Loneliest cruising altitude

And about to get even quieter.

I'm starting to get some interesting data, though I'd like larger numbers before reporting them. In the meanwhile, how does traffic density relate to the risk of collision? There's a huge body of literature about it which I don't currently have the time or academic privileges to wade through. Much of it is about simulating IFR commercial traffic in airways and around airports though, and not directly applicable to GA.

My reasoning, in the meanwhile, is that the risk of collision will go up approximately with the square of traffic density. Feel free to shoot down my reasoning: that's why I've put it there.

If you have two aircraft a and b flying random paths along a plane, you will have a chance of collision ab

If you have three aircraft a, b, c... you can have collisions ab, ac, bc (3)

If you have four aircraft a,b,c,d... you can have collisions ab, ac, ad, bc, bd, cd (6)

and so on...

The number of possible collisions between n aircraft can be calculated by the formula n(n-1).

If 'r' is the actual chance of two given aircraft flying around a finite area having a collision within a given timeframe, you can calculate the probability of n aircraft not having a collision by the formula

(1-r) to the power of (n(n-1))

or to put it another way, the chances of n aircraft having a collision will be 1-(1-r) to the power of (n(n-1)).

However, as the chances of having a collision are mercifully small it should be reasonable to assume that they are roughly proportional to the number of collision permutations.

As n becomes large, ((2n)/(2n-1))/(n x (n-1)) approaches 4, in other words the likelihood of having a collision is roughly proportional to the square of the number of aircraft flying at a particular level.

So... on a particular day, for the sake of argument, there are 1000 aircraft choosing to fly around 4,500 feet. They can either all fly at 4500 feet, or they can choose to spread themselves at 4000 to 4900 feet in 100 foot increments....

The chances of a collision occurring in the first instance will be about 10 times higher than the chances of a collision occuring in the second instance. (1000 x 9999) / (10 x 100 x 99).

I won't yet go into details about the altitudes that people have selected, but suffice to say that some numbers are turning out to be much more popular than others. It should be possible to work out both the relative risks of cruising at particular altitude settings, and the overall risk of collision relative to the theoretical minimums and maximums, as calculated above, for any distribution of aircraft cruising altitudes.

Another question relates to whether random cruising altitudes are a good idea anyway - as Genghis has pointed out, you can fly the quadrantal rule whilst VFR.

Flying in 500 foot increments depending on heading has two effects: firstly it means that anyone you meet is unlikely to be flying a reciprocal heading, giving you more time to see and avoid. Secondly, it corrals you into a relatively few altitude levels, increasing the potential for collisions. Whether the former effect outweighs the latter, is something that I can't answer.

You could also combine both strategies to fly 'at random' somewhat above or below the official 500 foot increments and still remain reasonably assured that if you meet anybody else flying the quadrantal rule, they are liable to be flying more or less the same heading as you.

Finally, most of our aircraft are less than 100 feet tall - even an a380 is only 80 feet tall, and the Hindenburg (girth 140 feet) isn't flying any more. So why limit ourselves to increments of 100 feet? Accurate piloting is a prerequisite to effective IFR aircraft separation, but for VFR flying, accurate piloting and altimeters might only serve to increase the likelihood of collisions, given the well known deficiencies of 'see and avoid'. Perhaps as I oscillate, wildly out of trim, thousands of feet from my preplanned altitude, I'm the safest pilot in the sky because I'm using the available airspace more effectively than any other.

I agree, but we still have all these little rules of thumb to stop it happening...

OK... I'm sorry for the delay in posting results. Sometimes life gets busy. I quite agree with the frustration some people feel when confronted with such an odd question. However, I did have a halfway sensible rationale behind it. It relates to a trick many pilots use to reduce the risk of a mid-air collision, which is to fly at a 'random' altitude, the assumption being that most other pilots will be flying at 'round' altitudes such as 2000 feet, or 1500 feet.

There's a children's game where if you ask someone to think of a vegetable under time pressure, almost everybody thinks of a carrot. If you ask people to think of a flower, most people think of a rose. But what if you ask people to think of a random number - for example a random altitude?

There's unfortunately surprisingly little research on this, though what little I've seen tends to suggest that we also have prototypical 'random' numbers that we prefer. And intuitively 2300 feet somehow seems more random than 2000 feet or 2500 feet, doesn't it? It does to me, though of course I know that 'randomness' occurs in the process of selecting an altitude, not in the individual result.

In my polls, I randomised the order of presentation of the altitudes to try and avoid any bias due to page position, and posted different versions of the same quizes on several fora. I'm not particularly serious about this question and I'm sure I could have carried out the questionnaires more rigorously, but it's a start. And the results were significant, both in practical and statistical terms.

This first graph shows the altitudes pilots select if they consciously choose to pick a 'random' cruising altitude.

"Many pilots try to pick 'random' cruising altitudes in order to reduce the risk of mid-air collisions.

Ignoring all other considerations, if you were trying to use this strategy on a VFR flight, which altitude would you be most likely to pick?"

182 responses - risk of collision 150% greater than the theoretical optimum.

The result is clearly biased towards '3' and '7', which I had predicted, though '2' and '8' surprised me in their popularity. '2' was only really popular on this forum, which made me wonder whether someone was possibly trying to fiddle the results. However '3' was popular with everyone on every forum where I posted the quiz, whatever order the options were presented in.

However, not every pilot knows the 'trick'. What altitudes do people actually fly at? The best way to do this would be to sit watching mode-C returns over Wales or Scotland - somewhere where people choose altitudes on the basis of preference rather than compulsion - as 2greens1red points out we often don't have much choice. However, that's not something I currently have the time to do so I posted a separate poll elsewhere asking people to pick an altitude.

*"It's a fine day for a VFR flight, and you can choose to fly at whatever level you want. Pick your favourite cruising altitude, and select the whole number of hundreds-of-feet on the poll below. Thanks!" *

112 responses altogether. Risk of collision 152% greater than the theoretical optimum.

The result - to my eye - shows a mix of 'random' cruising altitudes, with the big round numbers of thousands of feet, or five-hundred foot increments. Perhaps surprisingly here, people have tended towards lower numbers rather than higher numbers. i.e. x200 feet seems much more popular than x800 feet.

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Working from the fact that the risk of collision increases with the square of traffic density, it's possible to calculate a theoretical optimum if traffic were evenly spread throughout ten possible cruising levels offered as options... which begs the question 'why only ten options'. It's a good question to which I don't have a ready answer, except to point out that most of us are limited in our altitude-keeping ability. I realistically keep within about +/- 100 feet most of the time. Perhaps some of you do much better. However, there's no advantage to aiming to fly at 10 foot increments if we're straying so far from our allotted altitude. Really I picked hundreds of feet simply because that's how we normally report our altitude. I never heard anybody say they were currently at '1775 feet'.

Whenever traffic is spread unevenly, risk of collision increases above the optimum. e.g. if pilots fly at whole 1000s of feet, the risk is 10 times greater than the optimum (assuming 100 foot increments).

Genghis quite reasonably pointed out that there are advantages to flying according to quadrantal rules - though under EASA I understand that we're moving towards the semicircular rules in line with the rest of the world. If we restricted ourselves to 500 foot increments, our theoretical risk of collision would be 5 times greater than the 'optimal' risk. Of course it's mitigated by the fact that we are now less likely to meet another pilot on a reciprocal heading (though closing speeds may remain quite substantial). Does this mitigation outweigh the relatively large risk increase by confining ourselves to just two levels per 1000 feet? Quite possibly. I don't know.

Perhaps the ideal would be to combine the two techniques:

Take the last digit of the day of the month in your birthday - e.g. 16 february. Add or subtract 5 - whichever leaves you with a positive, single digit answer. You now have two single digits, 1 and 6. Fly quadrantal rules, but instead of flying at x000 feet, fly at x100 feet. Instead of flying at x500 feet, fly at x600 feet. If everybody did this you would be relatively unlikely to meet anybody going the opposite direction, and you would also spread out the traffic almost evenly over the available airspace.

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I think Peter hit on the most important point, which is simply that en-route mid-air collisions are vanishingly rare and not really something to worry about unduly. If this wikipedia page is truly complete:

http://en.wikipedia.org/wiki/ListofmidaircollisionsandmidairincidentsintheUnitedKingdom#cite_note-12

then there don't seem to have been any civil en-route vfr midairs in the UK... pretty much ever.

So fairly pointless, but I had fun anyway.