# Beyond Mapping III

Topic 26:  Assessing Spatially-Defined Neighborhoods

Map Analysis book with

companion CD-ROM for

hands-on exercises

discusses approaches for calculating slope and profile

describes a procedure for deriving a customer density surface

discusses techniques for smoothing “salt and pepper” results and deriving probability surfaces from aggregated incident records

describes a technique for “filling-in” information from surrounding data into no-data locations

Nearby Things Are More Alike  use of decay functions in weight-averaging surrounding conditions

Filtering for the Good Stuff  investigates a couple of spatial filters for assessing neighborhood connectivity and variability

discusses the three types of roving windows— fixed, weighted and dynamic.

Note: The processing and figures discussed in this topic were derived using MapCalcTM software.  See www.innovativegis.com to download a free MapCalc Learner version with tutorial materials for classroom and self-learning map analysis concepts and procedures.

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(GeoWorld, October 2005, pg. 18-19)

This and the following sections investigate a set of analytic tools concerned with summarizing information surrounding a map location.  Technically stated, the processing involves “analysis of spatially defined neighborhoods for a map location within the context of its neighboring locations.”  Four steps are involved in neighborhood analysis— 1) define the neighborhood, 2) identify map values within the neighborhood, 3) summarize the values and 4) assign the summary statistic to the focus location.  Then repeat the process for every location in a project area.

The neighborhood values are obtained by a “roving window” moving about a map.  To conceptualize the process, imagine a French window with nine panes looking straight down onto a portion of the landscape.  If your objective was to summarize terrain steepness from a map of digital elevation values, you would note the nine elevation values within the window panes, and then summarize the 3-dimensional surface they form.

Now imagine the nine values become balls floating at their respective elevation.  Drape a sheet over them like the magician places a sheet over his suspended assistant (who says GIS isn't at least part magical).  There it is— surface configuration.  All that is left is to ; summarize the lumps and bumps formed by the ghostly sheet by reducing the nine values to a single value characterizing the surface configuration.

Figure 1.  At a location, the eight individual slopes can be calculated for a 3x3 window and then summarized for the maximum, minimum, median and average slope.

Figure 1 shows a small portion of a typical elevation data set, with each cell containing a value representing its overall elevation.  In the highlighted 3x3 window there are eight individual slopes, as shown in the calculations on the right side of the figure.  The steepest slope in the window is 52% formed by the center and the NW neighboring cell.  The minimum slope is 11% in the NE direction.

To get an appreciation of this processing, shift the window one column to the right and, on your own, run through the calculations using the focal elevation value of 459.  Now imagine doing that a million times as the roving window moves an entire project area—whew!!!

But what about the general slope throughout the entire 3x3 analysis window?  One estimate is 29%, the arithmetic average of the eight individual slopes.  Another general characterization could be 30%, the median of slope values.  But let's stretch the thinking a bit more.  Imagine that the nine elevation values become balls floating above their respective locations, as shown in Figure 2.  Mentally insert a plane and shift it about until it is positioned to minimize the overall distances from the plane to the balls.  The result is a "best-fitted plane" summarizing the overall slope in the 3x3 window.

Figure 2.  Best-Fitted Plane and Vector Algebra can be used to calculate overall slope.

Techy-types will recognize this process as similar to fitting a regression line to a set of data points in two-dimensional space.  In this case, it’s a plane in three-dimensional space.  There is an intimidating set of equations involved, with a lot of Greek letters and subscripts to "minimize the sum of the squared deviations" from the plane to the points.  Solid geometry calculations, based on the plane's "direction cosines," are used to determine the slope (and aspect) of the plane.

Another procedure for fitting a plane to the elevation data uses vector algebra, as illustrated in the right portion of Figure 2.  In concept, the mathematics draws each of the eight slopes as a line in the proper direction and relative length of the slope value (individual vectors).  Now comes the fun part.  Starting with the NW line, successively connect the lines as shown in the figure (cumulative vectors).  The civil engineer will recognize this procedure as similar to the latitude and departure sums in "closing a survey transect."  The length of the “resultant vector” is the slope (and direction is the aspect).

In addition to slope and aspect, a map of the surface profiles can be computed (see figure 3).  Imagine the terrain surface as a loaf of bread, fresh from the oven.  Now start slicing the loaf and pull away an individual slice.  Look at it in profile concentrating on the line formed by the top crust.  From left to right, the line goes up and down in accordance with the valleys and ridges it sliced through.

Use your arms to mimic the fundamental shapes along the line.  A 'V' shape with both arms up for a valley.  An inverted 'V' shape with both arms down for a ridge.  Actually there are only nine fundamental profile classes (distinct positions for your two arms).  Values one through nine will serve as our numerical summary of profile.

Figure 3.  A 3x1 roving window is used to summarize surface profile.

The result of all this arm waving is a profile map— the continuous distribution terrain profiles viewed from a specified direction.  Provided your elevation data is at the proper resolution, it's a big help in finding ridges and valleys running in a certain direction.  Or, if you look from two opposing directions (orthogonal) and put the profile maps together, a location with an inverted 'V' in both directions is likely a peak.

There is a lot more to neighborhood analysis than just characterizing the lumps and bumps of the terrain.  What would happen if you created a slope map of a slope map?  Or a slope map of a barometric pressure map?  Or of a cost surface?  What would happen if the window wasn't a fixed geometric shape?  Say a ten minute drive window.  I wonder what the average age and income is for the population within such a bazaar window?  Keep reading for more on neighborhood analysis.

Milking Spatial Context Information

(GeoWorld, November 2005, pg. 18-19)

The previous discussion focused on procedures for analyzing spatially-defined neighborhoods to derive maps of slope, aspect and profile.  These techniques fall into the first of two broad classes of neighborhood analysis—Characterizing Surface Configuration and Summarizing map values (see figure 1).

Figure 1.  Fundamental classes of neighborhood analysis operations.

It is important to note that all neighborhood analyses involve mathematical or statistical summary of values on an existing map that occur within a roving window.  As the window is moved throughout a project area, the summary value is stored for the grid location at the center of the window resulting in a new map layer reflecting neighboring characteristics or conditions.

The difference between the two classes of neighborhood analysis techniques is in the treatment of the values—implied surface configuration or direct numerical summary.  Figure 2 shows a direct numerical summary identifying the number of customers within a quarter of a mile of every location within a project area.

Figure 2.  Approach used in deriving a Customer Density surface from a map of customer locations.

The procedure uses a “roving window” to collect neighboring map values and compute the total number of customers in the neighborhood.  In this example, the window is positioned at a location that computes a total of 91 customers within quarter-mile.

Note that the input data is a discrete placement of customers while the output is a continuous surface showing the gradient of customer density.  While the example location does not even have a single customer, it has an extremely high customer density because there are a lot of customers surrounding it.

The map displays on the right show the results of the processing for the entire area.  A traditional vector GIS forces the result into a set of 2D contour intervals stored as discrete polygon spatial objects—1-10 customer range, 10-20, 20-30, etc.  The 3D surface plot, on the other hand, shows all of the calculated spatial detail—mountains of high customer density and valleys of low density.  An importance difference is that the vector representation aggregates the results, whereas the grid representation contains all of the detailed information.

Figure 3.  Calculations involved in deriving customer density.

Figure 3 illustrates how the information was derived.  The upper-right map is a display of the discrete customer locations of the neighborhood of values surrounding the “focal” cell.  The large graphic on the right shows this same information with the actual map values superimposed.  Actually, the values are from an Excel worksheet with the column and row totals indicated along the right and bottom margins.  The row (and column) sum identifies the total number off customers within the window—91 total customers within a quarter-mile radius.

This value is assigned to the focal cell location as depicted in the lower-left map.  Now imagine moving the “Excel window” to next cell on the right, determine the total number of customers and assign the result—then on to the next location, and the next, and the next, etc.  The process is repeated for every location in the project area to derive the customer density surface.

The processing summarizes the map values occurring within a location’s neighborhood (roving window).  In this case the resultant value was the sum of all the values.  But summaries other than Total can be used—Average, StDev, CoffVar, Maximum, Minimum, Median, Majority, Minority, Diversity, Deviation, Proportion, Custom Filters, and Spatial Interpolation.  The remainder of this series will focus on how these techniques can be used to derive valuable insight into the conditions and characteristics surrounding locations—analyzing their spatially-defined neighborhoods.

(GeoWorld, January 2006, pg. 16-17)

A couple of the procedures used in the wildfire modeling warrant “under-the-hood” discussion neighborhood operations—1) smoothing the results for dominant patterns and 2) deriving wildfire ignition probability based on historical fire records.

Figure 1.  Smoothing eliminates the “salt and pepper” effect of isolated calculations to uncover dominant patterns useful for decision-making.

Figure 1 illustrates the effect smoothing raw calculations of wildfire risk.  The map in the upper-left portion depicts the fragmented nature of the results calculated for a set individual 30m grid cells.  While the results are exacting for each location, the resulting “salt and pepper” condition is overly detailed and impractical for decision-making.  The situation is akin to the old adage that “you can’t see the forest for the trees.”

The remaining three panels show the effect of using a smoothing window of increasing radius to average the surrounding conditions.  The two-cell reach averages the wildfire risk within a 13-cell window of slightly more than 2.5 acres.  Five and ten-cell reaches eliminate even more of the salt-and-pepper effect.  An eight-cell reach (44 acre) appears best for wildfire risk modeling as it represents an appropriate resolution for management.

Another use of a neighborhood operator is establishing fire occurrence probability based on historical fire records.  The first step in solving this problem is to generate a continuous map surface identifying the number of fires within a specified window reach from each map location.  If the ignition locations of individual fires are recorded by geographic coordinates (e.g., latitude/longitude) over a sufficient time period (e.g., 10-20 years) the solution is straightforward.  An appropriate window (e.g., 1000 acres) is moved over the point data and the total number of fires is determined for the area surrounding each grid cell.   The window is moved over the area to allow for determination of the likelihood of fire ignition over an area based on the historic ignition location data.  The derived fire density surface is divided by the number of cells in window (fires per cell) and then divided by the number of years (fires per cell per year).  The result is a continuous map indicating the likelihood (annualized frequency) that any location will have a wildfire ignition.

The reality of the solution, however, is much more complex.  The relative precision of recording fires differs for various reporting units from specific geographic coordinates, to range/township sections, to zip codes, to entire counties or other administrative groupings.  The spatially aggregated data is particularly aggravating as all fires within the reporting polygon are represented as occurring at the centroid of a reporting unit.  Since the actual ignition locations can be hundreds of grid cells away from the centroid, a bit of statistical massaging is needed.

Figure 2 summarizes the steps involved.  The reporting polygon is converted to match the resolution of the grid used by the wildfire risk model and each location is assigned the total number of fires occurring within its reporting polygon (#Fires).  A second grid layer is established that assigns the total number of grid cells within its reporting polygon (#Cells).  Dividing the two layers uniformly distributes the number of fires within a reporting unit to the each grid cell comprising the unit (Fires/Cell).

Figure 2.  Fires per cell is calculated for each location within a reporting unit then a roving window is used to calculate the likelihood of ignition by averaging the neighboring probabilities.

The final step moves a roving window over the map to average the fires per cell as depicted on the right side of the figure.  The result is a density surface of the average number of fires per cell reflecting the relative size and pattern of the fire incident polygons falling within the roving window.  As the window moves from a location surrounded by low probability values to one with higher values the average probability increases as a gradient that tracks the effect of the area-weighted average.

Figure 3.  A map of Fire Occurrence frequency identifies the relative likelihood that a location will ignite based on historical fire incidence records.

Figure 3 shows the operational results of stratifying the area into areas of uniform likelihood of fire ignition.  The reference grid identifies PLSS sections used for fire reporting, with the dots indicating total number of fires for each section.  The dark grey locations identify non-burnable areas, such as open water, agriculture lands, urbanization, etc.  The tan locations identify burnable areas with a calculated probability of zero.  Since zero probability is a result of the short time period of the recorded data the zero probability is raised to a minimum value.  The color ramp indicates increasing fire ignition probability with red being locations having very high likelihood of ignition.

It is important to note that interpolation of incident data is inappropriate and simple density function analysis only works for data that is reported with specific geographic coordinates.  Spatially aggregated reporting requires the use of the area-weighted frequency technique described above.  This applies to any discrete incident data reporting and analysis, whether wildfire ignition points, crime incident reports, product sales.  Simply assigning and mapping the average to reporting polygons just won’t cut it as geotechnology moves beyond mapping.

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Author’s Note:  Discussion based on wildfire risk modeling by Sanborn, www.sanborn.com/solutions/fire_management.htm.  For more information on wildfire risk modeling, see GeoWorld, December 2005, Vol.18, No. 12, 34-37, posted online at http://www.geoplace.com/uploads/FeatureArticle/0512ds.asp or click here for article with enlarged figures and .pdf hardcopy.

Extending Information into No-Data Areas (GeoWorld, July 2011)

I am increasingly intrigued by wildfire modeling.  For a spatial analysis enthusiast, it has it all— headlines grabbing impact, real-world threats to life and property, action hero allure, as well as a complex mix of geographically dependent “driving variables” (fuels, weather and topography) and extremely challenging spatial analytics.

However with all of their sophistication, most wildfire models tend to struggle with some very practical spatial considerations.  For example, figure 1 identifies an extension that “smoothes” the salt and pepper pattern of the individual estimates of flame length for individual 30m cells (left side) into a more continuous surface (right side).  This is done for more than cartographic aesthetics as surrounding fire behavior conditions are believed to be important.  It makes sense that an isolated location with predicted high flame length conditions adjacent to much lower values is presumed to be less likely to attain the high value than one surrounded by similarly high flame length values.  Also the mixed-pixel and uncertainty effects at the 30m spatial resolution suggest using a less myopic perspective.

Figure 1. Raw Flame Length values are smoothed to identify the average calculated lengths within a specified distance of each map location— from point-specific condition to a localized condition that incorporates the surrounding information (smoothing).

The top right portion of the figure shows the result of a simple-average 5-cell smoothing window (150m radius) while the lower inset shows results of a 10-cell reach (300m).  Wildfire professionals seem to vary in their expert opinion (often in heated debate—yes, pun intended) of the amount and type of smoothing required, but invariably they seem to agree that none (raw data) is too little and a 10-cell reach is too much.  The most appropriate reach and the type of smoothing to use will likely keep fire scientists busy for a decade or more.  In the interim, expert opinion prevails.

An even more troubling limitation of traditional wildfire models is depicted as the “white region” in figure 1 representing urban areas as “no-data,” meaning they are areas of “no wildland fuel data” and cannot be simulated with a wildfire model.  The fuel types and conditions within an urban setting form extremely complex and variable arrangements of non-burnable to highly flammable conditions.  Hence, the wildfire models must ignore urban areas by assigning no-data to these extremely difficult conditions.

However all too often, wildfires ignore this artificial boundary and move into the urban fringe.  Modeling the relative venerability and potential impacts within the “no data” area is a critical and practical reality.

Figure 2 shows the first step in extending wildfire conditions into an urban area.  A proximity map from the urban edge is created and then divided into a series of rings.  In this example, a 180m overall reach into the urban “no-data” area uses three 2-cell rings.

Figure 2. Proximity rings extending into urban areas are calculated and used to incrementally “step” the flame length information into the urban area.

A roving window of 4-cells is used to average the neighboring flame lengths for each location within the First Ring and these data are added to the original data.  The result is “oozing” the flame lengths a little bit into the urban area.  In turn, the Second Ring’s average is computed and added to the Original plus First Ring data to extend the flame length data a little bit more.  The process is repeated for the Third Ring to “ooze” the original data the full 180 meters (6-cell) into the urban area (see figure 3).

It is important to note that this procedure is not estimating flame lengths at each urban location, but a first-cut at extending the average flame length information into the urban fringe based on the nearby wildfire behavior conditions.  Coupling this information with a response function implies greater loss of property where the nearby flame lengths are greater.  Locations in red identify generally high neighboring flame lengths, while green identify generally low locations—a first-cut at the relative wildfire threat within the urban fringe.

Figure 3. The original smoothed flame length information is added to the First Ring’s data, and then sequentially to the Second Ring’s and Third Ring’s data for a final result that extends the flame length information into the urban area.

What is novel in this procedure is the iterative use of nested rings to propagate the information—“oozing” the data into the urban area instead of one large “gulp.”  If a single large roving window (e.g., a 10-cell radius) were used for the full 180 meter reach inconsistencies arise.  The large window produces far too much smoothing at the urban outer edge and has too little information at the inner edge as most of the window will contain “no-data.”

The ability to “iteratively ooze” the information into an area step-by-step keeps the data bites small and localized, similar to the brush strokes of an artist.

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Author’s Note:  For more discussion of roving windows concepts, see the online book, Beyond Modeling III, Topic 26, Assessing Spatially-Defined Neighborhoods at www.innovativegis.com/Basis/MapAnalysis/Default.htm.

(GeoWorld, February 2006, pg. 16-17)

Neighborhood operations summarize the map values surrounding a location based on the implied Surface Configuration (slope, aspect, profile) or the Statistical Summary of the values.  The summary procedure, as well as the shape/size of the roving window, greatly affects the results.

The previous section investigated these effects by changing the window size and the summary procedure to derive a statistical summary of neighbor conditions.  An interesting extension to these discussions involves using spatial filters that change the relative weighting of the values within the window based on standard decay function equations.

Figure 1 shows graphs of several decay functions.  A Uniform function is insensitive to distance with all of the weights in the window the same (1.0).  The other equations involve the assumption that “nearby things are more alike” and generate increasingly smaller weights with greater distances.  The Inverse Distance Squared function is the most extreme resulting in nearly zero weighting within less than a 10 cell reach.  The Inverted D^2 function, on the other hand, is the least limiting function with its weights decreasing at a much slower rate to a reach of over 35 cells.

Figure 1. Standard mathematical decay functions where weights (Y) decrease with increasing distance (X).

Decay functions like these often are used by mathematicians to characterize relationships among variables.  The relationships in a spatial filter require extending the concept to geographical space.  Figure 2 shows 2D and 3D plots of the results of evaluating the Inverse Distance, Linear Negative-Slope, Inverted Distance-Squared and Uniform functions to the X,Y coordinates in a grid-based system.  The result is a set of weights for a roving window (technically referred to as a “kernel”) with a radius of 38 cells.

Note the sharp peak for the Inverse Distance filter that rapidly declines from a weight of 1.0 (blue) for the center location to effectively zero (yellow) for most of the window.  The Linear Negative-Slope filter, on the other hand, decreases at a constant rate forming a cone of declining weights.  The weights in the Inverted Distance-Squared filter are much more influential throughout the window with a sharp fall-off toward the edge of the window.  The Uniform filter is constant at 1.0 indicating that all values in the window are equally weighted regardless of their distance from the center location.

Figure 2.  Example spatial filters depicting the fall-off of weights (Z) as a function of geographic distance (X,Y).

These spatial filters are the geographic equivalent to the standard mathematical decay functions shown in figure 1.  The filters can be used to calculate a weighted average by 1) multiplying the map values times the corresponding weights within a roving window, 2) summing the products, 3) then dividing by the sum of the weights and 4) assigning the calculated value to the center cell.  The procedure is repeated for each instance of the roving window as it passes throughout the project area.

Figure 3 compares the results of weight-averaging using a Uniform spatial filter (simple average) and a Linear Negative-Slope filter (weighted average) for smoothing model calculated values.  Note that the general patterns are similar but that the ranges of the smoothed values are different as the result of the weights used in averaging.

The use of spatial filters enables a user to control the summarization of neighboring values.  Decay functions that match user knowledge or empirical research form the basis of distance weighted averaging.  In addition, filters that affect the shape of the window can be used, such as using direction to summarize just the values to the north—all 0’s except for a wedge of 1’s oriented toward the north.

Figure 3.  Comparison of simple average (Uniform weights) and weighted average (Linear weights) smoothing results.

“Dynamic spatial filters” that change with changing geographic conditions define an active frontier of research in neighborhood summary techniques.  For example, the shape and weights could be continuously redefined by just summarizing locations that are uphill as a function of elevation (shape) and slope (weights) with steep slopes having the most influence in determining average landslide potential.  Another example might be determining secondary source pollution levels by considering up-wind locations as a function of wind direction (shape) and speed (weights) with values at stronger wind locations having the most weight.

The digital nature of modern maps supporting such map-ematics is taking us well beyond traditional mapping and our paper-map legacy.  As GIS modeling transitions us from a “where is what” focus to focusing on “why and so what,” GIS technology is redefining what a map is and the procedures used for extracting useful information.

(GeoWorld, December 2005, pg. 18-19)

The last couple of sections discussed procedures for analyzing spatially-defined neighborhoods through the direct numerical summary of values within a roving window.  An interesting group of extended operators are referred to as spatial filters.

A useful example of a spatial filter involves analysis of a Binary Progression Window (BPW) that summarizes the diagonal and orthogonal connectivity within the window.  The left side of figure 1 shows the binary progression (multiples of 2) assignment for the cells in a 3 by 3 window that increases left to right, top to bottom.

Figure 1. Binary Progression Window summarizes neighborhood connectivity by summing values in a roving window.

The interesting characteristic of the sum of a binary progression of numbers is that each unique combination of numbers results in a unique value.  For example if a condition does not occur in a window, the sum is zero.  If all cells contain the condition, the sum is 511.  The four example configurations on the right identify the unique sum that characterizes the patterns shown.  The result is that all possible patterns can be easily recognized by the computer and stored as a map.

A more sophisticated example of a spatial filter is the Binary Comparison Matrix (BCM) technique that estimates neighborhood variability from a couple of perspectives.  Complexity measures for the entire roving window neighborhood are derived by evaluating three fundamental landscape analysis concepts— Diversity, Interspersion and Jutaposition.  A Comparison index identified the contrast between the center location and its surrounding conditions by calculating the Proportion of similar cells.

Consider the 3x3 window in figure 2 where "M" represents meadow classified locations and "F" represents forest.  The simplest summary of neighborhood variability is to note there are just two classes.  If there was only one class in the window, you would say there is no variability (boring); if there were nine classes, there would be a lot of different conditions (exciting).

The count of the number of different classes (Diversity) is the broadest measure of neighborhood variability.  The measure of the relative frequency of occurrence of each class (Interspersion) is a refinement on the simple diversity count and notes that the window contains less M’s than F’s.  If the example's three "M's" were more spread out like a checkerboard, you would probably say there was more variability due to the relative positioning of the classes (Juxtapositioning).  The final variability measure (Proportion) is two because there are 2 similar cells of the 8 total adjoining cells.

Figure 2. Binary Comparison Matrix summarizes neighborhood variability by summing various groups of matrix pairings identified in a roving window.

A computer simply summarizes the values in a Binary Comparison Matrix to categorize all variability that you see.  First, "Binary" means it only uses 0's and 1's.  "Comparison" says it will compare each element in the window with every other element.  If they are the same, a 1 is assigned; if different, a 0 is assigned.  "Matrix" dictates how the data the binary data is organized and summarized.

In figure 2, the window elements are numbered from one through nine.  In the window, is the class for cell 1 the same as for cell 2?  Yes (both are M), so 1 is assigned at the 1,2 position in the table.  How about elements 1 and 3?  No, so assign a 0 in the second position of column one.  How about 1 and 4?  No, then assign another 0.   Repeat until all of the combinations in the matrix contain a 0 or a 1 as depicted in the figure.

While you are bored already, the computer enjoys completing the table for every grid location… thousands and thousands of BCM tables as the window moves throughout the project area.  As the computer compares the window cells it keeps track of the number of different classes it encounters— Diversity= 2 as just M and F vegetation types in the example window.

Within the table there are 36 possible comparisons.  In our example, eighteen of these are similar by summing the entire matrix— Interspersion= 18.  Orthogonal adjacency (side-by-side and top-bottom) is computed by summing the vertical/horizontal cross-hatched elements in the table— Juxtaposition= 9.  Comparison of the center to its neighbors computes the sum for all pairs involving element 5 having the same condition (5,1 and 5,2 only)— Proportion= 2.

You can easily ignore the mechanics of the computations and still be a good user of GIS technology.  But can you ignore the new source of information contained in the indexes?  Does the spotted owl prefer higher or lower juxtapositioning values?  What about a pine martin?  Or a typical outdoor recreation enthusiast?

While BPW’s neighborhood connectivity and BMC’s neighborhood variability indices might be unfamiliar, they are not beyond you or beyond mapping …they are just a couple of new ways of looking at things through spatial filters.

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Author’s Note:  This and other “Beyond Mapping” columns have been compiled into an online book Map Analysis posted at www.innovativegis.com/basis/.   Student and instructor materials with hands-on exercises including software are available.

(GeoWorld, August 2012)

The use of “roving windows” to summarize terrain configuration is well established.  The position and relative magnitude of surrounding values at a location on an elevation surface have long been used to calculate localized terrain steepness/slope and orientation/aspect.

A search radius and geometric shape of the window are specified, then surface values within the window are retrieved, a summary technique applied (e.g., slope, aspect, average, coefficient of variation, etc.) and the resulting summary value assigned to the center cell.  The roving window is systematically moved throughout the surface to create a map of the desired surface summary.

The top portion of figure 1 illustrates the planimetric configuration of three locations of a circular fixed window with a radius of ten grid spaces.  When superimposed onto the surface, the shape is warped to conform to the relative elevation values occurring within the window.  Note that the first location is moderately sloped toward the south; second location is steeply sloped toward the west; and the third location is fairly flat with no discernible orientation.

What you eye detects is easily summarized by mathematical algorithms with the resultant values for all of the surface locations creating continuous maps of landform character, such as surface roughness, tilted area and convexity/concavity, as well as slope and aspect (see author’s note 1).

A weighted window is a variant on the simple fixed window that involves preferential weighting of nearby data values.  For example, inverse distance weighted interpolation uses a fixed shape/size of a roving window to identify data samples that are weight-averaged to favor nearer sample values more than distant ones.  Or a user-specified weighting kernel can be specified as a decay function (see author’s note 2) or any other weighting preference, such as assigning more importance to easterly conditions to account for strong and dry Santa Ana winds when modeling wildfire threat in southern California.  It is common sense that these easterly conditions are more influential than just a simple or distance-weighted average in all directions.

Figure 1. Fixed windows form circles in planimetric space but become warped when fitted to a three-dimensional surface.

Dynamic windows use the same basic processing flow but do not use a fixed reach or consistent geometric shape in defining a roving window.  Rather, the size and shape is dependent on the conditions at each map location and varies as the window is moved over a map surface.

For example, figure 2 depicts a roving window based on uphill, downhill and across slope movements from the center location.  Lots of spatial processes respond differently to these basic landform conditions.  For example, uphill conditions can contribute surface runoff to the center cell, downhill locations can receive flows from the center cell and sediment movement at the across slope locations is independent of the center cell. Wildfire movement, on the other hand, is most rapid uphill, particularly in steep terrain, due to preheating of forest fuels.  Hence, downhill conditions are more important in modeling threat at a location than either the across or uphill surrounding conditions.

Figure 2. Uphill, downhill and across portions of a roving window can be determined by considering the relative values on a three-dimensional surface.

Another dynamic consideration is effective distance (see author’s note 3).  For example, a window’s geographic reach and direction can be a function of intervening conditions, such as the relative habitat preference when considering the surroundings in a wildlife model.  The window will expand and contract depending on neighboring conditions forming an ameba-like shape to identify data values to be summarized—the pseudopods change shape and extent at each instantaneous location.  The result is a localized summary of data, such as proximity to human activity within preferential reach of each grid location to characterize animal/human interaction potential.

Or a combination of window considerations can be applied, such as (1) preferential weighting of the fuel loadings (2) along downhill locations (3) as a function of slope with steep areas reaching farther away than gently sloped areas.  In a wildfire risk model, the resultant “roving window” summary would favor the fuel conditions within the elongated pseudopods of the steeply sloped downhill locations.

A third type of dynamic consideration involves line-of-sight connectivity where the “viewshed” of a location within a specified distance is used to define a roving window (see figure 3).  In a military situation, this type of window might be useful in summarizing the likelihood of enemy activity that is visually connected to each map location.  Areas with high visual exposure levels being poor places to setup camp, but ideal places for establishing forward observer outposts.

A less war-like application of line-of-sight windows involves terrain analysis.  Areas not seen are “over the hill” in a macro-sense for ridge lines and “in a slight depression” in a micro-sense for potholes.  If all locations are seen then there is minimal macro or micro terrain variations.

Figure 3. A line-of-sight window identifies locations that are seen and not seen from the window’s focus.

The rub is that most of the user community and much of the vector-based GIS’ers are unaware of even fixed roving windows, much less weighted and dynamic windows.  However, the utility of these advanced procedures in conceptualizing geographic space within context of its surroundings is revolutionary.  The view through a dynamic window is as useful as it is initially mind-boggling …see you on the other side.

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Author’s Notes:  1) See Topic 11, Characterizing Micro-Terrain Features, “Characterizing Local Terrain Conditions”; 2) Topic 26, Characterizing Micro-Terrain Features, “Nearby Things Are More Alike”; and 3) Topic 25, Calculating Effective Distance and Connectivity, “Measuring Distance Is Neither Here nor There” in the online book Beyond Mapping III posted at www.innovativegis.com/basis/MapAnalysis/.