SPATIAL AND TEMPORAL ERRORS IN ERS-2 RADIAL POSITIONING P Moore Department of Geomatics University of Newcastle-upon-Tyne Newcastle, NE1 7RU UK email:
[email protected]
INTRODUCTION Applications involving multi-satellite altimetry (such as from ERS, TOPEX/Poseidon and, in the near future GFO) require that any differences between the altimetric databases are either negligible, resolvable by simple spatial approximations or eliminated by enhancement through dual satellite crossovers. Although orbit error for satellites such as ERS-2 has been considerably reduced over the years it still remains as a major component in the error budget. For ERS-2 with altitude near 800km both gravitational and surface-force mismodelling are significant factors. Several gravity field enhancements such as DGM-E04 [1] and AGM-98 [2] specific for the ERS orbit have been released. These two gravity fields are tuned to ERS through minimisation of single satellite crossovers (SXO) residuals (DGM-E04) or SXO and dual crossovers (DXO) residuals with TOPEX/Poseidon (AGM-98). In this paper we briefly review the AGM-98 orbits for ERS-2. The geographically correlated and anti-correlated error is assessed by using DXO residuals for ascending and descending arcs with a spectral analysis in the Southern Ocean identifying deficiencies of a particular order in the gravity field. Spatial and temporal differences between the ERS-2 and TOPEX/Poseidon data sets is quantified by analysing dual crossover residuals on a cycle by cycle basis, with signatures recovered from the launch of ERS-2 in 1995 through to the end of 1999. Finally, ERS-2 and TOPEX/Poseidon orbits connected by dual crossover data are derived for 5 ERS-2 cycles. The enhanced orbits are seen to reduce discrepancies between the altimetric data sets and show that further refinement is still possible.
PRECISION ERS-2 ORBIT DETERMINATION Precise orbits for ERS-2 have been determined from SLR, PRARE and SXO using the FAUST software [3] and AGM98 gravity field [2]. Multi-arc runs of, typically, seven 5 day arcs were processed in a single batch with the individual arcs connected by SXO data across the discontinuity to reduce deficiencies at the start and end of each arc. In this manner orbital positioning was recovered in a single batch for each ERS-2 cycle. An internal check on the accuracy of the orbits is supplied by the tracking residuals of Table 1. The SXO residuals, in particular, provide a measure of the geographically anti-correlated error and show that the accuracy of the orbits decrease from the early cycles in 1995-6 through to the cycles in 1999. This is almost certainly related to the increase in solar activity from solar minimum conditions in 1995/6 to solar maximum in 2000/1. F10.7, the standard index of solar activity at 10.7 cm wavelength, increased from 70 in 1995/6, to over 200 in 1999. In terms of atmospheric air-density this represents a ten-fold increase form 1996 to 1999. We also note that the number of PRARE stations increased to a maximum of 17 or more (several were not used within the orbit determinations) as stations came on line in 1996/7 but decreased from cycle 30 onwards as wear and tear depleted the numbers. A comparison of the AGM-98 and DGM-E04 orbits shows that agreement is near the 3cm level rms for the early cycles but increasing to over 5cm for cycles in 1999. Further quality control for the orbits has been undertaken below by reference to ERS-1 AGM-98 orbits utilising dual crossover residuals with TOPEX/Poseidon.
Cycle 5 10 15 20 25 30 35 40 45
SLR (cm) 5.32 6.17 6.67 7.50 5.28 7.66 6.02 5.97 6.25
-----------------------PRARE-------------------Range (cm) Doppler (mm/s) stations # 6.61 0.59 2 7.31 0.67 15 6.80 0.69 16 7.58 0.72 17 6.76 0.63 16 8.03 0.73 15 6.30 0.66 8 7.85 0.79 10 7.36 0.81 4
SXO (cm) 7.55 7.55 7.46 9.07 7.44 8.08 8.45 8.51 8.60
Table 1. ERS-2 SLR, PRARE and SXO tracking residuals.
DUAL CROSSOVER DATA SETS The ERS-2 altimetric range and geophysical corrections were, in the most part, taken from the Precise Ocean Product (OPR) with the following modifications and choice of corrections: • • • • • • •
Radiometric wet tropospheric correction (recomputed after pass 650 of cycle 12 – as recommended by CERSAT); otherwise rejected Pole tide: applied Sea state bias: 3 parameter model [4] as recommended by CERSAT Inverse barometric correction: applied ERSIN correction for ultra stable oscillator (USO) bias drift: applied ESRIN correction for bias jumps as characterised by SPTR: applied Radial orbit height: DPAF, AGM-98 and DGM-E04
Bias jumps occur after the altimeter is placed in its safe-mode and then reactivated. The temperature differential over this stand-by period leads to a jump in the range measurement as the clock stabilises to a different temperature regime. According to [5] the Single Point Target Response (SPTR) can quantify the jump. Normal orbital heights were taken from three sources; • • •
the DPAF orbital height on the CD ROM based on the EGM2 gravity model in house NCL orbits using AGM-98 orbits released by DEOS, University of Technology relative to the DGM-E04 gravity field [1].
For TOPEX/Poseidon the altimetric range and geophysical corrections were taken from the AVISO CD ROMs, version C, with the following constraints • • • • • •
TOPEX microwave radiometer wet tropospheric correction; otherwise rejected TOPEX, dual frequency ionospheric correction; otherwise rejected Poseidon, DORIS ionospheric correction Pole tide: applied Inverse barometric correction: applied NASA orbital height
No adjustment was made for the long-term drift in the TOPEX altimeter prior to analysis.
AGM-98 QUALITY CONTROL The accuracy of the AGM-98 gravity field has been undertaken by using ERS-1 orbits determined in an identical fashion to ERS-2 using SLR and SXO tracking. DXO residuals between ERS-1 and TOPEX/Poseidon have been
recovered for epochs differing by 5 days or less. By binning the residuals for ascending and descending arcs the geographically correlated and anti-correlated orbit error of gravitational origin can be recovered. Here we make use of the result that the radial orbit error, ∆r, of gravitational origin can be written as ∆r = ∆f ± ∆v where ∆f and ∆v are the geographically correlated and anti-correlated error respectively with sign change from ascending to descending arcs. Figure 1 plots the geographical distribution as inferred from the ERS-1 second multidisciplinary phase. The figure shows that there are still areas where improvement is possible. For example, the swath of relatively high geographically correlated error in the Atlantic can probably be reduced by extending the gravity field refinement to include ERS-2 PRARE data from Ascension etc. In addition, for latitudes south of 41°S, it is possible to consider bandwidths of 1° or 1.5°, depending on the latitude, to undertake a spectral analysis for the geographically correlated and anti-correlated errors. Given that a geopotential harmonic of order m gives rise to m cycles along a particular latitudinal band any deficiency in the underlying gravity field will be evident. Figure 2 plots the power of the error signal averaged over all bands.
Figure 2a. AGM-98 geographically correlated error from spectral analysis in the Southern Ocean
Figure 2b. AGM-98 geographically anti-correlated error from spectral analysis in the Southern Ocean
The spectral analysis shows that most of the residual signal is at longer wavelengths above m=5. Of particular interest is the lack of any peaks at the resonances associated with orders m=14, 28, 29, 43, 56 etc. Apparently, these resonances are well-modelled in AGM-98 for the ERS inclination. Some improvement in the gravitational field is still possible by analysis of the PRARE tracking residuals, however, the increase in the SXO rms away from the solar minimum identifies air-drag as more problematic than gravitational mismodelling. For conditions near the solar maximum, airdrag now appears as the largest error source in ERS-2 orbit determinations.
DUAL CROSSOVER ANALYSES DXO residuals between ERS-2 and TOPEX/Poseidon were determined for the period May 1995 to Dec 1999. In terms of the repeat cycles, the period covered the first 48 ERS-2 cycles and cycles 97-267 for TOPEX/Poseidon. The DXO residuals were recovered for epochs differing by 5 days or less, the time interval being chosen to reduce ocean variability between the epochs but not to be too small to limit the geographical distribution. For any time interval, DXO data between ERS and TOPEX/Poseidon is dominated by high latitudinal locations due to the turning points of the TOPEX/Poseidon ground tracks at latitude 66.1° N and S. The numbers per ERS-2 cycle are also affected by the seasonal distribution of sea-ice in the Southern Ocean with larger numbers during the southern summer. For the 48 cycles a total of over 159000 DXO data points were recovered for analysis with at least 21261 epochs for each ERS-2 cycle. In previous studies [2, 6] the dual crossover residuals have been fitted by ∆r = A1(T) + Α2(P) + A3 cos φ cos λ +A4 cos φ sin λ +A5 sin φ +A6 rτ + Α7 (1.5 sin2 φ - 0.5)
(1)
where φ is latitude; λ longitude; A1(T), A1(P) the relative bias between ERS-2 and TOPRX and Poseidon respectively; A3 – A5 coefficients of first order spherical harmonics; A6 = τ the ERS-2 time tag bias and A7 the coefficient of the second order zonal harmonic. rτ a function [7] of the Keplerian orbital elements is given by rτ = -n RE sin2 I [f + RE /2a ] sin 2u where a, e, n, and I are the usual Keplerian elements; u the argument of latitude; C2,0 the unnormalised second-order geopotential coefficient; f the Earth’s flattening and RE the Earth’s mean equatorial radius. The coefficients of (1) need some further explanation. The effect of τ on the ERS-2 radial error alters sign between ascending and descending arcs while all other coefficients are geographically invariant. Thus, all coefficients, with the exception of the time tag bias, are unobservable in single satellite crossover data. Although a time tag bias was recovered for each arc within the NCL orbit determinations, the altimetry was not adjusted for the value. A mean value of near –1.2ms is typical for most arcs. Separate relative biases, A1(T) and Α2(P)’ were recovered for TOPEX and Poseidon. The NASA altimeter, TOPEX, is operating for near 90% of the time with Poseidon for the remainder. As the two altimeters differ, the characteristics and, hence, altimeter bias will be different. No a priori assumption is made concerning the relative bias between the TOPEX and Poseidon altimeters. The first order spatial harmonics, A3 and A 4 have two potential factors. Any error in the first order geopotential gravity field harmonics will contribute a constant component which will be supplemented by any offset between the ERS-2 and TOPEX/Poseidon altimetric centres of figure. The latter will appear as small translations in the geocentre as implied by the centre of figure of the satellite tracking network, i.e. SLR and DORIS for TOPEX/Poseidon and SLR and PRARE for ERS-2. Since DORIS is effectively global in its geographic distribution while SLR and PRARE and dominated by European and North American stations it is conceivable that A3 – A 4 will exhibit signatures indicative of variations in the ERS-2 tracking network due to seasonal and other temporal effects. The coefficient A5, which represents a northsouth orbital shift, will similarly have contributions due to the odd zonal harmonics in the gravity field and any geocentre deficiency in the z direction. However, as sin φ =sin i sin u, A 5 will also absorb orbital error at 1cy/rev due to, say, non-gravitational force modelling such as air-drag and solar radiation pressure. Thus, unlike A3 – A4, A5 is likely to have strong temporal signatures due to orbital mismodelling. Finally, A7 represents an empirical second order zonal term to reflect deficiencies in the altimeter records between polar and equatorial regions. Several ERS-2 geophysical corrections are either modelled (e.g. ionospheric correction) or have been inferred previously from the altimetry (e.g. sea-state bias). The underlying parameters for these corrections, and hence the corrections themselves, have a zonal
distribution with a similar distribution expected for any error. In addition, the wet tropospheric correction, as inferred from the radiometer, may conceivably be poorly calibrated for very high or low values, again giving a zonal distribution due to the high equatorial humidity. Similar trends are possible with TOPEX except for the ionospheric correction which is inferred from the dual frequency altimeter for TOPEX and from DORIS for Poseidon. DXO data cannot discriminate between the relative sources of error or satellite. Thus, TOPEX deficiencies will be absorbed into the coefficients of (1). It may be correct to assume that ERS-2 dominates the orbital error component as the TOPEX/Poseidon error is typically quoted at 2-3cm rms [8] but one cannot have the same confidence for the geophysical corrections. Indeed, the TOPEX microwave radiometer is known to drift [9] with impact on the altimeter bias.
SPATIAL AND TEMPORAL SIGNATURES IN ERS-2 ALTIMETRY Equation (1) represents long-wavelength spatial signatures in the DXO residuals; however, by solving for A1 - A 7 over each ERS-2 cycle we can also observe any temporal variation in these coefficients. Considerations over full ERS-2 cycles has the advantage that gravitational errors in the radial positioning tend to average to zero as sinusoidal variations at frequencies of integral multiple of the base frequency (1/511cy/rev) will cover a complete number of cycles. Figure 3 illustrates the temporal variation in the coefficients A1(T), and A3 – A7 for the three orbital computations, AGM-98 (NCL); DGM-E04 (DEOS) and EGM2 (DPAF). With reference to these figures, consider the equatorial displacement between the TOPEX and ERS-2 centre of figure as represented by A3 and A4. The non-zero means reveal deficiencies in the first order gravity field coefficients. AGM-98, based on DXO data between ERS-2 and TOPEX, clearly is a better representation of the first order coefficients at the ERS inclination than DGM-E04 or EGM2. However, the temporal variability within the NCL and DEOS coefficients is similar. Some evidence of an annual trend is observed but not as strongly as for the DPAF orbits where a regular annual oscillation of over 1cm amplitude and mid-late year maximum is clearly evident in both coefficients. The north-south displacements of A 5 again reveal a small non-zero mean but more importantly show an annual trend of increasing amplitude for all orbits. This amplitude increase is also probably related to the increase in solar activity from solar minimum conditions in 1995/6 to solar maximum in 2000/1. All orbital computations recovered a number of scale factors for atmospheric density over each arc, but orbital fit to the predominance of tracking in the Northern hemisphere may distort the orbit. The absorption of air-drag mismodelling through the drag scale factors depends on the arc length, the number of coefficients deployed over each arc and the thermospheric model employed (i.e. DTM94 (NCL); DTM78 ( DEOS); CIRA86 (DPAF)). A similar trend is observed in the time tag bias, A6. All orbits exhibit an annual signature with NCL and DPAF, in particular, appearing to have amplitudes increasing with solar activity. The dependence of time tag bias on the orbital computation has been observed previously. This is connected to the 2 cy/rev signal in air-drag as the solar panel maintains its sun pointing attitude compared to the nadir pointing attitude of the main satellite body that offers a near constant area to the along-track direction. A7 shows identical temporal variability for all orbits although the values are offset by 1cm at maximum. The consistency between the orbital solutions appears to confirm that A7 absorbs geophysical errors in the altimetry as speculated previously. Finally, the relative bias A1(T), corrected for the observed TOPEX bias [2], reveals agreement between NCL and DPAF with DEOS offset by a centimetre. Variation in the relative bias is a consequence of orbital and geophysical correction errors including known deficiencies in the SPTR derived corrections for the bias jumps. Linear regression of the AGM-98 (NCL) bias reveals a slope of 1.8 ± 0.7 mm/yr. The bias is analysed in detail in [10]. Differences between the mean values for A1 and A 3 –A5 identify systematic differences between the two altimetric data sets which are eliminated in any application in which the geoid height is removed by averaging at a particular location, e.g. sea-surface variability studies. However, variability from the mean signal will be aliased into the sea-surface variability affecting, for example, the annual signature. In addition, net orbital specific contributions over the cycle will affect sea-surface heights. In particular, radial orbit errors in A 5 affect the relative bias as the DXO data is dominated by high latitude data. We might expect that all such error would be removed through A5 but the incomplete global coverage of the DXO results in non-orthogonality of data and correlation between A1(T) and A5.
MULTI-SATELLITE ERS-2 ORBITS Improved consistency between ERS-2 and TOPEX may be achieved by multi-satellite orbit computations where the two orbits are connected by dual crossover data [11]. Here, we use the procedure [3] to compute both satellite orbits
simultaneously to reduce differences between the two ephemerides. In particular, as DXO data will provide some measure of the geographically correlated errors that are unobservable in SXOs some reduction is expected in the coefficients of (1). To investigate the value of this methodology, orbits for 5 ERS-2 cycles were computed simultaneously with TOPEX. Details of the tracking data and a posteriori residuals for the 5 multi-arc multi-satellite computations are presented in Table 2. Each computation typically comprised 7 ERS-2 five-day arcs, spanned by 4 or 5 ten-day TOPEX arcs. The ERS-2 arcs, themselves, are interconnected by SXO data and connected to one or two TOPEX arcs by DXO data. A maximum time difference of 5 days was imposed on the epochs for the SXO and DXO data. The state vector for each complete ERS-2 cycle comprised • • • • • • • •
Initial position and velocity for each sub-arc: 5 days for ERS-2; 10 days T/P Drag scale factors: 6hr ERS-2; 24hr T/P 1 cy/rev empirical accelerations along-track and cross-track: daily for ERS-2 and T/P SRP coefficient: one per T/P arc, fixed for ERS-2 DORIS: frequency offset and tropospheric scaling factor per pass PRARE: range offset for each station per cycle; tropospheric scaling factor per pass Altimetric timing bias: ERS-2 Relative altimetric bias : separate ERS-2 to TOPEX and Poseidon
The data was weighted in the ration 16:4:1 for crossover data; SLR data; and microwave data respectively . Weighting reflected the numerical superiority of the microwave tracking (DORIS and PRARE) compared with SLR. Crossover data was assigned the largest weight to force the computations to adjust to the additional radial information. The DXO residuals of 7.3-7.5 cm rms are lower than the 7.5-7.8 cm rms for the ERS-2 SXO data, both of which can be compared to the TOPEX/Poseidon rms residuals of 7.0cm rms as typically quoted for the CD ROMs. From these results we infer that ERS-2 orbits are, as is to be expected given the relative altitudes, slightly inferior in accuracy to TOPEX/Poseidon orbits, with ERS-2 rms radial accuracy close to 5cm. The spatial and temporal variations of (1) were estimated relative to the multi-satellite orbital positioning for ERS-2 and TOPEX/Poseidon as derived by the FAUST software relative to AGM-98. Figure 4 plots the coefficients for the 5 ERS-2 cycles of Table 2. Given that the ideal values for A3-A5 and A 7 are zero, reduction in the recovered difference is observed in A5, in particular, and to a lesser extent A4. The multi-satellite derived time tag bias and relative bias are offset from the previous values with the bias showing less variation from cycle to cycle. However, of all parameters the clearest impact has been on the 1999 north-south displacement between the ERS-2 and TOPEX/Poseidon altimetric centres of figures. Apparently, TOPEX/Poseidon assists in reducing the ERS-2 orbital error.
ERS-2 Cycle #
T/P Cycle #
39
232235 235239 239242 242246 246250
40 41 42 43
ERS-2 SLR rms (cm) /# 5.93 4868 6.40 4779 6.33 6281 5.73 5481 6.57 7995
ERS-2 PRARE rms (cm) /# 7.36 16583 8.02 20438 7.92 14575 7.30 13704 7.39 11732
Table 2. Multi-arc tracking and residuals.
ERS-2 PRARE rms (mm/s) /# 0.74 16242 0.79 18606 0.77 12255 0.72 12517 0.75 10535
ERS-2 SXO rms (cm) /# 7.79 7400 7.76 8571 7.53 8588 7.68 6039 7.40 7687
T/P SLR rms (cm) /# 5.95 22176 5.69 24882 6.59 20682 6.21 27446 6.58 28438
T/P DORIS rms(mm/s) /# 0.47 41864 0.47 51059 0.49 48426 0.47 50625 0.48 42736
ERS-2 T/P DXO rms (cm) /# 7.44 32052 7.26 30666 7.42 32881 7.50 27704 7.30 33509
CONCLUSIONS Spatial and temporal variations in DXO data between ERS-2 and TOPEX/Poseidon have been identified for three orbital computations for ERS-2. Differences between the orbits and underlying gravity field result in non-zero means in the first-order spherical harmonic coefficients. Temporal variability is observed particularly at the annual cycle but the amplitude is orbital dependent with the DPAF orbits on the ERS-2 CD ROMs showing the largest annual signal. The north-south displacement between ERS-2 and TOPEX/Poseidon altimetry has a clear annual signal and with amplitude increasing with solar activity from the solar minimum in 1995/6 to 2000/1. A time tag bias of near -1.2 ms is observed for all 3 orbits; again an annual signal is evident. The second order zonal correction is near identical for all orbits with inference that the underlying cause is geophysical. Improvement in consistency between ERS-2 and TOPEX may be achieved by multi-satellite orbit computations where the two orbits are connected by dual crossover data. Derived ERS-2 and TOPEX/Poseidon for just 5 ERS-2 arcs indicate that the methodology is still a powerful concept reducing, in particular, the north-south displacement between the two orbits. Computations over the full ERS-2 period are clearly necessary. However, the early conclusions are that DXO data can have a beneficial effect on ERS-1 and TOPEX/Poseidon orbits facilitating the merging of data from the multi-satellite altimetry.
REFERENCES [1] R. Scharroo and P. Visser, “Precise orbit determination and gravity field improvement for the ERS satellites”, J. Geophys. Res., vol 103, 8113-8127, 1998. [2] P. Moore, “Gravity field enhancement using dual crossovers between ERS and TOPEX/Poseidon with application to the ERS-2 altimeter bias”, in press [3] P. Moore, H.J. Boomkamp, S. Carnochan , and R.J. Walmsley, “FAUST: Multi-satellite orbital dynamics software”, Adv. Space Res., vol 23, No 4, 785-795, 1999. [4] P.F. Gaspar and F. Ogor, “Estimation and analysis of the sea-state bias of the new ERS-1 and ERS-2 altimeter data”, Task 2 Rep. Contract 96/2.426 002/C, Inst. Fr. de Rech. Pour l’Exploit de la Mer, Brest, France, 1996. [5] M. Roca and C.R. Francis, “Identification and origin of the on-board bias jumps”, in Minutes of RA and MWR2 CWG(#9), ERSIN, ESA, 1996. [6] P. Moore, S. Ehlers, C.M. Murphy and M.D. Reynolds, “Investigation of the stability of the ERS-1 range bias through tide gauge augmented altimetry2, J. Geophys. Res., vol 104, 30021-30038, 1999. [7] C.A. Wagner and J. Klokocnik, “Making the connection between GEOSAT and TOPEX/Poseidon”, J. Geod, vol 71, 272-281, 1997. [8] B.D. Tapley et al, “Precision orbit determination for TOPEX/Poseidon”, J. Geophys. Res., vol 101, 28029-28049, 1996. [9] B.J. Haines and Y.E. Bar-Sever, “Monitoring the TOPEX microwave radiometer with GPS; Stability of columnar water vapour measurements”, Geophys. Res. Lett., vol 25, 3563-3566, 1998. [10] P. Moore and G.T. Kilby, “Absolute and relative calibration of ERS-2 with applications to ENVISAT”, this issue, 2000. [11] B.J. Kozel, C.K. Shum, J.C. Ries and B.D. Tapley, “Precision orbit determination using dual satellite altimeter crossover measurements”, paper presented at AAS/AIAA Space Flight Mechanics Conference, Florida, Feb. 1994.
Figure 1: AGM-98 geographically correlated and anti-correlated orbit error as inferred from DXO residuals between ERS-1 and TOPEX/Poseidon for the ERS-1 second multidisciplinary phase.
Figure 3. Temporal variations in coefficients A1 and A3-A7 of (1) using NCL orbits (black), DEOS orbits (red) and DPAF orbits (green). Upper figures show A1 and A3; middle A4 and A5; lower A6 and A7.
Figure4. Temporal variations in coefficients A1 and A 3 – A 7 of (1) using orbits derived solely from ERS-2 tracking and orbits from simultaneous TOPEX/Poseidon and ERS-2 tracking. Upper figures A1 and A 3; middle A 4 and A 5; lower A 6 and A7.
