Semi-independent Geopotential Solutions

While the "along track" structure of the SSC and DSC data in Figure 2a looks like geopotential orbit error and Klokocnik et al. (1999) confirmed that most of the detailed spectrum of it is indeed consistent with the Jgm3 covariance matrix, can we be sure our computed inverse of it achieves genuine geopotential corrections? Both the existence of systematic residuals (Fig. 2b) and the need for extensive downweighting of most of the DSC data argues for at least some serious aliasing in the solution. What we would like ideally is to prove the validity of our corrections on an independent data set sensitive to these corrections.

For this purpose, we chose two semi-independent solutions, one from exclusively SSC data (on all three altimeter satellites) and the other with all DSC observations on the 3 pairs (Geosat-Ers1, Geosat-T/P and Ers1-T/P). Both the location and the timing of the crossovers in the SSCs are distinct from those in the DSCs while the inclusion of all three orbits in each solution helps overcome the natural tendency of the strong a priori normal matrix to degrade with the addition of only one or two new satellites. In addition the DSC sets test all parts of the geopotential (zonals as well as non-zonals, the mean as well as the variable part of the radial error at a given location). They also involve different media corrections over two missions often separated by many years. As we have seen their residual characteristics are quite different from those for SSCs.

However, as we alluded to above, both these sub-set solutions as well as the complete ones are also strongly conditioned by the a priori common errors in the low degree and order and higher resonant orders of the Jgm3 geopotential (Figure 7). The separate sub-set solutions are not truly independent. How can we judge the influence of the a priori errors on these inversions? Theoretically, even a single observation added to Jgm3, results in an error ratio (solution/a priori) of less than 1.0 for all coefficients. But Figure 7 reveals that the addition of many thousands of these crossovers (most with strong signal/noise ratios) has only a mild impact beyond this. Why is the overall effect so muted?

Figure 7

Correction Error/Apriori for Three Adjustments to Jgm3 from Crossover Altimetry.
Solid = Combined SSC+DSC solution,
Heavy Dashed = SSC-only solution,
Light Dashed = DSC-only solution.
The range of ratio's shows the significant effect of the a apriori on the correction especially at low order where the number of parameters resolved (and correlations) are greatest. Note the additional degradation of the ratio at the resonant orders (12, 13, 29 and 43). The spike at order 43 is due to a complete cutoff of these 17-day effects in Geosat data (assumed to be completely absorbed in the 4-day orbit redetermination process for this satellite).

The fundamental reason has been alluded to before. The three orbits here do not have enough variety to improve the condition of Jgm3 globally without a reweighting of the data in that model. Overall, the correlations in the covariance matrix after the solution are greater than before. Still Figure 7 shows the impact of the a priori on the solution errors declines with order. Why? Recall from our discussion of Figure 4 that in a satellite geopotential the information essentially separates by order. In the recovery of a solid field (with the same maximum degree and order), at high order there are only a few degrees in the solution to absorb that information leading to low correlations between them and much better conditioning (less a priori impact) than at low order where the correlations are much higher between larger number of degree-terms. The poorer conditioning of the low order terms holds up their improvement over Jgm3. But we should likewise be skeptical of the high-order result as well since by truncating the solution at degree 50 we probably introduce excessive truncation error into the higher order coefficients.

Figure 8

Correlations in Two Semi-independent Corrections of Jgm3 from Noaa Crossover Altimetry. Jgm3 covariances are common constraints for both corrections.
Correction 1: from DSCs only: Geosat-T/P, Ers1-T/P.
Correction 2: from SSCs only: Geosat (GM+ERM), Ers1, T/P (cycle 2-142): green (by order), red (by degree). For the solid lines, data in each solution has been edited and (normally) weighted to yield a standard error of fit of 1.0. For dashed line (degree only shown) the data in each solution has been downweighted to calibrate with Jgm3 (see text and Figure 6). Notice the improvement in the comparison at almost all degrees. The overall correlation coefficient (2546 samples) for all terms r = 0.52 with normal weighting but 0.64 with the data downweighted for proper calibration. The largest disagreements, at degrees between 6-18, probably are the result of distortion in the DSC-only correction from the interannual oceanographic variation over the 8 year gap in the G-T DSCs (see text).

Notwithstanding these cautions, how do the two semi-independent SSC-only and DSC-only solutions actually compare? Figure 8 shows the correlations between the same terms of these solutions by order and degree. On the whole the agreement seems good, much of the independent crossover information certainly refers to the same geopotential adjustment. Notice that by order the correlations are fairly high uniformly while by degree they are only poor between about 10 and 20. It happens that the largest discrepancies between these solutions occur in the low orders (1-3) which, because they tend to share the same (lumped) information with the most degrees, have the highest correlations. For that reason it may seem reasonable to have this result. So we need to know if it is normal, that is whether we might expect it if the actual errors in our data were "normal", or randomly distributed, without systematic correlation as we had asssumed in our weighted least-squares formulation [Equation (13)]. In Appendix B, we work out the expectation for the difference of two such semi-independent solutions from data with normally distributed errors.

The expectation (in Appendix B) for the difference of two common parameters in these kinds of solutions is formally in terms of a composite covariance matrix made up of the covariance matrices of the three constituents, the two solution matrices as well as the a priori. Consider the square of any such difference divided by its expectation. If the data errors were normally distributed then each such statistic would be -squared (with one degree of freedom) but because the composite matrix is not diagonal, these statistics would not be independent. However, we can always diagonalize such a matrix, finding its independent eigenvectors, and we know from this process that the the sum of the diagonal elements of the composite is invariant under the required rotation. Further, we also know that the original three covariance matrices as well as the final composite is nearly block-diagonal in terms of the individual geopotential orders. Therefore, to simplify the interpretation, we show the result of the sums of these (squared) statistics by order, noting their normal expectations as independent -squared variates (merely the number of terms compared). In making this calculation we use the convenient approximation of the triple matrix product as the product of the variance elements only [Appendix B, Equation (B10)]. Figure 9 presents this result.

Figure 9

Comparison of Two Semi-independent Corrections to Jgm3 from Noaa DSC-only and SSC-only Altimetry.
The two correction fields (with normally weighted data) are specified in Figure 8. Shown are the sum of the squared ratio's of the difference of the same terms for each correction to the estimated "normal" error for these differences by order (see Appendix B): Red = C(L,M), Green = S(L,M) terms, Dashed = expected (see Appendix B and text). The abnormal comparisons are confined to orders less than 7.

Figure 9 shows that (fairly in line with our expectation from Fig. 6a) the comparison only for orders less than 8 is strongly abnormal (larger than expected), while most of the higher orders are well within expectations. As the residuals in Figure 2b clearly show, the abnormal error distributions are mainly the fault of the DSC sets but even the SSC sets show residuals with some systematic geographic abnormalities. In fact the final downweighting of the data in the comprehensive solution, as we saw, was a cumulative response to a similar lack of agreement between the corrections and their expectations based on more optimistic weights.

At this point in the analysis we can only say that by downweighting the data in the comprehensive solution (decreasing the precisions used) we reduce the chance that the non-geopotential biases can infect the geopotential corrections. Likewise we can say that the downweigting probably leaves the the residuals as clean as possible to represent these non-geopotential biases.

What are these biases? From Figure 2b, we see that the strongest of them are in the multi-year one's, Geosat-T/P and Geosat-Ers1. Further, in the transition from data to residuals we see how the trends go from North-South along the orbit traces, presumably geopotential anomalies, to East-West zonally for the most part, suggesting oceanic anomalies.

Though the multi-year gapped residuals in Figure 2b look oceanic how can we verify this conclusion?

We tried to do this here in 3 ways, (i) by comparison with sparse tide gauge records at stations in the Pacific, (ii) by gross comparisons with General Circulation Models (GCM), "global" in extent, and (iii) by testing the consistency of our two gapped periods with intermediate differences from independent colinear Ers2 altimetry.

In the next section, we compare our original crossover data and then the residuals after the inversion to tide gauge data in the Pacific to see to what extent the residuals improve the gauge results.