Okay, tell me where I'm wrong: AFAIK you can buy a few hundred to few thousand dollar receiver that tracks the carrier phase, let it sit around for a few hours, and then with aggressive post-processing you should be able to figure out the right phase and know your location within a few mm.

Edit: And worst-case, you should be able to do an exact survey once, and then you can remember that and know which phase to use in the future. (Barring sudden multi-cm shifts, but in that case you have bigger problems.) You still have atmospheric fluctuations to deal with, but that's why you're measuring all day.

Vertical accuracy is usually lower than horiztonal accuracy. It is more difficult to calculate elevation than lat/long.

Not by a huge amount. Should be less than 2x, at least at some point during the day.

Much worse that 2x. Closer to an order of magnitude worse.

Indeed. Just to anchor this with current state-of-the-art numbers, I took a look at the CSV data file linked to by the page referenced in my above comment:

https://news.ycombinator.com/item?id=13040765

The quoted horizontal standard errors of the daily position measurements are ~1.5mm. The vertical standard errors are ~5-7mm. (Not as bad as a factor of 10, but certainly worse than 2x.)

The vertical position measurement responds well to averaging of daily errors to beat down the RF propagation effects that cause them.

Huh, why's that?

It has to do with the geometry: GPS is measuring the differences between the distances to satellites which are mostly "overhead". In a sense it's similar to how our stereoscopic vision can resolve movement from side to side more accurately than movement towards or away from us.

If you're at the top of a skyscraper you can mitigate this somewhat by looking at satellites closer to the horizon as well -- this solves the geometry problem -- but then you run into increased noise from the larger amount of atmosphere you're looking through.

Hmm, I still don't see it. The comparison to stereoscopic vision seems misplaced: we judge side-to-side motion by an entirely different (and non-stereoscopic) mechanism than depth, and neither mechanism is mathematically very similar to that of GPS. And my understanding is that GPS receivers compute their distance to various satellites, which means if all the satellites were all nearly overhead, I would expect the vertical component to be the best understood.

I Googled and found some vague and unsatisfying explanations, but, luckily, wikipedia tells us how to do the math: https://en.wikipedia.org/wiki/Dilution_of_precision_(navigat...

You judge depth not from a single image but from the difference between two images. Similarly, GPS works not by measuring signal arrival times, but by measuring the differences between arrival times.

Good point, but I'm afraid the analogy still isn't spelled out enough for me to actually make sense of it. Like why does that make a horizontal/vertical discrepancy?

If you move horizontally, you're getting closer to one satellite and further away from another -- the changes in GPS signals add, just like if something you're looking at moves from side to side. If you move vertically, you're getting closer to or further away from all the satellites at once; but since GPS relies on the differences between signal arrival times, that's much harder to pick up.

Thanks I think that's the best explanation I've read!

Thanks! It's unfortunate that the diagram on wikipedia is actively misleading since it corresponds to known distances rather than known differences of distances.