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    An example of a slope
    From: Peter Fogg
    Date: 2007 Mar 10, 13:52 +1100

    Earlier I wrote, about comparing a pattern of sights taken of the same
    body over a five minute period:
    Comparing the slope is a kind of averaging, since with even random
    errors the adopted line will bisect the negative and positive values.
    I think having a graphical solution is intuitively more useful than
    just a row of numbers - you can evaluate the sights made and shift the
    line of best fit to suit the data by eye in a way that no mathematical
    number crunching process can achieve quite as well. In effect you can
    nominate the importance given each sight; how close to the adopted
    line you allow it to sit, perhaps in accordance with your memory of
    how clearly and how well you remember that sight being made, as well
    as in light of how the other sights fit the line.
    
    Attached is an example (worth a thousand words?) based on actual observations.
    
    The body is Canopus, the latitude 34d, the azimuth 149d, leading to a
    calculated slope of 32 minutes of arc over 5 minutes. Since the obs
    are to the east the body is apparently rising - if looking to the west
    the slope would descend from left to right.
    
    On the form (from "The Complete On-Board Celestial Navigator" by
    George Bennett) there are spaces for 5 sights, which is about how many
    one unaided person can make over this period. I must have had someone
    acting as scribe to manage nearly twice this number over the period.
    
    The slope should not be extended beyond 5 minutes of time since the
    body's apparent rise or fall actually describes an arc, not a straight
    line. In any case the opportunity for taking sights may be limited
    (eg; twilight), plus all sights comprising a fix need to be made
    within a shortish time-frame on a moving object using the one DR, or
    suffer the distorting effects of platform movement.
    
    The calculated slope is the dotted line at the bottom, the adopted
    line of best fit the solid line parallel to it. Any point along this
    line can be adopted. I have chosen 5h 29m 00s / 66d 30' but could as
    well have adopted 5h 27m 00s / 66d16', or 5h 30m 30s / 66d 40'
    (indicated by crossing lines). All will lead to the same position line
    via sight reduction.
    
    Did I put the line in the right place? Who knows! Can you find a
    better place for it, based on the sights? The first and third obs
    don't seem to fit, so have been given little weight - essentially
    judged to be outliers and discarded. The calculated slope is the fact
    that the sights should match as best they can.
    
    You can see how a process of averaging or linear regression would be
    influenced by these dodgy sights - inevitably a different slope would
    be generated. But the slope is a fact, it cannot be changed. If the
    outliers were more extremely wrong (as they easily can be) and the
    sights fewer, the unhelpful distortion via these inappropriate
    techniques would be greater.
    
    Hopefully this example shows how the adopted alt / time is an
    improvement over any of the random sights, via a mix of science and
    art; or to put this another way: via calculation, skill, experience,
    intuition and good ole common sense. Don't leave home without it.
    
    Alex wrote:
    "In principle I agree that graphic methods are
    better, especially to select the outliers.
    But they are very time consuming. And prone to additional
    blunders."
    
    This goes to what I wrote earlier about how, in my judgement, it is
    better to reduce random error before sight reduction than worry later
    about the indeterminate nature of a fix based on random sights. In my
    (ever so humble) opinion, the extra time and effort of analysing the
    raw data of sights made, as shown here, is recompensed adequately by
    the improved quality of the resulting position lines.
    
    An alternative to this is to make many individual sights of different
    bodies and reduce them all and then cope with all the resulting
    position lines. I know which I prefer, and also which is quicker.
    
    Like many others; these days when on-board there is usually at least
    one GPS unit, so my quaint efforts with sextant, watch and forms serve
    mostly to show up their errors compared to the electronic deity. But
    it also means that I know that this comparison of slope consistently
    produces a better position than any of the individual sights.
    
    It really is a practical and easily-worked method of producing good
    data from bad.
    
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