Residual Effect of Tidal Errors

The residuals of our solution (Figure 2b and Table 3) can be catogorized into three classes based on their power. The SSCs have the smallest power (1-3 cm rms), the "contemporaneous" DSCs are next (2-3.5 cm rms) and the multi-year-gapped DSCs the highest (4-7 cm rms), probably dominated (as we have seen) by interannual oceanography.

The SSC residuals in Figure 2b are seen to resemble "white" noise mostly, except at ocean/sea margins where there are anomalies and in certain deep like white noise with some large scale trends at a level of 1-3 cm. Recalling the error budget for the altimetry in these satellites, one of the possible causes of the small-power and scale unresolved residuals is luni-solar tidal error (Table 2).

Each bin average of a given data set (crossover type) is made up of (on average) a few hundred crossover differences amounting to over a thousand crossovers of all types for most bins (which are represented by more than one kind). But because of the aliasing inherent in the sampling from these exact repeat missions, the distribution of the time differences for these is far from random with respect to the various tide frequencies. Generally though, even in bins with good coverage (amount, time distribution and number of crossover locations) we find the averages for a single data type to be sensitive to tide errors at a level of 1-2 cm, with more severe changes in certain coastal areas where the tide errors are greater. However in extracting changes in these averages from unbalanced tide discrepancies we find the uncertainty of the determined changes to be at about the same level even with good coverage. (The best results were found in bins with all SSC and DSC types present over a wide range of times).

For example, in each bin k we adjusted the original (media corrected) Pathfinder altimeter crossover differences in terms of an error tide series (to the Pathfinder model of Schrama and Ray, 1996), with constant parameters across the bin:

where is the time for the pass of satellite 1 using data type j and is the time for the pass of satellite 2 using data type j at a crossover location in bin k, and where we resolved i over the 8 principal and secondary luni-solar tides (of diurnal and semi-diurnal frequencies : M2, S2, K1, O1, N2, K2, P1 and Q1) on fitting Equation (15) to all DSC and SSC differences in the bin by least-squares. [Note, we did not determine errors in the tides of fortnightly (Mf), monthly (Mm) or longer period because our crossover time differences were either predominantly much less than a month apart (SSCs and "contemporaneous" DSCs) or close to integer-years apart (other DSCs). In either case the resolution of these tide errors from our crossovers was found to be poor.] In the solution for each bin there could be up to 15 crossover types j (4 DSCs for Geosat-Ers1, 4 DSCs for Ers1-T/P, 4 DSCs for Geosat-T/P and 3 SSCs for Geosat, Ers1 and T/P) each with it's own constant (Eq. 4) resolved from all the crossovers in the bin. These type-constants were compared to the original averages in the bins (without tide correction) to assess the effect of the tide resolution on the stationary "geopotential" information for that type.

Figure 14

Comparison of (a) T/P SSC Pathfinder residuals (from 50x50 Geopotential solution), Rms = 1.3 cm, average SD of original data = 0.8 cm to (b) T/P SSC biases determined empirically from eight-tide correction to all Pathfinder crossovers (Rms = 0.6 cm, average SD of bias's = 0.7 cm).
The tide bias is a small but significant contributor to long-term averaged crossover altimetry.

An example of these error-tide-shifted type-averages is seen in Figure 14a where we compare the tide-only biases for T/P SSCs against actual residuals from an all-Pathfinder 50x50 geopotential correction of Jgm3 using all sets of SSC and DSC altimetry for the three satellites (Figure 14b). Clearly, even in the case of the low power residuals for T/P, the tide adjustment, though substantial and with fair correlation with the residuals, still accounts for only a fraction of the values in detail in most areas. So the comparison suggests averaged tide errors are not the only source of the T/P residuals. However, as the uncertainty of the tide bias on average is of the same order as the value determined (and also the same order as the T/P data uncertainties and the T/P residuals), we should not expect the correlation to be any better.

With regard to the other SSC data sets we expected to find even less relative tidal influence on the residuals since both Ers1 and Geosat have poorer media corrections than T/P (see Tables 1 and 2) and more non-tidal oceanic influences over the increased time differences between crossover passes. But for both we actually found the influence was just as substantial as for T/P, probably because the crossover data distribution was poorer, with more limited time sampling than for T/P.

We conclude that tide error accounts for an important part of the crossover information in the SSCs and "contemporaneous" DSCs not resolved by geopotential and other orbit and bias parameters. However, as the T/P analysis of the tide solution shows the tide errors are only a contributor to the remaining biases in the data. For example, the tide bias cannot explain much of the large scale biases in the near-contemporaneous Ers1-T/P DSCs from NOAA altimetry (Figure 2b), which is not likely to be oceanographic. Analysis in Klokocnik et al. (1999, 2000) with combinations of the DSCs show that the likely cause is a media correction model (probably in sea state bias) for T/P which is incompatible with the Ers1 passes.