Highpoints by Volume

While looking for various mountain metrics I found the ORS (Spire) Measure as well as Tim Worth’s Steepness Measure which is a simplified version of ORS. I decided to try an intermediate measure – measuring the volume of the peak near the summit (100m) and at its shoulder (800m). The thought being that spires/towers will have a high volume at 100m, but their 800m volume won’t be that much larger. Whereas a flat-topped mountain will have a low volume at 100m, but their 800m volume will be much larger.

Imagine taking a core sample (100m or 800m radius) centered on the summit. Lower/drill a cylinder down until the entire bottom edge is contacting rock. This will occur at the “Max Drop” value Tim Worth calculates. Cut the cylinder off at this depth and calculate the volume of rock in the cylinder (core sample).

The ORS/Spire Measure uses integrals and a complex function to calculate the volume of a peak. I simplified this by measuring discreet rings from the center to the cylinder’s radius. I used 20m increments for the 100m cylinder and 100m increments for the 800m cylinder. The first level is simply a right triangular pyramid (triangle in 2D). All subsequent layers are a horizontal triangular prism (wedge) (triangle in 2D) and a vertical triangular wedge (cheese wedge) (rectangle in 2D). For all angles except for the Max Drop (Max Bearing), I add in a final vertical triangular wedge (cheese wedge) (rectangle in 2D) which brings all angles down to Max Drop level. You can see this in the graph below for Borah Peak, Idaho at 100m. The left side is along the Min Drop bearing, while the right side is along the Max Drop bearing.
The image on the right shows an exaggerated view (30° angle instead of a 1° angle) from the top. The color-coded layers match in the two graphics. The shaded triangles would form small right triangular pyramids but these were not included in the volume measurements below.

We can compare the 100m and 800m volumes in a couple of ways:

Let’s look at Mount Greylock, Massachusetts which has a median 100m volume but a higher than median 800m volume:

while White Butte, North Dakota has a higher than median 100m volume but a lower than median 800m volume:

and Granite Peak, Montana which has a very high 100m volume but an mostly equivalent 800m volume to Mount Greylock:

This is where I throw up my hands. While I (Clint Kaul) consider Granite Peak to be “steep” at 800m (and at 100m), I personally wouldn’t call Mount Greylock “steep” at 800m. Although they have roughly equivalent 800m volumes. I have a feeling we need an additional factor to help determine “steepness”. Maybe factor in the peak’s Prominence or the Max Drop value? Or look at the ratio of Avg Drop to Max Drop (as seen in this graphic – AvgMax xxx is the (Avg Drop)/(Max Drop) at 100m and 800m). The idea being if the Avg Drop is close to the Max Drop we are looking at a “conical” peak, while an Avg Drop far from Max Drop might indicate a flatter peak with a steep ravine/cliff on one side.

Looking at the radar plots above, maybe count the number of lobes (number of ravines/cliffs) or the size of the lobes or the variance in the size of the lobes [see the “Max StdDev” columns in the 100m/800m specific tables below]? Or perhaps look at intermediate angles (for example, 0-20, 20-40, 40-60, 60-80 and 80-100) and compare them to the overall angle (at 100m in this example) [see the “Sub Angle” columns in the 100m/800m specific tables below]. Another crazy idea is documented in Appendix – Drops as a Waveform. If you have comments/thoughts on this, please contact me clintkaul@highpointers.org.

This table shows the summary “volume” metrics for 100m/800m
Max 100 – maximum drop in feet at 100m
Avg 100 – average drop in feet at 100m
Vol 100 – volume in cubic feet at 100m
Max 800 – maximum drop in feet at 800m
Avg 800 – average drop in feet at 800m
Vol 800 – volume in cubic feet at 800m

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* – Note that Connecticut is measured from Mount Frissell, MA (the summit) versus the contour line on the CT/MA border.
* – Note that Northern Marianas Islands has no spot elevations

These scatter plots show how the Max Drop (left) and Avg Drop (right) compare to the Volume at 100m. You’ll notice that the Max Drop is highly correlated to the Volume, but this is due to how we calculate the volume – down to Max Drop in all directions.

This heat map is color-coded based upon the Volume metric at 100m.

I’ve recently calculated the maximum sub-angle and standard deviation of the drops in all directions at 100m. This graphic summarizes this information. It seems like the standard deviation is a good indication of the steepness near the summit area.

This table shows the “volume” metrics for 100m
Max Drop – maximum drop in feet
Max Angle – maximum angle [atan(max drop/100m)] in degrees
Sub Angle – maximum sub-angle found (0-20m, 20-40m, 40-60m, 60-80m, 80-100m)
Max StdDev – sample standard deviation of the drops in all directions at 100m
Avg Drop – average drop in feet
Avg Angle – average angle [atan(avg drop/100m)] in degrees
Volume – volume in cubic feet

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* – Note that Connecticut is measured from Mount Frissell, MA (the summit) versus the contour line on the CT/MA border.
* – Note that Northern Marianas Islands has no spot elevations

This scatter plot compares the standard deviation (stddev) at 100m versus at 800m. While the comparison doesn’t mean much, it helps identify comparable peaks.

Let’s look at Massachusetts (left) and Oregon (right) which have equivalent stddev at 800m, but Oregon has a vastly greater 100m stddev. Looking at the 100m ring (blue) you’ll see Massachusetts is much flatter than Oregon.

Now let’s look at South Dakota (left) and Texas (right) which have equivalent stddev at 100m, but Texas has a vastly greater 800m stddev. Looking at the 800m ring (orange) you’ll see Texas has more ridges/valleys than South Dakota.

I’ve recently calculated the maximum sub-angle and standard deviation of the drops in all directions at 800m. This graphic summarizes this information. It seems like the standard deviation is a good indication of the steepness near the shoulder area.

This table shows the “volume” metrics for 800m
Max Drop – maximum drop in feet
Max Angle – maximum angle [atan(max drop/100m)] in degrees
Sub Angle – maximum sub-angle found (0-100m, 100-200m, …, 700-800m)
Max StdDev – sample standard deviation of the drops in all directions at 800m
Avg Drop – average drop in feet
Avg Angle – average angle [atan(avg drop/100m)] in degrees
Volume – volume in cubic feet

Only admnistrator owned posts can execute the [includeme] shortcode. This message is shown only to administrators.

* – Note that Connecticut is measured from Mount Frissell, MA (the summit) versus the contour line on the CT/MA border.
* – Note that Northern Marianas Islands has no spot elevations

Appendix – Drops as a Waveform

Looking at the lobes in the radar plots got me thinking. How do I calculate the area? It turns out to be "Avg Drop" * 360. Initially I was going to “unwrap” the circle into a straight line and figure the area under the line. For example, see Massachusetts (left) versus Montana (right) at 100m:

That triggered the crazy idea. What if we treat this as a [triangular] waveform? Maybe by looking at amplitudes or frequencies we can find a “steepness measure”? To test the idea I sub-sampled 256 points out of the 360 data points and applied a fast fourier transform to come up with the following table.

This table shows the “statistics” and “fft” metrics for 100m and 800m
Avg 100 – average drop at 100m in feet
StdDev 100 – sample standard deviation of the drops in all directions at 100m
NyqF 100 – Nyquist frequency at 100m
Freq 100 – maximum frequency at 100m
Avg 800 – average drop in feet at 800m
StdDev 800 – sample standard deviation of the drops in all directions at 800m
NyqF 800 – Nyquist frequency at 800m
Freq 800 – maximum frequency at 800m

Only admnistrator owned posts can execute the [includeme] shortcode. This message is shown only to administrators.

* – Note that Connecticut is measured from Mount Frissell, MA (the summit) versus the contour line on the CT/MA border.
* – Note that Northern Marianas Islands has no spot elevations