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Gary Schwartz asked In Navlist 1848, "sorry this time I attached the
file and sent it to the correct list", a threadname which I have now
renamed.

"...there are six objects. Maybe plotting 6 objects is too many.  This
plot is exercise 3-2 from the book 100 problems in celestial
navigation.

My fundamential question is which sights form the enclosure of my
position? I'm thinking Alpheratz, Venus, and Rasalhague, however I
have no basis as to why these and not any others."

And he provided a picture of the plot as an attachment.

=========================

This is a matter that crops up from time and shows up much
misunderstanding among even experienced navigators, textbook authors,
and tutors. So it's fine that Gary raises it again and provides an
excuse to give it another going over. However, some old hands will
have heard it all before.

The short answer to Gary's question is that NONE of these lines forms
the "enclosure" of his position. All that can be said about his
position is that it is somewhere in the vicinity of where the lines
cross, a patch covered by a broad thumbprint. For the sake of putting
a dot on his chart, he might take it to be, say, 40deg 12'N, 153deg
15'W, but it doesn't matter much exactly where, within a few miles.
What is really important is that he is aware that it's only a rough
guide to where he actually is; to within 7 miles or so in any
direction, by the look of it. And to recognise that it's quite likely
that his true position may be completely outside the area bounded by
any combination of those lines.

He wondered if too many objects were being plotted. Not at all. The
more objects plotted, and the more crossing-lines shown,  the better
he will be able to estimate the centre their crossings congregate
about, and the scatter of those position lines around it, which
provides some notion of how precise the observations actually are.

There are indeed computer programs which attempt to make a
"least-squares" statistical analysis of such a round of sights, to
provide a nominal centre-position and an "error-ellipse" surrounding
it. That can avoid the need for the graphical construction (the
program can do that for you) but in my view it will gain you little
over a commonsense view of a plot such as Gary provided; and it can
sometimes actually mislead.

The simplest situation to consider is that of two such position lines,
which cross at a point, and that point is what you plot as your best
estimate of position. But every navigator should be aware that no
observation is perfect, and that his position lines have an error-band
which widens them to a few miles across, depending on the
circumstances of the time, which only he knows best. Things such as
the size of his boat, the roughness of the sea, the sharpness of the
horizon line, all give rise to scatter in his result. These are
matters that the computer doesn't know about, and can only guess at
from the discordance between many observations. With just two, it has
nothing at all to go on. It will give the crossing-point, nothing
else, but the navigator, estimating roughly his confidence in each, is
in a much stronger position, and can sketch in a rough error-zone
around his crossing-point, which also depends on the angle of the
crossing.

A common situation is whan a third observation is taken, to give a bit
of extra confidence. The three resulting position lines cross to
create a "cocked hat" error triangle, and it is in discussing this, in
the past, that so much heat has been created. This is because
erroneous notions have been so strongly ingrained, as a resut of
faulty teaching. It has often been taught in navigation classes, and
probably still is to this day, that such a triangle embraces the
possible position of the vessel, and that to be safe, a mariner has to
assume that he is whatever part of that triangle is nearer to a
danger-point. Nothing could be further from the truth. It's profoundly
dangerous nonsense; that is not a safe assumption at all.

In fact, if any systematic errors have been properly corrected for,
and only random scatter remains, the simple truth is this. Only on one
time in 4 will the vessel be inside that triangle at all, and 3 times
in 4 it will be somewhere outside it, though in the vicinity. This is
a simple statistical truth, easily proved, but one that mariners are
most reluctant to accept, because it is so contrary to what they have
been taught. Surprisingly, this 1 in 4 rule applies to the most
skilled observer, just the same as it does to a novice. The difference
is that the expert's triangles will turn out to be smaller, but still,
only one in four of those smaller triangles will embrace the true
position.

Given such a triangle, a least-squares analysis program will do its
best to assess an error ellipse, based on its size. But you have to
take such findings with a pinch of salt. Because there's so much
variation between one such triangle and another, simply as a result of
random scatter, some will just happen to be tiny in area, just because
the lines happen to cross closely. When plotting out such a case, the
observer might think that he had made a particularly precise
observation, and the computer thinks the same. But an astute observer
realises that it's just the luck of the draw, whereas the computer has
no such insight. Only after assessing a run of many similar
observations can you get a good feel for the overall accuracy being
obtained in those conditions; not from just a single triangle.

With more observations crossing, such as the six in the example Guy
Schwarz has given us, a computer has a bit more information to work on
and can make a better shot at assessing the precision of its resulting
"fix". And in just the same way, you and I can eyeball those crossings
of the 6 lines and weigh up the resulting accuracy for ourselves, and
our intuition will probably arrive at about as useful an answer as the
computer's. But there's no way that you can draw a boundary-line on
that diagram and say that the true position must lie within it, which
is what Guy was asking for.

=======================
Systematic errors and their effects.

There's a complication. Above, it was assumed that the only errors
were random ones; that were equally likely to be one way as the other,
and that any systematic errors had been corrected for beforehand. That
may not be the case. A careless observer may have got his index
correction wrong, offsetting all his altitudes by a common amount.
More insidiously, anomalous dip could be affecting his horizon in an
unknown way, with a similar result. Various proposals have been made
for detecting and correcting such errors, but are unlikely to succed
(in my view) unless those errors happen to be dominant, overwhelming
the random scatter. It is difficult, often impossible, to unravel and
separate the effects of such random and systematic errors. But if
systematic errors are making a significant contribution, they will
tend to affect that 1 in 4 probability (for triangles) discussed
above; either increasing or decreasing it, depending on the geometry.
One reason for making widely-spead observations all round the horizon
is to average out such systematic errors.

And there can be other type of error, just as systematic as those
considered above, with a different effect. Such as a clock error,
which works differently on the altitudes of bodies that are rising,
compared with those that are falling.

Thanks to Gary for giving me the opportunity to trot out an old
warhorse and give it a bit of exercise.

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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