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    The rapid-fire fix
    From: Frank Reed
    Date: 2009 Apr 25, 19:16 -0700

    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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