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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Compare Methods: Lat/Lon Near Noon
From: Antoine Couëtte
Date: 2021 Oct 2, 04:47 -0700

Dear Ed,

Again: still referring to the Enclosure of a previous and recent post as a "Ref Document" here-after.

I am now better guessing why you wish attempting to further refine your LAN position once you first got it through a "curve fit".

You probably wish to "duplicate" the successive approximations permitted by the classical "Many-body Fix" of the Marcq Saint Hilaire LOP's, don't you ?

In the "many-body fix" method you start from a first "assumed position" to derive an "observed position" which in turn you can use as an "updated assumed position" to get an updated "refined observed position" ... and so on ... Such successive iterations are very seldom in use except if the first overall distance between the initial "assumed position" and its subsequent "observed position" exceeds some preset value, e.g. 30 NM or even 20 NM.

Anything similar with the "curve fit method", would you be asking ... ?

Well ... probaly not, or at least, not to the same extent.

The reason is that the "curve fit method" requires no explicit "assumed position". Since you do not explicitly need one, you are not in a position to use successive iterations like with the "Many-Body fix" iteration here-above.

Nonetheless you can still in some way "iterate" at least once with the "curve fit method" because you can perform the following:

- Start from the "standard value" given in Formula (1a) of the Ref Document : ΔUT = (UT = culm - UT tran) in seconds of time = (48 / π) * (tan Lat - tan Dec) * (μDec - NS)

- From the first LAN Fix thus obtained, you are now able to further refine ΔLHA in °/hour and use the following "refined formula":

ΔUT = (10800 / π * [ΔLHA]² ) * (tan Lat - tan Dec) * (μDec - NS)  which yields a refined ΔUT value.

However, you will get only very marginal "real world" improvement after this first iteration. For 2 reasons :

(1) - Even this "refined formula" remains only a first order formula which accounts for most the actual ΔUT, but not for all of it. See Ref Document Comment in 3.3.1.

(2)  In the "curve fit method" you always remain limited by the quality of the "curve fit method" you are using (see Ref Document Constraint 4.1.2.3). On the other hand in the "Many-Body fix" iteration you are not limited by anything except the quality of your observations themsleves.

In other words, only the "many-body fix" methods can get you the most from your LAN data. And any LAN method has to be already quite "smart" to attempt equalling the performance of the "many-body fix" method under all circumstances.

This is why, after the very first LAN result obtained by the "curve fit method" I suggested you to turn your attention towards the "Many-Body fix" iteration with the usal caveat about GDOP.

Final reminder ; the very same GDOP does also exist in your LAN fix result. It is simply more or less concealed / hidden in the LAN method and you cannot immediately see it. On the other hand in the "many-body fix" method while watching at your almost parallell LOP's you immediately start suspecting something adverse about your overall fix accuracy.

It is interesting to see that, from whichever method, you are getting almost exactly the very same SDEV on your samples (makes sense, no ???) . See Ref Document 5.3 with SDEV = 0.23 NM for the 6th Order LAN Fix and 0.22 NM for the Many Body fix.

This is about the best I can say, again hoping that it can help you clarifying the situation.

Best Regards,

Antoine

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