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The posting from Peter Fogg contains arguments we have heard, and tried to
correct, before. I am aware that attempts at technical discussion with Peter
have in the past degenerated into personal abuse, but will try once again,
hoping that he can constrain himself to navigational matters.

In his arguments, there are indeed several grains of truth and sense, but
from these is created a structure that doesn't stand up.

Let's examine those arguments in detail-
| As to the elimination of constant error, this was discussed in some
| detail, complete with diagrams, some time ago (thus findable in our
| archives).
|
| Yes, sights using opposing azimuths will lead (with constant error
| only present, ie; assuming no erratic error which could complicate
| things) to a 'box' shape.  Since with each pair of intersecting LOPs
| the true fix will lie along a line bisecting the angle formed by their
| intersection, with such a box shape the fix will be found ... at the
| centre of that shape!

Usually, observational errors get divided into the categories "random"
errors, which are different at each repetition, and "systematic" errors,
which are not. Instead, Peter Fogg uses the terms "erratic" and "constant"
errors. Let's leave aside, for now, the casual discarding of "erratic error
which could complicate things", to deal with later. But what exactly does he
mean by "constant" error?

There is one particular type of error (let's christen it "type A") that will
affect all position lines in exactly the same way, shifting each one toward
or away from the observer by exactly the same amount. I can think of two
likely sources for this type of error-
1. taking an incorrect value for index correction for a number of sights,
without rechecking for index error in between.
2. the dip taken from dip tables being in error, due to the refraction near
the sea surface differing from expectations, and making the (reasonable)
assumption that this discrepancy is the same at all azimuths.
Are there others sources of such error, that I've missed?

And there are other errors (call them type B) that are also "constant" or at
least "systematic", to that the extent that repeated measurements of the
same body will be affected by them in exactly the same way, but which do NOT
affect all position lines in the same way.
For example-
1. Calibration error of the sextant, which varies along the sextant arc, so
it's different when measuring one star at one altitude compared with another
star at a different altitude.
2. Timing error of the chronometer, which would cause all position line to
shift Eastwards or Westwards together.

If we KNEW that the only errors in a multi-body fix were of type A, and that
there were no type B errors and no "erratic" or random errors, then Peter
Fogg's assertion, that the true position must lie at the centre of a 4-body
rectangle, or at the bisectors of a three-body cocked hat, would indeed be
correct. That is one of the grains of truth in his claims, but it depends on
the validity of those assumptions. From it one can make a valid deduction,
that it's always worthwhile to balance pairs of observations of stars, in
opposing directions, where it's possible to do so, because then systematic
errors of type A (but only type A) can be balanced out. No quibble about
that.

Peter Fogg continues-|
| It is only when the assumption is made that the sights are somehow
| free of any constant error, and thus subject only to erratic error,
| that the  fix becomes 3 times more likely to lie outside than inside
| the shape.

Correct. And it should be noted that whenever such a claim is made, that the
true position is three times more likely to be outside rather than inside a
cocked hat, it is (or should be) noted that any systematic errors have been
first corrected for. Otherwise those systematic errors enlarge the cocked
hat and make it more likely to contain the true position.

| How can either presumption be made?  It seems to me that, a priori,
| any round of sights may contain some extent of both types of error.

I agree.

| Therefore the most useful approach is to eliminate erratic error at
| source.

If only...

| This can be done via the comparison of a number of sights of
| the same body taken over a few minutes with the slope caused by the
| apparent rise/fall of the body observed over that period.

No, it can't, as various listmembers have tried to convince Peter Fogg over
the years, without much success. I will try again.

When Peter Fogg refers to "erratic error", I will presume that it's what the
rest of us call "random error".

First, let's deal with measuring a Sun noon altitude, a simple case because
the slope is zero. There's a certain inevitable scatter in each observation,
which will depend on the motion of the vessel, the quality of the horizon,
the clarity of the air, and the skill of the observer. And repeating and
averaging many such observations will improve the resulting precision of the
result in a way that's governed by well-known and undisputed statistical
laws. The errors depend on the square-root of the number of repeated
observations. So to halve the inherent scatter of a single observation, you
must average 4 such observations, to quarter it, 16, and so on. There's no
way that you can "eliminate" the scatter, but you can reduce it, by adding
more and more observations, with diminishing returns.

Instead of numerical averaging, you could plot it out, and draw a horizontal
line that pases through between the points as well as possible, and by eye
it's possible to make a reasonable shot at it; but never will you do any
better than by taking that average.

But at times away from noon, there's an extra complication. The Sun's
altitude is steadily changing, so when you try to make repeated
measurements, you are aiming at a moving target. If you tried to simply
average the results, the result would be affected by the details of the
timing. One way to deal with that problem is to plot it out against time,
and then to find the best fit to a sloping line, as you shift that line up
and down. And that's done most accurately if you precalculate the slope from
the rate of change of altitude that you estimate the Sun to have around that
time. This is a perfectly valid procedure, that Peter has advocated before,
and I do not criticise it. It's the "grain of truth" in this part of his
argument. And it applies to each star, in a round of star sights, just as
well as to a Sun observation.

But all that's been done, by that "fitting to a slope" procedure, is to
eliminate the problem caused by the steadily changing altitude. The points
will scatter about that sloping line just as much as in our noon example,
where they scattered around the horizontal line. We haven't changed the
scatter, and the laws of statistics still apply just as they did in our
Sun-noon case. So that you still need to have four observations to halve the
scatter of one, and 16 to quarter it. And still, you can never "eliminate"
that random scatter, only reduce it.

Peter concludes

| Then, erratic error eliminated (to a usefully practical extent), any
| remaining error, now assumed to be the constant type, can be dealt
| with by bisecting those intersecting LOPs and finding the fix ... at
| the centre of the shape.

Now he qualifies the notion of "eliminating" any random error, saying only
"to a usefully practical extent". What does that mean, exactly? How many
repetitions does he propose, rembering that with the square-root law,
diminishing returns start very quickly?

Then, if he can succeed in eliminating all "erratic error", and only if
those "constant" errors are all of type A, and none of type B, only then
will his conclusion be valid.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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