As described above (in Section 3) we condition the 70,000 2x3 bin averaged
crossover data among the three missions by the following matrix equation
for o observations
linear in p true parameters 
connected by the design matrix 
:
where the geopotential coefficients 
in Equation (2) are
the true values of these minus Jgm3's, and are active in
each equation, and the 1 cpr, time-tag, altimeter bias and geocenter shift
parameters described above are "local" to a specific SSC or DSC observation.
is the true error in each observation for which an initial
statistical
estimate 
is supplied by the standard error of the average value
found for the crossover observations in the bins (Figure 2c).
The true error for each observation in Equation (9) obviously depends
on the model parameters p to represent it (the so-called "truncation" part
of the error since no model is ever perfect) as well as the inherent "noise"
due to natural variability in the measurements
(the so called "commission" part of the error estimated statistically by
).
Comparing Figure 2b with 2a we see that overall the observations
(largely Jgm3 geopotential-orbit errors as demonstrated in
Klokocnik et al., 1999), have a signal to noise ratio of about 5/1 except
notably for T/P SSCs which has benefitted from extensive tracking with
the Global Positioning System (GPS) as well as conventional (Doppler and Laser)
instruments (Tapley et al., 1996).
The original crossovers (except for T/P SSCs) had
average power of about 15 cm with about 10 cm "noise"
so that though the bin averages (with an order of 100 observations each)
have reduced the signal power to below 10 cm (eliminating
much of the fluctuating error sources), the process has also reduced the
estimated errors of these averages to less than 1 cm in most cases, a
large gain in signal to noise ratio.
What set of geopotential coefficient corrections to Jgm3 should we provide to resolve these crossover observations in the above equations? Since these coefficients are still substantially correlated (at least among those of the same order, e.g., Lerch et al., 1991; Klokocnik, 1988), it is difficult to give a precise answer. In previous studies of these crossovers (e.g., Klokocnik et al., 1998, 1999), it was shown that except for some higher order (m) resonances, the covariant projection of Jgm3 error (70x70) to these observations was negligible beyond about degree 50. In addition, as a practical matter, since we have chosen 3 degree bin spacing in longitude, our cutoff order for the geopotential should be less than 60.
Figure 3 illustrates some inversion experiments we have made with actual
SSC observations to determine the
truncation point for parameters in the above conditioning equations.
Since in all realistic cases we deal with many more observations than
solved-for parameters (o much greather than p),
we inverted the equations for the parameters according to a
least-squares criterion
for discrepancies between the observations and computations with the design
matrix , using the estimated parameters.
The simple least-squares criterion
(weighting each equation by the estimate of its observation
error) results in the familiar inversion:
where 
= 
is the weight matrix of the observations,
with E[.] = 
here taken as diagonal
since each observation is independently measured.
(The matrix inverted on the right in Equation (10) is called
the "normal" matrix). Formally, 
are the estimates of
which minimize the sums of squares of weighted residuals of
the actual observations 
with those computed from the design
matrix.
However, we recognized early in these experiments that such "free" (otherwise unconstrained) solutions result in unacceptably large geopotential corrections to original Jgm3 values. With only a few harmonics resolved the conditioning of the "normal" matrix for them may be reasonable but, as Figure 3 shows, the truncation errors in the equations are unacceptably large. The structure of the residuals in Figure 3 clearly shows that we need to resolve geopotential orders for Jgm3 beyond 30, as well as include time-tag corrections (with a north-south zonal signature in SSCs) but it does not reveal any detail of the individual harmonic (l,m) contributions.
In Figure 4, we supply this detail for each coefficient of the Jgm3 error matrix (variances only) using Equation (2). For Geosat, with heavy representation in Jgm3, ignoring the covariances exaggerates the projection of total error over its true result so our judgement of individual effects is conservative here and, as Figure 4 shows, well over half of the individual harmonic errors in Jgm3 fall below the precision of the SSC observations (for average 2x3 bins).
With the exception of the low and a few higher orders (resonant with the Earth's rotation), the observations (even of the heavily averaged bins) are not precise enough to affect more than a small fraction of a solid high degree and order field. (This is the typical problem in Satellite Geodesy due, as mentioned above, to the strong supression of the high degree geopotential with altitude). The error projections are similar on Ers1 which is at about the same altitude as Geosat. (The much higher orbit of T/P, well tracked in Jgm3, shows little likely error in these observations above their 0.5 cm precision at any order or degree).
However, the likely errors for the orders as a whole (with both Geosat and Ers1) tell a different story. Except for about 13 orders in the 30's and 40's, there is at least one harmonic whose error (in Jgm3) projects at or near the precision in the Geosat or Ers1 crossover observations. Even if the other degree terms of the sensitive orders project below this level the generally high correlation of the same-order terms suggests we should include all terms of this order in the solution (up to a reasonable maximum). The minimum number of harmonics we should resolve conservatively with our data would then be a substantial part of a solid 50x50 field (about 2600 geopotential parameters).
Inevitably then, if we would reduce the truncation error (for the two altimeter orbits most affected by Jgm3) by expanding the harmonics in the solution significantly, we degrade the condition of the normal matrix and (without additional constraints) the resulting inverse again yields unacceptably large corrections. The full story for why this occurs in Satellite Geodesy is only hinted at in Figure 4. It may be summarized by the generalization that the full spectrum of the geopotential present at sea level is so filtered at altitude that its action on any specific orbit contains only a few strong lines inadequate for full resolution. Indeed even with the large number of current satellite orbits in fields such as Jgm3, correlations among terms (of the same order principally) are still high (see Lerch et al., 1991 for a more complete discussion of this problem).
The classical technique to permit a more complete resolution of an incompletely observed (less than ideally conditioned) system of equations is known variously as "constrained" or "Bayesian" or a priori (modified) least-squares or colocation (e.g., Moritz, 1972).
The idea is simple. Before we even look at our data we know something about the limits of likely corrections to Jgm3. These limits are expressed quantitatively in its covariance matrix which has already been well calibrated on previous satellite observations (Tapley et al., 1996). The simplest expression of this "before" (new data) information is to include an additional set of artificial (zero) observations of the true parameters with errors whose statistical properties are given by the covariance matrix of Jgm3:
with 
a unit matrix and the weight matrix for these
artificial observation:
The constrained least squares estimate of the parameters combining (or colocating, collecting) the actual with the artificial observations is, in comparison with Equation (10), given as:
All of our estimates for corrections to Jgm3 have included these a priori constraints which in effect allows us to use all the observations on the satellites and from surface gravimetry that are in that model. (Technically this is true only for 70x70 solutions since that is the trucation in Jgm3, but approximately for lesser truncations. In addition, for convenience, we actually use only the m-order submatrices of the covariance matrix up to m = 50 since these are virtually uncoupled from each other with low cross order correlations in the full matrix).