To generate an interactive screen version of
this figure using pMAP, enter
C:\PMAP> pmap tutor25
CMD: display slope 
CMD: renumber slope for s-classes
assign 0 to 1 thru 
assign 0 to 3
assign 
CMD: display s-classes
CMD: display covertype
CMD: intersect covertype with s-classes
completely for cover-slope
CMD: display cover-slope

Figure A-3. Point-by-point overlay is location specific. Each map location is assigned a unique value identifying the cover type and slope conditions occurring at that location. (see TUTOR3.CMD)

COMPUTE AND AVERAGE. A specific function used to compute new map category values from those of existing maps being overlaid may vary according to the nature of the data being processed and the specific use of that data within a modeling context. Most typical environmental analyses are those involving the manipulation of quantitative values to generate new values that are also quantitative in nature. Among these manipulations are the basic arithmetic operations such as addition, subtraction, multiplication, division, roots, and exponentiation. For example, given maps of assessed land values in 1970 and 1990, a map that shows the percent change in land values over that period can be expressed in pMAP software syntax as follows:

COMPUTE 1990.MAP MINUS 1970.MAP FOR CHANGE.MAP
COMPUTE CHANGE.MAP TIMES 100 /
DIVIDEDBY 1970.MAP FOR PERCENT.CHANGE

Functions related to simple statistical parameters (maximum, minimum, median, mode, majority, standard deviation, or weighted average) may also be applied in this manner. Type of data being manipulated dictates the appropriateness of the mathematical or statistical procedure used. For example, the addition of qualitative maps, such as soils and land use, would result in meaningless sums because their numeric values have no mathematical relationship.

Other map overlay techniques can be used to process either quantitative or qualitative values and to generate like values. Among these are masking, comparison, calculation of diversity, and reclassification of cross-tabulated map categories. (Figure A-3 shows an example of the latter.)

More complex statistical techniques can also be applied in this manner assuming that the inherent interdependence among spatial observations can be taken into account. This approach treats each map as a variable, each point as a case, and each value as an observation. A predictive statistical model can then be evaluated for each location, resulting in a spatially continuous surface of predicted values. The mapped predictions contain additional information over traditional nonspatial procedures such as direct consideration of coincidence among regression variables and the ability to locate areas of a given prediction level.

INTERSECT AND CROSSTAB. Two maps also can be combined on a point-by-point basis depending on the pairwise combination of values. This procedure is, in effect, a two map "renumber." A new number is assigned to all joint occurrences that a user specifies. Figure A-3 shows the results of user assigned values to specified pairs of values on the COVERTYPE and SLOPE maps. A related operation can be used to generate a crosstab table which reports statistics on all pairwise combinations between two maps.

COVER. Covering, or "masking," one map with another identifies a process in which values on one map are replaced with those from another map. This process is similar to manual map overlaying using acetate sheets. The "clear" areas on the top map allow the information from the bottom map to show through. In computer processing, the value zero is used to represent clear areas and the values at such a locations are replaced (show through). If a nonzero value is present, the other maps value does not replace the current value.

COMPOSITE. An entirely different approach to overlaying maps involves category-wide summarization of values. Rather than combining information on a point-by-point basis, this group of operations summarizes the spatial coincidence of entire categories of two or more maps. Figure A-4 shows an example of a category-wide overlay operation using the same input maps as those in figure A-3. In this example, the categories of the COVERTYPE map are used to define an area for which the coincidental values of the SLOPE map are averaged. The computed values of average slope are then assigned to the cover-type categories (AVERAGE-SLOPE). Note that the Forest category has the highest average slope of 29%.

Summary statistics that can be used in this way include the total, average, maximum, minimum, median, mode, or minority value; the standard deviation, variance, or diversity of values;

To generate a hardcopy of this figure, print TEXT.FIG on a HP Laserjet III or compatible printer using WordPerfect 5.0 or compatible word processing program.

To generate an interactive screen version of
this figure using pMAP, enter
C:\PMAP> pmap tutor25
CMD: display covertype 
CMD: display slope 
CMD: composite covertype with slope 
average for avg-slope

Figure A-4. Category-wide overlays summarize the spatial coincidence of areal features. Each of the three cover types (OPEN WATER, MEADOW, FOREST) are assigned a value equal to their average slopes. In computing these averages the SLOPE values occurring within each COVERTYPE boundary are averaged separately. (see TUTOR4.CMD) and the correlation, deviation, or uniqueness of particular value combinations. For example, a map indicating the proportion of undeveloped land within each of several counties can be generated by superimposing a map of county boundaries on a map of land use and then computing the ratio of undeveloped land to total land area for each county. As with location-specific overlay techniques, data types must be consistent with the summary procedure used. The order of processing is also important. Operations such as addition and multiplication are independent of the order of processing. Other operations such as subtraction and division yield different results depending on the order in which they are processed. This latter type of operation, termed noncommutative, cannot be used for category-wide summaries.

MEASURING DISTANCE AND CONNECTIVITY:

DRAIN, RADIATE, SPAN, SPREAD, STREAM

Most geographic information systems contain analytic capabilities for reclassifying and overlaying maps. These operations address the majority of applications that parallel manual map analysis techniques. To more fully integrate spatial considerations into decision-making, however, new techniques are emerging. The concept of distance historically has been associated with the "shortest straight line between two points." This measure is both easily conceptualized and implemented with a ruler but is frequently insufficient in a decision-making context. A straight line route may indicate the distance "as the crow flies" but offer little information for a walking or hitchhiking crow, or other flightless individual. Equally important to most travelers is the measurement of distance expressed in more relevant terms such as time or cost. The group of operations concerned with measuring distance, therefore, are best characterized as "rubber rulers."

The basis of any system for the measurement of distance requires two components: a standard measurement unit and a measurement procedure. The measurement unit used in most computer-oriented systems is the "grid space" defined by superimposing an imaginary uniform grid over a geographic area. The distance from one location to another is computed as the number of intervening grid spaces. The measurement procedure always retains the requirement of the "shortest" connection between points, although the "straight-line" requirement may be relaxed.

SPREAD. A frequently employed measurement procedure involves expanding the concept of distance to one of proximity. Rather than sequentially computing the distance between pairs of locations, concentric equidistant zones are established around a location, or set of locations. In effect, a map of proximity to a "target" location is generated that indicates the shortest straight-line distance from any map cell to the nearest target area. Figure A-5(a) is an example of a map, SIMPLE-PROXIMITY, that indicates the shortest straight line distance from the RANCH to all other locations.

Within many application contexts, the shortest route between two locations may not always be a straight line. Even if it is straight, the Euclidean length of that line may not always

To generate a hardcopy of this figure, print TEXT.FIG on a HP Laserjet III or compatible printer using WordPerfect 5.0 or compatible word processing program.

To generate an interactive screen version of this figure using pMAP, enter C:\PMAP> pmap tutor25

Inset (a)

CMD: display locations
CMD: renumber locations for ranch
assign 0 to 2 thru 5
CMD: spread ranch to 35 for simple-prox

Inset (b)

CMD: display cover-slope (see Fig. A-3)
CMD: renumber cover-slope for friction
assign
assign
assign
assign
CMD: spread ranch to 75 thru friction
for weighted-prox

Figure A-5. Distance between locations can be determined as Euclidean length or as a function of the effect on implied movement of absolute or relative barriers. Inset (a) identifies equidistant zones around the ranch. Inset (b) is a travel-time map identifying time zones of hiking from the ranch. This travel-time map was generated by considering the relative ease of travel through various cover type and slope conditions (COVER-SLOPE) where flat meadows are the fastest to traverse, steeply forested areas intermediate, and flat water slowest. (see TUTOR5.CMD) reflect a meaningful measure of distance. Distance in these applications is best defined in terms of movement expressed as traveltime, cost, or energy consumed at rates that vary over time and space. Distance-modifying effects may be expressed cartographically as "barriers" located within the space in which the distance is being measured. This implies that distance is the result of some sort of movement over that space and through those barriers. How these barriers affect the implied movement identifies two major types. "Absolute barriers" are those that completely restrict movement and imply an infinite distance between the points they separate unless a path around the barrier is available. A river might be regarded as an absolute barrier to a nonswimmer. A swimmer or a boater, however, might regard the same river as a relative, rather than an absolute, barrier. "Relative barriers" are ones that are passable although only at a cost equated with an increase in effective distance.

Figure A-5(b) shows a map, WEIGHTED-PROXIMITY, of hiking time around the target location identified as the ranch. The map was generated by reclassifying the various cover/slope categories on the COVER-SLOPE map (figure A-3) in terms of their relative ease of foot-travel. The example uses two types of barriers. The lake is treated as an absolute barrier completely restricting hiking. The land areas represent relative barriers to hiking indicating varied impedance to movement for each point as a function of the cover/slope conditions occurring at that location. Consider a similar example. Movement by automobile may be effectively constrained to a network of roads (absolute barriers) of varying speed limits (relative barriers) to generate a riding travel-time map. An example from an even less conventional perspective might be a weighted distance expressed in terms of accumulated cost of powerline construction from an existing trunkline to all other locations in a study area. The cost surface developed can be a function of a variety of social and engineering factors, such as visual exposure and adverse topography, expressed as absolute and/or relative barriers.

The ability to move, whether physically or abstractly, may vary as a function of the implied movement and the static conditions at a location. Direction is one aspect that may affect the ability of a barrier to restrict that movement. A topographic incline, for example, generally impedes hikers differently according to whether their movement is uphill, downhill, or across slope. Accumulation is another possible modifying factor.

After hiking a certain distance, "molehills" tend to become disheartening "mountains," and movement is more restricted.

Momentum, or speed, is a third factor that may dynamically alter the effect of a barrier. A Volkswagen that has to stop for a red light on a steep hill may not be able to resume movement; however, if it were allowed to maintain its momentum (e.g., green light), it could easily reach the top. Similarly a highway impairment that effectively reduces traffic speeds from 55 to 40 miles per hour has little or no effect during rush hour when traffic is already moving at a much slower speed.

Another distance-related class of operations is concerned with the nature of connectivity among locations on an overlay. Fundamental to understanding these procedures is the conceptualization of an "accumulation surface." If the thematic value of a simple proximity map from a point is used to indicate the third dimension of a surface, a uniform bowl would be depicted. The surface configuration for a weighted proximity map would have a similar appearance, but the bowl would be warped with numerous ridges and pinnacles. Also the nature of the surface is such that it cannot contain saddle points (i.e., false bottoms). This bowl-like "topology" is characteristic of all accumulation surfaces and can be conceptualized as a warped football stadium with each successive ring of seats identifying concentric, equidistant halos about its center (the playing field).

Figure A-6(a) shows the surface configuration of the hiking WEIGHTED-PROXIMITY map from the previous figure. The accumulated distance surface is shown as a perspective plot in which the ranch is the lowest location and all other locations are assigned progressively larger values of the shortest, but not necessarily straight, distance to the ranch. When viewed in perspective, this surface resembles a topographic surface with familiar valleys and hills. In this case, however, the highlands indicate areas that are effectively farther away from the ranch. The tallest protrusions are the Open Water features forming absolute barriers in which traveltime is infinitely far away. This plot was created by transferring the WEIGHTED-PROXIMITY map to a 3-D graphics package for plotting (see Appendix B).

STREAM. In the case of simple distance, "connectivity," or the delineation of paths, locates the shortest straight line between two points considering only two dimensions. The STREAM command traces the steepest downhill path from a location on a complex three-dimensional surface. The steepest downhill path along a topographic surface indicates the route of surface runoff. For a surface represented by a travel-time map, this technique traces the optimal (e.g., the shortest or quickest) route. In effect, this route traces the path of the distance "wave front" that arrived first. Figure A-6(a) indicates the optimal hiking path (BESTPATH) from a nearby cabin to the ranch as the steepest downhill path over the accumulated hiking-time surface (WEIGHTED- PROXIMITY). If an accumulation cost surface is considered, such as the cost surface for powerline construction described earlier, the minimum cost route will be located.

DRAIN. If powerline construction to a set of dispersed locations is simultaneously considered, an "optimal path density" map can be generated that identifies the number of individual optimal paths passing through each location from the dispersed termini to the trunkline. Such a map would be valuable in locating major "feeder" lines (i.e., areas of high optimal path density) attaching to a central trunkline.

To generate a hardcopy of this figure, print TEXT.FIG on a HP Laserjet III or compatible printer using WordPerfect 5.0 or compatible word processing program.

To generate an interactive screen version of this figure using pMAP, enter C:\PMAP> pmap tutor25

Inset (a)

CMD: display locations
CMD: renumber locations for cabin
assign 0 to 1 
assign 0 to 3 thru 5
CMD: display weighted-prox (see Fig 5)
CMD: stream cabin over weighted-prox
for bestpath
CMD: display bestpath

Inset (b)

CMD: slice elevation for contours
CMD: display contours
CMD: display covertype
CMD: renumber covertype for trees
assign 0 to 1 thru 2
assign
CMD: display trees
CMD: display ranch
CMD: radiate ranch to 35 over elevation
thru trees for viewshed
CMD: display viewshed

Figure A-6. Connectivity operations characterize the nature of spatial linkages between locations. Inset (a) delineates the shortest (i.e., the quickest) hiking route from the cabin to the ranch. The route traces the steepest downhill path along a cumulative hiking-time surface. Inset (b) identifies the viewshed of the ranch. Topographic relief and forest cover act as absolute barriers when establishing visual connectivity. (see TUTOR6.CMD)

SPAN. Another connectivity operation determines the narrowness of features. The narrowness at each point within a map feature is defined as the length of the shortest line segment (i.e., cord) that can be constructed through that point to diametrically opposing edges of the feature. The result of this processing is a continuous map of features with lower values indicating relatively narrow locations. For example, small values on a narrowness map of forest stands would indicate interior locations with easy access to edges.

RADIATE. The process of determining viewsheds involves establishing intervisibility among locations. Straight lines in three-dimensional space connect locations forming the viewshed of an area to the "viewer" location or set of viewers. Topographic relief and surface objects form absolute barriers that preclude visual connectivity. If multiple viewers are designated, such as a road network, locations within the viewshed can be assigned a value indicating the number, or density, of visual connections to the "extended eyeball." A "weighted visual density" map can be generated by assigning relative weights indicating traffic flow for each road type. Figure A-6(b) shows a map of the simple viewshed of the ranch considering the terrain and forest canopy height as visual barriers.

CHARACTERIZING NEIGHBORHOODS:

INTERPOLATE, ORIENT, PROFILE, SCAN, SLOPE

The fourth and final group of operations includes procedures that create a new map in which the value assigned to a location is computed as a function of independent values within a specified distance and direction around that location (i.e., its cartographic neighborhood). This general class of operations can be conceptualized as "roving windows" moving throughout the mapped area. The summary of information within these windows can be based on the configuration of the surface (e.g., slope and aspect) or the statistical summary of thematic values.

Establishing neighborhood membership is the initial step in characterizing cartographic neighborhoods. The set of points that lies within a specified distance and direction around a target location uniquely defines the neighborhood, or roving window, for that location. In most applications, the window has a uniform geometric shape and orientation (e.g., a circle or square). As noted in the previous section, however, the distance may not necessarily be Euclidean or symmetrical. Examples of effective neighborhoods include "downwind" locations within a quarter mile of a smelting plant, or a neighborhood of "the ten- minute drive" along a road network.

The summary of information within a neighborhood can be based on the surface configuration implied by the set of values occurring within the window. This is true of operations that measure topographic characteristics such as slope, aspect, or profile, from elevation values.

SLOPE AND ORIENT. One such technique involves the "least squares fit" of a plane to adjacent elevation values. This process is similar to fitting a linear regression line to a series of points expressed in two-dimensional space. The inclination of the plane denotes terrain slope; its orientation characterizes the aspect. The window is successively shifted over the entire elevation map to produce a continuous slope map (inclination) or aspect map (direction).

Figure A-7(a) shows the derived ASPECT map for the area shown in previous figures. Note most of the area is generally oriented to the west. In computing this map, a 3x3 window was moved about the ELEVATION map of the area, and the direction of the best- fitted plane was assigned. The inclination of the plane would indicate terrain steepness, or "% slope."

A "slope map" of any surface represents the first derivative of that surface. For an elevation surface, slope depicts the rate of change in elevation. For an accumulation cost surface, its slope map represents the rate of change in cost (i.e., a marginal cost map). For a traveltime overlay, its slope map indicates relative change in speed and its "aspect map" identifies direction of travel at each location. Also the slope map of an existing topographic slope map (i.e., second derivative) characterizes surface roughness (i.e., areas where slope is changing).

PROFILE. The creation of a "profile map" uses a window defined as the three adjoining points along a straight line oriented in a particular direction. Each set of three values can be regarded as defining a cross-sectional profile of a small portion of a surface. Each line is successively evaluated for the set of windows along that line. This procedure may be conceptualized as slicing a loaf of bread, then removing each slice and characterizing its profile (as viewed from the side) in small segments along its upper edge.

The center point of each three-member neighborhood is assigned a value indicating the profile form at that location. The assigned value can identify a fundamental profile class (e.g., inverted "V" shape indicative of a ridge) or the magnitude, in degrees, of the "skyward angle" formed by the intersection of the two line segments of the profile. The result of this operation is a continuous map of a surface's profile as viewed from a specified direction. Depending on the resolution of an elevation map, its profile map could be used to identify gullies or valleys running east-west ("V" shape as viewed from the east or west profile) or depressions ("V" shape in multiple orthogonal profiles).

The previous discussion identified several operations which characterize the neighborhood by the configuration of the implied surface. Another group of neighborhood operations summarizes thematic values contained in the neighborhood.

To generate a hardcopy of this figure, print TEXT.FIG on a HP Laserjet III or compatible printer using WordPerfect 5.0 or compatible word processing program.

To generate an interactive screen version of this figure using pMAP, enter C:\PMAP> pmap tutor25

Inset (a)

CMD: display contours (see Fig. 6)
CMD: orient elevation for aspectmap
CMD: display aspectmap

Inset (b)

CMD: display covertype
CMD: scan covertype diversity 
for variety

Figure A-7. Neighborhood operations characterize map locations by summarizing the attributes of their surrounding locations. Inset (a) is a map of topographic aspect generated by successively fitting a plane to neighborhoods of adjoining elevation values. Inset (b) Map of cover-type diversity generated by computing the number of different cover types in the immediate vicinity of each map location. (see TUTOR7.CMD)

SCAN. The simplest operations involve the calculation of summary statistics associated with the map categories occurring within each neighborhood. These statistics might include maximum income level, minimum land value, or diversity of vegetation types within a quarter-mile walking radius of each target point. Figure A-7(b) shows the covertype diversity occurring within the immediate vicinity of each map location. Other thematic summaries might include the total, average, standard deviation, or median value occurring within each neighborhood; the standard deviation or variance of those values; or the difference between the value occurring at a target point itself and the average of those surrounding it.

None of the neighborhood characteristics described so far relate to the amount of area occupied by the map categories within each neighborhood. Similar techniques can be applied, however, to characterize neighborhood values weighted according to spatial extent (e.g., total land value, on a per-acre basis, within three miles of each target point). This consideration of size of the neighborhood components enables several additional neighborhood statistics including:

Mode, the value associated with the greatest proportion of neighborhood areas Minority value, the value associated with the smallest proportion of a neighborhood area Uniqueness, the proportion of the neighborhood area associated with the value occurring at the target point itself

INTERPOLATE. Another locational attribute that can be used to modify thematic summaries is the cartographic distance from the target point. While distance has already been described as the basis for defining a neighborhood's absolute limits, it can be used also to define the relative weights of values within the neighborhood. Noise level, for example, may be measured according to the inverse square of the distance from surrounding sources. The azimuthal relationship between neighborhood location and the target point can be used also to weight the value associated with that location.

In conjunction with distance weighing, weighing enables a variety of spatial sampling and interpolation techniques. For example, "weighted nearest-neighbors" interpolation of lake bottom temperature assigns a value to an unsampled location of the distance-weighted average temperature of a set of sampled points within its vicinity. Spatial interpolation is routinely used to create maps from point sampled data.

CARTOGRAPHIC MODELING

The preceding discussion developed a topology of fundamental map processing procedures and described a set of reclassifying, overlaying, distance, and neighborhood operations common to a broad range of higher techniques of map analysis. By systematically organizing these primitive operations, the basis for a generalized cartographic modeling approach is identified. This approach accommodates a variety of analytic procedures in a common, flexible, and intuitive manner that is analogous to the mathematical structure of conventional algebra.

Figure A-8 outlines an example of the way fundamental map processing operations can be combined to perform more complex analyses. The flowchart structure indicates the logical sequencing of operations on the mapped data that progresses to the desired final map. This simplified cartographic model depicts the siting of the optimal corridor for a highway considering only two criteria: an engineering concern to avoid steep slopes and a social concern to avoid visual exposure.

To generate a hardcopy of this figure, print TEXT.FIG on a HP Laserjet III or compatible printer using WordPerfect 5.0 or compatible word processing program.

LANDUSE LANDUSE LANDUSE ELEVATION ELEVATION
| | | | |
| | VIEWERS --| |
| | | |
| | VIEWSHED SLOPE
| | |............|
| | |
| THATPLACE COST
| |...........................|
| |
THISPLACE COSTZONES
|......................|
|
HIGHWAY

Figure A-8. This simplified cartographic model depicts the siting of "best" route for a highway considering only two criteria: an engineering concern to avoid steep slopes and a social concern to avoid visual exposure. In a manner similar to conventional algebra, this process uses a series of pMAP operations (indicated by lines) to derive intermediate maps (indicated by boxes), leading to a final map of the best route. (see TUTOR8.CMD)

Given maps of topographic ELEVATION and of LANDUSE, the model allocates a minimum-cost highway alignment between two predetermined termini isolated on the maps named THISPLACE and THATPLACE. Cost is not measured in dollars, but in terms of locational criteria. The right side of the model develops a "discrete cost surface" (the COST map) in which each location is assigned a relative cost based on the steepness/exposure combination occurring at each location. For example, flat areas that are also not visible from houses might be assigned low values; areas on steep slopes and visually exposed would be assigned high values. This discrete cost surface is used as a map of relative barriers for establishing an accumulated cost surface (the COSTZONES map) from one terminus to all other locations within the mapped area. The final step locates the other terminus on the accumulated cost surface and identifies the minimum cost route as the steepest downhill path along the COSTZONES surface from that point to the bottom (i.e., the other end point). An expanded model is implemented and explained in more detail in the TUTOR8.CMD tutorial command file.

In addition to the benefits of efficient data management and automated cartographic procedures, the modeling structure of computer-assisted map analysis has several other advantages. Foremost among these is the capability of "dynamic simulation" (i.e., spatial "what if" analysis). For example, the highway siting model could be executed for several different relative weightings of the engineering and social criteria. What if the terrain steepness is more important? What if the visual exposure is twice as important? Where does the minimum cost route change, or, just as important, where does it not change? From this perspective, the model "replies" to user inquiries, rather than "answering" them: the processing provides information for decision making, rather than making tacit decisions.

Flexibility is another advantage of cartographic modeling. New considerations can be added easily and existing ones refined. For example, nonavoidance of open water bodies in the highway model is a major engineering oversight. In its current form, the model favors construction on lakes because they are flat and frequently visually hidden. This new requirement can be incorporated by identifying open water bodies as absolute barriers (i.e., infinite cost) when constructing the accumulation cost surface. The result will be routing of the minimal cost path around these areas of extreme cost.

Cartographic modeling also provides an effective structure for communicating both specific application considerations and fundamental principles. The flowchart structure provides a succinct format for communicating the processing considerations of specific applications. Model logic, assumptions, and relationships are readily apparent in the "box and line" schematic of processing flow. The instructional approach in this section developed a general mathematical framework and then elaborated on specific primitive operations within this structure. Numerous examples identified the operations with potential applications. Such a deductive approach establishes a comprehensive understanding of concepts that stimulates the development of applications. CONCLUSION

Land management has always required spatial information as its cornerstone process. Physical description of a management unit allows the conceptualization of its potential usefulness and constrains the list of possible management practices. Once a decision has been made and a plan implemented, additional spatial information is needed to evaluate its effectiveness. This strong allegiance between spatial information and effective management policy has become increasingly apparent with the exploding complexity of issues confronting our world's communities.

Statistical sampling designs involving geographic stratification have attempted to reduce the complexity of spatial data. This aggregation retains the quantitative aspects of the data necessary for most decision-making models but lacks spatial continuity. Traditional drafting-oriented procedures retain the spatial continuity of the data but are nonquantitative and difficult to incorporate into the decision-making process. Neither approach has capabilities for sophisticated analysis of complex spatial interrelationships.

The analytical process of both conventional statistical and drafting approaches inherently are limiting. The traditional statistical approach seeks to constrain the spatial variability within the data. However, it is the spatially induced variance of mapped data and their interactions that most often concern the land manager. The drafting approach, while spatially precise, is limited by the laborious procedures involved. In most instances, a final drafted map represents an implicit decision considering only a few policy strategies rather than the presentation of useful information to be assimilated in the decision process.

Digital cartographic modeling using fundamental map analysis operations addresses the needs of today's professionals. This approach, embodied in pMAP, identifies primitive operations that are independent of data and application. Similar to the structure of traditional algebra, these operations are logically sequenced on map variables to form complex analytical techniques.

These techniques are, in turn, logically sequenced to develop models that address specific applications. The maps used in an analysis specify the unique spatial character of a particular area under evaluation. The model structure expresses the interrelationships incorporated in the analysis and documents the factors and assumptions used in the process.

Spatial statistics and cartographic modeling are expressions of spatially consistent data. The techniques also allow for ease of model modification and the ability to rapidly simulate a variety of possible strategies. Within this analytic context, mapped data truly becomes spatial information. The Professional Map Analysis Package (pMAP) provides the analytic sophistication needed for effective planning and management in today's complex decision-making environment.

Note: BEYOND MAPPING: CONCEPTS, ISSUES AND ALGORITHMS IN GIS," by Joseph K. Berry (GIS World,Inc., Fort Collins, Colorado, 1993) is a collection of articles from the byline of the same title in GIS World magazine.