Computer-Assisted Map Analysis: Potential andPitfallsJoseph K. BerrySchool of Forestry and Environmental Studies, Yale University, New Haven, CT 06511

ABSTRACf: Geographic information systems (GIS) are profoundly changing the ways in which we use maps. The per­spectiveof maps as Images of the phySical arrangement of features is being extended to one of mapped data expressingthe spatial relationships of complex physical and conceptual systems. Within this context, a mathematical structure forcartographic modeling can be identified. The quantitative treatment of geographic space has analogies in traditionalstatistics and alg~bra ..Quanhtative approaches to digital map processing are described and potentials and pitfalls ofmap analySIS are Identified. Digital map analysiS provides a means for evaluation of error introduced by both processingand modeling considerations.

INTRODUCTION

H ISTORICALLY, maps have been used for navigation throughunfamiliar terrain and seas. Within this context, prepara­

tion of maps accurately locating geographic features became theprimary focus of attention. More recently, analysis of mappeddata for decision-making has become an important part of re­source and land planning. During the 1960s, manual analyticprocedures for overlaying maps were popularized (McHarg,1969). These techniques mark an important turning point in theuse of maps-from one emphaSizing physical descriptors ofgeographic space, to one spatially characterizing appropriatema.na!Sement a~tions. This movement from descriptive to pre­scnpttve mappmg has set the stage for revolutionary conceptsof map structure, content, and use. Geographic informationsystems (GIS) technology provides a means for quantitative:no?e~ing of spatial relationships. In one sense, this technologyIS SimIlar to conventional map processing involving map sheetsand drafting aids such as pens, rub-on shading, rulers, plani­meters, dot grids, and acetate sheets for light-table overlays. Inanother sense, these systems provide an analytic "tool­box,"enabling managers to address complex issues in entirelynew ways (Dangermond, 1986).

Spatial analysis involves tremendous volumes of data. Man­ual cartographic techniques allow manipulation of these data;however, they are fundamentally limited by their non-digitalnature. Traditional statistics enable numerical analysis of thed~t~,. but the sh.e~r magnitude of mapped data becomes pro­hibitive. Recogmtion of these limitations led to "stratified sam­pling" techniques developed in the early part of this century.These techniques treat spatial considerations at the onset ofanalysis by dividing geographic space into homogenous re­sponse parcels. Most often, these parcels are manually deline­ated on an appropriate map, and the "typical" value for eachparcel is determined. The results of any analysis is assumed tobe uniformly distributed throughout each parcel. The area­weighted average of the parcels statistically characterizes thetypical response for an entire study area. Mathematical mod­eling of spatial systems has followed a similar approach of spa­tially aggregating variation in model variables. Most ecosystemmodels, for example, identify "level" and "flow" variables pre­sumed to be typical for vast geographic expanses.

The fundamental concepts of comprehensive spatial statisticsand mathematics have been described for many years by boththeorist and practitioner (Shelton and Estes, 1981; Tomlin andBerry, 1979; Lewis, 1977; Steinitz et aI., 1976; Brown, 1949). Therepresentation of spatial data as a statistical surface is indicatedm most cartograpny texts (Cuff and Matson, 1982; Robinson etaI., 1978). A host of quantitative and analytical techniques arepresent in manual cartography. Muehrcke and Muehrcke (1980),in an introductory text, outline "cartometric" techniques allow­ing quantitative assessments of spatial phenomena.

PHOTOGRAMMETRlC ENGINEERING AND REMOTE SENSINGVol. 53, No. 10, October 1987, pp. 1405-1410. '

Until modern computers and the digital map, many of theseconcepts were limited in their practical implementation. Thecomputer has provided the means for both efficient handlingof voluminous data and the numerical analysis capabilities thatare required. From this perspective, the current revolution inmapping is rooted in the digital nature of the computerizedmap. The two contemporary fields of remote sensing and geo­graphic information systems have greatly advanced this mapform. Remotely sensed data, and their derived classificationproducts, have contributed to acquisition and use of digital maps.In fact, some of the early development of GIS technology in­volved the merging of "ancillary" data, such as elevation, withspectral data to improve classification. This early merger of re­mote sensing and GIS focused on the preparation of traditionalmap products. Increasingly, the merger is being viewed as anintegral system providing capabilities for both rapid preparationof mapped data and spatially consistent analysis necessary toconvert physical maps into decision-making terms.

GEOGRAPHIC INFORMATION SYSTEMS

The main purpose of a geographic information system is toprocess spatial information. These systems can be defined asinternally referenced, automated, spatial information systemsdesigned for data management, mapping, and analysis (Figure1). Within this classification scheme, the first distinction is be­tween administrative information (e.g., payroll, personnel, oroffice management) and spatial information characterizing geo­graphic factors, such as natural resources. Spatial informationcan be subdivided into that which is spatially aggregated (e.g.,soils classification schemes and timber yield tables) and thatwhich is geographically referenced. This latter class of data istermed spatially coherent, and "maps" patterns or distributionsof variables (e.g., soils, forest cover, and elevation maps). Thesemapped data can be managed by non-automated or automatedmeans. The spatial reference may be external or internal to anautomated spatial information system. An externally referencedsystem uses a computer to store information on various geo­graphic units; however, geographic coordinates are not storedwithin the system. Delineations on map sheets or aerial pho­tographs provide geographic reference.

Internally referenced schemes identify a true GIS. These sys­tems have an automated linkage between the data (or thematicattribute) and the whereabouts (or locational attribute) of thosedata. The locational information may be organized as a collec­tion of line segments identifying the boundaries of point, linear,and areal features. An alternative organization, similar to re­mote sensing data, establishes an imaginary grid pattern overa study area, then stores values identifying the characteristic ateach grid space. Although there are Significant practical differ­ences in these data structures, the primary theoretical difference

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1406 PHOTOGRAMMETRlC ENGINEERING & REMOTE SENSING, 1987

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"' SPATIAL'\:--Aggregated

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~ExternallyReferenced

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FIG. 1. Classification scheme for spatial information systems. Geographic informationsystems (GIS) can be defined as internally referenced, automated, spatial in­formation systems designed for data management, mapping, and analysis.

is that the grid structure stores information on the interior ofareal features and implies boundaries whereas the line structurestores information about boundaries and implies interiors. Mostmodern systems contain programs for converting between thetwo data structures. Generally, the line segment structure isbest for inventory-oriented systems. The difficulty of these sys­tems to characterize spatial gradients and the necessity to com­pute interior characteristics preclude many of the advancedanalytic operations and data characterizations. These differ­ences affect both the potential and pitfalls of computer-assistedmap analysis.

Regardless of the data storage structure, a GIS contains hard­ware and software for data input, storage, processing, and dis­play of digital maps. The processing functions may be groupedinto four categories: computer mapping, spatial data base man­agement, spatial statistics, and cartographic modeling. Com­puter mapping, also termed automated cartography, involvesthe preparation of map products. The focus of these operationsis on the input and display of computerized maps. Spatial data­base management, on the other hand, focuses on the storagecomponent of GIS technology. Like non-spatial data-base sys­tems, these procedures efficiently organize and search large setsof data for frequency statistics and/or coincidence among vari­ables. For example, a spatial data base may be searched to iden­tify and generate a map of areas of silty-loam soil, moderateslope, and ponderosa pine forest cover. The mapping and data­base capabilities have proven to be the backbone of current GISapplications. Once a data base is compiled, they allow rapidupdating and examination of mapped data. Aside from the sig­nificant advantages of processing speed and the ability to han­dle tremendous volumes of data, such uses are similar to thoseof manual techniques. Here is where the parallels to mathe­matics and traditional statistics may be drawn. These concep­tual similarities provide a generalized framework, or "map­ematics," for discussing the various data types and advancedanalytic operations of GIS technology.

SPATIAL STATISTICS

The dominant feature of GIS technology is that spatial infor­mation is represented numerically rather than in an analog fash­ion as inked lines on a map. Because of the analog nature ofmap sheets, manual analytic techniques are limited in theirquantitative processing. Digital representation, on the other hand,makes a wealth of quantitative as well as qualitative processingpossible. Spatial statistics seek to characterize the geographicdistribution, or pattern, of mapped data. They describe the spa­tial variation in the data, rather than assuming a typical re­sponse to be uniformally distributed in space.

Examples of spatial statistical analysis are shown in Figures2 and 3. Figure 2 depicts density mapping of animal activityderived from typical field data. The grid on the upper left is amap of sample locations with 16 sensors positioned at 1000metre intersections. The tabular data identify both the location

and the number of animal encounters at each sensor for twotime periods.

Traditional statistical analysis involves fitting a numerical dis­tribution to the data to determine the typical response. Suchdensity functions as standard normal, binomial, and Poissoncould be tried, and the best fitting functional form chosen. Thestatistics at the bottom of the table note the average activity as18.75 (± 11.9) in period 1 and 22.56 (± 26.2) in period 2. Theseparameters describe the central tendency of the data in numer­ical data space and are assumed uniformally distributed in geo­graphic space. Inset (a) plots the period 2 data using a hectaregridding resolution. The X and Y axes identify geographic po­sitioning of the sensors, while the Z axis indicates measuredanimal activity. The horizontal plane at 22.56 depicts the aver­age activity. The geographic distribution of the standard devia­tion would form two planes (analogous to "error bars") aboveand below the average plane. This traditional approach concen­trates on characterizing the typical thematic response, and dis­regards the locational information in the sampled data.

Spatial statistics, on the other hand, incorporate locationalinformation in mapping the variation of thematic values. Anal­ogous to traditional statistics, a density function is fitted to thedata. In this instance, the distribution is characterized in geo­graphic space rather than numerical space. Inset (b) of Figure2 shows a continuous surface implied by the sampled data. Thisnonplaner surface was fitted by "inverse distance squaredweighted averaging" of the sample values. Other surfaoe fittingtechniques, such as Kriging, spline, or polynomial fur.ctions,could be tried and the best fitting functional form chosen. A.nalysisof the residuals, from comparisons of the various surfaces withthe measured values, provides an assessment of fit. Davis (1973)and Ripley (1981) provide good treatise of these techniques.Inset (c) shows the "masking" of the spatial distribution ac­counting for the fact that the animal never enters water (lakein the northwest portion). The final distribution shows consid­erable deviation from the average plane implied by the tradi­tional statistical analysis. Activity appears to be concentrated inthe eastern portion, tending to conflict with the assumption thatvariation is uniformally distributed in geographic space.

Figure 3 depicts the natural extension to multivariate statisticsfor spatial coincidence between the two periods. In traditionalstatistics this involves fitting of a density function in multivar­iate statistical space as shown in the upper left portion of Figure3. For the example, a standard normal surface was fitted, withits "joint mean" and "covariance matrix" describing the typicalpaired occurance. Generally speaking, there is an inCrease inactivity (18.75 versus 22.56) with a slight positive correlation(0.42) between the periods. The joint mean is assumed to beuniformally distributed in geographic space. The covariancematrix indicates the level of variation associated with Ithis as­sumption; however, it does not indicate where the variationfrom the joint mean is most likely to occur. The right portionof Figure 3 depicts a spatial analysis between the two data sets.The two spatial distributions of implied activity (inset (a)) are

COMPUTER-ASSISTED MAP ANALYSIS 1407

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FIG. 2. Spatially characterizing data variation. Whereas traditional statistics identifies the typical response and assumes this estimate to beuniformally distributed in space, spatial statistics seeks to characterize the geographic distribution, or pattern, of mapped data.

(new price - old price) * 100old price

percent change

could be used to calculate the mark-up in the price of a shirt.Or the same equation could be used to calculate the percent

among and within maps create new operators, such as prox­imity, spatial coincidence, and optimal paths. These operatorscan be accessed through general purpose map analysis pack­ages, similar to the numerous matrix algebra packages. Thewidely distributed Map Analysis Package (MAP) (Tomlin, 1983b)is an example. A similar system, the Professional Map AnalysisPackage (pMAP) (SIS, 1986), recently was released for personalcomputers. All of the processing presented in this paper wasdone with the pMAP system in a standard IBM PC environment.

The logical sequencing of map processing involves

• retrieval of one or more maps from the database;• processing those data as specified by the user;• creation of a new map containing the processing results; and• storage of the new map for subsequent processing.

This cyclical processing provides an extremely flexible structuresimilar to "evaluating nested parentheticals" in traditional al­gebra. Within this structure a mathematician first defines thevalues for each dependent variable and then solves the equationby performing the primitive mathematical operations on thosenumbers in the order prescribed by the equation. For example,the equation

)(, = 22.56

5, =t 26.2

compared (that is, one subtracted from the other) to generate acoincidence surface. A planimetric map of the surface, regis­tered to the original map of the study area, is shown in inset(b). The darkened area locates large increases in activity (thatis, more than the average difference plus the average standarddeviation) between the two periods.

CARTOGRAPHIC MODELING

Just as a spatial statistics may be identified, a spatial mathe­matics is emerging. This approach uses sequential processingof mathematical primitives to perform a wide variety of complexmap analyses (Berry, 1987; Berry and Reed, 1987; Tomlin, 1983a).By controlling the order in which the operations are executedand using a common database to store the intermediate results,a mathematical-like processing structure is developed. The "mapalgebra" is similar to traditional algebra where primitive oper­ations, such as add, subtract, or exponentiate, are logically se­quenced for specified variables to form equations. However, inmap algebra the variables are entire maps represented as or­ganized sets of numbers. Most of the traditional mathematicalcapabilities, plus an extensive set of advanced map processingprimitives, comprise this map analysis toolbox. As in matrixalgebra (a mathematics operating on groups of numbers defin­ing variables), new primitives emerge that are based on thenature of the data. Matrix algebra's transpose, inverse, and dia­gonalization are examples of these new primitives. Within mapanalysis, the spatial coincidence and juxtapositioning of values

1408 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1987

2

Difference (P2- PI)

lines. The model uses the distributions presented in Figure 3and the model described above to create a map of the percentchange in animal activity (inset(a)). Derivation of the humanactivity map uses proximity to roads and housing density asindicators of activity. Locations with numerous houses and closeto roads are assigned high levels, whereas remote areas areassigned low levels. A tabular summary of the coincidente be­tween changes in animal activity and human activity is shownin inset (b). The spatial analysis indicates that areas of low hu­man activity are strongly related to areas of high increases inanimal activity (fourth line of the table noting 87.4 percknt ofall high animal activity is spatially coincident with areas of lowhuman activity). This same information could be produced asa map isolating those areas.

The development of a generalized analytic structure for mapprocessing is similar to those of many other nonspatial systems.For example, the popular dBASE III package contains less than20 analytic operations, yet may be used to create models forsuch diverse applications as address lists, payroll, inventorycontrol, or commitment accounting. Once developed, these log­ical dBASE sentences are "fixed" into menus for easy end-userapplications. A flexible analytic structure provides for dynamicsimulation as well as data base management. For example, theLotus spreadsheet package allows users to define interrelation­ships among variables. A financial model of a company's pro­duction process may be established by specifying a logicalsequence of interrelationships and variables. By changing spe­cific values of the model, the impact of several fiscal scenarioscan be simulated. The advent of data base management and

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change in value of a parcel of land. In a similar manner, a mapof percent change in land values for an entire area can be ex­pressed in pMAP command language as

COMPUTE NEWVALUE. MAP MINUS OLDVALUE. MAPTIMES 100 DIVIDEDBY OLDVALUE.MAP

FOR PERCENTCHANGE.MAP

Within this model, data for current and past land values arecollected and encoded as computerized maps. These data areevaluated, as shown above, to form a solution map of percentchange. The simple model might be extended to provide coin­cidence statistics, such as

FIG. 3. Assessing coincidence among mapped data. Maps characterizing spatial variation among two or more variables can be compared andlocations of unusual coincidence can be identified.

CROSSTABULATE ZONING.MAP WITH PERCENTCHANGE.MAP

for a table summarizing the spatial relationship between thetype of zoning and the change in land value (that is, whichzones have experienced the greatest decline in market value; orwhich, the greatest increase?) The model might be further ex­tended to include geographic searches, such as

RENUMBER PERCENTCHANGE.MAPASSIGNING 0 TO -20 THRU 20

FOR BIGINCREASES.MAP

for a map isolating those areas experiencing large changes inland value (exceeding ± 20 percent).

Another simple model is outlined in Figure 4. It consists ofless than ten sentences, similar to those above. In the flowchartin the upper left portion of Figure 4, encoded and derived mapsare shown as boxes, with processing operations indicated as

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5 131 MODERATE 1 218 DECREASE 83 13.28 63.4 38.15 131 MODERATE 5 200 MODERATE 38 6.08 29.0 19.05 131 MODERATE 10 207 HIGH 10 1.60 7.6 4.8

10 84 HIGH 1 218 DECREASE 51 8.16 60.7 23.410 84 HIGH 5 200 MODERATE 17 2.72 20.2 8.510 84 HIGH 10 207 HIGH 16 2.56 19.1 7.7

FIG. 4. A simple cartographic model. The analysis derives maps of animal and human activity and then generates a summary of theirspatial coincidence.

spreadsheet packages have revolutionized the handling of non­spatial data. The potential of computer-assisted map analysispromises a similar revolution for spatial data processing.

PITFALLS

Three major pitfalls have affected this revolution: data avail­ability, data characterization, and modeling uncertainty. The"storehouse" of spatial data is analog and the conversion todigital maps has been a bottleneck in most GIS applications.Digital remote sensing data are the rare exception. Early use ofGIS technology required users to develop there own data basesat large expenditures of time and money. These individual ef­forts also proliferated radically different data structures andprocessing systems. More recently, standardized data bases arebecoming available (Geological Survey, 1983; Glen, 1982). Also,the storage requirements of these data are massive. A completedigital data base containing all 54,000 7.5 minute quadranglemaps covering the lower 49 U.S. states would require 10 14 bitsof data for a ground resolution of 1.7 metres (Light, 1986). Withcurrent technology, all those data could be stored on 4000 op­tical disks-smaller than a phonograph record library at a localradio station. The storage requirements at a hectare groundresolution for a similar spatial data base for the entire continentof Africa could be stored on five optical disks. Both the com­mitment to the task and advances in computer hardware areactively addressing the data availability pitfall.

Data characterization issues are not so well defined. Dataexchange formats and geographic considerations, such as pro­jections and resolution, are being addressed by several nationaland international committees, such as the U.S. Geological Sur-

vey led National Committee for Cartographic Data Standards(TeSelle, 1985). However, appropriate characterization of the­matic values has received less attention. Here too, digital re­mote sensing data are the rare exception. These data are truestatistical surfaces. First, a sensor samples the spectral responseby integrating the signals over an entire ground resolution ele­ment. The area-weighted data are then classified using statis­tical procedures to identify the most probable class at eachlocation. At this juncture, both the thematic value and the "con­fidence" of classification are known. Most classification prod­ucts contain maps of the assigned classes and an error matrixsummarizing classification accuracy over an entire area. As intraditional statistics, the error is assumed to be uniformally dis­tributed in space. However, the nature of the data and theprocessing procedure provides information on the nonplanerdistribution. It is this character that makes digital maps fullyquantitative mapped data, and distinguishes mapping from digitalmap analysis.

For example, consider a simple geographic search for loca­tions of ponderosa pine on cohassett soil. In overlaying thesemaps, it is assumed that the joint probability of coincidence isthe same at all locations. Or more likely, error propagation isignored and the results assumed inviolate. For inventory-ori­ented applications, these assumptions may be acceptable.Analysis-oriented applications, on the other hand, have fargreater demands. Suppose the composite cover/soil map is usedin a mathematical equation with other map variables derivinghabitat suitability indicies for wildlife. The accuracy of the pre­dictions are, in large part, a function of the thematic accuracyof the input maps. Consider further that the habitat map, inturn, becomes input to a management model weighing timber,

1410 PHOTOGRAMMETRIC ENGI EERI G & REMOTE SENSING, 1987

water, recreation, and other uses of the land. Data exchangeand geographic registration requirements may be met and thehabitat maps passed to the other model. A spatial characteri­zation of the accuracy of each variable is not part of the data.The old adage of "garbage in, garbage out" at first appearsappropriate. However, if a "shadow map" of thematic error isgenerated at each step of a cartographic model, the "garbage"can be spatially separated. Those locations that are confidentlyderived should be differentiated from those predictions for lo­cations less accurately defined. The pitfall of data characteri­zation requires a complete restructuring of maps into a fullyquantitative form.

Even if spatial data are readily available and in an appropriatequantitative form, the results of map analysis may still be ques­tionable. Many spatial relationships have not been adequatelyquantified. The modeling of uncertain relationships, even withthe best data, introduces ill-defined error. Many GIS applica­tions evaluate subjective considerations with minimal empiricaltesting. In part, this is the result of the tool getting ahead ofknowledge. Most contemporary research considers space in ag­gregate terms using traditional statistical procedures. Until re­cently, both the means and the concepts for rigorous spatialinvestigations were lacking. Even if relationships were quanti­fied, practical implementation of that theory did not have anoutlet. GIS technology is both the salvation and the source ofthis pitfall. It can be instrumental in managing and processingthe massive amounts of data involved in developing a spatiallyoriented body of knowledge. In the interim, however, it pro­vides a vehicle for generating seemingly accurate maps express­ing complex systems with minimal understanding of the actualspatial relationships.

The use of dynamic simulation may assist in modeling suchuncertain situations. For example, consider a spreadsheet modelof a company's production process. Best estimate for each var­iable, parameter, and functional form are entered and resultsare generated. However, the analysis does not stop here. Whatif raw material costs double; or half; or a faster machine is pur­chased? Are the results significantly changed? In a similar fash­ion, the definitions of a cartographic model can be altered. Whatif slope is twice as important; or half; or visual exposure isincreased to a kilometre? Where does the map of a powerlinecorridor significantly change? Within this context, the individ­ual maps produced are no longer the focus. By noting how oftenand where this change occurs as successive runs are made,information on the sensitivity of a specific area to a particularmanagement action is described. Within this context, maps aretransformed from images to spatial information, becoming anactive and integral part of the decision process.

CONCLUDING REMARKS

Geographic information systems technology is revolutioniz­ing how we handle maps. Since the 1970s an ever-increasingportion of mapped data is being collected and processed indigital format. Currently, this processing emphasizes computermapping and data-base management. These techniques allowusers to quickly update maps, generate descriptive statistics,make geographic searches for areas of specified coincidence,and display the results as a variety of colorful and useful prod­ucts. Newly developing capabilities extend this revolution tohow we analyze mapped data. These operations provide a meansfor expressing the spatial interrelationships of maps. Analogousto traditional statistics and mathematics, primitive operationsare logically sequenced on variables to form cartographic models;however, the variables are represented as entire maps. Such aquantitative approach is changing basic concepts of map struc­ture, content, and use.

Several pitfalls affect the practical implementation of rigorousmap analysis. Data availability is being addressed. Standardsfor data exchange and geographic attributes are being estab­lished. However, effective procedures to spatially characterize

map variance and model uncertainty have yet to be developed.Digital map analysis provides a means for evaluation of errorintroduced by both processing and modeling considerations.Spatial statistics is an important component in researching spa­tial distributions and their interrelationships. Simulation o~ car­tographic models can assist in assessing the effects of imperfectlydefined relationships. From this perspective, maps move fromimages describing the location of features to mapped informa­tion quantifying a phYSical or abstract system in prescriptiveterms. This radical change in our use of maps has the pot~ntial

for more complete integration of spatial information intQ thedecision-making process. I

REFERENCES

Berry, J. K., 1987. Fundamental Operations in Computer-Assisted MapAnalysis, International Journal of Geographic Information Systems; Tay-lor & Francis, Ltd., Basingstoke, England (in press). I

Berry, J. K., and K. L. Reed, 1987. Computer-Assisted Map An1lysis:A set of Primitive Operators for a Flexible Approach, Technicfl Pa­pers of the 53th Annual Convention, American Society for Photogram­metry and Remote Sensing, Falls Church, Virginia. Vol. 1, pp. 206­218.

Brown, L. A., 1949. The Story of Maps, Little, Brown and Co., Boston,Massachusetts.

Cuff, D. J., and M. T. Matson, 1982. Thematic Maps, Methuen, INewYork, New York.

Dangermond, J., 1986. The Software Toolbox Approach to Meeting theUser's Needs for GIS Analysis, Proceedings of Geographic InforrrationSystems Workshop, Atlanta, Georgia, American Society for Photo­grammetry and Remote Sensing, Falls Church, Virginia. pp. 66­75.

Davis, John c., 1973. Statistics and Data Analysis in Geology, John Wileyand Sons, Inc., New York, pp. 170-411.

Geological Survey, 1983. USGS Digital Cartographic Data Standardi' Ov­erview and USGS Activities, U.S. Geological Survey Circular 8 5A.

Glen, R., 1982. SCS Geographical Information Format-Standard V rsion,U.S. Soil Conservation Service, CGIS Doc. Number 2.

Lewis, P., 1977. Maps and Statistics, Halsted Press, Cambridge, England.Light, D. L., 1986. Mass Storage Estimates for the Digital Mapping Era,

Photogrammetric Engineering and Remote Sensing, Vol. 52, No.3, pp.419-425.

McHarg, I. L., 1969. Design with Nature, DoubledayfNatural rfistoryPress, Doubleday and Company, Garden City, New York. I

Muehrcke, P. c., and J. O. Muehrcke, 1980. Map Use: Reading, Analysisand Interpretation, J.P. Publications, Madison, Wisconsin, pp. 192­250.

Ripley, Brian D., 1981. Spatial Statistics, series in probability and math­ematical statistics, John Wiley & Sons, Inc., New York.

Robinson, A., R. Sale, and J. Morrison, 1978. Elements of Cartography,4th edition, John Wiley and Sons, New York, New York.

SIS, 1986. The Professional Map Analysis Package (pMAP) User's Manualand Reference, Spatial Information Systems, Inc., Omaha, Nebraska.

Shelton, R. L., and J. E. Estes, 1981. Remote Sensing and GeographicInformation Systems: An Unrealized Potential, Geo-Processing, Vol.1, No.4, pp. 395-420.

Steinitz, C. F., P. Parker, and L. Jordan, 1976. Hand-drawn Overlays:Their History and Prospective Users, Landscape Architecture, Vol.66, No.8, pp. 444-455.

TeSelle, G. W., 1985. Digital Cartography Data Standards Activities, U.S.Soil Conservation Service, CGIS Doc. Number 23.

Tomlin, C. D., 1983a. A Map Algebra, Harvard Computer Graphics Con­ference, Harvard University, Graduate School of Design, Cam­bridge, Massachusetts.

--, 1983b. Digital Cartographic Modeling Techniques in EnvironmentalPlanning. Doctoral Dissertation, Yale University, School of Forestryand Environmental Planning, New Haven, Connecticut.

Tomlin, C. D., and J. K. Berry, 1979. A Mathematical Structl.\re forCartographic Modeling in Environmental Analysis, Proceedings of39th Symposium of American Congress on Surveying and Mapping, FallsChurch, Virginia, pp. 269-284. I