In previous analyses (e.g., Wagner and Klokocnik, 1994), we used the sea height differences at crossovers to remove empirically the complex orbit error that occurs at or near one cycle per revolution (cpr) due fundamentally to the reinitialization of the orbit (e.g., Colombo, 1984), here at 4-10 day intervals (necessary to keep the trajectory errors within prescribed bounds). The principal reason for the periodic reinitializtion of the low orbits (Ers1 and Geosat) is poorly modeled atmospheric drag but the result of all empirical orbit adjustments is to absorb some of the errors in the trajectory due to the geopotential that would otherwise be seen in the altimetry. Likewise, using the altimetry to remove the time-varying orbit errors near 1 cpr every 4-10 days has the same geopotential degrading effect (e.g., Moore and Rothwell, 1990) and can remove significant oceanography as well (e.g., Wagner and Tai, 1994), though largely limited to tide errors as long as the times between the passes at a crossover are reasonably small.
In this analysis, to minimize absorbtion of both oceanographic and geopotential effects we chose to merely average the raw crossover differences (after application of the stated media corrections), leaving the long term averaging to reduce the significant 1 cpr orbit errors to small values. In Appendix A we show that this treatment indeed reduces these errors satisfactorily, leaving a small systematic effect which can be resolved simultaneously with the geopotential, and perhaps some added noise due to the timing and other circumstances of the orbit and state corrections of the satellites.
From the geographical form for 1 cpr error (appropriate to a near circular orbit) we have (Wagner and Klokocnik, 1994), for SSC data (see also Appendix A for more details):
or also
with 
For DSC data (subscripts applying to the respective satellites 1 and 2), where the crossover is taken as (pass of satellite 1 - pass of satellite 2):
We solve for the constants 
.
In addition to the four 1 cpr parameters for DSC data, we add a constant,
the same for all four types of a given DSC pair, to account for any
altimeter bias in one or both instruments not properly gauged in the
geophysical data records. These have been found to be of order
10 cm and easily removed in the solutions which follow.