On-orbit installation matrix calibration and its application on AGRI of FY-4A

Abstract. Image navigation is a primary process for on-orbit optical payloads involving environmental disaster monitoring, meteorological observation, and the positioning and tracking of space-aeronautics targets. However, because they are affected by solar illumination and orbital heat flux, as well as shock and vibration during launch, the installation structures between the instruments and satellite platform, especially for geostationary satellites, will inevitably generate a displacement resulting in the reduction of positioning accuracy. During the application of FengYun-4A (FY-4A), it is found that the further away from the subsatellite point, the greater the positioning error of Advanced Geostationary Radiation Imager (AGRI) on FY-4A will be. The positioning error can reach 14 pixels at the Arabian Peninsula in operational images. In addition, compared with orbital and attitude measurement errors, long-term observations show that the installation matrix is likely to be the most significant factor determining the navigation accuracy of AGRI. Therefore, an on-orbit installation matrix calibration approach as well as a high-precision navigation algorithm is proposed to modify the positioning error of AGRI. Experimental results show that the navigation error of the processed images corrected by the proposed method can be reduced to 1.3 pixels, which greatly improves the navigation procession of AGRI. In general, this method could be a supplement to the correction of positioning error for geostationary payloads.


Introduction
Image navigation (IN) is an exceedingly crucial and fundamental prerequisite for applications involving the monitoring, forecasting, and warning of devastating weather, climate, and natural disasters for meteorological satellites. 1,2 Usually, IN indicates the quantitative mapping relationship between the image points in the image coordinate system and the corresponding objects in the Earth-centered rotating (ECR) coordinate system. 3 The characteristics of the sensors, distortions of the optical system, alignment matrix from the instrument to the satellite, attitude derived from star trackers and gyros, and the satellite position and velocity vectors play a significant role in determining the navigation accuracy of satellites. [4][5][6] In addition, suffering from the spatial thermal elastic deformation, the geometric positioning model of geostationary satellites changes greatly, 1,2,7 which inevitably has a bad effect on the positioning accuracy. Hence, an efficient method for promoting navigation accuracy is very important for geostationary satellites.
Then, the simulative analysis is adopted to show that installation matrix is one of the most significant factors affecting the navigation accuracy of AGRI and must be calibrated. The database, including error simulation and on-orbit observation data, is used to demonstrate the efficiency of the proposed method. Finally, the conclusion and the prospects are given.

Methodology
This section elaborately describes the rigorous geometric imaging model of AGRI, the proposed installation matrix solving algorithm, and the assessment of navigation precision. Figure 1 shows the overall flowchart of the proposed method. First, after constructing the geometric positioning model, we will perform the landmark simulation and calculate the LOS for the preparation calibration. Specially, the landmarks are simulated and obtained based on the GSHHS and the coastline template matching. Then, the installation matrix is calibrated with the GCPs and the observation ephemeris data. Finally, the renavigation and the resamping are implemented based on the calculated installation matrix. In the following, the proposed method will be described in detail.

Rigorous Imaging Model
As shown in Figs. 2 and 3, the rigorous imaging model of AGRI describes the mapping relationship between the image points in the image coordinate system and the corresponding objects in the ECR coordinate system, which is a very crucial foundation for the on-orbit geometric calibration. 2,14,15 In general, the remote sensing satellite observes the Earth via a complex optical system. In terms of the obtained images, every pixel in the image coordinate system has a mapping area on the surface of Earth. For high accurate applications of remote sensing data, the relationship between the pixel location coordinate of ði; jÞ and the latitude and longitude (latitude, longitude) of the corresponding area on the Earth's surface is supposed to be known. The pixel location coordinate is determined by the angles between the LOS and primary optical axis, which means ði; jÞ ∝ ðα; βÞ. The α and β are the stepping angles of the East-West (EW) and North-South (NS) scanning mirrors, respectively.   The complete transformation from the image coordinate system of ði; jÞ to the geocentric geodetic coordinate system (GGCS) of (latitude, longitude) can be expressed by Eqs. (1)-(5) as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 6 9 9 ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 6 3 3 (2) where ð0;1; 0Þ T is the unit vector coinciding with the optical axis and ðvx ECR ; vy ECR ; vz ECR Þ T is the corresponding direction vector in the ECR coordinate system. A, B, C, and D are the pole motion matrix, sidereal time matrix, nutation matrix, and precession of the equinoxes matrix, respectively. R ORB2ECR , R ORB2ECI , R SAT2ORB , and R INS2SAT are the transformation matrices from the orbital coordinate system (OCS) to the ECR coordinate system, from the OCS to the Earthcenter inertial (ECI) coordinate system, from the satellite coordinate system (SCS) to the OCS, and from the instrument coordinate system (ICS) to the SCS, respectively. R ECR2ECI ∝ ðΩ; i; e; ax; ϖ; MÞ and ðΩ; i; e; ax; ϖ; MÞ are the six orbital elements. R EW and R NS are the reflective matrices of the EW and NS scanning mirrors, respectively. In particular, compared to the NS scanning mirror, the EW scanning mirror in AGRI is closer to the focal plane. Therefore, the order of R EW and R NS in Eq. (1) is not changeable. We have E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 4 9 0 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 6 ; 4 0 3 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 6 ; 3 2 7 where t is the scale factor and ðx; y; zÞ is the corresponding coordinate in the ECR coordinate system. a and b are the semimajor axis and semiminor axis of the Earth ellipsoid model, namely the Internal Terrestrial Reference System. ðB; L; HÞ is the coordinate in the GGCS. N is the ellipsoid radius of the curvature in the prime vertical and e is the ellipsoid eccentricity. 27 Based on Eq. (1), the observing vector in ICS can be transformed to the ECR coordinate system with the associated transformation matrices. The rotating matrix R A2B in Eq. (1) represents the transformation from A coordinate system to B coordinate system and can be described as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 6 ; 1 8 6 where θ, ϕ, ψ are the three Euler angles from A coordinate system to B coordinate system. Every rotation can be expressed as follows: As we know, during the process of observation, the infrared radiation of the targets will be captured by the optical system. Therefore, theoretically, there will be only one intersection between the LOS in the ECR coordinate system and the Earth's surface. According to the geometric relationship, the intersection point must be the solution of Eq. (3). Based on the analysis above, it is undoubtable that the instant installation angles of the camera could be calculated with the locations of the observed targets. In this paper, the installation matrix is solved by the obtained GCPs to modify the navigation error of the camera.

Installation Matrix Solving Algorithm
In this part, the installation matrix solving algorithm based on the GCPs is described in detail.
Factually, the orbit and attitude measurement errors are random errors, whereas the installation angle error is a systematic error. 14,15 Although calibrated in the laboratory before launch, the installation matrix usually varies during operation in orbit. Affected by solar illumination, orbital heat flux, and the shock and vibration during launch, the installation angles are not constant but have a complex variation tendency. 2,14 In addition, because of the strong correlation between the orbit and attitude measurement errors and installation angles, 14 the orbit and attitude measurement random errors are likely to be compensated when calibrating the installation angles by the obtained image. In this paper, we proposed a correction method by calculating the installation angles of θ, ϕ, ψ to modify the final navigation error. We have E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 ; 1 1 6 ; 5 2 2 Þ are the coordinates of the corresponding GCPs and satellite in the ECR coordinate system, respectively. According to Eqs. (1) and (8), we have E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 1 1 6 ; 4 2 8 Then, Eq. (9) can be presented as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 0 ; 1 1 6 ; 3 5 3 where ½ vx tr vy tr vz tr T ¼ W2 Ã ½ 0 1 0 T . As described above, θ, ϕ, ψ are the three installation angles between the AGRI and the satellite platform and R INS2SAT is the transformation matrix from the ICS to SCS. Consequently, we have E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 6 ; 2 5 3 where sin 2 ðγÞ þ cos 2 ðγÞ ¼ 1, γ ¼ θ, ϕ, ψ. Then where Based on the equations above, it could be noted that there are only three variables, namely the installation angles of θ, ϕ, ψ in Eq. (12). Because Eqs. (3) and (4) are quadratic equations, there is more than one solution for the coordinates of ðx; y; zÞ. Considering there is only one intersection between the LOS and the Earth's surface, the solutions against the practical application will be rejected. Theoretically, the more accurate landmarks adopted in the process of solving, the more accurate the θ, ϕ, ψ will be. For calculating the equations accurately, the gradient descent method based on the fact that only a little error exists between the real installation matrix and the theoretical installation matrix is adopted to search the optimal solution of the variables. As shown in Algorithm 1, the pseudocode of the solving algorithm is given in detail. 4. Calculate the total error as: ; t e m p : i n t r a l i n k -; t 0 0 2 ; 1 8 8 ; 2 8 6 Run steps 2 to 4 using θ τþ1 , ϕ τþ1 , and ψ τþ1 . end while 6. The final obtained installation matrix parameters: E Q -T A R G E T ; t e m p : i n t r a l i n k -; t 0 0 2 ; 1 8 8 ; 8 9 θ ¼ θ τþ1 ϕ ¼ ϕ τþ1 ψ ¼ ψ τþ1 :

Assessment of Navigation Precision
In this paper, the defined angle errors of the LOS are adopted to represent the navagation errror of the satellite. The angle errors between the calculated observing vector and the real observing vector are calculated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 1 1 6 ; 6 8 0 where LOS ! is the real observing vector based on the coordinates of the landmarks and LOS ! 0 is the calculated observing vector via the vector transformation equation based on the angles of scanning mirrors. A err is the intersection angle between LOS ! and LOS ! 0 .
According to the calculated A err , the average of the angle errors can be calculated for assessing the navigation error: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 1 1 6 ; 5 8 0 where A err_avg is the calculated average of the intersection angles. Based on the parameters of the instrument apparatus, the navigation error measured by pixel can be expressed as where IFOV is the instantaneous field of view of the AGRI.

Experimental Results and Discussion
In order to demonstrate the effectiveness of the proposed method, the simulative data and the onorbit observation images are adopted to calculate the installation matrix and the positioning error of the AGRI. First, the simulative measurement errors of orbit, attitude, and installation angles are used to demonstrate the considerable effects of installation angles on the navigation accuracy. Then, the simulative control points with random and systematic noise are employed to show the stability of the solving algorithm. Subsequently, more than one year of on-orbit observation images are used to analyze the navigation accuracy of AGRI. In this paper, the experimental results at 13:00 (satellite local time) on October 21, 2018, are randomly selected to show the navigation accuracy of the proposed method, because the sunlight at 13:00 (satellite local time) literally has little effect on the research of positioning precision.

Simulation Experiments
Based on the whole transformation process from the image coordinate of ði; jÞ to the geodetic coordinate of (latitude, longitude), we analyzed the potential errors during the navigation. As shown in Eq. (1), A, B, C, D are the polar motion matrix, sidereal time matrix, nutation matrix, and precession of the equinoxes matrix, respectively, which are the certain coefficients during the operation of AGRI. R ORB2ECI is a function of the orbital elements and R ORB2ECI is a function of the satellite attitude, which is measured by the star sensor, Earth sensor, and the gyroscope mounted on the satellite. The R INS2ORB is the installation matrix between the instrument and the satellite platform. For FY-4A, the orbit precision is 100 m, the attitude precision is 5 angle seconds, and the experience from the GEOS-8 geostationary meteorological satellite shows that the variation of the installation matrix can reach 1000 mrad. 28 In order to analyze the influences of the orbital measurement error, attitude error, and installation matrix error on navigation accuracy, all these parameters are amplified three times in the experiments. Figures (4)- (6) show the variation tendency of the navigation errors with the errors of orbit, attitude, and installation matrix. It should be noted from Fig. 4 that although the orbit error is 300 m, which is three times bigger than that of real orbit precision, the positioning error is <0.5 visible pixel. As shown in Fig. 5, because the x and y directions are symmetrical, only three lines can be seen. Although the attitude error is 15 angle seconds, which is three times larger than that of real attitude precision, the positioning error is less than four visible pixels. Similarly, in Fig. 6, there are also only three lines that could be seen. However, it should be noticed that the errors of the installation matrix could lead to a huge positioning error, which could never be negligible. According to the experimental results above, it is safe to conclude that compared with the errors of orbit and attitude, the bias of installation angles could have a more considerable effect on the navigation accuracy of AGRI.
Here, another simulative experiment is designed to analyze the precision and robustness of the proposed algorithm. The simulation experiments are operated with the following steps: (1) randomly generate 30, 50, 100, 300, 500, 1000, and 2000 control points from the observation   images and calculate the theoretical observing vectors with the orbital measurement data, the attitude, and the initial installation matrix of AGRI. (2) Add random noises of mean = 0, variance = 0, 0.2, 0.4, 0.6, and 0.8, and systematic noises of mean = 0, 0.2, 0.4, 0.6, variance = 0 to all the ground points sets correspondingly. (3) Calculate the actual observing vectors using the same orbital measurement data, attitude, and initial installation matrix in step (1). (4) Calculate the navigation error with the proposed method mentioned in Sec. 2.
As shown in Fig. 7, it should be noted that the navigation error caused by the control points with random noise is obviously decreased when the number of control points increases. More importantly, the navigation error will not decrease any more once the number of control points reaches 500 because the solving algorithm turns out to be steady. It should also be noted that the navigation error can be <0.5 pixels with enough control points despite the large random noise. However, as shown in Fig. 8, the navigation error derived from the control points with systematic noise cannot be reduced by using more control points and the navigation error can be really stable with enough control points. Consequently, it could be concluded that the installation matrix solving algorithm can be steady and precise enough even with random and systematic noises.

On-Orbit Experimental Results
In this part, different numbers of GCPs are adopted to demonstrate the effectiveness of the proposed installation matrix calibration method and to assess the influence of GCPs errors on the navigation accuracy of AGRI. Images at 13:00 (satellite local time) on October 21, 2018, in five different regions are selected to show the navigation accuracy of AGRI.
First, all 26 control points in Table 2 are used to calculate the theoretical installation matrix. Then, the navigation results derived from 5 to 26 control points are calculated correspondingly. As shown in Fig. 9, the points in the red line represent the navigation errors assessed by all Table 2 Navigation precision with quality control and without quality control. No.

Latitude
Longitude A errðiÞ (pixel) . First, the installation angles are calculated with the GCPs and then these GCPs are adopted to assess the navigation error of this process. All the navigation errors are calculated with Eqs. (13) and (14). From the changing tendency of the red and black lines in Fig. 9, it is not difficult to find that the accuracy of the GCPs will affect the assessment of the navigation accuracy. It should be noted that the points in the red line fluctuate slightly and turn out to be flat and stable, which indicates that there is a large error in the GCPs and the error is stable under different installation angles. Differently, the points in the black line imply that fewer GCPs could get better navigation results, which means the effects from GCPs with large errors are eliminated.
Since the random error of sampling is 0.5 pixels, it is considered that the navigation errors within 0.5 pixels (calculated with the corresponding GCPs) is the random error, which can be suppressed by means of a mean filter. Therefore, as the accuracy of GCPs is improved within 0.5 pixels, the θ, ϕ, and ψ solved by the proposed method tend to be efficient and stable. In the visible spectrum, the navigation error reaches the least of 1.3 pixels. However, the GCPs with   a location error of more than 0.5 pixel cannot be used to solve the θ, ϕ, and ψ because the location error of the GCPs will have a detrimental effect on the navigation result. Obviously, the more GCPs that are used with quality control of 0.5 pixel, the more robust the solving results are.
In order to show the experiment results more clearly, five typical areas including the Arabian Peninsula are selected to compare the navigation precision of the operational image and the image processed by the proposed method in Fig. 10. The selected five places are distributed intentionally in the North-East, North-West, South-West, South-East, and middle of the image, correspondingly. Thus, these areas can reveal the navigation precision of the whole image. As shown in Fig. 11, the GSHHS landmarks are almost near the observing landmarks. Specifically, the location errors between the landmarks are around 1 pixel, which is approximately the same as the numerical measurement results.

Conclusion
In this work, in order to improve the navigation precision of AGRI, a new on-orbit installation matrix calibration approach is suggested. Based on the GCPs, the rigorous imaging model and the associated installation matrix solving method are presented in detail. Accordingly, some simulation experiments based on the actual parameters of FY-4A have been made to validate the effectiveness and stability of the proposed method. More importantly, when applied in the on-orbit observation images of AGRI, the new parameters of the installation matrix calibrated can decrease the positioning error to around 1.3 pixels, which is better than that of 14 pixels before correction. Although this installation matrix calibration approach is proposed for the navigation of AGRI, it is still suitable and versatile for other high-precision positioning research of other satellites because of the imaging similarity of different cameras. Qiang Guo received his MS degree in signal and information processing and his PhD in electronic science and technology from Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, China, in 2001 and 2003, respectively. He is currently the second chief designer of the ground-segment systems for both Fengyun-2 and Fengyun-4 serial satellites. His research interests include on-orbit radiometric calibration, geolocation, and specification evaluation for spaceborne instruments. Fansheng Chen received his BS degree in optoelectronic information engineering and his PhD in physical electronics from Shandong University, Jinan, China, in 2002, and Shanghai Institute of Technical Physics of the Chinese Academy of Sciences, Shanghai, China, in 2007, respectively. His research interests include the design of spatial high-resolution remote sensing and detection payloads, high-speed and low noise information acquisition technology, and infrared dim small target detection technology.