similarity property as a null hypothesis. In of fractals- Empirical studies have shown that other work by Burrough (1981), it was sho\vn most real-world cur\-es and surfacesare not that many environmental variables are fractals pure fractals \\"ith a constantD at all scales. with varying fractal dimensions, and that the Instead, D varies across a range of scales examination of D values would be useful for (Goodchild 1980; Mark and Aronson 1984). separatingscales of variation that might be the These findings, however, can be interpreted result of natural processes. positively. Rather than using D in the strict Furthermore, fractals can help formulate senseas defined by Mandelbrot (1983), it is hypothesesconcerning the spatial scaleof pro- possibleto use the D parameterto summarize cess-pattern interactions (Krummel et al. the scalechanges of the spatial phenomenon. 1987). It was also concluded that fractal tech- This latter use of fractals is supported by the niques should be particularly applicable to results from Burrough's (1981)work, \vhere it analysis of remotely senseddata, since it pro- is suggestedthat ~Iandelbrot'sD value can be vides D as a simple measure that indicatesthe used as an indicator over many scalesfor nat- scale at which processesare occurring (i.e., ural phenomena.Through interpretationof D operational scale). Changes in D computed values, therefore, it may be possibleto relate from remote sensing data, therefore, have pri- or separatescales of variation that might be mary implications for changes in environmen-~ the result of particular natural processes. 94 Volume44, Nttmber1, Febrttary1992

The second problem of applying fractals re- variables influencing the process at different lates to the technical aspect of fractal mea- scales must be understood. Third, the appro- surement, The fact that self-similarity exists priate methods for translating the results from only within certain ranges of scales ~akes it one scale to another must be de\"eloped.Fi- difficult to determine a breaking point for self- nally, the methods and results must be tested similarity; i.e., a point at \\'hich self-similarity acrossscales. ends for a curve or surface (Fig. 3b). This Severalmethods have beenemployed to ex- affects the final D value, \vhich is then used amine issuesrelated to the scaleand resolution to characterizethe curve or surface (Butten- problem, including for example,spectral anal- field 1989). Furthermore, there is some ques- ysis (Moellering and Tobler 1972), moving tion as to whether maps or map products pro- standard deviation measures (Woodcock and vide sufficient detail to accuratelymeasure the Strahler 1987),and variance and spatial cor- fractal dimension (Carstensen1989). The cal- relation measures(Carlile et al. 1989). With culation of the fractal dimension for surfaces consideration of the conceptual and technical presents an added technical problem. Roy et uncertainties associated\\.ith their use,fractals al. (1987)have shown that for the same sur- could contribute to all four stepsnoted above. face, different D values could result from us- Fractals could be used to identify the spatial ing different algorithms, with a range as low and temporal scale of the process.The fractal as 2.01 to a high of 2.33. Finally, all the ex- plot of step size against line length or area isting methods have so far been applied only measurement can be used to determine the to regular grid data, such as digital terrain dimension valuesand their correspondingself- modeldata or Landsat TM data. Fractal mea- similarity ranges.These values can be utilized surement of many other socio-economicphe- to help identify the underlying processescre- nomena, such as population and disease dis- ating the patterns. Since most natural curves tributions, or environmental process-response are not strictly self-similar, the derivations of phenomena,presents another challenge. For the fractal dimension and associatedself-sim- example,identification of the patterns of pine ilarity range are often basedonly on the range bark beetle infestation\vithin a forest may re- of points sho\ving linearity (Fig. 3b). veal insight into possible causes-responsesof For example, a study of the Louisiana coast- forestlandscape dynamics asaffected by insect line by Qiu (1988) has demonstrated that infestation. These data are typically reported coasts of different origins and development in an aggregatepolygonal form, with irregular stages are distinguishable by their fractal di- boundariesand possibly holes (in the forms of mension values and associatedself-similarity lakesor islands)or missing data. The existing ranges (Lam and Qiu, forthcoming). Coasts algorithms used to delineate such features will dominated by deltaic processesgenerally have have to be modified and extra steps will have higher fractal dimensions (about 1.3) with a to be taken before actual measurementtakes narro\ver scale range of 0.~12.8 km, \vhile place (Krummel et al. 1987). coasts dominated by marine processeshave lo\ver dimensions (about 1.1) and a wider self- similarity range of 0.2-51.2 km (Fig. 3a and A Research Agenda b). In analyzing China's cancer mortality pat- As mentioned previously, the basic question terns using data by commune for the Taihu surrounding the issuesof scaleand resolution Region, Lam and Qiu (1990) have demon- is: can a study predicated on or founded at strated that distinct self-similarity ranges exist one scale be used to make inferences to the in the three leading cancer mortality patterns samephenomena under observation at differ- in China (stomach, esophagus,and liver). ent scales?To ans\ver this question, several Fractal dimensions and self-similar ranges for interrelated steps are involved (Turner et al. these cancer surfaces were computed by the 1989a).First, the spatial and temporal scaleof variogram method (Mark and Aronson 1984). the process must be identified. Second, the Figure 5 is the variogram plot in logarithmic importance or changes in importance of the form illustrating the relationship be~veendis- Scale,Resoltttion, and Fractal Analysis 95

scalesand so does the importance of controlling factors. Cancermortality stUdiescan again provide an examplein this context (Lam et al. 1989). It is possible that, as a first step, the broad environmentalcontext fo,certain cancer types 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, finer resolution study investigating local factors (e.g., industrial locations, \vater supply). An- other example that requires data of various scalesis in veterinary medicine and parasitol- ogy. Coarse resolution satellite data, such as those obtained from the AVHRR (Advanced Very High Resolution Radiometer) with an averagepixel resolutionof 1 km, could be used to monitor and identify broad areas of large animal parasitehabitats (e.g., snails, mosqui- tos). The broad areasof habitat could then be refined using finer resolution remote sensing tance and variance (or squared difference) for data, such asairborne scannerdata with pixel all three cancer patterns. Fractal dimensions resolutionof five metersor less. Fractals,along of these patterns can be determined by the with other statisticalmeasures, can be used to equation D = 3 -(B/2), where B is the slope determine the complexity and spatial cluster- of the individual regressionline through the ing of these patterns,thus providing clues on respective points in the variogram. In this the existenceof certain factors related to the study the mortality pattern of liver cancerwas distribution of parasitehabitatS. found to have the highest dimension (D = The capability of fractal models in simula- 2.86), followed by stomach (D = 2.76) and tion, or rather, interpolation or extrapolation, esophageal(D = 2.71). The linearity exhibited providesanother method for translating results on the variogram plot for all three patterns acrossscales. If the phenomenon of interest indicates that self-similarity exists within cer- displays a self-similarity property with a rel- tain scale ranges. The patterns behave differ- atively stable dimensionality, one can invert ently, however, when reaching a certain scale the processand re-generatethe pattern based or in this case,a distance range (i.e., the break on the fractal dimension value. The issue of point in the variogram plot). Esophagealand line generalizationand line generation, which liver cancers have similar distance limits of has been a subjectof intense researchin ana- about 5-150 km, while the limit is approxi- lytical cartography, is basically an issue of mately 5-90 km for stomachcancer. This sug- ttanslating results across scales. Fractals, al- gests that if the communes are aggregatedto though with relatively less success, have a distance range as large as the limit, the pat- played an important role in this realm of anal- terns would look very different and would ysis (e.g., Dutton 1981;Jiang 1984; Muller result in a different interpretation of the pro- 1986).More stUdiesin this area, however, are cessesunderlying the patterns as defined by needed. that levelof scaleand resolution. Furthermore, Finally, fractal analysis can be used as a it is suspected that whatever the underlying meansto testthe resultsacross multiple scales. controlling factors (e.g., climate, topography, In addition to the cancer stUdies mentioned pollution), they are likely to operate in scale earlier, researchis now underway in which ranges similar to those of the cancer mortali- digital remote sensingimages of different res- ties. olutions are being tested and examined by Different processes operate at different meansof fractals (Quattrochi and Lam 1991). 96 Volume44, Number1, February1992

The results will provide insights \vith regard Literature Cited to scale, resolution, accuracy, and landscape representation.Fractals may offer a usefulper- Burrough, P. A. 1981. Fractal dimension of land- spective on predicting spatial, environmental, scapes and other environmental data. Nature and ecological system dynamics by revealing 294(19):240-42. Buttenfield, B. 1989. Scale-dependence and self- the self-similarity of phenomena at different similarity in cartographic lines. Cartographica scales,in rum, providing better models in a 26(1):79-100. hierarchicalframe\vork for the analysisof scale Carlile, D. W., J. R. Skalski, J. E. Batker, J. M. (O'Neill et al. 1989). Thomas, and V. I. Cullinan. 1989. Determination The issues of scale and resolution are no\\" of ecological scale. LandscapeEcology 2(4):203-13. emergingas important aspectsof the mapping Carstensen, L. W. Jr. 1989. A fractal analysis of sciencesthat will have a significant place in cartographic generalization. The American Cartog- global change and global modeling studies rapher 16(3): 181-89. (NASA 1988;Wickland 1989).It is recognized Carpenter, L. C. 1981. Computer rendering of fractal curves and surfaces. Proceedings of ACM SIG- that more than a single scale of observation GRAPH Conference, Seattle, Washington, July, may exist, thereby necessitatingmeasurement 1980. at severallevels of resolution. One of the major De Cola, L. 1989. Fractal analysis of a classified analytical challenges in studying global pro- Landsat scene. Photogrammetric Engineering and Re- cessesis to develop measurementsampling mote Sensing55(;):601-10. techniques that \vill elucidate how these pro- Dutton, G. H. 1981. Fractal enhancement of car- cessesrange over broad spatial and temporal tographic line detail. The American Cartographer scales. Fractal analysis may become a major 8:23-40. tool for measuring,and ultimately predicting, Falconer, K. 1990. Fractal Geometry-Mathematical Foundations and Applications. Ne\v York: John ho\v landscapepatterns and distributions, as Wiley and Sons. important contributors to land-atmosphere Feder, J. 1988. Fractals. New York: Plenum Press. global processes,change through a hierarchy Fournier, A., D. Fussell, and L. C. Carpenter. of scales.Fractals may also be useful in iden- 1982. Computer rendering of stochastic models. tifying the break points in a pattern or distri- Communications, Associationfor Computing Machin- bution at a particular scalewhere the processes ery 25:371-84. contributing to thesepatterns becomeunstable Goodchild, M. F. 1980. Fractals and the accuracy (i.e., a homogeneouspattern becomeshetero- of geographical measures. ""fathematical Geology geneousand vice versa). This is important to 12:85-98. the isolation of selected interactive variables, Goodchild, M. F., and D. M. Mark. 1987. The fractal nature of geographic phenomena. Annals oj suchas the relationship of precipitation distri- the Association of American Geographers77(2):265- bution with other meteorologicaland hydrol- 78. ogical variables at a local or regional scale for Harvey, D. 1968. Processes, patterns and scale development of more reliable hydrometeor- problems in geographical research. Transactions of logical models at the global scale. Moreover, the Institute of British Geographers4;:71-78. it may not be economically or logistically fea- Harvey, D. 1969. Explanation in Geography. New sible to measurethe phenomenonat the most York: St. Martin's Press. appropriate scale, especially in the case of so- Jiang, R-S. 1984. An evaluation of three curve gen- cio-economic data that are already collected eration methods. "'I. S. Thesis. Columbus, OH: Department of Geography, Ohio State Univer- by pre-defined arealunits (e.g., censustracts). sity. Knowledge of the appropriate scale for a par- Krummel, J. R., R. H. Gardner,G. Sugihara, R. ticular phenomenon or a process will, there- V. O'Neill, and P. R. Coleman. 1987. Landscape fore, lead to efficient sampling that yields the pattern in a disturbed environment. Oikos 48:321- most information about the phenomena.With- 24. out a clearunderstanding of the effects of scale Lam, N. S-N. 1982. An evaluation of areal inter- and resolution on various models,and how the polation methods. Proceedings,Fifth International results are translated to other scales,interpre- Symposium on Computer-AssistedCartography (Auto- Carto 5) 2:471-79. tations based on the results and predictions Lam, N. S-N. 1983. Spatial interpolation methods: could be invalid.. A review. The American Cartographer 10: 129-49. Scale, Resoltttion,and Fractal Analysis 97

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II, Washington, sensingin terrestrialecological research. In Theory D.C. and Applicationsin OpticalRemote Sensing, ed. G. O'Neill, R. V., A. R. Johnson, and A. W. King. Asrar, 691-724. New York: John Witey and Sons. 1989. A hierarchical frame\\.ork for the analysis Woodcock,C. E. and A. H. Strahler. 1987. The of scale. LandscapeEcology 3(3/4):194-20;. factor of scalein remotesensing. Remote Sensing of O'Neill, R. V., J. R. Krummel, R. H. Gardner, G. Environment21:311-32. Sugihara, B. Jackson, D. L. DeAngelis, B. T. J..lilne,M. G. Turner, B. Zygmunt, S. W. Chris- tensen, V. H. Dale, and R. L. Graham. 1988. Indices of landscape pattern. LandscapeEcology 1(3):153-62. Openshaw, S. 1984. The Modifiable Areal Unit NINA SIU-;..;GAN LA.\I is AssociateProfessor of Problem. CAT,j,IOGSeries 38. Norwich, England: Geography at Louisiana State University, Baton Geo Abstracts. Rouge, LA 70803. Her current research interests 98 Volume44, Number1, February1992 are fractals, scale, and error issuesin the mapping at the John C. Stennis Space Center, MS 39529- sciencesand spatial modeling in medicalgeography. 6000. His researchinterests focus on the integration DALE A. QUATfROCHI is a Geographer with of multiple scaled remote sensing data with GIS, the National Aeronautics and Space Administra- spatial analysis in landscape ecolog}., and thermal tion, Scienceand Technology Laboratory. iocated remote sensing.