Beyond Mapping
|
Map
Analysis book with companion CD-ROM for hands-on exercises and further reading |
GIS Software's Changing
Roles — discusses the evolution of GIS software and
identifies important trends
Determining Exactly Where
Is What — discusses
the levels of precision and accuracy
Finding Common Ground in
Paper and Digital Worlds — describes
the similarities and differences in information and organization between
traditional paper and digital maps
Resolving Map Detail — discusses
the factors that determine the “informational scale” digital maps
Referencing the
Future — describes
current and alternative approaches for referencing geographic and abstract
space
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______________________________
(GeoWorld,
September 1998, pg. 28-30)
Although
This point was struck home at a recent visit to Disneyland. The newest ride subjects you to a seemingly endless
harangue about the future of travel while you wait in line for over an
hour. The curious part is that the
departed Walt Disney himself is outlining the future through video clips from
the 50’s. The dream of futuristic travel
(application) hasn’t changed much and the 90’s practical reality (tool), as
embodied in the herky-jerky ride, is a long way from fulfilling the vision.
What impedes the realization of a
technological dream is rarely a lack of vision, but the nuts and bolts needed in
its construction. In the case of
With the 80’s came the renaissance of modern computers and with it the hardware
and software environments needed by
Working within a
From an application developer’s perspective the floodgates had opened. From an end user’s perspective, however, a
key element still was missing—the gigabytes of data demanded in practical
applications. Once again
But another less obvious impediment hindered progress. As the comic strip character Pogo might say,
“we have found the enemy and it’s us.”
By their very nature, the large
The 90’s saw both the data logjam and
So where are we now? Has the role of
MapInfo’s MapX and ESRI’s MapObjects are tangible
Like using a Lego set, application developers can apply the “building
blocks” to construct specific solutions, such as a real estate application that
integrates a multiple listing geo-query with a pinch of spatial analysis, a dab
of spreadsheet simulation, a splash of chart plotting and a sprinkle of report
generation. In this instance,
In its early stages,
The distinction between computer and application specialist isn’t so much their
roles, as it is characteristics of the combined product. From a user’s perspective the entire
character of a
As the future of
Determining Exactly Where Is What
(GeoWorld, February 2008, pg. 14-15)
The Wikipedia defines Accuracy as “the degree of veracity” (exactness) while Precision as “the degree of
reproducibility” (repeatable). It uses an archery target as an analogy to
explain the difference between the two terms where measurements are compared to
arrows shot at the target (left side of figure 1). Accuracy describes the closeness of arrows to
the bull’s-eye at the target center (actual/correct). Arrows that strike closer to the bullseye are
considered more accurate.
Precision, on the
other hand, relates to the size of the cluster of arrows. When the arrows are grouped tightly together,
the cluster is considered precise since they all strike close to the same spot,
if not necessarily near the bullseye. The
measurements are precise, though not necessarily accurate.
Figure 1. Accuracy refers to
“exactness” and Precision refers to “repeatability”of data.
However, it is not
possible to reliably achieve accuracy in individual measurements without
precision. If the arrows are not grouped
close to one another, they cannot all be close to the bullseye. While their average position might be an accurate estimation of the
bullseye, the individual arrows are inaccurate.
So what does all
this have to do with
Whereas
Figure 2
illustrates the two-fold consideration of Precise
Placement of coordinate delineation and Accurate
Assessment of attribute descriptor for three photo interpreters. The upper-right portion superimposes three
parcel delineations with Interpreter B outlining considerably more area than
Interpreters A and C—considerable variation in precision. The lower portion of the figure indicates
differences in classification with Interpreter B assigning Ponderosa pine as
the vegetation type—considerable variation in accuracy.
Figure 2. In mapped data,
precision refers to placement whereas accuracy refers to classification.
Many
In addition, our
paper map legacy of visualizing maps frequently degrades precision/accuracy in
detailed mapped data. For example, a
detailed map of slope values containing decimal point differences in terrain
inclination can be easily calculated from a elevation surface. But the detailed continuous spatial data is
often aggregated into just a few discrete categories so humans can easily
conceptualize and “see” the information—such as areas of gentle, moderate and
steep terrain. Another example is the
reduction of the high precision/accuracy inherent in a continuous “proximity to
roads” map to that of a discrete “road buffer” map that simply identifies all
locations within a specified reach.
Further thought
suggests an additional consideration of
As you might
suspect, different groups have differing perspectives on the interpretation and
relative importance of the routing criteria.
For example, homeowners might be most concerned about Housing Density
and Visual Exposure; environmentalists most concerned about Road Proximity and
Sensitive Areas; and engineers most concerned about Housing Density and Road
Proximity. Executing the model for these
differences in perspective (relative importance of the criteria) results in
three different preferred routes.
Figure 3. Maps derived by
The
lower-left portion of figure 3 shows the
spread of the three individual solutions.
One isn’t more precise/accurate than another, just an expression of a particular
prespective of the solution. The lower-right side of the figure suggests yet
another way to represent the solution using the simple average of the three
preference surfaces to identify an overall route and its optimal corridor—sort
of analogous to averaging a series of
The take home
from this discussion is that precision and acurracy aren’t the same thing and that
the terms can take on differrent meanings for different types of maps and
application settings. There are at least
three different levels of precision/accuracy—1) “Where is Where” considering just precise placement, 2) “Where is What” considering placement and
classification, and 3) “Where is What, if
you assume…” considering placement, classification and interpretation/logic/understanding/judgement
ingrained in spatial reasoning. Before
_____________________________
Author’s Note: Related discussion of routing
model considerations and procedures is in Topic 8, Spatial Model Example in the
book Map Analysis (
Finding Common Ground in Paper and Digital Worlds
(GeoWorld, February 2007, pg. 28-30)
In the real world, landscapes are composed rocks, dirt, water, green stuff and furry/feathered friends. In a “paper world” these things are represented by words, tables and graphics. The traditional paper map is a graphical representation with inked lines, shadings and symbols used to locate landscape features using three basic building blocks— Points, Lines and Areas. For example, a typical water map might identify a well as a dot, a stream as a squiggle and a lake as a blue blob (figure 1). Each feature is considered a well-defined “discrete spatial object” with unique spatial character, positioning and dimension.
Figure 1. Traditional and Extended Map Features.
In geometry a point is considered dimensionless, however the corresponding concept in cartography is a dot of ink having a physical dimension of a few inches to several miles depending on the scale of a paper map. Similarly, a line in mathematical theory has only length but is manually mapped as a thin serpentining polygon of the pen’s width. An area feature has both length and width in two-dimensional space. The interplay of mapping precision and accuracy in a digital world involves a discussion of scale and resolution reserved for next month. For now, let’s consider the revolutionary changes in map form and content brought on by the digital map as outlined in the rest of figure 1.
For thousands of years, manual cartography has been limited to characterizing all geographic phenomena as discrete 2-dimensional spatial objects. However many map variables, such as elevation, change continuously and representation as contour lines suggests a nested series of flat layers like a wedding cake instead of the actual continuously undulating terrain. The introduction of a grid-based data structure provides for a new basic building block—a map Surface of continuously changing values throughout geographic space.
Another extension to the building blocks is Volumes that track length, width and depth in characterizing discrete or continuous variables in 3-dimensional space. For example, the L,W,D coordinates identify a specific location in a lake and a fourth value (attribute) can identify its temperature, turbidity, salinity or other condition.
A hyper-Volume (or hyper-point, -line, -area or -surface) introduces time as an additional abstract coordinate. For example, the weekly water volume of a reservoir might be tracked by L,W,D,T coordinates identifying a location in 3-dimensional space as well as time, combined with a fifth value indicating whether water is present or not. This conceptual extension is a bit tricky and provides discussion fodder about mixed referencing units (e.g., meters and minutes) for a later column. However, the result is a discrete volumetric map feature that shrinks and expands throughout a year—a dynamic spatial entity that at first appears to violate orthodox mapping commandments.
Another mind-bend brought on by the digital map is the concept of fuzzy-Features. This idea tracks the certainty of a feature or condition at each map location. For example, the boundary line of a soil polygon is a subjective interpretation, while its actual edge could be considerable distance away—“the boundary is likely here (high probability) but could be over there (low probability).” Another fuzzy example is a classified satellite image where statistical probabilities are used to determine which cover type is most likely.
Taken to the hilt, one can conceptualize a data structure that carries L,W,D,T and A,P (attribute and probability) descriptors that identify a location in space and time, as well as characterize its most likely condition, next most likely, and so on—sort of a sandwich of probable conditions. Such a representation challenges the infallible paradigm of mapping but opens a whole new world of error propagation modeling.
Figure 2. Basic Vector and Raster Data Structure Considerations.
Whereas volumes, hyper-volumes and fuzzy-features define the current realm of GIS researchers, an understanding of contemporary approaches for characterizing points, lines, and areas is necessary for all GIS users. Figure 2 outlines the two fundamental approaches—vector and raster (see Author’s Note).
A Point defined by X,Y coordinates in vector, and a Cell defined by Col,Row indices in raster, form the basic data structure units—smallest addressable unit of space. Lines are formed by mathematically connecting points (vector) or identifying all of the adjoining cells containing a line (raster). Areas are defined by a set of points that define a closed line encompassing a feature (vector) or by all of the contiguous cells containing a feature (raster).
While spatial precision is a major operational difference between vector and raster systems, how they characterize space is important in understanding limitations and capabilities. Vector precisely identifies critical points along a line, but connections are implied. Raster, on the other hand, identifies all pre-defined cells containing a line without any implied gaps. Similarly, vector precisely stores an area’s boundary but implies its interior (must calculate); raster stores the interior but implies the boundary (must calculate).
The differences in “what is defined” and “what is implied” determines just about everything in GIS technology except maybe the color pallet for display—data structure, storage requirements, algorithms, coding and ultimately appropriate use. Vector systems precisely and efficiently store traditional discrete map objects, such as underground cables and property boundaries (mapping and inventory). Raster systems, on the other hand, predefine continuous geographic space for rapid and enhanced processing of map layers (analysis and modeling).
So how do you think vector and raster systems store surfaces, volumes, hyper-volumes and fuzzy-features?… very poorly, or not at all for vector systems. However raster systems pre-define all of a project area (no gaps) by carrying a thematic value for each cell in a 2-dimensional storage matrix to form a continuous map surface. For volumes, a third geographic referencing index is added to extend the 2D cells to 3D cubes in geographic space defined by their X,Y,Z position in the storage matrix.
A similar expansion is used for hyper-volumes with four indices (X,Y,Z,T) identifying the “position,” except in this instance an abstract space is implied due to the differences in geographic and time units. Information about fuzzy-features can be coded into a compound attribute value describing any map feature, where the first few digits identify the character/condition at a location with the trailing two digits identifying the certainty of classification.
The bottom line is that tomorrow’s maps aren’t simply colorful electronic versions of your grandfather’s maps. The digital map is an entirely different beast supporting radically new mapping approaches, perspectives, opportunities and responsibilities.
_____________________________
Author’s Note:
Topic 6, Alternative Data Structures, in Spatial Reasoning for
Effective GIS (Berry 1995, Wiley) contrasts vector and raster data
structures and describes related alternative structures including TIN,
Quadtree, Rasterized Lines and Vectorized Cells.
(GeoWorld, March 2007, pg. 28-30)
One of the most fundamental concepts in the paper map world is Geographic Scale—the relationship between a distance on a map and its corresponding distance on the earth. In equation form, scaleratio= (map distance / ground distance) but is often expressed as a representative fraction (RF), such as scaleRF= 1:63,360 meaning 1 inch on the map represents 63,360 inches (or 1 mile) on the earth’s surface.
However in the digital map world, this traditional concept of scale doesn’t exist. While at first this might seem like geographical heresy, note that the “map distance” component of the relationship is assumed to be fixed as ink marks on paper. In a GIS, however, the map features are stored as organized sets of numbers representing their spatial position (coordinates for “where”) and thematic attribute (map values for “what”). One can zoom in and out on the data thereby creating a continuous gradient of geographic scales in the resulting display or hardcopy plot.
Hence geographic scale is a function of the display, not an inherent property of the digital mapped data. What is important is the implied concept of informational scale, or Resolution—the ability to discern detail. Traditionally it is implicit that as geographic scale decreases, resolution also diminishes since drafted feature boundaries must be smoothed, simplified or not shown at all due to the width of the inked lines.
However in a GIS, the concept of resolution is explicit. In fact there are five types of resolution
that need to be considered—Spatial, Map, Thematic, Temporal and Model. Spatial
Resolution is the most basic and identifies the “smallest addressable unit”
of geographic space (figure 1). For
point features, the X,Y coordinates (vector) and cell size (raster) determine
the smallest addressable unit.
Figure 1. Spatial Resolution describes the level of positional detail
used to track a geographic pattern or distribution.
For line features in vector, however, the smallest addressable unit is the line segment with larger segments capturing less detail as the implied straight line misses the subtle wiggles and waggles of a pattern. Similarly, large grid cells capture less linear detail than smaller cells.
For polygon features in vector, an entire polygon represents the smallest addressable unit as the boundary needs to be completed before the implied interior condition can be identified. In raster, the smallest addressable unit is defined by the cell size as the condition is carried for each of the cells comprising the interior and edge of a polygon feature.
The concept of spatial resolution easily extends to the level of spatial aggregation or Map Resolution that identifies the “smallest physical grouping” of a map theme (figure 2). For example, a high resolution forest map might identify individual trees (very small polygons of crown extent), whereas more generally, numerous trees are used to identify a forest parcel of several acres that ignores the scattered tree occurrences. The size of the minimum polygon is determined by the interpretation process with smaller groupings capturing more detail of the pattern and distribution.
Figure 2. Map Resolution describes the level of physical aggregation
used to depict a geographic pattern or distribution.
Thematic Resolution identifies the “smallest classification grouping” of a map theme. For example, a simple forest/non-forest map might provide a sufficient description of vegetation for some uses and this coarse classification has appeared for years as green on USGS topographic sheets. However, resource managers require a higher thematic resolution of vegetation cover and expand the classification scheme to include species, age, stocking level and other characteristics. The result is a finer subdivision of a generalized forest area into smaller more detailed parcels (figure 3).
Figure 3. Thematic Resolution describes the level of classification
aggregation used to depict a geographic pattern or distribution.
A forth consideration involves Temporal Resolution that identifies the frequency, or time-step of map update. Some data types, such as geological and landform maps, change very slowly and do not need frequent revision. A city planner, on the other hand, needs land use maps that are updated every couple of years and include future development sites. A retail marketer needs even higher temporal resolution and will likely update sales and projection figures on a monthly, weekly or even daily basis.
Model Resolution is the least defined and involves factors affecting the level of detail used in creating a derived map, such as an optimal corridor for an electric transmission line or areas of suitable wildlife habitat. Model resolution considers detail ingrained in 1) the interpretation/analysis assumptions (logic) and 2) the algorithms/procedures (processing) used in implementing a spatial model. For example, a proposed transmission line could be routed considering just terrain slope for a low model resolution, or extended to include other engineering factors (soils, road proximity, etc.), environmental concerns (wetlands, wildlife habitats, etc.) and social considerations (visual exposure, housing density, etc.) for much higher model resolution.
So why should we care about digital map resolution? Because accounting for informational scale is just as important as adjusting for a common geographic scale and projection when interacting with a stack of maps. Our paper map heritage focused on descriptive mapping (inventory of physical phenomena) whereas an increasing part of the GIS revolution focuses on prescriptive mapping (spatial relationships of physical and cognitive interactions). This “thinking with maps” requires a thorough understanding of the spatial, map, thematic, temporal and model resolutions of the maps involved or you will surely be burned.
(GeoWorld, April 2007, pg. 28-30)
Geo-referencing is the cornerstone of GIS. In the mid-1600s the French mathematician, René Descartes established the Cartesian
coordinate system that is still in use today.
The system determines the
location of each point in a plane as defined by two numbers—its x-coordinate
and y-coordinate. A third z-coordinate is used to extend the
system to 3-dimensional geographic space.
In mapping, these coordinates reference a refined ellipsoid
(geodetic datum) that can be
conceptualized as a curved surface approximating the mean ocean surface of the
earth.
The location and
shape of map features can be established by X and Y distances measured along
flattened portions of the reference surface (figure 1). The familiar Universal Transverse
Mercator (UTM) coordinates represent
E-W and N-S movements in meters along the plane. The rub is that UTM zones are need to break
the curved earth surface into a series of small flat, projected subsections
that are difficult to edge-match.
Figure 1. Geographic referencing uses three coordinates to locate map
features in real world space.
A variant of the traditional referencing system uses spherical coordinates that are based on solid angles measured from the center of the earth. This natural form for describing positions on a sphere is defined by three coordinates—an azimuthal angle (θ) in the X,Y plane from the x-axis, the polar angle (φ) from the z-axis, and the radial distance (r) from the earth’s center (origin). The advantage of a spherical referencing system is that it is seamless throughout the globe and doesn’t require projecting to a localized flat plane.
Digital map storage is rapidly moving toward spherical referencing that
uses latitude and longitude in decimal degrees for internal storage and
on-the-fly conversion to any planar projection.
This radical change from our paper map heritage is fueled by ubiqutious use of GPS and a desire for
global databases that easily walk across political and administrative
boundaries.
Since the digital
map is a radical departure from the paper map, other alternative referencing
schemes are possible. For example,
hexagons can replace the Cartesian grid squares we have used for hundreds of
years (top portion of figure 2). The
hexagon naturally nests to form a continuous network like a beehive’s
honeycomb. An important property of a
hexagon grid is that it better represents curved surfaces than a square grid— a
socccer ball stiched from squares wouldn’t roll the same.
However the most
important property is that a hexagon has six sides instead of four. The added directions provide a foothold for
more precise measurement of continuous movement— one can turn right- and
left-oblique as well as just right and left.
Traditional routing moels using Least Cost Path would benefit greatly.
Expanding to 3-dimensional geographic space provides for polyhedrons to replace cubes. For example, a dodecahedron is a nesting twelve-sided object that can be used instead of the six-sided cube. Weather and ground water flow modeling could be greatly enhanced by the increased options for transfer from a location to its larger set of adjoining locations. The computations for cross-products of vectors, such as warp-speed cruise missiles, could be greatly assisted as they are affected by different atmospheric conditions and evasive trajectories.
Figure 2. Alternative
referencing systems and abstract space characterization are possible through
the digital nature of modern maps.
Another extension involves the use of abstract space (bottom portion of figure 2). For example, the Z-coordinate can be replaced with an attribute value to generate a map surface, such as customer density. In this instance, the abstract referencing is a mixture of spatial and attribute “coordinates” and doesn’t imply 3-dimensional, real word geographic occurrences. Instead, it relates geography and conditions in an extremely useful way for conceptualizing patterns. Normalization along the abstract coordinate axis is an important consideration for both visualization and analysis.
This brings us to space-time referencing. During a recent panel discussion I was challenged for suggesting such a combination is possible within a GIS. The idea has been debated for years by philosophers and physicists but H.G. Wells’ succinct description is one of the best—
'Clearly,' the
Time Traveller proceeded, 'any real body must have extension in four
directions: it must have Length, Breadth, Thickness, and - Duration. But
through a natural infirmity of the flesh, which I will explain to you in a
moment, we incline to overlook this fact. There are really four dimensions,
three which we call the three planes of Space, and a fourth, Time. There is,
however, a tendency to draw an unreal distinction between the former three
dimensions and the latter, because it happens that our consciousness moves
intermittently in one direction along the latter from the beginning to the end
of our lives.' (Chapter 1, Time
Machine).
The upshot seems to be that a fourth dimension exists (see Author’s Notes), it is just you can’t go there in person. But a GIS can easily take you there - conceptually. For example, an additional abstract “coordinate” representing time can be added to form a 3-dimentional data matrix. The GIS picks off the customer density data for the first “page” and displays it as in the figure. Then it uses the data on the on the second page (one time step forward) and displays it. This is repeated to cycle through time and you see an animation where the peaks and valleys of the density surface move with time.
So animation enables your to move around a city (X,Y) viewing the space-time relationship of customer density (A). In a similar manner you could evaluate a forest “green-up” model to predict re-growth at a series of time steps after harvesting to look into future landscape conditions. Or you can watch the progression over time of ground water pollutant flow in 3D space (4D data matrix) using a semi-transparent dodecahedron solid grid just for fun and increased modeling accuracy. In fact, it can be argued that GIS is inherently n-dimensional when you consider a map stack of multiple attributes and time is simply another abstract dimension.
My suspicions are that revolutions in referencing will be a big part of GIS’s frontier in the 2010s. See you there?
_____________________________
Author’s Notes: an excellent online reference for the
basic geometry concepts underlying traditional and future geo-referencing
techniques is the Wolfram MathWorld pages, such as the posting describing the dodecahedron at http://mathworld.wolfram.com/Dodecahedron.html;
a