Map Analysis
Topic 11: Characterizing Micro-Terrain Features |
© 2001 |
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Identifying Micro-Terrain
Features — describes techniques to
identify convex and concave features
Characterizing Local
Terrain Conditions — discusses the use of
"roving windows" to distinguish localized variations
Assessing Terrain
Slope and Roughness — discusses techniques for
determining terrain inclination and coarseness
Characterizing Surface
Flows — describes the use of optimal
path density analysis for mapping surface flows
Modeling Erosion and
Sediment Loading — illustrates a GIS model for assessing
erosion potential and sediment loading
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Identifying Micro-Terrain Features (return to top of Topic)
The past several columns investigated surface
modeling and analysis. The data
surfaces derived in these instances weren't the familiar terrain
surfaces you walk the dog, bike and hike on. None-the-less they form surfaces that contain all of the
recognizable gullies, hummocks, peaks and depressions we see on most
hillsides. The
"wrinkled-carpet" look in the real world is directly transferable to
the cognitive realm of the data world.
However, at least for the moment, let's
return to terra firma to investigate how micro-terrain features can be
characterized. As you look at a
landscape you easily see the changes in terrain with some areas bumped up
(termed convex) and others pushed down (termed concave). But how can a computer "see" the
same thing? Since its world is digital,
how can the lay of the landscape be transferred into a set of drab
numbers?
Figure
31-1. Identifying Convex and Concave features by deviation from the trend of
the terrain.
One of the most frequently used procedures
involves comparing the trend of the surface to the actual elevation
values. Figure 31-1 shows a terrain
profile extending across a small gully.
The dotted line identifies a smoothed curve fitted to the data, similar
to a draftsman's alignment of a French curve.
It "splits-the-difference" in the succession of elevation
values—half above and half below.
Locations above the trend line identify convex features while locations
below identify the concave ones. The
further above or below determines how pronounced the feature is.
In a GIS, simple smoothing of the actual
elevation values derives the trend of the surface. The left side of fig. 31-2 shows the actual and smoothed surfaces
for a project area. The flat portion at
the extreme left is an area of open water.
The terrain rises sharply from 500 feet to 2500 feet at the top of the
hill. Note the small "saddle"
(elevation dips down then up) between the two hilltops. Also note the small depression in the
relatively flat area in the foreground (SW) portion.
In generating the smoothed surface, elevation
values were averaged for a 4-by-4 window moved throughout the area. Note the subtle differences between the
surfaces—the tendency to pull-down the hilltops and push-up the gullies.
While you see (imagine?) these differences in
the surfaces, the computer quantifies them by subtracting. The difference surface on the
right contains values from -84 (prominent concave feature) to +94 (prominent
convex feature). The big bump on the
difference surface corresponds to the smaller hilltop on elevation surface. It's actual elevation is 2016 while the
smoothed elevation is 1922 resulting in 2016 - 1922 = +94 difference. In micro-terrain terms, this area is likely
drier that its surroundings as water flows away from it.
The other arrows on the surface indicate
other interesting locations. The
"pockmark" in the foreground is a small depression (764 - 796 = -32
difference) that is likely wetter as water flows into it. The "deep cut" at the opposite end
of the difference surface (539 - 623 = -84) suggests a prominent
concavity. However representing the
water body as fixed elevation isn't a true account of terra firma configuration
and misrepresents the true micro-terrain pattern.
Figure
31-2. Example of a micro-terrain
deviation surface.
In fact the entire concave feature in the
upper left portion of 2-D representation of the difference surface is suspect
due to its treatment of the water body as a constant elevation value. While a fixed value for water on a
topographic map works in traditional mapping contexts it's insufficient in most
analytical applications. Advanced GIS
systems treat open water as "null" elevations (unknown) and
"mirror" terrain conditions along these artificial edges to better
represent the configuration of solid ground.
The 2-D map of differences identifies areas
that are concave (dark red), convex (light blue) and transition (white portion
having only -20 to +20 feet difference between actual and smoothed elevation
values). If it were a map of a farmer's
field, the groupings would likely match a lot of the farmer's recollection of
crop production—more water in the concave areas, less in the convex areas.
A Colorado dryland wheat farmer knows that
some of the best yields are in the lowlands while the uplands tend to
"burn-out." A farmer in
Louisiana, on the other hand, likely see things reversed with good yields on
the uplands while the lowlands often "flood-out." In either case, it might make sense to
change the seeding rate, hybrid type, and/or fertilization levels within areas
of differing micro-terrain conditions.
The idea of variable rate response to spatial
conditions has been around for thousands of years as indigenous peoples
adjusted the spacing of holes they poked in the ground to drop in a seed and a
piece of fish. While the mechanical and
green revolutions enable farmers to cultivate much larger areas they do so in
part by applying broad generalizations of micro-terrain and other spatial
variables over large areas. The ability
to continuously adjust management actions to unique spatial conditions on
expansive tracks of land foretells the next revolution.
Investigate the effects of micro-terrain
conditions goes well beyond the farm.
For example, the Universal Soil Loss Equation uses "average"
conditions, such as stream channel length and slope, dominant soil types and
existing land use classes, to predict water runoff and sediment transport from
entire watersheds. These non-spatial
models are routinely used to determine the feasibility of spatial activities,
such as logging, mining, road building and housing development. While the procedures might be applicable to
typical conditions, they less accurately track unusual conditions clumped
throughout an area and provide no spatial guidance within the boundaries of the
modeled area.
GIS-based micro-terrain analysis can help us
be more like a "modern ancient farmer"— responding to site-specific
conditions over large expanses of the landscape. Calculation of a difference surface simply scratches the surface
of micro-terrain analysis. In the next
few columns we'll look other procedures that let us think like a raindrop while
mapping the micro-terrain.
Characterizing Localized Terrain Conditions (return to top of Topic)
Last month's column described a technique for
characterizing micro-terrain features involving the difference between the
actual elevation values and those on a smoothed elevation surface (trend). Positive values on the difference map
indicate areas that "bump-up" while negative values indicate areas that
"dip-down" from the general trend in the data.
Figure
31-3. Localized deviation uses a
spatial filter to compare a location to its surroundings.
A related technique to identify the bumps and
dips of the terrain involves moving a "roving window" (termed a
spatial filter) throughout an elevation surface. The profile of a gully can have micro-features that dip below its
surroundings (termed concave) as shown on the right side of
figure 31-3.
The localized deviation within
a roving window is calculated by subtracting the average of the surrounding
elevations from the center location's elevation. As depicted in the example calculations for the concave feature,
the average elevation of the surroundings is 106, that computes to a -6.00
deviation when subtracted from the center's value of 100. The negative sign denotes a concavity while
the magnitude of 6 indicates it's fairly significant dip (a 6/100= .06). The protrusion above its surroundings
(termed a convex feature) shown on the right of the figure has a
localized deviation of +4.25 indicating a somewhat pronounced bump (4.25/114=
.04).
Figure
31-4. Applying Deviation and
Coefficient of Variation filters to an elevation surface.
The result of moving a deviation filter
throughout an elevation surface is shown in the top right inset in figure
31-4. Its effect is nearly identical to
the trend analysis described last month-- comparison of each location's actual
elevation to the typical elevation (trend) in its vicinity. Interpretation of a Deviation Surface
is the same as that for the difference surface discussed last month—protrusions
(large positive values) locate drier convex areas; depressions (large negative
values) locate wetter concave areas.
The implication of the "Localized
Deviation" approach goes far beyond simply an alternative procedure for
calculating terrain irregularities. The
use of "roving windows" provides a host of new metrics and map
surfaces for assessing micro-terrain characteristics. For example, consider the Coefficient of Variation
(Coffvar) Surface shown in the bottom-right portion of figure
31-4. In this instance, the standard
deviation of the window is compared to its average elevation—small
"coffvar" values indicate areas with minimal differences in
elevation; large values indicate areas with lots of different elevations. The large ridge in the coffvar surface in
the figure occurs along the shoreline of a lake. Note that the ridge is largest for the steeply-rising terrain
with big changes in elevation. The
other bumps of surface variability noted in the figure indicate areas of less
terrain variation.
While a statistical summary of elevation
values is useful as a general indicator of surface variation or
"roughness," it doesn't consider the pattern of the differences. A checkerboard pattern of alternating higher
and lower values (very rough) cannot be distinguished from one in which all of
the higher values are in one portion of the window and lower values in
another.
Figure
31-5. Calculation of slope considers
the arrangement and magnitude of elevation differences within a roving window.
There are several roving window operations
that track the spatial arrangement of the elevation values as well as aggregate
statistics. A frequently used one is
terrain slope that calculates the "slant" of a surface. In mathematical terms, slope equals the
difference in elevation (termed the "rise") divided by the horizontal
distance (termed the "run").
As shown in figure 31-5, there are eight
surrounding elevation values in a 3x3 roving window. An individual slope from the center cell can be calculated for
each one. For example, the percent
slope to the north (top of the window) is ((2332 - 2262) / 328) * 100 =
21.3%. The numerator computes the rise
while the denominator of 328 feet is the distance between the centers of the
two cells. The calculations for the
northeast slope is ((2420 - 2262) / 464) * 100 = 34.1%, where the run is
increased to account for the diagonal distance (328 * 1.414 = 464).
The eight slope values can be used to
identify the Maximum, the Minimum and the Average slope as reported in the
figure. Note that the large difference
between the maximum and minimum slope (53 - 7 = 46) suggests that the overall
slope is fairly variable. Also note
that the sign of the slope value indicates the direction of surface
flow—positive slopes indicate flows into the center cell while negative ones
indicate flows out. While the flow into
the center cell depends on the uphill conditions (we'll worry about that in a
subsequent column), the flow away from the cell will take the steepest downhill
slope (southwest flow in the example… you do the math).
In practice, the Average slope can be
misleading. It is supposed to indicate
the overall slope within the window but fails to account for the spatial
arrangement of the slope values. An
alternative technique "fits a plane" to the nine individual elevation
values. The procedure determines the
best fitting plane by minimizing the deviations from the plane to the elevation
values. In the example, the Fitted
slope is 65%… more than the maximum individual slope.
At first this might seem a bit fishy—overall
slope more than the maximum slope—but believe me, determination of fitted slope
is a different kettle of fish than simply scrutinizing the individual
slopes. Next time we'll look a bit
deeper into this fitted slope thing and its applications in micro-terrain
analysis.
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Author's
Notes: An Excel worksheet investigating
Maximum, Minimum, and Average slope calculations is available online at the
"Column Supplements" page at http://www.innovativegis.com/basis.
Assessing Terrain Slope and Roughness (return to top of Topic)
The past few columns discussed several
techniques for generating maps that identify the bumps (convex features),
the dips (concave features) and the tilt (slope) of a terrain
surface. Although the procedures have a
wealth of practical applications, the hidden agenda of the discussions was to
get you to think of geographic space in a less traditional way—as an organized
set of numbers (numerical data), instead of points, lines and areas represented
by various colors and patterns (graphic map features).
A terrain surface is organized as a
rectangular "analysis grid" with each cell containing an elevation
value. Grid-based processing involves
retrieving values from one or more of these "input data layers" and
performing a mathematical or statistical operation on the subset of data to
produce a new set numbers. While computer
mapping or spatial database management often operates
with the numbers defining a map, these types of processing simply repackage the
existing information. A spatial query
to "identify all the locations above 8000' elevation in El Dorado
County" is a good example of a repackaging interrogation.
Map analysis operations, on the other hand, create
entirely new spatial information. For
example, a map of terrain slope can be derived from an elevation surface, then
used to expand the geo-query to "identify all the locations above 8000'
elevation in El Dorado County (existing data) that exceed 30% slope (derived
data)." While the discussion
in this series of columns focuses on applications in terrain analysis, the
subliminal message is much broader—map analysis procedures derive new spatial
information from existing information.
Now back to business. Last month's column described several
approaches for calculating terrain slope from an elevation surface. Each of the approaches used a "3x3
roving window" to retrieve a subset of data, yet applied a different analysis
function (maximum, minimum, average or "fitted" summary of the
data).
Figure
31-6. 2-D, 3-D and draped displays of terrain slope.
Figure 31-6 shows the slope map derived by
"fitting a plane" to the nine elevation values surrounding each map
location. The inset in the upper left
corner of the figure shows a 2-D display of the slope map. Note that steeper locations primarily occur
in the upper central portion of the area, while gentle slopes are concentrated
along the left side.
The inset on the right shows the elevation
data as a wire-frame plot with the slope map draped over the surface. Note the alignment of the slope classes with
the surface configuration—flat slopes where it looks flat; steep slopes where
it looks steep. So far, so good.
The 3-D view of slope in the lower left,
however, looks a bit strange. The big
bumps on this surface identify steep areas with large slope values. The ups-and-downs (undulations) are
particularly interesting. If the area
was perfectly flat, the slope value would be zero everywhere and the 3-D view
would be flat as well. But what do you
think the 3-D view would look like if the surface formed a steeply sloping
plane?
Are you sure? The slope values at each location would be the same, say 65%
slope, therefore the 3-D view would be a flat plane "floating" at a
height of 65. That suggests a useful
principle—as a slope map progresses from a constant plane (single value
everywhere) to more ups-and-downs (bunches of different values), an increase in
terrain roughness is indicated.
Figure
31-7. Assessing terrain roughness through the 2nd derivative of an
elevation surface.
Figure 31-7 outlines this concept by
diagramming the profiles of three different terrain cross-sections. An elevation surface's 2nd
derivative (slope of a slope map) quantifies the amount of ups-and-downs of the
terrain. For the straight line on the
left, the "rate of change in elevation per unit distance" is constant
with the same difference in elevation everywhere along the line—slope = 65%
everywhere. The resultant slope map
would have the value 65 stored at each grid cell, therefore the "rate of
change in slope" is zero everywhere along the line (no change in
slope)—slope2 = 0% everywhere.
A slope2 value of zero is interpreted as a
perfectly smooth condition, which in this case happens to be steep. The other profiles on the right have varying
slopes along the line, therefore the "rate of change in slope" will produce
increasing larger slope2 values as the differences in slope become increasingly
larger.
So who cares? Water drops for one, as steep smooth areas are ideal for downhill
racing, while "steep 'n rough terrain" encourages more infiltration,
with "gentle yet rough terrain" the most.
Figure
31-8. 2-D, 3-D and draped displays of terrain roughness.
Whew! That's a lot of map-ematical explanation for a couple of pretty simple concepts—steepness and roughness. Next month we'll continue the trek on steep part of the map analysis learning curve by considering "confluence patterns" in micro terrain analysis.
Characterizing
Surface Flows (return to top of Topic)
Last month's discussions focused on terrain
steepness and roughness. While the concepts
are simple and straightforward, the mechanics of computing them are a bit more
challenging. As you hike in the
mountains your legs sense the steepness and your mind is constantly assessing
terrain roughness. A smooth,
steeply-sloped area would have you clinging to things, while a rough
steeply-sloped area would look more like stair steps.
Figure
31-9. Map of surface flow confluence.
Water has a similar vantage point of the
slopes it encounters, except given its head, water will take the steepest
downhill path (sort of like an out-of-control skier). Figure 31-91 shows a 3-D grid map of an elevation surface and the
resulting flow confluence. It is based
on the assumption that water will follow a path that chooses the steepest
downhill step at each point (grid cell "step") along the terrain
surface.
In effect, a drop of water is placed at each
location and allowed to pick its path down the terrain surface. Each grid cell that is traversed gets the
value of one added to it. As the paths
from other locations are considered the areas sharing common paths get
increasing larger values (one + one + one, etc.).
The inset on the right shows the path taken
by a couple of drops into a slight depression.
The inset on the left shows the considerable inflow for the depression
as a high peak in the 3-D display. The
high value indicates that a lot of uphill locations are connected to this
feature. However, note that the
pathways to the depression are concentrated along the southern edge of the
area.
Now turn your attention to figure 31-10. Ridges on the confluence density surface
(lower left) identify areas of high surface flow. Note how these areas (darker) align with the creases in the
terrain as shown on the draped elevation surface on the right inset. The water collection in the
"saddle" between the two hills is obvious, as are the two westerly
facing confluences on the side of the hills.
The 2-D map in the upper left provides a more familiar view of where not
to unroll your sleeping bag if flash floods are a concern.
Figure
31-10. 2-D, 3-D and draped displays of
surface flow confluence.
The various spatial analysis techniques for
characterizing terrain surfaces introduced in this series provide a wealth of
different perspectives on surface configuration. Deviation from Trend, Difference Maps and Deviation
Surfaces are used to identify areas that "bump-up" (convex) or
"dip-down" (concave). A Coefficient
of Variation Surface looks at the overall disparity in elevation values
occurring within a small area. A Slope
Map shares a similar algorithm (roving window) but the summary of is
different and reports the "tilt" of the surface. An Aspect Map extends the analysis to
include the direction of the tilt as well as the magnitude. The Slope of a Slope Map (2nd
derivative) summarizes the frequency of the changes along an incline and
reports the roughness throughout an elevation surface. Finally, a Confluence Map takes an
extended view and characterizes the number of uphill locations connected to
each location.
The
coincidence of these varied perspectives can provide valuable input to
decision-making. Areas that are smooth,
steep and coincide with high confluence are strong candidates for gully-washers
that gouge the landscape. On the other
hand, areas that are rough, gently-sloped and with minimal confluence are
relatively stable. Concave features in
these areas tend to trap water and recharge soil moisture and the water
table. Convex features under erosive
conditions tend to become more prominent as the confluence of water flows
around it.
Similar interpretations can be made for
hikers, who like raindrops react to surface configuration in interesting
ways. While steep, smooth surfaces are
avoided by all but the rock-climber, too gentle surfaces tend too provide
boring hikes. Prominent convex features
can make interesting areas for viewing—from the top for hearty and from the
bottom for the aesthetically bent.
Areas of water confluence don't mix with hiking trail unless a
considerable number of water-bars are placed in the trail.
These "rules-of-thumb" make sense
in a lot of situations, however, there are numerous exceptions that can undercut
them. Two concerns in particular are
important— conditions and resolution.
First, conditions along the surface can alter the effect of terrain
characteristics. For example, soil
properties and the vegetation at a location greatly effects surface runoff and
sediment transport. The nature of
accumulated distance along the surface is also a determinant. If the uphill slopes are long steep, the
water flow has accumulated force and considerable erosion potential. A hiker that has been hiking up a steep
slope for a long time might collapse before reaching the summit. If that steep slope is southerly oriented
and without shade trees, then exhaustion is reached even sooner.
In addition, the resolution of the elevation
grid can effect the calculations. In
the case of water drops the gridding resolution and accurate "Z"
values must be high to capture the subtle twists and bends that direct water
flow. A hiker on the other hand, is
less sensitive to subtle changes in elevation.
The rub is that collection of the appropriate elevation is prohibitively
expensive in most practical applications.
The result is that existing elevation data, such as the USGS Digital
Terrain Models (DTM), are used in most cases by default. Since the GIS procedures are independent of
the gridding resolution, inappropriate maps can be generated and used in
decision-making.
The recognition of the importance of spatial
analysis and surface modeling is imperative, both for today and into the
future. Its effective use requires
informed and wary users. However, as
with all technological things, what appears to be a data barrier today, becomes
routine in the future. For example, RTK
(Real Time Kinematic) GPS can build elevation maps to centimeter accuracy— it's
just that there are a lot of centimeters out there to measure.
The more important limitation is
intellectual. For decades, manual
measurement, photo interpretation and process modeling approaches have served
as input for decision-making involving terrain conditions. Instead of using GIS to simply automate the
existing procedures our science needs to consider the new micro-terrain
analysis tools and innovative approaches they present.
_______________________
Author's
Notes: The figures presented in
this series on "Characterizing Micro-Terrain Features" plus several
other illustrative ones are available online as a set of annotated PowerPoint
slides at the "Column Supplements" page at http://www.innovativegis.com/basis.
Modeling Erosion and Sediment Loading (return to top of Topic)
Last month's discussions suggested that
combining derived maps often is necessary for a complete expression of an
application. A simple erosion potential
model, for example, can be developed by characterizing the coincidence of a Slope
Map and a Flow Map (see the previous two columns in this
series). The flowchart in the figure
31-11 identifies the processing steps that form the model— generate slope and
flow maps, establish relative classes for both, then combine.
Figure
31-11. A simple erosion potential model
combines
While a flowchart of the processing might
appear unfamiliar, the underlying assumptions are quite straightforward. The slope map characterizes the relative
"energy" of water flow at a location and the confluence map
identifies the "volume" of flow.
It's common sense that as energy and volume increase, so does erosion
potential.
The various combinations of slope and flow
span from high erosion potential to deposition conditions. On the map in the lower right, the category
"33 Heavy Flow; Steep" (dark blue) identifies areas that are steep
and have a lot of uphill locations contributing water. Loosened dirtballs under these circumstances
are easily washed downhill. However,
category "12 Light Flow; Moderate" (light green) identifies locations
with minimal erosion potential. In
fact, deposition (the opposite of erosion) can occur in areas of gentle slope,
such as category "11 Light Flow; Gentle" (dark red).
Before we challenge the scientific merit of the
simplified model, note the basic elements of the GIS modeling approach— flowchart
and command macro. The flowchart
is used to summarize the model's logic and processing steps. Each box represents a map and each line
represents an analysis operation. For
example, the first step depicts the calculation of a SlopeMap from a
base map of Elevation. The
actual command for this step, "Slope Elevation for SlopeMap," forms
the first sentence in the command macro (see author's note).
The remaining sentences in the macro and the
corresponding boxes/lines in the flowchart complete the model. The macro enables entering, editing,
executing, storing and retrieving the individual operations that form the
application. The flowchart provides an
effective means for communicating the processing steps. Most "GIS-challenged users" are
baffled by the detailed code in a command macro but readily relate to flowchart
logic. In developing GIS applications,
the user is the expert in the logic (domain expertise) while the developer is
the expert in the code (GIS expertise).
The explicit linkage between the macro and the flowchart provides a
common foothold for communication between the two perspectives of a GIS
application.
It provides a starting place for model
refinement as well. Suppose the user
wants to extend the simple erosion model to address sediment loading potential
to open water. The added logic is
captured by the additional boxes/lines shown in figure 31-12. Note that the upper left box (Erosion_classes)
picks up where the flowchart in figure 31-11 left off.
Figure
31-12. Extended erosion model
considering sediment loading potential considering intervening terrain
conditions.
A traditional approach would generate a simple
buffer of a couple of hundred feet around the stream and restrict all dirt
disturbing actives to outside the buffered area. But are all buffer-feet the same? Is a simple geographic reach on either side sufficient to hold
back sediment? Do physical laws apply
or merely civil ones that placate planners?
Common sense suggests that the intervening
conditions play a role. In areas that
are steep and have high water volume, the setback ought to be a lot as erosion
potential is high. Areas with minimal
erosion potential require less of a setback.
In the schematic in figure 31-12, a dirt disturbing activity on the
steep hillside, though 200 feet away, would likely rain dirtballs into the
stream. A similar activity on the other
side of the stream, however, could proceed almost adjacent to the stream.
The first step in extending the erosion
potential model to sediment loading involves "calibrating" the
intervening conditions for dirtball impedance.
The friction map identified in the flowchart ranges from 1 (very low
friction for the 33 Heavy Flow: Steep condition) to 10 (very high friction for
11 Light Flow; Gentle). A loose
dirtball in an area with a high friction factor isn't going anywhere, while one
in an area of very low friction almost has legs of its own.
The second processing step calculates the
effective distance from open water based on the relative friction. The command, "SPREAD Water TO 50 THRU
Friction OVER Elevation Uphill Across For Water_Prox," is entered simply
by completing a dialog box. The result
is a variable-width buffer that reaches farther in areas of high erosion
potential and less into areas of low potential. The lighter red tones identify locations that are effectively
close to water from the perspective of a dirtball on the move. The darker green tones indicate areas
effectively far away.
But notice the small dark green area in the
lower left corner of the map of sediment loading potential. How can it be effectively far away though
right near a stream? Actually it is a
small depression that traps dirtballs and can't contribute sediment to the
stream— effectively infinitely far away.
Several other real-world extensions are
candidates to improve model. Shouldn't
one consider the type of soils? The
surface roughness? Or the time of
year? The possibilities are
numerous. In part, that's the trouble
with GIS— it provides new tools for spatial analysis that aren't part of our
traditional procedures and paradigms.
Much of our science was developed before we had these spatially-explicit
operations and is founded on simplifying assumptions of spatial independence
and averaging over micro conditions.
But now the "chicken or egg" parable is moot. Spatial analysis is here and our science
needs updating to reflect the new tools and purge simplifying assumptions about
geographic relationships.
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Author's Note: The MapCalcTM program by Red Hen Systems, Inc., was used for processing and display of the examples presented in this series. An online tutorial extending the discussions is available at www.redhensystems.com.