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Well, looking at position finding as an optimization problem, where the 
model is highly non linear and the functional to minimize is not simply 
quadratic, but shows several local minima, mathematically nothing can be 
sayd other than, if you starts relatively close to one of these local minima 
you will find it as solution.
Usually in these cases, after finding a minimum, you apply a random 
perturbation to the solution and try again to see if other minima (or a best 
minimum) exist:
In this case I assume that all (two) minima have identical functional value 
so are undistinguishable just from the fitting.
Giuseppe
----- Original Message ----- 
From: 
To: 
Sent: Wednesday, July 16, 2008 6:32 AM
Subject: [NavList 5858] Re: My first Lunar



Giuseppe, you wrote:
"Dear Frank,
using my clearing algorithm I found:
time 22:19:38 GMT, pos 14�N 35.3', 61�W 41.3'
LD 68� 13.13' (sextant LD 68� 19.40')
The position is roughly 5 miles from yours
Giuseppe"

Sounds like a near-perfect match. Incidentally, the "best" GMT would appear
to be about 22:19:34 according to my calculation, but 0.1' difference in the
clearing process would correspond to 12 seconds difference in time so I
don't consider a difference smaller than 6 seconds in time, in any analysis
of lunars, to be meaningful.

The "fix" from the two altitudes comes from two altitudes in nearly opposite
azimuths so that the position is relatively indeterminate along azimuth
340/160. That is, you can shift the fix five miles or more along that
direction and there would be little difference in the result. So again, that
means your result is a near-perfect match with mine, which is re-assuring!

And this brings up an interesting question. How can this analysis be
producing different longitudes? Viewing a lunar as a sight for longitude, as
in traditional, historical lunars, how can there be any ambiguity in the
final longitude? The few miles difference that we're seeing here might be
excused but what about the big difference in longitude between the position
(presumably the correct one) west of Martinique and the other position
inland in Guyana? The answer, of course, is that the longitude resulting
from a lunar depends also on the local time. If these sights had been worked
in the early 19th century, and there was no reliable time kept by a common
watch, the altitude of Jupiter would probably have been worked to get the
local apparent time. But that calculation depends on the latitude. So if you
assume a different latitude, you get a different LAT and when combined with
the Greenwich Time that comes from clearing the lunar, you would end up with
a different longitude.

 -FER






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