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Re: Compare Methods: Lat/Lon Near Noon
From: Antoine Couëtte
Date: 2021 Oct 3, 11:07 -0700

Dear Peter,

Very Brilliant idea of yours to have "four iterations of the algorithm already hardwired in your many-body Fix."

This is an excellent reminder that in essence the "Marcq-Saint Hilaire LOP many-body Fix" is an iterative process. Most of the time if its first pass is more than sufficient, its convergence rate is also amazing, with in general each iteration ending up at least 10 times closer from the ultimate "perfect position" than the previous one.

Since we are Frank's initial example, and referring to the enclosure to this post, I have decided to compare the results of the Real World Data LAN results already published there with the results of the "Fictitious Navigator and Fictitious Body" Method also described in it.

*******

Derivation of the Fictitious Observations set

Use of option 4.0 from UT = 16:17:20 with initial (μDec - NS) = + 4.4710 ‘/h

As per 5.1.4 here-below the Fictitious Navigator is sailing to the East at a speed of +1.6514 kt with current forced to 0.0000 kt.

Likewise the Fictitious Sun Declination is constant and equal to its value at 16:17:20 (N20°28.3161’).

Hence for the Fictitious Navigator and the Fictitious Sun: (μDec - NS) = + 0.0000 ’/h.

We then easily derive our following Fictitious Observations set corrected from UT = 16:17:20 which is our chosen starting point. Hence at this specific time, the real World and the Fictitious Observations are identical in value and start gradually diverging afterwards at a rate of  + 4.4710 '/h .

NOTE : As earlier : many digits - totally non-significant in any Navigational sense - are published here to ensure that we are not at all concerned by any round-off error adverse effect or equivalent.

1 - 16:17:20 / 72°28.5000'    2 - 16:22:40 / 72°44.1026'     3 - 16:36:45 / 73°02.5531'     4 - 16:39:15 / 73°02.8668'              5 - 16:51:25 / 72°46.4602'     6 - 16:56:20 / 72°32.5939'

As per Section 4.0, we should expect the results of the Fictitious Data set to show the following:

1 - UT culm - UT tran = 0, or some value very close to 0. And:

2 - The Fictitious Observations set should be [almost] symmetrical relatively to (UT - UT culm), i.e. with odd powers in UT [almost] equal to zero in the development of Hc as a function of (UT- UT culm). Accordingly a 2nd order regression should better perform given that only symmetrical terms are remaining here.

3 - As a consequence, and by comparison to the Real World Data set, the Fictitious Data set should behave as follows:

- A 2nd order regression LAN Fix should be closer to its Fictitious many body Fix. And also:

- Just for fun, a 6th order regression LAN Fix position should also be closer to its Fictitious many body Fix.

4 - As per 4.0.1.3UT tran = UT culm of the Fictitious Heights set, and H tran = H culm of the Real World Heights set.

5.1 - PRELIMINARY RESULTS

In the Fictitious data set, SUN Dec = N20°28.3161' from 16:17:20 and throughout the entire observations set.

Speed to the East (unchanged): +1.6514 kt

Fictitious (μDec - NS) = + 0.0000 ’/h

5.2 - RESULTS OF 6th ORDER LAN SOFTWARE (see 4.4.3)

RESULTS FROM THE REAL WORLD DATA SET

UT tran = 16h37m30.7s and LAN fix at N37°11.3’ / W070°12.2’

UT culm - UT tran =+ 26.15 s  (vs. +26.4s at 5.1.8)

H culm - H tran = 0.0164’ (vs. 0.0164’ at 5.1.9)

SDEV = 0.23 NM (0.8’ on Lon)

Method Validity time span (16h11m - 17h04m : OK)

Many Body Fix Method : From the LAN fix and at the LAN time get a Many Body Fix at

N37°11.3’ / W070°12.2’, and SDEV = 0.22 NM.

The Many Body Fix is only 34 meters - i.e. less than 120 ft - from the LAN fix.

17h Fix at N37°09.8’ W070°11.4’

RESULTS FROM THE FICTITIOUS DATA SET

UT tran = 16h37m29.9s and Fictitious LAN fix at N37°12.6’ / W070°12.0’

UT culm - UT tran =+ 00.000 s (lucky me !!)

H culm - H tran = 0.0000’

SDEV = 0.246’ NM (0.9’ on Lon)

Method Validity time span (16h11m - 17h03m : OK)

Many Body Fix Method : From the Fictitious LAN fix and at the LAN time get a Fictitious Many Body Fix at N37°12.6’ / W070°12.0’, and SDEV = 0.240 NM.

The fictitious Many Body Fix is only 22 meters - i.e. less than 71 ft - from the Fictitious LAN fix.

Fictitious 17h Fix at N37°12.6’ W070°11.2’

As per 4.0.1.3 and to derive our Real world position from our Fictitious Data set results:

UT tran = UT culm of the Fictitious Data set. Hence : UT tran = 16h37m29.9s . And:

The Fictitious Lan Fix gives the Latitude at the time of the First Observation. To get the Real World Latitude at UT tran, we need to DR the Fictitious Lan Fix Lat during a time interval equal to (UT tran - UT 1st observation).

In the Real World and with (NS) -3.9848 kt the ship Latitude has decreased as follows:

-          Between the 1st Observation and (UT tran) : (16h37m29.9s - 16h17m20s ) * 3.9848 = -1.3382 ’

Hence the Real World Latitude at UT culm is equal to: N 37°11.3’

-          Between the 1st Observation and 17:00 : (17h00m00s - 16h17m20s ) * 3.9848 = -2.8336 ’

Hence the Real World Latitude at 17h00m00s is equal to: N 37°09.8’

CONCLUSION

Comparison between Real World positions derived from (1) the Real World Data set and (2) the Fictitious Data set :

The overall results are almost identical:

(1)    - UT tran = 16h37m30.7s and LAN fix at N37°11.3’ / W070°12.2’

The Many Body Fix is only 34 meters - i.e. less than 120 ft - from the LAN fix.

17h Fix at N37°09.8’ W070°11.4’

(2)    - UT tran = 16h37m29.9s and LAN fix at N37°11.3’ / W070°12.0’

The Many Body Fix is only 22 meters - i.e. less than 75 ft - from the LAN fix.

17h Fix at N37°09.8’ W070°11.3’

As we could expect, the Fictitious Data set LAN Fix is even closer from the many-body Fix (22 m vs. 34 m). Only a mathematical check, and absolutely no navigational significance whatsoever.

*******

Peter, if you could derive your own fictitious LAN fix position from the Fictitious Data set here-above and see how it now compares with your own Fictitious many-body fix derived from it, that would be great.

I am almost sure that from this now quite symmetrical Fictitious Data set your 2nd order regression will perform even better.

To make things easier onto you: again this Fictitious Data set is as follows, with ship East speed at  +1.6514 kt with current equal 0.0000 kt

1 - 16:17:20 / 72°28.5000'    2 - 16:22:40 / 72°44.1026'     3 - 16:36:45 / 73°02.5531'     4 - 16:39:15 / 73°02.8668'              5 - 16:51:25 / 72°46.4602'     6 - 16:56:20 / 72°32.5939'

Give a try ?

Hint : you need to "use" both a constant Latitude Navigator (rather easy to "enter") but also a Fictitious Sun with a constant declination equal to Dec at 16:17:20. This one might be a bit more delicate to "enter".

Thanks in advance for your results and Best Local Apparent Noon Regards,

Antoine

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