GPS Vectors, OPUS and OPUS-RS Observations in a Unified Adjustment

 

I recently observed and adjusted a three station network that included both GPS vectors and coordinate observations.  Each station in the network had at least one set of observed coordinates and was connected to the other two stations by GPS vectors.  No individual station was held fixed.  Coordinate observations provided the ties to the national network.  The adjustment process for this network highlights the advantages of including coordinates as observed variables rather than as fixed quantities.

 

Most commercial least squares adjustment software uses a mathematical model known as the “observation model.”  In the observation model the adjusted observations are functions of parameters which, are varied as part of the adjustment computation.  In most cases involving surveying adjustments the parameters are coordinates.  Because of that, the observation model is sometimes referred to as “variation of coordinates.”   This is a misnomer as the parameters to be varied in a least squares adjustment are not limited to coordinates.  Other parameters could include scale and orientation parameters.

 

In adjustment notation the observation model is written,.  Where is a vector of adjusted observations and are the adjusted parameters.  When coordinates are used as observationsand the observations,  are written as with being the individual observed coordinates.

 

In this case of this network the observed coordinates are OPUS and OPUS Rapid Static (OPUS-RS) results.  Although the OPUS-RS website (http://www.ngs.noaa.gov/OPUS/OPUS-RS.html) states that OPUS-RS should be treated as an operational prototype, this network marks my first production use of OPUS-RS.  The key component that makes it possible to use OPUS and OPUS-RS coordinates in a least squares adjustment is the inclusion of the covariance matrix in the OPUS / OPUS-RS extended data report.

 

GPS Observation Scheme

 

A schematic diagram of the network under discussion is included below. 

 

The distances involved are:

 

From

To

Distance

(kilometers)

66

79

1.6

79

90

1.5

90

66

2.9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Two receivers were used to observe this network.  Station 66 was occupied for two hours.  During this time stations 79 and 90 were occupied for 45 minutes each, resulting in two GPS vectors (66->79 and 66->90).  Station 90 was then occupied for two hours.  During this time station 79 and 90 were occupied a second time for 45 minutes each, resulting in two more GPS vectors (90->79 and 90->66).   These session lengths are much longer than required however I needed the two-hour sessions to submit to OPUS for stations 66 and 90.  The 45 minute sessions ensured that the vectors would be well determined.  The closed loop and the repeat vector from station 66 to 90 provide adequate redundancy.

 

Processing the GPS Data

 

After waiting for the release of the rapid ephemeris, the two hour data files for stations 66 and 90 were submitted to OPUS.  The two 45 minute sessions were windowed using TEQC from UNAVCO into six 15 minute sessions.  The six 15-minute sessions were submitted to OPUS-RS.  The OPUS and OPUS-RS results can be examined here:

 

OPUS: Station 66 Station 90

OPUS-RS: Station 79 B1, Station 79 B2, Station 79 B3, Station 79 C1, Station 79 C2, Station 79 C3

 

The alphanumeric combination after the station for the OPUS-RS results represents the GPS session and 15-minute segment within the 45 minute observation time span.  The OPUS-RS results from the same session are strictly speaking not independent results.  They share a common setup and very similar satellite constellation.  Session B and C are independent sessions with different antennas occupying the station with a 36 minute time difference from the end of windowed session B3 to the start of windowed session C1.

 

The coordinates reported in the OPUS solution were used as seed coordinates to process GPS vectors using the WAVE processor in Trimble Geomatics Office (TGO).  The entire 45-minute observation sessions were used to process the baselines.

 

 

Although the reference variance values are a bit larger than ideal, the ratio and RMS values are very good.  The difference in magnitude between the repeated vectors is only 3 millimeters.  The vectors along with their covariance matrices were exported from TGO to be adjusted using Columbus Network Adjustment Software.

 

Network Adjustment

 

A minimally constrained adjustment of the four vector network was performed using Columbus.  Setup errors were modeled at 1.5 millimeters horizontally and vertically.  This adjustment passed the chi square test without scaling the adjustment covariance matrix.  This is unusual.  The Columbus processing summary is included below.

 

 

OPUS Coordinate Observations

 

The next step is to add the coordinate observations from the two OPUS results.  Columbus makes this very simple.  Not only does Columbus now directly support coordinate observations but Columbus will read an OPUS or OPUS-RS report and convert the data into Columbus format.

 

 

If your least squares adjustment software does not support coordinate observations, the methods detailed at this site can be applied.

 

After importing the OPUS report the coordinates from the OPUS results and the covariance matrix are included as observations in the Columbus document.  Below is the Geodetic Coordinate Observation Set dialog box.  The covariance matrix is read directly from the OPUS extended data report.

 

 

GPS vector covariance matrices are notoriously optimistic.  An OPUS point solution covariance matrix being derived from three GPS vector covariance matrices is also, as a rule, optimistic.  The preferred method to objectively determine the scale factor for an OPUS covariance matrix is to combine multiple OPUS solutions, for the same station, in a least squares adjustment and use the a posteriori variance of unit weight from the least squares adjustment as the scale factor.  In this case there is only one OPUS solution at each station so an alternate method is used. 

 

Including the OPUS coordinate observations in the adjustment causes the chi square test to fail with an a posteriori variance of unit weight of 4.30.  The processing summary is included below.

 

 

Since the minimally constrained adjustment passed the chi square test, the OPUS coordinate observations have to be the cause of the adjustment failing the chi square test.  Failing the chi square test on the high side is indicative of the covariance matrix being optimistic as suspected.  Columbus tabulates the residuals, degrees of freedom and a posteriori variance of unit weight by observation type.  Looking at the geodetic coordinate observations the a posteriori variance of unit weight is 7.98.  Based on experience, this is not an unreasonable scale factor.  Scaling the two OPUS coordinate covariance matrices by the same scale factor is also reasonable.  Both stations were observed under similar conditions for the same session length and tied to the same CORS.  Scaling the covariance matrices of the coordinate observations by 7.98 and readjusting gives the following results:

 

 

The adjustment now passes the chi square test.  The sum of the square of the weighted residuals for the GPS observations is 8.9.  This is slightly larger than the value for the GPS only observations in the minimally constrained GPS only adjustment.  This indicates that the GPS vectors are absorbing slightly more of the residuals than in the GPS only adjustment.  The coordinate observations are exerting an influence in the estimation of the adjusted coordinates.

 

Advantages of Coordinate Observations

 

If the OPUS coordinates for stations 66 and 90 were held as fixed values, rather than as observations subject to adjustment, the two vectors from station 66 to station 90 would be redundant.  Being constrained between two fixed coordinates, the GPS vectors between stations 66 and 90 would have adjusted to conform to the inverse between the fixed coordinates; these two vectors would not have contributed to the network and rightly should have been removed from the adjustment.  Treating the OPUS coordinates as observations allows the two GPS vectors to contribute to the determination of the coordinates for stations 66 and 90.

 

In addition, using the OPUS coordinates as observations, allows for a more accurate estimate of the external accuracy of the network with reference to the national network.  All coordinates have some associated error.  Treating the OPUS coordinates as observations rather than fixed values reflects that reality.

 

OPUS-RS Coordinate Observations

 

There still remain the six OPUS-RS solutions for station 79.  Since there are six solutions we can combine them into a single solution in a least squares adjustment using the coordinates as observations.  The a posteriori variance of unit weight of that adjustment can then be used to objectively scale the OPUS-RS covariance matrices to more realistically estimate the observation errors.  Below is the adjustment summary:

 

 

There are 18 coordinate observations included in this adjustment, three for each OPUS-RS solution.  One OPUS-RS solution provides sufficient observations to uniquely define the coordinates of station 79 in three dimensions.  The other five OPUS-RS solutions provide the 15 degrees of freedom.  After scaling the covariance matrix by 4.58 the adjustment passes the chi square test with an a posteriori variance of unit weight of 1.000.

 

Final Combined Adjustment

 

We can now include the OPUS-RS coordinate observations with the OPUS coordinate observations and GPS vectors.  In this unified adjustment the covariance matrix for the GPS vectors will be unscaled, the covariance matrix for the OPUS coordinate observations will be scaled by 7.98, and the covariance matrix for the OPUS-RS observations will be scaled by 4.58.  The adjustment summary is included below.

 

 

This adjustment passes the chi square test indicating that the residuals are consistent with the assumptions made.  In this adjustment the sum of the squares of the weighted residuals for the GPS observations is 9.9.  This is greater than either of the two previous adjustments that include GPS vectors.  This indicates that the GPS vectors are absorbing a still greater share of the adjustment than before.  The coordinate observations are exerting greater influence on the adjusted coordinates for the network stations.

 

The final adjusted coordinates with standard deviations, in meters, are:

 

 

Conclusion

 

OPUS-RS allows centimeter level accuracy with observation spans as short as 10 to 15 minutes (Kashani et al, 2005).  With repeat occupations, as part of a normal network observation scheme, multiple OPUS-RS solutions can be included in a network adjustment to add strength to the adjustment. 

 

Kashani, Wielgosz, Grejner-Brzezinska, Mader, A New Network-Based Rapid-Static Module for the NGS Online Positioning User Service – OPUS-RS, ION 61st Annual Meeting, 2005