This is something a bit out of the ordinary. The table below consists of the results of observations that Major William Emory of the U.S. and Mexican Boundary Commission made in July and August of 1852 at a monument on the U.S. side of the Rio Grande River opposite presentday Ojinaga, Chihuahua.
Major William Emory
Major Emory observed the zenith angles of pairs of stars transiting his meridian, one star North of his station and one South, selected so that the altitudes of the stars were approximately equal in order to try to cancel the effects of atmospheric refraction in the value of the station latitude calculated from them.
The value that Emory adopted for the latitude of the station at which he made his observations was:
29°34'07.13" N
For the sake of compactness, the values I've tabulated are just the seconds fractions of the latitudes that Emory derived from each pair of stars on each night, a total of 122 separate estimates of the latitude of his station.
Here is a link to a pdf of the above table that is a bit easier to read.
Table of Emory's Observed Latitudes (10k pdf)
When Emory left on the extended field work that the survey and mapping of the boundary between the U.S. and Mexico required, he took along the best star catalogue available to him at the time. One of the provisions of the treaty under which the commissioners representing the United States and Mexico worked was in effect that the commissioners were to make their best efforts to determine the latitudes and longitudes of points to be fixed as the boundary and by mutual agreement to declare their determinations to be the latitudes or longitudes contemplated by the treaty.
Later, however, after the fieldwork was completed, Emory was able to use an improved star catalogue subsequently published and correct the latitude determinations he had made while in the field engaged in the work.
The net result of using the improved values of the declinations of the stars in some of the pairs that Emory had observed was to correct the latitude values for a particular pair of stars by the quantity in the "Decl. Corr." column of the table above.
Those same declination corrections were applied to the results of the star pairs that went into computing the means of each night's work. I've listed those corrections in the row labeled "Decl. Corr."
Note that each night a different number of star pairs were observed. Note also that some of the stars evidently did not have improved values of their declinations published in the later catalogues that Emory used to revise his calculations. These are the pairs for which a declination correction of 0 is listed.
So, the interesting question is what the best estimate of the latitude of the station opposite Ojinaga would be from Emory's 122 observed latitude values for the pairs of stars on various nights and what the estimated uncertainty of that estimate would be.
The way that I think this problem needs to be worked is to recognize that the observations reported by Major Emory fall into fundamentally two different groups.
The first groups are those made on star pairs 1, 6, 10, 11, 13, 14, 15, 16, 17, 19, 19, and 22. These are the observations that Emory subsequently corrected using an improved star catalogue that he didn't have available when he was in the field.
The second group consists of the remaining observations for which Emory had to rely upon the earlier catalogue for the apparent places of the stars observed.
The two groups of observations should have significantly different characteristics that are directly attributable to the accuracy of the source of declinations that were used in reducing the observations of the pairs of stars.
The way that I'm proceeding to work this problem is to consider the two groups separately and:
- estimate the standard deviation of a single night's latitude observation from a star pair, exclusive of ephemeris errors, i.e. just considering the scatter of results on the same star pair over a series of nights,
- identify any outliers that ought to be tossed out.
- estimate the uncertainty in the corrections from the "improved" star catalog that Emory used, relying upon analysis of variance alone.
- form the weighted mean of each group,
- estimate the uncertainty of the mean,
- decide how to combine the results of the two groups (or if they should be combined).
This looks like a feasible approach, subject to several considerations. It might be a little over-thought for the value of the tweaks you can make to the data. If you are trying to think of everything, be sure you have considered the following issues. You may have to make a best guess on some of them.
-Several pairs do not have enough data to make a reliable estimate of the variance for that pair. Is there a theoretical reason why it would be different for different pairs? Maybe a model with variance of sighting modified by trig functions of altitude angle and azimuth, and use that formula to weight each and all observations?
-To look for outliers, you will have to consider the set of all the observations, because there aren't enough for most star pairs to be considered individually.
- estimate the uncertainty in the corrections from the "improved" star catalog
Not sure how that would be done. Can you elaborate?
It seems to me that the corrections (std dev 1.17 sec) are indicative of the uncertainty in the first catalog, assuming the second catalog is really significantly better.
Is it safe to assume a correction of 0 means no correction was available, or might the new catalogs actually have verified the same value for one?
- form the weighted mean of each group ... combine results
Is there a bias between groups? The mean non-zero correction is -0.32 second.
If not, why not just use the appropriate weighting factor on each value and not process them in groups?
Kent: have you done a comparison by computing an "astronomic" latitude/longitude from geodetic positions and deflec09?
I thought of DEFLEC also, but didn't bother to figure out where Kent was going with this computation. I think the other thread included the original estimate of lat/lon for this point.
There is no way you are going to tweak his reduction of observations to do any better than saying the IBC point does or does not seem to agree within the observational error.
The main problem in that thread seems to be how to relate the old estimates of difference between astronomic positions for the church and the astro station to present day difference of positions. To do that it is hardly relevant how the old star observations were processed or could best be processed. What is important there is how the guy did the computation of relative position.
Several Considerations.
In reviewing the observations I see that the Aug. 12 observation for star pair 22 is almost 2" from the mean. That is a clear outlier and should be discarded. Some of the other pairs have large variances but no other outliers stick out. It is probably more difficult to get as good an observation on a lesser magnitude star.
I see no problem with the smaller sets if one weighs the means based on number of observations with some discount of the value for pairs with larger variance. The single observations of pairs 2 and 5 on July 18 look like outliers from the day's mean, but once the individual declination corrections are subtracted the data appears as normal as a single set can be.
Based on 1852 knowledge of the atmosphere a North South pair based on equal angles from the ecliptic to balance out refractive corrections is acceptable. Today we know that the atmosphere is not a sphere but is considerably thicker at the equator. This affects GPS signals because of the depth of atmosphere to be traversed but also affects refraction because the atmospheric level surfaces are not normal to one's zenith.
I surmised the time gap in the middle of the observations was due to Full Moon. I checked with the Naval Observatory and in fact there was a Blue Moon on July 31, following a totally eclipsed Full Moon on July 1.
Paul in PA
> -Several pairs do not have enough data to make a reliable estimate of the variance for that pair. Is there a theoretical reason why it would be different for different pairs? Maybe a model with variance of sighting modified by trig functions of altitude angle and azimuth, and use that formula to weight each and all observations?
Yes, I've tossed the observations for pairs 2 and 5 out immediately. There is only one observation on one night for each and there is plenty of data remaining without them.
> -To look for outliers, you will have to consider the set of all the observations, because there aren't enough for most star pairs to be considered individually.
Yes, my approach is to:
1) apply the declination corrections to all of the observations, not just the means.
This should reduce the bias between means of the observations for the star pairs (it does).
2) calculate the standard error of the series of observations made on each particular pair over various nights and then make a pooled estimate from them. The pooled estimate is s = 0.74" for the standard error of a single observation, exclusive of the contribution of ephemeris errors.
3) calculate the standard error of all 66 observations as if they were from the same population, for all 66, s = 1.35"
4) estimate the contribution of the remaining ephemeris errors from the last two variances, SQRT[1.35^2 - 0.74^2] = s = 1.13"
I don't see any reason why the second group of observations reduced by Emory in the field with the less accurate ephemeris couldn't also be used to produce a pooled estimate of the standard error of a single observation, exclusive of the contribution of ephemeris errors and use it as well in (2) above.
> - estimate the uncertainty in the corrections from the "improved" star catalog
> Not sure how that would be done. Can you elaborate?
Just by the difference in the variance of the whole population as in (3)less the variance of a single observation as in (1)
> It seems to me that the corrections (std dev 1.17 sec) are indicative of the uncertainty in the first catalog, assuming the second catalog is really significantly better.
I think that way I'd estimate the uncertainty of the declinations in the first catalog would be by (4) above, i.e. just subtracting the variance of an observation independent of declination errors (the pooled estimate from the standard errors of the series on the same pair) from the variance of all observations.
> Is it safe to assume a correction of 0 means no correction was available, or might the new catalogs actually have verified the same value for one?
I think it isn't reasonable to think that new observations at an observatory produced exactly (to the nearest 0.01 arc-second) the same declinations for both stars. That is what makes me conclude that it is more likely that the declinations of those stars weren't based on new observations, but were carried over from the old catalog that Emory had in the field.
> Kent: have you done a comparison by computing an "astronomic" latitude/longitude from geodetic positions and deflec09?
Yes, I have, but I've been trying not to look at it in order to keep from steering the analysis of Emory's work.
Several Considerations.
The two outliers that I've rejected in the first group of observations (reduced with the improved ephemeris data) are both from August 12, those on Star Pairs 15 and 22.
Several Considerations.
Your explanation of determining the variances makes sense.
The older ephemeris data for star pair 9 looks like an outlier, and so that pair's data might be rejected by some criteria.
My first run at an average of the last column of the table did not quite match the latitude given in the other thread. Do you have a method that matches his result?
Star Pair 15
I would not consider rejecting the 1.64 low for Aug 12 without also rejecting the 1.20 and 1.44 highs on July 16 and 21. Overall it is a larger spread than most but that may be due to decreased intensity of the stars making precise observation more difficult.
I agree, in looking at all the data for pair 9, that in some way the published data for one or both stars was off, so reject the whole set.
Paul in PA
Star Pair 15
> I would not consider rejecting the 1.64 low for Aug 12 without also rejecting the 1.20 and 1.44 highs on July 16 and 21. Overall it is a larger spread than most but that may be due to decreased intensity of the stars making precise observation more difficult.
The method that I'm using for determining rejection limits is to use the estimate of the variance taken as described above by pooling from the variance of each series on a star pair to eliminate the influence of ephemeris errors and to use that overall value to form the rejection limits from the mean of the series on a particular star pair over various nights.
My Calculation Points To The Steeple
I rejected all observations for pair 9 and the Aug 12 observation for pair 22. I took each corrected mean weighted by the number of pair observations and came up with 7.33". That puts the original point 0.21" North or 35' which puts the Church coordinates on the tower. Without a distinguishing tower in place it is unlikely he would have made an observation.
Interesting exercize.
Paul in PA
just curious
What is the difference between where a simple mean would place the station and where the station would be located based upon a detailed statistical reckoning? Is there corner evidence that exists from surveys based upon your location in question? Did they use the same instrument and crew on each trip? Would you or some knowledgable responder to this thread walk the rest through the math.
Thanks very much for your help
My Calculation Points To The Steeple
>Without a distinguishing tower in place it is unlikely he would have made an observation.
Actually, what is most likely is that there was a wooden cross at the peak of the roof. There was, of course, no bell tower in 1852. The bells were on a rack in front of the church. I believe there is a mission church in California that may have a similar arrangement. The direction from the Astro Point to the church is nearly on axis with the church, so the Mexican commissioner, Mr. Salazar, who carried out the triangulated tie could have picked nearly any object on axis and it would fit.
Keep in mind that the *relative* positions of the Astro Point and the "Cathedral" are mostly independent of any small revisions to the latitude of the Astro Point. The question is whether Major Emory's observations also support the now destroyed Boundary Commission monument "RP 27 A" as being in the position of Emory's monument, as it is reported to be on the NGS datasheet.
just curious
> What is the difference between where a simple mean would place the station and where the station would be located based upon a detailed statistical reckoning? Is there corner evidence that exists from surveys based upon your location in question? Did they use the same instrument and crew on each trip? Would you or some knowledgable responder to this thread walk the rest through the math.
Sure, I'll describe the calculations in a bit more detail when I get a chance. This was just to block out the method of solution to be pursued.
BTW thanks
These types of posts are really terrific, something that many will never run across in their own practice, but well worth following.
Thanks again