Probability of being in a Cocked Hat

In 1986 I bought a Decca Mk III Navigator and a posh new RDF set to replace my old Seafix, which had always seemed somewhat approximate. Sailing down west I found myself in the middle of Lyme Bay in a flat calm, blue sky, perfect for sunbathing - and practising with my new RDF.

I sat in front of the mast and got good signals from three stations, easy to take bearings when everything was so still. Wrote them down carefully, took them below and plotted them and got a reassuringly small cocked hat. Really pleased. Then I checked the actual position according to the Decca, which wasn't as close as GPS but a damn sight closer than I'd ever had before, and found that we weren't actually inside the cocked hat at all, but just outside it.

I never bothered with the RDF again.
 
That rather encourages my method. Take three careful bearings and plot them with maximum possible accuracy. Now ignore them and look at the GPS or the plotter on my phone :)

Yes, I know GPS could be jammed, my leccy could fail, etc, but with four independent, and independently powered, GPS receivers on board and, for trips out of sight of land, a position plotted every hour, the likelihood of that causing a problem is significantly lower than the probability of bumping into something while I'm down below doing the maths.
 
Bi111ion said:
Interestingly the discussion of cocked hats still continues, not just in pubs but also in the Journal of Navigation.

George Kaplan recently published this paper in which he does a simulation to find probabilities
Kaplan GH, Fix Probabilities from LOP Geometry DOI: Fix Probabilities from LOP Geometry | The Journal of Navigation | Cambridge Core

Robin Stuart managed to integrate normal distributions over triangles giving an explicit answer for the probabilities of being in a given cocked hat (rather than a random one), that is the paper I mentioned at the start of this thread
Stuart, RG. Probabilities in a Gaussian cocked hat. DOI: Probabilities in a Gaussian Cocked Hat | The Journal of Navigation | Cambridge Core

My own contribution is for people who really like ruler and compass constructions! (If you dont like that kind of old school Geometry you will not like the paper, but I really enjoyed the geometry of the horizontal sextant angle fix when I learnt it as a kid, and that obviously affected me!)
Lionheart, William RB, Peter JC Moses, and Clark Kimberling. The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position. DOI: The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position | The Journal of Navigation | Cambridge Core

I also made some (pretty bad) YouTube videos of me doing the construction for the probability ellipse. Demonstrating I think that while it is possible even dedicated scientifically trained naval officers in the age of sail would not have been bothered to construct an elliptical probability contour at sea unless it involved a bet with a barrel of rum as the stakes. I wonder if some secret archives of the Napoleonic navy would reveal they actually knew of this technique though and it was a military secret? Navigation and Geometry

A new paper has just come out in preprint
Bárány, Imre, William Steiger, and Sivan Toledo. "The cocked hat." arXiv preprint [2007.06838] The cocked hat
They consider the case where the LOPs are bearings, so "rays" not lines. So if you were really bad at taking bearings you could be on the wrong side of the landmark. Interestingly, with very general assumptions very carefully stated they still come out with 1/4.

I suppose I should not just cock but doff my hat to them!
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Are any of the papers, or accompanying presentations available without subscription? The look interesting.
 
Oh, goodie gum drops! Another 'blast from the past' that won't die gracefully, but comes around again and again - like PBO articles and 'which anchor' threads.
It's reassuring, in an odd sort of way....

These issues were thrashed out at RAF 'Nav School' half a lifetime ago and, no doubt, at one or other of the RN's nav training establishments since Pontius was a ( harbour ) pilot...
They formed part of the larger picture that pro navs needed to acquire so as to make 'best decisions' about the 'Safe Conduct of the Vessel/Aircraft'.

One of the important concepts that took some rearranged thinking was that there were no 'absolutes' in position-fixing; there was always our interpretation of navigation information and the inherent errors/ inexactitudes and the changing requirements of the flight/passage. The concept of Gaussian distribution of variance was all very well, but in practice position-line information was always subject to one or more 'biases', usually fleeting. Determination of an RDF 'nul', for example, to guestimate bearing within 5 degrees also was a 'movable feast', even in optimum reception conditions.

It remains VIP to retain awareness of the relative positions of hazard - perhaps more significant at times than good knowledge of one's own accurate location. It is also VIP to be aware of when a safety-significant change of course is necessary. Those are issues of focus. It is remarkable how frequently a yottie will today 'stand on into danger' when s/he knows full well s/he is uncertain of position and proximity to hazard - yet ploughs on....

'Motorway mindset' seems endemic, when the option to simply stop the boat is almost always available.

Lateral thinking can mean, quite literally, "I may not know where exactly I am, but I do know where exactly I'm not.... and that's OK!"
 
Catalina36 said:
In 1986 I bought a Decca Mk III Navigator and a posh new RDF set to replace my old Seafix, which had always seemed somewhat approximate. Sailing down west I found myself in the middle of Lyme Bay in a flat calm, blue sky, perfect for sunbathing - and practising with my new RDF.

I sat in front of the mast and got good signals from three stations, easy to take bearings when everything was so still. Wrote them down carefully, took them below and plotted them and got a reassuringly small cocked hat. Really pleased. Then I checked the actual position according to the Decca, which wasn't as close as GPS but a damn sight closer than I'd ever had before, and found that we weren't actually inside the cocked hat at all, but just outside it.

I never bothered with the RDF again.
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Well, you might say. "What are the chances of that?" The answer is 3 in 4 averaged over all cocked hats and a little different depending on its shape (and Robin Stuard and George Kaplan's papers can tell you). So it is not a surprise. On average you will typically be outside cocked hats 3 times out of 4.

I remember chartering a Contessa 32 from Lymington in 1982 with a Decca. Amazing.

I never got a good RDF with a Seafix.

I alsways wanted to play on a prooer rig with an Adcock andtenna and a goniometer. I do have a B&G Herron DF antenna and that should do the business but there are only air beacons now.
 
Bi111ion said:
Well, you might say. "What are the chances of that?" The answer is 3 in 4 averaged over all cocked hats and a little different depending on its shape (and Robin Stuard and George Kaplan's papers can tell you). So it is not a surprise. On average you will typically be outside cocked hats 3 times out of 4.

I remember chartering a Contessa 32 from Lymington in 1982 with a Decca. Amazing.

I never got a good RDF with a Seafix.

I alsways wanted to play on a prooer rig with an Adcock andtenna and a goniometer. I do have a B&G Herron DF antenna and that should do the business but there are only air beacons now.
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Even when the Seafix and it's likes were current equipment (late 60's!), its use was limited by the number of radio beacons - on the East coast of Scotland, they were few and far between! And I don't recall ever being within range of enough to get a three-point fix. I did use it once or twice to cross with a visual bearing, and on one memorable occasion used cross-bearings from the Seafix to monitor progress during a foggy passage from Arbroath to Dunbar! That one occasion was when it earned it's keep - without it, dead reckoning over that distance wasn't going to be at all reliable, even though I did all the tidal corrections - very carefully, as we didn't want to run into Bell Rock! The Seafix told me when we were safely past Bell Rock and allowed me to correct the distance run. Fortunately it cleared before we got near Dunbar - entering Dunbar harbour would have been impossible in poor visibility.
 
AntarcticPilot said:
Even when the Seafix and it's likes were current equipment (late 60's!), its use was limited by the number of radio beacons - on the East coast of Scotland, they were few and far between! And I don't recall ever being within range of enough to get a three-point fix. I did use it once or twice to cross with a visual bearing, and on one memorable occasion used cross-bearings from the Seafix to monitor progress during a foggy passage from Arbroath to Dunbar! That one occasion was when it earned it's keep - without it, dead reckoning over that distance wasn't going to be at all reliable, even though I did all the tidal corrections - very carefully, as we didn't want to run into Bell Rock! The Seafix told me when we were safely past Bell Rock and allowed me to correct the distance run. Fortunately it cleared before we got near Dunbar - entering Dunbar harbour would have been impossible in poor visibility.
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AIR I had Berry Head, Portland and Casquettes.
 
SS Darina - sorry mixed up my ships ... it was SS Donax

WKiNA2q.jpg


.... sailing Singapore to Arabian Gulf .... overcast and strong gales .... this is before GPS ... sextant no use as no clear sky ...

3 days of just plugging on .... and on 3rd day - I play with the Marconi RDF .... note we are out in the middle of the oggin ... I'm bored ... I get a signal and id .... I get the bearing ...
I then draw on the large area chart ... take last known position and run on at estimated speed adjusting till the two intersect ... WOW .. we cannot be that far south of the charted course. I pass it on to 2nd Mate at midnight ... he smiles and you can see he's not impressed .. 3rd Mates gone nuts ....
Next day Chief Mate gets clear sky and stars .... I take watch at 0800 and hand-over goes something like this ....

Got stars this morning ... gales must have blown us way south of intended ... ignore charted course .. we've altered to reach next waypoint ...... new course pencilled onto chart ... OK Nigel ?

Roy .. did you notice the X on the chart from last night ...

Yes actually 2nd Mate laughed when he showed me ...

...................

I think you know whats coming ... that RDF crossed by distance run was actually not so far wrong ... even though that RDF signal was travelling halfway across the ocean .....
I was quite chuffed with that .....
 
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A navigational principle:

"Ignore NO nav information which come your way/is observed. DO NOT reject it unless prepared to log the reason why - and sign it.
Every item of acquired navigation information MUST be used, either as stand-alone or as confirmation of something else. "

I still have an functional Consol chart, which I used to good effect sliding about The Channel on RAF 'windfall' yachts. Yes, there was often a Seafix RDF. No, I never got an RDF bearing I would have bet my shirt on..... except Round Island ( 200 nm range ) and we knew where we were(n't) at that point.
 
In my career at sea ... 1973 - 1989 .... I saw some really crap gear foisted onto ships for evaluation or actual use.

But in terms of Electronic navigation I used :

Sextant
Omega
Loran A
Loran C
Decca
Transit
Early evaluation GPS
RDF

Additional to augment :

Echo sounder
Doppler
Radar

After coming ashore - because Navigation was more than just a job choice for me - it was also a passion, as it was for my Father .. I continued my studies and use ... GPS and various other forms that have become common place today.

My present work still involves worldwide shipping ... consultative work for shipping comanpys and traders as well as my main field of Fuels and Oil market.

Do I have a view on anything related today .... one yes : I am really sad to have seen Decca stopped. It had evolved from flickering needles to digital displays to Lat Long auto calculation ... its accuracy was enough that commercial shipping accepted its position determination without need for secondary verification. In many companys - it even had its own position symbol a Capital D around the X ...

I'll let all return to thread !
 
At the risk of receiving abuse from the esteemed mathematicians and statisticians who have contributed at length to this thread...

As a YM(Ocean) who lives in Gosport, the probability of me being found inside the Cocked Hat is much higher on a Friday or Saturday night, especially since Boris re-opened the pubs...

Cocked Hat Pub in Gosport Flaming Grill
 
Amulet said:
The 1/4 value is conceptually easy to justify. Having drawn the first line there is a 50% chance of each subsequent line erring in the direction which puts you outside of the cocked-hat. !/2 x 1/2 = 1/4. Simple.
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I had to smile when TC's Social Media Guy's Facebook post reminded me of these threads - placing the position bang in the middle of the cocked hat.

https://www.facebook.com/100039882247624/posts/pfbid033Bo2g71tRZkkyLMDbg1i1yUF3jKrvSah1pzNEEs37iA7RK3rLDDMkDBPA97wiyasl


PS: I couldn't resist checking Chat GPT's opinion. 😁

The probability of being on one side of each line of position (LOP) is indeed 50%. However, when calculating the overall probability of being inside the cocked hat, it's not simply a multiplication of these probabilities. The situation is more complex because it involves the geometry of the LOPs and their intersections.
When three LOPs intersect to form a cocked hat, the probability of being inside it depends on the shape and size of that particular cocked hat. It is not a straightforward 50% chance for each LOP. In fact, the probability distribution of the actual position within the cocked hat is not uniform.
In practice, mariners and aviators use techniques like weighted least squares to calculate the most probable position within the cocked hat, taking into account the accuracy of the LOPs and their angles of intersection. This method provides a more accurate estimate of the position rather than simply assuming equal probabilities for each side of each LOP.
So, while the chance of being on one side of an individual LOP is 50%, the overall probability of being inside the cocked hat is determined by the specific geometry of the LOPs and their respective accuracies, making it more complex to calculate than a simple multiplication of probabilities.
 
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Oh, dear, Mark-1.

What a can of worms you've opened.

'Back in the day' there was a brigade of wizened RAF Staff Navigators arguing away incessantly about the optimum way to use Statistics and Probability Theory* to refine a position indicated by multiple Lines of Position ( LOPs ). In some circumstances, this mattered. For example, the sub-hunter Nimrod's tactical navigator may have several LOPs displayed on a very large 'tactical screen' derived from sources as abstruse as a vague bearing from a passive sonobuoy, a visual bearing on what MAY have been a fleeting periscope wake, a range from an active sonobuoy that was 12 minutes and 36 flight-miles old and not repeated. The question is 'Where exactly does one drop a depth charge?'

The need for accuracy in the answer may vary considerably if the depth charge intended is a nuclear one......

I have it on very good authority that the best sub hunter tac navigators were masters of the Stats arguments and Bands of Error calcs, but would use their experience and judgement to put a fingertip or cursor on a point and state 'There'. It would be a brave 'noobie' who would challenge them.

The WavyNavy types could get up before dawn and do a 'round of sights' using up to 7 Selected Stars, then go and quibble together about tenths of a nautical mile ( 'chains' for them wot's done the long course ) for half an hour before boiled eggs and coffee in the wardroom. Sweaty navs in a V-bomber 7 miles higher up would 'shoot' multiple sights on just two stars ( day or night ) - one ahead, one to the side - and use that to tweak the autopilot to better than one-tenth of an nm..... then continue doing that kind of star fixing in a continuous twelve-minute cycle.

The objective was to guarantee to bring the aircraft, after a 2000nm transit, down a half-mile wide 'alley in the sky' within 30 seconds of pre-briefed time. The good 'uns could do this within a quarter-mile and 15 seconds - and did, once every month. The very best crews, with a lot of tweaking and years of practice, could reliably do even better, regularly demonstrated in NATO Bombing Competitions with sub-200 metre CEPs and within 5 seconds of planned time.

The Americans simply couldn't believe it. The Russians did.

* I understand ( almost ) that the deeper depths of Probability Theory tells us that our position in space may be 'here' or may be 'there', but when we try to measure that precisely, we change it. And when we try to measure it twice, or go for 'best of three', we change it again. Cue bits of the Uncertainty Principle.....

So perhaps we should stop worrying about whether we're just inside a 'cocked hat' or just outside it, move along a bit and take another fix..... and later still, another. They're likely ALL right, enough for our purposes. And we should always end up following a Single Line of Position - a Leading Line - into our safe haven. That's usually unambiguous.
 
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Bi111ion said:
I wonder if every navigation class has at some point exited to the pub still debating if the probability of being in a cocked hat from a three line position fix is really 1/4 - surely it depends on the size of the hat!

The age old conundrum has been settled by Robin Stuart in the Journal of Navigation Probabilities in a Gaussian Cocked Hat | The Journal of Navigation | Cambridge Core

In a nut shell, before you know where the lines are it is 1/4. Once you have drawn the lines it depends, and he works it out.

The admiralty manual of navigation from over 100 years ago suggest picking the point of the triangle closest to danger.

A better idea might be to consider the ellipse of uncertainty and pick the point of that closest to danger.

Anyway such debates abound on NavList, where we still discuss the best use of our sextants and chronometers, and maybe not practical enough for PBO forum!
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£26 to read the answer. No thanks. I'm well aware of the concept of Estimate of Position Uncertainty (EPU) and apply it to my navigation.
As for cocked hats, Great if you can see land. Can't remember the last time I needed to draw one.
 
The probability of being on one side of each line of position (LOP) is indeed 50%. However, when calculating the overall probability of being inside the cocked hat, it's not simply a multiplication of these probabilities. The situation is more complex because it involves the geometry of the LOPs and their intersections.
When three LOPs intersect to form a cocked hat, the probability of being inside it depends on the shape and size of that particular cocked hat. It is not a straightforward 50% chance for each LOP. In fact, the probability distribution of the actual position within the cocked hat is not uniform.
In practice, mariners and aviators use techniques like weighted least squares to calculate the most probable position within the cocked hat, taking into account the accuracy of the LOPs and their angles of intersection. This method provides a more accurate estimate of the position rather than simply assuming equal probabilities for each side of each LOP.
So, while the chance of being on one side of an individual LOP is 50%, the overall probability of being inside the cocked hat is determined by the specific geometry of the LOPs and their respective accuracies, making it more complex to calculate than a simple multiplication of probabilities.
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What a load of ********

The standard action of a navigator when confronted by the cocked hat - is to assume the point nearest danger as the position and take action accordingly.

All that is talking about is if any LOP's get increasingly shallow angle to another - the point of intersection becomes more in error.
 
My early sailboat-nav excursions around the Western Channel were as 'last minute supernumery' and on sufference, on Windfall Series wooden yachts taken as loot - aka War Reparations - from the Germans and kept afloat by a turgid mix of faith, hope and not knowing any better.

I had the benefit of being a newly-minted 'professional nav' who knew One Over The Square Root of Not Much At All about making boats go and keeping them afloat, but brought the valuable benefit of being able to cadge from the Base Met Officer a detailed weather forecast for the intended race/route. That was the sum of my useful contribution....

I did learn, however, to put to use some of the pro skills which HerMaj has graciously paid for me to learn, especially in stormy weather when I was able always to produce a Fix Of Sorts ( not recognised by the gurus of the RYA ) which helped my crew-colleagues with the confidence to continue when prudence might suggest an early Retirement. For this purpose I carried inter alia an old Consol Chart. This quaint system dated from WW2 and was intended for use by U-Boat Wolf Packs. in Biscay and the Western Approaches. On one celebrated occasion I crossed an isogonal with a Meridian, transferred up a Consol LOP from earlier in the day, and (over) confidently declared "We are here. The next Mark of the Course is there. Our Course To Steer is so-and-so...." For want of any better naviguesses, the skipper indulged my temerity and a couple of hours later there was the Wolf Rock - just where it should be.

The ould Windfall Yachts are long past their 'Sell By Date', and probably so am I. I've survived military flying and RORC racing and have learned that it's not quite so important knowing where I/we are, as knowing where the hazards are - and being somewhere else.
 
The more useful question is not :
How likely are we to be in that little triangle?'
It's more like:
How big a circle do we need to draw around that point to be effectively certain we're in the circle?

A very crude precis goes something like:
Assume the angular errors are random and apply equally randomly to all three position lines
Draw a circle that encompasses the triangle. You have about a 50% chance of being in the circle
Double the circle's radius " about 95%
Treble the original circle's radius about 99%



Think about the random case where your first two position lines have some error in them, they cross a little way from your position.
There's a random error in your third bearing, there's a chance your cocked hat will be tiny.
 
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