Terrestrial Reference Frame Effects on Global Sea Level Risedetermination from TOPEX/Poseidon altimetric data

Laurent Morela, Pascal Willisb,c

aEcole Supérieure des Géomètres Topographes,1, boulevard Pythagore,72000 Le Mans, France. bInstitut Géographique National, Direction Technique, 2, avenue Pasteur, BP 68, 94160 Saint-Mandé, FrancecJet Propulsion Laboratory, California Institute of Technology,Pasadena,California 91109, USA

AbstractIn the past recent few years, satellite altimetry has produced several significant improvements in our scientific understanding of the oceans. However, several results related to global or regional sea level changes still too often rely on the assumption that orbit errors coming from stations coordinates adoption can be neglected in the total error budget. The goal of this poster is to study this general assumption in more details. In the case of the TOPEX/POSEIDON satellite, we will characterize orbital errors coming from the adoption of a specific Terrestrial Reference Frame and estimate simple transfer functions that can be used for several purposes. Simulations derived from actual DORIS data using the GIPSY/OASIS II software show that the present main source of errors comes from current imprecision in the Z-translation. Consequently, it can create systematic errors in the derived mean sea level due to the North-South non-symetrical distribution of the oceans all over the world. Furthermore, in the case of the TOPEX/Poseidon mission, consequences of such orbital errors on the estimation of Mean Sea Level and of seal level rise derived from altimetric data are discussed. Finally, taking into account the present geodetic accuracy of recent Terrestrial Reference Frame realizations a realistic budget error is derived for this specific source of error.

IntroductionFor oceanographic missions like Jason, the goal in terms of orbit accuracy (1 cm) induces that the aspects of Terrestrial Reference Frames (TRF) cannot be underestimated anymore. For every satellite altimetry missions, the relevant Precise Orbit Determination groups adopt the best possible conventions for the satellite. However, oceanographers must be aware that such a choice is not unique. Moreover, the geophysical velocities of the tracking stations are not perfectly known, especially for newly installed stations. In consequence, station positions accuracies naturally tend to slowly deteriorate with time. It is then possible that the precision of long term monitoring of geophysical parameters, such as long term sea level rise could be affected by these uncertainties. The long-term maintenance of the TRF is as critical as its initial realization.The goal of this poster is to characterize the systematic errors coming from the adoption of TRF. We will first focus on the satellite orbit itself and then on the sea level determination as well as its evolution in time. In particular, we have done a sensitivity study using actual DORIS from the TOPEX/Poseidon mission at different epoch in time. Finally we extend the discussion and try to assess the actual errors coming from our present knowledge of the TRF taking as example some latest realizations of TRF, ITRF97. We have processed in two different ways using the GIPSY/OASIS II software developed at JPL (Webb et al., 1995; Willis et al., 2004).

Preliminary analysis

Let us first try to characterize how systematic errors in the TRF will propagate in the estimated orbit. An error in the realization of TRF will affect the 7-parameter transformation relating two different frames:

where :-Tx, Ty, Tz are the translations between the two frames along the X, Y, Z axes,-K is the scale factor,-R is the rotation matrix between the two different frames (3 by 3 matrix).

In order to assess the consequences of errors in the TRF realization, we must understand how a global error in the coordinates of the tracking stations transfers into an orbital error. By choosing the parameter we want to perturb and by changing the values of the error level, we will perform a sensitivity analysis, allowing us to assess the consequences of an error on this specific parameter into the estimated orbit.At first, an orbit is estimated using the exact ITRF97 coordinates for the DORIS tracking stations. This orbit will now serve as reference orbit in all our future comparisons and we will compare all our estimated orbits obtained from a different TRF to this one.In the second step, we have changed all the tracking stations coordinates by applying one of the 7 parameters to the ITRF97. For each of these 7 parameters, we have used several values (from extremely small values to values that are basically 100 times larger than actual expected errors on this parameter). We have then compared our newly estimated orbit (perturbed ITRF97) with the reference one (original ITRF97). We can now compute the differences and conduct a sensitivity analysis for each of the 7 parameters independently.Let us do the study for the translation parameters. The following figure is showing results of such comparison when exact translation of 1 cm in TX (resp. TY and TZ) has been applied to all the stations coordinates. We show some period variations but no significant offset can be found in those results ( less than 0.1 mm for TX, TY and TZ).

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On these plots, standard deviation correspond to the magnitude of the radial orbit error at the orbital period because the main error goes into this period. These standard deviation are less than 3 mm for TX and TY and greater than 5 mm for TZ. So, it seems that the Z-shift on TRF leads a systematic effect on the radial component at the orbital period. In case of the scale factor parameter, the differences also show a non significant systematic effect on the radial orbit component when a constant scale factor of 1 ppb is applied to the TRF.

Transfer function from TRF parameters to derived orbit

Let us now extend our simulation study by changing the adopted value of the TRF scale factor or of the translations one after another. For each TRF parameter, we used several values (from extremely small values to extremely large values). Then, we compared the perturbed orbits with the estimated orbit with our reference (based on actual ITRF97). For each comparison, we derived the mean value and the standard deviation of the differences as described previously. We then performed a linear regression of these values. The slope obtained corresponds to the derived transfer function relating the observed variation on the orbit to the considered variation in the station coordinates. The table below presents the estimated transfer functions for each couple of considered TRF parameter and observed orbital parameter as well as an internal precision derived from the slope estimation (for the radial component (H) in the satellite frame and for the components (X,Y,Z) in the geocentric frame).

DH(mm)DX(mm)DY(mm)DZ(mm)K0.01 +/- 0.340.05 +/- 2.900.04 +/- 2.960.07 +/- 2.10TX0.00 +/- 1.170.53 +/- 1.950.02 +/- 1.810.14 +/- 0.99TY0.02 +/- 1.370.12 +/- 3.270.78 +/- 3.18-0.45 +/- 1.64TZ0.01 +/- 3.800.16 +/- 2.600.57 +/- 2.807.36 +/- 1.78

No significant mean radial orbit error can be observed from simulated errors in the TRF parameters: K, TX, TY or TZ. Similarly, no significant error orbit can be observed from simulated errors in the TRF parameters K, TX or TY, either in the satellite frame nor in the Terrestrial frame. However, for a simulated Z-translation of the TRF, a significant mean orbit error can be observed but only on the Z-component in the TRF (1 cm error in the Z-component gives 7.4 mm offset in the Z-component of the orbit). This systematic effect is also observed on the radial component at the orbital period. A 1 cm shift on the TRF leads to radial orbit error of 3.80 mm (which is three times more than the others parameters).It can be also be seen that the TX and the TY parameters of the TRF have no significant effect on the orbit (less than 1 mm). As the orbit is naturally computed in the inertial frame, there is no privileged direction in space except that the one given by the Earth rotation axis. The X and Y coordinates of the satellite have equivalent roles and any effect would be mostly cancelled by symmetry when using a complete daily data set that spans 13 orbital periods in the case of T/P. In contrary, TZ is a particular direction for both in the inertial and the geocentric frames. A shift of all the tracking stations in this direction in the geocentric frame almost corresponds to a similar shift in the inertial frame and would not cancel by the Earth’s rotation effect as TX and TY would. So a systematic effect coming from this Z shift can be observed in the orbit results. The systematic effect is extremely linear and could be predicted by a simple linear formula: TZ(cm) mm/cm 7.4DZ(mm) ×=

CONCLUSION

Consequently, we have estimated the transfer function from TRF parameters to mean sea level by appling a similar study than for the orbit but only for points above the oceans. We deduced the following transfer functions:

From orbit errors to derived mean sea level errors

We must now extend our simulation study by obtaining the systematic effect not only for all the orbit points over the globe (as done previously) but only over the ocean surfaces where altimetric measurements could be performed. The figure below represents the geographical distribution of the predicted errors coming from a Z-bias in station coordinates.

Positive error are in the northern hemisphere and negative errors in the southern hemisphere. So, when we only consider the radial orbit errors over oceans (to assimilate them to mean sea level errors as deduced from radar measurement assimilation), the positive errors are less than the negatives ones creating a negative bias on the observed mean sea level.

No similar effect can be observed for the scale factor or for the TX or TY translations because the radial orbit errors at the orbital period don’t depend on the parameter value (for TX and TY the radial orbit errors at the orbital period are three times less than TZ). The figure above represents the geographical distribution of the predicted errors coming from an X-bias in station coordinates. Whereas we observe a symmetrical error orbit distribution in longitude, no bias is finally observed principally because the radial orbit errors are weak.So, the more important systematic error of the TRF comes from a Z-bias in the TRF realization.

Finally, realistic uncertainties in the current realizations of the TRF established by comparing recent realizations have been used:

K is determined at the 1.5 ppb and maintained in time at the 0.5 ppb/year level,TX is determined at the 1 cm and maintained in time at the 0.4 cm/year level,TY is determined at the 2 cm and maintained in time at the 0.5 cm/year level,TZ is determined at the 2.5 cm and maintained in time at the 0.3 cm/year level)

we can derive using the previous equations the following error estimations in the mean sea level:

(cm) TZ -1.20(mm) MSLTY(cm) -0.14(mm) MSLTX(cm) -0.14(mm) MSL

×=×=×=

mm/yr 0.37)3.02.1()5.014.0()4.014.0((mm/yr) rate MSL

mm -3)5.22.1()0.214.0()0.114.0((mm) MSL222

222

=×−+×−+×−=

=×−+×−+×−=

In conclusion, we have derived for the TOPEX/Poseidon mission simple transfer functions that can be used to predict the systematic error in the mean sea level and in the sea level rise that would be seen by the oceanographers as coming from TRF realizations.

ReferencesWEBB FH, ZUMBERGE JF (1995) An Introduction to GIPSY/OASIS, JPL D-11088.WILLIS P, BAR-SEVER YE, TAVERNIER G (2004) DORIS as a Potential Part of a Global Observing System, J. Geodyn., 2004, in press..

The larger effects come from the Z-translation and the Z-translation evolution in time of the TRF. In order to meet users requirement of more recent oceanographic missions such as Jason-1. A factor of improvement of 2 seems to be needed if geodesists want to fulfill these long-term very demanding scientific needs.