Sunday, July 20, 2014

House Site Plan

I don't even want to think about how many hours it took me to put this together.

The county building permit application, the power company service application, the health department application and probably a few others have all required a site plan for my new house. Drawing a simple plan is easy, but the county wants detailed, two-foot contour lines. Obviously, there is no place I can go to get a two-foot contour plot of my property. I contacted a surveyor, but he wanted almost $2K to make me a contour plot.

I figured, how hard can it be?

Well, hard. But it's done. Here's the final version:

To get contour lines, I took a handheld GPS unit with a barometric altimeter and walked the property. And walked, and walked, and walked. As you can see from the final diagram, the land is far from flat. In fact I now know that the total elevation change is 160 feet. But to gather data, I needed to cris-cross the property many, many times, ideally gathering a position and elevation data point every few feet. I walked the whole thing four times, and the last time I walked it I covered six miles and climbed over 2000 feet.

The reason I walked it multiple times was partly because I was learning how to do it, and partly because GPS receivers aren't that accurate. I needed lots of data to help correct for the inaccuracy. GPS is particularly inaccurate when it comes to elevation, but that's why the units I used contain an electronic barometer. The barometer doesn't give them much information about absolute elevation (you can estimate elevation from air pressure, but it's even less precise than GPS elevation), but it provides them fairly precise information about relative elevation. That's what I need. I don't care how far above sea level the property is, I care about how much higher this part is than that part.

GPS receivers also have a fair amount of latitude and longitude error as well. I couldn't find any detailed information about the distribution of the error -- which may actually be different for different devices -- so I assumed it's probably normally distributed and can be reduced by averaging multiple samples together. Because GPS is based on very precise measurements of time signals, if you have multiple GPS receivers you can be guaranteed that they have synchronized time. It may not match up perfectly to wall clock time due to minor adjustments that get made to our clocks, but you can be sure the GPS units all agree.

So, I carried two identical Garmin Rino 130 receivers while I walked. The reason I had two identical ones is because the Rinos are actually walkie-talkies, not just GPS receivers, so it's quite useful to have more than one. But I carried both of them side by side.

After I got back and got the data off them (from all of the multiple runs), I first edited the track logs to remove extraneous data from before and after I started walking. Then I imported the mess into Mathematica. Next, I translated all of the points to try to correct for the fact that every time, every receiver, had the whole property shifted by a few feet in one direction or another. That was actually pretty easy to handle; I took waypoints at each of the corners, on each GPS, during each walk. I picked one of them (the westmost corner) to be my point of reference, and translated each log so that those points matched up.

Actually, I went one further, and translated all of them so that that corner point is (0,0). But more data munging was required. GPS coordinates are recorded as (latitude, longitude), which means they're (y,x), rather than (x,y). So I had to rotate all of the points 90 degrees around my origin to get to (x,y).

Next, I want all the distances on my map to be scaled so that measurements in any direction are valid. That means that one inch on the map should represent a fixed number of feet, no matter whether your ruler is measuring right to left (west to east) or bottom to top (south to north). But latitude and longitude aren't "square" like that in most places. In the area around my property, a millidegree of latitude is approximately 111 meters, and a millidegree of longitude is approximately 84 meters. So the "aspect ratio", the non-squareness, of my coordinate system is about 1.32. So I also scaled all of the y coordinates to "square up" my coordinates. As a result the Euclidean distance between any two points on the map provides a pretty good estimate of the straight-line distance between the real-world points they represent. This sort of scaling to square up only works across relatively small areas, but it's fine for my purposes.

For each walk, I now had two normalized logs, one from each GPSr. I combined them by matching timestamps and averaged the (x,y,z) coordinates from the pair of logs. Mostly. I found a few cases where one of the two devices wandered off and started giving very strange data, so I went back and deleted those points. There was actually a lot of hand-editing of the data to clean up various problems. Here's what I had at this point:

The result is obviously still spotty in some areas, which led to odd-looking contour plots. To fix that, I resampled the entire area at consistent intervals, and for each resampled point I used the mean of the nearest (in (x,y) only) five points. This actually helped with another problem: The fact that the data ends abruptly at the property lines tends to cause the contour plot generation algorithm to do funky things near the edge. My resampling technique resampled the area around the property as well, using the nearest data points. This "extra" data is, of course, complete nonsense, but it's nonsense which is much closer to reality than nothingness, which made the contour plotter happier.

Here's the resampled data, resampled at 200 points on a side, so 40K points total:

There's some waviness towards the center which isn't actually there. I started trying to figure out where it came from and how to smooth it back out, but ultimately decided it didn't matter, because I found that to get nice contour plots I really needed to scale back the level of detail, otherwise the lines were too "jagged". Using a resampling with 30 points on each side, 900 points, here's a contour plot, with the number of contour lines set to provide a vertical distance of two feet per line:

This isn't bad, but the contour lines are still a little too jagged. Applying some rather arbitrary smoothing to the lines, removing the color and cropping to the property lines gives this:

That actually looks really good. I have no doubt that there are some small errors, in fact I can point out some oddities which are almost certainly wrong, but it's the large features match up perfectly.

Good enough. What came next was really time-consuming. I had to add all of the additional details. Among other things, that meant taking the measurements from the house plans and putting them in to create an accurate rendition of the shape of the house, and then placing it in what seems like the right place on the map. Here's the house, already rotated to the right angle for placement on the map. Note that although this image is scale-free, the measurements are actually in my squared-up millidegrees, so when it's superimposed on the contour map, the measurements are all correct.

The enormous gray "turn pad", BTW, is a feature demanded by the fire marshal. I have to provide a hard surface large enough to turn a fire engine around, and that's it. The marshal said I needed 45 feet on a side, that pad is 45x55. I had to lengthen it a bit because of the decorative extension which comes 9 feet off the house. So actual clear turning space is 45x46.

Then I had to add in the well and septic field, barn, driveway, road and all of the "setback" and building size measurements, to produce the final version above.

Mathematica is an awesome tool, and I don't think I could have accomplished this without it. But it was still a lot of work. Okay, so I have thought about how many hours I put in and I'm sure it's around 60. I suppose that's better than spending $2,000. I'll keep telling myself that. Probably 20 of those hours were fun. The rest, not so much.

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