3. Nominal gradiometer data processing and analysis 25
3.4. Angular rate reconstruction
found by Cesare et al. (2008) by comparison of the angular acceleration error spectrum, coming either from the gradiometer or from the star sensor. The error spectrum of the star sensor angular accelerations was obtained by deriving two times the error spectrum of the star sensor attitude information (cf. Appendix B).
10−4 10−3 10−2 10−1 100
10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4
Frequency [Hz]
Error root PSD of ang. acc. [rad/s2/sqrt(Hz)]
STR EGG
(a) X-direction
10−4 10−3 10−2
10−13 10−12 10−11 10−10 10−9 10−8 10−7
Frequency [Hz]
Error root PSD of ang. acc. [rad/s2/sqrt(Hz)]
STR EGG
(b) Y-direction
10−4 10−3 10−2 10−1 100
10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4
Frequency [Hz]
Error root PSD of ang. acc. [rad/s2/sqrt(Hz)]
STR EGG
(c) Z-direction
Figure 3.17.:Concept of hybridization between STR and EGG measurements at the level of angular accelerations.
In Fig. 3.17 the concept of hybridization between STR and EGG measurements at the level of angular accelerations
3.4. Angular rate reconstruction 43 is shown for all three components. The blue lines indicate the error root PSD of the STR angular accelerations from numerical simulation, whereas the green lines show the error root PSD of the EGG angular accelerations from simulation (including EGG noise only). The corresponding hybridization frequencies between the STR and the EGG error curves are highlighted in red. For the angular acceleration x-component the crossover takes place at about 1 mHz, for the y- and z-component it takes place at about 0.2 and 0.5 mHz, respectively.
Since the hybridization frequencies are well below the gradiometer MB, the angular rate within the MB can be reconstructed by numerical integration of the gradiometer angular accelerations. The star sensor measurements are used below the hybridization frequencies for the determination of the absolute angular rate (and attitude) values (i.e. for the DC part), to compensate the effects of the accelerometer drift with increasing noise towards the lower frequencies and to counteract the drift of the numerical integration of the gradiometer angular accelerations.
From a processing point of view, the angular rate reconstruction is performed by Kalman-filtering in the time domain (Cesare and Catastini, 2008b; Stummer et al., 2011). In principle, the method consists of a prediction step and a correction step, see Fig. 3.18 and cf. also the processing flowchart in Fig. 3.16.
Figure 3.18.:Principle of Kalman-filtering for angular rate reconstruction.
The prediction starts with the computation of the one-step angular rate variation (between epochskandk+ 1), which is obtained by interpolation (i.e. one-step integration) of the gradiometer angular accelerations, cf. Fig. 3.19.
The integration is performed using a Lagrange interpolator with the four coefficientsck=−1 =ck=+2 =−0.0417 andck=0=ck=1= 0.5417, according to Cesare et al. (2008). This means that for the computation of the angular rate variation between epochskandk+ 1, the weighted sum of the gradiometer angular accelerations from epochs k−1 tok+ 2 is built.
Figure 3.19.:Integration of gradiometer angular accelerations to derive the one-step angular rate variation.
The attitude quaternions of the next epoch (k+ 1) are predicted by rotation of the attitude quaternions from the current epoch (k) to the next, using the one-step angular rate variation (between epochskandk+ 1), according to Eq. B.21. Also the angular rates for the next epoch (k+ 1) are predicted. Thus, for epoch k+ 1, besides the measured STR quaternion, also a predicted attitude quaternion is available. From the difference between these two sets of quaternions, the correction for the current attitude quaternion and angular rate is found. The gains of the estimator are derived from pre-launch measurement performance models of the gradiometer and the star sensor. Note that the estimator gains are not adapted to the current error behavior of the real GOCE data during the L1b processing. This is different from a classical Kalman filter approach, where also a correction of the estimator gains is foreseen. The transient of the Kalman filter is about half a day long. Hence, in the case that a re-initialization of the filters is necessary, like e.g. after a calibration phase, at least half a day of data is
lost. The new L1b implementation for the ARR, as derived in Sect. 6.2 is based on FIR (finite impulse response) filtering, where the used filters have a length of only 8401 seconds. In case of gaps, at least four times less data is lost with respect to the original implementation.
In the original processing scheme the measurements of only one star sensor, namely the one which is provided in theSTR VC2 1bproduct, are used within the ARR. In Sect. 6.4 it is shown that a combination of all simultaneously available star sensor data is possible and helps to improve the quality of several L1b products, like in particular the gravity gradients, cf. also Sect. 7.
In the following the data associated with the ARR are shown for the example day 11 November 2009. The two main input data sets are the calibrated gradiometer angular accelerations, as already discussed in the previous section with Fig. 3.15(a), and the star sensor attitude quaternions, as given in Fig. 3.20. As output the ARR provides merged inertial attitude quaternions (IAQ), cf. Fig. 3.21 and merged angular rates, cf. Fig. 3.22.
Fig. 3.20 shows the measured quaternions of star sensor 2, which are provided in theSTR VC2 1bfiles of 11 Novem-ber 2009. q0 denotes the scalar part of the quaternions. The values of the quaternions have to be within± 1.
In the time domain, Fig. 3.20(a), we observe a change in sign in all quaternion components, once per orbital revolution. Since quaternions have a sign ambiguity this change in sign has no impact on the contained attitude information. The reason for the change once per revolution is that the L1b data are provided in orbit-wise files. Hence, at every beginning of a new file the sign switch can happen. In Fig. 3.20(b) the root PSDs of the quaternions are given. Note that for the computation of the PSDs a continuous quaternion data set without sign changes has to be used. We observe an approximately continuous decay in magnitude up to a frequency of about 2 mHz, followed by an increase with its peak at about 4 mHz, and a further decay which is reached at about 50 mHz. For higher frequencies the root PSDs are approximately flat with a level of about 2·10−5 rad/√
Hz for the componentsq0 andq2 and of about 3·10−5 rad/√
Hz for q1 andq3. Further analyses of the GOCE star sensor data are made in Sect. 5.2.
In Fig. 3.21 the IAQs are shown. In the time domain, Fig. 3.21(a), we observe that these quaternions have no sign changes. They can be avoided within the original ARR implementation, because of a successive processing strategy. The corresponding root PSDs, Fig. 3.21(b), are at the low frequencies similar to the root PSDs of the star sensor quaternions in Fig. 3.20(b). For higher frequencies, where the two times integrated gradiometer angular accelerations are used, we observe a continuous decay in magnitude. The high frequencies of the star sensor only quaternions are dominated by noise, Fig. 3.20(b), whereas the root PSD of the IAQs follows (also) for high frequencies the expected behavior, cf. the root PSD of the simulated attitude quaternions in Fig. 2.5(b).
In Sect. 6.3 an improved method for the determination of the IAQs is introduced, and further quality assessment is made.
The main output of the ARR are the merged angular rates, called gradiometer angular rates (GAR) in the L1b processing, as illustrated in Fig. 3.22, which should be of good quality for all frequencies. The mean angular velocity about the y-axis is about 1.2·10−3 rad/s, Fig. 3.22(a), due to the rotation of the satellite about this axis once per revolution. ωx and ωz have a mean value of approximately zero and variations of up to about 1.2·10−4rad/s. In the corresponding root PSDs, Fig. 3.22(b), the x-component (blue) has the largest magnitude in the MB. The reason is again, as already pointed out for the calibrated gradiometer angular accelerations, which are the basis for the GAR in the high frequencies, the sensitivity of GOCE for rotations around its slight x-axis.
For further analyses of the GAR see Sect. 5.1.
3.4. Angular rate reconstruction 45
0 4 8 12 16 20 24
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Time [s]
Quaternion [rad]
q0 q1 q2 q3
(a)
10−3 10−2 10−1 100
10−6 10−5 10−4 10−3 10−2 10−1
Frequency [Hz]
Root PSD of quaternion [rad/sqrt(Hz)]
q0 q1 q2 q3
(b)
Figure 3.20.:Star sensor attitude quaternions (a) and corresponding root PSD (b) of 11 November 2009.
0 4 8 12 16 20 24
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Time [s]
Quaternion [rad]
q0 q1 q2 q3
(a)
10−3 10−2 10−1 100
10−10 10−8 10−6 10−4 10−2 100
Frequency [Hz]
Root PSD of quaternion [rad/sqrt(Hz)]
q0 q1 q2 q3
(b)
Figure 3.21.:Merged inertial attitude quaternions (a) and corresponding root PSD (b) of 11 November 2009.
0 4 8 12 16 20 24
−12
−10
−8
−6
−4
−2 0 2x 10−4
Time [s]
Angular rate [rad/s]
x y z
(a)
10−3 10−2 10−1 100
10−12 10−10 10−8 10−6 10−4
Frequency [Hz]
Root PSD of angular rate [rad/s/sqrt(Hz)]
x y z
(b)
Figure 3.22.:Merged angular rate (a) and corresponding root PSD (b) of 11 November 2009.