When performing GPS surveys in the United States, we are normally stuck with using one datum for the ground control and another for the GPS satellite positions. This is because we typically want to establish our ground control referenced to NAD83 but NGS and IGS both issue ephemerides referenced to a realization of the International Terrestrial Reference System (ITRS). The most rigorous method of processing GPS baselines is to use a consistent datum for ground control and satellite positions. As a rule this means using ITRF## ground control with the satellite ephemeris. The most recent ephemerides from NGS and IGS reference IGb 2000 (IGb00). IGb 2000 is for all intent and purpose equivalent to ITRF00. More information about the various ITRS realizations used for NGS ephemerides can be found at: http://www.ngs.noaa.gov/CORS/coordinates/.
Various agencies publish GPS ephemerides. Among them are NGS, IGS, Geodetic Survey Division Natural Resources Canada (GSD NRCan) and the National Geospatial - Intelligence Agency (NGA) formerly known as the National Imagery and Mapping Agency (NIMA). The IGS precise ephemeris is the gold standard of ephemerides. It is derived from the weighted average of the ephemerides of eight Analysis Centers. NGS and GSD NRCan are among the analysis centers.
Using ground control and satellite positions reference to two different systems introduces systematic error into the GPS baselines. The difference between NAD83 and IGb00 are very small. The differences in orientation are measured in milliarc seconds and the scale factor is measured in parts per billion. For short baselines, the differences are negligible but as baseline lengths increase the differences can be observed. The usual method to account for the differences between the ephemeris datum and NAD83 is to process GPS baselines using ground control in the same datum as the ephemeris and then transform the resulting vectors to align them with NAD83. These transformed vectors are added to the ECEF coordinates of the reference station to derive the ECEF coordinates of the forward station. Tomas Soler, Neil D. Weston and Heeyul Han describe this method in Computing NAD83 Coordinates Using ITRF-Derived Vector Components (ACSM-ASPRS 2002 Annual Conference Proceedings, Washington, DC, April 2002.). OPUS uses this technique.
It would be convenient to derive NAD83 vectors directly from commercial GPS baseline processing software. This would be possible using an ephemeris referenced to NAD83. GSD NRCan does offer such and ephemeris; however, it is derived from their precise ephemeris alone. There is no NAD83 equivalent to IGS precise ephemeris. As an experiment, I wrote a program to transform a GPS ephemeris, referencing one of the realizations of ITRS, to reference NAD83. I used this program to transform the IGS precise ephemeris to NAD83 and process baselines directly in NAD83.
The following example illustrates the systematic error introduced into a survey by mixing the IGS precise ephemeris with NAD83 ground control. For this example, four baselines were computed on two different days using five National CORS. Using CORS allows us to compare the position of the CORS derived from the processed baselines with the published CORS position. The misclosures act as a measure of the accuracy of the baseline computations. The five selected CORS are:
GODE - Goddard Space Center, Maryland CORB - Corbin, Virginia UVFM - University of Virginia Fan Mountain WES2 - Westford, Massachusetts NLIB - North Liberty, Iowa
Baselines radiate from GODE to form four baselines:
GODE - CORB 103 km GODE - UVFM 207 km GODE - WES2 601 km GODE - NLIB 1298 km
Each baseline was processed using twenty-four hours of data for April 20, 2003 (Day 110) and October 13, 2003. Baselines were processed using the Weighted Ambiguity Vector Estimator (WAVE) processor included in Trimble Geomatics Office (TGO) v. 1.61. No attempt was made to fix integers. Three combinations of ground control and ephemeris were used.
ITRF00 Ground Control with IGS ITRF00 Precise Ephemeris (The control group)In each case, the published CORS positions were updated to the epoch of observation. A fourth combination used the ITRF00 oriented vectors from scheme one. These vectors were manually transformed to NAD83, using MathCAD, and then combined with the ECEF NAD83 coordinates of GODE to derive coordinates for the other CORS. The table below summarizes the misclosures using the four baseline processing schemes. Scheme one acted as the control for the experiment.
Misclosures Difference from Control Group
To Km Day. dN dE dU RMS N E U
ITRF00 Ground Control IGS ITRF00 Ephemeris
CORB 103 110 -0.002 0.006 0.000
CORB 103 286 0.001 0.010 -0.009 0.006
UVFM 207 110 -0.005 0.009 0.003
UVFM 207 286 0.001 0.017 -0.007 0.009 Control Group
WES2 601 110 0.008 0.009 0.009
WES2 601 286 0.012 0.003 0.006 0.008
NLIB 1298 110 0.004 -0.011 0.018
NLIB 1298 286 0.008 -0.013 0.007 0.011
NAD83 Ground Control IGS ITRF00 Ephemeris
CORB 103 110 -0.001 0.011 -0.006 0.001 0.005 -0.006
CORB 103 286 0.002 0.016 -0.015 0.010 0.001 0.006 -0.006
UVFM 207 110 -0.008 0.021 -0.003 -0.003 0.012 -0.006
UVFM 207 286 -0.001 0.030 -0.014 0.016 -0.002 0.013 -0.007
WES2 601 110 0.006 -0.036 0.031 -0.002 -0.045 0.022
WES2 601 286 0.010 -0.043 0.025 0.028 -0.002 -0.046 0.019
NLIB 1298 110 -0.052 0.048 0.045 -0.056 0.059 0.027
NLIB 1298 286 -0.045 0.043 0.020 0.043 -0.053 0.056 0.013
NAD83 Ground ITRF00 Vectors Transformed to NAD83
CORB 103 110 -0.002 0.006 0.001 0.001 0.000 0.001
CORB 103 286 0.001 0.010 -0.008 0.006 0.000 0.000 0.001
UVFM 207 110 -0.006 0.008 0.004 -0.001 -0.001 0.001
UVFM 207 286 0.002 0.017 -0.006 0.009 0.001 0.000 0.001
WES2 601 110 0.007 0.009 0.011 -0.001 0.000 0.002
WES2 601 286 0.011 0.004 0.007 0.009 -0.001 0.001 0.001
NLIB 1298 110 0.002 -0.011 0.019 -0.002 0.000 0.001
NLIB 1298 286 0.005 -0.013 0.007 0.011 -0.003 0.000 -0.001
NAD83 Ground IGS ITRF00 Ephemeris Transformed to NAD83
CORB 103 110 -0.002 0.005 0.000 0.000 -0.001 0.000
CORB 103 286 0.001 0.010 -0.009 0.006 0.000 0.000 0.000
UVFM 207 110 -0.006 0.007 0.003 -0.001 -0.002 0.000
UVFM 207 286 0.000 0.017 -0.006 0.008 -0.001 0.000 0.001
WES2 601 110 0.008 0.009 0.011 0.000 0.000 0.002
WES2 601 286 0.012 0.003 0.006 0.009 0.000 0.000 0.000
NLIB 1298 110 0.004 -0.012 0.018 0.000 -0.001 0.000
NLIB 1298 286 0.008 -0.013 0.006 0.011 0.000 0.000 -0.001
As expected, when using the IGS ephemeris with NAD83 ground control, accuracy decreases with increasing baseline length. Between the 100-kilometer and 200-kilometer range the differences approach the centimeter level. Using the NAD83 ephemeris with NAD83 control directly produces NAD83 results with accuracy rivaling ITRF00 ground control with the IGS precise ephemeris.
The main problem is the differences in scale between the NAD 83 frame and the ITRF00. If you check our Coordinates page, mentioned in your article, you will notice that this difference amounts to 0.62 x 10-9. Considering that the satellites are approximately 26,000 km from the Earth's geocenter, this difference in scale will represent a bias of 1.61 cm in the geocentric position of the satellite. We at NGS believe that the ITRF scale is more accurate than the NAD 83 which was originally derived from a mixed bag of Doppler and classical geodetic observations. Consequently, our approach has been always to recommend GPS users to do all GPS processing on the ITRF00 frame and at the very end transform to the NAD 83.
Mathematical models need to recreate the physical world as close as possible. The scale of our 3D space is defined by the velocity of light. Consequently, we agree that the scale of ITRF which is based on GPS observations is closer to the truth. The real distance between two points depends on the scale of the space. The scale of the transformed NAD83-defined space is not 0 although very close to it.
As a consequence of this discrepancy the geocentric position of the satellites in the NAD83 ephemeris is wrong by about 2 cm. We also know that the origin of the NAD83 frame is about 2 meters away from the geocenter but this error does not affect relative GPS positioning. The problem is that the distance between the satellite and an arbitrary point on the ground is unique and it cannot have two values. When we observe using GPS our observations should be as close as possible to the real value of the distance. Assuming no atmosphere, when you use the 14 parameter transformation to NAD83 you change the relative position between point and satellite by a factor of 0.62x10-9. Consequently, the real observations and the modeled distance between the two points at the ends of the baseline are systematically biased. This primarily distorts the distance from the points you fixed during processing, the others are free to adjust although introducing the 2cm bias in the location of the satellite --fixed in the solution-- which affects every pseudorange. I repeat, it is simply a question of conceptual rigorousness.
To demonstrate the scale factor introduced into the baseline solutions I computed Station to Satellite ranges for the seven satellites tracked at GODE during the first observation epoch. Station GODE is the reference station for the relative positioning of the other four stations in this experiment. In relative positioning this distance is computed for each epoch and held as a constant for that epoch. Any corruption of the Station to SV range at GODE propagates through the baselines radiating from GODE.
ITRF
X (m) Y (m) Z (m) Sta. SV Range
GODE 1,130,773.777 -4,831,253.630 3,994,200.455
PRN 6 -3,351,496.955 16,524,282.228 -20,299,916.363 32,655,072.725
PRN 7 -14,089,012.251 -20,356,802.790 10,258,649.033 22,625,823.288
PRN 8 -1,120,346.749 -24,698,358.581 9,082,679.389 20,631,578.239
PRN 11 14,284,150.560 -14,987,103.936 16,595,743.998 20,855,491.265
PRN 27 4,251,899.130 -25,862,322.229 -1,305,528.829 21,911,969.340
PRN 28 -4,282,439.118 -14,727,353.162 21,847,114.067 21,117,816.749
PRN 31 20,005,168.058 -602,326.210 17,308,075.743 23,481,606.882
NAD83
X (m) Y (m) Z (m) Sta. SV Range Diff PPB
GODE 1,130,774.431 -4,831,255.084 3,994,200.575
PRN 6 -3,351,494.629 16,524,277.900 -20,299,919.051 32,655,072.705 -0.020 -0.61
PRN 7 -14,089,012.566 -20,356,802.604 10,258,650.818 22,625,823.275 -0.013 -0.55
PRN 8 -1,120,347.284 -24,698,359.250 9,082,682.000 20,631,578.227 -0.012 -0.57
PRN 11 14,284,150.389 -14,987,104.491 16,595,745.683 20,855,491.252 -0.012 -0.60
PRN 27 4,251,898.744 -25,862,324.518 -1,305,525.952 21,911,969.327 -0.014 -0.62
PRN 28 -4,282,439.373 -14,727,352.033 21,847,115.329 21,117,816.737 -0.012 -0.57
PRN 31 20,005,168.653 -602,326.994 17,308,075.708 23,481,606.868 -0.015 -0.62
Here we see two different values for the Station to SV range depending on the reference frame. This is what Dr. Soler refers to when he states, "the distance between the satellite and an arbitrary point on the ground is unique and it cannot have two values." Because the "scale of ITRF which is based on GPS observations is closer to the truth" the ITRF00 Station to Satellite range is a better estimate than is the NAD83 Station to Satellite range. All the NAD83 Station to Satellite ranges are short by 0.6 PPB. This is a very small systematic error but using ITRF00 ground control with an ITRF00 (IGb00) ephemeris avoids this systematic error entirely. Thus using ITRF00 ground control with an ITRF00 (IGb00) ephemeris is the more rigorous method.
All this discussion only comes to play with long baselines. Using an ephemeris referenced to NAD83 with NAD83 ground control is still better than using an ITRF00 (IGb00) ephemeris with NAD83 ground control; it is certainly adequate for all but the most rigorous uses. In fact, for short baselines one can mix NAD83 ground control with ITRF00 SV positions with little effect. For example, with the 103 km GODE - CORB baseline, the difference between the NAD83 ground - ITRF00 SV combination and the ITRF00 ground - ITRF00 SV combination is only 6 mm horizontal and 6mm vertical. However, Dr. Soler emphasizes, "As a matter of scientific principle it is important to avoid as much as possible systematic errors. These are always the major cause of undetected problems because, unknowingly, the solution may converge the wrong answer making resulting statistics useless."
Other more rigorous methods to determine NAD83 station coordinates while using an ITRF00 (IGb00) ephemeris include the following:
Both these methods rely on ties to ITRF00 ground control. The most practical access to ITRF00 ground control is through the CORS. This leads to a paradigm shift where active control stations (CORS or CACS) define the national network. Monuments on the ground will always be necessary to define boundaries or to control engineering surveys; ties to the active control stations will define their positions relative to the national network. As software and hardware advances squeeze the last millimeter of accuracy out of positioning technology, it is important to adopt methods that take advantage of these advances.
Peter Lazio