 | Distance and neighborhood |
In effect, the procedure is like throwing a handful of rocks into pond. Each set of concentric rings grows until the wave fronts from other locations meet; then they stop. The result is a map indicating the shortest straight-line distance to the nearest target area (house) for each non-target area. In the figure, the red tones indicate locations that are close to a house, while the green tones identify areas that are far from a house.
In a similar fashion, a proximity map to roads is generated by establishing data zones emanating from the road networksort of like tossing a wire frame into a pond to generate a concentric pattern of ripples (middle portion of figure 5.4-2). The same result is generated for a set of areal features, such as sensitive habitat parcels (right side of figure 5.4-2).
It is important to note that proximity is not the same as a buffer. A buffer is a discrete spatial object that identifies areas that are within a specified distance of map feature; all locations within a buffer are considered the same. Proximity is a continuous surface that identifies the distance to a map feature(s) for every location in a project area. It forms a gradient of distances away composed of many map values; not a single spatial object with one characteristic distance away.
The 3D plots of the proximity surfaces in figure 5.4-2 show detailed gradient data and are termed accumulated surfaces. They contain increasing distance values from the target point, line or area locations displayed as colors from red (close) to green (far). The starting features are the lowest locations (black= 0) with hillsides of increasing distance and forming ridges that are equidistant from starting locations.
In many applications, however, the shortest route between two locations might not always be a straight-line (or even a slightly wiggling set of grid steps). And even if it is straight, its geographic length may not always reflect a traditional measure of distance. Rather, distance in these applications is best defined in terms of movement expressed as travel-time, cost or energy that is consumed at rates that vary over time and space. Distance modifying effects involve weights and/or barriers concepts that imply the relative ease of movement through geographic space might not always constant.
Effective proximity responds to intervening conditions or barriers. There are two types of barriers that are identified by their effects absolute and relative. Absolute barriers are those completely restricting movement and therefore imply an infinite distance between the points they separate. A river might be regarded as an absolute barrier to a non-swimmer. To a swimmer or a boater, however, the same river might be regarded as a relative barrier identifying areas that are passable, but only at a cost which can be equated to an increase in geographical distance. For example, it might take five times longer to row a hundred meters than to walk that same distance.
In the conceptual framework of tossing a rock into a pond, the waves can crash and dissipate against a jetty extending into the pond (absolute barrier; no movement through the grid spaces). Or they can proceed, but at a reduced wavelength through an oil slick (relative barrier; higher cost of movement through the grid spaces). The waves move both around the jetty and through the oil slick with the ones reaching each location first identifying the set of shortest, but not necessarily straight-lines among groups of points.
The shortest routes respecting these barriers are often twisted paths around and through the barriers. The GIS database enables the user to locate and calibrate the barriers; the wave-like analytic procedure enabling the computer to keep track of the complex interactions of the waves and the barriers. For example, figure 5.4-3 shows the effective proximity surfaces for the same set of starter locations shown in the previous figure 5.4-2 expressed as simple proximity.
The point features in the left inset respond to treating flowing water as an absolute barrier to movement. Note that the distance to the nearest house is very large in the center-right portion of the project area (green) although there is a large cluster of houses just to the north. Since the water feature cant be crossed, the closest houses are a long distance to the south.
Terrain steepness is used in the middle inset to illustrate the effects of a relative barrier. Increasing slope is coded into a friction map of increasing impedance values that make movement through steep grid cells effectively farther away than movement through gently sloped locations. Both absolute and relative barriers are applied in determining effective proximity sensitive areas in the right inset.
Compare these results in figures 5.4-2 and 5.4-3 and note the dramatic differences between the concept of distance as the crow flies (simple proximity) and as the crow walks (effective proximity). In many practical applications, the assumption that all movement occurs in straight lines disregards reality. When traveling by trains, planes, automobiles, and feet there are plenty of bends, twists, accelerations and decelerations due to characteristics (weights) and conditions (barriers) of the movement.
Figure 5.4-4 illustrates how the splash algorithm propagates distance waves to generate an effective proximity surface. The Friction Map locates the absolute (blue/water) and relative (light blue= gentle/easy through red= steep/hard) barriers. As the distance wave encounters the barriers their effects on movement are incorporated and distort the symmetric pattern of simple proximity waves. The result identifies the shortest, but not necessarily straight distance connecting the starting location with all other locations in a project area
Note that the absolute barrier locations (blue) are set to infinitely far away and appear as pillars in the 3-D display of the final proximity surface. As with simple proximity, the effective distance values form a bowl-like surface with the starting location at the lowest point (zero away from itself) and then ever-increasing distances away (upward slope). With effective proximity, however, the bowl is not symmetrical and is warped with bumps and ridges that reflect intervening conditions the greater the impedance the greater the upward slope of the bowl. In addition, there can never be a depression as that would indicate a location that is closer to the starting location than everywhere around it. Such a situation would violate the ever-increasing concentric rings theory and is impossible.
The past series of four sections have focused on how simple distance is extended to effective proximity and movement in a modern GIS. Considerable emphasis was given to the calculations involving a propagating wave of increasing distance (algorithm) instead of our more familiar procedures of measuring with a ruler (manual) or solving the Pythagorean Theorem (mathematical).
While the computations of simple and effective proximity might be unfamiliar and appear complex, once programmed they are easily and quickly performed by modern computers. In addition, there is a rapidly growing wealth of digital data describing conditions that impact movement in the real world. It seems that all is in place for a radical rethinking and expression of distancecomputers, programs and data are poised.
However, what seems to be the major hurdle for adoption of this new way of spatial thinking lies in the experience base of potential users. Our paper map legacy suggests that the shortest straight line between two points is the only way to investigate spatial context relationships and anything else is wrong (or at least uncomfortable).
This restricted perspective has lead most contemporary GIS applications to employ simple distance and buffers. While simply automating traditional manual procedures might be comfortable, it fails to address the reality of complex spatial problems or fully engage the potential of GIS technology.
The first portion of figure 5.4-5 identifies the basic operations described in the previous sections. These procedures have set the stage for even more advanced distance operations, as outlined in the lower portion of the figure. For example, a Guiding Surface can be used to constrain movement up, down or across a surface. For example, the algorithm can check an elevation surface and only proceed to downhill locations from a feature such as roads to identify areas potentially affected by the wash of surface chemicals applied.
The simplest Directional Effect involves compass directions, such as only establishing proximity in the direction of a prevailing wind. A more complex directional effect is consideration of the movement with respect to an elevation surfacea steep uphill movement might be considered a higher friction value than movement across a slope or downhill. This consideration involves a dynamic barrier that the algorithm must evaluate for each point along the wave front as it propagates
Accumulation Effects account for wear and tear as movement continues. For example, a hiker might easily proceed through a fairly steep uphill slope at the start of a hike but balk and pitch a tent at the same slope encountered ten hours into a hike. In this case, the algorithm merely carries an equation that increases the static/dynamic friction values as the movement wave front progresses. A natural application is to have a user enter their gas tank size and average mileage so it would automatically suggest refilling stops along a proposed route.
A related consideration, Momentum Effects, tracks the total effective distance but in this instance it calculates the net effect of up/downhill conditions that are encountered. It is similar to a marble rolling over an undulating surfaceit picks up speed on the downhill stretches and slows down on the uphill ones.
The remaining three advanced operations interact with the accumulation surface derived by the wave fronts movement. Recall that this surface is analogous to football stadium with each tier of seats being assigned a distance value indicating increasing distance from the field. In practice, an accumulation surface is a twisted bowl that is always increasing but at different rates that reflect the differences in the spatial patterns of relative and absolute barriers.
Stepped Movement allows the proximity wave to grow until it reaches a specified location, and then restart at that location until another specified location and so on. This generates a series of effective proximity facets from the closest to the farthest location. The steepest downhill path over each facet, as you might recall, identifies the optimal path for that segment. The set of segments for all of the facets forms the optimal path network connecting the specified points.
The direction of optimal travel through any location in a project area can be derived by calculating the Back Azimuth of the location on the accumulation surface. Recall that the wave front potentially can step to any of its eight neighboring cells and keeps track of the one with the least friction to movement. The aspect of the steepest downhill step (N, NE, E, SE, S, SW, W or NW) at any location on the accumulation surface therefore indicates the direction of the best path through that location. In practice there are two directionsone in and one out for each location.
An even more bazaar extension is the interpretation of the 1st and 2nd Derivative of an accumulation surface. The 1st derivative (rise over run) identifies the change in accumulated value (friction value) per unit of geographic change (cell size). On a travel-time surface, the result is the speed of optimal travel across the cell. The second derivative generates values whether the movement at each location is accelerating or decelerating.
Chances are these extensions to distance operations seem a bit confusing, uncomfortable, esoteric and bordering on heresy. While the old straight line procedure from our paper map legacy may be straight forward, it fails to recognize the reality that most things rarely move in straight lines.
Effective distance recognizes the complexity of realistic movement by utilizing a procedure of propagating proximity waves that interact with a map indicating relative ease of movement. Assigning values to relative and absolute barriers to travel enable the algorithm to consider locations to favor or avoid as movement proceeds. The basic distance operations assume static conditions, whereas the advanced ones account for dynamic conditions that vary with the nature of the movement.
So whats the take home from the discussions involving effective distance? Two points seem to define the bottom line. First, that the digital map is revolutionizing how we perceive distance, as well as how calculate it. It is the first radical change since Pythagoras came up with his theorem a couple of thousand years ago. Secondly, the ability to quantify effective distance isnt limited by computational power or available data; rather it is most limited by difficulties in understanding accepting the concept.
5.5 Summarizing Neighbors
Analysis of spatially defined neighborhoods involves summarizing the context of surrounding 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. The process is then repeated for every location in a project area.
Summarizing Neighbors techniques fall into two broad classes of analysisCharacterizing Surface Configuration and Summarizing Map Values (see figure 5.5-1). 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 is in the treatment of the valuesimplied surface configuration or direct numerical summary. For example, figure 5.5-2 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 min
