The new generation Chinese high-resolution three-line stereo-mapping satellite Ziyuan 3 (ZY-3) is equipped with three sensors (nadir, backward, and forward views). Its objective is to manufacture the 1 : 50000 topographic map and revise and update the 1 : 25000 topographic map. For the push-broom satellite, the interpolation accuracy of orbit and attitude determines directly the satellite’s stereo-mapping accuracy and the position accuracy without ground control point. In this study, a new trajectory model is proposed for ZY-3 in this paper, according to researching and analyzing the orbit and attitude of ZY-3. Using the trajectory data set, the correction and accuracy of the new proposed trajectory are validated and compared with the other models, polynomial model (LPM), piecewise polynomial model (PPM), and Lagrange cubic polynomial model (LCPM). Meanwhile, the differential equation is derivate for the bundle block adjustment. Finally, the correction and practicability of piece-point with weight polynomial model for ZY-3 satellite are validated according to the experiment of geometric correction using the ZY-3 image and orbit and attitude data.
Most high-resolution remote-sensing satellites are the near polar satellite; these satellites generally run on their trajectory below 1000 km in order to acquire the higher resolution for earth observation [
For the linear push-broom satellite, every acquired image line has different data of orbit and attitude, and the instrument just records the data at regular intervals, but not all. The unrecorded data at a certain time therefore needs to be interpolated using exterior orientation model [
Generally, the satellite running trajectory is relatively stable in a short period so that the orbit and attitude in a short interval trajectory can be modeled with the polynomial, therefore avoiding the complex stress analysis of the satellite [
For satellite ZY-3, a new trajectory model (piece-point polynomial with weight model) is proposed to acquire the higher interpolation’s accuracy in this paper, based on analyzing the orbit and attitude data of ZY-3. In addition, the data set of ZY-3 is used to validate the correction of piece-point with weight polynomial model and compare it with the other trajectory models, LPM, PPM, and LCPM, used for other satellites. Meanwhile, the differentiation equation of the proposed trajectory model is derivate and it is validated with the block bundle adjustment. According to geometric correction experiment with the different ground control points, the correctness and applicability of piece-point with weight polynomial model are validated and assessed to ensure and improve the high accuracy of geometric correction for ZY-3 satellite.
Trajectory model is a mathematic relationship elucidating the orbit and attitude of satellite vary with the different time in its track. For the push-broom satellite, every acquired image line has a different data of orbit and attitude, and the instrument just records the data at regular intervals, but not all the data. The unrecorded data at a certain time thereby needs to be interpolated using exterior orientation model.
Piece-point with weight polynomial model (PWPM) is proposed according to the researching and analyzing of the data of ZY-3’s orbit and attitude in the long and short period. In comparison with the other models, LPM, PPM, and LCPM, the weight value is used in the new model to perform interpolation’s calculation. The new model therefore can acquire a higher accuracy, have a better flexibility, and can reduce the correlation among the exterior orientation elements. The PWPM is an interpolation model with weight value. Using the model to interpolate satellite’s orbit and attitude, the weight is calculated by the difference from any interpolation’s time to the time assumed as known. Through the least square method, the polynomial parameters are solved, and then the exterior orientation at any time on the orbit can be acquired with the polynomial parameters. The PWPM is represented by (
Diagram of piece-point with weight polynomial model.
Assuming that the four known times
Due to the four known times, the four error equations can be established and the coefficient matrix
According to the equation of weight, the weights
According to the least square method, normal equation coefficient matrix
For convenience, the matrices
Through least square adjustment, the polynomial parameters
Afterwards, the Kappa value at
In the process of interpolation, the PWPM can solve the different parameters corresponding to the different attitude and orbit at any time with the different weight value. In this paper, the two weight equations are given out, the reciprocal of the absolute value of time difference and the reciprocal of the square of the time difference. The selection of weight equation has a large impact for the interpolation’s accuracy of orbit and attitude. According to analysis and research, the reciprocal of the square of the time difference is adopted when the trajectory is relatively unstable. On the contrary, the reciprocal of the absolute value of time difference is utilized.
For the PWPM, the new trajectory has two kinds of interpolation methods to acquire the data set of orbit and attitude at any time. One is using the several known times selected round the unknown time to interpolate the orbit and attitude at the unknown time. The other is using all the selected known times to interpolate the orbit and attitude at the unknown time. The impaction from the selected known time for the orbit and attitude at the unknown time is measured and assessed according to the weight value. In other words, the time difference between the unknown time and the known time is more far, and the impaction from the unknown time is more great. The interpolation accuracies with the two methods are different, which is determined by the stability of the satellite trajectory, the number of selected known times, and the location of the selected known time. In the practical application, the two methods of PWPM are utilized together or respectively, which is determined by the stability of orbit and attitude.
Sensor’s imaging model describes the mathematic transformation relationship between the coordinate of image point
For ZY-3 satellite, the data received from dual-frequency GPS represents the location of the phase center of GPS and the attitude data from star sensor is measured in the J2000 coordinate [
For the push-broom high-resolution satellite, the objective of the high-accuracy trajectory model is to acquire the accurate elements of exterior orientation,
Assuming that the n known times are selected in a scene image of satellite and the orbit and attitude at time
In the calculation of bundle block adjustment, the differential expression of
Similarly, the differential expression of the other elements of exterior orientation, (Roll, Yaw) and
The systematic error model for the interior orientation is to describe the various distortions from satellite’s sensor such as the CCD-array distortions, the distortions of optic lenses, and principal point’s distortion. In order to realize the high-precision geometric correction for ZY-3 image, it is very necessary to establish the various error models based on the analysis of satellite’s structural parameters; thus the system error coming from the interior orientation, radial direction, and tangential direction distortion of optics lens and CCD-line’s distortion and rotation will be modeled [
According to the analysis of the correlations among the model’s parameters in the block bundle adjustment, the correlation between the principal point and focal length is very strong, so that the parameters are combined in order to reduce the parameters correlation and improve the stability and accuracy of the block bundle adjustment. Equation (
In this paper, data set of orbit and attitude used to validate the correction and accuracy of piece-point polynomial model is acquired from 609th track of ZY-3. In order to validate the high accuracy of the new proposed trajectory model, the LPM, PPM, LCPM, piece-point with weight polynomial with four known times model (PWP4M) and piece-point with weight polynomial model with all known times (PWPM) are utilized to interpolate and compare the interpolation accuracy. In the process, the different numbers of the known times, 10, 15 and 20, are selected from trajectory data set, and are used to interpolate the other unknown times’ orbit and attitude data with the different models, respectively. Finally, the result of interpolation is represented by the table and curve, and the advantage of PWPM is illuminated according to researching and analyzing the result.
In order to validate the correction and accuracy of PWPM, ZY-3 orbit, and attitude data, ground control point (GCP) and systematic error model of interior orientation are used in the bundle block adjustment of geometric correction. Based on the nadir image of ZY-3, the 74 GCPs are picked up from the image, and 27 GCPs are selected as check points (CPs) that do not take part in the block bundle adjustment. For validating the correction and stability of the proposed models, the 16, 26, 36, and 46 GCPs are performed, respectively, in the geometric correction experiment. Figure
(a) Diagram of the distribution of GCPs; (b) the error’s distribution of image points corresponding to GCPs.
According to analyzing the stability of the orbit and attitude of ZY-3, it can be seen obviously that the orbit and attitude angles of Yaw of ZY-3 are very stable, but the attitude angles of Pitch and Roll are unstable relatively. The curves of attitude angles in 10 seconds are shown in Figures
The attitude angle (Roll) curve in 10 seconds.
The attitude angle (Pitch) curve in 10 seconds.
The attitude angle (Yaw) curve in 10 seconds.
In Figure
The fitting curves with the different trajectory models.
From Figure
The fitting accuracy comparison of the different attitude and orbit modes selecting the different known times on the orbit (unit: degree).
|
10 | 15 | 20 | ||||||
---|---|---|---|---|---|---|---|---|---|
Angle | Roll | Pitch | Yaw | Roll | Pitch | Yaw | Roll | Pitch | Yaw |
1-LPM | 64.610 | 12.772 | 15.100 | 47.859 | 9.103 | 15.297 | 47.362 | 8.569 | 15.122 |
2-PPM | 63.136 | 12.237 | 16.210 | 47.768 | 8.737 | 15.027 | 47.281 | 8.532 | 14.948 |
3-LCPM | 63.523 | 11.955 | 13.919 | 51.758 | 8.275 | 10.257 | 45.901 | 5.946 | 9.554 |
4-PWP4M | 63.668 | 11.732 | 13.956 | 34.760 | 8.034 | 7.7186 | 15.628 | 5.727 | 4.539 |
5-PWPM | 60.287 | 10.387 | 12.534 | 32.837 | 7.648 | 7.5958 | 22.261 | 5.593 | 5.140 |
The new proposed trajectory model (PWPM) has higher interpolation’s accuracy and more flexibility than the other models according to the upper experiment and analysis. In order to validate the correction and accuracy of PWPM in the bundle block adjustment, the geometric correction experiment is performed using the data set of ZY-3. Before the process, which one interpolation’s method of PWPM is utilized according to the analysis of the orbit and attitude corresponding to the used image range? Thus, geometric correction is performed and the result of correction is represented by Figure
Diagram of geometric correction: (a) the residuals of GCP; (b) assessment with CP.
In Figure
The assessment of accuracy for geometric correction (unit: pixel).
Number of GCPs |
|
|
|
---|---|---|---|
46 | 0.0767 | 0.0215 | 0.0797 |
36 | 0.0706 | 0.0217 | 0.0739 |
26 | 0.0508 | 0.0146 | 0.0529 |
16 | 0.0725 | 0.0161 | 0.0743 |
10 | 0.0795 | 0.0276 | 0.0841 |
Analyzing and comparing Table
In the geometric correction experiment based on the PWPM, the accuracies with the different number of GCPs also reach a high level totally, which is represented by Table
In this study, the new trajectory model, PWPM, is proposed according to the researching and analyzing of the data of ZY-3’s orbit and attitude in the long and short period. By comparison with the other trajectory models, the PWPM can acquire a higher interpolation’s accuracy and has more flexibility. Meanwhile, the differentiation equation of the proposed trajectory model is derivate and it is validated through the bundle block adjustment. In the geometric correction experiment based on the PWPM, the accuracies of geometric correction with the different number of GCPs also reach a high level totally. According to the analyzing and researching of the assessment results with GCPs and CPs, the correctness and applicability of the PWPM are validated and assessed to ensure and improve the high accuracy of geometric correction for ZY-3 satellite. The further study will be performed to experiment with the real image data of ZY-3 and GCP to research better systematic error model for interior orientation, in order to explore the potentials of using ZY-3 data for stereo mapping.
The authors declare that there is no conflict of interests regarding the publication of this paper.