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George Huxtable wrote:

> Consider sights of 3 objects at azimuths of 0 deg, 120 deg, 240 deg equally
> spaced around the horizon.
>
> I'm not entirely certain whether Bill is considering (a) systematic errors
> combined with observational random scatter, or (b) systematic errors on
> their own. The latter case is rather simple to deal with first.
>
> Case b.(in which there is no  random scatter).
>
> If the systematic errors also were zero, then every round of sights would
> give three perfect position lines which intersect at a point, that of the
> true position. The triangle is always of zero size.
>
> Now consider a non-zero systematic sextant error, common to all three
> observations. If each sextant altitude shows a systematically high reading,
> due perhaps to a constant tilt-error or to an uncorrected on-the-arc index
> error, then for each minute of such error, each of the three position lines
> will be displaced by a mile from the true position, and toward the body
> being observed. However small the systematic errors are, as long as they
> are non-zero, there can be only TTT triangles; no others are possible. All
> these triangles will embrace the true position of the observer. If the
> systematic errors had been always the other way, such as an uncorrected
> off-the-arc index error, then every triangle would be an AAA triangle,
> again always embracing the true position.


One point that could be stressed more is that the larger the systematic
error, the larger the cocked hat. In this case, with no random error,
the true position always falls within the 'hat but that is a less and
less useful property with larger systematic errors. Thus, systematic
error is not desirable -- an issue that becomes more relevant below.

> Case a (in which observational random scatter is taken into account).
>
> If the systematic errors were zero, we will get get, as we have considered
> before at some length, a randomly varying set of triangles, such that only
> one in four of those triangles will embrace the true position. Those will
> be the TTT or AAA triangles
>
> But now consider adding some systematic error, in the same sense to each
> sight. This will shift the position lines, to enhance the size of the
> triangles we label TTT (if the systematic errors are such as to always
> increase the sextant reading), and enhance the probability that such a
> triangle will embrace the true position. For other combinations, the
> probability will be correspondingly reduced. If the systematic error
> becomes big enough to overwhelm the random scatter, then the TTT triangles
> become the only ones that are possible, and will always contain the true
> position.
>
> The same applies if the systematic errors are such as to always decrease
> the sextant reading, when a preponderance of AAA triangles will result,
> which also embrace the true position.


In this (more realistic) case, systematic error can appear desirable
since it increases the chance that the true position will fall within
the cocked hat. But it continues to do so by enlarging that 'hat.

We would probably do better to eliminate systematic error (so far as we
can) and thus improve our estimation of the MPP, while working on
estimating the standard errors of our observations so as to be able to
plot a rough confidence limit around that position.

> =========
>
> The situation, as outlined above, applies only when the three sights
> surround the observer, as at azimuths 0 deg, 120 deg, 240 deg. With a
> different geometry, in which the sights were to all one side, such as at 0
> deg, 60 deg, 120 deg, then that would cause a common systematic error in
> sextant readings (or, similarly, in compass bearings) to push the
> boundaries of the triangle in a different combination of directions. The
> effect would be to reduce the probability of the triangle containing the
> true position, and if the systematic error was large enough to overwhelm
> the random scatter, the triangle would then NEVER contain the true
> position.
>
> =========
>
> As I have said in an earlier mailing, I think (but haven't attempted to
> prove) these two different geometrical situations above are distinguished
> by the true position lying within, or without, the triangle connecting the
> three landmarks.


That seems to be the case from sketching a few examples but I would not
know how to go about proving the point mathematically.

Interestingly, if the true position lies exactly on the line between two
landmarks, any systematic error sufficient to overwhelm random errors
would prevent the two LOPs from cutting one another (at least in the
region between the landmarks). The same would apply with systematic
celestial errors, unless those were errors away and one went to the
trouble of plotting the curvature in the LOPs.

That suggests to me that George is right. If there is a phase change in
the behaviour of the cocked hat, in response to systematic errors,
depending on whether the true position is inside or outside the
triangle, it makes sense that the cocked hat is undefined in the
presence of such an error when the position is neither inside nor outside.

> In the case of a celestial fix, this (I think) would
> correspond to testing whether the true position was inside or outside the
> triangle formed by drawing great circles between the geographical positions
> of the three celestial objects. These suggestions in this paragraph are put
> forward rather tentatively.
>
> George Huxtable.



Trevor Kenchington


--
Trevor J. Kenchington PhD                         Gadus@iStar.ca
Gadus Associates,                                 Office(902) 889-9250
R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555

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