State borders are not as simple as they look

[November 1st, 2009]

When I was a kid, I would look at maps of the American West and be fascinated by those enormous states. Wyoming! Utah! Colorado! They looked like giant mathematical figures, crisply defined in an otherwise chaotic world. I also thought that the borders really were a dotted line on the ground, but that misconception was quickly disspelled by my parents.

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Nevertheless, I always carried in my mind the idea that western states were mostly defined by straight lines of whole latitude and longitude (parallels and meridians).

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So it was with some surprise that, while working on a <a href=”http://www.integrity-logic.com”>GIS app for the iPhone</a>, I discovered that these cherished beliefs were in fact plain wrong.

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The first misconception was that these straight lines run mostly along whole meridians and parallels. It certainly looks that way at first sight. For instance, the border between Arizona and New Mexico really does look like it’s smack on the 109th west meridian. But when you zoom in, you see that it’s actually off, and by more than a rounding error. What happened? Did the surveyors of yore goof? As a <a href=”http://www.amerisurv.com/content/view/6057/”>fascinating article in The American Surveyor</a> reveals:

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“In a case of national pride edging out common sense, the dividing line was defined as the 32nd meridian west of Washington”

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which turns out to be almost three minutes of arc west of the 109th meridian, or roughly three miles. Goodness forbid we should keep things simple. At least we can blame this one on the politicians.

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OK, so the border between Arizona and New Mexico is a bit strange. But that’s the exception, right?

<p/>

Well, no. The border between Colorado and New Mexico (and a bunch of other states) was originally defined as the 37th parallel. But man proposes, and the surveyor disposes. As it turns out, the surveyors did make a small mistake of 365 feet — not bad at all for late-19th century techniques, but enough to create a problem now. So, as described in <a href=”http://www.waymarking.com/waymarks/WM72RM_Latitude_37_North_the_border_between_six_US_States_Four_Corners”>Waymarking</a>:

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“Instead of a costly reassessment of the border line, the states affected and the US Supreme Court amended the definition of the state’s border, so now, the border between Utah, Colorado and Kansas in the North and Arizona, New Mexico and Oklahoma in the South is no longer at 37 N but at 36 59’56.34”N. “

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My belief in the cosmic order thus shaken, you can imagine how perturbed I was when I realized that, not only are these lines not exactly on whole meridians and parallels, but in fact they’re not even straight at all.

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Colorado, for instance, looks like it’s a perfect rectangle (if we ignore the spheroidal surface of the earth, of course). But if you examine closely the border with Utah, you’ll notice that there are <a href=”http://maps.google.com/?ie=UTF8&ll=38.155617,-109.008408&spn=0.478381,0.632401&z=11″>a few squiggles</a>. Small to be sure, but it is not a straight line. Same thing for the <a href=”http://maps.google.com/?ie=UTF8&ll=40.997537,-108.629014&spn=0.014349,0.019763&z=16″>border with Wyoming</a>. All these slight deviations from the geometrically ideal go back to the 19th century when surveying, although remarkably precise considering the technology available at the time, was not what it is today.

<p/>

In the case of the Colorado/Utah border, it was a <a href=”http://geology.utah.gov/surveynotes/gladasked/gladkink.htm”>surveying oopsie</a> of just over a mile. Still not bad for 1879. The border between California and Nevada was defined as a straight line from what turned out to be the middle of lake Tahoe to the intersection with the Colorado river. But wouldn’t you know it, the river moved, and the surveyor, rather than start from scratch, fudged the line. Ah, contractors!

<p/>

In fact, the closer you look, the worse it gets. All these borders are ultimately defined by monuments which were erected by surveying teams. As Ivars Peterson says in his blog, <a href=”http://mathtourist.blogspot.com/2007/08/rectangular-states-and-kinky-borders.html”>The Mathematical Tourist</a>:

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“once a border is defined on the ground and accepted by the interested parties, it becomes official, even if it doesn’t follow the written description.”

<p/>

So it’s not just that there are a few errors here and there — these seemingly-rectilinear states are actually polygons with hundreds of side!

<p/>

These vagaries can sometimes have serious consequences. A mini-war was actually fought between California and Nevada over this. Judges were thrown in jail, guns were shot in anger, and blood was shed during the <a href=”http://www.nevadacas.com/rccsawar.htm”>”Sagebrush War”</a> (more of a skirmish, really) in 1863, fought over which state Susanville belonged to. Maybe they felt like they were missing out on the Civil War and needed some excitement of their own. At one point, Susanville was the seat of two counties, one in Nevada, the other in California. They even got to elect a representative in both states.

<p/>

This all seems very quaint and amusing in this era of GPS and laser ranges, so it’s hard to believe that it’s not yet over. As recently as 1980, there was a <a href=”http://www.altlaw.org/v1/cases/384566″>lawsuit between the states of California and Nevada</a> regarding the exact location of the border. That is a long-running feud between the two states, mostly because the line has been <a href=”http://www.waymarking.com/waymarks/WM2X8Z_California_Arizona_and_Nevada”>redefined so many times</a>. And even today, in 2009, there are <a href=”http://www.nobleco.org/surveyor/index.php?q=all-in-the-meridian”>legal maneuverings between the U.S. and Canada</a> over the exact location of the Alaska border. It looks like the surveyors were a few hundred feet off (and, really, would could blame them?). A few hundred feet doesn’t sound like much until you multiply it by 647 miles. Add some oil and gas, and you can keep a sizeable herd of lawyers fat and happy for a long time.

<p/>

So this is what I’ve learned: borders are not on neat meridians and parallels, they’re not straight lines, and sometimes we’re not even completely sure where they are. Apparently, the world is not as neat and orderly as I imagined when I was a child.

When I was a kid, I would look at maps of the American West and be fascinated by those enormous states. Wyoming! Utah! Colorado! They looked like giant mathematical figures, crisply defined in an otherwise chaotic world. I also thought that the borders really were a dotted line on the ground, but that misconception was quickly dispelled by my parents.

Nevertheless, I always carried in my mind the idea that western states were mostly defined by straight lines of whole latitude and longitude (parallels and meridians).
So it was with some surprise that, while working on a GIS app for the iPhone (now defunct), I discovered that these cherished beliefs were in fact plain wrong.

The first misconception was that these straight lines run mostly along whole meridians and parallels. It certainly looks that way at first sight. For instance, the border between Arizona and New Mexico really does look like it’s smack on the 109th west meridian. But when you zoom in, you see that it’s actually off, and by more than a rounding error. What happened? Did the surveyors of yore goof? As a fascinating article in The American Surveyor reveals:

In a case of national pride edging out common sense, the dividing line was defined as the 32nd meridian west of Washington

which turns out to be almost three minutes of arc west of the 109th meridian, or roughly three miles. Goodness forbid we should keep things simple. At least we can blame this one on the politicians.

OK, so the border between Arizona and New Mexico is a bit strange. But that’s the exception, right?

Well, no. The border between Colorado and New Mexico (and a bunch of other states) was originally defined as the 37th parallel. But man proposes, and the surveyor disposes. As it turns out, the surveyors did make a small mistake of 365 feet — not bad at all for late-19th century techniques, but enough to create a problem now. So, as described in Waymarking:

Instead of a costly reassessment of the border line, the states affected and the US Supreme Court amended the definition of the state’s border, so now, the border between Utah, Colorado and Kansas in the North and Arizona, New Mexico and Oklahoma in the South is no longer at 37 N but at 36 59’56.34”N.

My belief in the cosmic order thus shaken, you can imagine how perturbed I was when I realized that, not only are these lines not exactly on whole meridians and parallels, but in fact they’re not even straight at all.

Colorado, for instance, looks like it’s a perfect rectangle (if we ignore the spheroidal surface of the earth, of course). But if you examine closely the border with Utah, you’ll notice that there are a few squiggles. Small to be sure, but it is not a straight line. Same thing for the border with Wyoming. All these slight deviations from the geometrically ideal go back to the 19th century when surveying, although remarkably precise considering the technology available at the time, was not what it is today.

In the case of the Colorado/Utah border, it was a surveying oopsie of just over a mile. Still not bad for 1879. The border between California and Nevada was defined as a straight line from what turned out to be the middle of lake Tahoe to the intersection with the Colorado river. But wouldn’t you know it, the river moved, and the surveyor, rather than start from scratch, fudged the line. That’s contractors for you.

In fact, the closer you look, the worse it gets. All these borders are ultimately defined by monuments which were erected by surveying teams. As Ivars Peterson says in his blog, The Mathematical Tourist:

once a border is defined on the ground and accepted by the interested parties, it becomes official, even if it doesn’t follow the written description.

So it’s not just that there are a few errors here and there — these seemingly-rectilinear states are actually polygons with hundreds of side!

These vagaries can sometimes have serious consequences. A mini-war was actually fought between California and Nevada over this. Judges were thrown in jail, guns were shot in anger, and blood was shed during the “Sagebrush War” (more of a skirmish, really) in 1863, fought over which state Susanville belonged to. Maybe they felt like they were missing out on the Civil War and needed some excitement of their own. At one point, Susanville was the seat of two counties, one in Nevada, the other in California. They even got to elect a representative in both states.

This all seems very quaint and amusing in this era of GPS and laser ranges, so it’s hard to believe that it’s not yet over. As recently as 1980, there was a lawsuit between the states of California and Nevada regarding the exact location of the border. That is a long-running feud between the two states, mostly because the line has been redefined so many times. And even today, in 2009, there are legal maneuverings between the U.S. and Canada over the exact location of the Alaska border. It looks like the surveyors were a few hundred feet off (and, really, would could blame them?). A few hundred feet doesn’t sound like much until you multiply it by 647 miles. Add some oil and gas, and you can keep a sizable herd of lawyers fat and happy for a long time.

So this is what I’ve learned: borders are not on neat meridians and parallels, they’re not straight lines, and sometimes we’re not even completely sure where they are. Apparently, the world is not as neat and orderly as I imagined when I was a child.

Reading the world in Braille

January 3rd, 2010

Ten years ago, in February 2000, NASA mapped the entire world in eleven days.

It’s true: the mission was called the Shuttle Radar Topography Mission (SRTM), and over the course of eleven days, it used a big radar attached to space shuttle Endeavour to get elevation data from the vast majority of solid Earth; practically all land between 60 degrees North and 56 degrees South was included, with a resolution of 30 meters (100 feet). Over 9 terabytes of data were captured. It then took two years to process that data and make it usable (and it’s still being refined to this day).

This data is freely available to anyone, and the number of possible applications is almost infinite. It’s been used in GIS, cartography, environmental planning, weather modeling (weather patterns are enormously influenced by the topography), flight simulators, Google Earth, and the list goes on.

I’ve become quite familiar with this data because we used it as one of the base layers in a suite of GIS applications for the iPhone (now defunct). We simply represented it using color-coding, like in most maps, and adding a bit of shading for effect.

In this short article, I’d like to give you a quick tour of the kinds of things this data can reveal. My hope is to get you thinking about what else could be done with this incredible resource.

So what does the data look like?

Imagine being a blind giant, feeling the surface of the Earth with your fingers and trying to guess what you’re touching — that’s what browsing this data is like.

The big mountains, such as Mt. Rainier in Washington state, look pretty much as you’d expect:

Even modest mountains, such as those of southern Ohio, look crisp and present a clear picture of the region’s topography, with water erosion clearly visible.

Distinct features

Some features are quite noticeable. For instance, can you guess what the following is (it’s in southern California)?

If you guessed that it was some sort of fault, don’t feel bad: that’s what I thought too. But it turns out that it’s just a line of sand dunes east of Mexicali:

Which teaches us that not all linear features are faults, even in California. Though a lot of them are. Here’s a portion of central California showing the San Andreas fault:

And here’s the same area, with an extra layer showing currently active faults (color shows recency of movement, line width shows slip rate):

Down to the bare essentials

Seeing nothing but the terrain turns out to be surprisingly useful. Without the distraction of all the other information that is usually displayed on a map, one can really focus on the shape of the landscape. For instance, central Wisconsin shows a subtle contrast between east and west, with the eastern part being more bumpy, and the western part seeming noticeably smoother (look carefully).

This is the limit of the last glaciation. When the glaciers retreated, they left all kinds of material behind, causing the bumpiness of the terrain in that part of the state. Here’s the same location with an additional layer showing the glacial limit:

Cities and buildings

It should come as no surprise that the radar cannot distinguish between natural and man-made features.

Let’s take a look at downtown Boston — an area not known to be particularly hilly.

These bumps are the downtown skyscrapers — averaged out, since the resolution in this picture is 90 meters (300 feet). The small cluster in the lower left is the John Hancock tower. I added a hydrology layer to make it easier to see, but obviously radar does not know water from land.

Big holes in the ground

Looking at Chicago, I was worried when I first saw what looks like a big hole just outside of the city. Could it be a flaw in the data? Why would there be a 250-foot (~75 meters) deep hole at this location?

But it turned out to really exist – it’s the McCook quarry, the second largest quarry in Illinois.

It is so large, in fact, that it’s going to be used to store runoff and sewage from the city. There’s one quarry you may not want to go swimming in.

Florida

Florida presented some special problems. We thought about not having a terrain layer at all, since the state is so flat (highest point: 345 feet/105 meters, on the border with Alabama). But in some ways, that makes smaller features that much more noticeable. For instance, I was puzzled by this strange hill southwest of Boca Raton (resolution 180 meters/600 feet):

Its top elevation is 177 feet/54 meters in the SRTM data, and that’s practically Everest in Florida. A look at the Google Maps view reveals what it is: a landfill.

In fact, it’s quite notorious. Dubbed Mount Trashmore by the locals, the Broward County Landfill has now reached a height of 225 feet/68 meters, and is the subject of much debate in the area. There is even talk of growing it to 280 feet. Why not take it all the way to 346 feet and make it the highest point in Florida? Perhaps our landfills will be the solution to rising sea levels. I have so many creative solutions like this, but no one will listen to me.

Conclusion

I hope to have given you a quick sense of the potential applications for the data from SRTM. It’s up for grabs. Wikipedia has quite a bit of information about it. What creative use will you make of it?

Geology as art

August 29th, 2011

When I see a geological map, it doesn’t make a whole of of sense to me. Perhaps that helps me appreciate their abstract beauty. The fact that they are in fact not at all abstract, and represent the very ground we stand on, makes them that much more appealing to me.

Recently I’ve had the opportunity to look at a lot of these maps. I have learned a little bit about geology, but I have also learned to appreciate the sheer visual quality of these works of art.
One striking example is this snapshot from the USGS geological map of Virginia, showing a portion of the Appalachian mountains:

If you’re a geologist, then you’ll know that greens and blues tend to represent sedimentary rocks, whereas purples are for metamorphic rocks. If you’re a geological rube like me, you’ll see an interesting composition with a clear northeast/southwest slant, which presumably tells us something about the formation of this place. The northwest half of the picture looks like billows of smoke, perhaps because the Great Smoky Mountain National Park is not far.

This color scheme is soft and soothing. The Appalachian mountains are old and tired, and they are slowly eroding away. But in northern California, where volcanic processes are still very much alive, the picture looks rather different:

This is the land of earthquakes, hot springs and active volcanoes. Red is the dominant color. The deep red represents rock that has come out of the bowels of the earth as magma. The lighter red shows areas of diorite, an extremely hard rock that was highly prized in ancient times. Around here, things are still moving, and are never at rest.

Sometimes the old and the new can mix, like in central Colorado, where we can see just about every color on the palette:

Here the Great Plains meet the Rocky Mountains (Denver is just off the upper right corner). This picture seems almost random, as if an artist had simply thrown paint blobs at a canvas. Things have been churned around quite a bit, and there is no clear picture — just an image of chaos.

Other regions are more organized, and sometimes even tortured, such as this area of western Massachusetts:

Most of the rocks here are quite old, a billion years and more, and have been worked and reworked by geological processes. Their shape reminds me of a pastry crust, folded and refolded until the layers can no longer be counted. This is busy, worried, anxious.

A more soothing image can be found in central North Carolina:

The general impression is much calmer, friendlier, with colors that seem almost bubble-gummy. Here the plains of the Atlantic coast rise slowly to meet the mountains. There are no volcanoes anywhere, and earthquakes are uncommon. This seems like a relaxed, genteel place.

I will end with one of my favorite places in the world: the Cascade mountains. They are young and vigorous. They are still growing. And yet…

If you look carefully, you’ll see a certain amount of veining. These are the mountains eroding away. Even though they are young, the weather is already working on them. It rains a lot on these slopes, and the water trickling down carries with it bits of rock. Yes, even the mighty Cascades are being worn down by drops of water. There is probably some philosophical lesson in there somewhere, but I will leave that to you, kind reader.

Notes

Many thanks to Dr. Janine Weber for her gracious assistance with geological concepts.