Appendix A: One Cycle Per Revolution Error in Crossover Altimetry

In previous studies using the crossovers of the relatively low orbits of Geosat and Ers1, significant one cycle per revolution signals have been found empirically (e.g., Wagner and Klokocnik, 1994; Moore et al., 1998). (See also Sanchez and Cartwright, 1988 applied to Seasat altimetry). These arise from the reinitialization of the orbit parameters approximately every four days, the process absorbing inexactly mismodeled (generally non-conservative) forces on the satellite (principally atmospheric drag).

Thus for Geosat during the the first two years of its ERM (November 1986-October 1988) we determined independent cosine and sine coefficients of one cycle per revolution variation for each orbit reinitialization period using all crossover differences within each repeat cycle (Figure  A1). The reference for these coefficients is the orbit's argument of latitude and since they are determined from crossovers we constrain one sine coefficient per repeat cycle to zero to account for the singularity in the resolution (e.g., Tai and Fu, 1986).

Figure A1

One Cycle per Revolution Errors (from 4-day arcs) in Geosat ERM (Cy. 1-44) with Jgm3, Using Altimeter Sea Height Crossovers
(a) Cosine term, (b) Sine term; error = C cos u + S sin u, u = orbit argument of latitude.
These errors, determined empirically, seem random but C (without constraint) has a positive bias probably due to tracking system errors repeating with each cycle. The larger values in late 1988 are due to increased drag errors in the approach to the solar cycle high in 1990.

It was expected that such 1 cpr errors would have only long period variation over many repeat cycles if the reinitializations were only absorbing long period trajectory errors (e.g., Colombo, 1984). But these independent 1 cpr terms generally have good signal/noise ratios so that their considerable variation from arc to arc is evidence that our empirical evaluation of them here are not just from orbit errors but also from sea surface effects which vary over the separate 4-day tracks. Part of that seasurface effect comes from the orbit-geopotential error which we want to resolve later in a global analysis. Another part may come from tidal errors which we might also like to retain for later analysis.

But the dominant effect is still orbital; there is a general rise in the power of these terms starting in 1988 as the drag on Geosat increases due to the atmospheric warming towards the solar cycle high in 1990. In addition (and most important) the Cosine term has a clear bias which would be expected from orbital error due to reinitializing at roughly the same places geographically cycle after cycle. The geographic representation -of this bias is easily found from the relation between the orbit argument, inclination and the geocentric latitude:

(A1)

where u is the orbit's argument of latitude (the central angle along track from the ascending equator crossing), I is the orbit's inclination and is the geocentric latitude. (The positive root is taken for ascending tracks, the negative for descending). C and S are empirical constants to be determined from observations.

From this form it is seen that the SSC height difference (ascending-descending) is:

= (A2)

This strictly zonal variation has no counterpart in orbit-geopotential SSCs [Equation (1)].

Figure A2

Average AD (and residuals) from Geosat ERM 1 cpr Errors (Cy. 1-44):
(a) Average Bin A-D differences: Rms = 2.4 cm
(b) Residuals (Avg. bin - global 1 cpr estimate): Rms = 1.3 cm.
The average 1 cpr error clearly has a global bias as hinted from Figure 14. But the resolution of this bias does not exhaust the contribution of either global or local (cycle specific) 1 cpr error to the alitmetric crossovers. The global contribution here appears to be due to Sine components which have this peculiar distribution because the first arc's Sine term in each cycle has been constrained to zero (there are 4-5 such arc's in each cycle).

Figure A2a shows the 2 3 bin averages of the 550,000 crossover differences in the Geosat ERM (cycles 1-44) represented by the empirical C and S 1 cpr terms shown in Figure A1.

Figure A2b shows the residuals of these effects with respect to a global cosine term of 1.4  cm determined from fitting Equation (A2) by least-squares to the data in Fig. A2a. Not only is a global cosine term a fairly good fit for these empirical 1 cpr errors but the residuals to this term clearly show the pattern of the 4 or 5 orbit arcs that make up each Geosat ERM period which roughly repeat over the same areas from cycle to cycle. But this pattern might also be due to aliasing of orbit-geopotential errors into these 1 cpr terms. Thus, the residuals in Figure 1 show no such pattern after removal of the global 1 cpr term (included in the geopotential inverse).

Without overinterpreting these results (e.g., we doubt that Fig. A2a data is purely from 1 cpr orbit error) they certainly suggest the conservative strategy we adopted in this study, namely to avoid the removal of 1 cpr orbit errors empirically until the final stage of the analysis when it is resolved simultaneously with the geopotential.

Appendix B: Expectation for Semi-independent Solutions

Suppose we have two solution vectors derived by constrained least-squares [e.g., Equation (13)] from a combination of independent data and a common a priori asumption that the true parameters of the problem are zero with covariances . Comparing these vectors what can we say about their likely difference?

Using the notation of Equations (12) and (13), let the formal covariances of the two semi-independent solutions be:

(B1)

Assuming that the observations are given in terms of the true parameters with normally distributed independent errors:

(B2)

we see that the expected constrained least-squares solutions [from Equation (13)] in terms of the true parameters are:

(B3)

with a unit matrix, since .
Thus, the expected difference of the two solutions ( ) is given by:

(B4)

We do not know the true values of the parameters but, except for cases where the covariances are the same or the true parameters are zero, we see that the differences of these solutions will be biased statistics. However, the expectation of the ratios of these differences (or their squares) to their expected values will still be 1.0 and in particular, as we shall soon see, the expectation of the squares (of the differences) may be well approximated by using the a priori covariances of the parameters in their conventional interpretation.

Thus, as a full matrix of difference products, we want to find:

(B5)

For example, for , from Equation 13, we have:

since and are always symmetric matrices. Then since

and ,

then

and

Working out the expectations for the other three solution products above in terms of the true parameters in the same way and noting that the expectation of products of errros is zero when , we find that:

(B6)

Approximating by , the covariance matrix of Jgm3 for example, or equivalently, taking the expectation of the expectation above over many random samples of "true parameters" all having the same covariance matrix, we find:

(B7)

The extremes of this expectation matrix should be noted. If there is no or very weak common a priori information , and Equation (B7) reduces to the usual expectation for the difference of two independent solutions:

(B8)

If one of the two "independent" solutions adds nothing to the common information (e.g., has very weak data) then that solution (e.g., 2) will be the same as the a priori (here zero) and its covariances then will be , unchanged from the a priori. Under these circumstances the difference of solutions is merely solution 1 itself as an adjustment from the a priori (here, zero). Its expectation matrix, modified from (B7), is:

(B9)

Note that the diagonal elements of this matrix are always positive since independent data always results in a formal reduction of the a priori variances in a combined solution. The expectation in Equation (B9) is another statement of the result in Lerch et al (1991) for the difference between full and subset solutions. Here the full solution includes the a priori information as common data (taken as zero, the reference parameters); in Lerch (ibid), the common data were actual observations.

A further simplification (or approximation) can be made to the general result [Equation (B7)] for cases where the three covariance matrices (for the two semi-independent solutions and the a priori information) are nearly diagonal. Notice that from the triple matrix products of Equation (B7) only the resulting diagonal terms contain products of the three diagonal elements of these matrices. Further, of the other triple product parts of these diagonal terms, all contain at least two off-diagonal elements of the three matrices. Since these off-diagonal elements represent covariances between different parameters likely to be small relative to the diagonal elements we may say that these triple products are second-order small relative to the the dominant product of diagonal elements only.

In particular Equation (B7), counting diagonal terms only, becomes in this approximation:

(B10)

where are the a priori and and the variances of the two solutions for parameter x.

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