We have examined over 10,000,000 DSC and SSC altimetric differences on the three satellites Geosat (March 1985 - November 1988), Ers1 (April 1992 - December 1993), and T/P (October 1992 - December 1996). As outlined above the different passes in each observation are either contemporaneous, for SSCs within each repeating orbit cycle (17 days for Geosat, 10 days for T/P and 35 days for Ers1), or within the same months for the DSCs of Ers1 and T/P (October 1992 - December 1993). Here have been two sources of these observations (involving principally the corrections to the same altimeter observations), one from NOAA during the early stages of the project (prior to 1997) and later from NASA (Pathfinder altimetry), incorporating updated media corrections after 1996). The original measurements are given in both NOAA and NASA versions as sea heights above a reference surface (for NOAA the reference is a detailed geoid above a certain ellipsoid, for Pathfinder a specific mean sea surface above another ellipsoid), in both cases the reference sea surface is the same for all altimeter measurements on the three satellites. Thus ideally in either case (not mixing the two sources) the crossover difference is indifferent to the reference surface.
The geopotential orbit model for all satellites except Pathfinder for Ers1 (Scharoo and Visser, 1998), is Jgm3 (Tapley et al., 1996). Our altimeter observations are independent of the data used to determine this model from combined satellite tracking and surface gravity anomalies. In particular Jgm3 has no tracking data for Ers1 (but some on the nearby orbit of Spot 2), considerable Doppler tracking information on Geosat and extensive observations (from Laser, Dopper and GPS) on T/P. We note also that most of the more precise marine surface gravity anomalies in Jgm3 were derived from large amounts of direct Geosat altimetry. However, it appears that these did not account for the geopotential effects on the radius of the Geosat orbit which we consider here.
In our application, we convert the measurement legs of Pathfinder Ers1 crossovers to Jgm3 basis by analytically projecting the radial effect of the difference of Jgm3 and the Pathfinder Geopotential for Ers1 (Dgm-E4, Scharroo and Visser, 1998) pass (either ascending or descending). Table 1 summarizes the media corrections to the altimeter observations that were applied to the source altimeter heights as well as their reference orbit and sea surface models. Notice that the only dynamic ocean surface models we applied initially were the so-called inverse barometer correction, the potential response of the water column to atmospheric pressure changes, and a tide model (both for the solid earth, the ocean itself and the effect of its load on the bottom). Table 1 also gives the time periods and amounts of SSC measurements used and their average point power (after only the media corrections) before binning.
Table 1
Corrections and Statistics for NOAA and NASA-Pathfinder Altimetry
used in Geopotential Solutions.
| Correction | Geosat | Geosat | T/P | T/P | ERS 1 | ERS 1 | |
| Data source | NOAA | NASA | NOAA | NASA | NOAA | NASA | |
| wet troposphere | TOVS+SSMI | SSMI (Kalnay) | TMR | TMR | SSMI | SSMI (Bernard) | |
| dry troposphere | ECMWF | ECMWF (S72) | ECMWF | EGMWF | NMC | ECMWF (S72) | |
| ionosphere | DORIS+T/P | DORIS+T/P | DORIS+T/P | DORIS+T/P | tuned Bent | IRI 95 | |
| EMB/sea state bias | 3% SWH | G 94 | FG91 | FG91 | 5% SWH | GO 96 | |
| ocean tides | CSR 3.0 | SR94+pt | CSR3.0+pt | SR94 | CR91+W94 | SR94+pt | |
| ocean load tides | RS89 | RS89 | RS89 | RS89 | RS89 | RS89 | |
| solid (body) tides | CE73 | CE73 | CE73 | CE73 | CE73 | CE73 | |
| 1 cpr (empirical) | none | none | none | none | none | yes | |
| # of data | 1,100,000 XD | 600,000 | 850,000 XD | 400,000 | 400,000 XD | 350,000 | |
| data span | Mar85-Sep86 Nov86-Nov88 | Nov86-Nov88 | Oct92-Dec96 | Oct92-Aug94 | Apr92-Dec93 | Apr92-Dec93 | |
| x-over rms long-term bin avgs.[cm]: | 7.8 | 9.1 | 2.9 | 1.9 | 7.4 | 6.2 |
EMB ...... Electromagnetic Bias of radar sea level,
sea state bias
SWH ..... Significant Wave Height ( 
),
CR91 .... Cartwright D.E., Ray R.D. (1991): JGR 96, 16897-16912,
CE73 .... Cartwright D.E., Edden, A.C., (1973): J. R. Astron. Soc. 23,
253-264
TMR ..... TOPEX/Poseidon multichanel Microwave Radiometer,
SSMI ..... Special Sensor Microwave Imager,
TOVS ..... Tiros Operational Vertical Sounder,
ECMWF .. European Centre for Medium-range Weather
Forecasting,
NMC ..... the US National Meteorological Center (model),
IRI 90 ..... International Reference Ionosphere,
Belitza D. et al (1993): Adv. Space Res. 13(3), 3-23,
IRI 95 ..... International Reference Ionosphere,
Belitza D., priv. commun.
W 94 ...... Wagner C.A. (1994), JGR 99 (C12), 24853-24865,
pt ........ pole tides (order 1 cm,
period 14 months (Chandler wobble),
Bent ...... after Llewellyn S. and Bent R. (1973):
Documentation and description of the "Bent" ionosphere,
Air Force Cambridge Research
Laboratory-TR-73-0657; "tuned" to 1992-93 ionosphere conditions
FG91 ..... Fu L., Glazman R. (1991):
The effect of the degree of wave development on
the sea state bias
in radar altimetry, JGR 96, 829-834
CSR 3.0 ... Center for Space Reseach tidal model, Texas Univ.,
R. Eanes, priv. commun., 1996
RS89 .... Ray R, and Sanchez B (1989): Radial Deformation of
the Earth by Oceanic Tidal Loading,
NASA Techn. Memo. 100743, GSFC, Greenbelt, Md.,
SR94 ..... Schrama and Ray (1994), JGR 99, 24994;
with improvements in 1995/96
S 72 . ...... Saastamoinen J, Geophys. Monography 15, AGU
1972
Bernard .... Bernard et al., IEEE Trans. Geosci. and
Remote Sensing 31(6),1186, 1993
Kalnay ... (SSMI/TOVS) Kalnay et al., Bull. Ame. Meteo. Soc.
77, 437, 1996
G 94 ....... Gaspar et al., "Analysis and Estim. of Geosat
Sea State Bias, 1996 (http://neptune.gsfc.nasa.gov)
GO 96 .... Gaspar and Ogor "Rep. on task 2 of IFREMER",
1996 (http://neptune.gsfc.nasa.gov)
XD ....... cross-over height difference data.
The bins were sized to recover a 50x50 geopotential and the power of these averages (also in Table 1) are a substantial reduction from that of the original observations which were (as will be seen) affected by substantial time-varying errors.
The analytic transformation for converting the Dgm-E4 base original Ers1 sea heights to Jgm3-basis is given in the next section but here we compare Ers1 SSCs (long term 2x3 degree bin averages) from the NOAA data (derived from Jgm3 orbits) with two of these conversions of original Pathfinder Ers1 SSCs (Figure 1). The first conversion (Figure 1b) is based just on the difference between the Jgm3 and Dgm-E4 geopotential's effect on the radius of Ers1, the second (Figure 1c) on a straight differencing of the radial ephemerides of the two orbit models from April 1992 to December 1993.
First notice the power of the Dgm-E4 basis SSc (Fig. 1.a) are significantly smaller than the two conversions. The reason is simply that this geopotential model was tuned to Ers1 with both ordinary tracking and crossover altimetry (to control 1 cpr error). While the two conversions of Pathfinder SSCs to Jgm3-basis appear equally close to the NOAA version (Fig. 1d), the trajectory differenced one is slightly superior in this example. However, we cannot be certain this slight superiority is not due to small differences in the media corrections for this particular Pathfinder and NOAA data set (Table 1) and also because the analytic conversion does not account (at this stage) for other time-varying orbit differences (at 1 cpr primarily) which will be resolved in the final inversion. For these reasons we chose to use the simple and flexible analytic conversion for all Ers1 Pathfinder sea heights.
Table 1 also gives the time periods, amounts and power of the DSC measurements used (point and bin averages), namely for Geosat-T/P, Geosat-Ers1 and Ers1-T/P.
What are the errors likely in a typical altimetric sea height "measurement" for the three satellites examined here? There are three generic sources of these associated with nature of the measurement itself (a subtraction of an altimeter distance from a trajectory height). The trajectory height can be in error, the measurement height along its path to the ocean can be in error and the estimation of the ocean's mean surface in the radar's footprint can be in error due to so-called electromagnetic bias (EMB) from waves.
Table 2 is a summary of the error budget for altimeter heights estimated for both NOAA and NASA Pathfinder sources (reduced to a Jgm3-basis for Ers1), following Wagner and Klokocnik (1994) and Klokocnik et al. (1996). The largest uncertainties are for the Geosat heights, 11 cm rms, due principally to the geopotential on the low orbit but also from relatively weak environmental corrections (ionosphere and wet troposphere, not as well known in 1985-88). Ers1 heights we assess as in error by roughly 9 cm rms, a relatively low orbit again dominated by geopotential uncertainty with better control over the environmental corrections and, for Pathfinder, use of altimeter crossovers in orbit determination. The T/P heights appear to be uncertain at a level of about 6 cm rms with low geopotential error on its relatively high orbit (about 1 cm) and well measured ionosphere and wet troposphere environmental corrections with on-board instruments.
Note that the principal orbit error component after the geopotential is due to incorrect initial conditions (mainly the 6 fundamental state components of the trajectory) which manifests itself at 1 cpr. In past analyses of satellite altimetry (e.g., Wagner and Klokocnik, 1994; Moore and Ehlers, 1990) this was removed empirically by use of crossovers exclusively within each orbit-arc of data spanning about 4 days (for Geosat and Ers1) and 10 days for T/P. Here, we take a different approach and initially leave this error unresolved as a part of the bin averages. Our reason is that the 1 cpr effect over the long period analysed will generally consist of a small constant and a larger component which fluctuates from one orbit cycle to the next. The fluctuating part tends to cancel in the average and the mean can be closely represented by a simple zonal function of the latitude which can be resolved in the global anlysis along with the geopotential (see below, also Appendix A).
We treat the time-tag error in a similar fashion, only as part of the global analysis, instead of on an arc by arc basis. In this way, not only is the preprocessing of the observations greatly simplified but the removal of important oceanographic and geopotential information in the large number of these short-term empirical parameters is avoided.
As alluded to above, we sought to minimize the errors in these complex media models by choosing crossovers with a little time between passes as possible. For the SSCs, we took those only in each natural orbit repeat cycle (17 days for Geosat, 35 for Ers1 and 10 for T/P) though the average time difference in these were less than half of these figures. For the DSCs, we segregated all passes for measurements by months, contemporaneously for pairs involving Ers1 and T/P (between October 1992 and December 1993), and with various multi-year gaps for Geosat and Ers1, Geosat and T/P (Table 1).
Notice there is a natural order for these media errors
which we anticipate: the smallest should be found for T/P SSCs
(with 
measured ionosphere and
wet troposphere delays, as well as the shortest cycle time),
the next largest from Ers1 SSCs
(with more current delay models than for Geosat), the next
from Geosat SSCs (at least small non-tidal ocean dynamics and other media
model errors within the few days between ascending and descending passes),
the next from Ers1-T/P DSCs (with model errors peculiar to two missions, not
one), and finally the DSCs from Geosat-T/P and Geosat-Ers1 with ocean
interannual effects also affecting the crossovers. It is the last of these
error sources which has proved most surprising (Klokocnik el al.,
1998, 2000).
We knew interannual ocean dynamics would affect our measurements (e.g.,
Wagner et al., 1997) but we hoped that after averaging our
crossover measurements in both space and time these errors as
well as from the other sources would be both reduced in amount and would
lose much of their systematic character in space, crucial to geopotential
recovery. Perhaps most surprising was not that the interannual ocean dynamics
would retain its character after this averaging treatment (interannual
implies long-term) but that we could distinguish this dynamic action from
the orbit-geopotential in our measurements. We will have much more to say
about this aspect later (see also Klokocnik et al., 2000.)