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    Re: Position lines, crossing.
    From: Guy Schwartz
    Date: 2006 Dec 9, 12:26 -0800

    Big thank you to George and Geoffrey.
    Geroge you are right I was taught that the position was located in the
    middle of a three point fix. After reading your responce I highlighted my
    position lines with a orange highlighter and BINGO I could see the
    convergance of the lines. Thank you.
    According to the book they say the answer is 40deg 01'N 153deg09'W, I'm sure
    they are using a computer for their answer. I agree with Geoffery's answer
    40deg 17' N 153 deg 13' W.
    Thank you gentleman,
    ----- Original Message -----
    From: "George Huxtable" 
    Sent: Saturday, December 09, 2006 8:52 AM
    Subject: [NavList 1850] Position lines, crossing.
    > 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.
    > >
    > --
    > No virus found in this incoming message.
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    > Version: 7.5.432 / Virus Database: 268.15.15/581 - Release Date: 12/9/2006
    > 3:41 PM
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