** Improved star camera attitude data 5**

**5.4 Effect of the improved SCA data on the GRACE fundamental observationsobservations**

**5.4 Effect of the improved SCA data on the GRACE fundamental**

0 500 1000 1500 2000 2500

−0.4

−0.2 0 0.2 0.4 0.6

GRACE time: 281374100 + x [s]

[mrad]

diff roll diff pitch diff yaw

(a)

10^{−3} 10^{−2} 10^{−1}

10^{−7}
10^{−6}
10^{−5}
10^{−4}
10^{−3}
10^{−2}

frequency [Hz]

[rad/sqrt(Hz)]

diff roll diff pitch diff yaw

(b)

**Figure 5.10:**Differences of the inter-satellite pointing angles derived from the "SCA1B IfE" and SCA1B RL02
star camera data in both time (a) and frequency domain (b). The differences of the roll, pitch and yaw angles

are shown for a part of the orbit of GRACE-A on 2008-12-01

0 500 1000 1500 2000 2500

−2

−1.5

−1

−0.5 0 0.5 1 1.5

2x 10^{−7}

GRACE time: 281374100 + x [s]

[ms−1]

JPL RL02 IfE

(a)

10^{−4} 10^{−3} 10^{−2} 10^{−1}

10^{−8}
10^{−7}
10^{−6}
10^{−5}

frequency [Hz]

[m.s−1/sqrt(Hz)]

JPL RL02 IfE diff

KBR error model

(b)

**Figure 5.11:**The KBR antenna offset correction for range rate derived from the official SCA Level-1B Release 02
solution (red) and from the "SCA IfE" solution (black) in time domain (a) and in frequency domain (b). The
differences of these two solutions (light blue) are compared to the KBR system error (blue), which is modeled

as white noise of 1*µm/*√

Hz at the range level

**5.4.2 Effect on the ACC observations**

The linear accelerations sensed by the accelerometer provide information about the non-gravitational forces acting on the satellite. Originally, they are provided in the Accelerometer Frame (see Appendix A.4), which is practically identical with the Science Reference Frame.

For the gravity field recovery, these linear acceleration are required in the inertial frame or as
in case of the Celestial mechanics approach (Beutler*et al., 2010) in the so called True Radial*
Reference Frame (TRRF).

The TRRF axes are defined as:

**x*** _{T RRF}* =

**y**

*×*

_{T RRF}**z**

_{T RRF}**y**

*=*

_{T RRF}**r**×

**v**

|r×**v|**

**z*** _{T RRF}* =

**r**

|r|

(5.16)

where**r**and **v** are the satellite’s position and velocity vectors in the inertial frame.

The rotation matrix rotating the inertial frame into TRRF (R* _{iner→T RRF}*) is then obtained
as

**R*** _{iner→T RRF}* =

(x* _{T RRF}*)

*(y*

^{T}*)*

_{T RRF}*(z*

^{T}*)*

_{T RRF}

^{T}

(5.17)

The direction-cosine matrix rotating the Science Reference Frame into True Radial Reference Frame is obtained as

**R**_{SRF}_{→T RRF} =**R*** _{iner→T RRF}* ·

**R**

^{T}

_{SRF}_{→iner}(5.18) where the latter rotation matrix,

**R**

_{SRF}_{→iner}, is derived from the SCA1B quaternions according to Equation B.7.

The comparison of the linear accelerations rotated into TRRF using the**R**_{SRF}_{→T RRF} matrix
derived from the "SCA1B RL02" and the "SCA1B IfE" is shown in Figure 5.12. The differences
of the rotated ACC data reach up to 1.5·10^{−8}ms^{−2}, cf. Figure 5.12(a). Figures 5.12(b)-5.12(d)
show the comparison of the ACC differences with the expected error model of the ACC
measurement. Although the ACC error models were originally defined for the Accelerometer
Frame, they can be still considered as true in TRRF because of the very small differences in
the mutual alignment of the Accelerometer Frame and TRRF along the orbit. In case of the
high sensitive axes, i.e. radial and along-track axes, the differences reach up to two orders of
magnitude above the expected error level. In case of the less accurate axis, i.e. the cross-track
axis, the differences are smaller, but still above the expected error level.

These results show that the ACC sensor measurement accuracy cannot be fully exploited as the effects caused by the imperfect star camera data combination exceed the ACC measurement accuracy by up to two orders of magnitude. Moreover, this is even more critical for the periods when the attitude data are obtained from single camera solution, which for GRACE is inevitable due to SCA blinding by the Sun and the Moon, cf. Figure 5.7.

1000 1500 2000 2500 3000 3500 4000 4500

−1

−0.5 0 0.5 1

x 10^{−8}

GRACE time: 281361600 + x [s]

[ms−2]

radial along−track cross−track

(a)

10^{−4} 10^{−3} 10^{−2} 10^{−1}

10^{−12}
10^{−11}
10^{−10}
10^{−9}
10^{−8}
10^{−7}

[Hz]

m.s−2/sqrt(Hz)

(b) radial

10^{−4} 10^{−3} 10^{−2} 10^{−1}

10^{−12}
10^{−11}
10^{−10}
10^{−9}
10^{−8}
10^{−7}

[Hz]

m.s−2/sqrt(Hz)

(c) along-track

10^{−4} 10^{−3} 10^{−2} 10^{−1}

10^{−12}
10^{−11}
10^{−10}
10^{−9}
10^{−8}
10^{−7}

[Hz]

m.s−2/sqrt(Hz)

(d) cross-track

**Figure 5.12:**Effect of the improved SCA data on ACC linear accelerations. Figure (a) shows the differences of
the linear accelerations along the particular axes after their rotation into TRRF based on the SCA1B RL02
and "SCA1B IFE" data. In figures (b-d) these differences are shown in frequency domain (blue curves) and

compared to the ACC error models (black curves). Based on GRACE-A data from Dec 1st, 2008

**5.4.3 Effect on the GPS observations**

Similarly to the KBR observations, which originally are related to the phase centers of the
KBR antennas, also the original GPS observations are carried out between the phase center
of the transmitter antenna onboard the GPS satellites and the phase center of the receiver
antenna onboard GRACE. The main GPS navigation antenna is located on the zenith panel of
each GRACE satellite, cf. Figure 2.8. As the final orbit solutions are required to be related to
the satellite’s center of mass, a geometric correction for the offset of the GPS antenna phase
center from the CoM has to be applied during the GPS data processing. The location of GPS
antenna phase center was determined from on-ground and in-flight calibration (Montenbruck
*et al., 2008; Jäggi* *et al., 2009). The PhC coordinates are then provided in SRF. The entire*
GPS processing is performed in ITRF, therefore the offset vector of the GPS antenna phase
center (pc* ^{GP S}*) needs to be rotated first from the SRF into inertial frame using the SCA1B
data, and from the inertial frame into the terrestrial frame using the IERS conventions:

**pc**^{GP S}* _{IT RF}* =

**R**

*·*

_{iner→IT RF}**R**

_{SRF}_{→iner}·

**pc**

^{GP S}*(5.19) The values for*

_{SRF}**pc**

^{GP S}*were obtained from the VGN1B data product, which represent the mean phase center location, i.e. phase center variation are not considered here. Figure 5.13 shows the differences of the*

_{SRF}**pc**

^{GP S}*vector components for both L1 and L2 carrier frequencies, which was rotated using the SCA1B RL02 and "SCA1B IfE" attitude data. The differences reach up to 0.2 mm, while most of the values are well below 0.1 mm. According to Montenbruck*

_{IT RF}*et al.* (2008), the phase center offset can be estimated with an accuracy between 0.1 mm and
0.5 mm. Because the magnitude of the **pc**^{GP S}* _{IT RF}* differences are very close to the currently
achieved accuracy of the phase center location, they cannot be neglected.

0 2000 4000 6000 8000 10000

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

GRACE time: 281361600 + x [s]

[mm]

dx dy dz

(a) L1

0 2000 4000 6000 8000 10000

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

GRACE time: 281361600 + x [s]

[mm]

dx

d_{y}
dz

(b) L2

**Figure 5.13:** The effect of the improved SCA data on the GPS observations. The figures show the differences
of the GPS antenna phase center offset vector**pc**^{GP S}* _{IT RF}* which was rotated using the SCA1B RL02 and "SCA1B
IfE" attitude data. Shown for both L1 (a) and L2 (b) carrier frequencies. Based on GRACE-A data from Dec

1st, 2008