Introduction

Using the inverse procedure (with a priori Jgm3 covariances) described above, we made many adjustments to Jgm3 with the 19 sets of crossovers in Figure 2, subsets of these and additional crossover data derived from Pathfinder altimetry. For a summary of the data statistics and parameter results in the solution whose residuals are illustrated in Figure 2c, see Table 3 and Figures 5-7.

Table 3. Statistics for Comprehensive 50 x 50 SSC - DSC Solutions
D = Down weighted solution, N = Normally weighted solution

X - Set (Orig.type name)	# of used obs.	Rms (cm) Orig. X	Rms (cm) Resids N/D	Precision (cm) MIN	Precision (cm) Avg	Precision (cm) MIN used N/D	Comment (periods)
NOAA SSC GSATX	6196	6.1	2.6/2.6	0.10	1.6	1.0/1.0	Apr85-Nov88 (GM+ERM)
NOAA SSC ERSX	6608	5.7	2.8/2.9	0.25	1.7	1.5/1.5	Apr92-Dec93 (Cy 1-18)
NOAA SSC TPX	4193	1.7	1.3/1.3	0.16	0.8	1.0/1.3	Oct92-Jan95 (Cy 2-142)
NOAA DSC ERSTOP AA	5642	4.9	2.5/2.8	0.33	1.30	2.0/6.0	Oct92-Dec93
NOAA DSC ERSTOP AD	4196	5.6	3.0/3.3	0.35	1.70	2.5/7.5	"
NOAA DSC ERSTOP DD	5648	5.1	2.6/3.0	0.25	1.30	2.0/6.0	"
NOAA DSC ERSTOP DA	4184	5.0	2.8/3.0	0.40	1.70	2.0/6.0	"
NOAA DSC GSATOP AA	5647	8.4	4.0/4.1	0.15	1.30	3.3/9.9	8yr.gap(85Apr93-88Oct96)
NOAA DSC GSATOP AD	1939	8.4	6.4/6.4	0.10	3.50	4.0/12.0	"
NOAA DSC GSATOP DD	5624	8.0	4.2/4.3	0.28	1.30	3.5/10.5	"
NOAA DSC GSATOP DA	1582	7.2	5.2/5.2	0.10	3.20	3.5/10.5	"
Path. DSC GSATOP AA	5535	9.4	5.3/5.3	0.08	1.40	3.5/10.5	", (86Nov94-88Oct96)
Path. DSC GSATOP AD	1225	8.6	6.9/6.9	0.30	2.40	4.0/12.0	" "
Path. DSC GSATOP DD	5543	8.5	5.6/5.6	0.35	1.50	3.5/10.5	" "
Path. DSC GSATOP DA	1213	8.6	7.2/7.2	0.08	2.30	4.0/12.0	" "
Path. DSC GSATERS AA	4427	8.0	4.9/5.2	0.44	2.50	3.5/10.5	5 yr.gap (87Apr92-88Oct93)
Path. DSC GSATERS AD	5850	7.9	4.4/4.6	0.22	1.80	3.5/10.5	"
Path. DSC GSATERS DD	4415	7.4	5.0/5.3	0.52	2.50	3.8/11.4	"
Path. DSC GSATERS DA	5865	7.3	4.4/4.7	0.11	1.80	3.8/11.4	"
OVERALL	85532	7.0	4.3/4.2

Avg condition of Normal matrix		Before Inversion Jgm3	After Inversion - N/D
		0.58	0.14/0.17

Note:
Orig. X .... original crossover data (SSC or DSC)
Resids ...... residuals of SSC or DSC after the inversion
Precisions: MIN = minimum sd original average X data, MIN used = assigned minimum precision with account for data biases
Periods ..... notation explained and already used in the text (see Sects. 6.2., 6.3.).

We have checked our least squares adjustment by means of independent software using the same data and as similar conditions for the adjustment as possible. The software package has been worked out in DGFI Munich (Bosch, 1997). The proper inversion is performed by the orthogonal transformation using the Q-R factorization by fast Givens (providing higher numerical stability than the normal equation approach). Preliminary results for the harmonic geopotential coefficients and the offset parameters were presented by Bosch et al (1998). The German software will include the complex variance-covariance matrix (in comparison with submatrices of the covariance matrix of Jgm3 for the separate orders used here), but till now it can work only with the variances of the full variance-covariance matrix. There is nevertheless an interesting finding on the a priori conditioning (artificial data): as the SSCs do not contain geographically mean part of the radial orbit component and thus, they are not sensitivite to its error, and as the DSCs of all 4 types are sensitive only to (the same) difference of that component between the two orbits, we can expect an instability of the inversion procedure. An additional condition of the type with beeing an error derived from the projection of the full covariance matrix is proved to stabilize the adjustment significantly (mainly for higher degree solutions).

Now, to our inversion (Figure 2a-c, Table 3). The process of achieving a satisfactory constrained least-squares adjustment with the bin-averaged data would be straightforward if the data contained only the signals we had anticipated (geopotential, geodetic, time-tag and 1 cpr orbit error). But, as Figure 2b shows, the data also contains considerable power in other non-random, broad scale signals which required a large reduction in the precisions of the bins used to weight observations (Table 3). (The cause of these systematic errors, not resolved by the model equations, will be discussed in more detail below and elsewhere: Klokocnik et al., 2000).

We found that using the measured precisions resulted not only in much too large adjustments (compared to the well calibrated variances of Jgm3) but standard errors of fit (for data weighted by these measured precisions) of from 2-3 unless excessive data showing large residuals were rejected from the solution.

To achieve a "normal" standard error of fit of 1.0 to the full solution as well as 1.0 weighted residuals (rms) for each data set we were obliged to add bias errors to the measured precisions of from about 1 cm (for the SSCs) to as much as 4 cm for some DSC sets (see Figure 5a).

In Fig. 5a, we are looking at about 2600 ratio's which (if the influence of the a priori could be ignored, and assuming the Jgm3 variances are valid) could reasonably show about 32% or 840 of these solution/Jgm3 errors outside of , and none outside of . Both of these expectations are far from realized in the "normally weighted" result. The high concentration of small ratio's shows that the a priori constraints exert a strong influence on this adjustment (shown in more detail later). But the preponderance of excessive ratio's in this "normally weighted" solution, mostly at low degree, is a clue that a fair number of geopotential coefficients are absorbing (aliasing) non-geopotential signals at broad scale in the binned data.

Figure 5ab

Correction/Jgm3 Standatd Deviations (SD) in Two DSC-SSC Combination Geopotential Solutions. Red = Cosine coefficient (L,M), Blue = Sine:
a. With DSCs normally weighted (note the large number of excessive ratio's expecially at low degree and order.
b. With DSCs downweighted to achieve adequate calibration with Jgm3. The number of excessive ratio's have been significantly reduced while the the correlation between all geopotential terms in the two solutions is 0.85.

Figure 5c

Correction/SD in Downweighted Combination Geopotential Solution.
These Signal/Noise ratio's are strong at low degree because of the enhanced sensitivity of the geopotential there but also at high degree and order because the conditioning of high order terms is superior (with fewer terms competing for the same number of frequencies). However these high degree and order corrections are also more likely to be aliased from neglected higher degree terms ( ) truncated from the solution.

In fact the discrepancies in the solution/a priori seen in Figure 5a for the "normal" correction show even worse when the ratio is taken properly (for calibration purposes), accounting for the significant effect of the a priori on the adjustment.

Recall that Lerch et al., (1991) showed that the expected error for the difference of two solutions dependent on the same subset of data, one exclusive so, is the difference: subset errors (in our case, the Jgm3 apriori) - full set errors (in our case the errors of our constrained adjustment of Jgm3). The difference of these two dependent solutions, Jgm3 and our adjustment of it is just our adjustment since the a priori Jgm3 solution we started with was zero for all coefficients].

Figure 6

Correction/Normal Error Statistics
for two 50x50 comprehensive adjustments of Jgm3 from NOAA and Pathfinder crossover altimetry (SSCs and DSCs).
Correction = 50x50 adjustment of Jgm3 constrained with a priori covariances.
Normal error = (a priori - correction variance) = estimated error of correction with normally distributed data errors.
Red = C(L,M), Green = S(L,M) terms.
Dashed = expected with normal errors.
a. For corrections with DSCs normally weighted (by order).
b. For corrections with DSCs normally weighted (by degree).
c. For corrections with DSCs downweighted for calibration with Jgm3 (by order).
d. For corrections with DSCs downweighted for calibration with Jgm3 (by degree).
Note the excessive sums for the normally weighted solution at low orders and for all degrees and the more reasonable calibration for the downweighted solution.

Figures 6a,b show this more realistic assessment of the "normal" solution by order and degree as the sum of the squares of the ratio's (solution/expected error), each one treated as an independent -squared statistic (with expected value of 1). For the "normally weighted" result note the hugely excessive sums for all orders less than about 5. The near universal excesses at every degree generally are due to the few anomalous low orders which always produce excessive ratio's for many high degree terms.

The failure of correct calibration with Jgm3 (Figure 6a,b) means that even with the bias errors degrading the precisions, aliasing from these broad scale effects (Figure 2b) still affects the adjustment. To remedy this miscalibration we further downweighted the DSC precisions mainly in the 19-sets of crossovers (Table 3) to produce a more reasonable calibration.

Figures 5b and 6c,d show the original and modified (correction/error) ratio's in the downweighted solution. The corrections themselves are now considerably reduced and only occasionaly beyond 1 from Jgm3 (Fig. 5b) but most important the compatibility with the a priori errors for all the orders (which control the satellite geopotential information) are now well within expectations (Fig. 6c). As Figure 6d shows however, some low degree terms are still excessive (likely aliased) in this solution. But since our weighting has already accomplished all it can and more for the other terms, to reduce the corrections of these few of low degree by further downweighting would go too far to the other extreme and leave too much geopotential signal in the residuals from the bulk of the field. In spite of the downweighting the corrections in this milder geopotential adjustment are still significant compared to their estimated errors from the inversion (Fig. 5c).

All the discussion to this point of the reasons behind the weight decisions leading to the preferred solution have been largely theoretical. What independent tests can we make which would support our conclusion that the original signals contain a mixture of genuine geopotential as well as significant bias information? It is clear from Figure 2b and Table 3 that the sets causing the most problems are the DSCs and particularly the one's with multi-year gaps. Noting the striking change in the structure of the large time-gapped DSCs from data to residuals, from North-South trending to East-West, we contend this change is due to strong interannual ocean signals in these DSCs which are not absorbed by the geopotential adjustment. (We will return to this aspect of the solution later). Now we want to examine just the geopotential information in our data in more detail. For example we would like to know whether the large-scale unresolved residuals in the DSCs, even for the near-contemporaneous Ers1-T/P has compromised the geopotential recovery from those dual-mission sets.