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    Re: Venus mer-pass for Lat, Sun time sight for Lon.
    From: Antoine Couëtte
    Date: 2020 Apr 13, 10:13 -0700

    Hello Tony,

    Yes, your set of Venus and Sun Observations is really excellent ! Well done again !

    Your enthusiasm convinced me to continue our exchanges. After hesitations prompted by some adverse environment I eventually thought that you do deserve a comprehensive reply on NavList. 


    Accordingly, and since I had time, I re-computed all your observations twice and updated my previously published REV1 document into REV 2 document here-enclosed.

    First : I treated your 15 Venus Observations on their own as a LAN through duly considering that your published "Ha" pertain to Sextant Heights corrected for both Instrument Errors and for Horizon Dip. This final updated position (N60°10.5' - E029°46.9' with Transit Time at 12:56:45.0) is only 0.8 NM away from your GPS position (N60°10.3 E029°48.5'), which is an absolutely remarkable achievement for a LAN set of observations.

    Second : from this updated Venus LAN position which I used as a DR position, I recomputed all Venus + Sun observations as classical Marcq de Saint Hilaire's LOP's and then processed them as earlier (averaged sets with weights equal to their number of observations). Compared to REV1, my REV2 Final Fix (N60°10.6' - E 029°48.8') has remained unchanged since this method is so insensitive to such a small change in the DR position. Likewise my "Hg (AMC)" values in REV2 have remained 100% unchanged since they are derived from your same "Ha's". 


    Now let le 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. As a well to often forgotten mathematical result : the averaged values, both for UT and Height do belong to the LR line. Hence using averaged values does not degrade information and it also avoids computing the LR.

    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 " 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.

    To answer to your question, we now just need to find the maximum elapsed time reckoned from the averaged time at which the averaged distances between the "LR line" and the "bow curve" reach some acceptable difference.

    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.

    If one agrees that the acceptable averaged absolute difference between the "bow curve" and the "LR line" should not exceed 1 arc minute for instance, then for heights equal or inferior to 80°, the worst case happens with : Dec 10°, LHA = 1°, Lat = 0° (with culmination height = 80°). LHA absolute value equal to 1° means that meridian passage time is 4 minutes away. We should also remember that culmination heights reaching or exceeding 80° become quite difficult and sometimes challenging to accurately observe.

    Hence for Heights below 80° using averaged values during a time span of 8 minutes will result in an extra error equal to 1 arc minute at the most. Whenever you stay away from such extreme case, the errors resulting from using averaged values stay negligible most of the time.

    In your example, you "worst case" happens for " Venus", in which the maximum error resulting from using averaged values is 0.3' . For  " Venus", the maximum error brought by using averaged values is 0.2' . And for " Sun", the maximum error is 0.02' .

    For additional reasons and whenever possible I generally prefer using sets of 5 observations recorded over a 5 to 6 minute time span.


    Some Extra points:

    1 - 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.

    2 - Since we have time here, I took good care to reprocess all your results and have come to the conclusion that your set of observations is absolutely remarkable. Processing it through "approximations" such as from a more distant DR position would not have brought this to our attention and to your satisfaction.

    4 - In immediate relation to point 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 the same DR position as mine (N60°10.5' - E029°46.9') . Very likely such final result[s] should be extremely close. Peter, or Andrés, are you here ?

    5 - Giving each averaged observation a weight equal to its number of observations should yield exactly the same position (to within less than 100 feet) as if had computed every LOP on its own. I have always observed this in the past. I am simply short of time to once again process all your observations individually this time. Hence the superiority of averaging and weighting methods in such cases.

    6 - To conclude I do thank you Tony for having readily and accurately specified the meanings of "Ha" and "Ho". Again, 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 or REV2) for any extra clarification on the methods I described here-above.

    Best Friendly and Easterly Regards,

    Antoine M. "Kermit" Couëtte

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