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    Re: The rapid-fire fix
    From: George Huxtable
    Date: 2009 Apr 28, 10:44 +0100

    Some of Frank Reed's claims in [8049] make good sense. Others do not. Let's
    try to choose between them.
    
    Describing increasing the number of observations, producing a tangle of
    intersecting position lines, he writes- "This tangle obscures the fact that
    numerous sights produces a better fix than fewer sights." Of course; who
    would disagree?
    
    But then he continues- "But as we increase the number of sights taken, the
    error in both directions decreases rapidly." Not exactly "rapidly", surely.
    As you add more observations, the error decreases, but more and more slowly.
    To halve the scatter that exists in a single observation, you need to take
    4. To quarter it, you need 16. Diminishing-returns sets in. When the random
    part of the scatter has been reduced, by such repetition, to the level of
    any systematic error, there's little point in taking it much further. "We
    get a good position fix in short order." he concludes. Not an obvious
    conclusion, and one that calls out for some justification. Unfortunately,
    it's expressed in such fuzzy terms, that it isn't easy to prove or disprove.
    We need to retain some scepticism about such arguments.
    
    The crucial paragraph is this one-
    "So how good is a "rapid-fire fix"? If I shoot N altitudes of a single body
    over a time interval T, how good is the fix in the position? If each
    altitude has a standard deviation error S_i, then what is the standard
    deviation of the fix in the direction perpendicular to the mean azimuth of
    the body, and what is the standard deviation of the fix in the direction
    towards the mean azimuth? Anyone care to work it out? What you may find is
    that the fix is suprisingly accurate, within a couple of miles, even when
    the total time interval is just ten or fifteen minutes (see last paragraph).
    Also, the result may be easier to express by changing variables and using
    the total range of azimuth A from the beginning to the end of the time
    interval T."
    
    If Frank wishes to convince doubters, he would do better to work it out
    himself, and offer a bit of proof. As for the claim that the fix is
    "surprisingly accurate, within a couple of miles, even when the total time
    interval is just ten or fifteen minutes...", I don't believe it. Mind you,
    he says "may find", not "will find". Let's see an example, then. Not the
    special case where the body passes close to the zenith, but a real-world
    example. And let's discover what scatter Frank is assuming in his individual
    sextant observations, to arrive at that result.
    
    "For example, if I take twelve equally-spaced sights of the Sun over the
    hour from 0900 to 1000 in the morning, I could group them into sets of four
    and average each set. Each average would presumably be better than any of
    the individual altitudes taken alone."
    
    Yes, of course. The more observations that are taken, (and the wider the
    spread of azimuths, too) the better the answer will be. Let's allow that 12
    observations are to be made, and allow an hour to spread them over. Then is
    Frank actually claiming that spreading the 12 equally in time (call it
    method A) is somehow BETTER than method B, which would take 6 sights, as
    closely spaced as possible, at the beginning of the hour, and the other 6 at
    the end? On what basis, as no evidence or arguments are provided? My own
    guess is that the advantage would be the other way round. It's a claim to
    which we could apply a statistical test, if Frank would care to specify some
    conditions for such a test. I'll be happy to deduce longitudes from
    simulated (or even better, if available, real) test data, by method B, if
    Frank, or anyone else, would offer to do the same by method A. Observer
    speed could be taken into account, or taken to be zero, whichever. Method B
    boils down, of course, to the traditional method of generating two position
    lines, as far apart in time, and azimuth, as possible.
    
    The only piece of hard evidence Frank has put forward, in justification, is
    in a posting by Jeremy at-
     http://www.fer3.com/arc/m2.aspx?i=105416  (Note, if you look at that
    posting, that its times were Zone Time, not UT as stated). This was a series
    of Moon altitudes, nicely taken over 8 minutes. The result was posted in-
    http://www.fer3.com/arc/m2.aspx?i=106066
    From that data, Jeremy (or actually, his navigational computing program,
    SkyMatePro) deduced a position that turned out to be within 1.7 miles of his
    GPS position, his comment being "...it is fairly accurate, certainly
    accurate enough for a deep sea position". That accuracy of 1.7 miles seems
    hard to believe, from just 8 minutes of observation, even though the Moon
    was rather high.
    
    Two comments here-
    1. As we've seen before, when looking at scatter, a single result tells us
    almost nothing. It may "hit the spot" just by chance. Consistency is what's
    needed.
    2. We don't know in detail how that program handled the data it was given.
    Jeremy has told us it was fed with a GPS position as DR, with known course
    and speed, and he "  plugged each moonline ... as individual moon LOP's.".
    Without understanding what went on in detail, let me make a guess. Given an
    individual position line, and a DR position, what could a computer make of
    it? It might well do its best, by finding the most probable spot on that
    LOP, which would be the nearest point on that line to the DR position. In
    which case, it would inevitably end up with a position very close to that
    DR. And then, it would do the same for all the other position lines, and
    somehow average the result. End result, a position close to the DR! I've
    suggested to Jeremy that next time he tries such an exercise, he plugs in
    deliberately-different trial-positions for his DR, to see if the answer
    changes. I wonder, if he had offered the computer just one "moonline", what
    would the result have been then?
    Now, I don't know how that program operates, internally, and I doubt whether
    Jeremy or Frank know either. It seems worthwhile, then, to ask some probing
    questions about the value of such an exercise, before building on to such an
    insecure foundation, as Frank has done. Healthy scepticism is called for,
    here as elsewhere.
    
    George.
    
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    ----- Original Message -----
    From: 
    To: 
    Sent: Sunday, April 26, 2009 3:16 AM
    Subject: [NavList 8049] The rapid-fire fix
    
    
    
    A statistical fix from a series of sights around noon is a special case of,
    what I have called in the absence of another name, a "rapid-fire fix".
    Consider the general case...
    
    At any time of day or night, I take a series of sextant sights of a single
    body in relatively rapid succession. From the perspective of a traditional
    LOP plot, each of these yields a line of position and when plotted on a
    chart and advanced/retarded for the motion of the vessel, they should cross
    at some single point, and that is the fix. Of course, the body's azimuth
    will not change much in a short period of time, so the individual LOPs cross
    at shallow angles with respect to each other, and careful plotting is
    essential. This is generally considered a "bad thing". In the real world,
    the sights are "noisy" --each has some small error arising from various
    causes (just random error --presumably systematic errors have been
    eliminated). So if we attempt to plot the LOPs, they no longer cross in a
    single point but instead in some tangled spider web of crossing lines. This
    tangle obscures the fact that numerous sights produces a better fix than
    fewer sights.
    
    The error in the fix resulting from the noise in the sights is the usual
    reason that navigators are advised to take sights of different objects at
    widely separated azimuths, or with a single body you should wait a few hours
    so that a later sight of that same body would be taken at a significantly
    different azimuth. This necessity for a wide separation in azimuth is easy
    to see from simple geometric arguments when we look at a single pair of
    LOPs. If they cross at a shallow angle, and we shift either of them slightly
    in the direction perpendicular to the LOP (corresponding to an error in the
    sight as taken), the crossing point moves by a distance that is
    approximately inversely proportional to the angle between the two LOPs, for
    small angles.
    
    What happens when we take multiple sights? If we shoot a dozen or even a
    hundred sights in rapid succession, what happens to the error in the fix?
    Clearly the error in the fix is minimized in the direction towards the
    celestial body. The error is largest in the direction perpendicular to the
    body. We get an "error ellipse". But as we increase the number of sights
    taken, the error in both directions decreases rapidly. We get a good
    position fix in short order.
    
    For a practical case, imagine spending forty-eight hours under stormy skies.
    You don't get any sights for two days. Then in the morning of the next day,
    the clouds begin to break. Let's say that you are able to shoot the Sun once
    every five to ten minutes as the clouds break up. How long does it take to
    get a usable fix? From the very first sight, you can plot an LOP, though it
    will not in general be a pure latitude or a pure longitude. Every navigator
    knows this, and indeed it's the basis of Sumner's original explanation of
    his discovery of the Sumner line of position. But what happens as you
    continue taking sights every few minutes? Most practitioners of standardized
    20th century celestial navigation would see little merit in this and might
    simply average the altitudes to get a refined (single) LOP. A modern
    celestial navigator with sight reduction software can enter each sight as
    taken and the software can quickly generate a least squares position
    combining all of the data in a way that is nearly impossible to see in a
    traditional plot of LOPs. This is an aspect of sight reduction that is
    generally known to navigators --most people are aware that this feature
    exists in software-- but the fact that it produces a good fix in latitude
    and longitude quickly is not widely appreciated. And by the way, "better"
    software will also plot an error ellipse which directly shows how the fix is
    improving despite the increasing area of the "tangle" made by the crossing
    LOPs (in fact one of the earliest binary attachments for the group, just
    over SEVEN years ago, demonstrated this:
    http://www.fer3.com/arc/m2.aspx?i=006308).
    
    So how good is a "rapid-fire fix"? If I shoot N altitudes of a single body
    over a time interval T, how good is the fix in the position? If each
    altitude has a standard deviation error S_i, then what is the standard
    deviation of the fix in the direction perpendicular to the mean azimuth of
    the body, and what is the standard deviation of the fix in the direction
    towards the mean azimuth? Anyone care to work it out? What you may find is
    that the fix is suprisingly accurate, within a couple of miles, even when
    the total time interval is just ten or fifteen minutes (see last paragraph).
    Also, the result may be easier to express by changing variables and using
    the total range of azimuth A from the beginning to the end of the time
    interval T.
    
    It's certainly true in celestial navigation that there is rarely anything
    "new under the Sun". It may be worth mentioning that we can understand in a
    very general sense what's going on by mapping a rapid-fire fix onto some of
    the earliest methods for determining a vessel's position: a time sight for
    longitude and a "double altitude" sight for latitude. For example, if I take
    twelve equally-spaced sights of the Sun over the hour from 0900 to 1000 in
    the morning, I could group them into sets of four and average each set. Each
    average would presumably be better than any of the individual altitudes
    taken alone. We would then have three averaged altitudes at effective times
    of 0910, 0930, and 0950. The middle sight can be treated as a time sight. We
    calculate local apparent time from it using standard methods, compare that
    with the chronometer time, and we have the vessel's longitude. Then we take
    the averaged sights from 0910 and 0950 and, using the rather complicated
    methods found in most 19th century navigation manuals, we would get latitude
    from the two altitudes and the time interval between them. So there's the
    fix. Similarly, for a 20th century LOP plot, we could average the sights
    just the same way in sets of four, and then cross the three resulting LOPs
    and place the fix somewhere inside the small triangle formed where they
    cross. By averaging sets of four sights, we have achieved some of the
    statistical least squares fitting in a modern rapid-fire fix. In fact, in
    some cases, the results should be identical.
    
    You can advance/retard the LOPs in a statistical fix just the same way you
    would with a Noon Sun fix. That is, rather than attempting to slide each LOP
    around on the chart, you can adjust the altitudes before you plot. You find
    the component of the vessel's true velocity that is along the mean azimuth
    to the celestial body and add that in proportion to the time from the middle
    sight (or any sight you choose). So for example, if the Sun is in the
    southeast at azimuth 135 in the morning and I am shooting a series of sights
    in rapid succession while travelling due south at 8 knots, then the
    component of my velocity towards the southwest is 5.7 knots so for every ten
    minutes, I should subtract just about 1 minute of arc from the altitudes of
    the later sights (I'm moving towards the GP of the celestial body at that
    rate). The correction for the changing position of the object (changing SHA
    and Dec) is more complicated in this case, so let's just assume it's small
    enough to be ignored.
    
    Finally, I hope it's clear that taking a series of sights around noon for
    latitude and longitude is simply a special case of this "rapid-fire fix".
    It's unique for a couple of reasons. First, the corrections for motion are a
    little easier since it's relatively easy to understand the north-south
    component of a vessel's motion as opposed to the components relative to some
    other bearing. Second, the correction for the changing position of the
    celestial body (doesn't have to be the Sun, by the way) are simpler and
    effectively limited to changing declination. That correction just adds or
    subtracts from the north-south component of the vessel's motion. And last,
    but most importantly, there are simple graphical techniques which can do an
    excellent job replacing the math of a statistical fix (like the "paper
    folding" method that I've described). This means that we get all the
    advantages of using all of the available sight data to get our vessel's
    position without requiring some calculating device. Perhaps someone can
    discover a graphical (or other) technique to get a statistical fix in other
    special cases, like sights on the Prime Vertical, or perhaps even in the
    general case...
    
    Almost a year ago, when I mentioned this idea of a rapid-fire fix, NavList
    member Jeremy Allen did what any good navigator should do: he got out his
    sextant and tried it out (no spreadsheet simulations required!). He shot
    eleven altitudes of the Moon over a period of just eight minutes. The sight
    data was in message 5416: http://www.fer3.com/arc/m2.aspx?i=105416  Most
    other NavList members either didn't know what to do with this set of sights
    or simply assumed that the altitudes were meant to be averaged to get a
    single line of position. Instead, using software, this set of sights yielded
    a true (statistical) fix in both latitude and longitude with very good
    accuracy. Jeremy's solution was posted in NavList 6066:
    http://www.fer3.com/arc/m2.aspx?i=106066
    
    -FER
    
    
    
    
    
    
    
    
    
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