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    Analysis of Meridian Transit Sights
    From: David Fleming
    Date: 2017 Mar 31, 20:21 -0700

    Recentlly Frank presented data for a series of Noon sights and asked for a discussion of how to analyze such data.  In that spirit I dreged up an old set of data.  The data and analysis of that data are in the attached Open Office spreadsheet.

    There are three methods investigated on separate sheets within that document.

    The first sheet (Data), is the classic method used by many to analyze Frank's data.  Plot the raw data, look at the plot and read a time and Ho off the plot and convert to a Longitude and Latitude.  For the case however of a sun with changing Dec of a moving observer, we seek not the point on the curve ( Ho versus time ) tangent to a horizontal line, the peak of the curve but rather the point of tangency to a sloped straight line were the slope is determined by the rate of change of the difference between Lat and Dec.  In the specific example in the spreadsheet the Latitude obtained by this proceedure agreeded with GPS DR within .3 moa and the Longitude was 6 moa off.  This first order analysis clearly is adequate for Lat but we might like to do better in Long.  Hence a second worksheet.

    It presents a slope method to try and capture the time of Meridian transit by utilizing more of the data points.  This method I saw I'm not sure where.  Perhaps Ocean Navigator or maybe here at NavList.  My appologies to whoever devised it.  First looking the graph of data in first analysis identify any pints that appear out of place due to mistakes rather than noise and eliminate those points.  Then determine a new curve  given by the difference in Ho values for successive sights divided by their time differences, ie digitally differentiate the data and associate the slope with the midpoint time.  For a noiseless parabolic curve this will produce a straight line that intersects the time axis at the peak of the curve.  With noise we fit the scattered data with a straight line and look for the time value where the fitted curve has the slope characteristic of the changes in. Dec and Lat.  That is our estimate of MT time based on all the data utilized.  For the case in the spreadsheet the Longitude differed from GPS DR by 2.5 moa.  Some improvement over the graphical method, worth the effort?

    The final spreadsheet takes same data used in the slope method and does a Least Mean Square Fit to a parabola.  The curve of Ho versus time are of course only parabolic near peak hence the limited time for ex meridian sights and also changing Dec and Lat though that is small compared to changes in Ho with time ( not counting airplanes and noboy in an airplane would do this stuff anyway).  For a subset of the data close the peak, I found a Longitude difference of 7.5 moa and a latitude difference of .2 moa from the GPS DR values. Taking all the data into account I found a Longitude difference of 2moa and a Latitude difference of 1moa from the GPS DR.  The data are not actually parabolic and that is apparent the further we get from the peak.  So for good Latitude use data near the peak.  For Longitude you need data from wide range of times involving a wide range of azimuths.

    This is just one example of several analysis of a set of data.  It does not get to a comparative error analysis for these methods.  Right now I leave that to others. 



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