[teqc] the three corrections to teqc's qc antenna position calculation in the next release

Lou Estey lou at unavco.org
Tue Mar 13 13:56:48 MDT 2018


Richard, Geoff, and others:

Ok, I'll try this again, correcting the one error in my explanations and showing
the effects of the first two qc position corrections in teqc for the next release.

The main corrections added in early 2017 were:

- 1) the relativistic eccentricity correction to each SV position calculation,
     e.g. proportional to "r dot v" of the SV's position and velocity at the time
     of signal transmission: 20.3.3.3.3.1 of the IS-GPS-200 (*), or covered in
     https://www.govinfo.gov/content/pkg/GOVPUB-C13-83ec647d39931e27e1a786845bb825c2/pdf/GOVPUB-C13-83ec647d39931e27e1a786845bb825c2.pdf
     Marc Weiss and Neil Ashby, "Global Positioning System Receivers and Relativity",
     NIST Technical Note 1385, 1999, p. 10 (PDF p. 16), Eq. 5 and Appendix A
     pp. 35-35 (PDF pp. 41-42),

- 2) accounting for the rotation of the Earth during the travel-time of signals
     from each SV to the antenna, which is on the order of 0.1 second (down
     to about 0.064 seconds for a GLONASS SV overhead, 0.11+ seconds for a
     Galileo SV near the horizon, etc.), and

- 3) accounting for the time differences between the different time systems
     used by different SV constellations (when delta UTC model information is
     is available).

Correction (3) only comes into play when there are data and nav information
for more than one constellation.  But when the different time systems for the
different constellations are reasonably close (e.g. on the order of nanoseconds),
this correction is negligible (since, say, GPS SVs orbit at about 3.6 km/s,
which are between GLONASS and Galileo SVs in orbital height).  But this correction
is in place in the code now, just in case, assuming the nav information contains
up-to-date UTC model parameters for the constellations being done.  (Neglecting
leap seconds, the UTC model parameters are used to compute a proxy common time base
to use for the different constellations.)

Correction (2) normally has the largest effect for most sites, except those
at very high latitudes.  I've known this correction was missing for over two
decades, but there were simply too many other things in teqc to worry about and
my reasoning for ignoring it was: The qc in teqc was to replace UNAVCO's original
QC program, and that program ignored the Earth's rotation.  The goal then was
to quickly get the antenna position to within about 100 meters to get on with
the rest of the qc of the observables.  With a signal travel-time of 0.1 second,
ignoring the Earth's rotation would give an offset of about 40 meters at the
equator, or about 40 m * cos(latitude) elsewhere -- well within the 100-meter
target.  The effect of ignoring this correction can readily be seen in a set
of our 1600+ daily solutions (as explained on Friday, comparing the qc solutions
to our "visit" database values, some of which are in error) in the first attachment
(no_Earth_rotation.txt).

You can see that solutions get smeared out to the east (because the Earth is
not being rotated east to the correct location) and are mostly 20+ to 35 meters
to the east from the correct position, about as one might expect.  (The qc
solutions that are still fairly close to being correct when ignoring the Earth's
rotation in this set are probably mostly from polar sites, but this guess
was not verified.)  In hindsight, this rotation correction is trivial to include,
so it really should have been done years ago.  (In fact, I'd periodically get
an email from a teqc user who would point out the teqc solutions were
statistically to the east of where they should be and they'd either suggest
why or I'd explain to them why.  It probably would have quicker to deal with
this issue long ago and have been done with it.)

After getting the corrections (2) and (3) in place last year, correction (1)
was the next one I really wanted to nail down, but as stated on Friday, doing
so was not possible with the original treatment of interpolating GLONASS SV
positions in teqc, because one either needed certain Keplerian orbital parameters
(none of which are not broadcast for GLONASS or SBAS) or needed both simultaneous
positions and velocities of each SV.  Hence, including correction (3) had to wait
until I had replaced the GLONASS position interpolation methodology (which had other
shortcomings anyway) with the ICD numerical integration of position, velocity,
and acceleration of the SV equations of motion.  (And the same numerical
integration methodology was then also applied to SBAS positions.)  The second
attachment, no_eccen_corr.txt, shows the effect of restoring correction (1)
but now ignoring the relativistic eccentricity correction to our daily batch
of 1600+ sites.  Again, the qc positions are smeared out to the east, mostly
in the 1.25 - 6.25 meter range.  So, as guessed beforehand, this is a smaller
effect than ignoring the Earth's rotation, but still pertinent.

For completeness, I've attached a third histogram showing the effect of
ignoring both corrections (1) and (2), ignoring_both.txt, for our 1600+ daily
site solutions.  As one might expect, the smearing of positions to the east
is roughly additive when missing both (1) and (2).  This also demonstrates
that all versions of teqc from 2016 Nov 7 or earlier would have had qc solutions
probably be off by 0-50 meters to the east (with north-south scatter) and roughly
be 14.7-16.7 meters too high.  Teqc versions 2017 Jul 3, 2017 Sep 1 and 15, and
2018 Jan 11 have various incomplete (or buggy) versions of implementing
corrections (1) and (2).

At any rate, Richard and Geoff, sorry about accidentally calling correction (1)
the Sagnac relativistic correction (see * below) and for giving the wrong Ashby
reference on Friday afternoon.

As for correction (1), the first and only derivation of it I've been able to
locate is:
https://ia800108.us.archive.org/23/items/DTIC_ADA154620/DTIC_ADA154620.pdf
L. Ralph Gibson, "A Derivation of Relativistic Effects in Satellite Tracking,"
Naval Surface Weapons Center TR 83-55, (April 1983),
pp. 3-7 (PDF pp. 9-13) Eqs. 6 - 20 and Appendix A, p. 12 (PDF p. 18).
Gibson calls it the "periodic relativistic effect", I suppose because of the
symmetry of the effect about the apogee and perigee of the SV's orbit. (Note
that "r dot v" of the SV is zero at both the apogee and perigee.)  Gibson also
credits the equivalent "r dot v" version of the correction to a derivation by
John T. Carr of the Space and Surface Systems Division, Strategic Systems Dept.

Two points about correction (1), the first mentioned, e.g. in 20.3.3.3.3.1 of the
IS-GPS-200:

(a) "It is immaterial whether the vectors R_vector and V_vector are expressed
in earth-fixed, rotating [i.e. ECEF] coordinates or in earth-centered, inertial
[i.e. ECI] coordinates." ... because the result is only a scalar dot product of
the r and v vectors times a constant.

The second point I've only found in one reference:
http://tycho.usno.navy.mil/ptti/1996papers/Vol%2028_16.pdf
Henry F. Fliegel and Raymond S. DiEsposti, "GPS and Relativity: An Engineering
Overview", 1996.  On p. 193 (PDF p. 5), after Eq. 10:

(b) "[The relativistic eccentricity correction] is appropriate for users on or
near the earth's surface, but not users in space, who should apply the frequency
correction equations given above [in the paper], or their integrals to transform
to the time domain."

... although I cannot find anything in Gibson's derivation that seems to
confirm this statement.  Perhaps someone would like to address this point.

At any rate, with the next teqc release I'm certain that for daily data files
you'll all see a substantial improvement in the qc positions, with a 1-sigma
in the horizontal plane of about 2.5-3 meters, although the qc positions will
still be systematically high by about 14.7-16.7 meters (but probably more
typically, on average, high by 15.4-16.0 meters).

cheers,
--lou

* What a comedy of errors!  At one point last year I was actually looking at
including with correction (2) the additional relativistic Sagnac correction term,
which is a different "r dot v" term entirely.  Apparently, in the code, I latter copied
part of a comment which was correctly associated with the Sagnac correction term
to the relativistic eccentricity correction, even the though the rest of the comment
for the eccentricity correction referenced 20.3.3.3.3.1 of the IS-GPS-200.  Then on
Friday afternoon, tired, I went looking for the Ashby paper with the "r dot v"
Sagnac correction, and thus cited the wrong paper.  Richard, Geoff: little wonder
you were baffled!

-------------- next part --------------
 * qc elevations offset by -15.7 meters

  QC Histogram  N = 1646
distance   count  c/100m^3
--------   -----  --------
<  2.5 m      46     70.3
<  5.0 m      48     10.5
<  7.5 m      24      1.9
< 10.0 m      26      1.1
< 12.5 m      21      0.5
< 15.0 m      25      0.4
< 17.5 m      50      0.6
< 20.0 m      54      0.5
< 22.5 m      50      0.4
< 25.0 m     102      0.6
< 27.5 m     353      1.6
< 30.0 m     564      2.2
< 32.5 m     142      0.5
< 35.0 m     112      0.3
< 37.5 m      26      0.1
< 40.0 m       1      0.0
< 42.5 m       1      0.0

        Quadrant distribution:
(+|- = qc above|below visit elevation)

       up =  910                dn =  736
          21                       16
     \    N    /              \    N    /
        \ + /                    \ - /
   12 W+  x  +E  872        12  W- x -E   695
        / + \                    / - \
     /    S    \              /    S    \
           5                       13

      22  |   714              16  |   495
      NW+ | +NE                NW- | -NE
     -----+-----              -----+----
      SW+ | +SE                SW- | -SE
       5  |   169               6  |   219 
-------------- next part --------------
 * qc elevations offset by -15.7 meters

  QC Histogram  N = 1637
distance   count  c/100m^3
--------   -----  --------
<  1.2 m      92   1124.5
<  2.5 m     304    530.8
<  3.8 m     398    256.0
<  5.0 m     402    132.8
<  6.2 m     184     36.9
<  7.5 m      80     10.7
<  8.8 m      56      5.4
< 10.0 m      36      2.6
< 11.2 m      31      1.7
< 12.5 m      25      1.1
< 13.8 m       7      0.3
< 15.0 m       4      0.1
< 16.2 m       0      0.0
< 17.5 m       2      0.0
< 18.8 m       0      0.0
< 20.0 m       1      0.0
< 21.2 m       0      0.0
< 22.5 m       0      0.0
< 23.8 m       0      0.0
< 25.0 m       2      0.0
< 26.2 m       0      0.0
< 27.5 m       1      0.0
< 28.8 m       1      0.0
< 30.0 m       5      0.0
< 31.2 m       1      0.0
< 32.5 m       1      0.0
< 33.8 m       1      0.0
< 35.0 m       1      0.0
< 36.2 m       1      0.0
< 37.5 m       1      0.0

        Quadrant distribution:
(+|- = qc above|below visit elevation)

       up = 1097                dn =  540
         215                      207
     \    N    /              \    N    /
        \ + /                    \ - /
   69 W+  x  +E  694        50  W- x -E   234
        / + \                    / - \
     /    S    \              /    S    \
         119                       49

     121  |   620              99  |   332
      NW+ | +NE                NW- | -NE
     -----+-----              -----+----
      SW+ | +SE                SW- | -SE
      58  |   298              35  |    74
-------------- next part --------------
 * qc elevations offset by -15.7 meters

  QC Histogram  N = 1638
distance   count  c/100m^3
--------   -----  --------
<  2.5 m      20     30.6
<  5.0 m      41      8.9
<  7.5 m      24      1.9
< 10.0 m      38      1.6
< 12.5 m      24      0.6
< 15.0 m      22      0.4
< 17.5 m      44      0.5
< 20.0 m      56      0.5
< 22.5 m      49      0.3
< 25.0 m      28      0.2
< 27.5 m     210      1.0
< 30.0 m     330      1.3
< 32.5 m     405      1.3
< 35.0 m     231      0.6
< 37.5 m      83      0.2
< 40.0 m      28      0.1
< 42.5 m       4      0.0
< 45.0 m       1      0.0

        Quadrant distribution:
(+|- = qc above|below visit elevation)

       up = 1040                dn =  598
          11                        7
     \    N    /              \    N    /
        \ + /                    \ - /
    9 W+  x  +E 1009         5  W- x -E   579
        / + \                    / - \
     /    S    \              /    S    \
          11                        7

      11  |   630               7  |   428
      NW+ | +NE                NW- | -NE
     -----+-----              -----+----
      SW+ | +SE                SW- | -SE
       5  |   394               2  |   161


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