# Distance between two locations

Hello Everyone:
I am hoping that someone from the Flight Aware community will be able to answer a question I have about determining the distance between two locations of a plane and then using that distance to determine the speed of the plane. The details of what I am looking for can be found in a video posted at

Thank you.

Ron

If you would prefer to have a link to watch the video, go to

tinyurl dot com/distancequestion

First for the easy one - the arrows by the course number in the tracklog are just showing you which octant the direction falls in, assuming North is up. (292.5 - 337.5 is â€śup and leftâ€ť, 337.5 - 22.5 is â€śupâ€ť, etc)

[Some context on the next question for other readers so you donâ€™t need to trawl through the video - the short version of the question is â€śwhy does the tracklog speed not match the speed I compute from taking the distance between points?â€ť. The flight in question is https://flightaware.com/live/flight/CGZPT/history/20200419/2235ZZ/CYHZ/CYHZ]

The speed question is mostly â€śknow your data sourceâ€ť. The speeds quoted are indeed groundspeeds. If you look at the tracklog details for that flight, all the data is coming from Canadian radar facilities (actually via a FAA feed) - e.g. â€śMoncton Centerâ€ť / CCZM. We donâ€™t have good low level terrestrial ADS-B coverage in some parts of Canada unfortunately.

The gotcha here is that the radar positions we receive (this is specific to the Canadian radar feed) are heavily quantized; they are only provided to arc-minute precision, i.e. 1/60th of a degree. At that latitude, that means something like a 0.7km average error in each position. Itâ€™s actually visible in the plotted track, in the â€śright-left-rightâ€ť case the aircraft was probably actually flying straight and itâ€™s just the quantization that makes it appear differently.

IIRC the speeds are also provided directly as part of the radar feed and theyâ€™re likely to be a more complex filtered output (often something like a Kalman filter is involved that directly produces velocity as part of the filter state), using the higher precision data that the radar has (but which we donâ€™t), and not simply dx/dt.

(edit: Iâ€™d also be suspicious about quantization in the reported speeds; all the radar speeds are suspiciously exact multiples of 10kt, so those might be getting rounded too, which would mean another +/- 5kt error to deal with)

The overall effect of the quantization is that the aircraft appears to take a longer path than itâ€™s actually flying, so your computed speed is higher than the true groundspeed.

On the topic of â€śwhich website should I believe?â€ť, the Movable Type equations are generally reliable, and Iâ€™d assume that Wolfram Alpha would be doing the right thing here; they agree with what I get with our internal tools. Possibly that other website is using spherical cosines rather than haversines, which doesnâ€™t behave so well at small angles.

tl;dr: You got unlucky with your choice of flight. You might want to look for a flight that has good ADS-B coverage, as the position precision is much better there. (But do note that ADS-B speeds have a similar thing going on, theyâ€™re usually the velocity output of the GPS Kalman filter and not simply dx/dt)

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Thank you so much for your quick response and for providing detailed answers to my questions. I also appreciate you providing a summary for members so they do not need to watch the video I prepared.

I have very little knowledge about aviation but I am keen to make connections between the high school mathematics curriculum and topics from aviation. Having students calculate the distance between two locations seems like a great idea but I have to make sure that students have some basic understanding of the context without having to spend a number of classes trying to explain all the technical terms. I also have to make sure I have some basic understanding about what is going on so that I can provide a response when students ask me why the distance between the locations that they calculated is not the same as what is posted at flightaware.

Unfortunately I found it hard to understand the information you provided (this is not a criticism of you) and I am wondering if you might be able to provide more details so a newbie like me can understand the technical terms. If you do not have the time to do this I will certainly understand.

Here are some specific things I need help with:

We donâ€™t have good low level terrestrial ADS-B coverage in some parts of Canada unfortunatelyâ€¦
Question: What is ADS-B coverage? How do I know where it is good and where it is bad?

The gotcha here is that the radar positions we receive (this is specific to the Canadian radar feed) are heavily quantizedâ€¦
Question: What does quantized mean?

Itâ€™s actually visible in the plotted track, in the â€śright-left-rightâ€ť case the aircraft was probably actually flying straight and itâ€™s just the quantization that makes it appear differently
I am not sure what this means.

IIRC the speeds are also provided directly as part of the radar feed and theyâ€™re likely to be a more complex filtered output (often something like a Kalman filter is involved that directly produces velocity as part of the filter state), using the higher precision data that the radar has (but which we donâ€™t), and not simply dx/dt.
I am quite lost here. Can you provide more details?

You got unlucky with your choice of flight. You might want to look for a flight that has good ADS-B coverage, as the position precision is much better there.
How do I know if a flight that has good ADS-B coverage?

After reading your message, it seems to me that the speed that is posted for a plane at flightaware is not unreliable. Why is the speed even given if it is often incorrect?

Ron

The one thing that I would really like to know is how do I know which source is providing the correct answer. The websites are giving me two different distances that lead to lead to two different speeds of the plane that are not the same as what is posted at flightaware. Is one of these sources always right? If not, how do I know which one provides the correct distance?

The short version is that ADS-B is an automated system where aircraft continuously send radio broadcasts to report their own position, speed, callsign, and so on. FlightAware has a network of ground-based receivers (many run by hobbyists) that listen for these broadcasts and send the collected information to the FlightAware servers. This data is one of the sources that makes up the aircraft tracks seen on the website. There are other data sources such as a data feed from the FAA that contains radar data, but generally we will prefer to use the ADS-B data where available as it is better quality.

The ADS-B radio signals are microwave-frequency (around 1GHz) and at that frequency, radio reception is essentially line-of-sight - the receiving antenna needs a clear path to the aircraft. For aircraft at low altitudes, only very nearby receivers can see the aircraft - for more distant receivers, the aircraft is blocked by the horizon and the ADS-B signals canâ€™t be heard. So there are some areas where we canâ€™t see aircraft because thereâ€™s no suitable nearby receiver.

There is a map of coverage here: https://flightaware.com/adsb/coverage#data-coverage
(turn off everything except â€śADS-Bâ€ť and â€śADS-B UATâ€ť; the altitude slider at the top left lets you look at coverage for e.g. aircraft flying at 0-10,000ft)

Quantization is when the data can only take certain discrete values / steps. For example, the true latitude of a position might be 45.00833 (45Â° 0â€™ 30â€™â€™) , but the radar feed can only report it as either 45.0 (45Â° 0â€™ 0â€™â€™) or 45.0166 (45Â° 1â€™ 0â€™â€™). You can think of it as all positions being snapped to a fixed grid with 1 arcminute spacing.

This introduces a large error if you try to use closely-spaced positions to determine a speed. A +/- 0.7km error per position is very large if youâ€™re looking at a distance of only 4.5kmâ€¦

Wikipedia has a decent introduction to Kalman filtering if youâ€™re interested. The short version is that the radar is likely to be using a system where it feeds raw measurements into a model of the aircraft state which directly produces both position and velocity; the velocity is not calculated from the positions as a separate step.

Look at the â€śreporting facilityâ€ť column of the tracklog. ADS-B data will be shown as â€śFlightAware ADS-Bâ€ť.

The speed is usually reliable; itâ€™s just specifically the Canadian radar feed that is worse quality than youâ€™d usually expect from ADS-B. Given the rounding to 10kt, Iâ€™d say that the speeds reported in this case would be accurate to +/- 5% or so. ADS-B groundspeeds would be more like +/- 1%.

As I said above, one of the websites you looked at seems incorrect (or at least numerically unstable for small distances). For the others, the inconsistency between the reported speed and a naive dx/dt is a combination of the error in positions due to the quantization described above, and the ~5% error in speed due to the radar feed apparently rounding to the nearest 10kt.

This is definitely more towards the physics/engineering end of things than the pure mathematics side; real world data is messy and imperfect, and understanding the types and sizes of measurement errors youâ€™re dealing with is important. I mean, given a 5% error in your speed measurement and a large (maybe 20-30%) error in your distance measurement, the differences you see are within those error bounds, right?

Keeping with the tradition that â€śthereâ€™s an xkcd for everythingâ€ť, I give you: https://xkcd.com/2295/
(in particular: â€śgarbage - garbage = much worse garbageâ€ťâ€¦)

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You have gone above and beyond in answering my questions. Thank you very much.

I am going to explore the information that you have provided me with and see how I can use it with my students.

I like what you said about real world data being messy and love the xkcd comic.

Ron

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