The content of this article comes from "Journal of Surveying and Mapping" 2022 Issue 2 (GS (2022) No. 540)
GOCE gravitational gradient internal calibration method
Pan Juanxia , Zou Xiancai ,2
. School of Surveying and Mapping, Wuhan University, , Wuhan, Hubei 430079;2. Key Laboratory of Earth Space Environment and Geometric Surveying, Wuhan, Hubei, Wuhan 430079Fund project: National Natural Science Foundation of China (41874021; 42192532; 41721003); Civil Aerospace "13th Five-Year Plan" Technology Preliminary Research Project
Abstract : The accurate calibration of GOCE satellite gravitational gradient is one of the prerequisites for inverting high-precision gravity field. This paper uses the gravitational gradient meter and star sensor data in GOCE satellite L1b data to realize the internal calibration of satellite gravitational gradient. The angular velocity used for internal calibration is determined using least squares combined with multiple star sensor observation data, effectively avoiding the impact of the low-precision angular velocity component of a single star sensor on the coordinate conversion process. Taking into account the change of the rotation matrix between the stellar sensor coordinate system and the gradient meter coordinate system over time, this paper proposes an internal calibration model that takes into account the calibration parameters of the rotation matrix, and uses GOCE actual measured data from November 2009 to verify the effect of the method. The results show that the calibration parameters of the rotation matrix are about 100″, and there is a drift of 3″~30″ in that month. Compared with the official internal calibration method of GOCE, according to the results of the official internal calibration method of the satellite gravity gradient accuracy, the calibration parameters of the rotation matrix and the internal calibration model of the calibration parameters of the 3 accelerometer pairs in the gradient meter are better than models that only consider the calibration parameters of the accelerometer; in addition, this paper discusses possible methods for data calibration of GOCE gradient meter based on this model, providing a reference for the data processing of GOCE and subsequent gravity satellites.
keywords: GOCE Gravitational gradient Accelerometer Internal calibration Posetic reconstruction
Citation format: Pan Juanxia, Zou Xiancai. GOCE gravity gradient internal calibration method[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(2): 192-200. DOI: 10.11947/j.AGCS.2022.20210067
PAN Juanxia, ZOU Xiancai. Internal calibration method of GOCE gravity gradients[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(2): 192-200. DOI: 10.11947/j.AGCS.2022.20210067
Read the full text : http://xb.sinomaps.com/article/2022/1001-1595/2022-2-192.
Introduction
gravity measurement satellite GOCE (gravity field and steady-state ocean circuit explorer) is a low-orbit gravity detection satellite developed and launched by the European Space Agency (ESA). Its scientific goal is to recover a global gravity field model of more than 200 orders) in 2 The accuracy of CM is measured global geoid level and the accuracy of 10-5m/s is measured to determine gravity anomalies. Compared with the CHAMP (challenging minisatellite payload ) and GRACE (gravity recovery and climate experiment) satellites, the GOCE key load is a high-precision gravitational gradient meter. It is symmetrically placed on 3 orthogonal axes by 6 accelerometers. The baseline of the three pairs of accelerometers is about 50 cm long. The center of mass of the satellite coincides with the center of mass of the gradient meter. Therefore, the second-order gradient tensor of the gravitational position is directly measured using the differential observations of the accelerometer, and the non-conservative force acting on the satellite is obtained using the accelerometer common mode observations. GOCE also has accurate high and low satellite tracking data and undamped control. Its unique advantage is to solve the gravity field with combined gravitational gradient data and high and low satellite tracking data.
GOCE gravitational gradient meter is affected by measurement errors, external observation conditions and other factors, resulting in systematic deviations, proportional errors and colored noises in the observation values. Therefore, calibration of GOCE observation data before inverting the gravity field is crucial. Different installation structures lead to the accelerometer calibration method of GRACE satellite no longer applicable to GOCE.Many scholars at home and abroad have done a lot of research on the use and calibration of GOCE data. Usually, the calibration of GOCE is divided into internal calibration and external calibration based on whether external auxiliary data such as reference gravity field models are introduced. The calibration model usually includes a scale factor and a deviation factor. Literature [2-3] proposes the use of stellar sensors to measure data and prior gravity field to estimate the scale factor. The model also calibrates the problem of unregistration between the gradient meter coordinate system (GRF) and the star sensor reference frame (SSRF). Literature [4] proposed to use satellite angular velocity reconstructed by stellar sensor combined with gradient meter measurement data for calibration models. Literature [5-6] proposes a similar method to use stellar sensor data and accelerometer measurements for internal calibration, and proposes solutions for the time correlation of observations. This method is the official ESA method. Literature [7-8] determines the scale factor and deviation factor based on precision orbits, and uses stellar sensor data to verify the calibrated gravitational gradient data. Literature [9-10] found that adding a secondary factor to the calibration model can weaken the influence of strong winds around the geomagnetic poles, and in 2018, an external gravity field model was introduced to reprocess the GOCE data. Many domestic studies include preprocessing research on GOCE satellite data, theoretical methods for determining the earth's gravity field by gradient data [11-13] and calibration methods based on accelerometer data. Literature [14-16] proposes to use the dynamic method to complete single accelerometer calibration and satellite non-conservative force determination by combining geometric method, and solve the gravity field model and calibration parameters at the same time, thereby reducing the impact of reference gravity field model error on calibration results, and discussing the compensation effect of satellite undamped control. Literature [17] uses external gravity field and stellar sensor data to initially verify the effectiveness of calibration within a certain frequency band. Literature [18] discusses the influence of calibration using different external gravity field models and different orders of the same gravity field model on calibration results.
At present, the processing and use of GOCE data in China is mainly based on the internal calibration of ESA. This article uses the methods officially released by ESA to realize the internal calibration of GOCE data. The main purpose is to improve the calibration processing of GOCE gradient meter observation data, and to make preliminary preparations for discussing the joint and comparison of internal calibration, external calibration methods and dynamic calibration. The internal calibration method released by ESA models the error factor into an inverse calibration matrix of three accelerometer pairs, and applies this matrix to the accelerometer to achieve calibration. On this basis, this paper proposes an improved model that takes into account the calibration parameters of the rotation matrix between SSRF and GRF, and discusses the changes that this change brings to the satellite’s gravitational gradient accuracy. The data involved include the measurement data of the gradient meter and stellar sensor mounted on the satellite. The accelerometer common mode observation value and differential observation value and the attitude data of the stellar sensor are used to establish a calibration model based on the principle of satellite gravity gradient measurement to obtain the gravitational gradient under the coordinate system of the gradient meter after calibration. In addition, restoring the gravity field also involves high-precision conversion of gradient meter coordinate system and inertia system, so this paper proposes a discussion on satellite attitude reconstruction.
Theory and methods
.1 Internal calibration
GOCE satellite is equipped with several important payloads. This article focuses on electrostatic gravity gradient meter (EGG) used to determine medium and short-wave gravity fields, and 3 star sensors that provide satellite attitudes. GOCE is the first satellite to use -damping control technology, and atmospheric resistance along the orbit is continuously compensated. During the actual operation of the satellite, there are many error factors. For this reason, GOCE designed to perform a specific vibration that lasts 1 d every two months for the calibration of the satellite. The data during this vibration is called shaking data, and other periods are called nominal data. Figure 1 shows three pairs of accelerometer placement structures in the gradient meter coordinate system. The solid line represents the super-sensitive axis, and the dotted line represents the non-sensitive axis [6].
Figure 1 Arrangement of the three pairs of accelerometers in the GRF
Figure 1 Arrangement of the three pairs of accelerometers in the GRF
Figure 1 Arrangement of the three pairs of accelerometers in the GRF
Figure 2 is the special structure of the three pairs of accelerometers in the gradient meter. According to the principle of satellite gravity gradient measurement, the acceleration value measured by the i accelerometers is
(1)
Figure 2 Source of error for a single accelerometer Fig. 2 Error sources of a single accelerometer Fig. 2 Error sources of a single accelerometer
diagram options
where i is the accelerometer number; V is the gravitational gradient; Ω and r are vectors from the satellite center of mass to the i centroid; Ωri is the centrifugal acceleration generated by the satellite rotation around its center of mass; d is the linear acceleration of the satellite center of mass. The specific definition of the above variables is shown in the literature [6]. Two types of observation modes of
gradient meter are differential mode acceleration (DM) and common mode acceleration (common mode) acceleration, CM)[6]
(2)
(3)
Since the GRF origin is about a few centimeters different from the satellite center of mass, and
is about 3 orders of magnitude lower than d in the value, then according to formula (1), combined with common mode acceleration Definition of degree,
(4)
(5)
Formula (5) is expressed as matrix form (6)
, where Ad=(ad, 14, ad, 25, ad, 36), and the matrix L[6]
(7)
, in the formula, Lx, Ly, L, L, z respectively represent the arm length of the gradient meter in the directions of x, y, and z, respectively. This article uses the official ESA value [6], Lx=0.514 0135 m, Ly=0.499 890 0 m, Lz=0.500 201 0 m. The relationship between the angular acceleration of satellite relative to the centroid of V and Ω and the symmetry of
(8) and
(9)
1 gives the relationship between the angular acceleration of satellite relative to the centroid of the center of mass and the differential acceleration. Equation (9) shows that the satellite gravity gradient tensor can be derived from the differential acceleration, baseline length and angular velocity, and is finally used to restore the gravity field. .1.2 Calibration model
In the actual operation of satellites, the following error factors are usually considered: ① The accelerometer center of mass deviates from the nominal position; ② The accelerometer axes are not strictly aligned with the corresponding coordinate axes of the gradient meter; ③ The accelerometer 3 axis is not completely orthogonal; ④ Due to the uncertainty of data output gain, the accelerometer scale factor [10] generated. The error is parameterized into a scale factor, a deviation factor and a quadratic term, where the quadratic term is eliminated in a physical vibration manner. Therefore, the internal calibration matrix should include a scale factor and a deviation factor.Figure 3 shows the accelerometer error factors other than linear quadratic terms. Each accelerometer includes 6 angle calibration parameters and 3 scale factors. A total of 54 calibration parameters need to be determined for the 6 accelerometers, and the calibration matrix [6]
(10)
Figure 3 Comparison of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz
where i accelerometer measurement values; a i is the corresponding real value; parameters s are the proportional factors in each direction; α, β, and γ are tangential factors respectively; δ, ε, and ζ are tiny tangential angles and tiny rotation angles respectively. In 2, K is used to characterize the scale factor, and θ is used to characterize the comprehensive influence of tangential factors and angles. For common mode and differential acceleration,
(11)
. In the formula, Mij is the calibration matrix, which defines the inverse calibration matrix Mij-1[6](inverse calibration matrices, ICM)
(12)
Equation (12) gives the relationship between the measured value, the true value and the inverse calibration matrix.
Equation (4) represents the common mode acceleration CM given by three pairs of accelerometer observations, that is, the non-conservative force exposed by the satellite in three directions. There are 6 independent conditions
(13)
angular acceleration
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The gravitational gradient V and the centrifugal acceleration term ΩS is much smaller than the differential acceleration observation signal Ad, see Figure 3 for details, using 0.05~0.1 bandpass filter of Hz filter equation (15), then
(16)
uses the gradient meter measurement data EGG_NOM_L1b and stellar sensor data STR_VC2_1b and STR_VC3_1b for the satellite nominal period officially released by ESA as input data to realize internal calibration. When establishing the above model, the following assumptions were made: ① The baseline length of the gravitational gradient meter given by ESA was Lx, Ly, L, Lz are accurate and fixed constants; ② The relative relationship between SSRF and GRF given by ESA is accurate and a constant matrix; this article explores whether these two assumptions are accurate and their impact on the accuracy of gravity gradients, and adds new parameters to the basic model to re-establish the model. Use tiny rotation angles rx, r, r, y, r, r to describe the comprehensive effect of the rotation matrix between the stellar sensor and the gradient meter coordinate system over time, use GRF' to represent the nominal gradient meter coordinate system, and GRF represents the coordinate system with deviation. The relationship between angular acceleration in the two coordinate systems is expressed as
(17)
(18 )
Formula (17), I represents unit array;
is the angular acceleration in GRF;
represents the angular acceleration in GRF' after calibration by ΔR, and change the formula (14) accordingly
(19)
write component form is
(20)
Since formula (17) is 0.05~0.1 Approximate treatment was performed in the Hz frequency band, and the baseline lengths Lx, Ly, and Lz were omitted on the denominator, resulting in less complete constraints on the estimate of baseline length. Therefore, this article will not discuss the changes in baseline length over time, and it is still considered that its value is a constant. Based on the ESA method, internal calibration is completed using formula (13), formula (20) and formula (16).
.1.3 Covariance matrix processing
This paper uses 2 d data to estimate a set of calibration parameters. According to the characteristics of GOCE sampling rate and colored noise, it is difficult to realize the storage and inversion of the matrix. References [6, 10], using symmetric sliding average decorrelation filtering, the corresponding frequency band filtering is realized while decorrelation processing is implemented to avoid inversion of large matrices. Formula (21) is the decorrelation filtering expression
(21)
In the formula, eni is the original error sequence containing color noise; n represents the nhtml in the sequence 18 elements; i represents the calibration conditions of i in Section 1.1.2; M is the order of filter ; fmi is the filter coefficient; eni is a sequence that contains only white noise after passing the filter. This article selects the filter order and the track period value. Readers who are interested in the calculation of filter coefficients can refer to the literature [6, 10].
.2 The three stellar sensors STR1, STR2 and STR3 installed on the combined
GOCE satellite provide satellite attitude information. One feature of the sensor is that the angular velocity around the viewing axis, that is, the angular velocity of the axis of z, that is, the angular velocity of the axis of z, the angular velocity of the axis of z, the y, the angular velocity of the axis of ωxSSRF, the ωySSR The accuracy of F, the error of ωzSSRF during coordinate conversion will be propagated to ωxGRF, ωyGRF and ωzGRF. According to the relative relationship between the star sensor and the gradient meter (Figure 4), this paper uses the least squares adjustment to combine the pose quaternions of the effective star sensor with [6, 10] to ensure that subsequent calibration results are more reliable and the angular velocity accuracy is higher.
Figure 4 Relative relationship between the gradient meter coordinate system and the coordinate system of 3 star sensors [6]Fig. 4 Relative orientation of the GRF and the SSRFs
Figure options
.3 Angular velocity and attitude reconstruction
angular velocity and attitude quaternion accuracy directly affect the gravitational gradient and the earth's gravity field inversion accuracy, and diagonal velocity and attitude reconstruction are necessary. The conversion of quaternions to angular velocity observed by the star sensor requires a numerical differentiation process, resulting in the amplification of the high-frequency band noise, and the angular velocity and quaternions derived from the integrated accelerometer's observations amplify the low-frequency noise. Literature [19] uses Kalman filtering to achieve angular velocity and posture reconstruction. Literature [20] uses the angular velocity of the stellar sensor and the angular velocity of the gradient meter to establish a weight model in the frequency domain, and reconstruct the angular velocity and attitude with Wiener filtering. Literature [10] proposes to use least squares fit to reconstruct the pose. Kalman filtering and least squares fitting are both unfolded in the time domain without using the frequency domain characteristics. The transient effect of Kalman filtering is severely caused by low data utilization. In this article, refer to [20] to use the advantages of Wiener filtering in the frequency domain to combine the advantages of two types of data in the frequency domain to achieve angular velocity and attitude reconstruction. The principle of
Wiener filtering is to realize the weighted average in the frequency domain based on the accuracy of the two types of observation values. The power spectrum density value of a certain frequency in the frequency domain (power spectral density, PSD) represents the accuracy here
(22)
In the formula, HSTR (f) and HGRAD (f) represent the weights of the sensor and the gradient meter, respectively; PSTR (f) and PGRAD (f) respectively represent the square root power spectral density of two types of angular velocities in the frequency domain, model reference [20]. Figure 5 shows two types of angular velocity noise PSD/2, where G represents the gradient meter and S represents the stellar sensor.
Figure 5 Two types of angular velocity noise PSD/2Fig. 5 PSD/2of two types of noise
Figure options
angular velocity was finally calculated by the product in the frequency domain and the Fourier forward and inverse transformation to obtain
(23)
In the formula, n represents x axis, y axis and z axis.Integrating the reconstruction angular velocity can obtain the quaternion qtGRAD. Also considering that numerical integration leads to amplification of low-frequency noise, Equation (24) uses Wiener filter consistent with the angular velocity reconstruction process and the quaternion qtSTR to reconstruct the posture [0] to reduce the influence of noise
(24)
In formula, t represents the observation time.
Data analysis
Selection November 2009 for every 2 consecutive days d's nominal measured data is a set of one set, with a total of 15 sets of data subjected to the above calibration and reconstruction process, and the inverse calibration matrix Mij-1 and the rotation matrix calibration parameters ΔR. The calibration model is divided into two categories to discuss the data processing results. The first category is consistent with the ESA internal calibration method and only consider Mij-1; the second category adds ΔR parameters. In the analysis of the results of this paper, in addition to using the Laplace equation , which is satisfactory with the gravitational gradient tensor gradient, the gravity field model EIGEN-5C is also used as the reference field. The data mainly used in the calculation of this model include satellite height measurement data, ground gravity data, GRACE and Lageos satellite data. The reference model is used to calculate the gravitational gradient reference value, and the power spectrum of the difference between the actual gravitational gradient data after calibration and the reference value is analyzed to analyze the impact of calibration parameters on the gradient results.
picture 6. Part of the Mij-1 sequence is given, and the 1st parameter of the 14 accelerometer pair C14 line 2 in the expression (12) is compared with the ideal situation (the proportional factor and deviation factor are 1 and 0 respectively). The results show that the matrix Cij, Dij, Dij (ij=14, 25, 36) The order of the non-diagonal difference is 10-5~10-4, and the order of the diagonal difference is 9×10-4~2.7×10-2. Considering that the calibration parameters have a linear trend of about 2×10-5 within one month, the parameters are linearly fitted and then calibrated. Table 1 is the numerical value of a set of scale factors obtained by fitting. Figure 7 shows the power spectral density of the gravitational gradient tensor trace after calibration of the first type of method in this paper, which matches the calibration results of ESA. Figure 8 shows PSD/2 of the stellar sensor combined with the front and back gravitational gradient tensor trace.
Figure 6 Part Mij-1 parameter sequence Fig. 6 Part parameters series of Mij-1
Figure options Table 1 Accelerometer comparative factor Tab. 1 Scale factors of accelerometer pairs
| accelerometer pair number | direction | ||
| x | y | y | z |
| 14 | 1.023 4 | 0.973 7 | 1.020 1 |
| 25 | 1.017 7 | 1.018 4 | 0.979 8 |
| 36 | 1.018 4 | 0.973 3 | 1.022 7 |
Table options
Figure 7 PSD/2Fig of the gravitational gradient tensor trace after calibration. 7 PSD/2of calibrated gravity gradient trace
Figure options
Figure 8 PSD/2Fig of the stellar sensor combined with the front and back gravitational gradient tensor trace. 8 PSD/2of calibrated gravity gradient trace before and after combination of star sensors6
Figure 8 and Figure 9 show the square root power spectrum density of the front and back gravitational gradient tensor trace and the square root power spectrum of the difference between each component and the reference component, indicating that combining multiple stellar sensors can increase below 0.005 The accuracy of the gravitational gradient in the Hz frequency band, the maximum improvement value of the square root power spectrum of the gradient tensor trace is 1.8×10mE/
, and the maximum improvement value range of each component is 3.8×10~1.2×10mE/
.In Figure 10, there is a clear sharp point at about 0.02~0.03 Hz for the V x component, and the other components also have this phenomenon at the corresponding frequency. By analyzing the power spectrum of the difference between the gravitational gradient observation value and the reference value difference between the gravitational field models of different types and different orders, it was found that the sharp point will disappear only when the GOCE gravity field model is used as the reference field, and sharp points will exist when the non-GOCE gravity field model is used. Therefore, it is speculated that the emergence of sharp points may reflect the contribution of GOCE gradient data. When the signal on the corresponding frequency band does not contain GOCE gradient data, the spherical harmonic coefficient error of the corresponding order of the gravity field model is relatively large. This phenomenon deserves further in-depth study.
Figure 9 PSD/2Fig of the difference between the gravitational gradient component and the model reference value before and after combination of star sensors 1 Figure 10 Parameter ΔR sequence Fig. 10 Time series of ΔR
Figure Option Based on the first class ESA model, this paper adds the parameter ΔR, which changes over time, the rotation matrix between SSRF and GRF. This change will affect the angular velocity and attitude reconstruction process, and the gravitational gradient component will also change accordingly. Here is a data sequence and linear fitting sequence with the newly added parameter ΔR. There are several obvious outliers in Figure 10. The working status of the three star sensors in the corresponding time period is analyzed. Statistics show that within the period when the outliers exist, the number of times available for only a single star sensor data is much larger than the corresponding values in other time periods, which also reflects the advantages and necessity of combining multiple star sensors. The absolute value of rx, ry, r, and rz is around 100″ and the monthly value shows a linear trend of about 3~30″. Therefore, the calibration parameters should not be ignored when establishing a calibration model.
applies ΔR to the angular velocity and quaternion of the star sensor, and then reconstructs the angular velocity and attitude to quantitatively analyze the impact of the newly added calibration parameters on the accuracy of the gravitational gradient. Figure 11 shows the square root power spectrum analysis of the gravitational gradient difference between the gravitational gradient component before and after the parameter ΔR and the reference model gravitational gradient difference. Table 2 is below 0.005 The two types of calibration models with Hz frequency correspond to the square root power spectrum statistics of the difference between the gravitational gradient components and the model reference value. Among them, "Method 1" means that it is consistent with the ESA model, and "Method 2" means that it is added to the new ΔR, 4 gravitational gradient components Vxx, Vxz, Vy, Vy, The square root power spectrum of the difference value corresponding to l17Vzz shows that the accuracy of the gravitational gradient component has been improved by adding new parameters. Among them, the accuracy of the component Vxz is the greatest improvement. This is because the newly added parameters in "Method 2" change the attitude of the satellite. Compared with other components, the Vx component is more sensitive to changes in posture, so its accuracy is maximized. Since the added parameters mainly affect the pose quaternions, the gradient tensor trace will not change with the quaternions [1-27], no corresponding comparison diagram is provided here.
Figure 11 PSD/2Fig of the difference between the gravity gradient components and the model reference value of the two types of calibration models. 11 PSD/2of differences of gravity gradients to EIGEN-5C model for the two calibration models
Figure options
Table 2 PSD/2 statistics of differences between the gravity gradient components and the model reference value of the two types of calibration models Tab. 2 PSD/2statistics of differences of gravity gradients to EIGEN-5C model for the two types of calibration models Tab. 2 PSD/2statistics of differences of gravity gradients to EIGEN-5C model for the two types of calibration models
| component | calibration model | maximum | minimum | mean | standard deviation |
| Vxxx | method 1 | 45 245 823.203 | 47.036 | 5 221 249.416 | 13 553 688.125 |
| method 2 | 45 244 950.335 | 43.651 | 5 220 653.314 | 13 553 591.064 | |
| Vxz | method 1 | 2 794 553.787 | 53.980 | 323 811.690 | 836 447.020 |
| method2 | 2 696 051.381 | 42.138 | 312 174.185 | 807 154.872 | |
| Vyy | method1 | 56 026 989.558 | 127.440 | 4 670 338.535 | 16 173 150.910 |
| Method 2 | 56 024 742.462 | 110.274 | 4 669 472.429 | 16 172 715.845 | |
| Vzzz | method1 | 18 562 356.933 | 94.722 | 2 142 510.910 | 5 560 223.548 |
| method2 | 18 561 125.427 | 74.125 | 2 141 971.948 | 5 560 046.890 |
table options
Conclusion and prospect
This paper starts from the principle of GOCE satellite gravity gradient measurement, and uses the gravitational gradient meter observation data and the stellar sensor attitude data in the nominal period in L1b data to internally calibrate the accelerometer measurement value, and combines the observation values of multiple stellar sensors with least squares to avoid the error propagation caused by the conversion between SSRF and GRF; in order to obtain a more accurate gravitational gradient tensor under the inertial coordinate system to restore the gravity field, the satellite angular velocity and attitude quaternion were reconstructed using Wiener filtering. Based on the ESA calibration method, an internal calibration model that takes into account the calibration parameters of the rotation matrix between SSRF and GRF was proposed. The results show that the absolute value of the calibration parameters of this group is about 100″, and a linear trend of about 3~30″ was shown in the month. At the same time, the internal calibration model of the inverse calibration matrix and the calibration parameters of the rotation matrix between SSRF and GRF can improve the accuracy of each component of the gravitational gradient in the frequency band below 0.005 Hz. The maximum improvement for the gradient component Vxzz is confirmed, which confirms the necessity of this calibration parameter and brings certain reference value to GOCE's own data processing and subsequent gravity satellite data processing.
Finally, based on this article, it proposes possible improvement directions for GOCE data processing: ① Due to the lack of external data constraints, the internal calibration method cannot avoid the impact of GOCE's own system deviation. Therefore, consider comparing the external calibration method and dynamic method to combine the advantages of various methods can promote the discussion of gradient data processing problems; ② Regarding the baseline length Lx, Lyhtm The determination of l18, Lz and ΔR and the compensation of quadratic term K2 are considered, and the introduction of a priori gravity field and combined with the stellar sensor data solution; ③ In the process of symmetric sliding average decorrelation filtering of the covariance matrix, this paper uniformly selects the filter order equal to the orbital period length, but the power spectrum estimated by this length is not optimal during the actual solution, and the compromise choice of power spectrum resolution and accuracy in the decorrelation process is worth in-depth discussion.
Author Profile
First Author Profile: Pan Juanxia (1996-), female, master, research direction is satellite gravity measurement.
The content of this article comes from "Journal of Surveying and Mapping" 2022 Issue 2 (GS (2022) No. 540)
GOCE gravitational gradient internal calibration method
Pan Juanxia , Zou Xiancai ,2
. School of Surveying and Mapping, Wuhan University, , Wuhan, Hubei 430079;2. Key Laboratory of Earth Space Environment and Geometric Surveying, Wuhan, Hubei, Wuhan 430079Fund project: National Natural Science Foundation of China (41874021; 42192532; 41721003); Civil Aerospace "13th Five-Year Plan" Technology Preliminary Research Project
Abstract : The accurate calibration of GOCE satellite gravitational gradient is one of the prerequisites for inverting high-precision gravity field. This paper uses the gravitational gradient meter and star sensor data in GOCE satellite L1b data to realize the internal calibration of satellite gravitational gradient. The angular velocity used for internal calibration is determined using least squares combined with multiple star sensor observation data, effectively avoiding the impact of the low-precision angular velocity component of a single star sensor on the coordinate conversion process. Taking into account the change of the rotation matrix between the stellar sensor coordinate system and the gradient meter coordinate system over time, this paper proposes an internal calibration model that takes into account the calibration parameters of the rotation matrix, and uses GOCE actual measured data from November 2009 to verify the effect of the method. The results show that the calibration parameters of the rotation matrix are about 100″, and there is a drift of 3″~30″ in that month. Compared with the official internal calibration method of GOCE, according to the results of the official internal calibration method of the satellite gravity gradient accuracy, the calibration parameters of the rotation matrix and the internal calibration model of the calibration parameters of the 3 accelerometer pairs in the gradient meter are better than models that only consider the calibration parameters of the accelerometer; in addition, this paper discusses possible methods for data calibration of GOCE gradient meter based on this model, providing a reference for the data processing of GOCE and subsequent gravity satellites.
keywords: GOCE Gravitational gradient Accelerometer Internal calibration Posetic reconstruction
Citation format: Pan Juanxia, Zou Xiancai. GOCE gravity gradient internal calibration method[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(2): 192-200. DOI: 10.11947/j.AGCS.2022.20210067
PAN Juanxia, ZOU Xiancai. Internal calibration method of GOCE gravity gradients[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(2): 192-200. DOI: 10.11947/j.AGCS.2022.20210067
Read the full text : http://xb.sinomaps.com/article/2022/1001-1595/2022-2-192.
Introduction
gravity measurement satellite GOCE (gravity field and steady-state ocean circuit explorer) is a low-orbit gravity detection satellite developed and launched by the European Space Agency (ESA). Its scientific goal is to recover a global gravity field model of more than 200 orders) in 2 The accuracy of CM is measured global geoid level and the accuracy of 10-5m/s is measured to determine gravity anomalies. Compared with the CHAMP (challenging minisatellite payload ) and GRACE (gravity recovery and climate experiment) satellites, the GOCE key load is a high-precision gravitational gradient meter. It is symmetrically placed on 3 orthogonal axes by 6 accelerometers. The baseline of the three pairs of accelerometers is about 50 cm long. The center of mass of the satellite coincides with the center of mass of the gradient meter. Therefore, the second-order gradient tensor of the gravitational position is directly measured using the differential observations of the accelerometer, and the non-conservative force acting on the satellite is obtained using the accelerometer common mode observations. GOCE also has accurate high and low satellite tracking data and undamped control. Its unique advantage is to solve the gravity field with combined gravitational gradient data and high and low satellite tracking data.
GOCE gravitational gradient meter is affected by measurement errors, external observation conditions and other factors, resulting in systematic deviations, proportional errors and colored noises in the observation values. Therefore, calibration of GOCE observation data before inverting the gravity field is crucial. Different installation structures lead to the accelerometer calibration method of GRACE satellite no longer applicable to GOCE.Many scholars at home and abroad have done a lot of research on the use and calibration of GOCE data. Usually, the calibration of GOCE is divided into internal calibration and external calibration based on whether external auxiliary data such as reference gravity field models are introduced. The calibration model usually includes a scale factor and a deviation factor. Literature [2-3] proposes the use of stellar sensors to measure data and prior gravity field to estimate the scale factor. The model also calibrates the problem of unregistration between the gradient meter coordinate system (GRF) and the star sensor reference frame (SSRF). Literature [4] proposed to use satellite angular velocity reconstructed by stellar sensor combined with gradient meter measurement data for calibration models. Literature [5-6] proposes a similar method to use stellar sensor data and accelerometer measurements for internal calibration, and proposes solutions for the time correlation of observations. This method is the official ESA method. Literature [7-8] determines the scale factor and deviation factor based on precision orbits, and uses stellar sensor data to verify the calibrated gravitational gradient data. Literature [9-10] found that adding a secondary factor to the calibration model can weaken the influence of strong winds around the geomagnetic poles, and in 2018, an external gravity field model was introduced to reprocess the GOCE data. Many domestic studies include preprocessing research on GOCE satellite data, theoretical methods for determining the earth's gravity field by gradient data [11-13] and calibration methods based on accelerometer data. Literature [14-16] proposes to use the dynamic method to complete single accelerometer calibration and satellite non-conservative force determination by combining geometric method, and solve the gravity field model and calibration parameters at the same time, thereby reducing the impact of reference gravity field model error on calibration results, and discussing the compensation effect of satellite undamped control. Literature [17] uses external gravity field and stellar sensor data to initially verify the effectiveness of calibration within a certain frequency band. Literature [18] discusses the influence of calibration using different external gravity field models and different orders of the same gravity field model on calibration results.
At present, the processing and use of GOCE data in China is mainly based on the internal calibration of ESA. This article uses the methods officially released by ESA to realize the internal calibration of GOCE data. The main purpose is to improve the calibration processing of GOCE gradient meter observation data, and to make preliminary preparations for discussing the joint and comparison of internal calibration, external calibration methods and dynamic calibration. The internal calibration method released by ESA models the error factor into an inverse calibration matrix of three accelerometer pairs, and applies this matrix to the accelerometer to achieve calibration. On this basis, this paper proposes an improved model that takes into account the calibration parameters of the rotation matrix between SSRF and GRF, and discusses the changes that this change brings to the satellite’s gravitational gradient accuracy. The data involved include the measurement data of the gradient meter and stellar sensor mounted on the satellite. The accelerometer common mode observation value and differential observation value and the attitude data of the stellar sensor are used to establish a calibration model based on the principle of satellite gravity gradient measurement to obtain the gravitational gradient under the coordinate system of the gradient meter after calibration. In addition, restoring the gravity field also involves high-precision conversion of gradient meter coordinate system and inertia system, so this paper proposes a discussion on satellite attitude reconstruction.
Theory and methods
.1 Internal calibration
GOCE satellite is equipped with several important payloads. This article focuses on electrostatic gravity gradient meter (EGG) used to determine medium and short-wave gravity fields, and 3 star sensors that provide satellite attitudes. GOCE is the first satellite to use -damping control technology, and atmospheric resistance along the orbit is continuously compensated. During the actual operation of the satellite, there are many error factors. For this reason, GOCE designed to perform a specific vibration that lasts 1 d every two months for the calibration of the satellite. The data during this vibration is called shaking data, and other periods are called nominal data. Figure 1 shows three pairs of accelerometer placement structures in the gradient meter coordinate system. The solid line represents the super-sensitive axis, and the dotted line represents the non-sensitive axis [6].
Figure 1 Arrangement of the three pairs of accelerometers in the GRF
Figure 1 Arrangement of the three pairs of accelerometers in the GRF
Figure 1 Arrangement of the three pairs of accelerometers in the GRF
Figure 2 is the special structure of the three pairs of accelerometers in the gradient meter. According to the principle of satellite gravity gradient measurement, the acceleration value measured by the i accelerometers is
(1)
Figure 2 Source of error for a single accelerometer Fig. 2 Error sources of a single accelerometer Fig. 2 Error sources of a single accelerometer
diagram options
where i is the accelerometer number; V is the gravitational gradient; Ω and r are vectors from the satellite center of mass to the i centroid; Ωri is the centrifugal acceleration generated by the satellite rotation around its center of mass; d is the linear acceleration of the satellite center of mass. The specific definition of the above variables is shown in the literature [6]. Two types of observation modes of
gradient meter are differential mode acceleration (DM) and common mode acceleration (common mode) acceleration, CM)[6]
(2)
(3)
Since the GRF origin is about a few centimeters different from the satellite center of mass, and
is about 3 orders of magnitude lower than d in the value, then according to formula (1), combined with common mode acceleration Definition of degree,
(4)
(5)
Formula (5) is expressed as matrix form (6)
, where Ad=(ad, 14, ad, 25, ad, 36), and the matrix L[6]
(7)
, in the formula, Lx, Ly, L, L, z respectively represent the arm length of the gradient meter in the directions of x, y, and z, respectively. This article uses the official ESA value [6], Lx=0.514 0135 m, Ly=0.499 890 0 m, Lz=0.500 201 0 m. The relationship between the angular acceleration of satellite relative to the centroid of V and Ω and the symmetry of
(8) and
(9)
1 gives the relationship between the angular acceleration of satellite relative to the centroid of the center of mass and the differential acceleration. Equation (9) shows that the satellite gravity gradient tensor can be derived from the differential acceleration, baseline length and angular velocity, and is finally used to restore the gravity field. .1.2 Calibration model
In the actual operation of satellites, the following error factors are usually considered: ① The accelerometer center of mass deviates from the nominal position; ② The accelerometer axes are not strictly aligned with the corresponding coordinate axes of the gradient meter; ③ The accelerometer 3 axis is not completely orthogonal; ④ Due to the uncertainty of data output gain, the accelerometer scale factor [10] generated. The error is parameterized into a scale factor, a deviation factor and a quadratic term, where the quadratic term is eliminated in a physical vibration manner. Therefore, the internal calibration matrix should include a scale factor and a deviation factor.Figure 3 shows the accelerometer error factors other than linear quadratic terms. Each accelerometer includes 6 angle calibration parameters and 3 scale factors. A total of 54 calibration parameters need to be determined for the 6 accelerometers, and the calibration matrix [6]
(10)
Figure 3 Comparison of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz Figure 3 Comparison of PSD/2 of centrifugal acceleration, differential acceleration and gravity gradients in the 0.05~0.1 Hz
where i accelerometer measurement values; a i is the corresponding real value; parameters s are the proportional factors in each direction; α, β, and γ are tangential factors respectively; δ, ε, and ζ are tiny tangential angles and tiny rotation angles respectively. In 2, K is used to characterize the scale factor, and θ is used to characterize the comprehensive influence of tangential factors and angles. For common mode and differential acceleration,
(11)
. In the formula, Mij is the calibration matrix, which defines the inverse calibration matrix Mij-1[6](inverse calibration matrices, ICM)
(12)
Equation (12) gives the relationship between the measured value, the true value and the inverse calibration matrix.
Equation (4) represents the common mode acceleration CM given by three pairs of accelerometer observations, that is, the non-conservative force exposed by the satellite in three directions. There are 6 independent conditions
(13)
angular acceleration
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The gravitational gradient V and the centrifugal acceleration term ΩS is much smaller than the differential acceleration observation signal Ad, see Figure 3 for details, using 0.05~0.1 bandpass filter of Hz filter equation (15), then
(16)
uses the gradient meter measurement data EGG_NOM_L1b and stellar sensor data STR_VC2_1b and STR_VC3_1b for the satellite nominal period officially released by ESA as input data to realize internal calibration. When establishing the above model, the following assumptions were made: ① The baseline length of the gravitational gradient meter given by ESA was Lx, Ly, L, Lz are accurate and fixed constants; ② The relative relationship between SSRF and GRF given by ESA is accurate and a constant matrix; this article explores whether these two assumptions are accurate and their impact on the accuracy of gravity gradients, and adds new parameters to the basic model to re-establish the model. Use tiny rotation angles rx, r, r, y, r, r to describe the comprehensive effect of the rotation matrix between the stellar sensor and the gradient meter coordinate system over time, use GRF' to represent the nominal gradient meter coordinate system, and GRF represents the coordinate system with deviation. The relationship between angular acceleration in the two coordinate systems is expressed as
(17)
(18 )
Formula (17), I represents unit array;
is the angular acceleration in GRF;
represents the angular acceleration in GRF' after calibration by ΔR, and change the formula (14) accordingly
(19)
write component form is
(20)
Since formula (17) is 0.05~0.1 Approximate treatment was performed in the Hz frequency band, and the baseline lengths Lx, Ly, and Lz were omitted on the denominator, resulting in less complete constraints on the estimate of baseline length. Therefore, this article will not discuss the changes in baseline length over time, and it is still considered that its value is a constant. Based on the ESA method, internal calibration is completed using formula (13), formula (20) and formula (16).
.1.3 Covariance matrix processing
This paper uses 2 d data to estimate a set of calibration parameters. According to the characteristics of GOCE sampling rate and colored noise, it is difficult to realize the storage and inversion of the matrix. References [6, 10], using symmetric sliding average decorrelation filtering, the corresponding frequency band filtering is realized while decorrelation processing is implemented to avoid inversion of large matrices. Formula (21) is the decorrelation filtering expression
(21)
In the formula, eni is the original error sequence containing color noise; n represents the nhtml in the sequence 18 elements; i represents the calibration conditions of i in Section 1.1.2; M is the order of filter ; fmi is the filter coefficient; eni is a sequence that contains only white noise after passing the filter. This article selects the filter order and the track period value. Readers who are interested in the calculation of filter coefficients can refer to the literature [6, 10].
.2 The three stellar sensors STR1, STR2 and STR3 installed on the combined
GOCE satellite provide satellite attitude information. One feature of the sensor is that the angular velocity around the viewing axis, that is, the angular velocity of the axis of z, that is, the angular velocity of the axis of z, the angular velocity of the axis of z, the y, the angular velocity of the axis of ωxSSRF, the ωySSR The accuracy of F, the error of ωzSSRF during coordinate conversion will be propagated to ωxGRF, ωyGRF and ωzGRF. According to the relative relationship between the star sensor and the gradient meter (Figure 4), this paper uses the least squares adjustment to combine the pose quaternions of the effective star sensor with [6, 10] to ensure that subsequent calibration results are more reliable and the angular velocity accuracy is higher.
Figure 4 Relative relationship between the gradient meter coordinate system and the coordinate system of 3 star sensors [6]Fig. 4 Relative orientation of the GRF and the SSRFs
Figure options
.3 Angular velocity and attitude reconstruction
angular velocity and attitude quaternion accuracy directly affect the gravitational gradient and the earth's gravity field inversion accuracy, and diagonal velocity and attitude reconstruction are necessary. The conversion of quaternions to angular velocity observed by the star sensor requires a numerical differentiation process, resulting in the amplification of the high-frequency band noise, and the angular velocity and quaternions derived from the integrated accelerometer's observations amplify the low-frequency noise. Literature [19] uses Kalman filtering to achieve angular velocity and posture reconstruction. Literature [20] uses the angular velocity of the stellar sensor and the angular velocity of the gradient meter to establish a weight model in the frequency domain, and reconstruct the angular velocity and attitude with Wiener filtering. Literature [10] proposes to use least squares fit to reconstruct the pose. Kalman filtering and least squares fitting are both unfolded in the time domain without using the frequency domain characteristics. The transient effect of Kalman filtering is severely caused by low data utilization. In this article, refer to [20] to use the advantages of Wiener filtering in the frequency domain to combine the advantages of two types of data in the frequency domain to achieve angular velocity and attitude reconstruction. The principle of
Wiener filtering is to realize the weighted average in the frequency domain based on the accuracy of the two types of observation values. The power spectrum density value of a certain frequency in the frequency domain (power spectral density, PSD) represents the accuracy here
(22)
In the formula, HSTR (f) and HGRAD (f) represent the weights of the sensor and the gradient meter, respectively; PSTR (f) and PGRAD (f) respectively represent the square root power spectral density of two types of angular velocities in the frequency domain, model reference [20]. Figure 5 shows two types of angular velocity noise PSD/2, where G represents the gradient meter and S represents the stellar sensor.
Figure 5 Two types of angular velocity noise PSD/2Fig. 5 PSD/2of two types of noise
Figure options
angular velocity was finally calculated by the product in the frequency domain and the Fourier forward and inverse transformation to obtain
(23)
In the formula, n represents x axis, y axis and z axis.Integrating the reconstruction angular velocity can obtain the quaternion qtGRAD. Also considering that numerical integration leads to amplification of low-frequency noise, Equation (24) uses Wiener filter consistent with the angular velocity reconstruction process and the quaternion qtSTR to reconstruct the posture [0] to reduce the influence of noise
(24)
In formula, t represents the observation time.
Data analysis
Selection November 2009 for every 2 consecutive days d's nominal measured data is a set of one set, with a total of 15 sets of data subjected to the above calibration and reconstruction process, and the inverse calibration matrix Mij-1 and the rotation matrix calibration parameters ΔR. The calibration model is divided into two categories to discuss the data processing results. The first category is consistent with the ESA internal calibration method and only consider Mij-1; the second category adds ΔR parameters. In the analysis of the results of this paper, in addition to using the Laplace equation , which is satisfactory with the gravitational gradient tensor gradient, the gravity field model EIGEN-5C is also used as the reference field. The data mainly used in the calculation of this model include satellite height measurement data, ground gravity data, GRACE and Lageos satellite data. The reference model is used to calculate the gravitational gradient reference value, and the power spectrum of the difference between the actual gravitational gradient data after calibration and the reference value is analyzed to analyze the impact of calibration parameters on the gradient results.
picture 6. Part of the Mij-1 sequence is given, and the 1st parameter of the 14 accelerometer pair C14 line 2 in the expression (12) is compared with the ideal situation (the proportional factor and deviation factor are 1 and 0 respectively). The results show that the matrix Cij, Dij, Dij (ij=14, 25, 36) The order of the non-diagonal difference is 10-5~10-4, and the order of the diagonal difference is 9×10-4~2.7×10-2. Considering that the calibration parameters have a linear trend of about 2×10-5 within one month, the parameters are linearly fitted and then calibrated. Table 1 is the numerical value of a set of scale factors obtained by fitting. Figure 7 shows the power spectral density of the gravitational gradient tensor trace after calibration of the first type of method in this paper, which matches the calibration results of ESA. Figure 8 shows PSD/2 of the stellar sensor combined with the front and back gravitational gradient tensor trace.
Figure 6 Part Mij-1 parameter sequence Fig. 6 Part parameters series of Mij-1
Figure options Table 1 Accelerometer comparative factor Tab. 1 Scale factors of accelerometer pairs
| accelerometer pair number | direction | ||
| x | y | y | z |
| 14 | 1.023 4 | 0.973 7 | 1.020 1 |
| 25 | 1.017 7 | 1.018 4 | 0.979 8 |
| 36 | 1.018 4 | 0.973 3 | 1.022 7 |
Table options
Figure 7 PSD/2Fig of the gravitational gradient tensor trace after calibration. 7 PSD/2of calibrated gravity gradient trace
Figure options
Figure 8 PSD/2Fig of the stellar sensor combined with the front and back gravitational gradient tensor trace. 8 PSD/2of calibrated gravity gradient trace before and after combination of star sensors6
Figure 8 and Figure 9 show the square root power spectrum density of the front and back gravitational gradient tensor trace and the square root power spectrum of the difference between each component and the reference component, indicating that combining multiple stellar sensors can increase below 0.005 The accuracy of the gravitational gradient in the Hz frequency band, the maximum improvement value of the square root power spectrum of the gradient tensor trace is 1.8×10mE/
, and the maximum improvement value range of each component is 3.8×10~1.2×10mE/
.In Figure 10, there is a clear sharp point at about 0.02~0.03 Hz for the V x component, and the other components also have this phenomenon at the corresponding frequency. By analyzing the power spectrum of the difference between the gravitational gradient observation value and the reference value difference between the gravitational field models of different types and different orders, it was found that the sharp point will disappear only when the GOCE gravity field model is used as the reference field, and sharp points will exist when the non-GOCE gravity field model is used. Therefore, it is speculated that the emergence of sharp points may reflect the contribution of GOCE gradient data. When the signal on the corresponding frequency band does not contain GOCE gradient data, the spherical harmonic coefficient error of the corresponding order of the gravity field model is relatively large. This phenomenon deserves further in-depth study.
Figure 9 PSD/2Fig of the difference between the gravitational gradient component and the model reference value before and after combination of star sensors 1 Figure 10 Parameter ΔR sequence Fig. 10 Time series of ΔR
Figure Option Based on the first class ESA model, this paper adds the parameter ΔR, which changes over time, the rotation matrix between SSRF and GRF. This change will affect the angular velocity and attitude reconstruction process, and the gravitational gradient component will also change accordingly. Here is a data sequence and linear fitting sequence with the newly added parameter ΔR. There are several obvious outliers in Figure 10. The working status of the three star sensors in the corresponding time period is analyzed. Statistics show that within the period when the outliers exist, the number of times available for only a single star sensor data is much larger than the corresponding values in other time periods, which also reflects the advantages and necessity of combining multiple star sensors. The absolute value of rx, ry, r, and rz is around 100″ and the monthly value shows a linear trend of about 3~30″. Therefore, the calibration parameters should not be ignored when establishing a calibration model.
applies ΔR to the angular velocity and quaternion of the star sensor, and then reconstructs the angular velocity and attitude to quantitatively analyze the impact of the newly added calibration parameters on the accuracy of the gravitational gradient. Figure 11 shows the square root power spectrum analysis of the gravitational gradient difference between the gravitational gradient component before and after the parameter ΔR and the reference model gravitational gradient difference. Table 2 is below 0.005 The two types of calibration models with Hz frequency correspond to the square root power spectrum statistics of the difference between the gravitational gradient components and the model reference value. Among them, "Method 1" means that it is consistent with the ESA model, and "Method 2" means that it is added to the new ΔR, 4 gravitational gradient components Vxx, Vxz, Vy, Vy, The square root power spectrum of the difference value corresponding to l17Vzz shows that the accuracy of the gravitational gradient component has been improved by adding new parameters. Among them, the accuracy of the component Vxz is the greatest improvement. This is because the newly added parameters in "Method 2" change the attitude of the satellite. Compared with other components, the Vx component is more sensitive to changes in posture, so its accuracy is maximized. Since the added parameters mainly affect the pose quaternions, the gradient tensor trace will not change with the quaternions [1-27], no corresponding comparison diagram is provided here.
Figure 11 PSD/2Fig of the difference between the gravity gradient components and the model reference value of the two types of calibration models. 11 PSD/2of differences of gravity gradients to EIGEN-5C model for the two calibration models
Figure options
Table 2 PSD/2 statistics of differences between the gravity gradient components and the model reference value of the two types of calibration models Tab. 2 PSD/2statistics of differences of gravity gradients to EIGEN-5C model for the two types of calibration models Tab. 2 PSD/2statistics of differences of gravity gradients to EIGEN-5C model for the two types of calibration models
| component | calibration model | maximum | minimum | mean | standard deviation |
| Vxxx | method 1 | 45 245 823.203 | 47.036 | 5 221 249.416 | 13 553 688.125 |
| method 2 | 45 244 950.335 | 43.651 | 5 220 653.314 | 13 553 591.064 | |
| Vxz | method 1 | 2 794 553.787 | 53.980 | 323 811.690 | 836 447.020 |
| method2 | 2 696 051.381 | 42.138 | 312 174.185 | 807 154.872 | |
| Vyy | method1 | 56 026 989.558 | 127.440 | 4 670 338.535 | 16 173 150.910 |
| Method 2 | 56 024 742.462 | 110.274 | 4 669 472.429 | 16 172 715.845 | |
| Vzzz | method1 | 18 562 356.933 | 94.722 | 2 142 510.910 | 5 560 223.548 |
| method2 | 18 561 125.427 | 74.125 | 2 141 971.948 | 5 560 046.890 |
table options
Conclusion and prospect
This paper starts from the principle of GOCE satellite gravity gradient measurement, and uses the gravitational gradient meter observation data and the stellar sensor attitude data in the nominal period in L1b data to internally calibrate the accelerometer measurement value, and combines the observation values of multiple stellar sensors with least squares to avoid the error propagation caused by the conversion between SSRF and GRF; in order to obtain a more accurate gravitational gradient tensor under the inertial coordinate system to restore the gravity field, the satellite angular velocity and attitude quaternion were reconstructed using Wiener filtering. Based on the ESA calibration method, an internal calibration model that takes into account the calibration parameters of the rotation matrix between SSRF and GRF was proposed. The results show that the absolute value of the calibration parameters of this group is about 100″, and a linear trend of about 3~30″ was shown in the month. At the same time, the internal calibration model of the inverse calibration matrix and the calibration parameters of the rotation matrix between SSRF and GRF can improve the accuracy of each component of the gravitational gradient in the frequency band below 0.005 Hz. The maximum improvement for the gradient component Vxzz is confirmed, which confirms the necessity of this calibration parameter and brings certain reference value to GOCE's own data processing and subsequent gravity satellite data processing.
Finally, based on this article, it proposes possible improvement directions for GOCE data processing: ① Due to the lack of external data constraints, the internal calibration method cannot avoid the impact of GOCE's own system deviation. Therefore, consider comparing the external calibration method and dynamic method to combine the advantages of various methods can promote the discussion of gradient data processing problems; ② Regarding the baseline length Lx, Lyhtm The determination of l18, Lz and ΔR and the compensation of quadratic term K2 are considered, and the introduction of a priori gravity field and combined with the stellar sensor data solution; ③ In the process of symmetric sliding average decorrelation filtering of the covariance matrix, this paper uniformly selects the filter order equal to the orbital period length, but the power spectrum estimated by this length is not optimal during the actual solution, and the compromise choice of power spectrum resolution and accuracy in the decorrelation process is worth in-depth discussion.
Author Profile
First Author Profile: Pan Juanxia (1996-), female, master, research direction is satellite gravity measurement.E-mail: [email protected]
correspondence author: Zou Xiancai, E-mail: [email protected]
preliminary review: Zhang Yanling
review: Song Qifan
final review: Jin Jun
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