# Is there a way to quantify how hilly an area is? March 19, 2013 8:49 PM   Subscribe

Is there some measure or standard way of quantifying how hilly or a flat a city is, or some other area is? Can you say San Francisco is 4.9 hillybits while Houston is only 0 hillybits?
posted by sien to Science & Nature (18 answers total) 9 users marked this as a favorite

Topographical prominence comes to mind.
posted by dfriedman at 9:06 PM on March 19, 2013

It seems this question at GIS stackexchange is roughly what I was after.

Sorry, couldn't find this looking around previously.

But if anyone knows more that would be much appreciated.
posted by sien at 9:23 PM on March 19, 2013

Average slope seems like the right idea. Calculating it wouldn't be hard given the data in the right format, but I don't know anything about GIS so I don't know how to get the data.
posted by madcaptenor at 9:28 PM on March 19, 2013

Informally at work, I just say there is "a lot of topography" when I'm dealing with a hilly area. Formally you could describe the size of the hilly area and the spacing, character, and contrast between topo highs and lows. Check out a description of the basin and range geologic province or the sandhills area in nebraska for a good idea of that. Geology can be a very descriptive art because it is hard to compare apples and oranges that are colloquially in the same category!

That link just tells you the overall slope of an area... On a small scale it could tell you the slope of one hill, but over a area with more than one hill you dont get the height, frequency of peaks and valleys, or anything that would be very useful in characterizing "hilliness" for comparison.
posted by cakebatter at 9:30 PM on March 19, 2013 [2 favorites]

One way to measure it would be to consider the ratio of the surface area to the circumference. A hillier area would have a larger surface area bounded with in a given circumference than a perfectly flat region would. That would depend on how you're sampling it, though.
posted by empath at 9:31 PM on March 19, 2013 [2 favorites]

empath : Yep, very true.

It would suffer from what cakebatter points out as well though, it wouldn't capture the total height difference.

cakebatter : Yep, all very true. Interesting.

posted by sien at 9:38 PM on March 19, 2013

I think empath has it but another term is to search for is "total relief".
posted by fshgrl at 9:59 PM on March 19, 2013

I googled "quantify hilliness" and came up with some bicycle sites that look at this as part of describing bicycle rooutes. This link shows a list of terms used (flat, rolling, hilly, etc) and it is the closest thing I have seen to a system of classification.

I am not sure what you are trying to do though. In searching on "quantify hilliness toppography" I also found an excerpt from a book called Golfonomics where they did an objective measure of actual elevation differences and a subjective measure via interviews of golfers as to how hilly a course was.

I also tripped across an ESRI article on surface creation and analysis, but I am assuming that it isn't really what you need.
posted by Michele in California at 10:19 PM on March 19, 2013 [1 favorite]

Not entirely a serious answer, but something amusing and related: Kansas is flatter than a pancake.
posted by thack3r at 10:30 PM on March 19, 2013

Fractal dimension was created for pretty much this purpose (characterizing roughness).

You could also assess the average difference between the average height and the actual height.

You could analyze the kurtosis or variance of the elevation data in the region.
posted by pmb at 10:32 PM on March 19, 2013 [1 favorite]

If you want to quantify this, perhaps consider sampling some number of polygons at random from three sets of regions, calculating the mean and variance of elevations sampled in each set.

The first set would be "background", the elevations across, say, samples of polygonal regions randomly picked from over the entire country. The second set would be elevations in polygons sampled over San Francisco. The third set would be elevations sampled in polygons from Houston.

The first set of elevations gives you the baseline or expected mean and variance of elevations. The second and third sets give you the observed deviation from this baseline.

On a qualitative level, greater variance is greater "hilliness", while lower variance would suggest the sampled region is "flatter". On a quantitative level, you can calculate a z-score from the expected and observed parameters which shows how much San Francisco and Houston deviate from the country-wide baseline.
posted by Blazecock Pileon at 10:45 PM on March 19, 2013 [1 favorite]

empath : Great find on that GIS plugin. It gets quite a few different measures that seems to be well worth it.

You could combine these and then weight them as desired.
posted by sien at 12:00 AM on March 20, 2013

If you can get your hands on a contour map, you could base hilliness on lots of touching isopleths.
posted by oceanjesse at 6:30 AM on March 20, 2013

You haven't said why you are interested in this, but as a person who rides a bicycle in a city, I can tell you that even if you measure the hilliness of the terrain itself, that's not enough to tell whether your ride will be pleasant or not. You have to take into account any road crossings which are built over another road, since you will have to pedal mightily to get to the top and then glide down the other side.
posted by CathyG at 9:56 AM on March 20, 2013

There's a concept in mathematical analysis called "total variation (there's a wikipedia article, but beyond the first equation that defines it, the article is more confusing than illuminating).

The essential idea is that you add up all the changes in altitude in going from one point to another, but you ignore the sign, i.e. up 10 feet and then down 10 feet totals to 20 feet of change. It's nice because, unlike that stackexchange question, you don't have to calculate slopes, which are very sensitive. It's pretty easy to understand what it means in one dimension, but a little harder to understand the two dimensional case (the equation is much uglier too). But once you calculate it you could divide by the underlying area and get a "feet per square mile" measurement suitable for comparison.

That said, going from this concept to a useful GIS implementation would be a lot of work.
posted by benito.strauss at 3:47 PM on March 20, 2013

CathyG - I was interested in this because a city in Australia, Brisbane, is said to be 'hillier' than other major cities and I was curious as to how you would quantify this.

I've also wondered about it from riding bikes around too.

But it just seemed like an interesting question as well really.
posted by sien at 4:24 PM on March 20, 2013

In case anyone comes across this later, if you read down further in that Wikipedia article on Total Variation, you can see that in most cases the total variation just finds the total slope over the surface, which is what was originally asked. So nothing really new there. (Though if you're feeling really nerdy, I think it becomes interesting again if you think about using the ℓ1 norm instead of the ℓ2 norm specified.)
posted by benito.strauss at 11:38 PM on March 29, 2013

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