Beyond Mapping III

Topic 3 – Basic Techniques in Spatial Analysis (Further Reading)

 

Book_GeoTec

 

Map Analysis book

 

 

(Spatial Coincidence)

Key Concepts Characterize Unique Conditions — describes a technique for handling unique combinations of map layers (April 2006)

Use “Shadow Maps” to Understand Overlay Errors — describes how shadow maps of certainty can be used to estimate error and its propagation (September 2004)

 

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Key Concepts Characterize Unique Conditions

(GeoWorld, April 2006)  

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Back in the days of old GIS (a couple of decades ago) computer power was but a fraction of what sits on most desks today.  As a result, CPU cycles and online storage for map analysis were severely rationed.  Maps of a megabyte were relegated to super computers behind glass windows in air conditioned rooms. 

 

In these austere conditions programmers searched for algorithms and data structures that saved nanoseconds and kilobytes.  The concept of a Universal Polygon Coverage was a mainstay in vector-based processing.  By smashing a stack of relatively static map layers together a single map was generated that contained all of the son and daughter polygons.  This “compute once/use many” approach had significant efficiency gains as coincidence overlay was solved once then table query and math simply summarized the combinations as needed.

 

The top portion of figure 1 shows a simplified schematic of the Universal Polygon approach.  The lines defining the individual polygons on the three vector maps are intersected (analogous to throwing spaghetti on a wall) to identify various combinations of the input conditions as depicted in the large map in the middle.          

 

For example, consider the unique combination of A2, B1 and C1.  The intersections of the lines identify nodes that split the parent polygon boundaries into the segments defining the six son and daughter polygons of the combined conditions.  The result is a single spatial table containing all of the original delineations plus the derived coincidence overlay information.  Its corresponding attribute table can be easily searched for any combination of conditions and/or mathematically manipulated to generate a new field in the table—fast and efficient.

 

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Figure 1.  Comparison of the related concepts of Universal Polygon Coverage (vector) and Unique Conditions Matrix (grid).

 

This pre-processing technique also works for grid-based data (bottom portion of figure 1).  The Unique Conditions Matrix smashes a stack of grid layers together assigning a unique value for each possible combination of conditions.  For example, the A2, B1 and C1 combination (termed a cohort) is defined by the grid cell block shown in the right portion of the figure.  Grid location column= 12 and row= 12 is one of the forty member cohort.

 

Figure 2 depicts a natural extension of the procedure involving two tables—a Value Attribute Table and a Cohort Membership Table.  For example, given grid layers for travel-time for six stores classified into proximity classes (1= close, 2, 3 and 4= far) results in 4096 possible combinations.  Each combination, in turn, forms a single record (row) in the Value Attribute Table (VAT) file.  The record contains a unique cohort ID number and a field (column) for each of the base map conditions (figure 2, step 1).  The user can operate on this table to derive new information, such as the minimum travel-time to the closest store, which is appended as a new field (step 2). 

 

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Figure 2.  Steps linking unique conditions and cohort membership tables for efficient and fast coincidence analysis.

 

Note that the VAT records identify unique conditions occurring anywhere a project area whether the grid cells occur in few large clumps or widely dispersed.  The Cohort Membership Table (CMT) contains a sting of values identifying the column/row coordinates for each grid cell in the list.  Thoughtful organization of the list can implicitly carry information about the spatial patterns within and among the cohorts of cells.

 

Step 3 links the attribute and spatial tables through their common ID numbers.  Any of the fields in the VAT can be mapped by assigning the derived values to the cells defining each cohort in the CMT (step 4). 

 

Sounds easy and straight forward but there are a few caveats.  First, the technique only addresses point-by-point coincidence overlay and myopically ignores surrounding conditions.  Also, the conditions need to be fairly stable, such as proximity and slope.  Finally, it requires continuous data to be reassigned into few discrete classes on a relatively small number of map layers or the number of combinations explodes.  For example, 20 map layers with only four classification categories on each, results in over sixteen million possible cohorts; a number that challenges most database tables.

 

The advantage of the Unique Conditions technique in appropriate application settings is that you calculate once (VAT) and use many (CMT)—efficient and fast grid processing.  In the early years this was imperative for even the basic GIS models.  Today it is becoming imperative again due to the extent and complexity of sophisticated models.  For example, deriving a wildfire risk map for California at a 25 meter scale involves over 500 million grid cells.  Saving a few billion redundant computations and a few hundred gigabytes or so of storage can actually can add up—even in today’s “super computer on a desk” environment.

 

 

Use “Shadow Maps” to Understand Overlay Errors

(GeoWorld, September 2004)  

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Previous discussions in Topic 3 have focused on map overlay by describing some of the procedures, considerations and applications.  However, now is the moment of atonement—the pitfalls of introduced error and its propagation.  Keep in mind that there are two broad types of errors in GIS: those that are present in the encoded base maps and those that arise during map analysis.  Better field collection, encoding and handling of base data address the former and improved processing algorithms and procedures address the latter. 

 

Be realistic—soil or forest maps are just estimates of the actual conditions and geographic patterns of these features.  No one used a transit to survey the precise and sharp boundary lines that delineate the implied distinct spatial objects.  Soil and forest parcels aren’t discrete things in either space or time and significant judgment is used in forcing boundaries around them.  Under some conditions the guesses are pretty good; under other conditions, they can be pretty bad.  So how can the computer “see” where things are good and bad? 

 

That’s where the concept of a “shadow map of certainty” comes in and directs attention to both the amount and pattern of map certainty.  The left side of figure 1 depicts such an information sandwich of a typical soil map with a shadow map of certainty glued to its bottom.  In this way the relative certainty of the classification is known for each map location—look at the top map to see the soil classification, then peer through to the bottom map to see how likely that classification is at that location. 

 

In this instance, any location with 100 meters of a soil boundary is assigned only 50% certainty (orange) of correct classification while interior areas of large features are assigned 100% certainty (grey).  The assumption reflects the thought that “there is a soil boundary around here somewhere, but I am just not sure exactly where.” 

 

A similar shadow map of certainty can be developed for the forest information (right side of figure 1).  This simple estimate of certainty accounts for one photo interpreter reaching out to add a tree along the edge of a parcel, and another interpreter deciding not to.  Both interpreters “see” the interior of the forest parcel (certain) but discretion is used to form its border (less certain).  The 100m certainty buffer reflects a bit of wiggle-room for interpretation.

 

 

Figure 1.  Thematic maps with their corresponding shadow maps of certainty. The orange areas indicate “less certain” areas (.50 probability) that are adjacent to a boundary.

 

Remote sensing classification, however, provides a great deal more information about map certainty.  For example, the “maximum likelihood classifier” determines the probability that a location is one of a number of different forest types using a library of “spectral signatures” and multivariate statistics.  In a sense the computer thinks “given the color pattern for an area and knowledge of the various color patterns for forest types in the area, which color most closely matches the appearance of the area”—analogous to the manual interpretation process. 

 

After a nanosecond or so, the computer decides which known pattern is the closest, classifies it as that forest type, and then moves on to consider the next grid cell.  For example, riparian and aspens tend to reflect lighter greens while pines and firs tend to reflect darker greens. The import point is that the computer isn’t “seeing” subtle color differences; it is analyzing numerical values to calculate the probability that a location is one of any of the possible choices. 

 

Traditionally, the approach simply chooses the most likely spectral signature, classifies it, and then throws away the information on relative certainty of the classification.  Heck, the historical objective was to produce a map, not data.  In addition, a spline function often is used to inscribe a seemingly precise line around groups of similar classified cells so the results look more like a manually drafted map.  The result is a set of spatially discrete objects implying perfect data for a warm and fuzzy feeling, but disregarding the spatial resolution and classification probabilities employed—sort of like throwing the bathwater out with the baby.

 

As GIS technology matures, the ability to discern map certainty will become increasingly important.  There are several mechanisms for viewing a shadow map of certainty information.  The simplest is to include the certainty value as a mouse-over that displays the certainty value as the cursor moves about a map.  Another sets the hue/color for different forest types (i.e., red for pine) then drives the brightness gun based on the certainty value at each location.  You see a gradient of washed-out pink for areas of very uncertain pine classification through a bright red for areas whose classification are very certain.

 

 

Figure 2.  A shadow map of certainty for map overlay is calculated by multiplying the individual certainty maps to derive a value (joint probability) indicating the relative confidence in the coincidence among map layers.

 

However, the real power of a shadow map of certainty is in addressing processing errors that occur during map overlay.  Common sense suggests that if you overlay a fairly uncertain map with another fairly uncertain map, chances are the resulting coincidence map is riddled with even more uncertainty.  But the problem is more complex, as it is dependent on the intersection of the unique spatial patterns of certainty of the two (or more) maps.

 

Consider overlaying the soils and forest maps as depicted in figure 2.  A simple map overlay considers just the thematic map information and summarizes the coincidence between the two maps.  Note that both of the “speared” locations identify Lowland soils occurring with Aspen forest cover. 

 

Now consider the effect of certainty propagation.  The location on the left is certain on both maps, so the coincidence is certain (1.00 *1.00= 1.00).  However, the location on the right is less certain on both maps, so the chance that it contains both lowland soils and aspen is very uncertain certain (.50 * .50= .25).

 

The ability to quantify and account for map certainty is a critical step in moving GIS beyond mapping.  As map analysis becomes more entrenched in advanced applications, the ability to tell where GIS products are good or bad becomes at least as important as their graphical display and Internet availability.  Heck, there might a bunch of good looking maps out there but can you risk your professional life on them?

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