Proposed JASON-1 Orbit Comparison Campaign

A JASON-1 Orbit Comparison Campaign will be initiated in November 2002.

 

The comparison campaign for CHAMP started as a project for assisting the IGS LEO analysis centers in setting up their CHAMP data processing. Interest in this campaign turned out to be greater than expected, while precision levels of the contributions soon improved from about 30 cm RMS early on, to about 5 cm RMS at present. The emphasis of the campaign has shifted accordingly, from being a practical guide on CHAMP POD to being a more careful geodetic analysis.

 

The main purpose of the JASON campaign will not be to assist groups in getting JASON POD started, because for JASON the highest precision levels can be expected from the beginning. To make sure that the campaign provides the most precise results this document aims at summarizing the analysis that will be done as independent check of orbit quality. Based on that, some requirements are derived for the contributed orbits.

1 Objectives

The main objectives of the JASON comparison campaign will be

These objectives are similar to those of the CHAMP campaign, but the priorities have clearly shifted from basic POD, to high-precision analysis.

 

For CHAMP some effort has been spent on determining LEO combination solution strategies based on the contributed orbits, similar to what IGS does for the GPS constellation. So far the results have never been very satisfactory: it is virtually impossible to assess the true precision of such a combination solution, and therefore it is not clear what the best combination strategy would be. Something similar might be attempted again for JASON, but it is not considered a main objective of the campaign.

 

The available orbit analysis methods are as always (a) comparisons between independent orbit solutions for the same period, and (b) analysis of tracking data residuals.

2 Orbit comparisons and overlap comparisons

The usual measure for orbit accuracy is the RMS of differences between the orbit of interest and the true satellite orbit. Unfortunately the latter is not available, and the differences with the true orbit must be inferred from comparisons between the available solutions, each with its own error. If two solutions are perfectly independent the RMS of differences between these two orbits is equal to the RSS of the two separate error signals. The RMS of differences between any independent pair sets the upper limit to the orbit error for each of the two compared orbits (i.e. if one orbit would be free of error the RMS of orbit differences is only due to the error in the second orbit).

 

In practice orbits are never perfectly independent, and for the CHAMP campaign some dependencies were readily observed (e.g., between centers using the same POD software). JASON solutions based on completely different data sets - say GPS-only versus DORIS-only - and obtained from different software packages will be more independent than solutions based on the same tracking data. Nonetheless, identical gravity field models, inhomogeneous geographical distribution of land-based tracking stations, etc. will always result in common orbit error signals in different solutions, and the comparison of orbits should be treated with care.

 

Three types of orbit comparisons can be useful:

  1. Comparisons between two full solution arcs (solutions from different centers, or solutions based on different data sets)
  2. Overlap comparisons between consecutive arcs from a single solution
  3. Overlap comparisons between consecutive arcs from two different solutions

 

On the basis of experience with the CHAMP campaign, a few things should be changed for JASON:

 

A Some orbits show incidental error peaks, typically at the start or end of an arc, or around manoeuvres. These peaks tend to dominate the RMS of orbit differences, which means that this RMS value is not representative for the actual orbit errors. It will be better to reject a short orbit interval around known anomalies. In case of arc boundary problems, solution arcs should have a margin that absorbs the effect so that the actual arc of interest is clean. In case of anomalies due to orbit events, the same interval should be excluded from all compared arcs, so that the set of sample points remains identical for all pairwise comparisons.

 

B Separate analysis can be presented for the norm of the difference vector and for the radial component. The groups that work on JASON altimetry will be mainly interested in the radial orbit precision. For GPS groups, the 3-D orbit error is more relevant but it is still useful to have separate results for the radial direction (vertical station position component for the LEO). For terrestrial GPS stations the vertical position component is the least accurate, but for a LEO it is probably the most precisely known, as long as the orbit is obtained from a dynamic solution. This property is relevant to IGS applications of LEO data, so that separate radial results will be of interest to everybody.

 

C If centers contribute new orbit solutions at a later stage, these should be clearly identified as different solutions, in order to avoids confusion at this point as seen with CHAMP.

3 Collocation of reference frames

A 7-parameter transform (translation, rotation, scale) is estimated for each comparison pair to detect reference frame deficiencies between the two orbits. In the CHAMP campaign the information from these transformation parameters is only used as a crude check for obvious frame errors. The results on the CHAMP website are all based on the pre-transform orbits. However, for JASON small reference frame deficiencies will be of particular interest and it will be a good idea to use the information from the estimated transform parameters in some way. This makes it also important to remove any anomalies mentioned under A above, so that the 7 estimated parameters are meaningful.

 

All N orbit solutions can be compared to the (N-1) other solutions, which results in (N-1) sets of 7 parameters for each solution. These (N-1) transforms must be defined in the same direction, e.g. they always define the transform from the solution of interest into one of the N-1 others. All pairs are simply compared twice, leading to two contrary transforms that are associated unambiguously with either of the two orbits.

 

From the (N-1) transforms for a single solution a mean 7-parameter transform Tmean for that solution can be computed. The global mean over all N*(N-1) translations and rotations is necessarily zero, due to the way in which they are estimated. This means that the distribution of the values Tmean over the N separate solutions can provide useful insight in offsets of individual orbit frames, e.g. different Z-translations.

 

In principle the orbit solutions define N different reference frames, which are implied by the orbits but not known explicitly (i.e. in terms of transform parameters with respect to a known external frame like ITRF2000). Any of the N frames might be chosen as a common frame for collocating all orbits, but the global mean reference frame can also be found via a straightforward collocation method:

  1. The mean transform Tmean of an orbit k is called Tk. This transform can be scaled by a factor (N-1)/N and applied to the associated orbit k.
  2. The opposite transform of Tk is called Tk-1. This opposite transform is scaled by a factor 1/N and applied to each of the (N-1) other orbits.
  3. The global mean reference frame of all N orbits remains the same under the above transforms, because the single transform over (N-1)/N Tk is balanced by the N-1 opposite transforms over 1/N Tk-1.
  4. If new transforms Tmean are now computed for all N orbits, the orbit k will necessarily have a mean transform of zero. The global mean over the N*(N-1) transforms is zero by default. From these two basic truths it follows that the mean over the other (N-1) transforms Tmean must be zero. This implies that the new frame of orbit k is the global mean frame.
  5. The N-1 other solutions can now easily be collocated with the global mean frame via their respective 7-parameter transforms into the frame of orbit k.

 

An example to illustrate this collocation process is given in Annex A. The orbit k can be chosen arbitrarily from the set of N solutions, the global mean frame that comes out will always be the same. If the N possibilities to realize the global mean reference are averaged, the frame will be very well established.

 

Three of the included JASON orbits should be determined by DORIS data only, by GPS data only and by SLR only, respectively. Hopefully this is possible without losing accuracy, meaning in this case that the resulting orbit should accurately reflect the chosen tracking data reference frame. The transforms to and from the SLR, DORIS and GPS reference frames will then form part of the set of collocation transforms to and from the global mean.

 

For all solutions, the 7-parameter transforms into the three main reference frames (SLR, DORIS, GPS) can be listed on the web page. The SLR analysis (see below) can be repeated after collocating each orbit frame with the SLR frame, for maximum precision. The a priori orbits should still be processed as well, in any case.

4 Tracking data analysis

While the orbit comparisons set an upper limit to the orbit error, the tracking data sets a lower limit to the orbit error, assuming of course that the tracking data residuals are primarily due to orbit error. Keeping the a priori orbits fixed, tracking residuals to at least nine observation types could be computed for JASON, but not all of these will be useful as an independent check of orbit quality. The data sets and observation types are

 

SLR two-way range

Altimeter altimeter height

single satellite crossover differences (JASON-JASON)

crossover differences with other satellites (TOPEX, ERS-2)

DORIS range-rate

GPS pseudo range

phase

single differences

double differences

 

SLR

The laser data remains the most important data set for absolute orbit assessment. This is not only because of its high precision, but also because it will typically not be the dominant data set in the JASON POD process so that it remains relatively independent. For processing the SLR observations to fixed a priori orbits, we should clearly define

1.      The station coordinates to be used. This will be ITRF2000, but some stations in the ESOC set may need to be updated with respect to the original set (I have doubts about our values for Arequipa and Cadiz, for instance).

2.      The data points or passes that need to be rejected, for whatever reason.

What we did for CHAMP at some point was to compare the set of included SLR observations between ESOC and CSR. After tuning the rejection criteria used for the CHAMP campaign, the two data sets were virtually identical. For JASON we should do something similar. Poor SLR observations are relatively easy to identify, especially if the a priori orbits are accurate to the level of a few cm as can be expected for JASON.

 

Altimetry

Direct altimeter observations depend on many a priori models (geoid, tides, ocean currents, instrument bias, atmosphere) and may therefore not be very suitable for the purpose of absolute orbit precision analysis. Altimeter crossover residuals will be more useful, they should depend mainly on orbit error as long as the maximum time between crossings is kept short (1 day or so). However, at ESOC we can not process altimetry crossovers at present. If this analysis is considered relevant, some assistance from another center will be required.

 

DORIS

The DORIS data tends to show very stable and virtually identical residual values under all circumstances. Even with fixed a priori orbits it is still necessary to estimate the pass-dependent biases, which further levels off the residuals to their usual values. The usefulness of DORIS data as an independent orbit check will therefore be limited. On the other hand, it is easy to process this data so that we might as well do it. Perhaps the estimated bias parameters contain more relevant information than the residuals themselves (e.g. geographical distribution of bias values may hint at north-south reference frame differences).

 

GPS

The GPS data again depends on a variety of a priori models, in particular GPS reference orbits and clocks. JASON clocks will have to be computed together with the residuals, because no set of a priori clock values can really be decorrelated from its associated orbit solution. In any case, it will be good if the centers that provide GPS-based orbit solutions also send their JASON clock values, so that these clocks can be compared as well. Single and double difference combinations are more suitable as an independent check, although at present the ESOC software does not work very well for LEO-based difference data (we never use it for CHAMP). This situation is being corrected right now, and the corresponding software delivery is expected for November. Considering the time required to collect contributions for the JASON campaign, I presume that it will be possible to process double differences at ESOC for the JASON campaign.

 

Considering the options, I propose to process the SLR data, the DORIS data, and GPS double differences (using exactly the same data sets for all orbits of course). The altimeter crossovers - single sat and dual sat - would be useful as well, in particular they can be used to derive geographically correlated and anti-correlated orbit errors. If somebody else also thinks that this should form part of the JASON campaign analysis, please make a suggestion on what center could do this analysis. At ESOC we have no crossover software at all.

 

The other data sets do not seem to be appropriate for absolute orbit precision assessment, but if somebody has a different opinion on that, please comment.

 

5 Summary of proposed campaign analysis

For the CHAMP campaign, the absolute orbit was estimated on the basis of theoretical proportionality between pair-wise RMS of SLR residuals and pair-wise RMS of orbit error. The first can be constructed very easily from the SLR residuals of individual orbit solutions, while the second is given in theory - by the orbit difference RMS. This analysis depends on the independence between solutions, which is a dubious assumption. The analysis is easy to run, however, so that it will be done for JASON as well. In fact it can be done for SLR and GPS double difference data separately, which should in theory provide the same absolute orbit precision estimates if the theory holds. The DORIS residuals could also be used, but will probably not show much variability.

 

Considering the above, the proposed analysis for the JASON campaign consists of this:

1.      SLR residuals are computed to the fixed a priori orbit solutions

2.      DORIS residuals and pass-dependent biases are computed to all solutions

3.      GPS double-difference residuals are computed to all solutions

4.      GPS clocks are compared between solutions that provide them.

5.      Pairwise comparisons are run between all possible solution pairs

6.      Overlap comparisons are run between consecutive arcs of a single solution

7.      Overlap comparisons are run between consecutive arcs of different solutions

8.      7-parameter transforms are estimated between the orbits and the SLR, GPS and DORIS frames (including the three frames separately).

9.      The orbits are collocated with the SLR reference frame, and new SLR residuals are computed. The same can be done for GPS double difference data.

10.  Pairwise RMS values are constructed for SLR and GPS-DD residuals, using the values from the collocated frames (8).

11.  Proportionality factors are computed for pairwise orbit difference and pairwise residuals. Separate factors can be considered for SLR and GPS. In theory they should be the same, so that comparison of SLR results with GPS results can indicate the quality of the analysis.

12.  Absolute orbit accuracy can be estimated by applying the proportionality factor to the individual solutions. The validity of this precision estimate is open to debate, but at least the number that comes out will be consistent both with the pairwise comparisons and with the SLR / GPS residuals. It must be considered as the best available estimate of absolute accuracy.

6 Requirements to contributed orbits

On the basis of the proposed analysis and earlier experience with CHAMP, the following requirements can be formulated for the contributed orbit solutions:

 

Until now, the total number of contributed orbit solutions to the CHAMP campaign has been 26, from 14 different IGS associated analysis centers. Interest in the JASON campaign has been expressed from outside IGS as well, and the number of participating centers will therefore probably be larger. At the same time the variety of tracking data will lead to more solutions per center, for instance separate orbits based on DORIS+SLR, DORIS+GPS, GPS+DORIS+SLR, etc. The campaign analysis will be the same for all contributions, but it will be easy to also include some statistics per solution type.

 

A JASON orbit repository can be installed at CDDISA but if an alternative is already in place (e.g. for the CalVal period) this could also be used.

 


Annex A example of mean frame collocation

The collocation proposed in Section 3 will be illustrated on the basis of a simple two-dimensional case. The points in Figure A.1 represent origins of 6 different reference frames that need to be collocated in a mean reference frame. In this example the absolute coordinates are known with respect to frame OXY (see Table A.1) but in the case of pair-wise orbit comparisons the only available information is formed by the estimated transform parameters between each pair. The main issue is therefore that the collocation method can only make use of the relative information.

4

 
 

 

 


point

X

Y

1

1

3

2

3

3

 
1

3

4

1

 
4

4

5

5

 
6

5

7

2

 

X

 

O

 

k

 
2

k

2

5

 

Table A.1 : a priori coordinates Figure A.1: distribution of frame origins

 

In this case five translations are found for k: (-1, -2), (+1,-4), (+2,-1), (+3,+1), (+5,-3). The mean transform for k is then Tk = (+2, -1.8). With N = 6, the transform (N-1)/N Tk becomes (5/6) * (+2, -1.8) = (+1.6667, -1.5). Applying this transform to k provides the mean origin of all frames, which is the point (3.6667, 3.5). It can be readily verified that any other choice of k would lead to the same mean origin. The origins 1 to 5 must be shifted over -1/6 * (+2, -1.8) = (-0.3333, +0.3000) to maintain the global mean frame.

 

The new coordinates for all origins, as well as the new transforms from k into the other five frame origins are listed in Table A.2. Obviously, the mean transform for frame k is now zero both in X and Y direction. Using the opposite transforms the frames 1 to 5 can immediately be collocated with the new origin k, which defines the global mean frame.

 

point

new X

New Y

new transform Xk

new transform Yk

1

0.6667

3.3

-3

-0.2

2

2.6667

1.3

-1

-2.2

3

3.6667

4.3

0

+0.8

4

4.6667

6.3

+1

+2.8

5

6.6667

2.3

+3

-1.2

k

3.6667

3.5

0

0

 

Table A.2: Frame origins after collocating point k with the common mean origin