J. Cent. South Univ. (2014) 21: 2397−2403 DOI: 10.1007/s11771-014-2193-z

GOCE kinematic orbit adjustment for EGM validation and accelerometer calibration

XU Tian-he(徐天河)1, 2, 3, SUN Zhang-zhen(孙张振)4, JIANG Nan(江楠)4, CHEN Kang-kang(陈康康)4, LI Min(李敏)4

1. State Key Laboratory of Geo-information Engineering (Changan University), Xi’an 710054, China;

2. Xi’an Research Institute of Surveying and Mapping (Changan University), Xi’an 710054, China; 3. State Key Laboratory of Geodesy and Earth’s Dynamics, Wuhan 430077, China;

4. College of Geology Engineering and Surveying, Changan University, Xi’an 710054, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2014

Abstract: The main principle and mathematical model of GOCE kinematic orbit adjustment for Earth gravity field model (EGM) validation and accelerometer calibration are presented. Based on 60 days GOCE kinematic orbits with 1−2 cm accuracy and accelerometer data from 2009−11−02 to 2009−12−31, the RMS-of-fit (ROF) of them using EGM2008, EIGEN-5C, ITG- GRACE2010S and GOCO01S up to 120, 150 and 180 degree and order (d/o) are evaluated and compared. The scale factors and biases of GOCE accelerometer data are calibrated and the energy balance method (EBM) is performed to test the accuracy of accelerometer calibration. The results show that GOCE orbits are also sensitive to EGM from 120 to 150 d/o. The ROFs of EGMs with 150 and 180 d/o are obviously better than those of EGMs with 120 d/o. The ROFs of GOCO01S and ITG-GRACE2010S are almost the same up to 120 and 150 d/o, which are about 3.3 cm and 1.8 cm, respectively. They are far better than those of EGM2008 and EIGEN-5C with the same d/o. The ROF of GOCO01S with 180 d/o is about 1.6 cm, which is the best one among those EGMs. The accelerometer calibration accuracies (ACAs) of ITG-GRACE2010S and GOCO01S are obviously higher that those of EGM2008 and EIGEN-5C. The ACA of GOCO01S with 180 d/o is far higher than that of EGMs with 120 d/o, and a little higher than that of ITG-GRACE2010S with 150 d/o. It is suggested that the newest released EGM such as GOCO01S or GOCO02S till at least 150 d/o should be chosen in GOCE precise orbit determination (POD) and accelerometer calibration. Key words: GOCE; orbit adjustment; Earth gravity field model; kinematic orbit; accelerometer calibration

1 Introduction

The gravity field and steady-state ocean circulation explorer (GOCE) satellite was successfully launched on 17 March 2009. It is the first mission using gradiometer measurements to recover the Earth gravity field model (EGM) at least up to 220 degree and order (d/o) with high accuracy and resolution [1]. From 30 September 2009, GOCE data have been released and users can acquire the data through the official website of European Space Agency (ESA). Several EGMs based on GOCE data have been computed in the last three years. According to processing strategies, these models fall into three types, time-wise GOCE gravity model (GOCE- TIM), space-wise GOCE gravity model (GOCE-SPW)

and direct-method GOCE gravity model (GOCE-DIR) using QR factorization and parallel computation [2−3]. All these models give quality information in terms of variance-covariance matrices, which brings a good impression about the performance and error characteristics of the models, but does not provide a final answer about the precision that one can reach using GOCE data.

Aiming at testing GOCE-based EGM, many researchers have done a lot of work and proposed three types of methods. The first one is to compare GOCE-based EGM with other EGMs without GOCE data such as EGM2008 [4], EIGEN-5C [5], and ITG-GRACE2010S [6]. The second one is to use the terrestrial gravity measurements to validate GOCE-based EGM [7−8]. The third one is to use LAGEOS, GRACE

Foundation item: Project(41174008) supported by the National Natural Science Foundation of China; Project(SKLGED2013-4-2-EZ) supported by the

Open Foundation of State Key Laboratory of Geodesy and Earth’s Dynamics, China; Project(2007B51) supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China

Received date: 2013−01−17; Accepted date: 2013−07−03 Corresponding author: XU Tian-he, Associate Professor, PhD; Tel: +86−13991944148; E-mail: thxugfz@163.com

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or GOCE orbits to evaluate the GOCE-based EGM by orbit adjustment [8]. Most discussions focus on the former two methods for the evaluation of GOCE-based EGM. Another important work focuses on the spectrum analysis of gradiometer measurements and common- mode accelerometer data to obtain the characteristics of their noise and their realistic accuracy levels [2, 9]. The results show that the error of VXX and VYY is approximately at the level of the requirement on the gravitational gradient trace, whereas the VZZ error is a factor of 2−3 above the requirement for higher frequencies [7, 9]. GOCE common-mode accelerometer data are provided as well as their calibration parameter by ESA. GOCE users have to recalibrate them by setting scale and bias parameters when they perform the GOCE SST-only gravity field recovery using dynamic integral method (DIM), energy balance method (EBM) or acceleration method (AM). A lot of work on GOCE calibration can be found for gradiometer data or differential accelerometer [9−11]. For GOCE common-mode accelerometer, VISSER [12] provided a calibration method of using GOCE POD which can be called dynamic calibration [12]. An improved method based on dynamic orbit fit was developed [13].

The released GOCE orbits include reduced-dynamic orbits and kinematic orbits with 1−2 cm in accuracy, much better than CHAMP orbits and a little superior to GRACE orbits [14−15]. The reduced-dynamic orbits are inevitably influenced by the reference model used in GOCE precise orbit determination (POD). For this reason, kinematic orbits are usually used for GOCE SST-only gravity field recovery [14, 16]. Considering higher accuracy and lower attitude of GOCE kinematic orbits, they may be used to validate the EGMs as well as to analyze the sensitive maximum d/o of EGM. At the same time, if the newest released EGM is used in GOCE kinematic orbit adjustment, the scale factors and biases of accelerometer data can be determined. Based on the above idea, GOCE kinematic orbit adjustment is studied for EGM validation and accelerometer calibration in this work. 2 Principle of GOCE kinematic orbit

adjustment The main principle of GOCE kinematic orbit

adjustment is to use dynamic orbit determination to obtain optimal orbit ROF based on GOCE kinematic orbits as well as to calibrate the common-mode acceleration data. The detailed procedure is described as follows.

The equation of movement for GOCE satellite can

be expressed as [17]

3( , , , )g t α

rr F r r α

r (1)

where r and r are position and velocity vector, respectively, g is the gravity constant of the Earth, t is the time, and α is the dynamic model parameter vector. There are two parts in the right side of Eq. (1), where the first one is the acceleration vector caused by two-body central gravitation and the second one is the disturbing acceleration vector. The disturbing forces acting on satellite can be classified into two types, the conservative forces and the non-conservative forces. The conservative forces include Earth non-spherical gravitation, N-body gravitation, solid earth tide, ocean tide, pole tide and general relativity effect, which can be computed precisely by models. The non-conservative forces include solar and Earth radiation pressure, air drag, etc, which are measured by common-mode accelerometer in the drag-free control (DFC) for GOCE.

The observational equation of orbit adjustment can be expressed as

kin( )i i it r r V (2)

where

T( ) ( )i i i it x y zr is the satellite position vector of ti time determined by satellite dynamic model,

kin kin kin kin T( )i i i ix y zr is the satellite kinematic position, and

T( )i i ii x y zV V VV is the orbit fitting

residual vector. The principle of least square (LS) is

T minV PV (3)

where P is the weight matrix related to the accuracy of kinematic orbits. The initial state vector and the according force model parameters can be estimated by batch processing. Then, the orbit fitting residuals can be obtained.

From the view of processing mode, there is no difference between orbit adjustment and reduced- dynamic orbit determination. They both use batch processing based on LS. The main differences of them are the following two aspects. The first one is the observation difference. Orbit adjustment uses the kinematic orbit as observation, while reduced-dynamic orbit determination uses the carrier phase and pseudo- range as observation. Therefore, observation equation does not need to be linearized in orbit adjustment. The second one is the difference of the considered dynamic models. Pure dynamic models are adopted in orbit adjustment without introducing any empirical acceleration or velocity impulse in order to reasonably evaluate the OFA and accelerometer calibration accuracy

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by EGM. In reduced-dynamic orbit determination, empirical accelerations or velocity impulses are usually introduced in order to improve the orbit solution accuracy [18].

3 Accelerometer evaluation on orbit fitting

and calibration accuracy

The root mean square (RMS) of orbit fitting residuals is used to validate the ROF of different EGMs. Its formula can be expressed as

2 2 2

RMS x y zV V V V (4)

where kin 2

1

1ˆ( )

n

x i ii

V x xn

is the RMS of orbit

fitting residual of x component. kin 2

1

1ˆ( )

n

y i ii

V y yn

is the RMS of orbit fitting residual of y component. Vz=

kin 2

1

1ˆ( )

n

i ii

z zn

is the RMS of orbit fitting residual of

z component. ˆ ˆ ˆ( )i i ix y z is the estimated satellite position vector. Using the above formula, the OFA by different EGMs with different d/o can be obtained.

The differences between the resulting potential from EBM and the one computed from EGM directly are used to evaluate the accelerometer calibration accuracy (ACA). Energy balance equation in inertial frame can be expressed as [19]

2EBMt 0 0

1d ( )

2 i i i i i i iT t x y x y V U E x x f

(5) where

T( )i i i ix y zx represents the satellite position vector, fi is the non-conservative force vector measured by accelerometer, ω is the Earth rotation angle velocity, Vt is the non-Earth gravitational potential caused by N-body gravitation, solid earth tide, ocean tide ect, and U0 is the EGM normal potential (without centrifugal potential), which is usually chosen as GRS80.

Using the calibrated parameters of accelerometer by orbit adjustment, the disturbing potential can be derived through Eq. (5). Considering that the EBM is widely used in CHAMP and GRACE gravity field recovery [20], the computational procedure in details is not shown here. The computed potentials from a newest released EGM GOCO02S [21] are regarded as reference. The derived potentials are compared with them, and the RMS is used to evaluate the ACA, namely

EBM Model 2RMS

1

1( )

n

i ii

T T Tn

(6)

where EBM

iT is the disturbing potential determined from EBM, and

ModeliT the one computed from GOCO02S

directly. 4 Computations and comparisons

Two types of GOCE orbits, kinematic orbits (SST_PKI) with 1 s sample interval and reduced- dynamic orbits (SST_PRD) with 10 s sample interval are provided by ESA. It is more suitable to use the kinematic orbits for EGM validation than to use the reduced- dynamic orbits since the latter is influenced seriously by the reference EGM in GOCE POD. Outlier detection should be performed since there are inevitably outliers in kinematic orbits. A simple method comparing kinematic orbits with high-precision reduced-dynamic orbits is used to detect the outliers. The Newton’s interpolation procedure is adopted to produce the satellite positions using reduced-dynamic orbits at the epoch of kinematic orbits since they are not synchronized with integer second. More details about the above procedure can be seen in Ref. [20]. The preprocess of attitude quaternion and common-mode acceleration data is focused on filling the gap and getting the data of non-integer second corresponding to kinematic orbits by using a LS linear interpolation.

GOCE kinematic orbits, the common-mode acceleration (EGG-CCD) and the satellite attitude quaternion (EGG-IAQ) data from 2009−11−02 to 2009−12−31 [16] corresponding to 60 days of data are used in orbit adjustment. Four representative EGMs, EGM2008, EIGEN-5C, ITG-GRACE2010S and GOCO01S till 120, 150 and 180 d/o are adopted to evaluate the OFA [4−6, 16]. In orbit adjustment, two biases are estimated in radial (R), along-track (T) and normal (N) directions per revolution and one scale factor in R and N directions per day for the common-mode acceleration. The reason that we don’t estimate the scale factor of T direction is that the common mode accelerometer along the flight direction is the main direction of non-conservative forces and has much higher stability and reliability than the other two directions. Real computation proves that if the scale factor of T direction is also estimated, then the ROF will decrease.

The orbit fitting residuals are obtained using the orbit adjustment and their RMSs per day are shown in Fig. 1 for different EGMs up to 120, 150 and 180 d/o. The average RMSs of two months are listed in Table 1. The estimated scale factors and biases are shown in Figs. 2−3 (for the limitation of length, only the calibration results of EGMs with 180 d/o are shown), and their statistical results are listed in Table 2 and Table 3, respectively. In order to evaluate the accuracy and

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Fig. 1 Orbit fitting results by EGMs with different d/o values:

(a) 120; (b) 150; (c) 180

Table 1 Statistical results of orbit fitting by different EGMs with different d/o (cm)

EGM 120 d/o 150 d/o 180 d/o

Mean RMS Mean RMS Mean RMS

EGM2008 4.00 1.44 2.89 1.35 2.80 1.31

EIGEN-5C 3.45 0.95 2.31 0.67 2.20 0.66

GRACE2010S 3.27 0.97 1.85 0.69 1.74 0.61

GOCO01S 3.28 0.97 1.80 0.65 1.65 0.61

Fig. 2 Bias parameters: (a) R direction; (b) N direction; (c) T

direction

reliability of accelerometer calibration, the differences between the resulting potentials from EBM and those computed from GOCO02S [21] are computed and their RMSs are shown in Table 4. The computation of the disturbing potential using EBM from kinematic orbits implies the need of estimating satellite velocities from satellite positions. The polynomial interpolating error of satellite velocity from kinematic orbit will be introduced in the resulting potential. In order to avoid this influence, the high-precision velocity of reduced-dynamic orbit is

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Fig. 3 Scale parameters: (a) R direction; (b) N direction Table 2 Statistical results of bias parameter by EGMs of different d/o (10−7m/s2)

EGM d/o R direction T direction N direction

Mean RMS Mean RMS Mean RMS

EGM2008

120 −0.2000 1.3425 −1.8759 0.2798 3.3402 0.9983

150 −0.0254 1.1448 −1.8756 0.2031 3.3501 0.7284

180 −0.0307 1.0987 −1.8757 0.1962 3.3534 0.7048

EIGEN-5C

120 −0.2622 1.0270 −1.8758 0.2483 3.2516 0.8785

150 −0.1408 0.7095 −1.8752 0.1500 3.3047 0.5915

180 −0.1317 0.6512 −1.8752 0.1381 3.3070 0.5491

GRACE2010S

120 −0.1193 0.9833 −1.8763 0.2349 3.2080 0.7979

150 −0.0623 0.6071 −1.8758 0.1278 3.2443 0.4675

180 −0.0530 0.5473 −1.8759 0.1176 3.2417 0.4248

GOCO01S

120 −0.2417 0.9690 −1.8761 0.2310 3.2157 0.8046

150 −0.0652 0.5289 −1.8756 0.1159 3.2519 0.4557

180 −0.0651 0.4618 −1.8757 0.0982 3.2530 0.4043

Table 3 Statistical results of scale parameter by EGMs of different d/o

EGM d/o R direction N direction

Mean RMS Mean RMS

EGM2008

120 0.8633 0.9922 0.9166 0.2449

150 0.7779 0.7481 0.9397 0.2156

180 0.7857 0.7186 0.9419 0.2116

EIGEN-5C

120 0.9015 0.9320 0.8912 0.2223

150 0.7861 0.5895 0.9254 0.1436

180 0.7705 0.5632 0.9300 0.1249

GRACE2010S

120 0.7602 0.8649 0.8928 0.1950

150 0.6752 0.5079 0.9228 0.1018

180 0.6924 0.4560 0.9258 0.0989

GOCO01S

120 0.9414 0.8534 0.8817 0.1965

150 0.8595 0.4032 0.9119 0.0966

180 0.8647 0.3560 0.9145 0.0938

Table 4 Statistical results of disturbing potential by energy balance method (m2/s2)

EGM d/o

120 150 180

EGM2008 1.245 1.127 1.089

EIGEN-5C 1.113 1.092 0.974

GRACE2010S 1.074 0.926 0.845

GOCO01S 1.075 0.873 0.802

directly used to produce velocities at the epoches of kinematic orbits using Newton’s interpolation procedure. The following conclusions can be drawn from the above results. GOCE orbits are also sensitive to the EGM above 120 d/o because of its low orbit attitude and high accuracy. With the increase of d/o, the OFA is improved. The 1.8 cm OFA can be obtained using ITG- GRACE2010S and GOCO01S up to 150 d/o. GOCO01S with 180 d/o has the best OFA, which is about 1.65 cm.

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The second one is ITG-GRACE2010S with 180 d/o, which is about 1.74 cm. They are obviously superior to those of EGM2008 and EIGEN-5C with the same d/o. Therefore, it is suggested that the d/o of EGM should be chosen as at least 150 in GOCE POD.

The calibrated parameters of GOCE accelerometer have large differences in R and N direction, and smaller differences in T direction (Tables 2−4). This implies that the measured acceleration along the flight direction has much higher stability and reliability than the other two directions. It is the reason that the scale factor of T direction does not need to be estimated in the orbit adjustment. The series of scale factors and biases from GOCO01S till 180 d/o has the best stability, and the second one is that from ITG-GRACE2010S with 180 d/o. They are obviously superior to those of EGM2008 and EIGEN-5C with the same d/o. By comparing the curve of OFA with that of calibrated parameters, it can be seen that the lower the OFA is, the worse the stability and reliability of scale and bias parameters are.

The tests of EBM show that the accelerometer calibration using ITG-GRACE2010S or GOCO01S has higher accuracy and reliability than EGM2008 and EIGEN-5C. The derived disturbing potentials from ITG-GRACE2010S and GOCO01S are more accurate. The accelerometer calibration using GOCO01S up to 180 d/o has the best accuracy, and the second one is that using ITG-GRACE2010S up to 180 d/o, which are obviously superior to those of EGM2008 and EIGEN-5C with the same d/o. It should be pointed out that the accuracy of disturbing potential will be lower than that shown in Table 8 if the derived velocity from kinematic orbit rather than RDO velocity is used in EBM. Some initial results imply that the disturbing potential accuracy is about 1.5−1.7 m2/s2 when only using GOCE kinematic orbits [22]. 5 Conclusions

1) The kinematic orbit adjustment with strict theory and simple computation can be used as a good method for EGM accuracy evaluation and common-mode accelerometer calibration as well as to determine the sensitive maximum d/o of EGMs on GOCE orbits.

2) GOCE orbit is sensitive to EGM till at least 150 d/o. The OFAs from GOCO01S with 120, 150 and 180 d/o are about 3.3, 1.8 and 1.6 cm, respectively, and they are a little superior to those of GRACE2010S and obviously higher than those of EGM2008 and EIGEN-5C.

3) The accelerometer calibration using ITG- GRACE2010S or GOCO01S has higher accuracy and

reliability than EGM2008 and EIGEN-5C. The derived disturbing potentials from ITG-GRACE2010S and GOCO01S are more accurate. Therefore, it is better to choose the newest released EGMs such as GOCO01S or GOCO02S as reference model in GOCE POD or accelerometer calibration, and the maximum d/o is not less than 150. Acknowledgments

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