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SLOPE ANGLE—ARE YOU IN THE DANGER ZONE?

November 17, 2021
Backcountry

You have your beacon, shovel, probe, and maybe even an avalanche airbag, but
what are you using for route planning and terrain assessment? If you’re new to
digital maps and curious about the tools this technology can unlock, read on. 

Slope Angle is a critical component when assessing avalanche danger. It is
defined as the angle a slope makes with the horizon, where completely flat is 0
degrees and vertical is 90 degrees. Slab avalanches occur most frequently on
slopes between 30 and 45 degrees from horizontal. Anything above 45 degrees is
generally too steep for a slab to form and anything less than around 30 degrees
is too flat for a slab to slide. Knowing how to navigate in this danger zone can
help us travel safer in the backcountry. Thanks to the NASA-funded Shuttle Radar
Topography Mission (SRTM), we now have the data we need to calculate and
visualize avalanche danger zones (based on slope angle) on a digitized map,
anywhere on the planet.



In the past, we’ve relied on inclinometers—an in-field measurement to assess
slope angle. While this type of measurement is an absolute must when traveling
in the backcountry, a digitized slope angle layer can help with route planning
both before a trip and in the field. In the onX Backcountry and Offroad apps, we
display slope angle as a color-coded overlay that you can take with you, even
when you don’t have an internet connection. 


WHERE DOES THIS DATA COME FROM?


Slope angles, and most terrain analyses, rely on Digital Elevation Models (DEMs)
which are digital representations of the height of the earth’s surface at any
given location. There are two parameters we consider when using DEMs—spatial
resolution and vertical accuracy. Spatial resolution is the area covered by a
measurement. For example, a 30 meter spatial resolution means that a measurement
of the height at a given point covers a 30 meter by 30 meter square on the
ground. Vertical accuracy refers to the accuracy of the height measurement at
that location. The 3D Elevation Program (3DEP) claims a vertical accuracy of
3.04 meters (95% confidence level) for the conterminous United States. There are
two common techniques used to obtain this data: interferometry and Light
Detection and Ranging (LIDAR). Interferometry is a method that relies on
interference of radio, light, or UV waves to measure displacement, while LIDAR
relies on differences in the time it takes a beam of coherent light (laser) to
bounce off a surface. LIDAR-derived elevation data are typically higher
resolution and more accurate than interferometric elevation data, with both
vertical accuracy and horizontal resolutions below 3 meters. RADAR instruments
on satellites hundreds of kilometers above the earth’s surface provide the
source data for interferometric elevation, while LIDAR data is typically
collected via fixed-wing aircraft. At onX, we’ve sourced high resolution terrain
data, giving us a resolution for the U.S. in the range of 3 to 10 meters (and 25
meters for most of Canada) [3DEP, ArcticDEM, CDEM, and SRTM]. 

We provide slope angle data for four different horizontal resolutions or “zoom
levels” [slippy map tile resolutions]: 11, 12, 13 and 14. Horizontal resolution
refers to the number of measurements taken per meter along the surface of the
earth. At each of these zoom levels a sample point (where a height measurement
was recorded) represents a different horizontal distance. At zoom level 11, for
example, a single measurement (one pixel) covers an area of 76 meters by 76
meters. As we zoom in (get closer to earth) we pass through each of the above
zoom levels dividing the square’s side by 2. At level 12 our horizontal distance
(one pixel) becomes 38 meters by 38 meters, while zoom level 14 gives us a pixel
size of 9.6 meters by 9.6 meters. It is important to note that our vertical
resolution remains constant (between 3 and 10 meters).

Takeaway: Zoom level 14 corresponds to a distance of about 9.6 meters, the slope
angle at any point is the average slope covering an area measuring 9.6 meters by
9.6 meters. In our field testing, we found this resolution provided the right
amount of information to aid in terrain assessment without overwhelming map
download sizes; and yes, you can and should take this data offline. 

The author field testing the Slope Angle Layer.


LET’S DEFINE SLOPE ANGLE

Consider a simple two-dimensional line:



Recall from basic geometry, we can define this line as follows:

y = mx + b

where y is the height, b is y-intercept and m is our slope. Some simple algebra
allows us to rearrange this equation to solve for the slope as follows:

m = (y – b) / x

Alternatively, we can find the slope from any two points on the line, p1 and p2,
as follows:

m = (y2 – y1) / (x2 – x1) = dy / dx

This equation allows us to find a slope angle, 𝛳, given any two points:

𝛳 = atan(m)

Note, if our height value (y2 – y1) is big and our distance value (x2 – x1) is
small, we get a large slope value (steep). Conversely, if our height value is
small and our distance is big, we get a small slope value (flat). 

This works great for a line but what about curved terrain, like mountains? For
any point on the curve, if we place a ski on it such that the ski touches the
curve only once, we would find the tangent of the curve (the tangent represents
the instantaneous slope at that point). The angle the ski makes with the horizon
is the slope angle. 

With this in mind, we can now move the ski around adjusting it so that it
touches the curve only once, giving us the slope at any point on the curve. If
we moved the ski every meter or so along the curve and recorded the angle it
makes with the horizon, we would get a pretty good picture of how the slope
varies along the length of the curve (our terrain). We can achieve this by
applying the equation above to solve for the slope using a series of discrete
points as shown below:



Now, it’s important to note here that the distance, x2 – x1, matters. If we
choose an interval that is too big we might skip over a bunch of important
information. On the other hand, if the distance is too small, we would end up
spending the weekend calculating all the slopes and at some point the extra
detail would do us little good. This could also result in a dataset that is just
too large to take offline. 

Takeaway: Picking the right parameters for our slope angle calculation is
critical for an accurate and portable representation of the terrain around us. 




So, this works fine for two dimensions, but we care about the real world and our
world is three dimensional. Fortunately, we can extend this calculation to three
dimensions as follows:

m = atan(sqrt(dy / dx ^ 2 + dy / dz ^ 2)) 

Again, all we’re really doing here is approximating the slope at a given point;
and, if we have a good 3D representation of the earth’s surface, we can easily
calculate slope angle anywhere on the planet. 


THE LEGEND MATTERS

While this tool is incredibly powerful, it doesn’t substitute good
decision-making. Slope angle visualization is just one tool in your toolbox, and
like any tool, it is important that you learn how to use it safely. Slope angle
visualizations can be misleading if we don’t understand the legend. For example,
a slope angle color gradient not made for backcountry skiing might simply go
from green to red, 0 to 90 degrees. While this is a perfectly accurate way to
visualize slope, it doesn’t provide the best visualization for avalanche danger
for recreationalists. We’ve chosen a gradient that we think intuitively
highlights the danger zone for our users. In our Apps, green areas are 20-25
degrees—generally safer terrain. Yellow is 25-30 degrees, which is also
generally safe. Once we start to approach the 30 degree mark, shown in red,
in-field observation and careful terrain choice is recommended. 

This visualization allows us to create routes that minimize exposure to slopes
in an avalanche zone; and when we’re in the field, it allows us to quickly
determine if the line we’re about to ski is in a dangerous area. 

Takeaway: Slope angle visualization is a powerful tool but you must know how to
use it properly. This starts by understanding exactly what the different colors
indicate. 


CONCLUSION 

In this article we showed you how we calculate slope angle, where the data comes
from, its accuracy, and how we’ve chosen to visualize it on the map. While onX
has sourced the highest-resolution terrain data and double- (nay, triple-)
checked our math to create this powerful tool for your adventure, Slope Angle is
just one instrument in your arsenal to help you assess safety in the
backcountry. As always, it is your responsibility to ensure that you are
prepared for dangerous backcountry conditions, including checking avalanche and
weather forecasts, knowing avalanche safety protocols, staying alert to changing
conditions, and, of course, downloading Offline Maps. 

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JEFF LUTZENBERGER

Jeff Lutzenberger is VP of Research at onXmaps. He earned a Ph.D. in Electrical
Engineering from Montana State University, where he specialized in micro-optics.
After completing his doctorate, he shifted his focus to software engineering,
which eventually led him to digital mapping. Jeff is passionate about using
technology to make the world a healthier place. He spent the last 25 years in
Bozeman, Montana, raising a family, snowboarding, briefly tele-skiing, dirt
biking, mountain biking, and fly fishing. He recently traded mountains for water
and moved to Hood River, Oregon, where he now obsesses over his new outdoor
passion, wing foiling.
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