Tuesday, September 22, 2015

Answer: How to describe the hills?

It's interesting to answer.. 

.. an SRS question about hills when I'm back in Bonaire, spending most of my free time underwater.  As you probably remember from the parrotfish Challenge of 2 years ago, this is a fascinating place.  

But we were talking about hills, like this one: 

Or this one... The key feature is that the hills repeat periodically.  

(These are both from the Pt. Reyes area.)  

Recall that the Challenge was this: 

1.  Does the term "frequency" make sense when applied to repeating hills?  Frequency is usually defined as cycles / unit-time.  Can you figure out how to apply the concept of "frequency" to hills?  
2.  What's the frequency (however you define it for hills) of this stretch of hills above?  You'll have to find it, measure it, and then figure out the "frequency," if you can!  
3.  Can you find a stretch of the Earth that has a nicely repeating pattern to it similar to the one above?  If so, where is it?  (Give us the lat/long in your answer.)  

Here's how I thought about the Challenges.  

1.  Frequency for repeating hills?  This Challenge came about when I thought about my travel uphill and downhill as though I were on a sine wave.  When I wanted to write about the experience of running on the trails in the hills, I really wanted to say  "the hills are high frequency hills."  The idea being that they go up and down in rapid succession, just as a high frequency sound wave would.  

As several people noted, the idea of frequency generally involves a time basis.  That is, frequency is usually expressed in "cycles per second."  At the beginning of an orchestral concert, the concertmaster would often play the note of A, which has a frequency of 440 hertz (or cycles-per-second).  

But what about hills?  More generally, what about space?  

I figured that I must not have been the first guy to think about this, so I did a search for: 

     [ spatial frequency ] 

and found, much to my surprise, that I'd accidentally found that this is exactly the right term for what we seek.  The Wikipedia article defines spatial frequency as "a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components ... of the structure repeat per unit of distance. The SI unit of spatial frequency is cycles per meter."   (cpm)  

Ah ha!  So if we measure the peak-to-peak distance of the waves of the hills, we can figure out the "cycles / meter" to get the spatial frequency of those particular Pt. Reyes hills.  

So.. how do we do measure the peak-to-peak distance?  

You COULD get the "Terrain map" view (aka "topographic map" and just measure the elevation of the peaks and the valleys.  But that's pretty slow.  Could we find a better way? 

A query like this: 

     [ Google Earth elevation data

tells you that you can use Google Earth to draw a path anywhere on Earth, and ask for the "elevation profile."  

That's exactly what we want.  In Google Earth I fly to that part of the Bolinas Ridge, and draw a path, then ask for the elevation profile.  

(Let me know if you want more details on how to do this.  I can always add an appendix with details about how to open Earth, draw a path, and get the Elevation Profile.)  

2.  What's the frequency Kenneth?  From this diagram, I just read off the first set of distances of the peaks from the origin.  

I dropped the numbers into a spreadsheet, then averaged the distances together (not a perfect solution, but good enough for this use case). 

Since the definition of a cycle is the distance between peaks, we can see that the average distance between these regular hills is 576 meters.  

Now imagine that you've got hills that are regularly spaced, like this:  

Suppose that the red line is 1 m long.  These hills (well, speed bumps, more like) would be at 1 cycle/meter--or 1 cpm.  If they're 2 meters apart, they'd be 0.5 cpm, and so on.  

Our Point Reyes hills are 576 meters apart, so that's 1/576th of a cycle per meter, or (using Google Calculator)  0.001736 cpm.  

At this point, it probably would make sense to convert the measure to kilometers.  (So I multiply our number by 1000.)  1.736 c/km.  That's a much more understandable measure:  it means you go up and down once in just under 2 km.  (Or, in English units, one cycle / mile.)  

Now we have it!  

3. Other parts of the world with repeating hills? Sure.  This is a great way to spend time in Google Earth.  My favorites are the Sand Hills of Nebraska: 

and some nicely regular places in the Rockies: 

Search Lessons 

1. Looking for concepts you sometimes find you what you need...  In the "spatial frequency" case, we found the idea we were looking for.  In the process, we learned how to define cycles-per-meter, which is what we needed. 

2.  Look for tools.  As we've mentioned before, searching for a tool (in this case to find elevation profiles) was straightforward.  In this case, I just created a spreadsheet to compute the average of the inter-peak distances, but if you want to get fancy, you could create a path with waypoints on each of the peaks, then export the KML file and get the elevations automatically--if enough people write in, I'll show how to do that particular piece of data munging as well)  

In the week ahead... Search on! 

My crack SearchResearch Dive Team hunting for the next SRS Challenge


  1. Hello Dr. Russell. It was as always a great Challenge.

    I don't know why didn't think about [Spatial frequency-] Maybe because it sounds too obvious or because I was looking for something else. In any case, one more lesson in Challenge for me.

    Elevation profile is a great tool, thanks for telling us more about it. I never thought about using this tool for the answer. And, yes, I didn't know how to answer q1 and q2. Glad that you told us how.

    Beautiful hills, Dr. Russell, thanks for sharing.

    I hope you enjoying in Bonaire. I can not believe there are 2 years already since ParrotFish Challenge.

    One question, I didn't understand this:"What's the frequency Kenneth?" so searched [ frequency kenneth] is it about This ?

    Looking forward for the New MOOC and new challenges. Enjoy trip!

    1. Glad you enjoyed it, Ramón!

      You're correct.. that's exactly what my "what is the frequency Kenneth" comment was in reference to. It was a fairly bizarre episode.

    2. a bit of elaboration…
      where's the density,Dan? (little known photo of the 2nd man possibly involved - not William Tager)

  2. Hello Dan,

    "1.736 c/km. That's a much more understandable measure: it means you go up and down once in just under 2 km." I'm not sure I understand. 1.736 c/km means that when on the map you cover 1 km, you go up and down about once and 3/4 (you're halfway on the second descending slope). If you want to go up and down just once, you have to cover 576 m i.e the wavelength of the pattern. I think this is why "spatial frequency" isn't used but wavelength may be.