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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Antoine Couëtte
Date: 2020 Apr 14, 07:22 -0700
Hello again Tony,
Referring to my last post, I decided to significantly rework the reply to your question about the maximum time-span validity in using averaging techniques.
No change in the results, but I made it more pragmatic and hopefully easier to figure out and to understand.
The very last part “Some Extra points” also needed correction since one paragraph was totally missing.
Finally I corrected REV2 into an "improved" REV3 here-enclosed since I just discovered to-day some significant inconsistencies definitely requiring corrections. These corrections have been entered just in case somebody might wish to recheck my numbers.
Last but not least, I discovered that you live in the vicinity of Saint Petersburg Russia. Could we privately exchange on this ? My e-mail address is on the enclosed document.
*******
Now let me reply to your question " how far apart samples may be (time-wise) for their (arithmetic) average to be meaningful?".
Using averaged values for both UT and sextant heights is exactly equivalent to performing Linear Regression (LR) and picking up one point of such LR. A well too often forgotten mathematical result shows that the averaged UT and Height Values do belong to the LR line. Hence using averaged values does not waste any information whatsoever while saving the LR computation.
If we continuously plot the successive values of Ha's vs. UT's on a two dimension diagram, we can draw a continuous curve. If we limit such curve to the time span of each set of observations (e.g. 12:40:30 UT - 12:46:33 UT for "⑥ Venus" in the enclosed Document) we obtain some kind of a "bow shaped curve" with more or less curvature. If we also draw the "LR line" on this diagram we can see that such "LR straight line" is somewhat parallel to the "bow curve" general direction with the central part of the "bow curve "most often above/sometimes below the "LR line" and the extreme parts of the "bow curve" most often below/sometimes above the "LR line".
The further away we get from the averaged time value, the farther the "bow curve" starts diverging from the "LR straight line". Aside note : such "bow curvature" is immediately related to the second derivative of H vs. UT. i.e. to d²Ho/dUT².
Treating this on a general stand-point involves 3 variables : (1) Observer's Latitude, and : (2) Body Declination and (3) : Body Local Hour Angle.
Nonetheless this complex problem can be greatly simplified since Math also shows us that the maximum curvature values occur : (1) : At, or close to meridian passages, i.e. at or close from culminations. And (2) : For important culmination heights. Note : The second derivative becomes infinitely negative for culminations at Heights equal to 90°. And finally : (3) For observers on the Equator.
Let us then study such a “worst case” situation involving equally spaced LHA values centered around Meridian passage of a Body with Declination 10° seen from an Observer on the Equator. This specific configuration implies that at Meridian passage UT₀ the culmination Height Hculm is equal to 80°00’0, which is already becoming difficult if not challenging to observe in a Sextant.
UT-UT₀ | -3m36s | -2m24s | -1m12s | 0m00s | +1m12s | +2m24s | +3m36s |
LHA | -0°54.0’ | -0°36.0’ | -0°18.0’ | 0°00.0’ | +0°18.0’ | +0°36.0’ | +0°54.0’ |
H | 79.57.6’ | 79.58.9’ | 79°59.7’ | 80°00.0’ | 79°59.7’ | 79°58.9’ | 79°57.6’ |
Averaged UT : UTₐ = 0m00.0s Averaged H : Hₐ = 79°58.9’ True H value at UTₐ : H = Hculm = 80°.00.0’ Time span of the observations 7m12s Error between true value Hculm and averaged value Hₐ used in the computations : Hₐ - Hculm = -1.1’ Note : this example can also be readily solved on a plan with the “Rectangular to Polar” function |
Hence in the extreme case here-above, the systematic error provoked by the averaging method is equal to 1.1’ over a time span of 7.2 minutes, rounded up as 1’ and 8 minutes. Hint ! The averaging method requires observations more or less evenly spaced. If you were only retaining the extreme values of this example, the involved “time gap” would be too important and would cost you a systematic error of 2.4’. It is much better in this case to treat each observation separately..
As a summary, if we consider that such an “extra maximum systematic 1’ error” is acceptable, we can simply remember that observations should be more or less evenly spaced (no time gap exceeding 2 minutes) over a maximum time span of 8 minutes. |
In your example, the "worst case" happens for "⑧ Venus", in which the maximum error resulting from using averaged values is 0.3' . For "⑥ Venus", the maximum error brought by using averaged values is 0.2' . And for "⑤ Sun", the maximum error is 0.02'
For additional reasons and whenever possible I generally prefer using sets of 5 observations recorded over a 4 to 5 minute time span.
*******
Some Extra points (See enclosed REV3)
1 - Tony, your example is extremely interesting since it shows that one should work from data as close as possible from "raw" data. Dave Walden initially started from your "Ho's" values, but unfortunately some of them have some typos. On my side, I discovered some inconsistencies in REV2, this is why I enclosed a corrected and “improved" REV3 (see 5 here-under), hopefully errors free at last.
2 - I took very special care to best reprocess all your results and have come to the conclusion that your set of observations is truly remarkable. Had I processed them through "approximations" such as from a more distant DR position, it would not have been possible to bring this nice feature to our attention and to your satisfaction.
3 - Giving to each averaged observation a weight equal to its number of observations should yield exactly the same position (to within less than 100 feet) as the position obtained through processing every single individual LOP. I have always observed this in the past. Hence the superiority of averaging and weighting methods in such cases.
4 - In immediate relation to points 2 and 3 here-above it would be quite interesting to compare my Final Fix (N60°10.6' - E 029°48.8') with the one to be obtained by another software using this time the very same DR position as mine (N60°10.5' - E029°46.9') and processing each LOP separately. Very likely such final result[s] should be extremely close from this Final Fix. Peter, Paul, or Andrés, are you here ?
5 - Also, using the Final Fix (N60°10.6’ - E029°48.8’) as DR to process only the observations of Venus as individual LOP’s should put you back almost exactly onto the Venus LAN fix (N60°10.5' - E029°46.9'). In this case, the Venus LOP’s Fix puts me back at less than 0.05 NM from the LAN Fix. This constitutes a good check of the LOP Fix, and thus confirms that this Venus LAN set is excellent.
6 - To conclude I do thank you again Tony for having readily and accurately specified the meanings of "Ha" and "Ho". We constitute an international group, and accurately tagging one's data can save a lot of guessing if not frustration to colleagues. Farewell to all and feel free anybody to privately email me (see REV1,2 or 3) for any extra clarification on the methods I described here-above.
Best Friendly and Easterly Regards,
Antoine M. "Kermit" Couëtte