Computer-Assisted Map Analysis: Potential and Pitfalls

Transcription

1 Computer-Assisted Map Analysis: Potential and Pitfalls Joseph K. Berry School of Forestry and Environmental Studies, Yale University, New Haven, CT 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 expressing the spatial relationships of complex physical and conceptual systems. Within this context, a mathematical structure for cartographic modeling can be identified. The quantitative treatment of geographic space has analogies in traditional statistics and alg~bra..quanhtative approaches to digital map processing are described and potentials and pitfalls of map analysis are Identified. Digital map analysis provides a means for evaluation of error introduced by both processing and modeling considerations. H INTRODUCTION ISTORICALLY, maps have been used for navigation through unfamiliar terrain and seas. Within this context, preparation of maps accurately locating geographic features became the primary focus of attention. More recently, analysis of mapped data for decision-making has become an important part of resource and land planning. During the 1960s, manual analytic procedures for overlaying maps were popularized (McHarg, 1969). These techniques mark an important turning point in the use of maps-from one emphasizing physical descriptors of geographic space, to one spatially characterizing appropriate ma.na!sement a~tions. This movement from descriptive to prescnpttve mappmg has set the stage for revolutionary concepts of map structure, content, and use. Geographic information systems (GIS) technology provides a means for quantitative :no?e~ing of spatial relationships. In one sense, this technology IS SimIlar to conventional map processing involving map sheets and drafting aids such as pens, rub-on shading, rulers, planimeters, dot grids, and acetate sheets for light-table overlays. In another sense, these systems provide an analytic "toolbox,"enabling managers to address complex issues in entirely new 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-digital nature. Traditional statistics enable numerical analysis of the d~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 of analysis by dividing geographic space into homogenous response parcels. Most often, these parcels are manually delineated on an appropriate map, and the "typical" value for each parcel is determined. The results of any analysis is assumed to be uniformly distributed throughout each parcel. The areaweighted average of the parcels statistically characterizes the typical response for an entire study area. Mathematical modeling of spatial systems has followed a similar approach of spatially aggregating variation in model variables. Most ecosystem models, for example, identify "level" and "flow" variables presumed to be typical for vast geographic expanses. The fundamental concepts of comprehensive spatial statistics and mathematics have been described for many years by both theorist and practitioner (Shelton and Estes, 1981; Tomlin and Berry, 1979; Lewis, 1977; Steinitz et ai., 1976; Brown, 1949). The representation of spatial data as a statistical surface is indicated m most cartograpny texts (Cuff and Matson, 1982; Robinson et ai., 1978). A host of quantitative and analytical techniques are present in manual cartography. Muehrcke and Muehrcke (1980), in an introductory text, outline "cartometric" techniques allowing quantitative assessments of spatial phenomena. Until modern computers and the digital map, many of these concepts were limited in their practical implementation. The computer has provided the means for both efficient handling of voluminous data and the numerical analysis capabilities that are required. From this perspective, the current revolution in mapping is rooted in the digital nature of the computerized map. The two contemporary fields of remote sensing and geographic information systems have greatly advanced this map form. Remotely sensed data, and their derived classification products, 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, with spectral data to improve classification. This early merger of remote sensing and GIS focused on the preparation of traditional map products. Increasingly, the merger is being viewed as an integral system providing capabilities for both rapid preparation of mapped data and spatially consistent analysis necessary to convert physical maps into decision-making terms. GEOGRAPHIC INFORMATION SYSTEMS The main purpose of a geographic information system is to process spatial information. These systems can be defined as internally referenced, automated, spatial information systems designed for data management, mapping, and analysis (Figure 1). Within this classification scheme, the first distinction is between administrative information (e.g., payroll, personnel, or office management) and spatial information characterizing geographic factors, such as natural resources. Spatial information can be subdivided into that which is spatially aggregated (e.g., soils classification schemes and timber yield tables) and that which is geographically referenced. This latter class of data is termed spatially coherent, and "maps" patterns or distributions of variables (e.g., soils, forest cover, and elevation maps). These mapped data can be managed by non-automated or automated means. The spatial reference may be external or internal to an automated spatial information system. An externally referenced system uses a computer to store information on various geographic units; however, geographic coordinates are not stored within 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 thematic attribute) and the whereabouts (or locational attribute) of those data. 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 over a study area, then stores values identifying the characteristic at each grid space. Although there are Significant practical differences in these data structures, the primary theoretical difference PHOTOGRAMMETRlC ENGINEERING AND REMOTE SENSING Vol. 53, No. 10, October 1987, pp ' /87/ $02.25/ American Society for Photogrammetry and Remote Sensing

2 1406 PHOTOGRAMMETRlC ENGINEERING & REMOTE SENSING, 1987 INFORMATION MANAGEMENT '\:-- Non-Spatial "' SPATIAL '\:--Aggregated "' COHERENT ~Non-Automated -GIS - inlemalty referenced.automated. spatial inlormation system:; wrch are designed for data management maop"l. am analysis "' AUTOMATED~ Externally Referenced INTERNALLY REFERENCED ~INVENTORY "' ANALYSIS FIG. 1. Classification scheme for spatial information systems. Geographic information systems (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 of areal features and implies boundaries whereas the line structure stores information about boundaries and implies interiors. Most modern systems contain programs for converting between the two data structures. Generally, the line segment structure is best for inventory-oriented systems. The difficulty of these systems to characterize spatial gradients and the necessity to compute interior characteristics preclude many of the advanced analytic operations and data characterizations. These differences affect both the potential and pitfalls of computer-assisted map 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 grouped into four categories: computer mapping, spatial data base management, spatial statistics, and cartographic modeling. Computer mapping, also termed automated cartography, involves the preparation of map products. The focus of these operations is on the input and display of computerized maps. Spatial database management, on the other hand, focuses on the storage component of GIS technology. Like non-spatial data-base systems, these procedures efficiently organize and search large sets of 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, moderate slope, and ponderosa pine forest cover. The mapping and database capabilities have proven to be the backbone of current GIS applications. Once a data base is compiled, they allow rapid updating 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 those of 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 advanced analytic 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 of map sheets, manual analytic techniques are limited in their quantitative processing. Digital representation, on the other hand, makes a wealth of quantitative as well as qualitative processing possible. Spatial statistics seek to characterize the geographic distribution, 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 Figures 2 and 3. Figure 2 depicts density mapping of animal activity derived from typical field data. The grid on the upper left is a map of sample locations with 16 sensors positioned at 1000 metre intersections. The tabular data identify both the location and the number of animal encounters at each sensor for two time periods. Traditional statistical analysis involves fitting a numerical distribution to the data to determine the typical response. Such density functions as standard normal, binomial, and Poisson could be tried, and the best fitting functional form chosen. The statistics at the bottom of the table note the average activity as (± 11.9) in period 1 and (± 26.2) in period 2. These parameters 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 hectare gridding resolution. The X and Y axes identify geographic positioning of the sensors, while the Z axis indicates measured animal activity. The horizontal plane at depicts the average activity. The geographic distribution of the standard deviation would form two planes (analogous to "error bars") above and 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 locational information in mapping the variation of thematic values. Analogous to traditional statistics, a density function is fitted to the data. In this instance, the distribution is characterized in geographic space rather than numerical space. Inset (b) of Figure 2 shows a continuous surface implied by the sampled data. This nonplaner surface was fitted by "inverse distance squared weighted averaging" of the sample values. Other surfaoe fitting techniques, such as Kriging, spline, or polynomial fur.ctions, could be tried and the best fitting functional form chosen. A.nalysis of the residuals, from comparisons of the various surfaces with the 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 (lake in the northwest portion). The final distribution shows considerable deviation from the average plane implied by the traditional statistical analysis. Activity appears to be concentrated in the eastern portion, tending to conflict with the assumption that variation is uniformally distributed in geographic space. Figure 3 depicts the natural extension to multivariate statistics for spatial coincidence between the two periods. In traditional statistics this involves fitting of a density function in multivariate statistical space as shown in the upper left portion of Figure 3. For the example, a standard normal surface was fitted, with its "joint mean" and "covariance matrix" describing the typical paired occurance. Generally speaking, there is an increase in activity (18.75 versus 22.56) with a slight positive correlation (0.42) between the periods. The joint mean is assumed to be uniformally distributed in geographic space. The covariance matrix indicates the level of variation associated withithis assumption; however, it does not indicate where the variation from the joint mean is most likely to occur. The right portion of Figure 3 depicts a spatial analysis between the two data sets. The two spatial distributions of implied activity (inset (a)) are

3 COMPUTER-ASSISTED MAP ANALYSIS 1407 Sensor X Y Penodl " ~ TOTAL 300 M(AN AVERAGE 1875 STD DEV (1\ ""CHANGE. - Penod2 4 9 o o !'.!L ~ 19% SOO lel~~ m GEOGRAPHIC SPACE (a) Period 2 Data ~ NUMERIC SPACE )(, = , =t 26.2 FIG. 2. Spatially characterizing data variation. Whereas traditional statistics identifies the typical response and assumes this estimate to be uniformally distributed in space, spatial statistics seeks to characterize the geographic distribution, or pattern, of mapped data. compared (that is, one subtracted from the other) to generate a coincidence 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 (that is, more than the average difference plus the average standard deviation) between the two periods. CARTOGRAPHIC MODELING Just as a spatial statistics may be identified, a spatial mathematics is emerging. This approach uses sequential processing of mathematical primitives to perform a wide variety of complex map analyses (Berry, 1987; Berry and Reed, 1987; Tomlin, 1983a). By controlling the order in which the operations are executed and using a common database to store the intermediate results, a mathematical-like processing structure is developed. The "map algebra" is similar to traditional algebra where primitive operations, such as add, subtract, or exponentiate, are logically sequenced for specified variables to form equations. However, in map algebra the variables are entire maps represented as organized sets of numbers. Most of the traditional mathematical capabilities, plus an extensive set of advanced map processing primitives, comprise this map analysis toolbox. As in matrix algebra (a mathematics operating on groups of numbers defining variables), new primitives emerge that are based on the nature of the data. Matrix algebra's transpose, inverse, and diagonalization are examples of these new primitives. Within map analysis, the spatial coincidence and juxtapositioning of values among and within maps create new operators, such as proximity, spatial coincidence, and optimal paths. These operators can be accessed through general purpose map analysis packages, similar to the numerous matrix algebra packages. The widely distributed Map Analysis Package (MAP) (Tomlin, 1983b) is an example. A similar system, the Professional Map Analysis Package (pmap) (SIS, 1986), recently was released for personal computers. All of the processing presented in this paper was done 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 structure similar to "evaluating nested parentheticals" in traditional algebra. Within this structure a mathematician first defines the values for each dependent variable and then solves the equation by performing the primitive mathematical operations on those numbers in the order prescribed by the equation. For example, the equation percent change (new price - old price) * 100 old price 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

4 1408 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1987 SPIlSO'., '0 "'C '1 '5 x y f'('r;odt 10DO 1000 '000 ',00 "'9 '000 2DOO ' :; < 000 ' 'SOD '0, ' :'000? " "0 c :'': '2 =' ~ o 6 22 '2 33 '6 4' 87 ~ TOTAL ;" AVEAAGl SToDEV,"9.262 CHANG[ 19 ~ , rn GEOGRAPHIC SPACE 2 NUMERIC SPACE M = ;>= r =.42 " (b) Difference (P2-PI)... IHf-U '" UI~FU;- Nl.l I IHh"\J 10 ><bo I 1 I~*,U -:iof _I lhf,;u ;'" FIG. 3. Assessing coincidence among mapped data. Maps characterizing spatial variation among two or more variables can be compared and locations of unusual coincidence can be identified. change in value of a parcel of land. In a similar manner, a map of percent change in land values for an entire area can be expressed in pmap command language as COMPUTE NEWVALUE. MAP MINUS OLDVALUE. MAP TIMES 100 DIVIDEDBY OLDVALUE.MAP FOR PERCENTCHANGE.MAP Within this model, data for current and past land values are collected and encoded as computerized maps. These data are evaluated, as shown above, to form a solution map of percent change. The simple model might be extended to provide coincidence statistics, such as CROSSTABULATE ZONING.MAP WITH PERCENTCHANGE.MAP for a table summarizing the spatial relationship between the type of zoning and the change in land value (that is, which zones have experienced the greatest decline in market value; or which, the greatest increase?) The model might be further extended to include geographic searches, such as RENUMBER PERCENTCHANGE.MAP ASSIGNING 0 TO -20 THRU 20 FOR BIGINCREASES.MAP for a map isolating those areas experiencing large changes in land value (exceeding ± 20 percent). Another simple model is outlined in Figure 4. It consists of less than ten sentences, similar to those above. In the flowchart in the upper left portion of Figure 4, encoded and derived maps are shown as boxes, with processing operations indicated as lines. The model uses the distributions presented in Figure 3 and the model described above to create a map of the percent change in animal activity (inset(a)). Derivation of the human activity map uses proximity to roads and housing density as indicators of activity. Locations with numerous houses and close to roads are assigned high levels, whereas remote areas are assigned low levels. A tabular summary of the coincidente between changes in animal activity and human activity is shown in inset (b). The spatial analysis indicates that areas of low human activity are strongly related to areas of high increases in animal activity (fourth line of the table noting 87.4 percknt of all high animal activity is spatially coincident with areas of low human activity). This same information could be produced as a map isolating those areas. The development of a generalized analytic structure for map processing is similar to those of many other nonspatial systems. For example, the popular dbase III package contains less than 20 analytic operations, yet may be used to create models for such diverse applications as address lists, payroll, inventory control, or commitment accounting. Once developed, these logical dbase sentences are "fixed" into menus for easy end-user applications. A flexible analytic structure provides for dynamic simulation as well as data base management. For example, the Lotus spreadsheet package allows users to define interrelationships among variables. A financial model of a company's production process may be established by specifying a logical sequence of interrelationships and variables. By changing specific values of the model, the impact of several fiscal scenarios can be simulated. The advent of data base management and

5 , " ~! ~~--''-:.J(Vr~ PERIOD I ~ DATA E ~l ~~ PERIOD 2 l ANUUl DATA! ACTIVIT'!' 2,, ọ, U,, ~ IOO " ASSISTED MAP ANALYSIS (a) LlLLl'>b"~,.., IHMJ I". r~ I"IUll f<ail,0 IHh,U _"q". Hll... "0 THl'.U tol:" (b) crosstabulate human-activity with percent-change WORK I NG COINCIDENCE TABLE FOR MAPI = HUMAN-ACTI VITY WITH MAP2 = PERCENT-CHANGE MAP 1 It OF LABEL MAP2 It OF It OF 'l. OF 'l. OF 'l. OF VALUE CELLS VALUE CELLS LABEL CROSS TOTAL MAPI MAP2 0 B2 OPEN WATER MODERATE LOW ACTIVITY DECREASE B LOW ACTIVITY MODERATE LOW ACTIVITY HIGH MODERATE DECREASE MODERATE MODERATE MODERATE HIGH HIGH DECREASE HIGH MODERATE HIGH HIGH FIG. 4. A simple cartographic model. The analysis derives maps of animal and human activity and then generates a summary of their spatial coincidence. spreadsheet packages have revolutionized the handling of nonspatial data. The potential of computer-assisted map analysis promises 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 to digital maps has been a bottleneck in most GIS applications. Digital remote sensing data are the rare exception. Early use of GIS technology required users to develop there own data bases at large expenditures of time and money. These individual efforts also proliferated radically different data structures and processing systems. More recently, standardized data bases are becoming available (Geological Survey, 1983; Glen, 1982). Also, the storage requirements of these data are massive. A complete digital data base containing all 54, minute quadrangle maps covering the lower 49 U.S. states would require bits of data for a ground resolution of 1.7 metres (Light, 1986). With current technology, all those data could be stored on 4000 optical disks-smaller than a phonograph record library at a local radio station. The storage requirements at a hectare ground resolution for a similar spatial data base for the entire continent of Africa could be stored on five optical disks. Both the commitment to the task and advances in computer hardware are actively addressing the data availability pitfall. Data characterization issues are not so well defined. Data exchange formats and geographic considerations, such as projections and resolution, are being addressed by several national and international committees, such as the U.S. Geological Survey 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 true statistical surfaces. First, a sensor samples the spectral response by 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 each location. 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 matrix summarizing classification accuracy over an entire area. As in traditional statistics, the error is assumed to be uniformally distributed in space. However, the nature of the data and the processing procedure provides information on the nonplaner distribution. It is this character that makes digital maps fully quantitative mapped data, and distinguishes mapping from digital map analysis. For example, consider a simple geographic search for locations of ponderosa pine on cohassett soil. In overlaying these maps, it is assumed that the joint probability of coincidence is the same at all locations. Or more likely, error propagation is ignored and the results assumed inviolate. For inventory-oriented applications, these assumptions may be acceptable. Analysis-oriented applications, on the other hand, have far greater demands. Suppose the composite cover/soil map is used in a mathematical equation with other map variables deriving habitat suitability indicies for wildlife. The accuracy of the predictions are, in large part, a function of the thematic accuracy of the input maps. Consider further that the habitat map, in turn, becomes input to a management model weighing timber,

6 1410 PHOTOGRAMMETRIC ENGI EERI G & REMOTE SENSING, 1987 water, recreation, and other uses of the land. Data exchange and geographic registration requirements may be met and the habitat 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 appears appropriate. However, if a "shadow map" of thematic error is generated at each step of a cartographic model, the "garbage" can be spatially separated. Those locations that are confidently derived should be differentiated from those predictions for locations less accurately defined. The pitfall of data characterization requires a complete restructuring of maps into a fully quantitative form. Even if spatial data are readily available and in an appropriate quantitative form, the results of map analysis may still be questionable. Many spatial relationships have not been adequately quantified. The modeling of uncertain relationships, even with the best data, introduces ill-defined error. Many GIS applications evaluate subjective considerations with minimal empirical testing. In part, this is the result of the tool getting ahead of knowledge. Most contemporary research considers space in aggregate terms using traditional statistical procedures. Until recently, both the means and the concepts for rigorous spatial investigations were lacking. Even if relationships were quantified, practical implementation of that theory did not have an outlet. GIS technology is both the salvation and the source of this pitfall. It can be instrumental in managing and processing the massive amounts of data involved in developing a spatially oriented body of knowledge. In the interim, however, it provides a vehicle for generating seemingly accurate maps expressing complex systems with minimal understanding of the actual spatial relationships. The use of dynamic simulation may assist in modeling such uncertain situations. For example, consider a spreadsheet model of a company's production process. Best estimate for each variable, parameter, and functional form are entered and results are generated. However, the analysis does not stop here. What if 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. What if slope is twice as important; or half; or visual exposure is increased to a kilometre? Where does the map of a powerline corridor significantly change? Within this context, the individual maps produced are no longer the focus. By noting how often and where this change occurs as successive runs are made, information on the sensitivity of a specific area to a particular management action is described. Within this context, maps are transformed from images to spatial information, becoming an active and integral part of the decision process. CONCLUDING REMARKS Geographic information systems technology is revolutionizing how we handle maps. Since the 1970s an ever-increasing portion of mapped data is being collected and processed in digital format. Currently, this processing emphasizes computer mapping and data-base management. These techniques allow users 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 to how we analyze mapped data. These operations provide a means for expressing the spatial interrelationships of maps. Analogous to traditional statistics and mathematics, primitive operations are logically sequenced on variables to form cartographic models; however, the variables are represented as entire maps. Such a quantitative approach is changing basic concepts of map structure, content, and use. Several pitfalls affect the practical implementation of rigorous map analysis. Data availability is being addressed. Standards for 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 error introduced 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 imperfectly defined relationships. From this perspective, maps move from images describing the location of features to mapped information quantifying a physical or abstract system in prescriptive terms. This radical change in our use of maps has the pot~ntial for more complete integration of spatial information intq the decision-making process. I REFERENCES Berry, J. K., Fundamental Operations in Computer-Assisted Map Analysis, International Journal of Geographic Information Systems; Taylor & Francis, Ltd., Basingstoke, England (in press). I Berry, J. K., and K. L. Reed, 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 Brown, L. A., The Story of Maps, Little, Brown and Co., Boston, Massachusetts. Cuff, D. J., and M. T. Matson, Thematic Maps, Methuen, INew York, New York. Dangermond, J., The Software Toolbox Approach to Meeting the User's Needs for GIS Analysis, Proceedings of Geographic Inforrration Systems Workshop, Atlanta, Georgia, American Society for Photogrammetry and Remote Sensing, Falls Church, Virginia. pp Davis, John c., Statistics and Data Analysis in Geology, John Wiley and Sons, Inc., New York, pp Geological Survey, USGS Digital Cartographic Data Standardi' Overview and USGS Activities, U.S. Geological Survey Circular 8 5A. Glen, R., SCS Geographical Information Format-Standard V rsion, U.S. Soil Conservation Service, CGIS Doc. Number 2. Lewis, P., Maps and Statistics, Halsted Press, Cambridge, England. Light, D. L., Mass Storage Estimates for the Digital Mapping Era, Photogrammetric Engineering and Remote Sensing, Vol. 52, No.3, pp McHarg, I. L., Design with Nature, DoubledayfNatural rfistory Press, Doubleday and Company, Garden City, New York. I Muehrcke, P. c., and J. O. Muehrcke, Map Use: Reading, Analysis and Interpretation, J.P. Publications, Madison, Wisconsin, pp Ripley, Brian D., Spatial Statistics, series in probability and mathematical statistics, John Wiley & Sons, Inc., New York. Robinson, A., R. Sale, and J. Morrison, Elements of Cartography, 4th edition, John Wiley and Sons, New York, New York. SIS, The Professional Map Analysis Package (pmap) User's Manual and Reference, Spatial Information Systems, Inc., Omaha, Nebraska. Shelton, R. L., and J. E. Estes, Remote Sensing and Geographic Information Systems: An Unrealized Potential, Geo-Processing, Vol. 1, No.4, pp Steinitz, C. F., P. Parker, and L. Jordan, Hand-drawn Overlays: Their History and Prospective Users, Landscape Architecture, Vol. 66, No.8, pp TeSelle, G. W., 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 Environmental Planning. Doctoral Dissertation, Yale University, School of Forestry and Environmental Planning, New Haven, Connecticut. Tomlin, C. D., and J. K. Berry, A Mathematical Structl.\re for Cartographic Modeling in Environmental Analysis, Proceedings of 39th Symposium of American Congress on Surveying and Mapping, Falls Church, Virginia, pp I