Effect of orbit precision on ERS Sea State Bias models
This page presents the results of a small study into the possible aliasing of orbit errors into parameters of sea state bias models correcting ERS1 and ERS2 altimeter range measurements depending on sea state and wind speed.
The data
The results obtained here are based on the latest Version 6 Ocean Product (OPR) provided by CERSAT. Corrections include (corrections in bold face are not provided on the OPR data sets):
 Time tag correction of 1.5 and 1.3 ms for ERS1 and ERS2, respectively
 GFZ/DPAF PGM055 or DUT/DEOS DGME04 precise orbits
 Radiometer wet tropospheric correction
 ECMWF Dry tropospheric and Bent ionospheric corrections
 Gaspar and Ogor [1996] Sea state bias corrections:
SSB(ERS1)=SWH(0.047  0.0035 U + 0.000160 U^{2})
SSB(ERS2)=SWH(0.048  0.0026 U + 0.000126 U^{2})
 Grenoble FES95.2.1 ocean tide and loading
 Solid earth and pole tide
 OSUMSS95 mean sea surface
After editing all measurements are converted to singlesatellite crossover height differences spanning the period of 20 June 1995 till 2 June 1996 and with a maximum time interval of 17.5 days between ascending and descending tracks.
The approach
From this data we estimate the coefficient b in the simple BM1type sea state bias model in addition to the models given above:
SSB = b * SWH
In fact, we have enhanced this model by estimating one coefficient b_{i} for each SWH interval of 1 meter. Assuming the crossover height differences are due to errors in the b_{i} coefficients, this leads to observation equations:
Crossover difference = SSH_{1}  SSH_{2} = SSB_{1}  SSB_{2} = b_{i} * SWH_{1}  b_{j}* SWH_{2}
The coefficients b_{i} (i=1...8) are solved for in a least squares minimisation of all crossover height differences.
The two alternative orbit solutions have fairly different radial precision (GFZ/DPAF approximately 67 cm and DUT/DEOS approximately 45 cm) and differ radially by about 5 cm rms. This provides the opportunity to test the effect of orbit errors on the sea state bias estimation. At the same time, we can validate the original Gaspar and Ogor [1996] BM3type model.
The results
The table below lists the b_{i} coefficients (in percents) obtained from the analysis of the crossover height differences, for ERS1 and ERS2 and based on DUT and GFZ orbits. The onebutlast line gives the effect of the corrections to the SSB model: the variance of the changes to the derived sea height implied by these corrections. The bottom line gives the overall value in case only one b coefficient was estimated, as in a true BM1 model.
SWH  ERS1  ERS2

(m)  number  DUT  GFZ  number  DUT  GFZ

01  90709  0.595  0.415  42695  0.103  0.066

12  254463  0.484  0.450  198712  0.278  0.099

23  215105  0.450  0.432  207745  0.263  0.141

34  115812  0.380  0.368  122849  0.276  0.137

45  52027  0.216  0.237  62725  0.256  0.141

56  20668  0.040  0.006  28233  0.162  0.086

67  7343  0.033  0.130  11542  0.065  0.003

78  2471  0.119  0.227  4307  0.117  0.040

Var (cm^{2})   0.376  0.352   0.192  0.050

08  758598  0.370  0.355  678808  0.244  0.118

The values are presented in graphical form below. Circles are for ERS1, diamonds for ERS2. Blue (light grey) markers are for DUT orbits, red (dark grey) markers for GFZ orbits.
The conclusions
 The adjustments to the Gaspar and Ogor sea state bias models are of the order of 0.2 to 0.4% of SWH and would result in a 0.05 to 0.38 cm^{2} correction to the sea surface height. Naturally, the adjustments do not include effects of windspeed, but results show a clear tendency that the current sea state bias is underestimated for the range of 15 meters of SWH.
 Both the trend of the SSB as a function of SWH and the overall implied variance are similar to the differences found between a BM3 and BM4 model. Yet, the variances appear too small to warrent the selection of a BM4 over a BM3 model.
 Remarkably, the selection of the orbit (DUT or GFZ) does have little effect on the ERS1 sea state bias adjustments, but certainly has a considerable impact on the ERS2 results. In the later case the estimated adjustments are about 0.1% for SWH up to 5 meter.
 The effect of orbit error on the SSB coefficient also urge for caution in generating SSB models and prevailing one over the other. The SSB estimation still is quite sensitive to many external effects.
Acknowledgements
Thanks to Philippe Gaspar for his comments.
Questions or comments:
Remko Scharroo, remko.scharroo@lr.tudelft.nl.
20 May 1997.