B Calculating 'Typical' Grade of a Hiking Trail

[Elquery]

TL;DR Summary

Is there a difference between 'average' slope and 'typical' slope, and can both be calculated from the same limited information?

Hi all.
I am tasked with quantifying hiking trail grades (percent grade) and their 'typical' grade.
I am not sure if 'typical' is really a math term, but my first inclination was that it equates to 'average.' However, I now think there are two ways to approach this with different results.

Is it accurate to say that the average slope of a function is the rise over run of the straight line drawn between the beginning and end point?

In the case of trail grades, I would take the absolute value of slope values such that negative slopes translate into positive slope, because even a 'negative' slope is a grade as far as a hiker is concerned.

It is therefore possible to calculate this 'average slope' by dividing total elevation change (absolute value of all line segment rises) by horizontal trail distance (run). This is mathematically the same as taking the slope of each line segment (I am modeling this as a collection of straight line segments, no curves. See attached photo), and doing a weighted average based on the run of each segment. So you don't need to know the run of each segment; you only need to know the total change and the total horizontal run.

The issue is that in this application, it seems 'typical' grade should really account for the distance of the actual trail—in other words, the distance of the sloping line and not just the horizontal component (run). You cannot simply divide the total rise by the sloping line distance though; that doesn't give slope of any kind as far as I can tell. I know I can calculate the slope of each line segment and then do a weighted average weighted based on the linear distance (the hypotenuse of each segment) as opposed to weighted by the run. But this requires knowing the distance values of each of these segments and punching a long string of numbers out.

With the second definition of 'typical' slope, is there any way to calculate this value from only:

• total elevation change
• total linear distance (sum of all segment's hypotenuse)
• total horizontal distance (run)

without having to know either distance value for each given segment?

Or is the only way to weight these values based on knowing the distance of each unique slope segment?

Thanks for any thoughts!

 

[hutchphd]

What form is the data at your sidosal? Do you have just a topo map or do you have access to isomething better? This will drive the method you wish to use.
An Excel spreadsheet could easilly slice and dice a good data set. n But data is data and averages are averages.

 

[Elquery]

I can use ArcGis, however the 'data' is yet to be collected. ArcGis can actually perform slope calculation based on digital elevation models and trail line data, but it will not be very accurate vs on-the-ground measurments. I have yet to collect any on the ground data, and that is part of the reason for my question.

data is data and averages are averages.

I have no idea what this means. Do you agree or disagree with my assessment above that there are 2 distinct ways to calculate 'average' or 'typical' grade?

I am specifically interested if the second 'typical grade' method can be calculated with the 3 data items I listed in bullet point form in the OP, or if I will need much more granular data.

 

[hutchphd]

The term "typical grade" could mean any of a number of things, each calculated differently. You could, for instance define it in such a way that it is the same as the average grade. I do not know what definition you will choose nor why you will choose it. For instance I don't know the intended use of the information. The size of the granularity will be determined by these requirements.
I would think the mean value of the magnitude of the altitude change at a granularity of ~20 ft might work. Or maybe that number for only the "nonflat" parts of the trail. But better data will give you more options. I am unaware of any standard definition.

 

[jbriggs444]

The term "typical grade" could mean any of a number of things

"Typical" could refer to "most common" -- a mode. You sort your sample values into bins (0 to 1 percent, 1 to 2 percent, etc) and see which bin(s) have the most sample values. You may want to take absolute values before sorting.

"Typical" could refer to a median. You sort your sample slopes in order and select the value that is at the 50'th percentile.

You could get fancier and use something like "the mean of the absolute value of the slopes for all samples between the 10th and the 90th percentile". So your measure ignores the stairs at the beginning and the flat part down by the lake.

 

[A.T.]

With the second definition of 'typical' slope, is there any way to calculate this value from only:

• total elevation change
• total linear distance (sum of all segment's hypotenuse)
• total horizontal distance (run)

without having to know either distance value for each given segment?

I don't think so. You can use limiting cases to see that this information is not sufficient to give a unique solution: Note that this "hypotenuse weighed average slope" goes to infinity as soon as one segment of non-zero length tends towards vertical. So for the same values of the 3 parameters above you can construct 2 similar paths, but with vastly different "hypotenuse weighed average slopes".