SH degree

0 20 40 60 80

SH degree amplitudes in mm geoid height_{10}^{-4}
10^{-3}
10^{-2}
10^{-1}
10^{0}
10^{1}
10^{2}

mean AOHIS

AOHIS residuals: Pendulum OTD res.: Pendulum formal errors: Pend.

AOHIS res.: in-line OTD res.: in-line formal errors: in-line

SH degree

0 20 40 60 80

SH degree amplitudes in mm geoid height_{10}^{-4}
10^{-3}
10^{-2}
10^{-1}
10^{0}
10^{1}
10^{2}

SH degree

0 20 40 60 80

SH degree amplitudes in mm geoid height_{10}^{-4}
10^{-3}
10^{-2}
10^{-1}
10^{0}
10^{1}
10^{2}

*Figure 8.12:*SH degree amplitudes in mm geoid height up to lmax = 80 of the residuals of the AOHIS and the OTD
(EOT08a−FES2004) analyses wrt. the mean AOHIS for the polar Pendulum pairs in comparison with the mean AOHIS
signal, the polar in-line pair residuals and the formal errors, respectively. From left to right there are the results for
ORB1p, ORB2p, and ORB3p. The zonal coefficients forl∈ {2,4,6}have been excluded from the Pendulum results.

0 10 20 30 40 50 60
10^{−2}

10^{−1}
10^{0}
10^{1}

SH degree

SH degree amplitudes in mm geoid

mean AOHIS (31 days) mean AOHIS (0.5, 1, 2 days) AOHIS aliasing 0.5 days AOHIS aliasing 1 day AOHIS aliasing 2 days OTD aliasing 0.5 days OTD aliasing 1 day OTD aliasing 2 days

*Figure 8.13:*SH degree amplitudes in mm geoid height of the residuals of the AOHIS and the OTD (EOT08a−FES2004)
analyses wrt. the mean AOHIS for the double pair ORB1p + ORB1i in comparison with the mean AOHIS signal for the
three short periods of 0.5, 1, and 2 days.

In Fig. 8.14 the estimated mean gravity field (row 2) and the residuals for AOHIS (row 3) and OTD
(row 4) is displayed in comparison with the mean AOHIS signal (row 1) in terms of global geoid heights
computed up to the differentl_{max}. The mean AOHIS signal shows geoid heights of more than 50 mm for
the areas with large hydrological variations, especially for the Congo region. It changes from 1 January
(0.5 days period) to 10 January (1 day period) and 20 January (2 days period) with a global RMS of 3
and 4 mm (maximum geoid height changes of 15 and 25 mm), respectively.

The residuals for AOHIS and OTD are one order of magnitude smaller than the mean AOHIS signal.

For the 0.5 days period the characteristic of the geoid height residuals are similar for AOHIS and OTD.

A possible reason for this is spatial aliasing due to large gaps. It has to be analysed how the double pair orbit configuration, especially the ascending node separation, could be optimized in order to reduce spatial aliasing in this case. However, the effects are small compared to the mean AOHIS signal. For 1 and 2 days the geoid height residuals for AOHIS and OTD show the same behaviour as the monthly retrievals in Sec. 8.1 with lower spatial resolution (cf. Figs. 8.7 and 8.11).

The global RMS values of the geoid heights in Fig. 8.14 are given in Tab. 8.2. The residuals show global RMS values of around 1 mm and the mean AOHIS signals of around 8 mm. This confirms that a global gravity field can be observed in short periods of 0.5, 1, and 2 days. In the following it is discussed how this estimation reduces the temporal aliasing errors of the NGGM.

The main advantage of co-estimating short period gravity field parameters together with one set of high resolution gravity field parameters for the whole monthly period is temporal aliasing reduction.

The signals above the Nyquist period of the short period estimates do not alias into the high resolution solution, because they are observed with high temporal resolution. This effect is analysed in Wiese et al. (2011) for a single polar pair and two double pairs. Without post-processing filtering but with AO de-aliasing the short period gravity estimates improve the results without additional short period gravity estimates approximately by a factor of 2. However, Wiese et al. (2011) estimate the smallest improvements for the Bender-type double pair. This improvement rate agrees with the results of the following assessment for the estimation of 0.5, 1, and 2 days gravity retrievals.

Analogously to Fig. 6.7 the amplitudes of the non-tidal AOHIS variations are shown in Fig. 8.15

de-mean AOHIS

0.5 days (l max=15)

0 180 360

60 0

−60

−50 0 50

estimated AOHIS 0 180 360

60 0

−60

−50 0 50

AOHIS aliasing

0 180 360

60 0

−60

−1 0 1

OTD aliasing

0 180 360

60 0

−60

−5 0 5

1 day (l max=30)

0 180 360

60 0

−60

−50 0 50

0 180 360

60 0

−60

−50 0 50

0 180 360

60 0

−60

−5 0 5

0 180 360

60 0

−60

−5 0 5

2 days (l max=60)

0 180 360

60 0

−60

−50 0 50

0 180 360

60 0

−60

−50 0 50

0 180 360

60 0

−60

−5 0 5

0 180 360

60 0

−60

−5 0 5

*Figure 8.14:*Global geoid heights in mm computed on a1^{◦}×1^{◦}grid of mean AOHIS, estimated AOHIS, and the residuals
of OTD (EOT08a−FES2004) and AOHIS wrt. the mean AOHIS for the 3 short periods of 0.5 (l^{max}= 15), 1 (l^{max}= 30),
and 2 (lmax= 60) days, respectively.

pending on the period between 0.5 and 30 days and on the SH degree 0≤l≤100. The amplitudes for each period are computed from the Fourier series of the AOHIS time series for January 2005 (Gruber et al., 2011). The signals which are observed with the three short period retrievals are marked with blue boxes. Except for the AOHIS signals with periods below 1 day the largest amplitudes are covered by the three short period retrievals.

All AOHIS signal contents outside the blue boxes in Fig. 8.15 alias into the monthly high resolution solution. By summing up the total RMS for each period the total temporal aliasing error RMS of AOHIS can be estimated. This is done for 2≤l≤140 for the three short periods separately and for the combination of the three short periods. Figure 8.16 shows the cumulative geoid heights over the periods of the total AOHIS variations (gray) in comparison with the variations without the 4 short period cases (blue) and the reduction (red).

As the short period estimates reduce the AOHIS signal contents with the largest amplitudes the RMS of the reduced variations shows no significant increase above each Nyquist period. Especially the signal contents with periods larger than 3 days and amplitudes larger than 0.1 mm are observed (cf. Fig. 8.15).

The total RMS of the AOHIS variations (gray) for this month sums up to 2.3 mm geoid height. With the combined short period retrievals (0.5, 1, and 2 days) the RMS for this month is reduced to 1 mm.

*Table 8.2:*Global geoid height RMS (weighted with cosine of latitude) in mm for the mean AOHIS signal and for the
non-tidal (AOHIS) and non-tidal (OTD) residuals for the double pair ORB1p + ORB1i for 0.5, 1, and 2 days periods, respectively
(cf. Fig. 8.14).

Period in days 0.5 (l_{max}= 15) 1 (l_{max}= 30) 2 (l_{max}= 60)

mean AOHIS signal 7.0 7.6 8.4

AOHIS residuals 0.35 1.7 2.1

OTD residuals 1.3 1.1 0.89

AOHIS

SH degree

0 10 20 30 40 50 60 70 80 90 100

30 days 10 days 6 days 3 days 2 days

1 day

12 hours -5

-4
-3
-2
-1
0
l_{max} double

0.5 days retrieval 1 day retrieval 2 days retrieval

SH degree amplitudes in log10(mm)

*Figure 8.15:*SH degree amplitudes in log10(mm)geoid heights of non-tidal AOHIS variations depending on the period in
days. The amplitudes for different periods are computed from the Fourier series of the AOHIS time series for January
2005 from Gruber et al. (2011). The red line represents the maximum possible SH degree of expansion for a double pair.

The blue lines mark the amplitudes which are observed with the three short periods of 0.5, 1, and 2 days, respectively.

With the 1 day retrieval, for example, the signals above the Nyquist period of 2 days are observed up tol^{max}= 30.

This analysis applies only for AOHIS and not for ocean tides. But as Wiese et al. (2011) include ocean tide de-aliasing errors in the analyses, and a similar reduction is shown, it can be assumed for the ocean tide aliasing reduction as well.

The largest part is reduced already with the 12 hours estimates. Therefore it is recommended to take this period into account for the NGGM. As described above, the orbit configuration of the double pair has to be opimized wrt. optimal sampling in each 12 hours in order to avoid spatial aliasing and to increase the spatial resolution. The main conclusions of this section are used for the discussions of the final high resolution results in Secs. 8.3 and 8.4.

10^{−1} 10^{0} 10^{1} 10^{2}
0

0.5 1 1.5 2 2.5

period in days

cumulative geoid height in mm

total AOHIS

reduced AOHIS (0.5,1,2 days) reduced AOHIS (0.5 days) reduced AOHIS (1 day) reduced AOHIS (2 days) reduction (0.5,1,2 days) reduction (0.5 days) reduction (1 day) reduction (2 days)

*Figure 8.16:*Cumulative geoid heights in mm over the periods of the total AOHIS variations (gray) in comparison with the
variations without the 3 short period retrievals separately and without the combination of the three short period retrievals,
respectively. The geoid heights are computed from the total RMS for each period of AOHIS for 2≤l≤140 with and
without the blue boxes in Fig. 8.15. The red curves represent the reduction of AOHIS variations with each of the 4 short
period cases.