With the recent additions to the extended output I would like to suggest a method to combine multiple OPUS solutions into a single solution with quantifiable precision. This method will also allow one to combine the OPUS output with other measurements in a unified solution.
The new OPUS extended output includes the G file for each individual PAGES solution. This G file contains the GPS vector information along with the standard deviations for the individual components and the correlation matrix. Many least squares adjustment programs can read a G file and incorporate the data into an adjustment. Here is an example of using the extended output in a least squares adjustment to combine OPUS solutions. Columbus Network Adjustment software was used in this example, but any adjustment software that uses a 3d geodetic model and will accept an NGS G file should work.
This example uses National CORS NLIB as the unknown station. NLIB is also an IGS station so it has well defined coordinates referenced to both ITRF00 and NAD83. This experiment was done using NAD83 coordinates and vectors. The published NAD83 coordinates for NLIB, transformed to April 5, 2004 are:
Latitude Longitude h
41 46 17.70041 91 34 29.59226 208.078
Four hours of data were downloaded for April 4, 5, 6, 2004 and submitted to OPUS for processing. The following NAD83 results were returned:
Day Latitude Longitude h dN dE dh
95 41 46 17.70051 (0.020) 91 34 29.59230 (0.026) 208.080 (0.064) 0.003 -0.001 0.002
96 41 46 17.70028 (0.007) 91 34 29.59241 (0.018) 208.094 (0.016) -0.004 -0.004 0.016
97 41 46 17.70027 (0.007) 91 34 29.59258 (0.015) 208.103 (0.014) -0.004 -0.004 0.025
Numbers in parenthesis are the peak to peak spread and the dN, dE and dh are computed - published coordinate. Naturally for an actual project we would not have such good a priori knowledge of the station position.
The first step in the adjustment process is to compare the baseline repeatability. For this go to the section of the extended output entitled Derivation of NAD 83 Position. There you will find the NAD83 vectors from the remote station to each base station. These vectors can be cut and pasted into a spread sheet such as Excel. Click here for Baseline Comparisons The baseline from NLIB to BLRW on day 95 was rejected based on the 4.6 and 6.4 cm misclosures in the Y component.
The next step in combining these solutions is to extract the control station coordinates from the OPUS extended output. The NAD83 base station coordinates are also found in the section entitled Derivation of NAD 83 Position. Here you will find the NAD83 ECEF cartesian coordinates of the reference stations at the epoch of observation. Columbus will directly accept ECEF cartesian coordinates. If your least squares program will accept ECEF cartesian coordinates you can cut and paste them into your program; otherwise convert them to latitude, longitude and ellipsoid height and add them to your adjustment file.
Next cut and paste the G file from the OPUS extended output. The G files are found in the G - FILES section of the extended output. See GLOBAL POSITIONING SYSTEM DATA TRANSFER FORMAT for a detailed explanation of the G file format. There are three G files in each extended output. Cut and paste the G files into it a text file. These G files have ITRF00 vector information. We can convert these ITRF00 G files to NAD83 by replacing the ITRF00 vectors with NAD83 vectors. Copy and paste the Vectors from unknown station monument to reference station monument rotated and scaled into NAD83 into the G file. The vectors in the G file show four decimal places while the vectors in the OPUS output show five. Trim one decimal place from the OPUS output. GPS vectors in the G file have an implied decimal point. Remove the decimal point from the OPUS output before you paste it into the G file. The G file also contains the correlation matrix for each vector. The conversion of the ITRF00 correlation matrix to NAD83 is so slight that the numbers do not change at the number of significant figures shown so no action is needed.
Columbus has a conversion routine to convert G files into its proprietary input format. Other least squares programs have similar capabilities. The G file refers to a Station Serial Number to identify the Origin and Differential stations. The OPUS extended output uses the same serial number for all three base CORS in the three G files. The Station Serial Numbers must be edited so that each station has a unique Station Serial Number. CORS corresponding to the Station Serial Number can be identified by looking at the end of the third line of the G file. There you will see the CORS names embedded in the Meta Identifier of the Vector Record. The Columbus conversion routine will read what is called a SERFIL (Serial Number File) to make the conversion from Serial Numbers to station names in the input file. One could manually edit the Serial Numbers to station names otherwise.
Once the base station coordinates and the GPS observations with covariance matrix have been imported into your least squares adjustment program, run the adjustment. In this case adjustments of each group of repeat baselines from NLIB to BLRW, SLAI and WNCI were separately adjusted in a minimally constrained adjustment. This allowed for a further check on outlier observations and the a posteriori variance factor from each individual adjustment was then used to scale the error estimates provided in the G file. Once each group of repeat vectors was adjusted, and the covariance matrices scaled to pass the chi square test the combined eight vectors were adjusted in a minimally constrained adjustment. This adjustment passed the chi square test with no intervention.
For the initial constrained adjustment all three CORS were constrained in three dimensions. This adjustment failed the chi square test indicating a potential problem with the constrained coordinates. Since the control stations are CORS we can check the 60-day time series for the three CORS. On April 25, 2004 the 60-day series information was:
CORS N cm E cm U cm BLRW 0.01 (0.22) -0.46 (0.28) -0.46 (0.68) SLAI 0.36 (0.17) 1.56 (0.26) -1.70 (0.44) WNCI 0.90 (0.21) -0.49 (0.23) -2.66 (0.80)
These give us a clue why the constrained adjustment failed the chi square test. Both SLAI and WNCI have large vertical displacements. In addition SLAI has a large east displacement. Any solution fixing SLAI horizontally fails the chi square test. Constraining BLRW and WNCI in three dimensions and SLAI vertically passes the chi square test indicating that this combination of control is consistent with the accuracy of the measurements. The final adjusted coordinates are:
Latitude Longitude h dN dE dh
41 46 17.70026 (0.002) 91 34 29.59224 (0.001) 208.098 (0.004) -0.005 0.000 0.020
In this case the numbers in parenthesis are the standard deviation of the solution. In this solution the coordinates of SLAI shift 1.6 cm which is consistent with the 60-day series information.
The same adjustment could be done referenced to ITRF00 and converted to NAD83 using HTDP. Working in ITRF00 is actually simpler in some respects as one avoids editing the G files.
Combining OPUS solutions in this manner allows one to take advantage of the NGS' PAGES baseline processor without learning new software. One maintains control of the adjustment of the survey data. The PAGES derived baselines could be combined with shorter occupations on other stations, L1 only data or even conventional observations in a unified adjustment. OPUS is now more than just a positioning service, it now provides all the information you would get from your baseline processor.