Optimal Inter-Satellite Distance and Inclinations

3.3.1 Selection of optimal Inter-Satellite Distance

The noise PSDs for SST and accelerometer, eq. (3-1), were defined by the project team as a result from earlier and ongoing studies. Figure 3-7 shows the PSDs for different intersatellite distances and Figure 3-8 shows results obtained for different intersatellite distances  for a single polar pair obtained with the semi-analytic QLT (Sneeuw, 2000). The following simulation parameters have been assumed:

inclination I = 89.5°, orbit height h = 420 km, maximum spherical harmonic resolution L = 130,

Figure 3-7: PSDs of ll-SST system, ACC and total noise (SST-ACC) for two inter-satellite distance scenarios ( = 10 km/100 km)

Figure 3-8: Performance of single polar pairs (I = 89.5°, h = 420 km, L = 130, T = 30 d) with different intersatellite distances; degree-RMS (left) and accumulated geoid errors (right).

From these results the following conclusions can be drawn:

 All satellite distances ≥ 50 km fulfill the mission requirements

 The larger the satellite distance the lower the formal gravity field errors. The relation is almost linear with satellite distance . This is clear from sensitivity point of view, but since the SST error increases with  it was expected that this effect is less prominent.

 No significant improvements were found for satellite distances  > 70 km. Here, this is different, as mentioned before. However, a satellite distance of  = 100 km is a good

compromise between sensitivity and technological instrument issues (higher error bounds are possible for shorter distance) and is suggested for e².motion. Furthermore full-scale simulations in (Reubelt, et al., 2014) haven’t shown much benefit by using larger distances of 150 km/200 km.

From the analysis and the conclusions a satellite distance of  = 100 km is suggested as some kind of optimal choice.

3.3.2 Selection of optimal Inclination

The performance of Bender constellations consisting of a polar pair and an inclined pair with different inclinations I  [55°,80°] have been investigated (outside this limits the Bender constellations begin to perform worse). The following simulation parameters have been assumed: inclination of first satellite I1 = 89.5°, satellite distance  = 100 km, orbit height h = 420 km, maximum spherical harmonic resolution L = 130, observation interval T = 30 d and PSDs (dtotal) from eq. (3-1) for observation noise.

Figure 3-9 shows the performance of various Bender constellations (compared to the single polar pair) in terms of different performance measures (degree-RMS, accumulated geoid errors, geoid error per latitude, spatial correlations (at equator) w.r.t North azimuth) and Figure 3-10 displays formal error triangle plots and spatial covariance functions. In Table 3-2 values for different performance criteria are specified. These criteria contain on the one hand geoid errors (long-wavelength (L = 50) and short wavelength (L = 130)), quasi local criteria as geoid error over ice/equatorial regions (60°    85°

and 0°    30°, respectively) and homogeneity measures as normalized geoid error variation along the meridian (variation of geoid errors per latitude) and average correlation w.r.t. North azimuth (at equator). The best performances w.r.t. this criteria and tolerances within 22.5° have been identified and finally a best Bender constellation was found which stays within this tolerances. The gain compared to a single polar pair is also declared in this table.

Figure 3-9: Performance of Bender constellations (I₁ = 89.5°,  = 100 km, h = 420 km, L = 130, T = 30 d) with different inclinations of the 2^nd pair; degree-RMS (top left), accumulated geoid errors (top right), geoid errors per latitude (bottom left) and spatial correlations (at equator) w.r.t North azimuth (bottom right).

Table 3-2: Evaluation of Bender pairs with different inclinations of the 2^nd pair w.r.t. different performance criteria; best mission’s values are in red and tolerances within 22.5% are in blue. The best constellation (I = 70°/89.5°) fulfilling all tolerances is marked in the red box.

inclination

(***) average correlation w.r.t. North azimuth (at equator)

(****) gain (factor) of optimal Bender pair (I = 89.5°/70°) w.r.t. single pair (I = 89.5°) single polar pair (I = 89.5 °)

Bender constellation (I = 70°/89.5°)

Bender constellation (I = 80°/89.5°)

Figure 3-10: Formal error triangle plots and spatial covariance functions for Bender pairs with different inclinations vs. a single polar pair.

From these results the following conclusions can be drawn:

 All investigated Bender mission lead to improvements compared to a single polar pair; the improvement is dependent on the spherical resolution and the inclination and lies mainly between a factor of 3 and 10

 All Bender constellations fulfill the science requirements

 For higher SH resolutions l > 70 all Bender constellations lead to similar geoid accuracy (and degree-RMS). However, for longer wavelengths, especially between degrees 5  l  50 Bender missions with lower inclinations perform better (a factor of 5-6 between the pairs I = (60°/90°) and I = (80°/90°) and a factor of 2-3 between pairs I = (60°/90°) and I = (75°/90°))

 Lower inclinations of the 2^nd pair lead to a higher (equatorial) isotropy; for inclinations I2  70°

it is more than 95%, for I2 = 80° it drops significantly to 72% (for I2 = 75° it is still 91%);

compare with the single pair, which isotropy measure is only 23% (→ trackiness!)

 Geoid errors per latitude show which latitude regions will benefit most from the Bender design;

these are the low and mid latitude regions with ||  I; the smallest errors in these regions are produced by the lower inclinations, but the difference between the different Bender missions varies not much (less than 25%)

 The largest geoid homogeneity (i.e. the smallest variation of geoid errors along the latitude) is

the important ice-mass loss observations. Thus low geoid errors for these latitudes are welcome.

The lowest errors in these regions are produced by higher inclinations, especially for I2 = 80°.

For inclinations of I2 = 60° and 70° this error is increased by approx. 50% and 7%, respectively.

The analysis shows that it is important to evaluate degree-RMS or accumulated geoid errors, but also homogeneity measures as isotropy and regional measures as geoid errors for certain latitude bands are important. Such criteria are evaluated numerically in Table 3-2. As can be seen, lower inclinations (I2

≈ 60°) show a better performance for most of the criteria except for ice latitudes, where a higher latitude (I2 ≈ 70°- 80°) is beneficial. Since the latter is an important topic for future missions, an inclination of I2 = 70° is regarded as optimal. Such an inclination balances the criteria and lies within the tolerance of 22.5% w.r.t. the optimal performance for each criteria. The average gain compared to GRACE is approx. a factor of 3.5.

All in all one can conclude that inclinations of I = 70°/89.5° are suggested for a Bender type mission.

3.3.3 Stand-alone Performance of inclined Satellite Pair

An important aspect is that each of the two pairs of the Bender constellation is a valuable mission on its own. The reason for this is that very likely one agency will be able to launch one pair due to the enormous costs, and this single pair must already be a valuable mission. For a polar pair this is clear, since it covers the whole Earth and will outperform GRACE due to sophisticated instruments, background models and analysis methods (apart from the fact of a continuation of time-variable gravity time series). However it has to be investigated what the benefit is of a stand-alone inclined satellite pair. Due to the large polar gap problems will arise as e.g. missing observability of polar areas and its phenomena (mainly ice) and missing observability a large part of the spherical harmonic spectrum (low orders). In contrast, some benefits can be found, as mentioned below. Figure 3-11 displays formal error triangle plots and spatial covariance functions for the polar and inclined pair and in Figure 3-12 geoid errors per latitude and correlations w.r.t. the North azimuth (at the equator) are compared. To account for the polar gap the inclined solution was regularized by the covariance matrix generated by a polar pair with 1000 times larger measurement noise.

single polar pair (I = 89.5°)

Single inline pair (I = 70°)

Figure 3-11: Single polar vs. single inclined pair; formal error triangle plots (left) and spatial covariance functions (right); inclined pair regularized by covariance matrix of single polar pair with 1000 times larger noise (i.e. PSD multiplied by 1000)

Figure 3-12: Single polar vs. single inclined pair; geoid errors per latitude (left) and spatial correlations (at equator) w.r.t North azimuth (right)

Pros and cons of single inclined pairs (compared to single polar pairs):

Cons:

 Low order SH coefficients cannot be determined due to polar data gap, thus a complete SH spectrum is not available

 Due to the polar data gap time variable gravity phenomena in this regions cannot be investigated; this means in principle no ice mass studies are possible

 For SH analysis regularization is necessary; this might be avoided by alternative parameterizations (local methods as wavelets, Slepian functions, …)

 Leakage effects of the polar gap to lower latitudes (mainly 5° - 10° outside the gap) Pros:

 Much larger isotropy for all latitudes inside the inclination; thus GRACE type North-South striping is reduced

 Much higher accuracy for low and mid latitude regions

 Probably better recovery of low and mid-latitude geophysical effects, e.g. hydrology in Amazon and Africa

In conclusion one can state that an inclined pair might be a valuable mission on its own; low and mid-latitudes geophysical signals (mainly hydrology) will benefit from the much higher isotropy.